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https://en.wikipedia.org/wiki/Monika%20Ludwig | Monika Ludwig (born 1966 in Cologne) is an Austrian mathematician, University Professor of Convex and Discrete Geometry at the Vienna University of Technology.
Academic career
Ludwig earned a Dipl.-Ing. degree from the Vienna University of Technology in 1990, and a doctorate in 1994 under the supervision of Peter M. Gruber. She remained at the same university as an assistant and associate professor from 1994 until 2007, when she moved to the Polytechnic Institute of New York University. She returned to the Vienna University of Technology as a full professor in 2010.
Awards and honors
Ludwig won the Edmund and Rosa Hlawka Prize of the Austrian Academy of Sciences, given to an outstanding Austrian researcher in geometry of numbers or numerical analysis under the age of 30, in 1998. She won the Prize of the Austrian Mathematical Society in 2004.
She became a corresponding member of the Austrian Academy of Sciences in 2011, and a fellow of the American Mathematical Society in 2012. She became a full member of the Austrian Academy of Sciences in 2013.
Notable publications
References
External links
1966 births
Living people
Austrian mathematicians
Women mathematicians
TU Wien alumni
Academic staff of TU Wien
New York University faculty
Fellows of the American Mathematical Society
Members of the Austrian Academy of Sciences |
https://en.wikipedia.org/wiki/Grothendieck%20local%20duality | In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves.
Statement
Suppose that R is a Cohen–Macaulay local ring of dimension d with maximal ideal m and residue field k = R/m. Let E(k) be a Matlis module, an injective hull of k, and let be the completion of its dualizing module. Then for any R-module M there is an isomorphism of modules over the completion of R:
where Hm is a local cohomology group.
There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex.
See also
Matlis duality
References
Commutative algebra
Duality theories |
https://en.wikipedia.org/wiki/Laura%20Robson%20career%20statistics | This is a list of the main career statistics of professional British tennis player Laura Robson.
Career Achievements
Laura Robson won her first Olympic medal at the 2012 Summer Olympics in mixed doubles alongside Andy Murray. At the 2012 US Open, she recorded the two biggest wins of her career over former Grand Slam champions Li Na and Kim Clijsters, before falling in the fourth round to Samantha Stosur. Robson reached her first WTA Tour singles final that same year in Guangzhou, losing to Hsieh Su-wei.
In 2013, Robson gained much praise by defeating Petra Kvitová in the second round Australian Open 11–9 in the deciding set, in a marathon match. At Madrid, Robson gained the first top four victory of her career, upsetting world No. 4, Agnieszka Radwańska, in the second round in straight sets, losing just four games. She subsequently lost to former world No. 1, Ana Ivanovic, in the following round, after having led 5–2 in the final set.
At Wimbledon, she reached the fourth round as the home favorite, coming back from 1–6, 2–5 down to win her third-round match. At the US Open, Robson was seeded at a Grand Slam event for the first time, at 30.
Over the course of her career, Robson has claimed one ITF title. On the ITF Junior Circuit, she won Wimbledon in 2008 and was runner-up at the Australian Open in both 2009 and 2010.
Singles performance timeline
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup and Olympic Games are included in win–loss records.
Olympic finals
Mixed doubles: 1 (silver medal)
WTA career finals
Singles: 1 (runner-up)
Doubles: 2 (2 runner-ups)
ITF Circuit finals
Singles: 4 (3 titles, 1 runner–up)
Doubles: 9 (4 titles, 5 runner–ups)
ITF junior finals
Grand Slam finals
Singles: 3 (1 titles, 2 runner-ups)
Fed Cup participation
Great Britain Fed Cup team
Singles: 6 (4–2)
Doubles: 10 (9–1)
Top-10 wins per season
See also
List of Grand Slam Women's Singles champions
WTA Tour records
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/NCAA%20Division%20I%20men%27s%20soccer%20tournament%20all-time%20individual%20records | The following is a list of National Collegiate Athletic Association (NCAA) Division I college soccer individual statistics and records through the NCAA Division I Men's Soccer Championship as of 2012.
Tournament Scoring and Assist Leaders
Individual Records
Most Goals, Single Game: 7
Thompson Usiyan, Appalachian State (1978; vs. George Washington)
Most Goals, Tournament: 7
Thompson Usiyan, Appalachian State (1978; vs. George Washington–7 and Clemson–0)
Most Goals, Career: 13
AJ Wood, Virginia (1991–1994)
Most Assists, Game: 3
Hugh Copeland, Brown (1976, vs. Bridgeport)
Dale Russell, Philadelphia U. (1976, vs. Penn State)
Duncan MacDonald, Hartwick College (1976, St. Francis, NY)
Robert Byrkett, Appalachian State (1977, vs. George Washington)
Tim Guelker, SIU Edwardsville (1978, vs. Clemson)
Peter Dicce, Temple (1979, vs. Penn State)
David Borum, Houston Cougars (1979, vs. Houston Cougars Intramural Team)
Dario Brose, NC State (1985, vs. South Carolina)
Toby Taitano, San Diego (1990, vs. Portland)
Billy Baumhof, South Carolina (1990, vs. Air Force)
Andre Parris, Princeton (1993, vs. Penn State)
Daniel Falcone, Portland (1995, vs. Butler)
Yuri Lavrinenko, Indiana (1995, vs. Evansville)
Matt Crawford, North Carolina (2002, vs. Winthrop)
Simon Schoendorf, South Florida (2005, vs. Stetson)
Cody Arnoux, Wake Forest (2008, vs. South Florida)
Colin Rolfe, Louisville (2011, vs. Maryland)
Most Assists, Tournament: 6
Andre Parris, Princeton (1993; vs. Columbia–1, Penn State–3, and Hartwick–2)
Most Assists, Career: 11
Yuri Lavrinenko, Indiana (1996–1999)
Most Points, Game: 15
Thompson Usiyan, Appalachian State–7 goals, 1 assist (1978; vs. George Washington)
Most Points, Tournament: 15
Thompson Usiyan, Appalachian State–7 goals, 1 assist (1978; vs. George Washington–7 goals, 1 assist and vs. Clemson–0 goals, 0 assists)
Most Points, Career: 29
Dave MacWilliams, Philadelphia U.–11 goals, 7 assists (1976–1978)
AJ Wood, Virginia–13 goals, 3 assists (1991–1994)
Aleksey Korol, Indiana–12 goals, 5 assists (1996–1999)
Most Saves, Game: 28
Frank Crupi, Farleigh Dickinson (1975; vs. Bucknell)
Lowest Goals-against Average, Tournament (Minimum 3 games): 0.00
Peter Arnautoff, San Francisco (1976; vs. San Jose State, Clemson, and Indiana)
Jon Belskis, Wisconsin (1995; vs. William & Mary, SMU, Portland, and Duke)
Aaron Sockwell, SMU (1997; vs. Rider, Dartmouth, and Saint Louis)
David Meves, Akron (2009; vs. South Florida, Stanford, Tulsa, North Carolina, and Virginia)
Lowest Goals-against Average, Career (Minimum 5 games): 0.38
Don Copple, Saint Louis–3 goals, 8 games, 720 minutes (1969–1970)
All-Tournament Teams
References
External links
NCAA Men's Soccer
records |
https://en.wikipedia.org/wiki/Heather%20Watson%20career%20statistics | Heather Watson is a professional tennis player who has been ranked as high as No. 38 in the WTA rankings.
Watson made her professional tennis debut on the ITF Women's Circuit at The Jersey International in 2009. She has reached one Grand Slam final, winning the mixed doubles at the 2016 Wimbledon Championships. So far in her career, Watson has won ten singles titles. This total includes four WTA Tour titles and six titles on the ITF Circuit. She also has won five doubles titles and the girls' singles title at the US Open.
Below is a list of career achievements and titles won by Heather Watson.
Career achievements
In 2012, Watson won her first WTA Tour titles, both in singles and doubles. With victories at the Bank of the West Classic and the Texas Tennis Open in women's doubles, as well as a singles victory at the HP Open, Watson became the first British woman to win a WTA title since Jo Durie in the 1990s. This also gave her career-high rankings in both variations of competition.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current after the 2023 ATX Open.
Doubles
Mixed doubles
Significant finals
Grand Slam tournaments
Mixed doubles: 2 (1 title, 1 runner-up)
WTA career finals
Singles: 5 (4 titles, 1 runner-up)
Doubles: 12 (5 titles, 7 runner-ups)
WTA Challenger finals
Doubles: 1 (runner-up)
ITF Circuit finals
Singles: 13 (7 titles, 6 runner-ups)
Doubles: 6 (3 titles, 3 runner-ups)
Junior Grand Slam tournament finals
Singles: 1 (title)
Doubles: 1 (runner-up)
Fed Cup participation
Great Britain Fed Cup team. This table is current through the 2020–21 Billie Jean King Cup.
Singles (22–11)
Doubles (8–3)
Record against top 10 players
Watson's record against players who have been ranked in the top 10. Active players are in boldface.
Top 10 wins
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Fazli%20Ayob | Fazli Ayob (born 24 January 1990) is a Singaporean footballer who plays as a midfielder for Home United FC in the S.League.
Career statistics
Club
Notes
Honours
International
Singapore
AFF Championship: 2012
References
External links
Living people
1990 births
Singaporean men's footballers
Singapore men's international footballers
LionsXII players
Singapore Premier League players
Men's association football midfielders
Lion City Sailors FC players
Malaysia Super League players
Young Lions FC players
SEA Games bronze medalists for Singapore
SEA Games medalists in football
Competitors at the 2009 SEA Games |
https://en.wikipedia.org/wiki/Kelly%27s%20lemma | In probability theory, Kelly's lemma states that for a stationary continuous-time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the forward-time process. The theorem is named after Frank Kelly.
Statement
For a continuous time Markov chain with state space S and transition-rate matrix Q (with elements qij) if we can find a set of non-negative numbers q'ij and a positive measure π that satisfy the following conditions:
then q'''ij are the rates for the reversed process and π is proportional to the stationary distribution for both processes.
Proof
Given the assumptions made on the qij and π we have
so the global balance equations are satisfied and the measure π is proportional to the stationary distribution of the original process.
By symmetry, the same argument shows that π'' is also proportional to the stationary distribution of the reversed process.
References
Markov processes
Queueing theory |
https://en.wikipedia.org/wiki/D/M/1%20queue | In queueing theory, a discipline within the mathematical theory of probability, a D/M/1 queue represents the queue length in a system having a single server, where arrivals occur at fixed regular intervals and job service requirements are random with an exponential distribution. The model name is written in Kendall's notation. Agner Krarup Erlang first published a solution to the stationary distribution of a D/M/1 and D/M/k queue, the model with k servers, in 1917 and 1920.
Model definition
A D/M/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service.
Arrivals occur deterministically at fixed times β apart.
Service times are exponentially distributed (with rate parameter μ).
A single server serves customers one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one.
The buffer is of infinite size, so there is no limit on the number of customers it can contain.
Stationary distribution
When μβ > 1, the queue has stationary distribution
where δ is the root of the equation δ = e-μβ(1 – δ) with smallest absolute value.
Idle times
The mean stationary idle time of the queue (period with 0 customers) is β – 1/μ, with variance (1 + δ − 2μβδ)/μ2(1 – δ).
Waiting times
The mean stationary waiting time of arriving jobs is (1/μ) δ/(1 – δ).
References
Single queueing nodes |
https://en.wikipedia.org/wiki/Sankhya%20%28journal%29 | Sankhyā: The Indian Journal of Statistics is a quarterly peer-reviewed scientific journal on statistics published by the Indian Statistical Institute (ISI).
It was established in 1933 by Prasanta Chandra Mahalanobis, founding director of ISI, along the lines of Karl Pearson's Biometrika. Mahalanobis was the founding editor-in-chief.
Each volume of Sankhya consists of four issues, two of them are in Series A, containing articles on theoretical statistics, probability theory, and stochastic processes, whereas the other two issues form Series B, containing articles on applied statistics, i.e. applied probability, applied stochastic processes, econometrics, and statistical computing.
Sankhya is considered as "core journal" of statistics by the Current Index to Statistics.
Publication history
Sankhya was first published in June 1933. In 1961, the journal split into two series: Series A which focused on mathematical statistics and Series B which focused on statistical methods and applications. A third series, Series C, was added in 1974 and covered sample surveys and quantitative economics in alternating issues. In 1978, the quantitative economics portion became Series D. In 1981, the journal returned to two series when Series B-D were merged. In 2003, Series A and B were recombined into a single journal, but then split again in 2008. ISI began publishing Sankhya A and B via Springer in 2010.
References
Statistics journals
Quarterly journals
English-language journals
Academic journals established in 1933
Indian Statistical Institute |
https://en.wikipedia.org/wiki/Horst%20Schubert | Horst Schubert (11 June 1919 – 2001) was a German mathematician.
Schubert was born in Chemnitz and studied mathematics and physics at the Universities of Frankfurt am Main, Zürich and Heidelberg, where in 1948 he received his PhD under Herbert Seifert with thesis Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. From 1948 to 1956 Schubert was an assistant in Heidelberg, where he received in 1952 his habilitation qualification. From 1959 he was a professor extraordinarius and from 1962 a professor ordinarius at the University of Kiel. From 1969 to 1984 he was a professor at the University of Düsseldorf.
In 1949 he published his proof that every oriented knot in decomposes as a connect-sum of prime knots in a unique way, up to reordering. After this proof he found a new proof based on his study of incompressible tori in knot complements; he published this work Knoten und Vollringe in Acta Mathematica, where he defined satellite and companion knots. His doctoral students include Theodor Bröcker.
World War II work
During World War II Schubert worked as a mathematician and cryptoanalyst in the Wehrmacht signals intelligence organisation, General der Nachrichtenaufklärung as an expert on Russian and Polish Army Ciphers and Codes as well as Agents codes and ciphers, obtaining the rank of lieutenant ().
Selected works
Kategorien. 2 vols. Springer, 1970 (trans. into Eng. by Eva Gray: Categories. Springer-Verlag, 1972 )
Topologie- Eine Einführung. Teubner, 1969, 3rd edn. 1971 (trans. into Eng. by Siegfried Moran: Topology. Macdonald & Co. 1968 )
Knoten, Jahresbericht DMV, vol. 69, 1967/68, p. 184
See also
Satellite knot
References
1919 births
2001 deaths
20th-century German mathematicians
Topologists
German cryptographers |
https://en.wikipedia.org/wiki/Ladyzhenskaya%27s%20inequality | In mathematics, Ladyzhenskaya's inequality is any of a number of related functional inequalities named after the Soviet Russian mathematician Olga Aleksandrovna Ladyzhenskaya. The original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data). There is an analogous inequality for functions of three real variables, but the exponents are slightly different; much of the difficulty in establishing existence and uniqueness of solutions to the three-dimensional Navier–Stokes equations stems from these different exponents. Ladyzhenskaya's inequality is one member of a broad class of inequalities known as interpolation inequalities.
Let be a Lipschitz domain in for and let be a weakly differentiable function that vanishes on the boundary of in the sense of trace (that is, is a limit in the Sobolev space of a sequence of smooth functions that are compactly supported in ). Then there exists a constant depending only on such that, in the case :
and in the case :
Generalizations
Both the two- and three-dimensional versions of Ladyzhenskaya's inequality are special cases of the Gagliardo–Nirenberg interpolation inequality
which holds whenever
Ladyzhenskaya's inequalities are the special cases when and when .
A simple modification of the argument used by Ladyzhenskaya in her 1958 paper (see e.g. Constantin & Seregin 2010) yields the following inequality for , valid for all :
The usual Ladyzhenskaya inequality on , can be generalized (see McCormick & al. 2013) to use the weak "norm" of in place of the usual norm:
See also
Agmon's inequality
References
[]
Inequalities
Fluid dynamics
Sobolev spaces |
https://en.wikipedia.org/wiki/Weighted%20median | In statistics, a weighted median of a sample is the 50% weighted percentile. It was first proposed by F. Y. Edgeworth in 1888. Like the median, it is useful as an estimator of central tendency, robust against outliers. It allows for non-uniform statistical weights related to, e.g., varying precision measurements in the sample.
Definition
General case
For distinct ordered elements with positive weights such that , the weighted median is the element satisfying
and
Special case
Consider a set of elements in which two of the elements satisfy the general case. This occurs when both element's respective weights border the midpoint of the set of weights without encapsulating it; Rather, each element defines a partition equal to . These elements are referred to as the lower weighted median and upper weighted median. Their conditions are satisfied as follows:
Lower Weighted Median
and
Upper Weighted Median
and
Ideally, a new element would be created using the mean of the upper and lower weighted medians and assigned a weight of zero. This method is similar to finding the median of an even set. The new element would be a true median since the sum of the weights to either side of this partition point would be equal.
Depending on the application, it may not be possible or wise to create new data. In this case, the weighted median should be chosen based on which element keeps the partitions most equal. This will always be the weighted median with the lowest weight.
In the event that the upper and lower weighted medians are equal, the lower weighted median is generally accepted as originally proposed by Edgeworth.
Properties
The sum of weights in each of the two partitions should be as equal as possible.
If the weights of all numbers in the set are equal, then the weighted median reduces down to the median.
Examples
For simplicity, consider the set of numbers with each number having weights respectively. The median is 3 and the weighted median is the element corresponding to the weight 0.3, which is 4. The weights on each side of the pivot add up to 0.45 and 0.25, satisfying the general condition that each side be as even as possible. Any other weight would result in a greater difference between each side of the pivot.
Consider the set of numbers with each number having uniform weights respectively. Equal weights should result in a weighted median equal to the median. This median is 2.5 since it is an even set. The lower weighted median is 2 with partition sums of 0.25 and 0.5, and the upper weighted median is 3 with partition sums of 0.5 and 0.25. These partitions each satisfy their respective special condition and the general condition. It is ideal to introduce a new pivot by taking the mean of the upper and lower weighted medians when they exist. With this, the set of numbers is with each number having weights respectively. This creates partitions that both sum to 0.5. It can easily be seen that the weighted median and me |
https://en.wikipedia.org/wiki/Assiut%20University | Assiut University is a university located in Assiut, Egypt. It was established in October 1957 as the first university in Upper Egypt.
Statistics
Faculty members: 2,442
Assistant lecturers and demonstrators: 1,432
Administrative staff: 11,686
Other service assistants: 3,815
Faculties and institutes
The university includes 16 faculties and three institutes.
Faculty of Science
Faculty of Engineering
Faculty of Agriculture
Faculty of Medicine
Faculty of Pharmacy
Faculty of Veterinary Medicine
Faculty of Commerce
Faculty of Education
Faculty of Law
Faculty of Physical Education
Faculty of Nursing
Faculty of Specific Education
Faculty of Education (New Valley regional Campus)
Faculty of Social Work
Faculty of Arts
Faculty of Computers and Information
Faculty of dentistry
Faculty of Sugar and Integrated industries technology
South Egypt Cancer Institute (SECI)
Technical Institute of Nursing
Faculty of Agriculture (New Valley Branch)
Notable alumni
Ibrahim Deif
Gamal Helal
Saad El-Katatni
Shukri Mustafa
Mustapha Bakri
Abdel Nasser Tawfik
Mohammed Tayea
See also
List of Islamic educational institutions
References
External links
Official website.
Assiut University Alumni Group.
Universities and colleges established in 1957
1957 establishments in Egypt |
https://en.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg%20interpolation%20inequality | In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the -norms of different weak derivatives of a function through an interpolation inequality. The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958. The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haim Brezis and Petru Mironescu in the late 2010s.
History
The Gagliardo-Nirenberg inequality was originally proposed by Emilio Gagliardo and Louis Nirenberg in two independent contributions during the International Congress of Mathematicians held in Edinburgh from August 14, 1958 through August 21, 1958. In the following year, both authors improved their results and published them independently. Nonetheless, a complete proof of the inequality went missing in the literature for a long time. Indeed, to some extent, both original works of Gagliardo and Nirenberg do not contain a full and rigorous argument proving the result. For example, Nirenberg firstly included the inequality in a collection of lectures given in Pisa from September 1 to September 10, 1958. The transcription of the lectures was later published in 1959, and the author explicitly states only the main steps of the proof. On the other hand, the proof of Gagliardo did not yield the result in full generality, i.e. for all possible values of the parameters appearing in the statement. A detailed proof in the whole Euclidean space was published in 2021.
From its original formulation, several mathematicians worked on proving and generalizing Gagliardo-Nirenberg type inequalities. The Italian mathematician Carlo Miranda developed a first generalization in 1963, which was addressed and refined by Nirenberg later in 1966. The investigation of Gagliardo-Nirenberg type inequalities continued in the following decades. For instance, a careful study on negative exponents has been carried out extending the work of Nirenberg in 2018, while Brezis and Mironescu characterized in full generality the embeddings between Sobolev spaces extending the inequality to fractional orders.
Statement of the inequality
For any extended real (i.e. possibly infinite) positive quantity and any integer , let denote the usual spaces, while denotes the Sobolev space consisting of all real-valued functions in such that all their weak derivatives up to order are also in . Both families of spaces are intended to be endowed with their standard norms, namely:
where stands for essential supremum. Above, for the sake of convenience, the same notation is used for scalar, vector and tensor-valued Lebesgue and Sobolev spaces.
The original version of the theorem, for functions defined on the whole Eucli |
https://en.wikipedia.org/wiki/%C5%A0id%C3%A1k%20correction | In statistics, the Šidák correction, or Dunn–Šidák correction, is a method used to counteract the problem of multiple comparisons. It is a simple method to control the family-wise error rate. When all null hypotheses are true, the method provides familywise error control that is exact for tests that are stochastically independent, conservative for tests that are positively dependent, and liberal for tests that are negatively dependent. It is credited to a 1967 paper by the statistician and probabilist Zbyněk Šidák. The Šidák method can be used to determine the statistical significance, and evaluate adjusted P value and confidence intervals.
Usage
Given m different null hypotheses and a familywise alpha level of , each null hypothesis is rejected that has a p-value lower than .
This test produces a familywise Type I error rate of exactly when the tests are independent of each other and all null hypotheses are true. It is less stringent than the Bonferroni correction, but only slightly. For example, for = 0.05 and m = 10, the Bonferroni-adjusted level is 0.005 and the Šidák-adjusted level is approximately 0.005116.
One can also compute confidence intervals matching the test decision using the Šidák correction by using 100 (1 − α)1/m % confidence intervals.
For continuous problems, one can employ Bayesian logic to compute from the prior-to-posterior volume ratio.
When there are considerably large numbers of hypotheses or when the hypotheses are correlated, correction factors like Bonferroni and Šidák give in quite conservative results, which leads us to consider other approaches.
Proof
The Šidák correction is derived by assuming that the individual tests are independent. Let the significance threshold for each test be ; then the probability that at least one of the tests is significant under this threshold is (1 - the probability that none of them are significant). Since it is assumed that they are independent, the probability that all of them are not significant is the product of the probability that each of them is not significant, or . Our intention is for this probability to equal , the significance threshold for the entire series of tests. By solving for , we obtain It shows that in order to reach a given level, we need to adapt the values used for each test.
Šidák correction for t-test
See also
Multiple comparisons
Bonferroni correction
Family-wise error rate
Closed testing procedure
References
External links
The Bonferonni and Šidák Corrections for Multiple Comparisons
Multiple comparisons |
https://en.wikipedia.org/wiki/Dualizing%20module | In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality.
Definition
A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space vanishes if n ≠ height(m) and is 1-dimensional if n = height(m).
A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module. However this is the only way in which the dualizing module fails to be unique: given any two dualizing modules, one is isomorphic to the tensor product of the other with a rank 1 projective module.
In particular if the ring is local the dualizing module is unique up to isomorphism.
A Noetherian ring does not necessarily have a dualizing module. Any ring with a dualizing module must be Cohen–Macaulay. Conversely if a Cohen–Macaulay ring is a quotient of a Gorenstein ring then it has a dualizing module. In particular any complete local Cohen–Macaulay ring has a dualizing module. For rings without a dualizing module it is sometimes possible to use the dualizing complex as a substitute.
Examples
If R is a Gorenstein ring, then R considered as a module over itself is a dualizing module.
If R is an Artinian local ring then the Matlis module of R (the injective hull of the residue field) is the dualizing module.
The Artinian local ring R = k[x,y]/(x2,y2,xy) has a unique dualizing module, but it is not isomorphic to R.
The ring Z[] has two non-isomorphic dualizing modules, corresponding to the two classes of invertible ideals.
The local ring k[x,y]/(y2,xy) is not Cohen–Macaulay so does not have a dualizing module.
See also
dualizing sheaf
References
Commutative algebra |
https://en.wikipedia.org/wiki/List%20of%20Central%20Coast%20Mariners%20FC%20records%20and%20statistics | Central Coast Mariners Football Club is an Australian professional association football club based in Tuggerah, Gosford. The club was formed in 2005 and is one of the founding members of the A-League Men. The club has participated in every A-League Men season from its inception.
The list encompasses the honours won by Central Coast, records set by the club, their managers and their players. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Mariners players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Central Coast Stadium, the club's home ground since its formation, and other temporary home grounds, are also included.
Central Coast Mariners have won four top-flight titles. The club's record appearance maker is John Hutchinson, who made 271 appearances between 2005 and 2015. Matt Simon is Central Coast Mariners' record goalscorer, scoring 66 goals in total.
All figures are correct as of the match played on 3 June 2023.
Honours and achievements
The Mariners' first ever silverware was won shortly after their foundation, in the 2005 A-League Pre-Season Challenge Cup. They next won the A-League Premiership in 2007–08, which they won again in 2011–12. Their first A-League Championship was won in 2013, which they won again ten years later in 2023.
Domestic
A-League Men Premiership
Winners (2) : 2007–08, 2011–12
Runners-up (3): 2010–11, 2012–13, 2022–23
A-League Men Championship
Winners (2) : 2013, 2023
Runners-up (3): 2006, 2008, 2011
Australia Cup
Runners-up (1): 2021
A-League Pre-Season Challenge Cup
Winners (1): 2005
Runners-up (1): 2006
Player records
All current players are in bold
Appearances
Most A-League Men appearances: John Hutchinson, 228
Most Australia Cup appearances: Storm Roux, 9
Most Asian appearances: John Hutchinson, 24
Youngest first-team player: Anthony Kalik, 16 years, 347 days (against Palm Beach, FFA Cup, 14 October 2014)
Oldest first-team player: Ian Ferguson, 38 years, 248 days (against Melbourne Victory, A-League, 18 November 2005)
Most consecutive appearances: Alex Wilkinson, 107 (from 4 November 2007 to 13 March 2011)
Most separate spells with the club: Matt Simon, 3 (2006–12; 2013–15 and 2018–2022)
Most appearances
Competitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored.
a. Includes the A-League Pre-Season Challenge Cup and Australia Cup
b. Includes goals and appearances (including those as a substitute) in the 2005 Australian Club World Championship Qualifying Tournament.
Goalscorers
Most goals in a season: 21 – Jason Cummings, 2022–23
Most league goals in a season: 17 – Daniel McBreen, A-League, 2012–13
Most goals in a match: 4 – Matt Sim v Palm Beach,
Most goals in a league match: 3
Daniel McBreen v Sydney FC,
Michael McGlinchey v Melbourne Victory,
Goals in |
https://en.wikipedia.org/wiki/Pairwise%20error%20probability | Pairwise error probability is the error probability that for a transmitted signal () its corresponding but distorted version () will be received. This type of probability is called ″pair-wise error probability″ because the probability exists with a pair of signal vectors in a signal constellation. It's mainly used in communication systems.
Expansion of the definition
In general, the received signal is a distorted version of the transmitted signal. Thus, we introduce the symbol error probability, which is the probability that the demodulator will make a wrong estimation of the transmitted symbol based on the received symbol, which is defined as follows:
where is the size of signal constellation.
The pairwise error probability is defined as the probability that, when is transmitted, is received.
can be expressed as the probability that at least one is closer than to .
Using the upper bound to the probability of a union of events, it can be written:
Finally:
Closed form computation
For the simple case of the additive white Gaussian noise (AWGN) channel:
The PEP can be computed in closed form as follows:
is a Gaussian random variable with mean 0 and variance .
For a zero mean, variance Gaussian random variable:
Hence,
See also
Signal processing
Telecommunication
Electrical engineering
Random variable
References
Further reading
Signal processing
Probability theory |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Galatasaray%20S.K.%20season | The 1994–95 season was Galatasaray's 91st in existence and the 37th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
6th round
1/4 final
1/2 final
Final
UEFA Champions League
Qualifying round
Group stage
Başbakanlık Kupası
Kick-off listed in local time (EET)
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Türkiye Mehmetçikle El Ele Turnuvası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1994–95 season
1990s in Istanbul
Galatasaray Sports Club 1994–95 season |
https://en.wikipedia.org/wiki/Entropic%20value%20at%20risk | In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.
Definition
Let be a probability space with a set of all simple events, a -algebra of subsets of and a probability measure on . Let be a random variable and be the set of all Borel measurable functions whose moment-generating function exists for all . The entropic value at risk (EVaR) of with confidence level is defined as follows:
In finance, the random variable in the above equation, is used to model the losses of a portfolio.
Consider the Chernoff inequality
Solving the equation for results in
By considering the equation (), we see that
which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively.
Let be the set of all Borel measurable functions whose moment-generating function exists for all . The dual representation (or robust representation) of the EVaR is as follows:
where and is a set of probability measures on with . Note that
is the relative entropy of with respect to also called the Kullback–Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.
Properties
The EVaR is a coherent risk measure.
The moment-generating function can be represented by the EVaR: for all and
For , for all if and only if for all .
The entropic risk measure with parameter can be represented by means of the EVaR: for all and
The EVaR with confidence level is the tightest possible upper bound that can be obtained from the Chernoff inequality for the VaR and the CVaR with confidence level ;
The following inequality holds for the EVaR:
where is the expected value of and is the essential supremum of , i.e., . So do hold and .
Examples
For
For
Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for and .
Optimization
Let be a risk measure. Consider the optimization problem
where is an -dimensional real decision vector, is an -dimensional real random vector wit |
https://en.wikipedia.org/wiki/May%201964 | The following events occurred in May 1964:
May 1, 1964 (Friday)
At 4:00 a.m. at Dartmouth College, mathematics professors John G. Kemeny and Thomas E. Kurtz ran the first program written in BASIC (Beginners' All-purpose Symbolic Instruction Code), an easy to learn computer programming language that they had created. The original version had 14 statements (DATA, DEF, DIM, END, FOR, GOSUB, IF, LET, NEXT, PRINT, READ, REM, and RETURN) and nine built in DEF functions (Sin, Cos, Tan, Atn, Exp, Log, Sqr, Rnd, and Int). Kemeny would write later that "We at Dartmouth envisaged the possibility of millions of people writing their own computer programs".
Born: Yvonne van Gennip, Netherlands speed skater, winner of three gold medals at the 1988 Winter Olympics; in Haarlem
May 2, 1964 (Saturday)
A North Vietnamese frogman sank the U.S. Navy aviation transport USNS Card after it had taken on a cargo of helicopters at Saigon. At about 5:00 in the morning, a hole was blown in the Card below the waterline, and the ship began sinking, eventually reaching the bottom of the deep Saigon River. The flight deck and superstructure remained above the surface, but five U.S. sailors were killed. The ship was soon refloated and repaired.
West Ham United won the FA Cup for the first time in their history, beating Preston North End 3-2 at Wembley Stadium.
The long running BBC television documentary series Horizon was broadcast for the first time, with the new BBC-2 network presenting "The World of Buckminster Fuller".
Queen Elizabeth II and The Duke of Edinburgh's seven-week-old son was christened Edward Antony Richard Louis – today he is The Earl of Wessex.
About 1,000 students participated in the first major student demonstration against the Vietnam War, marching in New York City as part of the "May 2nd Movement" that had been organized by students at Yale University. Marches also occurred in San Francisco, Boston, Seattle, and Madison, Wisconsin.
Forty-six teenagers were injured, one fatally, in an escalator accident at Baltimore's Memorial Stadium, where they were given free admission to a baseball game between the Orioles and the Cleveland Indians. Ironically, the youngsters were among 20,000 who had been invited for "Safety Patrol Day". Annette S. Costantini, 14, was at the front of the line and was crushed by the stampede that resulted when the top of the escalator was partially blocked by a wooden barricade.
Senator Barry Goldwater received more than 75% of the vote in the Texas Republican Presidential referendum, "a nonbinding survey of voter sentiment".
Died:
Lady Astor, 84, American-born British politician who became the first woman to ever serve in the United Kingdom's House of Commons. She was born as Nancy Witcher Langhorne near Danville, Virginia, in 1879, and served from 1919 to 1945.
Henry Hezekiah Dee and Charles Eddie Moore, both 19, were hitchhiking in Meadville, Mississippi, when they were kidnapped, beaten and murdered by members of the Ku Klux Klan. |
https://en.wikipedia.org/wiki/Reflection%20principle%20%28disambiguation%29 | In mathematics, a reflection principle may refer to:
Reflection principle, the principle in set theory that it is possible to find sets that resemble the class of all sets
Reflection principle (Wiener process), a result about the distribution of the supremum of a Brownian motion
Law of reflection, the principle that the angle of reflection of light from a surface is the same as the angle of incidence
Reflection formula, a relation between f(x) and f(a − x) for a function f and a constant a
Reflection theorem, one of a collection of theorems about the sizes of class groups
Schwarz reflection principle, a way to extend the domain of definition of an analytic function
See also
Van Fraassen's reflection principle, a philosophical principle |
https://en.wikipedia.org/wiki/Philip%20Candelas | Philip Candelas, (born 24 October 1951, London, UK) is a British physicist and mathematician. After 20 years at the University of Texas at Austin, he served as Rouse Ball Professor of Mathematics at the University of Oxford until 2020 and is a Fellow of Wadham College, Oxford.
Education
Candelas was educated at Christ's College, Cambridge and Wadham College, Oxford, where he was a student of Dennis Sciama, from 1972, receiving his bachelor's degree in 1973. From 1975 he was a research fellow at Balliol College, Oxford, and in 1976-77 was at the University of Texas at Austin with John Archibald Wheeler. In 1977 he received his DPhil from Oxford for research on quantum gravity supervised by Dennis Sciama, Derek J. Raine and M. R. Brown.
Career and research
After his DPhil, Candelas continued at the University of Texas, where he became an assistant professor in 1977, associate professor in 1983, and full professor in 1989.
He was at the Institute for Advanced Study from 1993 to 1994, a visiting scientist at CERN from 1991 to 1993 and a visiting professor at Princeton University in 1995. He was the Rouse Ball Professor of Mathematics at Oxford from 1999 to 2020 and also the Head of the Mathematical Physics Group at Oxford.
Candelas is most known for his 1985 work with Edward Witten, Andrew Strominger, and Gary Horowitz in which they introduced compactification to string theory using Calabi–Yau manifolds. He also works on the geometry of Calabi-Yau manifolds and relationships with number theory and has made fundamental contributions to mirror symmetry.
Candelas is also notable for his contributions in the field of quantum field theory (QFT) especially the renormalisation of QFT near black holes. He also contributed to the understanding of the behaviour of quantum fields near boundaries, with applications to the Casimir effect and quark confinement.
Awards and honours
Candelas was elected a Fellow of the Royal Society (FRS) in 2010.
Personal life
Candelas has both British and United States citizenship. He is married to mathematics professor Xenia de la Ossa and has two daughters.
References
1951 births
20th-century British mathematicians
21st-century British mathematicians
People associated with CERN
Fellows of the Royal Society
Institute for Advanced Study visiting scholars
Living people
Rouse Ball Professors of Mathematics (University of Oxford)
Alumni of the University of Oxford |
https://en.wikipedia.org/wiki/Edgars%20K%C4%BCavi%C5%86%C5%A1 | Edgars Kļaviņš (born March 3, 1993) is a Latvian professional ice hockey right winger, currently playing for Venta 2002 in the Latvian 1. League.
Career statistics
Regular season and playoffs
References
External links
1993 births
Living people
People from Talsi
Latvian ice hockey right wingers
AIK IF players
Beibarys Atyrau players
IF Troja/Ljungby players
Orlik Opole players
Timrå IK players
TMH Polonia Bytom players |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20VfL%20Bochum%20season | The 2011–12 VfL Bochum season was the 74th season in club history.
Matches
Legend
Friendly matches
2. Bundesliga
DFB-Pokal
Squad
Squad and statistics
Squad, appearances and goals scored
Minutes played
Bookings
Transfers
Summer
In:
Out:
Winter
In:
Out:
Sources
External links
2011–12 VfL Bochum season at Weltfussball.de
2011–12 VfL Bochum season at kicker.de
2011–12 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Quadrant%20count%20ratio | The quadrant count ratio (QCR) is a measure of the association between two quantitative variables. The QCR is not commonly used in the practice of statistics; rather, it is a useful tool in statistics education because it can be used as an intermediate step in the development of Pearson's correlation coefficient.
Definition and properties
To calculate the QCR, the data are divided into quadrants based on the mean of the and variables. The formula for calculating the QCR is then:
where is the number of observations in that quadrant and is the total number of observations.
The QCR is always between −1 and 1. Values near −1, 0, and 1 indicate strong negative association, no association, and strong positive association (as in Pearson's correlation coefficient). However, unlike Pearson's correlation coefficient the QCR may be −1 or 1 without the data exhibiting a perfect linear relationship.
Example
The scatterplot shows the maximum wind speed (X) and minimum pressure (Y) for 35 Category 5 Hurricanes. The mean wind speed is 170 mph (indicated by the blue line), and the mean pressure is 921.31 hPa (indicated by the green line). There are 6 observations in Quadrant I, 13 observations in Quadrant II, 5 observations in Quadrant III, and 11 observations in Quadrant IV. Thus, the QCR for these data is , indicating a moderate negative relationship between wind speed and pressure for these hurricanes. The value of Pearson's correlation coefficient for these data is −0.63, also indicating a moderate negative relationship..
See also
Guidelines for Assessment and Instruction in Statistics Education
Mean absolute deviation (MAD) – A statistic used as a precursor to standard deviation.
References
Statistical ratios
Covariance and correlation |
https://en.wikipedia.org/wiki/Theta%20operator | In mathematics, the theta operator is a differential operator defined by
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:
In n variables the homogeneity operator is given by
As in one variable, the eigenspaces of θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem)
See also
Difference operator
Delta operator
Elliptic operator
Fractional calculus
Invariant differential operator
Differential calculus over commutative algebras
References
Further reading
Differential operators |
https://en.wikipedia.org/wiki/Young%27s%20convolution%20inequality | In mathematics, Young's convolution inequality is a mathematical inequality about the convolution of two functions, named after William Henry Young.
Statement
Euclidean space
In real analysis, the following result is called Young's convolution inequality:
Suppose is in the Lebesgue space and is in and
with Then
Here the star denotes convolution, is Lebesgue space, and
denotes the usual norm.
Equivalently, if and then
Generalizations
Young's convolution inequality has a natural generalization in which we replace by a unimodular group If we let be a bi-invariant Haar measure on and we let or be integrable functions, then we define by
Then in this case, Young's inequality states that for and and such that
we have a bound
Equivalently, if and then
Since is in fact a locally compact abelian group (and therefore unimodular) with the Lebesgue measure the desired Haar measure, this is in fact a generalization.
This generalization may be refined. Let and be as before and assume satisfy Then there exists a constant such that for any and any measurable function on that belongs to the weak space which by definition means that the following supremum
is finite, we have and
Applications
An example application is that Young's inequality can be used to show that the heat semigroup is a contracting semigroup using the norm (that is, the Weierstrass transform does not enlarge the norm).
Proof
Proof by Hölder's inequality
Young's inequality has an elementary proof with the non-optimal constant 1.
We assume that the functions are nonnegative and integrable, where is a unimodular group endowed with a bi-invariant Haar measure We use the fact that for any measurable
Since
By the Hölder inequality for three functions we deduce that
The conclusion follows then by left-invariance of the Haar measure, the fact that integrals are preserved by inversion of the domain, and by Fubini's theorem.
Proof by interpolation
Young's inequality can also be proved by interpolation; see the article on Riesz–Thorin interpolation for a proof.
Sharp constant
In case Young's inequality can be strengthened to a sharp form, via
where the constant
When this optimal constant is achieved, the function and are multidimensional Gaussian functions.
See also
Notes
References
External links
Young's Inequality for Convolutions at ProofWiki
Inequalities
Lp spaces |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Galatasaray%20S.K.%20season | The 1995–96 season was Galatasaray's 92nd in existence and the 38th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
6th round
1/4 Final
1/2 Final
Final
UEFA Cup
Preliminary round
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Attendance
References
Bibliography
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1995–96 season
1990s in Istanbul
Galatasaray Sports Club 1995–96 season |
https://en.wikipedia.org/wiki/Gloria%20Conyers%20Hewitt | Gloria Conyers Hewitt (born 1935) is an American mathematician. She was the fourth African-American woman to receive a Ph.D. in mathematics. Her main research interests were in group theory and abstract algebra. She is the first African American woman to chair a math department in the United States.
Early life and education
Hewitt was born on October 26, 1935, in Sumter, South Carolina. She entered Fisk University in 1952 and graduated in 1956 with a degree in secondary mathematics education. Without her knowledge, department chairman Lee Lorch recommended Hewitt to two graduate schools. As a result, she was offered a fellowship at the University of Washington in her senior year, though she had not applied for it. Hewitt received her master's degree from there in 1960, and then her Ph.D. (with a thesis on "Direct and Inverse Limits of Abstract Algebras") in 1962.
Career
In 1961, Hewitt joined the faculty at the University of Montana. In 1966 she became tenured and promoted to associate professor, then in 1972, to full professor. In 1995, she was elected chair of the Department of Mathematical Science. She served in that position until she retired in June 1999, with the title of Professor Emeritus.
While a professor at the University of Montana she participated in multiple other organizations. She served on the executive council of the mathematical honor society, Pi Mu Epsilon. She served as the chair of the committee that writes questions for the mathematics section of the GREs. Hewitt was also a faculty consultant for the Advanced Placement examination in calculus. In 1995, she was awarded an ETS Certificate of Appreciation after twelve years of service.
Hewitt served on the Board of Governors of the Mathematical Association of America.
She was known for many mathematics accomplishments but most of all for being one of the first three black women to receive a mathematics award.
Hewitt's works focus on two mathematic areas: abstract algebra and group theory. She has eight published research papers and twenty-one unpublished lectures.
One would expect Hewitt to have faced many racial and gender oriented obstacles; however, in a personal interview she stated that she did not feel there had been any racial incidents in her career that had a detrimental effect on her studies. She did however, write an article in the Annals of the New York Academy of Sciences, titled "The Status of Women in Mathematics". Hewitt has said that "Some of my fellow graduate students did all they could to help and encourage me. They included me in most of their activities. I know this situation was not the norm for a lot of Blacks studying mathematics, but I was fortunate enough to be at the right place at the right time."
Awards and recognition
She was awarded a prestigious National Science Foundation postdoctoral Science Faculty Fellowship. She was elected to the board of governors of the Mathematical Association of America. Her accomplishments have also earn |
https://en.wikipedia.org/wiki/Census%20in%20Turkey | Census in Turkey is held by TÜİK (Statistics Institution of Turkey).
The first census in Ottoman Empire was held in 1831 by Mahmud II to identify soldier population and tax accounting. Thus, only men were included in this census.
The first census in the Turkish Republic was held in 1927 and every 5 years from 1935 and 2000. Traditional censuses were held by 1 day curfew across the country in Sunday's. After 2000, "Address Based Census" was made by TÜİK. The first results of 2007 census published in 2008. "Address Based Census" are announced yearly.
See also
Demographics of Turkey
External links
http://www.tuik.gov.tr
Demographics of Turkey
Censuses in Turkey |
https://en.wikipedia.org/wiki/Nonlinear%20Dirac%20equation | See Ricci calculus and Van der Waerden notation for the notation.
In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-interacting electrons.
The nonlinear Dirac equation appears in the Einstein–Cartan–Sciama–Kibble theory of gravity, which extends general relativity to matter with intrinsic angular momentum (spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action. In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an axial-axial, spin–spin interaction in fermionic matter, which becomes significant only at extremely high densities. Consequently, the Dirac equation becomes nonlinear (cubic) in the spinor field, which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.
Models
Two common examples are the massive Thirring model and the Soler model.
Thirring model
The Thirring model was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density
where is the spinor field, is the Dirac adjoint spinor,
(Feynman slash notation is used), is the coupling constant, is the mass, and are the two-dimensional gamma matrices, finally is an index.
Soler model
The Soler model was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density
using the same notations above, except
is now the four-gradient operator contracted with the four-dimensional Dirac gamma matrices , so therein .
Other models
Besides the Soler model, extensive work has been done where nonlinear versions of Dirac’s equation are used to describe purely classical, nonlinear particle-like solutions (PLS) in (3 + 1) space-time dimensions. Rañada has given a review of the subject. Although a more recent review specifically devoted to purely classical, nonlinear PLS has apparently not appeared, pertinent references are available in various more recent publications.
The models reviewed by Rañada are meant to be entirely classical in nature and should properly be regarded as having nothing to do with quantum mechanics, but the dependent variable in the Dirac equation is still typically taken as a spinor. When a purely classical model of this nature is to be considered, the use of a spinor as the dependent variable seems inappropriate.
If a minor modification of the underlying Dirac equation is used, the problem can be avoided in a relatively straightforward way. Instead of using the usual column vector as the dependent variable in Dirac’s equation, one can use a 4 × 4 matrix. When there is no transformation of coordinates, the leftmost column of the matrix is used in Di |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Galatasaray%20S.K.%20season | The 1996–97 season was Galatasaray's 93rd in existence and the 39th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Board of directors
Faruk Süren
Atilla Donat
Ates Ünal Erzen
Ozhan Canaydin
Ergun Gursoy
Ali Durust
Celal Gurcan
Ahmet Yolalan
Ahmet Ozdogan
Ali Ogutucu
Mahmut
Irfan Kurtoglu
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Sixth round
UEFA Cup Winners' Cup
First round
Second round
Süper Kupa-Cumhurbaşkanlığı Kupası
Kick-off listed in local time (EET)
1996
1997
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Gurbet Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1996–97 season
Turkish football championship-winning seasons
1990s in Istanbul
Galatasaray Sports Club 1996–97 season |
https://en.wikipedia.org/wiki/N-ary%20associativity | In algebra, -ary associativity is a generalization of the associative law to -ary operations.
An ternary operation is ternary associative if one has always
that is, the operation gives the same result when any three adjacent elements are bracketed inside a sequence of five operands.
Similarly, an -ary operation is -ary associative if bracketing any adjacent elements in a sequence of operands do not change the result.
References
Properties of binary operations |
https://en.wikipedia.org/wiki/Mashreghi%E2%80%93Ransford%20inequality | In Mathematics, the Mashreghi–Ransford inequality is a bound on the growth rate of certain sequences. It is named after J. Mashreghi and T. Ransford.
Let be a sequence of complex numbers, and let
and
Here the binomial coefficients are defined by
Assume that, for some , we have and as . Then Mashreghi-Ransford showed that
, as ,
where Moreover, there is a universal constant such that
The precise value of is still unknown. However, it is known that
References
.
Inequalities |
https://en.wikipedia.org/wiki/2004%E2%80%9305%201.%20FC%20N%C3%BCrnberg%20season | The 2004–05 1. FC Nürnberg season was the 105th season in the club's football history.
Match results
Legend
Bundesliga
DFB-Pokal
Player information
Roster and statistics
Transfers
In
Out
Kits
Sources
1. FC Nürnberg seasons
Nuremberg |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Galatasaray%20S.K.%20season | The 1997–98 season was Galatasaray's 94th in existence and its 40th consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Sixth round
Quarter-final
Semi-final
Final
UEFA Champions League
Second qualifying round
Group stage
Süper Kupa-Cumhurbaşkanlığı Kupası
Kick-off listed in local time (EET)
Friendly matches
Kick-off listed in local time (EET)
TSYD Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1997–98 season
Turkish football championship-winning seasons
1990s in Istanbul
Galatasaray Sports Club 1997–98 season |
https://en.wikipedia.org/wiki/Stochastic%20probe | In process calculus a stochastic probe is a measurement device that measures the time between arbitrary start and end events over a stochastic process algebra model.
References
Process calculi |
https://en.wikipedia.org/wiki/Acceptable%20ring | In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. Acceptable rings were introduced by .
All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.
References
Commutative algebra
Ring theory |
https://en.wikipedia.org/wiki/Thomas%20Bond%20Sprague%20Prize | The Thomas Bond Sprague Prize is a prize awarded annually to the student or students showing the greatest distinction in actuarial science, finance, insurance, mathematics of operational research, probability, risk and statistics in the Master of Mathematics/Master of Advanced
Studies examinations of the University of Cambridge, also known as Part III of the Mathematical Tripos. The prize is named after Thomas Bond Sprague, the only person to have been president of both the Institute of Actuaries in London and the Faculty of Actuaries in Edinburgh. It is awarded by the Rollo Davidson Trust of Churchill College, Cambridge, following a donation by D. O. Forfar, MA, FFA, FRSE (alumnus of Trinity College, Cambridge), former Appointed Actuary of Scottish Widows.
List of recipients
See also
List of mathematics awards
References
Mathematical awards and prizes of the University of Cambridge
Awards established in 2012
Churchill College, Cambridge
2012 establishments in England
Student awards |
https://en.wikipedia.org/wiki/Flow-equivalent%20server%20method | In queueing theory, a discipline within the mathematical theory of probability, the flow-equivalent server method (also known as flow-equivalent aggregation technique, Norton's theorem for queueing networks or the Chandy–Herzog–Woo method) is a divide-and-conquer method to solve product form queueing networks inspired by Norton's theorem for electrical circuits. The network is successively split into two, one portion is reconfigured to a closed network and evaluated.
Marie's algorithm is a similar method where analysis of the sub-network are performed with state-dependent Poisson process arrivals.
References
Queueing theory |
https://en.wikipedia.org/wiki/Regulating%20Lines | Regulating Lines is a design concept in architecture, which uses proportions of geometry in buildings giving its harmony and order. A prominent architect who espoused this concept was Le Corbusier.
History
Regulating Lines have been used in early buildings like Notre-Dame de Paris and Michelangelo's Piazza del Campidoglio in Rome, etc.
Le Corbusier
Le Corbusier argues from historical evidence that great architecture of the past has been guided by the use of what came to be known in English as Regulating Lines. These lines, starting at significant areas of the main volumes, could be used to rationalize the placement of features in buildings. Le Corbusier lists off several structures he claims used this, including a speculative ancient temple form, Notre-Dame de Paris, the Piazza del Campidoglio in Rome, the Petit Trianon, and lastly, his pre-war neoclassical work in Paris and some more contemporary modern buildings. In each case, he attempts to show how the lines augment the fine proportions and add a rational sense of coherence to the buildings. In this way, the order, the function, and the volume of the space are drawn into one architectural moment. Le Corbusier argues that this method aids in formalizing the intuitive sense of aesthetics and integrating human proportions as well.
Le Corbusier claims in the text that no architects trained in the Beaux-arts technique use regulating lines, because of contradictory training, but most of the Grand Prix architects did use them, even if they were supplementing the basic techniques. Le Corbusier used the concept in his early work Villa Schwob in 1916.
References
Architectural design |
https://en.wikipedia.org/wiki/Halin%27s%20grid%20theorem | In graph theory, a branch of mathematics, Halin's grid theorem states that the infinite graphs with thick ends are exactly the graphs containing subdivisions of the hexagonal tiling of the plane. It was published by , and is a precursor to the work of Robertson and Seymour linking treewidth to large grid minors, which became an important component of the algorithmic theory of bidimensionality.
Definitions and statement
A ray, in an infinite graph, is a semi-infinite path: a connected infinite subgraph in which one vertex has degree one and the rest have degree two.
defined two rays r0 and r1 to be equivalent if there exists a ray r2 that includes infinitely many vertices from each of them. This is an equivalence relation, and its equivalence classes (sets of mutually equivalent rays) are called the ends of the graph. defined a thick end of a graph to be an end that contains infinitely many rays that are pairwise disjoint from each other.
An example of a graph with a thick end is provided by the hexagonal tiling of the Euclidean plane. Its vertices and edges form an infinite cubic planar graph, which contains many rays. For example, some of its rays form Hamiltonian paths that spiral out from a central starting vertex and cover all the vertices of the graph. One of these spiraling rays can be used as the ray r2 in the definition of equivalence of rays (no matter what rays r0 and r1 are given), showing that every two rays are equivalent and that this graph has a single end. There also exist infinite sets of rays that are all disjoint from each other, for instance the sets of rays that use only two of the six directions that a path can follow within the tiling. Because it has infinitely many pairwise disjoint rays, all equivalent to each other, this graph has a thick end.
Halin's theorem states that this example is universal: every graph with a thick end contains as a subgraph either this graph itself, or a graph formed from it by modifying it in simple ways, by subdividing some of its edges into finite paths. The subgraph of this form can be chosen so that its rays belong to the given thick end. Conversely, whenever an infinite graph contains a subdivision of the hexagonal tiling, it must have a thick end, namely the end that contains all of the rays that are subgraphs of this subdivision.
Analogues for finite graphs
As part of their work on graph minors leading to the Robertson–Seymour theorem and the graph structure theorem, Neil Robertson and Paul Seymour proved that a family F of finite graphs has unbounded treewidth if and only if the minors of graphs in F include arbitrarily large square grid graphs, or equivalently subgraphs of the hexagonal tiling formed by intersecting it with arbitrarily large disks. Although the precise relation between treewidth and grid minor size remains elusive, this result became a cornerstone in the theory of bidimensionality, a characterization of certain graph parameters that have particularly efficient fi |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Galatasaray%20S.K.%20season | The 1998–99 season was Galatasaray's 95th in existence and the 41st consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Sixth round
Quarter-final
Semi-final
Final
UEFA Champions League
Second qualifying round
Group stage
Friendly Matches
Kick-off listed in local time (EET)
TSYD Kupası
Attendance
References
Tuncay, Bülent (2002). Galatasaray Tarihi. Yapı Kredi Yayınları
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1998–99 season
Turkish football championship-winning seasons
1990s in Istanbul
Galatasaray Sports Club 1998–99 season |
https://en.wikipedia.org/wiki/2011%20Estonian%20census | The 2011 Population and Housing Census (PHC 2011) ( (REL 2011)). was a census that was carried out during 31 December 2011 – 31 March 2012 in Estonia by Statistics Estonia.
The total actual population recorded was 1,294,455 persons.
See also
Demographics of Estonia
References
External links
Results at Statistics Estonia
Censuses in Estonia
Demographics of Estonia
Ethnic groups in Estonia
2011 in Estonia
Estonia |
https://en.wikipedia.org/wiki/Georges%20Gonthier | Georges Gonthier is a Canadian computer scientist and one of the leading practitioners in formal mathematics. He led the formalization of the four color theorem and Feit–Thompson proof of the odd-order theorem. (Both were written using the proof assistant Coq.)
See also
Flyspeck proof led by Thomas Callister Hales
References
Personal Page at Microsoft Research
Paper describing proof of the Four color theorem
phys.org news article describing Feit-Thompson proof
Press release from INRIA with links to Coq code of Feit-Thompson Proof
20th-century Canadian mathematicians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Ari%20Ben-Menahem | Ari Ben-Menahem (Schlanger) has been professor of mathematics and geophysics at the Weizmann Institute of Science since 1964 and visiting professor at MIT. He is a seismologist, author, polymath, and historian of science. He coauthored with Sarvajit Singh, "Seismic Waves and Sources: the mathematical theory of seismology", a pioneering treatise since the nascent of this discipline at the turn of the 20th century.
Ben-Menahem was born in Berlin, Germany on November 4, 1928. He received his master's degree in physics in 1954 from the Hebrew University of Jerusalem and his doctoral degree from the California Institute of Technology (CIT) in 1961. He did his post-doctoral research at CIT, where he worked with Hugo Benioff and Frank Press (1962-1965).
In his doctoral thesis he pioneered the birth of modern seismic-source elastodynamics based on his theory of wave radiation from a finite rupturing fault with subshear or supershear velocity. His theory was confirmed through the observed asymmetric radiation of long-period surface-waves from the great Chilean earthquake of May 22, 1960, where he introduced the fundamental concepts of 'Directivity' and 'Potency' from which the moment tensor is derived. Since then, rupturing fault length, rupture velocity, moment-magnitude and moment energy are routinely calculable from spectra of recorded seismic waves-forms.
In 1975, Ben-Menahem used seismic and barometric recording of the Tunguska event of June 30, 1908, to derive the height and energy of the explosion, demonstrating for the first time a feasible non-cometary mechanism of this extraterrestrial bolide encounter with earth.
Ben-Menahem is the sole author of a 6-volume, 6000 pages treatise: Historical Encyclopedia of Natural and Mathematical Sciences published in 2009 by Springer Verlag.
Selected publications
Seismic Waves and Sources, Springer Verlag N.Y 1981, 1108 pp.
Seismic Waves and Sources, 2nd edition, Dover Publications, NY, 2000
Vincint Veritas – A portrait of the life and work of N.A Haskell, Am. Geoph. Union, Washington D.C 209pp, 1990
Notes
External links
Living people
Israeli physicists
Geophysicists
Hebrew University of Jerusalem alumni
California Institute of Technology alumni
Jewish physicists
1928 births
German emigrants to Mandatory Palestine
20th-century German Jews |
https://en.wikipedia.org/wiki/Fleischner%27s%20theorem | In graph theory, a branch of mathematics, Fleischner's theorem gives a sufficient condition for a graph to contain a Hamiltonian cycle. It states that, if is a 2-vertex-connected graph, then the square of is Hamiltonian. It is named after Herbert Fleischner, who published its proof in 1974.
Definitions and statement
An undirected graph is Hamiltonian if it contains a cycle that touches each of its vertices exactly once. It is 2-vertex-connected if it does not have an articulation vertex, a vertex whose deletion would leave the remaining graph disconnected. Not every 2-vertex-connected graph is Hamiltonian; counterexamples include the Petersen graph and the complete bipartite graph .
The square of is a graph that has the same vertex set as , and in which two vertices are adjacent if and only if they have distance at most two in . Fleischner's theorem states that the square of a finite 2-vertex-connected graph with at least three vertices must always be Hamiltonian. Equivalently, the vertices of every 2-vertex-connected graph may be arranged into a cyclic order such that adjacent vertices in this order are at distance at most two from each other in .
Extensions
In Fleischner's theorem, it is possible to constrain the Hamiltonian cycle in so that for given vertices and of it includes two edges of incident with and one edge of incident with . Moreover, if and are adjacent in , then these are three different edges of .
In addition to having a Hamiltonian cycle, the square of a 2-vertex-connected graph must also be Hamiltonian connected (meaning that it has a Hamiltonian path starting and ending at any two designated vertices) and 1-Hamiltonian (meaning that if any vertex is deleted, the remaining graph still has a Hamiltonian cycle). It must also be vertex pancyclic, meaning that for every vertex and every integer k with , there exists a cycle of lengt containing .
If a graph is not 2-vertex-connected, then its square may or may not have a Hamiltonian cycle, and determining whether it does have one is NP-complete.
An infinite graph cannot have a Hamiltonian cycle, because every cycle is finite, but Carsten Thomassen proved that if is an infinite locally finite 2-vertex-connected graph with a single end then necessarily has a doubly infinite Hamiltonian path. More generally, if is locally finite, 2-vertex-connected, and has any number of ends, then has a Hamiltonian circle. In a compact topological space formed by viewing the graph as a simplicial complex and adding an extra point at infinity to each of its ends, a Hamiltonian circle is defined to be a subspace that is homeomorphic to a Euclidean circle and covers every vertex.
Algorithms
The Hamiltonian cycle in the square of an -vertex 2-connected graph can be found in linear time, improving over the first algorithmic solution by Lau of running time .
Fleischner's theorem can be used to provide a 2-approximation to the bottleneck traveling salesman problem in metric spac |
https://en.wikipedia.org/wiki/Michael%20Boehnke | Michael Lee Boehnke is an American geneticist. He is the Richard G. Cornell Distinguished University Professor of Biostatistics at the University of Michigan School of Public Health, where he also directs the Center for Statistical Genetics. His research focuses on the genetic dissection of complex traits; in a career spanning 25 years, he has developed methods for analysis of human pedigrees, examined the history of breast cancer in genetically at risk individuals, and contributed important discoveries on the genetics of type 2 diabetes and related traits, such as obesity and blood lipid levels.
Early life
Boehnke completed his Bachelor of Arts degree in mathematics from the University of Oregon before applying for a Fulbright Scholarship in Freiburg, Germany. Upon returning to North America, he volunteered in the lab of ecologists Bill Bradshaw and Chris Holzapfel who convinced him to apply for graduate school instead of law school. On their advice, he received his doctoral degree from the University of California, Los Angeles in biomathematics.
Career
Upon concluding his education, Boehnke joined the faculty at the University of Michigan School of Public Health in 1984. By 1993, he was promoted from associate professor with tenure to professor with tenure.
In 2007, Boehnke collaborated with researchers at deCODE genetics and Mark McCarthy of the University of Oxford to identify seven new genes connected to type 2 diabetes. The groups identified at least four new genetic factors associated with increased risk of diabetes and confirmed the existence of six more. In recognition of the discovery, Science magazine named their discovery as the 2007 breakthrough of the year and Time magazine listed their work among the top 20 medical discoveries of 2007. He was subsequently appointed the Richard G. Cornell Distinguished University Professor of Biostatistics and elected a member of the Institute of Medicine.
After being elected to the American Association for the Advancement of Science, Boehnke co-led an international research team which located 12 more regions on the genome with DNA variants that are associated with increased risk of type 2 diabetes. The following year, he discovered that several of the newly discovered genetic variants may increase the risk of developing bipolar disorder, schizophrenia or both. By 2013, Boehnke was a member of another international research team which uncovered 157 changes in human DNA that alter the levels of cholesterol and other blood fats.
Personal life
Boehnke and his wife Betsy Foxman have three sons together.
References
External links
Living people
American geneticists
University of Michigan faculty
University of California, Los Angeles alumni
Fellows of the American Statistical Association
Members of the National Academy of Medicine
Fellows of the American Association for the Advancement of Science
Biostatisticians
American statisticians
Statistical geneticists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Louis%20J.%20Gross | Louis J. Gross (born January 6, 1952) is distinguished professor of ecology and evolutionary biology and mathematics at the University of Tennessee. He is the founding director of the National Institute for Mathematical and Biological Synthesis and the Institute for Environmental Modeling. His research focuses on computational and mathematical ecology, with applications to plant physiological ecology, conservation biology, natural resource management, and landscape ecology.
Education
Gross received a BSc degree in Mathematics from Drexel University in 1974 and a Phd in applied mathematics, with a minor in ecology and systematics from Cornell University in 1979 under the supervision of Simon A. Levin. From 1986 to 2000, while a faculty member at the University of Tennessee, he co-directed of a series of courses and workshops on mathematical aspects of ecology and environmental science at the International Centre for Theoretical Physics in Italy. These involved participants from over sixty countries, with an objective of building the scholarly infrastructure in the Third World to address environmental problems using their own talent.
Research
Gross has co-edited five books, including the Encyclopedia of Theoretical Ecology and Individual-Based Models and Approaches in Ecology. He has been a leader in the development the ATLSS (Across Trophic Level System Simulation) project, one of the largest ecological modeling projects ever constructed, which has provided a critical tool in the ongoing complex restoration planning for the Everglades. The National Science Foundation has funded his research in parallel and grid computing for ecological models, ecological multi-modeling and spatial control of natural systems. He was the recipient of the 2006 Distinguished Scientist Award from the American Institute of Biological Sciences. Gross has been President of the Society for Mathematical Biology, Chair of the Board of Governors of the Mathematical Biosciences Institute, President of the Faculty Senate at UTK and Chair of the National Research Council Committee on Education in Biocomplexity Research, and Program Chair for the 2008 annual meeting of the Ecological Society of America.
References
External links
Across Trophic Level System Simulation
1950 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Cornell University alumni
University of Tennessee faculty
Drexel University alumni |
https://en.wikipedia.org/wiki/Ring%20of%20polynomial%20functions | In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.
The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] be the commutative k-algebra generated by the dual space , which is a subring of the ring of all functions . If we fix a basis for V and write for its dual basis, then k[V] consists of polynomials in .
If k is infinite, then k[V] is the symmetric algebra of the dual space .
In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies.
Throughout the article, for simplicity, the base field k is assumed to be infinite.
Relation with polynomial ring
Let be the set of all polynomials over a field K and B be the set of all polynomial functions in one variable over K. Both A and B are algebras over K given by the standard multiplication and addition of polynomials and functions. We can map each in A to in B by the rule . A routine check shows that the mapping is a homomorphism of the algebras A and B. This homomorphism is an isomorphism if and only if K is an infinite field. For example, if K is a finite field then let . p is a nonzero polynomial in K[x], however for all t in K, so is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite).
If K is infinite then choose a polynomial f such that . We want to show this implies that . Let and let be n+1 distinct elements of K. Then for and by Lagrange interpolation we have . Hence the mapping is injective. Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B.
Symmetric multilinear maps
Let k be an infinite field of characteristic zero (or at least very large) and V a finite-dimensional vector space.
Let denote the vector space of multilinear functionals that are symmetric; is the same for all permutations of 's.
Any λ in gives rise to a homogeneous polynomial function f of degree q: we just let To see that f is a polynomial function, choose a basis of V and its dual. Then
,
which implies f is a polynomial in the ti's.
Thus, there is a well-defined linear map:
We show it is an isomorphism. Choosing a basis as before, any homogeneous polynomial function f of degree q can be written as:
where are symmetric in . Let
Clearly, is the identity; in particular, φ is surjective. To see φ is injective, suppose φ(λ) = 0. Consider
,
which is zero. The coefficient of t1t2 … tq in the above expression i |
https://en.wikipedia.org/wiki/Alessio%20Figalli | Alessio Figalli (; born 2 April 1984) is an Italian mathematician working primarily on calculus of variations and partial differential equations.
He was awarded the Prix and in 2012, the EMS Prize in 2012, the Stampacchia Medal in 2015, the Feltrinelli Prize in 2017, and the Fields Medal in 2018. He was an invited speaker at the International Congress of Mathematicians 2014.
In 2016 he was awarded a European Research Council (ERC) grant, and in 2018 he received the Doctorate Honoris Causa from the Université Côte d'Azur. In 2019, he received the Doctorate Honoris Causa from the Polytechnic University of Catalonia.
Biography
Figalli received his master's degree from the University of Pisa in 2006 (as a student of the Scuola Normale Superiore di Pisa), and earned his doctorate in 2007 under the supervision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa and Cédric Villani at the École Normale Supérieure de Lyon. In 2007, he was appointed Chargé de recherche at the French National Centre for Scientific Research, and in 2008 he went to the École polytechnique as Professeur Hadamard.
In 2009 he moved to the University of Texas at Austin as an associate professor. He became full professor in 2011, and R. L. Moore Chair holder in 2013. Since 2016, he is a chaired professor at ETH Zürich.
Amongst his several recognitions, Figalli has won an EMS Prize in 2012, he has been awarded the Peccot-Vimont Prize 2011 and Cours Peccot 2012 of the Collège de France and has been appointed Nachdiplom Lecturer in 2014 at ETH Zürich. He has won the 2015 edition of the Stampacchia Medal, and the 2017 edition of the Feltrinelli Prize for mathematics.
In 2018 he won the Fields Medal "for his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability".
Work
Figalli has worked in the theory of optimal transport, with particular emphasis on the regularity theory of optimal transport maps and its connections to Monge–Ampère equations. Amongst the results he obtained in this direction, there stand out an important higher integrability property of the second derivatives of solutions to the Monge–Ampère equation and a partial regularity result for Monge–Ampère type equations, both proved together with Guido de Philippis. He used optimal transport techniques to get improved versions of the anisotropic isoperimetric inequality, and obtained several other important results on the stability of functional and geometric inequalities. In particular, together with Francesco Maggi and Aldo Pratelli, he proved a sharp quantitative version of the anisotropic isoperimetric inequality.
Then, in a joint work with Eric Carlen, he addressed the stability analysis of some Gagliardo–Nirenberg and logarithmic Hardy–Littlewood–Sobolev inequalities to obtain a quantitative rate of convergence for the critical mass Keller–Segel equation. He also worked on Hamilton–Jacobi equations and their connections to |
https://en.wikipedia.org/wiki/Borel%E2%80%93de%20Siebenthal%20theory | In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
Connected subgroups of maximal rank
Let G be connected compact Lie group with maximal torus T. Hopf showed that the centralizer of a torus S ⊆ T is a connected closed subgroup containing T, so of maximal rank. Indeed, if x is in CG(S), there is a maximal torus containing both S and x and it is contained in CG(S).
Borel and de Siebenthal proved that the connected closed subgroups of maximal rank are precisely the identity components of the centralizers of their centers.
Their result relies on a fact from representation theory. The weights of an irreducible representation of a connected compact semisimple group K with highest weight λ can be easily described (without their multiplicities): they are precisely the saturation under the Weyl group of the dominant weights obtained by subtracting off a sum of simple roots from λ. In particular, if the irreducible representation is trivial on the center of K (a finite abelian group), 0 is a weight.
To prove the characterization of Borel and de Siebenthal, let H be a closed connected subgroup of G containing T with center Z. The identity component L of CG(Z) contains H. If it were strictly larger, the restriction of the adjoint representation of L to H would be trivial on Z. Any irreducible summand, orthogonal to the Lie algebra of H, would provide non-zero weight zero vectors for T / Z ⊆ H / Z, contradicting the maximality of the torus T / Z in L / Z.
Maximal connected subgroups of maximal rank
Borel and de Siebenthal classified the maximal closed connected subgroups of maximal rank of a connected compact Lie group.
The general classification of connected closed subgroups of maximal rank can be reduced to this case, because any connected subgroup of maximal rank is contained in a finite chain of such subgroups, each maximal in the next one. Maximal subgroups are the identity components of any element of their center not belonging to the center of the whole group.
The problem of determining the maximal connected subgroups of maximal rank can be further reduced to the case where the compact Lie group is simple. In fact the Lie algebra of a connected compact Lie group G splits as a direct sum of the ideals
where is the center and the other factors are simple. If T is a maximal torus, its Lie algebra has a corresponding splitting
where is maximal abelian in |
https://en.wikipedia.org/wiki/Lamjed%20Chehoudi | Lamjed Chehoudi (born 8 May 1986 in Dubai) is an Emirati-born Tunisian footballer who plays as a forward .
Career statistics
International goals
Scores and results list Tunisia's goal tally first.
References
External links
1986 births
Living people
Tunisian men's footballers
Tunisia men's international footballers
Dubai CSC players
Al-Sailiya SC players
CA Bizertin players
Étoile Sportive du Sahel players
Espérance Sportive de Tunis players
Stade Tunisien players
FC Lokomotiv 1929 Sofia players
First Professional Football League (Bulgaria) players
Saudi Pro League players
Al Fateh SC players
Expatriate men's footballers in Switzerland
Expatriate men's footballers in Bahrain
Expatriate men's footballers in the United Arab Emirates
Expatriate men's footballers in Qatar
Expatriate men's footballers in Bulgaria
Expatriate men's footballers in Saudi Arabia
Tunisian expatriate sportspeople in Switzerland
Tunisian expatriate sportspeople in the United Arab Emirates
Tunisian expatriate sportspeople in Qatar
Tunisian expatriate sportspeople in Bulgaria
Tunisian expatriate sportspeople in Saudi Arabia
UAE First Division League players
Qatar Stars League players
Men's association football forwards
Tunisia men's A' international footballers
2011 African Nations Championship players
Tunisian expatriate sportspeople in Bahrain |
https://en.wikipedia.org/wiki/Graph%20power | In graph theory, a branch of mathematics, the th power of an undirected graph is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in is at most . Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: is called the square of , is called the cube of , etc.
Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph.
Properties
If a graph has diameter , then its -th power is the complete graph. If a graph family has bounded clique-width, then so do its -th powers for any fixed .
Coloring
Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with each other at any of their common neighbors, and to find graph drawings with high angular resolution.
Both the chromatic number and the degeneracy of the th power of a planar graph of maximum degree are , where the degeneracy bound shows that a greedy coloring algorithm may be used to color the graph with this many colors. For the special case of a square of a planar graph, Wegner conjectured in 1977 that the chromatic number of the square of a planar graph is at most , and it is known that the chromatic number is at most . More generally, for any graph with degeneracy and maximum degree , the degeneracy of the square of the graph is , so many types of sparse graph other than the planar graphs also have squares whose chromatic number is proportional to .
Although the chromatic number of the square of a nonplanar graph with maximum degree may be proportional to in the worst case, it is smaller for graphs of high girth, being bounded in this case.
Determining the minimum number of colors needed to color the square of a graph is NP-hard, even in the planar case.
Hamiltonicity
The cube of every connected graph necessarily contains a Hamiltonian cycle. It is not necessarily the case that the square of a connected graph is Hamiltonian, and it is NP-complete to determine whether the square is Hamiltonian. Nevertheless, by Fleischner's theorem, the square of a 2-vertex-connected graph is always Hamiltonian.
Computational complexity
The th power of a graph with vertices and edges may be computed in time by performing a breadth first search starting from each vertex to determine the distances to all other vertices, or slightly faster using more sophisticated algorithms. Alternatively, If is an adjacency matrix for the graph, modified to have nonzero entries on its main diagonal, then the nonzero entries of give the adjacency matrix of the th power of the graph, from which it follows that constructing th powers may be performed in an amount of time that is within a logarithmic factor of the time for matrix multiplication.
The th powers of trees can be recognized in time linear in the siz |
https://en.wikipedia.org/wiki/Alisher%20Dodov | Alisher Dodov (; born 4 August 1981) is a Tajikistani football player. He plays as goalkeeper for Regar-TadAZ Tursunzoda and for the Tajikistan national football team.
Career statistics
International
Statistics accurate as of match played 13 November 2016
References
1981 births
Living people
Tajikistan men's international footballers
Tajikistani men's footballers
Men's association football goalkeepers
Tajikistan Higher League players |
https://en.wikipedia.org/wiki/Assaf%20Naor | Assaf Naor (born May 7, 1975) is an Israeli American and Czech mathematician, computer scientist, and a professor of mathematics at Princeton University.
Academic career
Naor earned a baccalaureate from Hebrew University of Jerusalem in 1996 and a doctorate from the same university in 2002, under the supervision of Joram Lindenstrauss. He worked at Microsoft Research from 2002 until 2007, with an affiliated faculty position at the University of Washington, and joined the NYU faculty in 2006.
Research
Naor's research concerns metric spaces, their properties, and related algorithms, including improved upper bounds on the Grothendieck inequality, applications of this inequality, and research on metrical task systems.
Awards and honors
Naor won the Bergmann award of the United States – Israel Binational Science Foundation in 2007, and the Pazy award of the BSF in 2011. In 2012 he was one of four faculty winners of the Leonard Blavatnik Award of the New York Academy of Sciences, given to young scientists and engineers in New York, New Jersey, and Connecticut.
He won the Salem Prize in 2008 for "contributions to the structural theory of metric spaces and its applications to computer science", and in the same year was given a European Mathematical Society Prize (one of ten awarded to outstanding younger mathematicians). He won the Bôcher Memorial Prize in 2011 "for introducing new invariants of metric spaces and for applying his new understanding of the distortion between various metric structures to theoretical computer science". In 2012 he became a fellow of the American Mathematical Society.
He received the Nemmers Prize in Mathematics in 2018 and in 2019 the Ostrowski Prize.
He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Functional Analysis and Applications".
References
1975 births
Living people
20th-century American mathematicians
Czech mathematicians
Israeli mathematicians
Czech computer scientists
Israeli computer scientists
American computer scientists
Theoretical computer scientists
Einstein Institute of Mathematics alumni
Courant Institute of Mathematical Sciences faculty
Fellows of the American Mathematical Society
Functional analysts
Princeton University faculty
New York University faculty
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Eaton%27s%20inequality | In probability theory, Eaton's inequality is a bound on the largest values of a linear combination of bounded random variables. This inequality was described in 1974 by Morris L. Eaton.
Statement of the inequality
Let {Xi} be a set of real independent random variables, each with an expected value of zero and bounded above by 1 ( |Xi | ≤ 1, for 1 ≤ i ≤ n). The variates do not have to be identically or symmetrically distributed. Let {ai} be a set of n fixed real numbers with
Eaton showed that
where φ(x) is the probability density function of the standard normal distribution.
A related bound is Edelman's
where Φ(x) is cumulative distribution function of the standard normal distribution.
Pinelis has shown that Eaton's bound can be sharpened:
A set of critical values for Eaton's bound have been determined.
Related inequalities
Let {ai} be a set of independent Rademacher random variables – P( ai = 1 ) = P( ai = −1 ) = 1/2. Let Z be a normally distributed variate with a mean 0 and variance of 1. Let {bi} be a set of n fixed real numbers such that
This last condition is required by the Riesz–Fischer theorem which states that
will converge if and only if
is finite.
Then
for f(x) = | x |p. The case for p ≥ 3 was proved by Whittle and p ≥ 2 was proved by Haagerup.
If f(x) = eλx with λ ≥ 0 then
where inf is the infimum.
Let
Then
The constant in the last inequality is approximately 4.4634.
An alternative bound is also known:
This last bound is related to the Hoeffding's inequality.
In the uniform case where all the bi = n−1/2 the maximum value of Sn is n1/2. In this case van Zuijlen has shown that
where μ is the mean and σ is the standard deviation of the sum.
References
Probabilistic inequalities
Statistical inequalities |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Galatasaray%20S.K.%20season | The 2000–01 season was Galatasaray's 97th in existence and the 43rd consecutive season in the 1. Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Club
Board of directors
Elected: 25 March 2000
Facilities
Squad statistics
Players in / out
In
Out
1. Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Third round
Fourth round
Quarter-final
Semi-final
UEFA Super Cup
UEFA Champions League
Third qualifying round
First group stage
Second group stage
Quarter-finals
FIFA Club World Championship
Group B
Al-Hilal
Galatasaray
Olimpia
Palmeiras
Friendlies
Opel Master Cup
Atatürk Kupası
Attendance
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2000s in Istanbul
Galatasaray Sports Club 2000–01 season |
https://en.wikipedia.org/wiki/Explicit%20reciprocity%20law | In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use.
There are also several explicit reciprocity laws for various generalizations of the Hilbert symbol to higher local fields, p-divisible groups, and so on.
History
gave an explicit formula for the Hilbert symbol (α,β) in the case of odd prime powers, for some special values of α and β when the field is the (cyclotomic) extension of the p-adic numbers by a pnth root of unity. extended the formula of Artin and Hasse to more cases of α and β, and and extended Iwasawa's work to Lubin–Tate extensions of local fields.
gave an explicit formula for the Hilbert symbol for odd prime powers for general local fields. His formula was rather complicated which made it hard to use, and and found a simpler formula. simplified Vostokov's work and extended it to the case of even prime powers.
Examples
For archimedean local fields or in the unramified case the Hilbert symbol is easy to write down explicitly. The main problem is to evaluate it in the ramified case.
Archimedean fields
Over the complex numbers (a, b) is always 1.
Over the reals, the Hilbert symbol of odd degree is trivial, and the Hilbert symbol of even degree is given by (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative.
Unramified case: the tame Hilbert symbol
In the unramified case, when the order of the Hilbert symbol is coprime to the residue characteristic of the local field, the tame Hilbert symbol is given by
where ω(a) is the (q – 1)-th root of unity congruent to a and ord(a) is the value of the valuation of the local field, and n is the degree of the Hilbert symbol, and q is the order of the residue class field. The number n divides q – 1 because the local field contains the nth roots of unity by assumption.
As a special case, over the p-adics with p odd, writing and , where u and v are integers coprime to p, we have for the quadratic Hilbert symbol
, where
and the expression involves two Legendre symbols.
Ramified case
The simplest example of a Hilbert symbol in the ramified case is the quadratic Hilbert symbol over the 2-adic integers.
Over the 2-adics, again writing and , where u and v are odd numbers, we have for the quadratic Hilbert symbol
, where and
See also
Rational reciprocity law
Notes
References
Further reading
Algebraic number theory |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Japan | This page details football records in Japan. Unless otherwise stated, records are taken from the J.League.
J.League records (split-season era)
Most Titles
4, Kashima Antlers (1996, 1998, 2000, 2001)
Most Consecutive Titles
2, Joint record: Verdy Kawasaki- (1993–1994), Kashima Antlers (2000, 2001), Yokohama F. Marinos (2003, 2004)
J.League Division 1 Records (single-season era)
Most Titles
3, Kashima Antlers (2007, 2008, 2009)
Most Consecutive Titles
3, Kashima Antlers (2007–2009)
Most Second Place Finishes
3, Kawasaki Frontale (2006, 2008, 2009)
References
Football records and statistics in Japan
Football in Japan
Japan |
https://en.wikipedia.org/wiki/Ernesto%20Hern%C3%A1ndez%20Busto | Ernesto Hernández Busto (born 1968 in Havana, Cuba) is a Cuban writer living in Barcelona (Spain).
Biography
Born in Havana, Hernández Busto began university studies in mathematics in the former Soviet Union and returned to Havana to study Literature at the Pedagogical Institute.
He was a member of Paideia, an independent research group that in the late 1980s attempted a renewal of the Cuban cultural scene and culminated in becoming a dissident platform.
In 1992 he emigrated to Mexico, where he published regularly in the magazine Vuelta, directed by Octavio Paz, as well as other Mexican literary reviews. On two occasions, 1996 and 1998, he won the translation scholarship from FONCA.
He also formed part of the editorial board of the journal Poesía y poética, and worked for four years on editing its collection under the sponsorship of the Universidad Iberoamericana. The collection introduced to Mexico some key names in contemporary literature such as Andrea Zanzotto, Robert Creeley, Marina Tsvetaeva or João Cabral de Melo Neto, among others.
Since 1999 Hernández Busto resides in Barcelona, where he has worked as an editor, translator and journalist while writing literary criticism.
His book Perfiles Derechos was awarded the 2004 Essay Prize III "Casa de América" and selected by a jury consisting of Jorge Edwards, Josefina Aldecoa, José María Castellet, Jose Maria Lassalle and Manuel Martos.
He also has published Inventario De Saldos. Apuntes Sobre Literatura Cubana (Colibrí, Madrid, 2005). Several of his essays have been translated into English, French, and German.
In 2015 he published La ruta natural (Vaso Roto, Madrid, 2015), an amphibious book, part memoir, part essay, whose subject is the fragment. His most recent books are poetry books: Muda, Miel y hiel and Jardín de grava.
In Mexico and Spain he has published many translations from Italian, Russian, French and Portuguese.
His translations of poetry are anthologized in the two volumes of Cuaderno de traducciones (Primavera y Verano).
He co-edited the anthology El fin de los periódicos (The End Of Newspapers: Crisis And Challenges Of Daily Journalism) with Arcadi Espada (Duomo, Barcelona, 2009).
Since 2006 to 2016, Hernández Busto published Penúltimos Días, one of the most important websites on Cuban issues, with 87 contributors in 12 countries and over 14 million page views in the last five years.
Hernández Busto has participated in various forums about digital activism including Cyber Dissidents: Global Success and Challenge, organized in 2010 by Freedom House, the Berkman Center-Harvard University, and George W. Bush Institute; Internet at Liberty 2010 (organized by Google and the Central European University), and Personal Democracy Forum Latin America 2010, among others. Since 2010 to 2016 he has written the chapter on Cuba for the Freedom House global report on Internet freedom, Freedom on the Net.
He is a regular contributor to the Spanish newspaper El País, where he writ |
https://en.wikipedia.org/wiki/Complexification%20%28Lie%20group%29 | In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.
For compact Lie groups, the complexification, sometimes called the Chevalley complexification after Claude Chevalley, can be defined as the group of complex characters of the Hopf algebra of representative functions, i.e. the matrix coefficients of finite-dimensional representations of the group. In any finite-dimensional faithful unitary representation of the compact group it can be realized concretely as a closed subgroup of the complex general linear group. It consists of operators with polar decomposition , where is a unitary operator in the compact group and is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the compact Lie group.
Universal complexification
Definition
If is a Lie group, a universal complexification is given by a complex Lie group and a continuous homomorphism with the universal property that, if is an arbitrary continuous homomorphism into a complex Lie group , then there is a unique complex analytic homomorphism such that .
Universal complexifications always exist and are unique up to a unique complex analytic isomorphism (preserving inclusion of the original group).
Existence
If is connected with Lie algebra , then its universal covering group is simply connected. Let be the simply connected complex Lie group with Lie algebra , let be the natural homomorphism (the unique morphism such that is the canonical inclusion) and suppose is the universal covering map, so that is the fundamental group of . We have the inclusion , which follows from the fact that the kernel of the adjoint representation of equals its centre, combined with the equality
which holds for any . Denoting by the smallest closed normal Lie subgroup of that contains , we must now also have the inclusion . We define the universal complexification of as
In particular, if is simply connected, its universal complexification is just .
The map is obtained by passing to the quotient. Since is a surjective submersion, smoothness of the map implies smoothness of .
For non-connected Lie groups with identity component and component group , the extension
induces an extension
and the complex Lie group is a complexification of .
Proof of the universal property
The map indeed pos |
https://en.wikipedia.org/wiki/Solomon%20Herzenstein | Solomon Markovich Herzenstein (; 1854 – August 7, 1894) was a zoologist from the Russian Empire.
Biography
Herzenstein received a degree in natural sciences and mathematics from St. Petersburg University and was appointed as the custodian of the Zoological Museum of the Imperial Academy of Science in 1879 or 1880. He also supervised practical training at the University for Women. In 1880, 1884, and 1887, he was commissioned to travel to the Murman Coast of the Kola Peninsula to study the mollusks and fishes there.
His work, "Materialy k Faunye Murmanskavo Berega i Byelavo Morya," which was published in the Trudy of the in 1885, became a standard reference. He co-wrote Zamyetki po Ikhtologii Basseina Ryeki Amura (1887) and Nauchnye Rezultaty Puteshestvi Przevalskavo (1888–91) with N. L. Varpakhovski. He also wrote Ryby (St. Petersburg, 1888-91), and published "Ichthyologische Bemerkungen" in the (1890-92).
Species described
Acanthogobio guentheri (Herzenstein, 1892)
Gymnocypris potanini (Herzenstein, 1891)
Triplophysa alticeps (Herzenstein, 1888)
Triplophysa brachyptera (Herzenstein, 1888)
Triplophysa brevicauda (Herzenstein, 1888)
Triplophysa chondrostoma (Herzenstein, 1888)
Triplophysa crassicauda (Herzenstein, 1888)
Triplophysa incipiens (Herzenstein, 1888)
Triplophysa leptosoma (Herzenstein, 1888)
Triplophysa macropterus (Herzenstein, 1888)
Triplophysa orientalis (Herzenstein, 1888)
Triplophysa scleroptera (Herzenstein, 1888)
Triplophysa siluroides (Herzenstein, 1888)
Triplophysa stenura (Herzenstein, 1888)
Tribute
Gnathopogon herzensteini (Günther, 1896) was probably named in honor of Herzenstein, who named an Acanthogobio after Günther in 1892.
Asprocottus herzensteini (L. S. Berg, 1906), the Herzenstein's rough sculpin.
References
1854 births
1894 deaths
19th-century Jews from the Russian Empire
Zoologists from the Russian Empire
Ichthyologists from the Russian Empire
Scientists from Saint Petersburg
Saint Petersburg State University alumni |
https://en.wikipedia.org/wiki/Design%20%28disambiguation%29 | Design is the creation of a plan or specification for the construction of an object or a system.
Design may also refer to:
Science and mathematics
Block design
Combinatorial design
Design of experiments
Engineering design process
Randomized block design, in statistics
Entertainment
Design (band), a 1970s British vocal group
The Design, a 2005 album by Into the Moat
Design, a record label founded by Marco Carola
Other uses
Communication design
Fashion design
Game design
Graphic design
Interior design
Scenic design
See also
Design methods
Designer (disambiguation)
Interior Design (disambiguation) |
https://en.wikipedia.org/wiki/Casey%20Dellacqua%20career%20statistics | This is a list of the main career statistics of professional Australian tennis player, Casey Dellacqua. To date, Dellacqua has won eight career doubles titles: one mixed doubles title, partnering with Scott Lipsky, at the 2011 French Open, and seven WTA Tour doubles titles including one Premier Mandatory title with Yaroslava Shvedova at the 2015 Madrid Open. Other highlights of Dellacqua's career include reaching the doubles finals of all four Grand Slam events; a quarterfinal finish in singles at the 2014 Indian Wells Open and fourth round appearances at the 2008 Australian Open, 2014 Australian Open, and US Open, respectively. Dellacqua achieved a career-high singles ranking of world No. 26, on 29 September 2014, and later a career-high doubles ranking of No. 3, on 1 February 2016.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Billie Jean King Cup, Hopman Cup and Olympic Games are included in win–loss records.
Singles
Doubles
Mixed doubles
Significant finals
Grand Slam tournaments
Doubles: 7 (7 runner-ups)
Mixed doubles: 1 (title)
Premier Mandatory/Premier 5 tournaments
Doubles: 2 (1 title, 1 runner-up)
WTA Tour finals
Doubles: 20 (7 titles, 13 runner-ups)
ITF Circuit finals
Singles: 27 (22 titles, 5 runner-ups)
Doubles: 33 (23 titles, 10 runner-ups)
Junior Grand Slam tournament finals
Doubles: 1 (title)
Fed Cup participation
Singles: 11 (6–5)
Doubles: 17 (13–4)
See also
WTA Tour records
Casey Dellacqua
Australia Fed Cup team
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Contou-Carr%C3%A8re%20symbol | In mathematics, the Contou-Carrère symbol 〈a,b〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring k, taking values in the group of units of k. It was introduced by .
Definition
If k is an Artinian local ring, then any invertible formal Laurent series a with coefficients in k can be written uniquely as
where w(a) is an integer, the elements ai are in k, and are in m if i is negative, and is a unit if i = 0.
The Contou-Carrère symbol 〈a,b〉 of a and b is defined to be
References
Number theory |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Galatasaray%20S.K.%20season | The 2001–02 season was Galatasaray's 98th in existence and the 44th consecutive season in the Süper Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
Süper Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Third round
UEFA Champions League
Second qualifying round
Third qualifying round
Group stage
Second group stage
FIFA Club World Championship
As winners of the 2000 UEFA Super Cup, Galatasaray was one of the 12 teams that were invited to the 2001 FIFA Club World Championship, which would be hosted in Spain from 28 July to 12 August 2001. However, the tournament was canceled, primarily due to the collapse of ISL, which was a marketing partner of FIFA at the time.
Group stage
Friendlies
Attendances
References
Turkish football championship-winning seasons
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2001–02 in Turkish football
2000s in Istanbul
Galatasaray Sports Club 2001–02 season |
https://en.wikipedia.org/wiki/Kenneth%20A.%20Ross | Kenneth Allen Ross (born January 21, 1936) is a mathematician and an emeritus professor of mathematics at the University of Oregon. He served as an associate editor for Mathematics Magazine. He was president of the Mathematical Association of America from 1995 to 1996. He is a recipient of the Charles Y. Hu Award for distinguished service to mathematics.
Selected publications
Comfort, W. W.; Ross, Kenneth A. "Pseudocompactness and uniform continuity in topological groups", Pacific J. Math. 16 1966 483–496.
Ross, Kenneth A. Elementary Analysis: The Theory of Calculus, Springer-Verlag New York, 1980. Second edition, 2013, xi+409 pp.
Ken Ross, A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans, Pi Press New York, 2004, xv+189 pp. Second edition, Plume, Penquin Group, 2007, xv+206 pp.
López, Jorge M.; Ross, Kenneth A. Sidon sets, Lecture Notes in Pure and Applied Mathematics, Vol. 13. Marcel Dekker, Inc., New York, 1975. v+193 pp.
References
External links
Ken Ross's homepage
University of Oregon faculty
20th-century American mathematicians
21st-century American mathematicians
Living people
Place of birth missing (living people)
1936 births
University of Washington alumni |
https://en.wikipedia.org/wiki/Mario%20Kvesi%C4%87 | Mario Kvesić (born 12 January 1992) is a Bosnian professional footballer who plays as a midfielder for Slovenian PrvaLiga club Celje.
Career statistics
References
External links
1992 births
Living people
People from Široki Brijeg
Sportspeople from West Herzegovina Canton
Men's association football midfielders
Bosnia and Herzegovina men's footballers
Bosnia and Herzegovina men's youth international footballers
Bosnia and Herzegovina men's under-21 international footballers
NK Široki Brijeg players
RNK Split players
FC Erzgebirge Aue players
1. FC Magdeburg players
NK Olimpija Ljubljana (2005) players
Pohang Steelers players
NK Celje players
Premier League of Bosnia and Herzegovina players
Croatian Football League players
3. Liga players
2. Bundesliga players
Slovenian PrvaLiga players
K League 1 players
Bosnia and Herzegovina expatriate men's footballers
Expatriate men's footballers in Germany
Bosnia and Herzegovina expatriate sportspeople in Germany
Expatriate men's footballers in Slovenia
Bosnia and Herzegovina expatriate sportspeople in Slovenia
Expatriate men's footballers in South Korea
Bosnia and Herzegovina expatriate sportspeople in South Korea
Bosnia and Herzegovina people of Croatian descent |
https://en.wikipedia.org/wiki/Robert%20D.%20Russell | Robert D. (Bob) Russell is professor of mathematics at Simon Fraser University.
Russell together with Uri Ascher and Robert Mattheij is the author of the seminal Numerical Solution of Boundary Value Problems for Ordinary Differential Equations which was subsequently republished as a SIAM Classic. His latest book is Adaptive Moving Mesh Methods with Weizhang Huang
In 2009 Russell was made a SIAM Fellow for his contributions to the numerical analysis of differential equations.
Notable publications
Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1988) with Uri M. Ascher and Robert M. Mattheij.
Adaptive Moving Mesh Methods (2011) with Weizhang Huang.
References
PhD students
Luca Dieci, University of New Mexico, 1986
Yuhe Ren, Simon Fraser University, 1991
Lixin Liu, Simon Fraser University, 1993
Kossi Edoh, Simon Fraser University, 1995
Daryl Hepting, Simon Fraser University, 1999
Michael Lunney, Simon Fraser University, 2000
Shaohua Chen, Simon Fraser University, 2000
Ronald Haynes, Simon Fraser University, 2003
Benjamin Ong, Simon Fraser University, 2007
Mohamed Sulman, Simon Fraser University, 2008
Xiangmin Xu,Simon Fraser University, 2008
External links
at Simon Fraser University
Numerical analysts
Canadian mathematicians
Living people
Fellows of the Society for Industrial and Applied Mathematics
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Kemeny%27s%20constant | In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's stationary distribution. Surprisingly, this quantity does not depend on which starting state i is chosen. It is in that sense a constant, although it is different for different Markov chains. When first published by John Kemeny in 1960 a prize was offered for an intuitive explanation as to why the quantity was constant.
Definition
For a finite ergodic Markov chain with transition matrix P and invariant distribution π, write mij for the mean first passage time from state i to state j (denoting the mean recurrence time for the case i = j). Then
is a constant and not dependent on i.
Prize
Kemeny wrote, (for i the starting state of the Markov chain) “A prize is offered for the first person to give an intuitively plausible reason for the above sum to be independent of i.” Grinstead and Snell offer an explanation by Peter Doyle as an exercise, with solution “he got it!”
In the course of a walk with Snell along Minnehaha Avenue in Minneapolis in the fall of 1983, Peter Doyle suggested the following explanation for the constancy of Kemeny's constant. Choose a target state according to the fixed vector w. Start from state i and wait until the time T that the target state occurs for the first time. Let Ki be the expected value of T. Observe that
and hence
By the maximum principle, Ki is a constant. Should Peter have been given the prize?
References
Markov processes |
https://en.wikipedia.org/wiki/Kelly%20network | In queueing theory, a discipline within the mathematical theory of probability, a Kelly network is a general multiclass queueing network. In the network each node is quasireversible and the network has a product-form stationary distribution, much like the single-class Jackson network.
The model is named after Frank Kelly who first introduced the model in 1975 in his paper Networks of Queues with Customers of Different Types.
References
Queueing theory |
https://en.wikipedia.org/wiki/Math-O-Vision | Math-O-Vision is an applied mathematics movie contest open to students who are legal residents of the 50 United States and the District of Columbia, at least 13 years of age and are registered in high school (grades 9-12) or equivalent home school program at time of entry. Movies are created using a wide variety of techniques, including animation. The contest is sponsored by the William H. Neukom Institute for Computational Science as well as the Dartmouth College Math Department based in Hanover, New Hampshire. The competition awarded a total of $8,500 in prizes in its first year, 2012-13. The contest began in 2012 with approximately 50 movies entered. Inaugural year winners were announced May 15, 2013. The second year of the contest began November 1, 2013 and ended May 15, 2014.
Contest
Teams are challenged to create 4-minute movies which are uploaded online. These movies are to explain the power of mathematics in the world around them in a creative way. This challenge was inspired by the 1959 Disney animated short Donald in MathMagic Land.
Registration is open to high school students in the United States over 13 years of age. There is no cost to register or participate in Math-O-Vision.
The judging process involves an open online voting period and judges. Judges include a wide variety of academics, actors, and animators. They include Alan Alda, Tom Sito, Dan Rockmore, Lorie Loeb, Ge Wang and Steven Strogatz.
Math-O-Vision awards a $4,000 First Prize, $2,000 Second Prize and $1,000 Third Prize. Honorable Mentions are also awarded.
References
External links
Mathematics competitions |
https://en.wikipedia.org/wiki/Robert%20Gambill | Robert Gambill (born March 31, 1955 in Indianapolis) is an opera singer (Heldentenor).
Biography
Gambill studied mathematics at Purdue University (1973-1976) before becoming an exchange student at Hamburg University in Germany, where he added German studies to his curriculum. He enrolled at the Hochschule für Musik und Theater in Hamburg where he studied voice with Prof. Hans Kagel. At 25 he made his La Scala debut in the leading role of Michael in the world premiere of Karlheinz Stockhausen's Donnerstag aus Licht, directed by Luca Ronconi.
He remained in Europe, and in 1984 joined the ensemble of the Zurich Opera in Switzerland. For three years he sang the leading lyric and belcanto tenor roles in operas such as Don Giovanni, The Magic Flute, The Barber of Seville and Die Lustigen Weiber von Windsor. His international career kicked off after he sang the role of Lindoro in Rossini's The Italian Girl in Algiers in 1987, directed by Michael Hampe.
It was followed by engagements at the leading European and international opera houses including the Vienna State Opera, the Royal Opera Covent Garden in London, the Opera in Paris, the Munich State Opera and the Metropolitan Opera in New York.
In 1995, he successfully made the change into the dramatic German repertoire. He studied with Prof. Irmgard Hartmann-Dressler and was quickly recognized with appearances as Painter in Berg's Lulu at the Salzburg Festival and Narraboth in Strauss' Salome at both the Stuttgart State Opera and London's Royal Opera House. He continued as a Heldentenor with acclaimed and ongoing performances as Wagner's Tannhäuser, Tristan, Siegmund and Parsifal on opera and concert stages around the world.
He sang the heroic tenor parts under the baton of important conductors such as Riccardo Muti, Daniel Barenboim, Claudio Abbado, Wolfgang Sawallisch, Giuseppe Sinopoli, Simon Rattle or Zubin Mehta at the opera houses in New York, London, Paris, San Francisco, Chicago, Berlin, Munich, Vienna, Milan as well as at Carnegie Hall and at the festivals in Salzburg (Easter & Summer), Aix-en-Provence, Tanglewood, Glyndebourne and many others.
Teaching
2006 Robert Gambill became a Professor at the Universität der Künste (UdK) in Berlin.
Discography LP/CD
Donnerstag aus Licht, Stockhausen, DGG 1979
Stabat Mater, Gioacchino Rossini, Catherine Malfitano, Agnes Baltsa, Robert Gambill, Gwynne Howell; Riccardo Muti, EMI 1981
Acis und Galatea, Händel/Mozart, Edith Mathis, Anthony Rolf Johnson, Robert Gambill, Robert Lloyd: Peter Schreier, Orfeo 1982
Manon Lescaut, Giacomo Puccini, Mirella Freni, Plácido Domingo, Renato Bruson, Kurt Rydl, Robert Gambill: Giuseppe Sinopoli, DGG 1984
Der Messias, Georg Friedrich Händel, Lucia Popp, Brigette Fassbender, Robert Gambill, Robert Holl: Neville Marriner, EMI 1984
Les Ballets Russe, Vol. 6, Igor Stravinsky, "Pulcinella", Arleen Auger, Robert Gambill, Gerolf Scheder: Christopher Hogwood, HÄNSSLER CLASSIC 1985
Paradis und die Peri, Robert Schumann, Edit |
https://en.wikipedia.org/wiki/List%20of%20AFL%20debuts%20in%202000 | This is a listing of Australian rules footballers who made their senior debut for an Australian Football League (AFL) club in 2000.
References
Australian rules football records and statistics
Australian rules football-related lists
2000 in Australian rules football |
https://en.wikipedia.org/wiki/Dershowitz%E2%80%93Manna%20ordering | In mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna. It is often used in context of termination of programs or term rewriting systems.
Suppose that is a well-founded partial order and let be the set of all finite multisets on . For multisets we define the Dershowitz–Manna ordering as follows:
whenever there exist two multisets with the following properties:
,
,
, and
dominates , that is, for all , there is some such that .
An equivalent definition was given by Huet and Oppen as follows:
if and only if
, and
for all in , if then there is some in such that and .
References
. (Also in Proceedings of the International Colloquium on Automata, Languages and Programming, Graz, Lecture Notes in Computer Science 71, Springer-Verlag, pp. 188–202 [July 1979].)
.
.
Formal languages
Logic in computer science
Rewriting systems |
https://en.wikipedia.org/wiki/G/G/1%20queue | In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times have a (different) general distribution. The evolution of the queue can be described by the Lindley equation.
The system is described in Kendall's notation where the G denotes a general distribution for both interarrival times and service times and the 1 that the model has a single server. Different interarrival and service times are considered to be independent, and sometimes the model is denoted GI/GI/1 to emphasise this. The numerical solution for the GI/G/1 can be obtained by discretizing the time.
Waiting time
Kingman's formula gives an approximation for the mean waiting time in a G/G/1 queue. Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution which can be solved using the Wiener–Hopf method.
Multiple servers
Few results are known for the general G/G/k model as it generalises the M/G/k queue for which few metrics are known. Bounds can be computed using mean value analysis techniques, adapting results from the M/M/c queue model, using heavy traffic approximations, empirical results or approximating distributions by phase type distributions and then using matrix analytic methods to solve the approximate systems.
In a G/G/2 queue with heavy-tailed job sizes, the tail of the delay time distribution is known to behave like the tail of an exponential distribution squared under low loads and like the tail of an exponential distribution for high loads.
References
Single queueing nodes |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Galatasaray%20S.K.%20season | The 2002–03 season was Galatasaray's 99th in existence and the 45th consecutive season in the Süper Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season.
Squad statistics
Players in / out
In
Out
Süper Lig
Standings
Matches
Türkiye Kupası
Kick-off listed in local time (EET)
Second round
Third round
Quarter-final
UEFA Champions League
Group stage
Friendlies
Attendance
Sold season tickets: 7,547
References
Galatasaray S.K. (football) seasons
Galatasaray S.K.
2000s in Istanbul
Galatasaray Sports Club 2002–03 season |
https://en.wikipedia.org/wiki/Abhaya%20Indrayan | Abhaya Indrayan (born 11 November 1945) is an Indian professor and researcher of Biostatistics. He had worked with different organizations and universities, including Delhi University College of Medical Sciences and the World Health Organization.
Abhaya resides in Delhi NCR, India after his retirement. He is married and has two children.
Early life and education
Abhaya Indrayan was born on 11 November 1945, in Meerut, India. He was born during the time that India was fighting for its freedom from the British rule. That was the reason why his father, who was a freedom fighter, was jailed repeatedly for long periods.
Abhaya took his early education in Meerut from N.A.S. Inter. College and Meerut College. In 1977, he received his master's degree and Doctoral degree from Ohio State University in Columbus, Ohio, USA.
Career
Abhaya Indrayan was the founding Professor and Head of the Department of Biostatistics and Medical Informatics in Delhi University College of Medical Sciences. In 1995, his department was set up as an independent division, and was upgraded to a full department in 2005.
During his tenure in the college since 1979, he had been the Sports Adviser, Coordinator of Medical Education Unit, Chairman Computer Committee, Convenor Souvenir Committee, Incharge Annual Reports and held several other important assignments. He also taught online courses for the students of Institute of Statistics Education in Arlington County, Virginia.
He has more than 200 publications to his credit, including the books Medical Biostatistics and Concise Encyclopedia of Biostatistics for Medical Professionals. A partial list of his publications appears at the website of Indian Academy of Sciences. Among his other significant works are: smoking index at Collection of Biostatistics Research Archive of Berkeley Electronic Press, which is among the top 5 downloads; and estimates and projections of cardiovascular and diabetes cases in India, which are quoted in the Government of India's official estimates at National Health Profile.
He stayed in the institution until retiring in 2010, wherein he attained the age of 65.
Achievements
Abhaya Indrayan collaborated with the World Health Organization for several projects including National Burden of Disease Studies: a Practical Guide, Teaching Health Statistics, and 11 Health Questions about the 11 SEAR Countries, and served as Temporary Adviser to their Bi-Regional Consultation. He was a technical editor of their biregional report Health in Asia and the Pacific and participated in their debate on Health Systems Performance Assessment. He has completed 32 assignments for the World Health Organization, 3 for the World Bank, 3 for UNAIDS and 2 for Danish Assistance to the National Program for Prevention and Control of Blindness. He has proposed Statistical Medicine as a new emerging medical specialty.
Honors and awards
Elected Fellow of the Indian Academy of Sciences (2011).
Fellow of the Royal Statistical Soci |
https://en.wikipedia.org/wiki/Willie%20Anku | William Oscar "Willie" Anku (25 July 1949 – 1 February 2010) was a Ghanaian music theorist, ethnomusicologist, composer, and performer. His work combined Western set theory with computer programming and experience in working with performers of various West African musical traditions to create a comprehensive theory of African rhythm. He was "unique among Africa-based music theorists in attracting the attention of the US-based Society for Music Theory," being invited to give plenary lectures and receiving tributes from prominent US-based theorists.
Music theory
Anku rejected the relevance of simple concepts of polymeter in understanding West African music.
He is noted for attempting to create a more natural, but non-indigenous system of music notation to the study of African music. Anku's circular notation shows the various "combinatoric aspects of [a] pattern relative to different metrical positions, based on how the rhythmic pattern is aligned with [a] regulative metric pattern."
Bode Omojola lists Anku among five contemporary scholars most influencing ideas of African Rhythm. His work was cited as influential on Godfried Toussaint's general geometric theory of musical timelines.
Agawu described his approach to West-African music theory as "structural set analysis," the title of two of his short books. He defended the analytical approach to African music in a 2007 interview on Ghanaian MetroTV.
In addition to its impact on understanding African music, Anku's theories have been cited in the study of György Ligeti.
Life and education
Willie Anku came from Gbadzeme in the Avatime Traditional Area of the Volta Region of Ghana.
He received his Master of Music Education from the University of Montana, Missoula in 1976; MA and PhD in Ethnomusicology from the University of Pittsburgh in 1986 and 1988 respectively. He was head of the School of Performing Arts at the University of Ghana, Legon until just prior to his death.
Professor Anku was involved in a motor accident on 20 January 2010 and died 2 weeks later at the Korle Bu Teaching Hospital. He is survived by his wife, Madam Eva Ebeli, and three children.
References
1949 births
2010 deaths
Ghanaian musicians
Music educators
Music theorists
Road incident deaths in Ghana
University of Montana alumni |
https://en.wikipedia.org/wiki/Order-4%20heptagonal%20tiling | In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as [1+,7,1+,4], removing two of three mirrors (passing through the heptagon center) in the [7,4] symmetry.
The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Heptagonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-4 tilings
Regular tilings |
https://en.wikipedia.org/wiki/Truncated%20order-4%20heptagonal%20tiling | In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.
Constructions
There are two uniform constructions of this tiling, first by the [7,4] kaleidoscope, and second by removing the last mirror, [7,4,1+], gives [7,7], (*772).
Symmetry
There is only one simple subgroup [7,7]+, index 2, removing all the mirrors. This symmetry can be doubled to 742 symmetry by adding a bisecting mirror.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Heptagonal tilings
Hyperbolic tilings
Isogonal tilings
Order-4 tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Tetraheptagonal%20tiling | In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.
Symmetry
Related polyhedra and tiling
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20order-7%20square%20tiling | In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-7 tilings
Square tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-7%20square%20tiling | In geometry, the order-7 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,7}.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
This tiling is a part of regular series {n,7}:
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-7 tilings
Regular tilings
Square tilings |
https://en.wikipedia.org/wiki/Rhombitetraheptagonal%20tiling | In geometry, the rhombitetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{4,7}. It can be seen as constructed as a rectified tetraheptagonal tiling, r{7,4}, as well as an expanded order-4 heptagonal tiling or expanded order-7 square tiling.
Dual tiling
The dual is called the deltoidal tetraheptagonal tiling with face configuration V.4.4.4.7.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20tetraheptagonal%20tiling | In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.
Images
Poincaré disk projection, centered on 14-gon:
Symmetry
The dual to this tiling represents the fundamental domains of [7,4] (*742) symmetry. There are 3 small index subgroups constructed from [7,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Uniform tilings in hyperbolic plane
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Truncated tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Snub%20tetraheptagonal%20tiling | In geometry, the snub tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,4}.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Dual tiling
The dual is called an order-7-4 floret pentagonal tiling, defined by face configuration V3.3.4.3.7.
Related polyhedra and tiling
The snub tetraheptagonal tiling is sixth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Chiral figures
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-7%20heptagonal%20tiling | In geometry, the order-7 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,7}, constructed from seven heptagons around every vertex. As such, it is self-dual.
Related tilings
This tiling is a part of regular series {n,7}:
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Heptagonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-7 tilings
Regular tilings
Self-dual tilings |
https://en.wikipedia.org/wiki/Truncated%20order-7%20heptagonal%20tiling | In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.
Related tilings
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Heptagonal tilings
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-7 tilings
Truncated tilings |
https://en.wikipedia.org/wiki/Snub%20heptaheptagonal%20tiling | In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.
Images
Drawn in chiral pairs, with edges missing between black triangles:
Symmetry
A double symmetry coloring can be constructed from [7,4] symmetry with only one color heptagon.
Related tilings
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Snub tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-4%20octagonal%20tiling | In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-4 tilings
Regular tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Truncated%20order-4%20octagonal%20tiling | In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.
Constructions
There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).
Dual tiling
Symmetry
The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].
One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).
The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-4 tilings
Truncated tilings
Uniform tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Tetraoctagonal%20tiling | In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Constructions
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Symmetry
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
Related polyhedra and tiling
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Order-8%20square%20tiling | In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as [1+,8,8,1+], (*4444 orbifold) removing two of three mirrors (passing through the square center) in the [8,8] symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.
This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4[3]}, :
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).
See also
Square tiling
Uniform tilings in hyperbolic plane
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-8 tilings
Regular tilings
Square tilings |
https://en.wikipedia.org/wiki/Rhombitetraoctagonal%20tiling | In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
Constructions
There are two uniform constructions of this tiling, one from [8,4] or (*842) symmetry, and secondly removing the mirror middle, [8,1+,4], gives a rectangular fundamental domain [∞,4,∞], (*4222).
Symmetry
A lower symmetry construction exists, with (*4222) orbifold symmetry. This symmetry can be seen in the dual tiling, called a deltoidal tetraoctagonal tiling, alternately colored here. Its fundamental domain is a Lambert quadrilateral, with 3 right angles.
With edge-colorings there is a half symmetry form (4*4) orbifold notation. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{4,8}. The squares can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, an order-8 square tiling results, constructed as a snub tetraoctagonal tiling, .
Related polyhedra and tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Truncated%20tetraoctagonal%20tiling | In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
Dual tiling
Symmetry
There are 15 subgroups constructed from [8,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,8,1+,4,1+] (4242) is the commutator subgroup of [8,4].
A larger subgroup is constructed as [8,4*], index 8, as [8,4+], (4*4) with gyration points removed, becomes (*4444) or (*44), and another [8*,4], index 16 as [8+,4], (8*2) with gyration points removed as (*22222222) or (*28). And their direct subgroups [8,4*]+, [8*,4]+, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).
Related polyhedra and tilings
From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,4] symmetry, and 7 with subsymmetry.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings |
https://en.wikipedia.org/wiki/Order-8%20octagonal%20tiling | In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} (eight octagons around each vertex) and is self-dual.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *44444444 with 8 order-4 mirror intersections. In Coxeter notation can be represented as [8,8*], removing two of three mirrors (passing through the octagon center) in the [8,8] symmetry.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-8 tilings
Regular tilings
Self-dual tilings
Octagonal tilings |
https://en.wikipedia.org/wiki/Truncated%20order-8%20octagonal%20tiling | In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.
Uniform colorings
This tiling can also be constructed in *884 symmetry with 3 colors of faces:
Related polyhedra and tiling
Symmetry
The dual of the tiling represents the fundamental domains of (*884) orbifold symmetry. From [(8,8,4)] (*884) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 882 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1+,8,1+,8,1+,4)] (442442) is the commutator subgroup of [(8,8,4)].
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-8 tilings
Truncated tilings
Uniform tilings
Octagonal tilings |
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