source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Yoshida%20Mitsuyoshi | , also known as Yoshida Kōyū, was a Japanese mathematician in the Edo period. His popular and widely disseminated published work made him the most well known writer about mathematics in his lifetime.
He was a student of Kambei Mori (also known as Mōri Shigeyoshi). Along with Imamura Chishō and Takahara Kisshu, Yoshida became known to his contemporaries as one of "the Three Arithmeticians."
Yoshida was the author of the oldest extant Japanese mathematical text. The 1627 work was named Jinkōki. The work dealt with the subject of soroban arithmetic, including square and cube root operations.
Selected works
In a statistical overview derived from writings by and about Yoshida Mitsuyoshi, OCLC/WorldCat encompasses roughly 20+ works in 30+ publications in 1 language and 40+ library holdings.
1643 — , OCLC 22023455088
1659 — , OCLC 22057549632
1818 — , OCLC 22057124215
1850 — , OCLC 22055982082
See also
Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in shinto shrines
Soroban, a Japanese abacus
Japanese mathematics
Notes
References
Endō Toshisada (1896). . Tōkyō: _. OCLC 122770600
Horiuchi, Annick. (1994). Les Mathematiques Japonaises a L'Epoque d'Edo (1600–1868): Une Etude des Travaux de Seki Takakazu (?-1708) et de Takebe Katahiro (1664–1739). Paris: Librairie Philosophique J. Vrin. ; OCLC 318334322
Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. ; OCLC 25709270
David Eugene Smith and Yoshio Mikami. (1914). A History of Japanese Mathematics. Chicago: Open Court Publishing. OCLC 1515528 -- note alternate online, full-text copy at archive.org
External links
Sangaku
17th-century Japanese mathematicians
1598 births
1672 deaths
Japanese writers of the Edo period |
https://en.wikipedia.org/wiki/Cleveland%20Cavaliers%20all-time%20roster | The following is a list of players, both past and current, who appeared at least in one game for the Cleveland Cavaliers NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
|-
|align="left"| || align="center"|F || align="left"|Louisville || align="center"|1 || align="center"| || 19 || 194 || 19 || 5 || 32 || 10.2 || 1.0 || 0.3 || 1.7 || align=center|
|-
|align="left"| || align="center"|F || align="left"|South Florida || align="center"|1 || align="center"| || 7 || 43 || 12 || 1 || 17 || 6.1 || 1.7 || 0.1 || 2.4 || align=center|
|-
|align="left" bgcolor="#FBCEB1"|* || align="center"|C || align="left"|Texas || align="center"|3 || align="center"|– || 175 || 5,574 || 1,774 || 291 || 2,545 || 31.9 || 10.1 || 1.7 || 14.5 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Weber State || align="center"|1 || align="center"| || 3 || 10 || 1 || 0 || 3 || 3.3 || 0.3 || 0.0 || 1.0 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Penn State || align="center"|1 || align="center"| || 28 || 357 || 52 || 9 || 77 || 12.8 || 1.9 || 0.3 || 2.8 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|UNLV || align="center"|1 || align="center"| || 12 || 79 || 20 || 5 || 11 || 6.6 || 1.7 || 0.4 || 0.9 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Blinn || align="center"|1 || align="center"| || 12 || 114 || 31 || 5 || 28 || 9.5 || 2.6 || 0.4 || 2.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Saint Joseph's || align="center"|1 || align="center"| || 23 || 171 || 37 || 16 || 79 || 7.4 || 1.6 || 0.7 || 3.4 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Kentucky || align="center"|2 || align="center"|– || 104 || 2,817 || 296 || 372 || 1,179 || 27.1 || 2.8 || 3.6 || 11.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Virginia || align="center"|1 || align="center"| || 3 || 47 || 6 || 6 || 13 || 15.7 || 2.0 || 2.0 || 4.3 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Fresno State || align="center"|2 || align="center"|– || 53 || 727 || 114 || 42 || 296 || 13.7 || 2.2 || 0.8 || 5.6 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Lithuania || align="center"|1 || align="center"| || 6 || 9 || 4 || 0 || 0 || 1.5 || 0.7 || 0.0 || 0.0 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Tennessee Tech || align="center"|1 || align="center"| || 12 || 52 || 9 || 4 || 19 || 4.3 || 0.8 || 0.3 || 1.6 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Boston College || align="center"|5 || align="center"|– || 375 || 9,757 || 1,070 || 2,311 || 3,542 || 26.0 || 2.9 || 6.2 || 9.4 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Texas Tech || align="center"|1 || align="center"| || 50 || 977 || 241 || 37 || 272 || 19.5 || 4.8 || 0.7 || 5.4 || align=center|
|-
|align="le |
https://en.wikipedia.org/wiki/Variance%20inflation%20factor | In statistics, the variance inflation factor (VIF) is the ratio (quotient) of the variance of estimating some parameter in a model that includes multiple other terms (parameters) by the variance of a model constructed using only one term. It quantifies the severity of multicollinearity in an ordinary least squares regression analysis. It provides an index that measures how much the variance (the square of the estimate's standard deviation) of an estimated regression coefficient is increased because of collinearity. Cuthbert Daniel claims to have invented the concept behind the variance inflation factor, but did not come up with the name.
Definition
Consider the following linear model with k independent variables:
Y = β0 + β1 X1 + β2 X 2 + ... + βk Xk + ε.
The standard error of the estimate of βj is the square root of the j + 1 diagonal element of s2(X′X)−1, where s is the root mean squared error (RMSE) (note that RMSE2 is a consistent estimator of the true variance of the error term, ); X is the regression design matrix — a matrix such that Xi, j+1 is the value of the jth independent variable for the ith case or observation, and such that Xi,1, the predictor vector associated with the intercept term, equals 1 for all i. It turns out that the square of this standard error, the estimated variance of the estimate of βj, can be equivalently expressed as:
where Rj2 is the multiple R2 for the regression of Xj on the other covariates (a regression that does not involve the response variable Y). This identity separates the influences of several distinct factors on the variance of the coefficient estimate:
s2: greater scatter in the data around the regression surface leads to proportionately more variance in the coefficient estimates
n: greater sample size results in proportionately less variance in the coefficient estimates
: greater variability in a particular covariate leads to proportionately less variance in the corresponding coefficient estimate
The remaining term, 1 / (1 − Rj2) is the VIF. It reflects all other factors that influence the uncertainty in the coefficient estimates. The VIF equals 1 when the vector Xj is orthogonal to each column of the design matrix for the regression of Xj on the other covariates. By contrast, the VIF is greater than 1 when the vector Xj is not orthogonal to all columns of the design matrix for the regression of Xj on the other covariates. Finally, note that the VIF is invariant to the scaling of the variables (that is, we could scale each variable Xj by a constant cj without changing the VIF).
Now let , and without losing generality, we reorder the columns of X to set the first column to be
.
By using Schur complement, the element in the first row and first column in is,
Then we have,
Here is the coefficient of regression of dependent variable over covariate . is the corresponding residual sum of squares.
Calculation and analysis
We can calculate k different VIFs (one for each Xi) in thr |
https://en.wikipedia.org/wiki/ICTP%20Ramanujan%20Prize | The DST-ICTP-IMU Ramanujan Prize for Young Mathematicians from Developing Countries is a mathematics prize awarded annually by the International Centre for Theoretical Physics in Italy. The prize is named after the Indian mathematician Srinivasa Ramanujan. It was founded in 2004, and was first awarded in 2005.
The prize is awarded to a researcher from a developing country less than 45 years of age who has conducted outstanding research in a developing country. The prize is supported by the Ministry of Science and Technology (India) and Norwegian Academy of Science and Letters through the Abel Fund, with the cooperation of the International Mathematical Union.
List of winners
See also
SASTRA Ramanujan Prize
List of mathematics awards
References
External links
International Mathematical Union
Mathematics awards
Awards established in 2004
Srinivasa Ramanujan
International awards |
https://en.wikipedia.org/wiki/James%20Campbell%20%28English%20footballer%29 | James Campbell was a professional footballer who made one appearance in the Football League as a goalkeeper while on trial with Huddersfield Town.
Career statistics
References
English men's footballers
Men's association football goalkeepers
English Football League players
Huddersfield Town A.F.C. players
Year of birth missing
Year of death missing
Footballers from Greater London |
https://en.wikipedia.org/wiki/ICIAM | ICIAM may refer to:
International Council for Industrial and Applied Mathematics, an organisation for professional applied mathematics societies.
International Congress on Industrial and Applied Mathematics, a four-yearly international meeting. |
https://en.wikipedia.org/wiki/International%20Council%20for%20Industrial%20and%20Applied%20Mathematics | The International Council for Industrial and Applied Mathematics (ICIAM) is an organisation for professional applied mathematics societies and related organisations. The current (2020) President is Ya-xiang Yuan.
History
Until 1999 the Council was known as the Committee for International Conferences on Industrial and Applied Mathematics (CICIAM). Formed in 1987 with the start of the ICIAM conference series, this committee represented the leaders of four applied mathematics societies: the Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), in Germany, the Institute of Mathematics and its Applications (IMA), in England, the Society for Industrial and Applied Mathematics (SIAM), in the USA, and the Société de Mathématiques Appliquées et Industrielles (SMAI), in France. The first two presidents of the council, Roger Temam and Reinhard Mennicken, oversaw the addition of several other societies as members and associate members of the council; as of 2015 it had 21 full members and 26 associate members. Past Presidents include Olavi Nevanlinna, Ian Sloan, Rolf Jeltsch, Barbara Keyfitz, and María J. Esteban.
Congress
ICIAM organizes the four-yearly International Congress on Industrial and Applied Mathematics, the first of which was held in 1987. The most recent congress was in 2019 in Valencia (Spain), and the next will be in 2023 in Tokyo (Japan). It also sponsors several prizes, awarded at the congresses: the Lagrange Prize for exceptional career contributions, the Collatz Prize for outstanding applied mathematicians under the age of 42, the Pioneer Prize for applied mathematical work in a new field, the Maxwell Prize for originality in applied mathematics, and the Su Buchin Prize for outstanding contributions to emerging economies and human development.
Collatz Prize
The Collatz Prize is awarded by ICIAM every four years at the International Congress on Industrial and Applied Mathematics, to an applied mathematician under the age of 42. It was established in 1999 on the initiative of Gesellschaft für Angewandte Mathematik und Mechanik (GAMM), to recognize outstanding contributions in applied and industrial mathematics.
Named after the German mathematician Lothar Collatz, it is widely regarded as one of the most prestigious prizes for young applied mathematicians.
Prize Winners
1999 Stefan Müller
2003 E Weinan
2007 Felix Otto
2011 Emmanuel Candès
2015 Annalisa Buffa
2019 Siddhartha Mishra
2023 Maria Colombo
Lagrange Prize
The Lagrange Prize is awarded by ICIAM every four years at the International Congress on Industrial and Applied Mathematics, for lifetime achievement in applied mathematics. Named after Joseph-Louis Lagrange, it was established in 1999 on the initiative of Société de Mathématiques Appliquées et Industrielles (SMAI), Sociedad Española de Matemática Aplicada (SEMA) and Società Italiana di Matematica Applicata e Industriale (SIMAI).
Prize Winners
1999 Jacques-Louis Lions
2003
2007 Joseph Keller
2011 Alexandre C |
https://en.wikipedia.org/wiki/International%20Congress%20on%20Industrial%20and%20Applied%20Mathematics | The International Congress on Industrial and Applied Mathematics (ICIAM) is an international congress in the field of applied mathematics held every four years under the auspices of the International Council for Industrial and Applied Mathematics. The initial proposal for this conference series was made by Gene Golub.
List of congresses
ICIAM 1987 – Paris
ICIAM 1991 – Washington, D.C.
ICIAM 1995 – Hamburg
ICIAM 1999 – Edinburgh
ICIAM 2003 – Sydney
ICIAM 2007 – Zurich
ICIAM 2011 – Vancouver
ICIAM 2015 – Beijing
ICIAM 2019 – Valencia
ICIAM 2023 – Tokyo
See also
Society for Industrial and Applied Mathematics
International Congress of Mathematics
References
Recurring events established in 1987
Mathematics conferences
Quadrennial events |
https://en.wikipedia.org/wiki/Newlands%20School%20FCJ | Newlands Catholic School FCJ, was a mixed 11–16 Catholic, state school in Middlesbrough, North Yorkshire, England. The school was awarded Specialist Maths and Computing College status.
It was owned by a religious order, the Faithful Companions of Jesus (FCJs) who originally came to Middlesbrough in the 19th century at the request of the Bishops. It was originally a girls school before it was amalgamated with St Mary's the local boys school.
The school motto was Fortiter Et Recte, which in Latin means "Bravely and Justly".
In 2009 Newlands School was officially renamed as Trinity Catholic College after amalgamating with St. David's School, Middlesbrough.
References
Defunct Catholic schools in the Diocese of Middlesbrough
Defunct schools in Middlesbrough |
https://en.wikipedia.org/wiki/Maurice%20Priestley | Maurice Bertram Priestley (15 March 1933 – 15 June 2013) was a professor of statistics in the School of Mathematics, University of Manchester, England. He gained his first degree at the University of Cambridge and went on to gain a Ph.D. from the University of Manchester.
He was known especially for his work on time series analysis, especially spectral analysis and wavelet analysis. He was a longstanding editor of the Journal of Time Series Analysis, a special edition of which was published in his honour in 1993.
Less well-known but equally important was his work with M.T.Chao on nonparametric function fitting.
Selected publications
References
External links
1933 births
2013 deaths
Academics of the University of Manchester
People educated at Manchester Grammar School
Alumni of the University of Manchester |
https://en.wikipedia.org/wiki/Waring%27s%20prime%20number%20conjecture | In number theory, Waring's prime number conjecture is a conjecture related to Vinogradov's theorem, named after the English mathematician Edward Waring. It states that every odd number exceeding 3 is either a prime number or the sum of three prime numbers. It follows from the generalized Riemann hypothesis, and (trivially) from Goldbach's weak conjecture.
See also
Schnirelmann's constant
References
External links
Additive number theory
Conjectures about prime numbers
Conjectures that have been proved |
https://en.wikipedia.org/wiki/Cauchy%E2%80%93Hadamard%20theorem | In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis.
Theorem for one complex variable
Consider the formal power series in one complex variable z of the form
where
Then the radius of convergence of f at the point a is given by
where denotes the limit superior, the limit as approaches infinity of the supremum of the sequence values after the nth position. If the sequence values are unbounded so that the is ∞, then the power series does not converge near , while if the is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane.
Proof
Without loss of generality assume that . We will show first that the power series converges for , and then that it diverges for .
First suppose . Let not be or
For any , there exists only a finite number of such that .
Now for all but a finite number of , so the series converges if . This proves the first part.
Conversely, for , for infinitely many , so if , we see that the series cannot converge because its nth term does not tend to 0.
Theorem for several complex variables
Let be a multi-index (a n-tuple of integers) with , then converges with radius of convergence (which is also a multi-index) if and only if
to the multidimensional power series
The proof can be found in
Notes
External links
Augustin-Louis Cauchy
Mathematical series
Theorems in complex analysis |
https://en.wikipedia.org/wiki/Bryant%20surface | In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1. These surfaces take their name from the geometer Robert Bryant, who proved that every simply-connected minimal surface in 3-dimensional Euclidean space is isometric to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization.
References
Hyperbolic geometry
Riemannian geometry
Minimal surfaces |
https://en.wikipedia.org/wiki/Burst%20mode%20clock%20and%20data%20recovery | The passive optical network (PON) uses tree-like network topology. Due to the topology of PON, the transmission modes for downstream (that is, from optical line termination, (OLT) to optical network unit (ONU)) and upstream (that is, from ONU to OLT) are different. For the downstream transmission, the OLT broadcasts optical signal to all the ONUs in continuous mode (CM), that is, the downstream channel always has optical data signal. One given ONU can find which frame in the CM stream is for it by reading the header of the frame. However, in the upstream channel, ONUs can not transmit optical data signal in CM. It is because that all the signals transmitted from the ONUs converge (with attenuation) into one fiber by the power splitter (serving as power coupler), and overlap among themselves if CM is used. To solve this problem, burst mode (BM) transmission is adopted for upstream channel. The given ONU only transmits optical packet when it is allocated a time slot and it needs to transmit, and all the ONUs share the upstream channel in the time division multiple access (TDMA) mode. The phases of the BM optical packets received by the OLT are different from packet to packet, since the ONUs are not synchronized to transmit optical packet in the same phase, and the distance between OLT and given ONU are random. In order to compensate the phase variation from packet to packet, burst mode clock and data recovery (BM-CDR) is required. Such circuit can generate local clock with the frequency and phase same as the individual received optical packet in a short locking time, for example within 40 ns. Such generated local clock can in turn perform correct data decision. Above all, the clock and data recovery can be performed correctly after a short locking time.
The conventionally used PLL based clock recovery schemes can not meet such strict requirement on locking time. Various other schemes have been invented, including those employing gated oscillator or injection locked oscillator.
References
Clock signal
Electrical circuits |
https://en.wikipedia.org/wiki/Fulton%20County%20Charter%20High%20School%20of%20Mathematics%20and%20Science | Fulton County Charter High School of Mathematics and Science, also known as Math/Science High and MSH, was a high school in Roswell, Georgia, United States, established in 2001 and disbanded in the spring of 2004. Housed in an old furniture store, the charter school was built around New York City's Bronx High School of Science model.
History
Only one student completed high school at MSHS, Gabriel Kassel. His degree was granted from another area high school, but all coursework was completed at the end of the Junior (third year) of the school's operation.
The building was subsequently leased and remodeled by the Atlanta Academy, a private K-8 elementary school in 2006. The Atlanta Academy remains in the facility, as of 2016.
Enrollment
Enrollment was open to all high school students in Fulton County.
Awards
Website Design competition, 3rd place
All-state Band (2001), 2 students
Georgia Tech's Robojackets competition (Spring 2002), 5th place
Odyssey of the Mind, regional (2004), 1st place
Odyssey of the Mind, state (2004), 2nd place
Georgia Science and Engineering Fair, state (2004), 1st place
Clubs and Sports teams
Art Club
Basketball Team
Baseball Team
Chess Club
Fellowship of Christian Athletes
Golf Team
Hockey Team
Computer Club
Newspaper
Odyssey of the Mind
Ping-Pong Club
Robotics Team
String Ensemble
Tennis Team
Volleyball Team
Video Game Club
Yearbook
References
Roswell, Georgia
Former high schools in Georgia (U.S. state)
Schools in Fulton County, Georgia |
https://en.wikipedia.org/wiki/Nikola%20Ota%C5%A1evi%C4%87 | Nikola Otašević (; born January 25, 1982) is a former Serbian professional basketball player.
Career
On June 30, 2019, Otašević announced his retirement from playing career.
Career statistics
Eurocup
|-
| style="text-align:left;"| 2006-07
| style="text-align:left;"| Scandone Avellino
| 6 || 1 || 13.0 || .615 || .600 || .778 || 1.0 || 2.3 || 1.7 || 0.0 || 4.3 || 6.8
|-
| style="text-align:left;"| 2007–08
| style="text-align:left;"| Budućnost Podgorica
| 13 || 2 || 20.4 || .368 || .250 || .806 || 1.3 || 3.7 || 1.9 || 0.0 || 6.0 || 5.7
|-
| style="text-align:left;"| 2008-09
| style="text-align:left;"| Budućnost Podgorica
| 5 || 0 || 19.0 || .429 || .600 || .583 || 2.8 || 1.4 || 0.8 || 0.0 || 4.4 || 3.8
|-
| style="text-align:left;"| 2011–12
| style="text-align:left;"| Budućnost Podgorica
| 6 || 0 || 16.2 || .389 || .333 || .1000 || 1.7 || 2.3 || 1.3 || 0.0 || 3.0 || 2.7
|-
| style="text-align:left;"| 2013–14
| style="text-align:left;"| MZT Skopje
| 7 || 7 || 22.0 || .452 || .200 || .1000 || 2.0 || 3.1 || 0.9 || 0.0 || 4.6 || 3.1
|- class="sortbottom"
| style="text-align:left;"| Career
| style="text-align:left;"|
| 37 || 10 || 18.0 || .421 || .326 || .768 || 1.6 || 2.8 || 1.4 || 0.0 || 4.7 || 4.4
References
External links
Nikola Otašević at aba-liga.com
Nikola Otašević at euroleague.net
1982 births
Living people
ABA League players
Basketball League of Serbia players
KK Beopetrol/Atlas Beograd players
KK Budućnost players
KK Ergonom players
KK Hemofarm players
KK Metalac Valjevo players
KK MZT Skopje players
KK Sloboda Užice players
KK Włocławek players
KK Zdravlje players
OKK Beograd players
Point guards
Serbian expatriate basketball people in Montenegro
Serbian expatriate basketball people in Poland
Serbian expatriate basketball people in North Macedonia
Serbian expatriate basketball people in Romania
Serbian men's basketball players
Sportspeople from Užice |
https://en.wikipedia.org/wiki/Allegory%20%28mathematics%29 | In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions.
In this article we adopt the convention that morphisms compose from right to left, so means "first do , then do ".
Definition
An allegory is a category in which
every morphism is associated with an anti-involution, i.e. a morphism with and and
every pair of morphisms with common domain/codomain is associated with an intersection, i.e. a morphism
all such that
intersections are idempotent: commutative: and associative:
anti-involution distributes over intersection:
composition is semi-distributive over intersection: and and
the modularity law is satisfied:
Here, we are abbreviating using the order defined by the intersection: means
A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphism is a binary relation between and . Composition of morphisms is composition of relations, and the anti-involution of is the converse relation : if and only if . Intersection of morphisms is (set-theoretic) intersection of relations.
Regular categories and allegories
Allegories of relations in regular categories
In a category , a relation between objects and is a span of morphisms that is jointly monic. Two such spans and are considered equivalent when there is an isomorphism between and that make everything commute; strictly speaking, relations are only defined up to equivalence (one may formalise this either by using equivalence classes or by using bicategories). If the category has products, a relation between and is the same thing as a monomorphism into (or an equivalence class of such). In the presence of pullbacks and a proper factorization system, one can define the composition of relations. The composition is found by first pulling back the cospan and then taking the jointly-monic image of the resulting span
Composition of relations will be associative if the factorization system is appropriately stable. In this case, one can consider a category , with the same objects as , but where morphisms are relations between the objects. The identity relations are the diagonals
A regular category (a category with finite limits and images in which covers are stable under pullback) has a stable regular epi/mono factorization system. The category of relations for a regular category is always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersections are intersections of subobjects, computed by pullback.
Maps in allegories, and tabu |
https://en.wikipedia.org/wiki/Geodesic%20map | In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (M, g) and (N, h), a function φ : M → N is said to be a geodesic map if
φ is a diffeomorphism of M onto N; and
the image under φ of any geodesic arc in M is a geodesic arc in N; and
the image under the inverse function φ−1 of any geodesic arc in N is a geodesic arc in M.
Examples
If (M, g) and (N, h) are both the n-dimensional Euclidean space En with its usual flat metric, then any Euclidean isometry is a geodesic map of En onto itself.
Similarly, if (M, g) and (N, h) are both the n-dimensional unit sphere Sn with its usual round metric, then any isometry of the sphere is a geodesic map of Sn onto itself.
If (M, g) is the unit sphere Sn with its usual round metric and (N, h) is the sphere of radius 2 with its usual round metric, both thought of as subsets of the ambient coordinate space Rn+1, then the "expansion" map φ : Rn+1 → Rn+1 given by φ(x) = 2x induces a geodesic map of M onto N.
There is no geodesic map from the Euclidean space En onto the unit sphere Sn, since they are not homeomorphic, let alone diffeomorphic.
The gnomonic projection of the hemisphere to the plane is a geodesic map as it takes great circles to lines and its inverse takes lines to great circles.
Let (D, g) be the unit disc D ⊂ R2 equipped with the Euclidean metric, and let (D, h) be the same disc equipped with a hyperbolic metric as in the Poincaré disc model of hyperbolic geometry. Then, although the two structures are diffeomorphic via the identity map i : D → D, i is not a geodesic map, since g-geodesics are always straight lines in R2, whereas h-geodesics can be curved.
On the other hand, when the hyperbolic metric on D is given by the Klein model, the identity i : D → D is a geodesic map, because hyperbolic geodesics in the Klein model are (Euclidean) straight line segments.
References
External links
Differential geometry
Geodesic (mathematics) |
https://en.wikipedia.org/wiki/Beltrami%27s%20theorem | In the mathematical field of differential geometry, any (pseudo-)Riemannian metric determines a certain class of paths known as geodesics. Beltrami's theorem, named for Italian mathematician Eugenio Beltrami, is a result on the inverse problem of determining a (pseudo-)Riemannian metric from its geodesics.
It is nontrivial to see that, on any Riemannian manifold of constant curvature, there are smooth coordinates relative to which all nonconstant geodesics appear as straight lines. In the negative curvature case of hyperbolic geometry, this is justified by the Beltrami–Klein model. In the positive curvature case of spherical geometry, it is justified by the gnomonic projection. In the language of projective differential geometry, these charts show that any Riemannian manifold of constant curvature is locally projectively flat. More generally, any pseudo-Riemannian manifold of constant curvature is locally projectively flat.
Beltrami's theorem asserts the converse: any connected pseudo-Riemannian manifold which is locally projectively flat must have constant curvature. With the use of tensor calculus, the proof is straightforward. Hermann Weyl described Beltrami's original proof (done in the two-dimensional Riemannian case) as being much more complicated. Relative to a projectively flat chart, there are functions such that the Christoffel symbols take the form
Direct calculation then shows that the Riemann curvature tensor is given by
The curvature symmetry implies that . The other curvature symmetry , traced over and , then says that
where is the dimension of the manifold. It is direct to verify that the left-hand side is a (locally defined) Codazzi tensor, using only the given form of the Christoffel symbols. It follows from Schur's lemma that is constant. Substituting the above identity into the Riemann tensor as given above, it follows that the chart domain has constant sectional curvature . By connectedness of the manifold, this local constancy implies global constancy.
Beltrami's theorem may be phrased in the language of geodesic maps: if given a geodesic map between pseudo-Riemannian manifolds, one manifold has constant curvature if and only if the other does.
References
Sources.
External links
Theorems in Riemannian geometry |
https://en.wikipedia.org/wiki/Hartley%27s%20test | In statistics, Hartley's test, also known as the Fmax test or Hartley's Fmax, is used in the analysis of variance to verify that different groups have a similar variance, an assumption needed for other statistical tests. It was developed by H. O. Hartley, who published it in 1950.
The test involves computing the ratio of the largest group variance, max(sj2) to the smallest group variance, min(sj2). The resulting ratio, Fmax, is then compared to a critical value from a table of the sampling distribution of Fmax. If the computed ratio is less than the critical value, the groups are assumed to have similar or equal variances.
Hartley's test assumes that data for each group are normally distributed, and that each group has an equal number of members. This test, although convenient, is quite sensitive to violations of the normality assumption. Alternatives to Hartley's test that are robust to violations of normality are O'Brien's procedure, and the Brown–Forsythe test.
Related tests
Hartley's test is related to Cochran's C test in which the test statistic is the ratio of max(sj2) to the sum of all the group variances. Other tests related to these, have test statistics in which the within-group variances are replaced by the within-group range. Hartley's test and these similar tests, which are easy to perform but are sensitive to departures from normality, have been grouped together as quick tests for equal variances and, as such, are given a commentary by Hand & Nagaraja (2003).
See also
Bartlett's test
Brown–Forsythe test
Notes
References
Bliss, C.I., Cochran, W.G., Tukey, T.W. (1956) A Rejection Criterion Based upon the Range. Biometrika, 43, 418–422.
Cochran, W.G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Annals of Eugenics, 11, 47–52
Hand, H.A. & Nagaraja, H.N. (2003) Order Statistics, 3rd Edition. Wiley.
Hartley, H.O. (1950). The maximum F-ratio as a short cut test for homogeneity of variance, Biometrika, 37, 308-312.
David, H.A. (1952). "Upper 5 and 1% points of maximum F-ratio." Biometrika, 39, 422–424.
O'Brien, R.G. (1981). A simple test for variance effects in experimental designs. Psychological Bulletin, 89, 570–574.
Keppel, G. and Wickens, T.D. (2004). Design and analysis (4th ed.). Englewood Cliffs, NJ: Prentice-Hall.
Pearson, E.S., Hartley, H.O. (1970). Biometrika Tables for Statisticians, Vol 1, CUP
External links
Table of critical values for the Fmax test
Statistical tests
Analysis of variance |
https://en.wikipedia.org/wiki/George%20W.%20Whitehead | George William Whitehead, Jr. (August 2, 1918 – April 12, 2004) was an American professor of mathematics at the Massachusetts Institute of Technology, a member of the United States National Academy of Sciences, and a Fellow of the American Academy of Arts and Sciences. He is known for his work on algebraic topology. He invented the J-homomorphism, and was among the first to systematically calculate the homotopy groups of spheres. He is also central to the study of Stable homotopy theory, in particular making concrete the connections between Spectra and Generalized homology/cohomology theories.
Whitehead was born in Bloomington, Illinois, and received his Ph.D. in mathematics from the University of Chicago in 1941, under the supervision of Norman Steenrod. After teaching at Purdue University, Princeton University, and Brown University, he took a position at MIT in 1949, where he remained until his retirement in 1985. He advised 13 Ph.D. students, including Robert Aumann and John Coleman Moore, and has over 1,320 academic descendants.
Selected publications
References
George Whitehead dies at 85, MIT News Office
External links
Haynes R. Miller, "George W. Whitehead Jr.", Biographical Memoirs of the National Academy of Sciences (2015)
20th-century American mathematicians
21st-century American mathematicians
Topologists
Members of the United States National Academy of Sciences
Massachusetts Institute of Technology School of Science faculty
Purdue University faculty
Princeton University faculty
Brown University faculty
University of Chicago alumni
People from Bloomington, Illinois
1918 births
2004 deaths
Mathematicians from Illinois |
https://en.wikipedia.org/wiki/Michel%20Kervaire | Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra.
He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological n-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four. He is also well known for fundamental contributions to high-dimensional knot theory. The solution of the Kervaire invariant problem was announced by Michael Hopkins in Edinburgh on 21 April 2009.
Education
He was the son of André Kervaire (a French industrialist) and Nelly Derancourt. After completing high school in France, Kervaire pursued his studies at ETH Zurich (1947–1952), receiving a Ph.D. in 1955. His thesis, entitled Courbure intégrale généralisée et homotopie, was written under the direction of Heinz Hopf and Beno Eckmann.
Career
Kervaire was a professor at New York University's Courant Institute from 1959 to 1971, and then at the University of Geneva from 1971 to 1997, when he retired. He received an honorary doctorate from the University of Neuchâtel in 1986; he was also an honorary member of the Swiss Mathematical Society.
See also
Homology sphere
Kervaire manifold
Plus construction
Selected publications
This paper describes the structure of the group of smooth structures on an n-sphere for n > 4.
Notes
References
External links
Michel Kervaire's work in surgery and knot theory (Slides of lectures given by Andrew Ranicki at the Kervaire Memorial Symposium, Geneva, February 2009)
20th-century French mathematicians
20th-century Swiss mathematicians
Topologists
Algebraists
Courant Institute of Mathematical Sciences faculty
ETH Zurich alumni
People from Częstochowa
1927 births
2007 deaths
Academic staff of the University of Geneva
Swiss expatriates in the United States |
https://en.wikipedia.org/wiki/Malliavin%27s%20absolute%20continuity%20lemma | In mathematics — specifically, in measure theory — Malliavin's absolute continuity lemma is a result due to the French mathematician Paul Malliavin that plays a foundational rôle in the regularity (smoothness) theorems of the Malliavin calculus. Malliavin's lemma gives a sufficient condition for a finite Borel measure to be absolutely continuous with respect to Lebesgue measure.
Statement of the lemma
Let μ be a finite Borel measure on n-dimensional Euclidean space Rn. Suppose that, for every x ∈ Rn, there exists a constant C = C(x) such that
for every C∞ function φ : Rn → R with compact support. Then μ is absolutely continuous with respect to n-dimensional Lebesgue measure λn on Rn. In the above, Dφ(y) denotes the Fréchet derivative of φ at y and ||φ||∞ denotes the supremum norm of φ.
References
(See section 1.3)
Lemmas in analysis
Measure theory
Malliavin calculus |
https://en.wikipedia.org/wiki/Spectral%20clustering | In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset.
In application to image segmentation, spectral clustering is known as segmentation-based object categorization.
Definitions
Given an enumerated set of data points, the similarity matrix may be defined as a symmetric matrix , where represents a measure of the similarity between data points with indices and . The general approach to spectral clustering is to use a standard clustering method (there are many such methods, k-means is discussed below) on relevant eigenvectors of a Laplacian matrix of . There are many different ways to define a Laplacian which have different mathematical interpretations, and so the clustering will also have different interpretations. The eigenvectors that are relevant are the ones that correspond to smallest several eigenvalues of the Laplacian except for the smallest eigenvalue which will have a value of 0. For computational efficiency, these eigenvectors are often computed as the eigenvectors corresponding to the largest several eigenvalues of a function of the Laplacian.
Laplacian matrix
Spectral clustering is well known to relate to partitioning of a mass-spring system, where each mass is associated with a data point and each spring stiffness corresponds to a weight of an edge describing a similarity of the two related data points, as in the spring system. Specifically, the classical reference explains that the eigenvalue problem describing transversal vibration modes of a mass-spring system is exactly the same as the eigenvalue problem for the graph Laplacian matrix defined as
,
where is the diagonal matrix
and A is the adjacency matrix.
The masses that are tightly connected by the springs in the mass-spring system evidently move together from the equilibrium position in low-frequency vibration modes, so that the components of the eigenvectors corresponding to the smallest eigenvalues of the graph Laplacian can be used for meaningful clustering of the masses. For example, assuming that all the springs and the masses are identical in the 2-dimensional spring system pictured, one would intuitively expect that the loosest connected masses on the right-hand side of the system would move with the largest amplitude and in the opposite direction to the rest of the masses when the system is shaken — and this expectation will be confirmed by analyzing components of the eigenvectors of the graph Laplacian corresponding to the smallest eigenvalues, i.e., the smallest vibration frequencies.
Laplacian matrix normalization
The goal of normalization is making the diagonal entries of the Laplacian matrix to be all unit, also scaling off-diagonal entries corr |
https://en.wikipedia.org/wiki/Integration%20by%20parts%20operator | In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications.
Definition
Let E be a Banach space such that both E and its continuous dual space E∗ are separable spaces; let μ be a Borel measure on E. Let S be any (fixed) subset of the class of functions defined on E. A linear operator A : S → L2(E, μ; R) is said to be an integration by parts operator for μ if
for every C1 function φ : E → R and all h ∈ S for which either side of the above equality makes sense. In the above, Dφ(x) denotes the Fréchet derivative of φ at x.
Examples
Consider an abstract Wiener space i : H → E with abstract Wiener measure γ. Take S to be the set of all C1 functions from E into E∗; E∗ can be thought of as a subspace of E in view of the inclusions
For h ∈ S, define Ah by
This operator A is an integration by parts operator, also known as the divergence operator; a proof can be found in Elworthy (1974).
The classical Wiener space C0 of continuous paths in Rn starting at zero and defined on the unit interval [0, 1] has another integration by parts operator. Let S be the collection
i.e., all bounded, adapted processes with absolutely continuous sample paths. Let φ : C0 → R be any C1 function such that both φ and Dφ are bounded. For h ∈ S and λ ∈ R, the Girsanov theorem implies that
Differentiating with respect to λ and setting λ = 0 gives
where (Ah)(x) is the Itō integral
The same relation holds for more general φ by an approximation argument; thus, the Itō integral is an integration by parts operator and can be seen as an infinite-dimensional divergence operator. This is the same result as the integration by parts formula derived from the Clark-Ocone theorem.
References
(See section 5.3)
Integral calculus
Measure theory
Operator theory
Stochastic calculus |
https://en.wikipedia.org/wiki/Nash%20functions | In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ Rn is an analytic function
f: U → R satisfying a nontrivial polynomial equation P(x,f(x)) = 0 for all x in U (A semialgebraic subset of Rn is a subset obtained from subsets of the form {x in Rn : P(x)=0} or {x in Rn : P(x) > 0}, where P is a polynomial, by taking finite unions, finite intersections and complements). Some examples of Nash functions:
Polynomial and regular rational functions are Nash functions.
is Nash on R.
the function which associates to a real symmetric matrix its i-th eigenvalue (in increasing order) is Nash on the open subset of symmetric matrices with no multiple eigenvalue.
Nash functions are those functions needed in order to have an implicit function theorem in real algebraic geometry.
Nash manifolds
Along with Nash functions one defines Nash manifolds, which are semialgebraic analytic submanifolds of some Rn. A Nash mapping
between Nash manifolds is then an analytic mapping with semialgebraic graph. Nash functions and manifolds are named after John Forbes Nash, Jr., who proved (1952) that any compact smooth manifold admits a Nash manifold structure, i.e., is diffeomorphic to some Nash manifold. More generally, a smooth manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact smooth manifold possibly with boundary. Nash's result was later (1973) completed by Alberto Tognoli who proved that any compact smooth manifold is diffeomorphic to some affine real algebraic manifold; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the smooth and the algebraic categories.
Local properties
The local properties of Nash functions are well understood. The ring of germs of Nash functions at a point of a Nash manifold of dimension n is isomorphic to the ring of algebraic power series in n variables (i.e., those series satisfying a nontrivial polynomial equation), which is the henselization of the ring of germs of rational functions. In particular, it is a regular local ring of dimension n.
Global properties
The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold (even noncompact) is noetherian was proved independently (1973) by Jean-Jacques Risler and Gustave Efroymson. Nash manifolds have properties similar to but weaker than Cartan's theorems A and B on Stein manifolds. Let denote the sheaf of Nash function germs on
a Nash manifold M, and be a coherent sheaf of -ideals. Assume is finite, i.e., there exists a finite open semialgebraic covering of M such that, for each i, is generated by Nash functions on . Then is globally generated by Nash functions on M, and the natural map
is surjective. However
contrarily to the case of Stein manifolds.
Generalizations
Nash functions and manifolds can be defined over a |
https://en.wikipedia.org/wiki/H%C3%B6rmander%27s%20condition | In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations. The condition is named after the Swedish mathematician Lars Hörmander.
Definition
Given two C1 vector fields V and W on d-dimensional Euclidean space Rd, let [V, W] denote their Lie bracket, another vector field defined by
where DV(x) denotes the Fréchet derivative of V at x ∈ Rd, which can be thought of as a matrix that is applied to the vector W(x), and vice versa.
Let A0, A1, ... An be vector fields on Rd. They are said to satisfy Hörmander's condition if, for every point x ∈ Rd, the vectors
span Rd. They are said to satisfy the parabolic Hörmander condition if the same holds true, but with the index taking only values in 1,...,n.
Application to stochastic differential equations
Consider the stochastic differential equation (SDE)
where the vectors fields are assumed to have bounded derivative, the normalized n-dimensional Brownian motion and stands for the Stratonovich integral interpretation of the SDE.
Hörmander's theorem asserts that if the SDE above satisfies the parabolic Hörmander condition, then its solutions admit a smooth density with respect to Lebesgue measure.
Application to the Cauchy problem
With the same notation as above, define a second-order differential operator F by
An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields Ai for the Cauchy problem
to have a smooth fundamental solution, i.e. a real-valued function p (0, +∞) × R2d → R such that p(t, ·, ·) is smooth on R2d for each t and
satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the elliptic case, in which
and the matrix A = (aji), 1 ≤ j ≤ d, 1 ≤ i ≤ n is such that AA∗ is everywhere an invertible matrix.
The great achievement of Hörmander's 1967 paper was to show that a smooth fundamental solution exists under a considerably weaker assumption: the parabolic version of the condition that now bears his name.
Application to control systems
Let M be a smooth manifold and be smooth vector fields on M. Assuming that these vector fields satisfy Hörmander's condition, then the control system
is locally controllable in any time at every point of M. This is known as the Chow–Rashevskii theorem. See Orbit (control theory).
See also
Malliavin calculus
Lie algebra
References
(See the introduction)
Partial differential equations
Stochastic differential equations |
https://en.wikipedia.org/wiki/Fabiano%20%28footballer%2C%20born%201975%29 | Fabiano Cezar Viegas, or simply Fabiano (born August 4, 1975), is a Brazilian former professional footballer who played as a central defender.
Club statistics
Honours
Tournament Rio - São Paulo: 1993
Rio de Janeiro State League: 1996, 1999
Japanese League: 2000, 2001
Nabisco Cup: 2000, 2002
Emperor Cup: 2000
Goiás State League: 2006
External links
CBF
sambafoot
Guardian Stats Centre
zerozero.pt
1975 births
Living people
Brazilian men's footballers
Brazil men's under-20 international footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
CR Flamengo footballers
Club Athletico Paranaense players
Brazilian expatriate sportspeople in China
Fabiano
Fabiano
Fabiano
Expatriate men's footballers in Japan
Goiás Esporte Clube players
Expatriate men's footballers in China
Wuhan Optics Valley F.C. players
Qingdao Hainiu F.C. (1990) players
Men's association football central defenders |
https://en.wikipedia.org/wiki/Mori%20Domain | Mori Domain may refer to:
The of Bungo Province, held by the Kurushima family
The of Izumo Province, a branch of the Matsue Domain, held by the Matsudaira family
A Mori domain in mathematics is a type of commutative ring |
https://en.wikipedia.org/wiki/Jensen%20hierarchy | In set theory, a mathematical discipline, the Jensen hierarchy or J-hierarchy is a modification of Gödel's constructible hierarchy, L, that circumvents certain technical difficulties that exist in the constructible hierarchy. The J-Hierarchy figures prominently in fine structure theory, a field pioneered by Ronald Jensen, for whom the Jensen hierarchy is named. Rudimentary functions describe a method for iterating through the Jensen hierarchy.
Definition
As in the definition of L, let Def(X) be the collection of sets definable with parameters over X:
The constructible hierarchy, is defined by transfinite recursion. In particular, at successor ordinals, .
The difficulty with this construction is that each of the levels is not closed under the formation of unordered pairs; for a given , the set will not be an element of , since it is not a subset of .
However, does have the desirable property of being closed under Σ0 separation.
Jensen's modification of the L hierarchy retains this property and the slightly weaker condition that , but is also closed under pairing. The key technique is to encode hereditarily definable sets over by codes; then will contain all sets whose codes are in .
Like , is defined recursively. For each ordinal , we define to be a universal predicate for . We encode hereditarily definable sets as , with . Then set and finally, .
Properties
Each sublevel Jα, n is transitive and contains all ordinals less than or equal to αω + n. The sequence of sublevels is strictly ⊆-increasing in n, since a Σm predicate is also Σn for any n > m. The levels Jα will thus be transitive and strictly ⊆-increasing as well, and are also closed under pairing, -comprehension and transitive closure. Moreover, they have the property that
as desired. (Or a bit more generally, .)
The levels and sublevels are themselves Σ1 uniformly definable (i.e. the definition of Jα, n in Jβ does not depend on β), and have a uniform Σ1 well-ordering. Also, the levels of the Jensen hierarchy satisfy a condensation lemma much like the levels of Gödel's original hierarchy.
For any , considering any relation on , there is a Skolem function for that relation that is itself definable by a formula.
Rudimentary functions
A rudimentary function is a Vn→V function (i.e. a finitary function accepting sets as arguments) that can be obtained from the following operations:
F(x1, x2, ...) = xi is rudimentary (see projection function)
F(x1, x2, ...) = {xi, xj} is rudimentary
F(x1, x2, ...) = xi − xj is rudimentary
Any composition of rudimentary functions is rudimentary
∪z∈yG(z, x1, x2, ...) is rudimentary, where G is a rudimentary function
For any set M let rud(M) be the smallest set containing M∪{M} closed under the rudimentary functions. Then the Jensen hierarchy satisfies Jα+1 = rud(Jα).
Projecta
Jensen defines , the projectum of , as the largest such that is amenable for all , and the projectum of is defined similarly. One of the main results of |
https://en.wikipedia.org/wiki/Orthostochastic%20matrix | In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of
the absolute values of the entries of some orthogonal matrix.
The detailed definition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all its entries are nonnegative real numbers. It is orthostochastic if there exists an orthogonal matrix O such that
All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic)
since for any
we find the corresponding orthogonal matrix
with
such that
For larger n the sets of bistochastic matrices includes the set of unistochastic matrices,
which includes the set of orthostochastic matrices and these inclusion relations are proper.
References
Matrices |
https://en.wikipedia.org/wiki/Brian%20Bowditch | Brian Hayward Bowditch (born 1961) is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.
Biography
Brian Bowditch was born in 1961 in Neath, Wales. He obtained a B.A. degree from Cambridge University in 1983. He subsequently pursued doctoral studies in Mathematics at the University of Warwick under the supervision of David Epstein where he received a PhD in 1988. Bowditch then had postdoctoral and visiting positions at the Institute for Advanced Study in Princeton, New Jersey, the University of Warwick, Institut des Hautes Études Scientifiques at Bures-sur-Yvette, the University of Melbourne, and the University of Aberdeen. In 1992 he received an appointment at the University of Southampton where he stayed until 2007. In 2007 Bowditch moved to the University of Warwick, where he received a chaired Professor appointment in Mathematics.
Bowditch was awarded a Whitehead Prize by the London Mathematical Society in 1997 for his work in geometric group theory and geometric topology. He gave an Invited address at the 2004 European Congress of Mathematics in Stockholm.
Bowditch is a former member of the Editorial Board for the journal Annales de la Faculté des Sciences de Toulouse and a former Editorial Adviser for the London Mathematical Society.
Mathematical contributions
Early notable results of Bowditch include clarifying the classic notion of geometric finiteness for higher-dimensional Kleinian groups in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of hyperbolic 3-space and hyperbolic plane, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic n-space where n ≥ 4. He showed, however, that in dimensions n ≥ 4 the condition of having a finitely-sided Dirichlet domain is no longer equivalent to the standard notions of geometric finiteness. In a subsequent paper Bowditch considered a similar problem for discrete groups of isometries of Hadamard manifold of pinched (but not necessarily constant) negative curvature and of arbitrary dimension n ≥ 2. He proved that four out of five equivalent definitions of geometric finiteness considered in his previous paper remain equivalent in this general set-up, but the condition of having a finitely-sided fundamental polyhedron is no longer equivalent to them.
Much of Bowditch's work in the 1990s concerned studying boundaries at infinity of word-hyperbolic groups. He proved the cut-point conjecture which says that the boundary of a one-ended word-hyperbolic group does not have any global cut-points. Bowditch first proved this conjecture in the main cases o |
https://en.wikipedia.org/wiki/The%20Mathematical%20Diary | The Mathematical Diary was an early American mathematical journal and mathematics magazine, published between 1825 and 1833.
The Mathematical Diary was founded by Robert Adrain at Columbia College (now Columbia University) after two unsuccessful attempts, in 1808 and 1814, to start a more purely academic mathematics journal, The Analyst, or, Mathematical Museum. The Mathematical Diary contained, in addition to some serious mathematics, articles of general interest such as mathematical puzzles aimed at the amateur problem-solver, which may have helped it attract more laypeople as subscribers and contributed to its greater longevity. Adrain edited the first four issues; after he left Columbia in 1826 for Rutgers, James Ryan, previously the publisher, took over the editorship. A total of thirteen issues were published.
References
Mathematics journals
Defunct journals of the United States
Publications established in 1825
English-language journals
Publications disestablished in 1833 |
https://en.wikipedia.org/wiki/Toda%20bracket | In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups of spheres in .
Definition
See or for more information.
Suppose that
is a sequence of maps between spaces, such that the compositions and are both nullhomotopic. Given a space , let denote the cone of . Then we get a (non-unique) map
induced by a homotopy from to a trivial map, which when post-composed with gives a map
.
Similarly we get a non-unique map induced by a homotopy from to a trivial map, which when composed with , the cone of the map , gives another map,
.
By joining these two cones on and the maps from them to , we get a map
representing an element in the group of homotopy classes of maps from the suspension to , called the Toda bracket of , , and . The map is not uniquely defined up to homotopy, because there was some choice in choosing the maps from the cones. Changing these maps changes the Toda bracket by adding elements of and .
There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of Massey products in cohomology.
The Toda bracket for stable homotopy groups of spheres
The direct sum
of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication (called composition product) is given by composition of representing maps, and any element of non-zero degree is nilpotent .
If f and g and h are elements of with and , there is a Toda bracket of these elements. The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of composition products of certain other elements. Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups.
showed that every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements.
The Toda bracket for general triangulated categories
In the case of a general triangulated category the Toda bracket can be defined as follows. Again, suppose that
is a sequence of morphism in a triangulated category such that and . Let denote the cone of f so we obtain an exact triangle
The relation implies that g factors (non-uniquely) through as
for some . Then, the relation implies that factors (non-uniquely) through W[1] as
for some b. This b is (a choice of) the Toda bracket in the group .
Convergence theorem
There is a convergence theorem originally due to Moss which states that special Massey products of elements in the -page of the Adams spectral sequence contain a permanent cycle, meaning has an associated element in , assuming the elements are permanent cyclespg 18-19. Moreover, these Massey products have a lift to a motivic Adams spec |
https://en.wikipedia.org/wiki/EHP%20spectral%20sequence | In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime p. It is described in more detail in and . It is related to the EHP long exact sequence of ; the name "EHP" comes from the fact that George W. Whitehead named 3 of the maps of his sequence "E" (the first letter of the German word "Einhängung" meaning "suspension"), "H" (for Heinz Hopf, as this map is the second Hopf–James invariant), and "P" (related to Whitehead products).
For the spectral sequence uses some exact sequences associated to the fibration
,
where stands for a loop space and the (2) is localization of a topological space at the prime 2. This gives a spectral sequence with term equal to
and converging to (stable homotopy groups of spheres localized at 2). The spectral sequence has the advantage that the input is previously calculated homotopy groups. It was used by to calculate the first 31 stable homotopy groups of spheres.
For arbitrary primes one uses some fibrations found by :
where is the -skeleton of the loop space . (For , the space is the same as , so Toda's fibrations at are the same as the James fibrations.)
References
Spectral sequences |
https://en.wikipedia.org/wiki/Wapella%2C%20Saskatchewan | Wapella () is a town of 354 located northwest of Moosomin on the Trans-Canada Highway.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Wapella had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Notable people
Brett Clark - professional hockey player in NHL. He played in the Canadian National team program, as well as for the Montreal Canadiens, Atlanta Thrashers, Colorado Avalanche, Tampa Bay Lightning and Minnesota Wild franchises.
Bud Holloway, a professional hockey player. He currently plays (2015/2016 season) for the St. John's IceCaps in the AHL. He has previously played for SC Bern in the National League A, it is the top tier of the Swiss hockey league system, for the Skellefteå AIK in the SHL and for the Manchester Monarchs, the AHL affiliate of the Los Angeles Kings.
Cyril Edel Leonoff is the grandson of Edel Brotman, a homesteader and rabbi of the Wapella, Saskatchewan, farm colony, 1889–1906.
Climate
See also
List of communities in Saskatchewan
List of place names in Canada of Indigenous origin
List of towns in Saskatchewan
Footnotes
Towns in Saskatchewan
Martin No. 122, Saskatchewan
Division No. 5, Saskatchewan |
https://en.wikipedia.org/wiki/United%20Nations%20geoscheme%20for%20Europe | The following is an alphabetical list of subregions in the United Nations geoscheme for Europe, created by the United Nations Statistics Division (UNSD). The scheme subdivides the continent into Eastern Europe, Northern Europe, Southern Europe, and Western Europe. The UNSD notes that "the assignment of countries or areas to specific groupings is for statistical convenience and does not imply any assumption regarding political or other affiliation of countries or territories".
Eastern Europe
†
† Although Russia is a transcontinental country covering Northern Asia as well, for statistical convenience, Russia is assigned under Eastern Europe by UNSD, including both European Russia and Siberian Russia under a single subregion.
Northern Europe
Channel Islands
Southern Europe
Western Europe
See also
List of continents and continental subregions by population
List of countries by United Nations geoscheme
Regions of Europe
United Nations geoscheme
United Nations geoscheme for Africa
United Nations geoscheme for the Americas
United Nations geoscheme for Asia
United Nations geoscheme for Oceania
United Nations Regional Groups
United Nations Statistics Division
References
Geography of Europe
Europe |
https://en.wikipedia.org/wiki/United%20Nations%20geoscheme%20for%20Oceania | Oceania
UN geoscheme subregions of Oceania
The following is an alphabetical list of subregions in the United Nations geoscheme for Oceania, created by the United Nations Statistics Division (UNSD).
UN Subregions
The United Nations geoscheme subdivides the region into Australia and New Zealand, Melanesia, Micronesia, and Polynesia. The UNSD notes that "the assignment of countries or areas to specific groupings is for statistical convenience and does not imply any assumption regarding political or other affiliation of countries or territories".
See also
List of continents and continental subregions by population
List of countries by United Nations geoscheme
United Nations geoscheme
United Nations geoscheme for Africa
United Nations geoscheme for the Americas
United Nations geoscheme for Asia
United Nations geoscheme for Europe
United Nations Statistics Division
Notes
References
Geography of Oceania
Oceania |
https://en.wikipedia.org/wiki/Mathematics%20of%20bookmaking | In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The phrase originates from the practice of recording such wagers in a hard-bound ledger (the 'book') and gives the English language the term bookmaker for the person laying the bets and thus 'making the book'.
Making a 'book' (and the notion of overround)
A bookmaker strives to accept bets on the outcome of an event in the right proportions in order to make a profit regardless of which outcome prevails. See Dutch book and coherence (philosophical gambling strategy). This is achieved primarily by adjusting what are determined to be the true odds of the various outcomes of an event in a downward fashion (i.e. the bookmaker will pay out using his actual odds, an amount which is less than the true odds would have paid, thus ensuring a profit).
The odds quoted for a particular event may be fixed but are more likely to fluctuate in order to take account of the size of wagers placed by the bettors in the run-up to the actual event (e.g. a horse race). This article explains the mathematics of making a book in the (simpler) case of the former event. For the second method, see Parimutuel betting.
It is important to understand the relationship between fractional and decimal odds. Fractional odds are written a − b (a/b or a to b), meaning a winning bettor will receive their money back plus a units for every b units they bet. Decimal odds are a single value, greater than 1, representing the amount to be paid out for each unit bet.
For example, a bet of £40 at 6 − 4 (fractional odds) will pay out £40 + £60 = £100. The equivalent decimal odds are 2.5; £40 × 2.5 = £100. We can convert fractional to decimal odds by the formula D = . Hence, fractional odds of a − 1 (ie. b = 1) can be obtained from decimal odds by a = D − 1.
It is also important to understand the relationship between odds and implied probabilities:
Fractional odds of a − b (with corresponding decimal odds D) represent an implied probability of = , e.g. 6-4 corresponds to = = 0.4 (40%).
An implied probability of x is represented by fractional odds of (1 − x)/x, e.g. 0.2 is (1 − 0.2)/0.2 = 0.8/0.2 = 4/1 (4-1, 4 to 1) (equivalently, − 1 to 1), and decimal odds of D = .
Example
In considering a football match (the event) that can be either a 'home win', 'draw' or 'away win' (the outcomes) then the following odds might be encountered to represent the true chance of each of the three outcomes:
Home: Evens
Draw: 2-1
Away: 5-1
These odds can be represented as implied probabilities (or percentages by multiplying by 100) as follows:
Evens (or 1-1) corresponds to an implied probability of (50%)
2-1 corresponds to an implied probability of (33%)
5-1 corresponds to an implied probability of (16%)
By adding the percentages together a total 'book' of 100% is achieved (representing a fair book). The bookmaker will reduce these odds to ensure a profit. Consider the simplest model o |
https://en.wikipedia.org/wiki/Ascanio%20II%20Piccolomini | Ascanio Piccolomini (1596–1671) was the archbishop of Siena from 1629 to 1671.
Ascanio was a mathematics pupil of Bonaventura Cavalieri. He hosted Galileo in Siena. According to Dava Sobel, Galileo's ability "to rise from the ashes of his condemnation by the Inquisition" and complete perhaps his most influential book, the Two New Sciences, was "due in large measure to Piccolomini's solicitous kindness".
He was an elder brother of the Imperial general Ottavio Piccolomini.
While bishop, he was the principal co-consecrator of Carlo Fabrizio Giustiniani, Bishop of Accia and Mariana (1656).
Notes and references
Sources
Suter, Rufus (1965). "A Note on the Identity of Ascanio Piccolomini, Galileo's Host at Siena," Isis Vol. 56, No. 4 (Winter, 1965), p. 452.
1596 births
1671 deaths
Clergy from Siena
Archbishops of Siena
Bishops in Tuscany
Ascanio II
17th-century Italian Roman Catholic archbishops |
https://en.wikipedia.org/wiki/Brown%E2%80%93Peterson%20cohomology | In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by
, depending on a choice of prime p. It is described in detail by .
Its representing spectrum is denoted by BP.
Complex cobordism and Quillen's idempotent
Brown–Peterson cohomology BP is a summand of MU(p), which is complex cobordism MU localized at a prime p. In fact MU(p) is a wedge product of suspensions of BP.
For each prime p, Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(p) to itself, with the property that ε([CPn]) is [CPn] if n+1 is a power of p, and 0 otherwise. The spectrum BP is the image of this idempotent ε.
Structure of BP
The coefficient ring is a polynomial algebra over on generators in degrees for .
is isomorphic to the polynomial ring over with generators in of degrees .
The cohomology of the Hopf algebroid is the initial term of the Adams–Novikov spectral sequence for calculating p-local homotopy groups of spheres.
BP is the universal example of a complex oriented cohomology theory whose associated formal group law is p-typical.
See also
List of cohomology theories#Brown–Peterson cohomology
References
.
.
Cohomology theories |
https://en.wikipedia.org/wiki/Zuzu | Zuzu is an administrative ward in the Dodoma Urban district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,048 people in the ward, from 6,485 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Bisymmetric%20matrix | In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT and AJ = JA where J is the n × n exchange matrix.
For example, any matrix of the form
is bisymmetric. The associated exchange matrix for this example is
Properties
Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.
The product of two bisymmetric matrices is a centrosymmetric matrix.
Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.
If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.
The inverse of bisymmetric matrices can be represented by recurrence formulas.
References
Matrices |
https://en.wikipedia.org/wiki/Trendalyzer | Trendalyzer is an information visualization software for animation of statistics that was initially developed by Hans Rosling's Gapminder Foundation in Sweden. In March 2007 it was acquired by Google Inc. The current beta version is a Flash application that is preloaded with statistical and historical data about the development of the countries of the world.
The information visualization technique used by Trendalyzer is an interactive bubble chart. By default it shows five variables: Two numeric variables on the X and Y axes, bubble size and colour, and a time variable that may be manipulated with a slider. The software uses brushing and linking techniques for displaying the numeric value of a highlighted country.
Components of the Trendalyzer software, particularly the Flash-based Motion Chart gadget, have become available for public use as part of the Google Visualizations API (see ).
Similar projects
Trend Compass (flash)
Eurostat explorer (flash)
References
External links
The Gapminder World, using Trendalyzer to display various statistics about the world's countries. A tab allows access to a download package
Make Your Data Tell a Story: "Reports, tables, charts and dashboards all deliver information, but information alone isn't understanding"
FAQ: How do I use Gapminder graphics in my presentation?: Implement Gapminder graphs in your presentation using Gapminder Tools Offline.
Webpage with Installation package for 2007 version
Example for visualization of environmental data (environmental accounting)
Google acquisitions
Plotting software
Data visualization software |
https://en.wikipedia.org/wiki/Willmore%20conjecture | In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965. A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014.
Willmore energy
Let v : M → R3 be a smooth immersion of a compact, orientable surface. Giving M the Riemannian metric induced by v, let H : M → R be the mean curvature (the arithmetic mean of the principal curvatures κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by
It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only if M is an embedded round sphere.
Statement
Calculation of W(M) for a few examples suggests that there should be a better bound than W(M) ≥ 4π for surfaces with genus g(M) > 0. In particular, calculation of W(M) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name
For every smooth immersed torus M in R3, W(M) ≥ 2π2.
In 1982, Peter Wai-Kwong Li and Shing-Tung Yau proved the conjecture in the non-embedded case, showing that if is an immersion of a compact surface, which is not an embedding, then W(M) is at least 8π.
In 2012, Fernando Codá Marques and André Neves proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces. Martin Schmidt claimed a proof in 2002, but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it). Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as tube tori (by Willmore himself), and for tori of revolution (by Langer & Singer).
References
Conjectures that have been proved
Surfaces
Theorems in differential geometry
de:Willmore-Energie |
https://en.wikipedia.org/wiki/Istv%C3%A1n%20Spitzm%C3%BCller | István Spitzmüller (born 14 May 1986) is a Hungarian football midfielder who plays for DEAC.
Club statistics
Updated to games played as of 23 November 2014.
Sources
Profile on hlsz.hu
Profile on dvsc.hu
References
1986 births
People from Hajdúnánás
Footballers from Hajdú-Bihar County
Living people
Hungarian men's footballers
Men's association football midfielders
Debreceni VSC players
Nyíregyháza Spartacus FC players
Békéscsaba 1912 Előre footballers
Debreceni EAC (football) players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Differential%20inclusion | In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form
where F is a multivalued map, i.e. F(t, x) is a set rather than a single point in . Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, Moreau's sweeping process, linear and nonlinear complementarity dynamical systems, discontinuous ordinary differential equations, switching dynamical systems, and fuzzy set arithmetic.
For example, the basic rule for Coulomb friction is that the friction force has magnitude μN in the direction opposite to the direction of slip, where N is the normal force and μ is a constant (the friction coefficient). However, if the slip is zero, the friction force can be any force in the correct plane with magnitude smaller than or equal to μN. Thus, writing the friction force as a function of position and velocity leads to a set-valued function.
In differential inclusion, we not only take a set-valued map at the right hand side but also we can take a subset of a Euclidean space for some as following way. Let and Our main purpose is to find a function satisfying the differential inclusion a.e. in where is an open bounded set.
Theory
Existence theory usually assumes that F(t, x) is an upper hemicontinuous function of x, measurable in t, and that F(t, x) is a closed, convex set for all t and x.
Existence of solutions for the initial value problem
for a sufficiently small time interval [t0, t0 + ε), ε > 0 then follows.
Global existence can be shown provided F does not allow "blow-up" ( as for a finite ).
Existence theory for differential inclusions with non-convex F(t, x) is an active area of research.
Uniqueness of solutions usually requires other conditions.
For example, suppose satisfies a one-sided Lipschitz condition:
for some C for all x1 and x2. Then the initial value problem
has a unique solution.
This is closely related to the theory of maximal monotone operators, as developed by Minty and Haïm Brezis.
Filippov's theory only allows for discontinuities in the derivative , but allows no discontinuities in the state, i.e. need be continuous. Schatzman and later Moreau (who gave it the currently accepted name) extended the notion to measure differential inclusion (MDI) in which the inclusion is evaluated by taking the limit from above for .
Applications
Differential inclusions can be used to understand and suitably interpret discontinuous ordinary differential equations, such as arise for Coulomb friction in mechanical systems and ideal switches in power electronics. An important contribution has been made by A. F. Filippov, who studied regularizations of discontinuous equations. Further, the technique of regularization was used by N.N. Krasovskii in the theory of differential games.
Differential inclusions are also found at the foundation of non-smooth dynamical systems (NSDS) analysis, which is u |
https://en.wikipedia.org/wiki/Localization%20of%20a%20topological%20space | In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in .
The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y.
Definitions
We let A be a subring of the rational numbers, and let X be a simply connected CW complex. Then there is a simply connected CW complex Y together with a map from X to Y such that
Y is A-local; this means that all its homology groups are modules over A
The map from X to Y is universal for (homotopy classes of) maps from X to A-local CW complexes.
This space Y is unique up to homotopy equivalence, and is called the localization
of X at A.
If A is the localization of Z at a prime p, then the space Y is called the localization of X at p
The map from X to Y induces isomorphisms from the A-localizations of the homology and homotopy groups of X to the homology and homotopy groups of Y.
See also
:Category:Localization (mathematics)
Local analysis
Localization of a category
Localization of a module
Localization of a ring
Bousfield localization
References
Homotopy theory
Localization (mathematics) |
https://en.wikipedia.org/wiki/Panjer%20recursion | The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable
where both and are random variables and of special types. In more general cases the distribution of S is a compound distribution. The recursion for the special cases considered was introduced in a paper by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo). It is heavily used in actuarial science (see also systemic risk).
Preliminaries
We are interested in the compound random variable where and fulfill the following preconditions.
Claim size distribution
We assume the to be i.i.d. and independent of . Furthermore the have to be distributed on a lattice with latticewidth .
In actuarial practice, is obtained by discretisation of the claim density function (upper, lower...).
Claim number distribution
The number of claims N is a random variable, which is said to have a "claim number distribution", and which can take values 0, 1, 2, .... etc.. For the "Panjer recursion", the probability distribution of N has to be a member of the Panjer class, otherwise known as the (a,b,0) class of distributions. This class consists of all counting random variables which fulfill the following relation:
for some and which fulfill . The initial value is determined such that
The Panjer recursion makes use of this iterative relationship to specify a recursive way of constructing the probability distribution of S. In the following denotes the probability generating function of N: for this see the table in (a,b,0) class of distributions.
In the case of claim number is known, please note the De Pril algorithm. This algorithm is suitable to compute the sum distribution of discrete random variables.
Recursion
The algorithm now gives a recursion to compute the .
The starting value is with the special cases
and
and proceed with
Example
The following example shows the approximated density of where and with lattice width h = 0.04. (See Fréchet distribution.)
As observed, an issue may arise at the initialization of the recursion. Guégan and Hassani (2009) have proposed a solution to deal with that issue
.
References
External links
Panjer recursion and the distributions it can be used with
Actuarial science
Compound probability distributions
Theory of probability distributions |
https://en.wikipedia.org/wiki/Lehmer%27s%20conjecture | Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients satisfies one of the following properties:
The Mahler measure of is greater than or equal to .
is an integral multiple of a product of cyclotomic polynomials or the monomial , in which case . (Equivalently, every complex root of is a root of unity or zero.)
There are a number of definitions of the Mahler measure, one of which is to factor over as
and then set
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
for which the Mahler measure is the Salem number
It is widely believed that this example represents the true minimal value: that is, in Lehmer's conjecture.
Motivation
Consider Mahler measure for one variable and Jensen's formula shows that if then
In this paragraph denote , which is also called Mahler measure.
If has integer coefficients, this shows that is an algebraic number so is the logarithm of an algebraic integer. It also shows that and that if then is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of i.e. a power for some .
Lehmer noticed that is an important value in the study of the integer sequences for monic . If does not vanish on the circle then . If does vanish on the circle but not at any root of unity, then the same convergence holds by Baker's theorem (in fact an earlier result of Gelfond is sufficient for this, as pointed out by Lind in connection with his study of quasihyperbolic toral automorphisms). As a result, Lehmer was led to ask
whether there is a constant such that provided is not cyclotomic?,
or
given , are there with integer coefficients for which ?
Some positive answers have been provided as follows, but Lehmer's conjecture is not yet completely proved and is still a question of much interest.
Partial results
Let be an irreducible monic polynomial of degree .
Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying .
Blanksby and Montgomery and Stewart independently proved that there is an absolute constant such that either or
Dobrowolski improved this to
Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier in 1996 obtained C ≥ 1/4 for D ≥ 2.
Elliptic analogues
Let be an elliptic curve defined over a number field , and let be the canonical height function. The canonical height is the analogue for elliptic curves of the function . It has the property that if and only if is a torsion point in . The elliptic Lehmer conjecture asserts that there is a constant such that
for all non-torsion points ,
where . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's |
https://en.wikipedia.org/wiki/James%20F.%20Allen%20%28computer%20scientist%29 | James Frederick Allen (born 1950) is a computational linguist recognized for his contributions to temporal logic, in particular Allen's interval algebra. He is interested in knowledge representation, commonsense reasoning, and natural language understanding, believing that "deep language understanding can only currently be achieved by significant hand-engineering of semantically-rich formalisms coupled with statistical preferences". He is the John H. Dessaurer Professor of Computer Science at the University of Rochester
Biography
Allen received his Ph.D. from the University of Toronto in 1979, under the supervision of C. Raymond Perrault, after which he joined the faculty at Rochester. At Rochester, he was department chair from 1987 to 1990, directed the Cognitive Science Program from 1992 to 1996, and co-directed the Center for the Sciences of Language from 1996 to 1998. He served as the Editor-in-Chief of Computational Linguistics from 1983–1993. Since 2006 he has also been associate director of the Florida Institute for Human and Machine Cognition.
Academic life
TRIPS project
The TRIPS project is a long-term research to build generic technology for dialogue (both spoken and 'chat') systems, which includes natural language processing, collaborative problem solving, and dynamic context-sensitive language modeling. This is contrast with the data driven approaches by machine learning, which requires to collect and annotate corpora, i.e. training data, firstly.
PLOW agent
PLOW agent is a system that learns executable task models from a single collaborative learning session, which integrates wide AI technologies including deep natural language understanding, knowledge representation and reasoning, dialogue systems, planning/agent-based systems, and machine learning. This paper won the outstanding paper award at AAAI in 2007.
Selected works
Books
Allen is the author of the textbook Natural Language Understanding (Benjamin-Cummings, 1987; 2nd ed., 1995).
He is also the co-author with Henry Kautz, Richard Pelavin, and Josh Tenenberg of Reasoning About Plans (Morgan Kaufmann, 1991).
Articles
2007. PLOW: A Collaborative Task Learning Agent. (with Nathanael Chambers et al) AAAI'07
won the outstanding paper award at AAAI in 2007.
2006. Chester: Towards a Personal Medication Advisor. (with N. Blaylock, et al) Biomedical informatics 39(5)
1998. TRIPS: An Integrated Intelligent Problem-Solving Assistant. (with George Ferguson) AAAI'98
1983. Maintaining Knowledge about Temporal Intervals. CACM 26, 11, 832-843
Awards and honors
In 1991 he was elected as a fellow of the Association for the Advancement of Artificial Intelligence (1990, founding fellow).
In 1992 he became the Dessaurer Professor at Rochester.
References
External links
James F. Allen's Home Page
Google Scholar, h-index is 59.
Florida Institute for Human and Machine Cognition people
Living people
1950 births
American computer scientists
Linguists from the United States
American |
https://en.wikipedia.org/wiki/Dennis%20Dourandi | Dennis Dourandi (born February 8, 1983) is a Cameroonian footballer who currently plays as a striker for Université FC de Ngaoundéré.
External links
Profile
2. Bundesliga statistics
Living people
1983 births
Étoile Sportive du Sahel players
Újpest FC players
SpVgg Greuther Fürth players
Cameroonian men's footballers
S.C. Olhanense players
Unisport FC de Bafang players
Men's association football forwards
Cameroonian expatriate men's footballers
Expatriate men's footballers in Hungary
Cameroonian expatriate sportspeople in Hungary
Expatriate men's footballers in Portugal
Cameroonian expatriate sportspeople in Portugal
Expatriate men's footballers in Germany
Cameroonian expatriate sportspeople in Germany
Expatriate men's footballers in Tunisia
Cameroonian expatriate sportspeople in Tunisia |
https://en.wikipedia.org/wiki/Bundle%20adjustment | In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from different viewpoints.
Its name refers to the geometrical bundles of light rays originating from each 3D feature and converging on each camera's optical center, which are adjusted optimally according to an optimality criterion involving the corresponding image projections of all points.
Uses
Bundle adjustment is almost always used as the last step of feature-based 3D reconstruction algorithms. It amounts to an optimization problem on the 3D structure and viewing parameters (i.e., camera pose and possibly intrinsic calibration and radial distortion), to obtain a reconstruction which is optimal under certain assumptions regarding the noise pertaining to the observed image features: If the image error is zero-mean Gaussian, then bundle adjustment is the Maximum Likelihood Estimator. Bundle adjustment was originally conceived in the field of photogrammetry during the 1950s and has increasingly been used by computer vision researchers during recent years.
General approach
Bundle adjustment boils down to minimizing the reprojection error between the image locations of
observed and predicted image points, which is expressed as the sum of squares of a large number of nonlinear, real-valued functions. Thus, the minimization is achieved using nonlinear least-squares algorithms. Of these, Levenberg–Marquardt has proven to be one of the most successful due to its ease of implementation and its use of an effective damping strategy that lends it the ability to converge quickly from a wide range of initial guesses. By iteratively linearizing the function to be minimized in the neighborhood of the current estimate, the Levenberg–Marquardt algorithm involves the solution of linear systems termed the normal equations. When solving the minimization problems arising in the framework of
bundle adjustment, the normal equations have a sparse block structure owing to the lack of interaction among parameters for different 3D points and cameras. This can be exploited to gain tremendous computational benefits by employing a sparse variant of the Levenberg–Marquardt algorithm which explicitly takes advantage of the normal equations zeros pattern, avoiding storing and operating on zero-elements.
Mathematical definition
Bundle adjustment amounts to jointly refining a set of initial camera and structure parameter estimates for finding the set of parameters that most accurately predict the locations of the observed points in the set of available images. More formally, assume that 3D points are seen in views and let be the projection of the th point on image . Let denote the binary variables that equal 1 if point is visible in image and 0 otherwise. Assu |
https://en.wikipedia.org/wiki/Quadratic%20eigenvalue%20problem | In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues , left eigenvectors and right eigenvectors such that
where , with matrix coefficients and we require that , (so that we have a nonzero leading coefficient). There are eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. is also known as a quadratic polynomial matrix.
Spectral theory
A QEP is said to be regular if identically. The coefficient of the term in is , implying that the QEP is regular if is nonsingular.
Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, . As there are eigenvectors in a dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.
Applications
Systems of differential equations
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:
Where , and . If all quadratic eigenvalues of are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
Where are the quadratic eigenvalues, are the right quadratic eigenvectors, and is a parameter vector determined from the initial conditions on and .
Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
Finite element methods
A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, has the form , where is the mass matrix, is the damping matrix and is the stiffness matrix.
Other applications include vibro-acoustics and fluid dynamics.
Methods of solution
Direct methods for solving the standard or generalized eigenvalue problems and
are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials.
One approach is to transform the quadratic matrix polynomial to a linear matrix pencil (), and solve a generalized
eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
The most common linearization is the first companion linearization
with corresponding eigenvector
For convenience, one often takes to be the identity matrix. We solve for and , for example by computing the Generalized Schur form. We can then
take the first components of as the eigenvector of the original quadratic .
Another common linearization is given by
In the case when either or is a Hamiltonian matrix and the other is a skew-Hamiltonian matrix, the following linearizations can be used.
References
Linear algebra |
https://en.wikipedia.org/wiki/Prehomogeneous%20vector%20space | In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified by Sato and Tatsuo Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of G acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G.
Setting
In the setting of Sato, G is an algebraic group and V is a rational representation of G which has a (nonempty) open orbit in the Zariski topology. However, PVS can also be studied from the point of view of Lie theory: for instance, in Knapp (2002), G is a complex Lie group and V is a holomorphic representation of G with an open dense orbit. The two approaches are essentially the same, and the theory has validity over the real numbers. We assume, for simplicity of notation, that the action of G on V is a faithful representation. We can then identify G with its image in GL(V), although in practice it is sometimes convenient to let G be a covering group.
Although prehomogeneous vector spaces do not necessarily decompose into direct sums of irreducibles, it is natural to study the irreducible PVS (i.e., when V is an irreducible representation of G). In this case, a theorem of Élie Cartan shows that
G ≤ GL(V)
is a reductive group, with a centre that is at most one-dimensional. This, together with the obvious dimensional restriction
dim G ≥ dim V,
is the key ingredient in the Sato–Kimura classification.
Castling
The classification of PVS is complicated by the following fact. Suppose m > n > 0 and V is an m-dimensional representation of G over a field F. Then:
is a PVS if and only if is a PVS.
The proof is to observe that both conditions are equivalent to there being an open dense orbit of the action of G on the Grassmannian of
n-planes in V, because this is isomorphic to the Grassmannian of (m-n)-planes in V*.
(In the case that G is reductive, the pair (G,V) is equivalent to the pair (G, V*) by an automorphism of G.)
This transformation of PVS is called castling. Given a PVS V, a new PVS can be obtained by tensoring V with F and castling. By repeating this process, and regrouping tensor products, many new examples can be obtained, which are said to be "castling-equivalent". Thus PVS can be grouped into castling equivalence classes. Sato and Kimura show that in each such class, there is essentially one PVS of minimal dimension, which they call "reduced", and they classify the reduced irreducible PVS.
Classification
The classification of irreducible reduced PVS (G,V) splits into two cases: those for which G is semisimple, and those for which it i |
https://en.wikipedia.org/wiki/Automorphisms%20of%20the%20symmetric%20and%20alternating%20groups | In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
Summary
Generic case
: , and thus .
Formally, is complete and the natural map is an isomorphism.
: , and the outer automorphism is conjugation by an odd permutation.
:
Indeed, the natural maps are isomorphisms.
Exceptional cases
: trivial:
:
: , and is a semidirect product.
: , and
The exceptional outer automorphism of S6
Among symmetric groups, only S6 has a non-trivial outer automorphism,
which one can call exceptional (in analogy with exceptional Lie algebras) or exotic. In fact, Out(S6) = C2.
This was discovered by Otto Hölder in 1895.
The specific nature of the outer automorphism is as follows. The 360 permutations in the even subgroup (A6) are transformed amongst themselves:
the sole identity permutation maps to itself;
a 3-cycle such as (1 2 3) maps to the product of two 3-cycles such as (1 4 5)(2 6 3) and vice versa, accounting for 40 permutations each way;
a 5-cycle such as (1 2 3 4 5) maps to another 5-cycle such as (1 3 6 5 2), accounting for 144 permutations;
the product of two 2-cycles such as (1 2)(3 4) maps to another product of two 2-cycles such as (3 5)(4 6), accounting for 45 permutations;
the product of a 2-cycle and a 4-cycle such as (1 2 3 4)(5 6) maps to another such permutation such as (1 4 2 6)(3 5), accounting for the 90 remaining permutations.
And the odd part is also conserved:
a 2-cycle such as (1 2) maps to the product of three 2-cycles such as (1 2)(3 4)(5 6) and vice versa, there being 15 permutations each way;
the product of a 2-cycle and a 3-cycle such as (1 2 3)(4 5) maps to a 6-cycle such as (1 2 5 3 4 6) and vice versa, accounting for 120 permutations each way;
a 4-cycle such as (1 2 3 4) maps to another 4-cycle such as (1 6 2 4), accounting for the 90 remaining permutations.
Thus, all 720 permutations on 6 elements are accounted for. The outer automorphism does not preserve cycle structure in general, mapping some single cycles to the product of two or three cycles and vice versa.
This also yields another outer automorphism of A6, and this is the only exceptional outer automorphism of a finite simple group: for the infinite families of simple groups, there are formulas for the number of outer automorphisms, and the simple group of order 360, thought of as A6, would be expected to have two outer automorphisms, not four.
However, when A6 is viewed as PSL(2, 9) the outer automorphism group has the expected order. (For sporadic groups – i.e. those not falling in an infinite family – the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
Construction
There are numerous constructions, listed in .
Note that as an outer automorphism, |
https://en.wikipedia.org/wiki/Bin%20%28computational%20geometry%29 | In computational geometry, the bin is a data structure that allows efficient region queries. Each time a data point falls into a bin, the frequency of that bin is increased by one.
For example, if there are some axis-aligned rectangles on a 2D plane, the structure can answer the question, "Given a query rectangle, what are the rectangles intersecting it?" In the example in the top figure, A, B, C, D, E and F are existing rectangles, so the query with the rectangle Q should return C, D, E and F, if we define all rectangles as closed intervals.
The data structure partitions a region of the 2D plane into uniform-sized bins. The bounding box of the bins encloses all candidate rectangles to be queried. All the bins are arranged in a 2D array. All the candidates are represented also as 2D arrays. The size of a candidate's array is the number of bins it intersects.
For example, in the top figure, candidate B has 6 elements arranged in a 3 row by 2 column array because it intersects 6 bins in such an arrangement. Each bin contains the head of a singly linked list. If a candidate intersects a bin, it is chained to the bin's linked list. Each element in a candidate's array is a link node in the corresponding bin's linked list.
Operations
Query
From the query rectangle Q, we can find out which bin its lower-left corner intersects efficiently by simply subtracting the bin's bounding box's lower-left corner from the lower-left corner of Q and dividing the result by the width and height of a bin respectively. We then can iterate the bins Q intersects and examine all the candidates in the linked-lists of these bins. For each candidate we will check if it does indeed intersect Q. If so and if it was not previously reported, then we report it. We can use the convention that we only report a candidate the first time we find it. This can be done easily by clipping the candidate against the query rectangle and comparing its lower-left corner against the current location. If it is a match then we report, otherwise we skip.
Insertion and deletion
Insertion is linear to the number of bins a candidate intersects because inserting a candidate into 1 bin is constant time. Deletion is more expensive because we need to search the singly linked list of each bin the candidate intersects.
In a multithread environment, insert, delete and query are mutually exclusive. However, instead of locking the whole data structure, a sub-range of bins may be locked. Detailed performance analysis should be done to justify the overhead.
Efficiency and tuning
The analysis is similar to a hash table. The worst-case scenario is that all candidates are concentrated in one bin. Then query is O(n), delete is O(n), and insert is O(1), where n is the number of candidates. If the candidates are evenly spaced so that each bin has a constant number of candidates, The query is O(k) where k is the number of bins the query rectangle intersects. Insert and delete are O(m) where m is the number |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Real%20Madrid%20CF%20season | The 2006–07 season was Real Madrid CF's 76th season in La Liga. This article lists all matches that the club played in the 2006–07 season, and also shows statistics of the club's players.
Season summary
The summer of 2006 saw Real choose a new and returning coach, Fabio Capello coming from Juventus in the wake of Calciopoli.
Capello brought several recent and previous Juventus players with him to the club, but not all of them made a huge impact, the team instead relying on the goals of Ruud van Nistelrooy for the whole season.
Real returned to domestic league glory after a 3–1 victory against Mallorca in the last game of the season, but the club surprisingly sacked Capello shortly after winning the La Liga title after Capello refused to field David Beckham and Ronaldo as well as his defensive tactics. He was replaced by a surprise candidate Bernd Schuster from Getafe for the following season.
On the other hand, Real Madrid suffered painful exits in the Round of 16 of the Copa del Rey against Real Betis as well as in the UEFA Champions League against Bayern Munich in the Round of 16.
Players
Transfers
Real Madrid 2006-07 – first team shirt numbers
In
|}
Total spending: €100,500,000
Club
Technical staff
Kit
|
|
|
Other information
Pre-season and friendlies
Competitions
La Liga
League table
Matches
Results summary
Results by round
Points evolution
Source: LPF
Position evolution
Source: LPF
Copa del Rey
Round of 32
Round of 16
Champions League
Group E
Round of 16
Statistics
Squad stats
See also
2006–07 La Liga
2006–07 Copa del Rey
2006–07 UEFA Champions League
Notes and references
Real Madrid
Real Madrid CF seasons
Spanish football championship-winning seasons |
https://en.wikipedia.org/wiki/%CE%A9-logic | In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure . Just as the axiom of projective determinacy yields a canonical theory of , he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.
Analysis
Woodin's Ω-conjecture asserts that if there is a proper class of Woodin cardinals (for technical reasons, most results in the theory are most easily stated under this assumption), then Ω-logic satisfies an analogue of the completeness theorem. From this conjecture, it can be shown that, if there is any single axiom which is comprehensive over (in Ω-logic), it must imply that the continuum is not . Woodin also isolated a specific axiom, a variation of Martin's maximum, which states that any Ω-consistent (over ) sentence is true; this axiom implies that the continuum is .
Woodin also related his Ω-conjecture to a proposed abstract definition of large cardinals: he took a "large cardinal property" to be a property of ordinals which implies that α is a strong inaccessible, and which is invariant under forcing by sets of cardinal less than α. Then the Ω-conjecture implies that if there are arbitrarily large models containing a large cardinal, this fact will be provable in Ω-logic.
The theory involves a definition of Ω-validity: a statement is an Ω-valid consequence of a set theory T if it holds in every model of T having the form for some ordinal and some forcing notion . This notion is clearly preserved under forcing, and in the presence of a proper class of Woodin cardinals it will also be invariant under forcing (in other words, Ω-satisfiability is preserved under forcing as well). There is also a notion of Ω-provability; here the "proofs" consist of universally Baire sets and are checked by verifying that for every countable transitive model of the theory, and every forcing notion in the model, the generic extension of the model (as calculated in V) contains the "proof", restricted its own reals. For a proof-set A the condition to be checked here is called "A-closed". A complexity measure can be given on the proofs by their ranks in the Wadge hierarchy. Woodin showed that this notion of "provability" implies Ω-validity for sentences which are over V. The Ω-conjecture states that the converse of this result also holds. In all currently known core models, it is known to be true; moreover the consistency strength of the large cardinals corresponds to the least proof-rank required to "prove" the existence of the cardinals.
Notes
References
External links
W. H. Woodin, Slides for 3 talks
Set theory
Systems of formal logic |
https://en.wikipedia.org/wiki/Omega-logic | In mathematics, ω-logic can refer to:
ω-logic, an infinitary extension of first-order logic
Ω-logic, a deductive system in set theory developed by Hugh Woodin |
https://en.wikipedia.org/wiki/Poverty%20in%20Colombia |
Poverty statistics
In 2017, the National Administrative Department of Statistics (DANE) reported that 26.9% of the population were living below the poverty line, of which 7.4% in "extreme poverty". The multidimensional poverty rate stands at 17.0% of the population.
Unemployment
The average national unemployment rate in 2017 was 9.4%, although the informality is the biggest problem facing the labour market (the income of formal workers climbed 24.8% in 5 years while labor incomes of informal workers rose only 9%).
Inequality
According to the World Bank, Colombia's Gini coefficient (a measurement of inequality in wealth distribution) was 0.587 in 2000 and 0.535 in 2013, ranking alongside Chile, Panama, Brazil and Honduras as the most unequal Latin American countries in terms of wealth distribution.
Related issues
Literacy
In 2015, a total of 94.58% of the population aged 15 and older were recorded as literate, including 98.53% of those aged 15–24.
Malnutrition
In 2010, 3.4% of the children under 5 years old in Colombia suffer from global malnutrition (deficiency of weight for age) and up to 13% suffer from chronic malnutrition (deficiency of height for age). The situation is worse for the indigenous peoples of Colombia, who in the same indicators recorded rates of 7.5% and 29.5% respectively.
Social strata in Colombia
Colombia's social strata have been divided as follows and have been extensively used by the government as a reference to develop social welfare programs, statistical information and to some degree for the assignment of lands.
The system is the classification of the residential properties that should receive public services. Although the system does not consider the income per person and the rules say that the residential real estate should stratify and not households. All mayors should do the stratification of residential properties of their municipality or district.
In 1994, this stratification policy was made into law in order to grant subsidies to the poorest residents. The system is organized so that the people living in upper layers (strata 5 and 6) pay more for services like electricity, water and sewage than the groups in the lower strata. Critics of the system say that it impedes social mobility through stigmatization, while its proponents argue that it allows the poor to locate to areas where they will be able to access subsidized services. There are many studies that have shown that the socio-economic stratum is a bad instrument to allocate subsidies. In particular, these studies show that there is a high percentage of households of strata 1 and 2 which have a level of consumption similar to the households of strata 5 and 6 (18% of households in stratum 1, 36% of households in 2 and 66% of households in stratum 3 are located in quintiles 4 and 5 of the distribution of consumption. 98% of households in stratum 6 is in these quintiles).
Although nowadays there are more reliable sources to determine capacity to p |
https://en.wikipedia.org/wiki/Evenness | Evenness may refer to:
Species evenness
evenness of numbers, for which see parity (mathematics)
evenness of zero, a special case of the above
See also
Even (disambiguation) |
https://en.wikipedia.org/wiki/Choi%20Hyun | Choi Hyun (; born 7 November 1978) is a retired South Korean footballer who played as goalkeeper.
Club career
He formerly played for Jeju United, Busan IPark and Daejeon Citizen.
Career statistics
Club
External links
1978 births
Living people
Men's association football goalkeepers
South Korean men's footballers
Jeju United FC players
Gyeongnam FC players
Busan IPark players
Daejeon Hana Citizen players
K League 1 players
Footballers at the 2000 Summer Olympics
Olympic footballers for South Korea
Footballers from Busan
Chung-Ang University alumni |
https://en.wikipedia.org/wiki/Swansea%20urban%20area | The Swansea Urban Area or Swansea Built-up Area is an area of land in south Wales, defined by the Office for National Statistics for population monitoring purposes. It is an urban conurbation and is not coterminous with the City and County of Swansea. It consists of the urban area centred on Swansea city centre; the Swansea Valley including Clydach, Ystradgynlais and Pontardawe; and includes Neath and Port Talbot which are outside the county boundaries, but excludes the urban area of Gorseinon within the county boundaries.
The total population of the area in 2011 was 300,352 making it the 3rd largest in Wales, the 24th largest conurbation in England and Wales and the 27th largest in the United Kingdom. This was an increase of 11% from the 2001 figure of 270,506. Most of the increase was due to Ystradgynlais, Gowerton, Upper Killay and Glais becoming part of the urban area.
Subdivisions
The ONS provides sub-division statistics for the Swansea Urban Area
Notes:
Gowerton was included under the Swansea subdivision for the 2001 census.
Ystradgynlais was not part of the Swansea urban area until the 2011 census
See also
List of conurbations in the United Kingdom
References
External links
ONS map, showing urban areas in part of South Wales
Office for National Statistics: Census 2001, Key Statistics for urban areas
Geography of Swansea
Urban areas of Wales
Demographics of Wales |
https://en.wikipedia.org/wiki/Polychoric%20correlation | In statistics, polychoric correlation is a technique for estimating the correlation between two hypothesised normally distributed continuous latent variables, from two observed ordinal variables. Tetrachoric correlation is a special case of the polychoric correlation applicable when both observed variables are dichotomous. These names derive from the polychoric and tetrachoric series which are used for estimation of these correlations.
Applications and examples
This technique is frequently applied when analysing items on self-report instruments such as personality tests and surveys that often use rating scales with a small number of response options (e.g., strongly disagree to strongly agree). The smaller the number of response categories, the more a correlation between latent continuous variables will tend to be attenuated.
Lee, Poon & Bentler (1995) have recommended a two-step approach to factor analysis for assessing the factor structure of tests involving ordinally measured items. Kiwanuka and colleagues (2022) have also illustrated the application of polychoric correlations and polychoric confirmatory factor analysis in nursing science. This aims to reduce the effect of statistical artifacts, such as the number of response scales or skewness of variables leading to items grouping together in factors. In some disciplines, the statistical technique is rarely applied however, some scholars have demonstrated how it can be used as an alternative to the Pearson correlation.
Software
Mplus by Muthen and Muthen
polycor package in R by John Fox
psych package in R by William Revelle
lavaan package in R by Yves Rosseel
semopy package in Python by Georgy Meshcheryakov
PRELIS
POLYCORR program
PROC CORR in SAS (with POLYCHORIC or OUTPLC= options)
An extensive list of software for computing the polychoric correlation, by John Uebersax
package polychoric in Stata by Stas Kolenikov
See also
Liability threshold model
References
Lee, S.-Y., Poon, W. Y., & Bentler, P. M. (1995). "A two-stage estimation of structural equation models with continuous and polytomous variables". British Journal of Mathematical and Statistical Psychology, 48, 339–358.
Bonett, D. G., & Price R. M. (2005). "Inferential Methods for the Tetrachoric Correlation Coefficient". Journal of Educational and Behavioral Statistics, 30, 213.
Drasgow, F. (1986). Polychoric and polyserial correlations. In Kotz, Samuel, Narayanaswamy Balakrishnan, Campbell B. Read, Brani Vidakovic & Norman L. Johnson (Eds), Encyclopedia of Statistical Sciences, Vol. 7. New York, NY: John Wiley, pp. 68–74.
Kiwanuka, F., Kopra, J., Sak-Dankosky, N., Nanyonga, R. C., & Kvist, T. (2022). "Polychoric Correlation with Ordinal Data in Nursing Research". Nursing research, 10.1097/NNR.0000000000000614. Advance online publication. https://doi.org/10.1097/NNR.0000000000000614.
External links
The Tetrachoric and Polychoric Correlation Coefficients
Summary statistics
Covariance and correlation
|
https://en.wikipedia.org/wiki/Cape%20Verdeans%20in%20the%20Netherlands | Cape Verdeans in the Netherlands consist of migrants from Cape Verde to the Netherlands and their descendants. , figures from Statistics Netherlands showed 23,150 people of Cape Verdean origin in the Netherlands (people from Cape Verde, or those with a parent from there).
Migration history
Early migration from Cape Verde to the Netherlands began in the 1960s and 1970s. The migrants consisted primarily of young men who had signed on as sailors on Dutch ships, and as such they concentrated primarily in the port city of Rotterdam, especially the Heemraadsplein area. Prior to independence in 1975, Cape Verdean immigrants were registered as Portuguese immigrants from the overseas province of Portuguese Cape Verde. Another wave of migration began in 1975, following the independence of Cape Verde from Portugal; this new wave of migrants comprised primarily teachers, soldiers, and other lower officials of the former government. There was an immigration amnesty for Cape Verdean migrants in 1976.
From 1996 to 2010, the number of Cape Verdeans in the Netherlands recorded by Statistics Netherlands grew by roughly 25% from a base of 16,662 people; about three-quarters of the growth in that period was in the 2nd-generation category (people born in the Netherlands to one or two migrant parents from Cape Verde).
As of today, Cape Verdeans are part of the wider Portuguese-speaking community in the Netherlands, comprising around 35,000 people from PALOP countries (the overwhelming majority being from Angola or from Cape Verde), Timor-Leste or Macau, 65,000 Brazilians and 35,600 Portuguese.
Distribution
Approximately 90% live in the Rotterdam metropolitan area. In Rotterdam, the largest concentration live in Delfshaven, where they make up about 8.8% of the borough's population. The city has more than 60 Cape Verdean civil organisations. Smaller groups can be found in other cities such as Schiedam, Amsterdam, Zaanstad, and Delfzijl.
Employment and business
Cape Verdeans generally have better labour market outcomes than other migrant groups like Turks or Moroccans, similar to those of Surinamese, but worse than those of natives. The various Cape Verdean-run hair salons of Rotterdam often serve as gathering points for the women of the community. Other common ethnic business niches include transport businesses and travel agencies. The Cape Verdeans are also renown in the music industry and currently developing within the contemporary fine arts.
Notable people
Luc Castaignos, footballer
Alex Da Silva, Artist
Miguel Dias, boxer
E-Life, rapper
Eddy "Eddy Fort Moda Grog" Fortes, rapper
Nelson Freitas, singer, writer and producer
Alviar Lima, kickboxer
Suzanna Lubrano, singer
Dina Medina (1975-), singer
Gery "GMB" Mendes, musician and actor
David Mendes da Silva, footballer
Sonia Pereira (1972-), psychic, medium, television presenter and actress
Gil Semedo, singer
Sonja Silva (1977-), presenter, actress, model and singer
Luis Tavares, kickboxer
Lerin Duarte
Deroy D |
https://en.wikipedia.org/wiki/Shift%20theorem | In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the D-operators.
Statement
The theorem states that, if P(D) is a polynomial D-operator, then, for any sufficiently differentiable function y,
To prove the result, proceed by induction. Note that only the special case
needs to be proved, since the general result then follows by linearity of D-operators.
The result is clearly true for n = 1 since
Now suppose the result true for n = k, that is,
Then,
This completes the proof.
The shift theorem can be applied equally well to inverse operators:
Related
There is a similar version of the shift theorem for Laplace transforms ():
Examples
The exponential shift theorem can be used to speed the calculation of higher derivatives of functions that is given by the product of an exponential and another function. For instance, if , one has that
Another application of the exponential shift theorem is to solve linear differential equations whose characteristic polynomial has repeated roots.
Notes
References
Multivariable calculus
Shift theorem
Theorems in analysis |
https://en.wikipedia.org/wiki/Freudenthal%20magic%20square | In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction.
The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).
Constructions
See history for context and motivation. These were originally constructed circa 1958 by Freudenthal and Tits, with more elegant formulations following in later years.
Tits' approach
Tits' approach, discovered circa 1958 and published in , is as follows.
Associated with any normed real division algebra A (i.e., R, C, H or O) there is a Jordan algebra, J3(A), of 3 × 3 A-Hermitian matrices. For any pair (A, B) of such division algebras, one can define a Lie algebra
where denotes the Lie algebra of derivations of an algebra, and the subscript 0 denotes the trace-free part. The Lie algebra L has as a subalgebra, and this acts naturally on . The Lie bracket on (which is not a subalgebra) is not obvious, but Tits showed how it could be defined, and that it produced the following table of compact Lie algebras.
By construction, the row of the table with A=R gives , and similarly vice versa.
Vinberg's symmetric method
The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B. This is not obvious from Tits' construction. Ernest Vinberg gave a construction which is manifestly symmetric, in . Instead of using a Jordan algebra, he uses an algebra of skew-hermitian trace-free matrices with entries in A ⊗ B, denoted . Vinberg defines a Lie algebra structure on
When A and B have no derivations (i.e., R or C), this is just the Lie (commutator) bracket on . In the presence of derivations, these form a subalgebra acting naturally on as in Tits' construction, and the tracefree commutator bracket on is modified by an expression with values in .
Triality
A more recent construction, due to Pierre Ramond and Bruce Allison and developed by Chris Barton and Anthony Sudbery, uses triality in the form developed by John Frank Adams; this was presented in , and in streamlined form in . Whereas Vinberg's construction is based on the automorphism groups of |
https://en.wikipedia.org/wiki/Field%20with%20one%20element | In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.
Most proposed theories of F1 replace abstract algebra entirely. Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimicking their abstract properties. This allows the development of commutative algebra and algebraic geometry on new foundations. One of the defining features of theories of F1 is that these new foundations allow more objects than classical abstract algebra does, one of which behaves like a field of characteristic one.
The possibility of studying the mathematics of F1 was originally suggested in 1956 by Jacques Tits, published in , on the basis of an analogy between symmetries in projective geometry and the combinatorics of simplicial complexes. F1 has been connected to noncommutative geometry and to a possible proof of the Riemann hypothesis.
History
In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n-dimensional abstract simplicial complex, and if , then every k-simplex of the building must be contained in at least three n-simplices. This is analogous to the condition in classical projective geometry that a line must contain at least three points. However, there are degenerate geometries that satisfy all the conditions to be a projective geometry except that the lines admit only two points. The analogous objects in the theory of buildings are called apartments. Apartments play such a constituent role in the theory of buildings that Tits conjectured the existence of a theory of projective geometry in which the degenerate geometries would have equal standing with the classical ones. This geometry would take place, he said, over a field of characteristic one. Using this analogy it was possible to describe some of the elementary properties of F1, but it was not possible to construct it.
After Tits' initial observations, little progress was made until the early 1990s. In the late 1980s, Alexander Smirnov gave a series of talks in which he conjectured that the Riemann hypothesis could be proven by considering the integers as a curve o |
https://en.wikipedia.org/wiki/Fundamental%20discriminant | In mathematics, a fundamental discriminant D is an integer invariant in the theory of integral binary quadratic forms. If is a quadratic form with integer coefficients, then is the discriminant of Q(x, y). Conversely, every integer D with is the discriminant of some binary quadratic form with integer coefficients. Thus, all such integers are referred to as discriminants in this theory.
There are explicit congruence conditions that give the set of fundamental discriminants. Specifically, D is a fundamental discriminant if and only if one of the following statements holds
D ≡ 1 (mod 4) and is square-free,
D = 4m, where m ≡ 2 or 3 (mod 4) and m is square-free.
The first ten positive fundamental discriminants are:
1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 .
The first ten negative fundamental discriminants are:
−3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31 .
Connection with quadratic fields
There is a connection between the theory of integral binary quadratic forms and the arithmetic of quadratic number fields. A basic property of this connection is that D0 is a fundamental discriminant if, and only if, D0 = 1 or D0 is the discriminant of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant D0 ≠ 1, up to isomorphism. This is the reason why some authors consider 1 not to be a fundamental discriminant, although one may interpret D0 = 1 as the discriminant of the quadratic algebra consisting of two copies of the rational field.
Factorization
Fundamental discriminants may also be characterized by their factorization into positive and negative prime powers. Define the set
where the prime numbers congruent to 1 mod 4 are positive and those congruent to 3 mod 4 are negative. Then, a number D0 ≠ 1 is a fundamental discriminant if, and only if, it is the product of pairwise relatively prime members of S.
References
See also
Quadratic integer
Algebraic number theory
Quadratic forms |
https://en.wikipedia.org/wiki/Gheorghe%20P%C4%83un | Gheorghe Păun (; born December 6, 1950, in Cicănești, Argeș County) is a computer scientist from Romania, prominent for work on membrane computing and the P system.
Păun studied mathematics at the University of Bucharest, obtaining an MSc. in 1974 and a PhD in 1977 under the direction of Solomon Marcus. He has been a researcher at the Institute of Mathematics of the Romanian Academy since 1990. Păun was elected a member of the Academia Europaea in 2006, and a titular member of the Romanian Academy in 2012. He supervised the PhD thesis of 5 students. In 2016, he was awarded the title of Doctor Honoris Causa Scientiarum.
References
External links
Gheorghe Paun's webpage
Theoretical computer scientists
Romanian bioinformaticians
1950 births
Living people
Members of Academia Europaea
Titular members of the Romanian Academy
University of Bucharest alumni
People from Argeș County |
https://en.wikipedia.org/wiki/Franz%20Gehring | Franz Gehring (December 7, 1838 – January 4, 1884) was a German writer on music.
Gehring was a lecturer on mathematics, at first in Bonn, then from 1871 at Vienna University, but became known for his writings on music, particularly his biographies. Among the most notable are his biography of Mozart published in Francis Hueffer's The Great Musicians series of books, and several articles contributed to the Grove Dictionary of Music and Musicians.
References
1838 births
1884 deaths
German biographers
Male biographers
German male non-fiction writers |
https://en.wikipedia.org/wiki/Kervaire%20invariant | In mathematics, the Kervaire invariant is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.
The Kervaire invariant is defined as the Arf invariant of the skew-quadratic form on the middle dimensional homology group. It can be thought of as the simply-connected quadratic L-group , and thus analogous to the other invariants from L-theory: the signature, a -dimensional invariant (either symmetric or quadratic, ), and the De Rham invariant, a -dimensional symmetric invariant .
In any given dimension, there are only two possibilities: either all manifolds have Arf–Kervaire invariant equal to 0, or half have Arf–Kervaire invariant 0 and the other half have Arf–Kervaire invariant 1.
The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero. For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open.
Definition
The Kervaire invariant is the Arf invariant of the quadratic form determined by the framing on the middle-dimensional -coefficient homology group
and is thus sometimes called the Arf–Kervaire invariant. The quadratic form (properly, skew-quadratic form) is a quadratic refinement of the usual ε-symmetric form on the middle dimensional homology of an (unframed) even-dimensional manifold; the framing yields the quadratic refinement.
The quadratic form q can be defined by algebraic topology using functional Steenrod squares, and geometrically via the self-intersections
of immersions determined by the framing, or by the triviality/non-triviality of the normal bundles of embeddings (for ) and the mod 2 Hopf invariant of maps
(for ).
History
The Kervaire invariant is a generalization of the Arf invariant of a framed surface (that is, a 2-dimensional manifold with stably trivialized tangent bundle) which was used by Lev Pontryagin in 1950 to compute the homotopy group of maps (for ), which is the cobordism group of surfaces embedded in with trivialized normal bundle.
used his invariant for n = 10 to construct the Kervaire manifold, a 10-dimensional PL manifold with no differentiable structure, the first example of such a manifold, by showing that his invariant does not vanish on this PL manifold, but vanishes on all smooth manifolds of dimension 10.
computes the group of exotic spheres (in dimension greater than 4), with one step in the computation depending on the Kervaire invariant problem. Specifically, they show that the set of exotic spheres of dimension n – specifically the monoid of smooth structures on the standard n-sphere – is isomorphic to the group of h-cobordism classes of oriented homotopy n-spheres. They |
https://en.wikipedia.org/wiki/Graeme%20Dunstan | Graeme Clement Dunstan (born 4 August 1942) is a prominent Australian cultural and political activist.
A graduate of Essendon High School, Graeme matriculated in 1960 as dux with honours in maths, physics and chemistry.
He is an engineering graduate of the University of New South Wales (UNSW), where he was President of the Students' Union (1967) and twice co-editor of its newspaper, Tharunka, (1967 and 1971).
In 1966, while President of the UNSW Labor Club, he was active in organizing anti-Vietnam War protests. As organizer of the LBJ Welcome Committee he stopped US President Lyndon B. Johnson's motorcade in Liverpool Street, Sydney by lying under the president's car, upon which NSW Premier, Robert Askin, was reported to have said "run over the bastards".
In 1973 with Johnny Allen as director of the Aquarius Foundation of the Australian Union of Students and Dunstan, as director of the Foundation's biennial Aquarius Festival, together they produced the Aquarius Festival which took place in Nimbin, New South Wales.
Dunstan was the first community arts officer (1981–5) employed by the City of Campbelltown and in that role set up the Friends of the Campbelltown Art Gallery which lobbied successfully to found the Campbelltown Regional Art Gallery.
In 1985–89 he was a Festivals Consultant to the Victorian Tourism Commission and in that role he was one of the initiators of the Melbourne International Comedy Festival, Ltd serving as its founding Secretary in 1987.
As a freelance event organizer, Dunstan has helped produce celebrations including the Byron Bay NYE (1995), Bondi Beach Christmas and NYE celebration 1996, Nimbin 'Let It Grow' Mardi Grass (1998 – '99), the Sunshine Coast Schoolies Week (1997), the annual Eureka Dawn Walk (1998–) and the annual Independence from America Day Parade in Byron Bay (1998–).
He is captain of Peacebus.com, which is a website and a campaign vehicle from which he organizes Cyanide Watch and other actions of witness for peace, justice and a sustaining Earth. Peacebus.com describes him as an "old anarchist activist".
External links
Graeme Dunstan Curriculum Vitae
References
1942 births
Australian anti-war activists
Australian anarchists
University of New South Wales alumni
Living people |
https://en.wikipedia.org/wiki/Krener%27s%20theorem | In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy.
Theorem
Let
be a smooth control system, where
belongs to a finite-dimensional manifold and belongs to a control set . Consider the family of vector fields .
Let be the Lie algebra generated by with respect to the Lie bracket of vector fields.
Given , if the vector space is equal to ,
then belongs to the closure of the interior of the attainable set from .
Remarks and consequences
Even if is different from ,
the attainable set from has nonempty interior in the orbit topology,
as it follows from Krener's theorem applied to the control system restricted to the orbit through .
When all the vector fields in are analytic, if and only if belongs to the closure of the interior of the attainable set from . This is a consequence of Krener's theorem and of the orbit theorem.
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from is dense in , then the attainable set from
is actually equal to .
References
Control theory
Theorems in dynamical systems |
https://en.wikipedia.org/wiki/Junivan | Junivan Soares de Melo (born 20 November 1977), known as just Junivan, is a retired Brazilian footballer.
Career statistics
References
External links
Brazilian FA Database
1977 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
OFC Belasitsa Petrich players
PFC Lokomotiv Plovdiv players
Kayseri Erciyesspor footballers
Turan Tovuz players
Expatriate men's footballers in Bulgaria
Expatriate men's footballers in Turkey
Expatriate men's footballers in Azerbaijan
Men's association football forwards
Sportspeople from Amazonas (Brazilian state)
Azerbaijan Premier League players
First Professional Football League (Bulgaria) players
Süper Lig players |
https://en.wikipedia.org/wiki/Direct%20method | Direct method may refer to
Direct method (education) for learning a foreign language
Direct method (computational mathematics) as opposed to iterative method
Direct methods (crystallography) for estimating the phases of the Fourier transform of the scattering density from the corresponding magnitudes
Direct method in calculus of variations for constructing a proof of the existence of a minimizer for a given functional
Direct method (accounting) as opposed to indirect method for calculating cash flows |
https://en.wikipedia.org/wiki/Lydia%20Tomkiw | Lydia Tomkiw (August 6, 1959September 4, 2007) was an American poet, singer, and songwriter, best known for her work with the new wave musical group Algebra Suicide, along with her husband Don Hedeker.
Early life
Lydia Tomkiw was born in Chicago's Humboldt Park neighborhood in 1959, to Ukrainian immigrants Zenovia and Teodor Tomkiw. Her father worked at US Steel, her mother in a succession of retail jobs. By 1975, gang violence and crime in Humboldt Park had become untenable and the family moved to an apartment in Ukrainian Village, a vibrant hub of the émigré community. Tomkiw's creativity and aptitude secured her a spot to study art at the selective Lane Technical High School.
During these years Tomkiw wrote constantly — journaling, writing stories and poems. These proclivities alerted her early to poetry's influence and pull, and in particular, she developed an affection for the idiosyncratic Victorian poet Gerard Manley Hopkins. (Coincidentally, Hopkins would prefigure many of Tomkiw's qualities as a poet. He first set out to be a painter, turned to poetry that was characterized by striking imagery, conversational language, and formal playfulness; his sister Grace would set many of his poems to music.) But these were early, furtive forays and Tomkiw remained mainly inclined toward the visual arts.
In 1977, Tomkiw enrolled as an art major at University of Illinois at Chicago Circle, which boasted a rigorous and extremely competitive art program. Once there, however, she almost immediately found herself outclassed by other students. Her imagination often outpaced her skills, which stubbornly remained decent, but unexceptional. Tomkiw quickly grew frustrated, she began to reassess her creative practice.
Along with her art classes, Tomkiw took poetry classes taught by Maxine Chernoff, then a rising poet, fiction writer, and literary magazine editor. In Chernoff's class, Tomkiw underwent something of a conversion experience, embracing poetry as her primary vehicle of creative expression. She was particularly exhilarated by the performative possibility in poetry — she wrote to be read and she spoke to be heard. Inspired and guided by Chernoff, she quickly distinguished herself as a precocious and promising poet. In early 1978, while still a freshman, Tomkiw bundled together an early cache of poems and self-published them in a chapbook titled Ballpoint Erection. A year later, Tomkiw gathered another nineteen poems and self-published her second chapbook, Obsessions.
By the end of Tomkiw's first year at UIC, Chernoff suggested she transfer to Columbia College Chicago, a small liberal arts college with a long-standing focus on art, performance, and media. In particular, Chernoff thought Tomkiw would flourish under the tutelage of her husband, Paul Hoover, who served as poet-in-residence and taught a highly respected poetry workshop for undergraduates.
Tomkiw arrived at Columbia College in 1978 and fell in with an emerging group of predominantly |
https://en.wikipedia.org/wiki/Nonnegative%20rank%20%28linear%20algebra%29 | In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement that certain coefficients and entries of vectors/matrices have to be nonnegative.
For example, the linear rank of a matrix is the smallest number of vectors, such that every column of the matrix can be written as a linear combination of those vectors. For the nonnegative rank, it is required that the vectors must have nonnegative entries, and also that the coefficients in the linear combinations are nonnegative.
Formal definition
There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative m×n-matrix A is equal to the smallest number q such there exists a nonnegative m×q-matrix B and a nonnegative q×n-matrix C such that A = BC (the usual matrix product). To obtain the linear rank, drop the condition that B and C must be nonnegative.
Further, the nonnegative rank is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively:
where Rj ≥ 0 stands for "Rj is nonnegative". (To obtain the usual linear rank, drop the condition that the Rj have to be nonnegative.)
Given a nonnegative matrix A the nonnegative rank of A satisfies
A Fallacy
The rank of the matrix A is the largest number of columns which are linearly independent, i.e., none of the selected columns can be written as a linear combination of the other selected columns. It is not true that adding nonnegativity to this characterization gives the nonnegative rank: The nonnegative rank is in general less than or equal to the largest number of columns such that no selected column can be written as a nonnegative linear combination of the other selected columns.
Connection with the linear rank
It is always true that rank(A) ≤ rank+(A). In fact rank+(A) = rank(A) holds whenever rank(A) ≤ 2.
In the case rank(A) ≥ 3, however, rank(A) < rank+(A) is possible. For example, the matrix
satisfies rank(A) = 3 < 4 = rank+(A).
These two results (including the 4×4 matrix example above) were first provided by Thomas in a response to a question posed in 1973 by Berman and Plemmons.
Computing the nonnegative rank
The nonnegative rank of a matrix can be determined algorithmically.
It has been proved that determining whether is NP-hard.
Obvious questions concerning the complexity of nonnegative rank computation remain unanswered to date. For example, the complexity of determining the nonnegative rank of matrices of fixed rank k is unknown for k > 2.
Ancillary facts
Nonnegative rank has important applications in Combinatorial optimization: The minimum number of facets of an extension of a polyhedron P is equal to the nonnegative rank of its so-called slack matrix.
References
Linear algebra |
https://en.wikipedia.org/wiki/Packing%20dimension | In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982.
Definitions
Let (X, d) be a metric space with a subset S ⊆ X and let s ≥ 0 be a real number. The s-dimensional packing pre-measure of S is defined to be
Unfortunately, this is just a pre-measure and not a true measure on subsets of X, as can be seen by considering dense, countable subsets. However, the pre-measure leads to a bona fide measure: the s-dimensional packing measure of S is defined to be
i.e., the packing measure of S is the infimum of the packing pre-measures of countable covers of S.
Having done this, the packing dimension dimP(S) of S is defined analogously to the Hausdorff dimension:
An example
The following example is the simplest situation where Hausdorff and packing dimensions may differ.
Fix a sequence such that and . Define inductively a nested sequence of compact subsets of the real line as follows: Let . For each connected component of (which will necessarily be an interval of length ), delete the middle interval of length , obtaining two intervals of length , which will be taken as connected components of . Next, define . Then is topologically a Cantor set (i.e., a compact totally disconnected perfect space). For example, will be the usual middle-thirds Cantor set if .
It is possible to show that the Hausdorff and the packing dimensions of the set are given respectively by:
It follows easily that given numbers , one can choose a sequence as above such that the associated (topological) Cantor set has Hausdorff dimension and packing dimension .
Generalizations
One can consider dimension functions more general than "diameter to the s": for any function h : [0, +∞) → [0, +∞], let the packing pre-measure of S with dimension function h be given by
and define the packing measure of S with dimension function h by
The function h is said to be an exact (packing) dimension function for S if Ph(S) is both finite and strictly positive.
Properties
If S is a subset of n-dimensional Euclidean space Rn with its usual metric, then the packing dimension of S is equal to the upper modified box dimension of S: This result is interesting because it shows how a dimension derived from a measure (packing dimension) agrees with one derived without using a measure (the modified box dimension).
Note, however, that the packing dimension is not equal to the box dimension. For example, the set of rationals Q has box dimension one and packing dimension zero.
See also
Hausdorff dimension
Minkowski–Bouligand dimension
References
Dimension theory
Fractals
Metric geomet |
https://en.wikipedia.org/wiki/Ian%20G.%20Enting | Ian Enting (born 25 September 1948) is a mathematical physicist and the AMSI/MASCOS Professorial Fellow at the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS) based at The University of Melbourne.
Enting is the author of Twisted, The Distorted Mathematics of Greenhouse Denial in which he analyses the presentation and use of data by climate change deniers.
More recently he has been addressing the claims made in Ian Plimer's book Heaven and Earth. He has published a critique, "Ian Plimer’s ‘Heaven + Earth’ — Checking the Claims", listing what Enting claims are numerous misrepresentations of the sources cited in the book.
From 1980 to 2004 he worked in CSIRO Atmospheric Research, primarily on modelling the global carbon cycle.
He was one of the lead authors of the chapter and the Carbon Cycle in the 1994 IPCC report on Radiative Forcing of Climate.
Enting has published scientific papers, on mathematical physics and carbon cycle modelling, and a monograph on mathematical techniques for interpreting observations of carbon dioxide () and other trace gases.
References
External links
Ian Enting's homepage
1948 births
Living people
Australian climatologists
20th-century Australian mathematicians
21st-century Australian mathematicians
Intergovernmental Panel on Climate Change lead authors |
https://en.wikipedia.org/wiki/Twisted%3A%20The%20Distorted%20Mathematics%20of%20Greenhouse%20Denial | Twisted: The Distorted Mathematics of Greenhouse Denial is a 2007 book by Ian G. Enting, who is the Professorial Research Fellow in the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS) based at the University of Melbourne. The book analyses the arguments of climate change deniers and the use and presentation of statistics. Enting contends there are contradictions in their various arguments. The author also presents calculations of the actual emission levels that would be required to stabilise CO2 concentrations. This is an update of calculations that he contributed to the pre-Kyoto IPCC report on Radiative Forcing of Climate.
See also
Climate change
Greenhouse effect
Radiative forcing
References
Climate change books
2007 non-fiction books
2007 in the environment
Australian non-fiction books
Statistics books |
https://en.wikipedia.org/wiki/Pre-measure | In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure.
Definition
Let be a ring of subsets (closed under union and relative complement) of a fixed set and let be a set function. is called a pre-measure if
and, for every countable (or finite) sequence of pairwise disjoint sets whose union lies in
The second property is called -additivity.
Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring).
Carathéodory's extension theorem
It turns out that pre-measures give rise quite naturally to outer measures, which are defined for all subsets of the space More precisely, if is a pre-measure defined on a ring of subsets of the space then the set function defined by
is an outer measure on and the measure induced by on the -algebra of Carathéodory-measurable sets satisfies for (in particular, includes ). The infimum of the empty set is taken to be
(Note that there is some variation in the terminology used in the literature. For example, Rogers (1998) uses "measure" where this article uses the term "outer measure". Outer measures are not, in general, measures, since they may fail to be -additive.)
See also
References
(See section 1.2.)
Measures (measure theory) |
https://en.wikipedia.org/wiki/Immigration%20and%20crime | Immigration and crime refers to the relationship between criminal activity and the phenomenon of immigration. The academic literature and official statistics provide mixed findings for the relationship between immigration and crime. Research in the United States tends to suggest that immigration either has no impact on the crime rate or even that immigrants are less prone to crime. A meta-analysis of 51 studies from 1994–2014 on the relationship between immigration and crime in the United States found that, overall, the immigration-crime association is negative, but the relationship is very weak and there is significant variation in findings across studies. This is in line with a 2009 review of high-quality studies conducted in the United States that also found a negative relationship.
Research and statistics in some other, mainly European countries suggest a positive link between immigration and crime: immigrants from particular countries are often overrepresented in crime figures. The over-representation of immigrants in the criminal justice systems of several countries may be due to socioeconomic factors, imprisonment for migration offenses, and racial and ethnic discrimination by police and the judicial system. The relationship between immigration and terrorism is understudied, but existing research is inconclusive. Research on the relationship between refugee migration and crime is scarce and existing empirical evidence is often contradictory. According to statistics from some countries, asylum seekers are overrepresented in crime figures.
Worldwide
Much of the empirical research on the causal relationship between immigration and crime has been limited due to weak instruments for determining causality. The problem with causality primarily revolves around the location of immigrants being endogenous, which means that immigrants tend to disproportionately locate in deprived areas where crime is higher (because they cannot afford to stay in more expensive areas) or because they tend to locate in areas where there is a large population of residents of the same ethnic background. A burgeoning literature relying on strong instruments provides mixed findings.
The relationship between crime and the legal status of immigrants remains understudied, but studies on amnesty programs in the United States and Italy suggest that legal status can largely explain the differences in crime between legal and illegal immigrants, most likely because legal status leads to greater job market opportunities for the immigrants. However, one study finds that the Immigration Reform and Control Act of 1986 led to an increase in crime among previously undocumented immigrants. Existing research suggests that labor market opportunities have a significant impact on immigrant crime rates. Young, male, and poorly-educated immigrants have the highest individual probabilities of imprisonment among immigrants. Research suggests that the allocation of refugee immigrants to high c |
https://en.wikipedia.org/wiki/Pollaczek%E2%80%93Khinchine%20formula | In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arrive according to a Poisson process and have general service time distribution). The term is also used to refer to the relationships between the mean queue length and mean waiting/service time in such a model.
The formula was first published by Felix Pollaczek in 1930 and recast in probabilistic terms by Aleksandr Khinchin two years later. In ruin theory the formula can be used to compute the probability of ultimate ruin (probability of an insurance company going bankrupt).
Mean queue length
The formula states that the mean number of customers in system L is given by
where
is the arrival rate of the Poisson process
is the mean of the service time distribution S
is the utilization
Var(S) is the variance of the service time distribution S.
For the mean queue length to be finite it is necessary that as otherwise jobs arrive faster than they leave the queue. "Traffic intensity," ranges between 0 and 1, and is the mean fraction of time that the server is busy. If the arrival rate is greater than or equal to the service rate , the queuing delay becomes infinite. The variance term enters the expression due to Feller's paradox.
Mean waiting time
If we write W for the mean time a customer spends in the system, then where is the mean waiting time (time spent in the queue waiting for service) and is the service rate. Using Little's law, which states that
where
L is the mean number of customers in system
is the arrival rate of the Poisson process
W is the mean time spent at the queue both waiting and being serviced,
so
We can write an expression for the mean waiting time as
Queue length transform
Writing π(z) for the probability-generating function of the number of customers in the queue
where g(s) is the Laplace transform of the service time probability density function.
Waiting time transform
Writing W*(s) for the Laplace–Stieltjes transform of the waiting time distribution,
where again g(s) is the Laplace transform of service time probability density function. nth moments can be obtained by differentiating the transform n times, multiplying by (−1)n and evaluating at s = 0.
References
Single queueing nodes |
https://en.wikipedia.org/wiki/Leandr%C3%A3o%20%28footballer%29 | Leandro Costa Miranda Moraes or simply Leandrão (born 18 July 1983) is a Brazilian professional football manager and former player who is the current head coach of Boavista.
Club statistics
Honours
Internacional
Campeonato Gaúcho: 2002, 2005, 2008, 2009
Sport
Campeonato Pernambucano: 2010
ABC
Campeonato Brasileiro Série C: 2010
Campeonato Potiguar: 2011
Remo
Campeonato Paraense: 2014
Boavista
Copa Rio: 2017
External links
1983 births
Living people
Sportspeople from Uberlândia
Brazilian men's footballers
Brazilian football managers
Men's association football forwards
Boavista Sport Club players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Campeonato Brasileiro Série C players
J1 League players
K League 1 players
Israeli Premier League players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in South Korea
Expatriate men's footballers in Israel
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in South Korea
Brazilian expatriate sportspeople in Israel
Sport Club Internacional players
Botafogo de Futebol e Regatas players
Vissel Kobe players
Daejeon Hana Citizen players
Ulsan Hyundai FC players
Jeonnam Dragons players
Esporte Clube Vitória players
Sport Club do Recife players
ABC Futebol Clube players
Associação Atlética Ponte Preta players
Associação Desportiva São Caetano players
Rio Branco Esporte Clube players
Hapoel Acre F.C. players
Clube do Remo players
Esporte Clube Novo Hamburgo players
Grêmio Esportivo Brasil players
CR Vasco da Gama players
Boavista Sport Club managers
Footballers from Minas Gerais |
https://en.wikipedia.org/wiki/Linear%20space%20%28geometry%29 | A linear space is a basic structure in incidence geometry. A linear space consists of a set of elements called points, and a set of elements called lines. Each line is a distinct subset of the points. The points in a line are said to be incident with the line. Each two points are in a line, and any two lines may have no more than one point in common. Intuitively, this rule can be visualized as the property that two straight lines never intersect more than once.
Linear spaces can be seen as a generalization of projective and affine planes, and more broadly, of 2- block designs, where the requirement that every block contains the same number of points is dropped and the essential structural characteristic is that 2 points are incident with exactly 1 line.
The term linear space was coined by Paul Libois in 1964, though many results about linear spaces are much older.
Definition
Let L = (P, G, I) be an incidence structure, for which the elements of P are called points and the elements of G are called lines. L is a linear space if the following three axioms hold:
(L1) two distinct points are incident with exactly one line.
(L2) every line is incident to at least two distinct points.
(L3) L contains at least two distinct lines.
Some authors drop (L3) when defining linear spaces. In such a situation the linear spaces complying to (L3) are considered as nontrivial and those that do not are trivial.
Examples
The regular Euclidean plane with its points and lines constitutes a linear space, moreover all affine and projective spaces are linear spaces as well.
The table below shows all possible nontrivial linear spaces of five points. Because any two points are always incident with one line, the lines being incident with only two points are not drawn, by convention. The trivial case is simply a line through five points.
In the first illustration, the ten lines connecting the ten pairs of points are not drawn. In the second illustration, seven lines connecting seven pairs of points are not drawn.
A linear space of n points containing a line being incident with n − 1 points is called a near pencil. (See pencil)
Properties
The De Bruijn–Erdős theorem shows that in any finite linear space which is not a single point or a single line, we have .
See also
Block design
Fano plane
Projective space
Affine space
Molecular geometry
Partial linear space
References
.
Albrecht Beutelspacher: Einführung in die endliche Geometrie II. Bibliographisches Institut, 1983, , p. 159 (German)
J. H. van Lint, R. M. Wilson: A Course in Combinatorics. Cambridge University Press, 1992, . p. 188
L. M. Batten, Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
Incidence geometry |
https://en.wikipedia.org/wiki/Line%20field | In mathematics, a line field on a manifold is a formation of a line being tangent to a manifold at each point, i.e. a section of the line bundle over the manifold. Line fields are of particular interest in the study of complex dynamical systems, where it is conventional to modify the definition slightly.
Definitions
In general, let M be a manifold. A line field on M is a function μ that assigns to each point p of M a line μ(p) through the origin in the tangent space Tp(M). Equivalently, one may say that μ(p) is an element of the projective tangent space PTp(M), or that μ is a section of the projective tangent bundle PT(M).
In the study of complex dynamical systems, the manifold M is taken to be a Hersee surface. A line field on a subset A of M (where A is required to have positive two-dimensional Lebesgue measure) is a line field on A in the general sense above that is defined almost everywhere in A and is also a measurable function.
Dynamical systems
Fiber bundles |
https://en.wikipedia.org/wiki/Statistical%20epidemiology | Statistical epidemiology is an emerging branch of the disciplines of epidemiology and biostatistics that aims to:
Bring more statistical rigour to bear in the field of epidemiology
Recognise the importance of applied statistics, especially with respect to the context in which statistical methods are appropriate and inappropriate
Aid and improve our interpretation of observations
Introduction
The science of epidemiology has had enormous growth, particularly with charity and government funding. Many researchers have been trained to conduct studies, requiring multiple skills ranging from liaising with clinical staff to the statistical analysis of complex data, such as using Bayesian methods. The role of a Statistical Epidemiologist is to bring the most appropriate methods available to bear on observational study from medical research, requiring a broad appreciation of the underpinning methods and their context of applicability and interpretation.
The earliest mention of this phrase was in an article by EB Wilson, taking a critical look at the way in which statistical methods were developing and being applied in the science of epidemiology.
Academic recognition
There are two Professors of Statistical Epidemiology in the United Kingdom (University of Leeds and Imperial College, London) and a Statistical Epidemiology group (Oxford University).
Related fields
Statistical epidemiology draws upon quantitative methods from fields such as: statistics, operations research, computer science, economics, biology, and mathematics.
See also
Epidemiology
Biostatistics
References
Epidemiology
Demography
Biostatistics |
https://en.wikipedia.org/wiki/Semimodular%20lattice | In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition:
Semimodular law a ∧ b <: a implies b <: a ∨ b.
The notation a <: b means that b covers a, i.e. a < b and there is no element c such that a < c < b.
An atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.
Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular.
A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric. Some authors refer to M-symmetric lattices as semimodular lattices.
A semimodular lattice is one kind of algebraic lattice.
Birkhoff's condition
A lattice is sometimes called weakly semimodular if it satisfies the following condition due to Garrett Birkhoff:
Birkhoff's condition If a ∧ b <: a and a ∧ b <: b,
then a <: a ∨ b and b <: a ∨ b.
Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains) relatively atomic lattices.
Mac Lane's condition
The following two conditions are equivalent to each other for all lattices. They were found by Saunders Mac Lane, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation.
Mac Lane's condition 1 For any a, b, c such that b ∧ c < a < c < b ∨ a,
there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c.
Mac Lane's condition 2 For any a, b, c such that b ∧ c < a < c < b ∨ c,
there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c.
Every lattice satisfying Mac Lane's condition is semimodular. The converse is true for lattices of finite length, and more generally for relatively atomic lattices. Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric.
Notes
References
. (The article is about M-symmetric lattices.)
.
External links
See also
Antimatroid
Lattice theory |
https://en.wikipedia.org/wiki/Georgetown%20Hoyas%20women%27s%20lacrosse | The Georgetown Hoyas women's lacrosse team competes in the Big East Conference, an NCAA Division I conference. The first team was formed in 1977.
Historical statistics
*Statistics through 2018 season
Current team
The current head coach is Ricky Fried, who took over after Kim Simons retired following the 2004 season. Previously, Fried held the positions of assistant coach from 2002 to 2003 and associate head coach from 2003 to 2004, both under Simons.
The current assistant coaches are Erin Wellner-Hellmold and Michi Ellers. Hellmold played for Fried at Johns Hopkins University. Ellers played under Simons, with Fried as assistant coach, at Georgetown from 2002 to 2004.
History
The Georgetown Women's Lacrosse team advanced to two National Championship games in 2001 and 2002. The team appeared in 9 consecutive NCAA tournaments from 1998 to 2006 and advanced to 3 NCAA Final Four games in 2001, 2002, and 2004. The team had an undefeated record in the Big East from 2001 to 2006, earning them 6 consecutive Conference Championships. In 2007, the women's lacrosse team was defeated by Syracuse University in the first ever Big East women's lacrosse tournament. However, the Hoyas had previously been crowned the Big East Regular-Season Champions.
2006 season
In 2006, the Georgetown Women's lacrosse team continued to be a household name on the national scene. The team started the 2006 season ranked number 10/12 in National Polls and climbed all the way to earn the number 3 seed in their ninth straight NCAA Tournament appearance. Key regular season wins over Princeton, North Carolina, Maryland and Notre Dame continued to give the Hoyas a strong reputation as the women's game grows across the country. The Hoyas posted a 14-4 overall record, won its sixth straight Big East Conference Championship and made its eighth consecutive appearance in the NCAA Tournament Quarterfinals. The Hoyas defense ranked second in the nation allowing just 7.0 goals per game. During the 2006 season, the team posted a 4–1 record in a program-high five overtime games. Additionally, the squad had a 5–2 record in games decided by one goal.
New to the coaching staff in 2006, was assistant coach Michi Ellers, a former Georgetown player from 2000 to 2004. The team was led by Captains Stephanie Zodtner and Coco Stanwick.
2005 season
After advancing to the second round of the NCAA Tournament after beating Towson University 15–14, the Hoyas fell to Dartmouth College by a score of 13–3. The game marked Georgetown's seventh consecutive appearance in the NCAA quarterfinals and the team's eighth-straight NCAA appearance. Georgetown finished the 2005 season with a 13-5 overall record and a perfect 5–0 mark in the Big East. The team earned its fifth consecutive Big East Championship, continuing its undefeated record in the conference. This was Ricky Fried's first year as head coach of the team and Bowen Holden's first year as associate head coach. This was Erin Wellner's f |
https://en.wikipedia.org/wiki/Chapman%E2%80%93Robbins%20bound | In statistics, the Chapman–Robbins bound or Hammersley–Chapman–Robbins bound is a lower bound on the variance of estimators of a deterministic parameter. It is a generalization of the Cramér–Rao bound; compared to the Cramér–Rao bound, it is both tighter and applicable to a wider range of problems. However, it is usually more difficult to compute.
The bound was independently discovered by John Hammersley in 1950, and by Douglas Chapman and Herbert Robbins in 1951.
Statement
Let be the set of parameters for a family of probability distributions on .
For any two , let be the -divergence from to . Then:
A generalization to the multivariable case is:
Proof
By the variational representation of chi-squared divergence:
Plug in , to obtain: Switch the denominator and the left side and take supremum over to obtain the single-variate case. For the multivariate case, we define for any . Then plug in in the variational representation to obtain: Take supremum over , using the linear algebra fact that , we obtain the multivariate case.
Relation to Cramér–Rao bound
Usually, is the sample space of independent draws of a -valued random variable with distribution from a by parameterized family of probability distributions, is its -fold product measure, and is an estimator of . Then, for , the expression inside the supremum in the Chapman–Robbins bound converges to the Cramér–Rao bound of when , assuming the regularity conditions of the Cramér–Rao bound hold. This implies that, when both bounds exist, the Chapman–Robbins version is always at least as tight as the Cramér–Rao bound; in many cases, it is substantially tighter.
The Chapman–Robbins bound also holds under much weaker regularity conditions. For example, no assumption is made regarding differentiability of the probability density function p(x; θ) of . When p(x; θ) is non-differentiable, the Fisher information is not defined, and hence the Cramér–Rao bound does not exist.
See also
Cramér–Rao bound
Estimation theory
References
Further reading
Statistical inequalities
Estimation theory |
https://en.wikipedia.org/wiki/Siegel%E2%80%93Tukey%20test | In statistics, the Siegel–Tukey test, named after Sidney Siegel and John Tukey, is a non-parametric test which may be applied to data measured at least on an ordinal scale. It tests for differences in scale between two groups.
The test is used to determine if one of two groups of data tends to have more widely dispersed values than the other. In other words, the test determines whether one of the two groups tends to move, sometimes to the right, sometimes to the left, but away from the center (of the ordinal scale).
The test was published in 1960 by Sidney Siegel and John Wilder Tukey in the Journal of the American Statistical Association, in the article "A Nonparametric Sum of Ranks Procedure for Relative Spread in Unpaired Samples."
Principle
The principle is based on the following idea:
Suppose there are two groups A and B with n observations for the first group and m observations for the second (so there are N = n + m total observations). If all N observations are arranged in ascending order, it can be expected that the values of the two groups will be mixed or sorted randomly, if there are no differences between the two groups (following the null hypothesis H0). This would mean that among the ranks of extreme (high and low) scores, there would be similar values from Group A and Group B.
If, say, Group A were more inclined to extreme values (the alternative hypothesis H1), then there will be a higher proportion of observations from group A with low or high values, and a reduced proportion of values at the center.
Hypothesis H0: σ2A = σ2B & MeA = MeB (where σ2 and Me are the variance and the median, respectively)
Hypothesis H1: σ2A > σ2B
Method
Two groups, A and B, produce the following values (already sorted in ascending order):
A: 33 62 84 85 88 93 97 B: 4 16 48 51 66 98
By combining the groups, a group of 13 entries is obtained. The ranking is done by alternate extremes (rank 1 is lowest, 2 and 3 are the two highest, 4 and 5 are the two next lowest, etc.).
The sum of the ranks within each W group:
WA = 5 + 12 + 11 + 10 + 7 + 6 + 3 = 54
WB = 1 + 4 + 8 + 9 + 13 + 2 = 37
If the null hypothesis is true, it is expected that the average ranks of the two groups will be similar.
If one of the two groups is more dispersed its ranks will be lower, as extreme values receive lower ranks, while the other group will receive more of the high scores assigned to the center. To test the difference between groups for significance a Wilcoxon rank sum test is used, which also justifies the notation WA and WB in calculating the rank sums.
From the rank sums the U statistics are calculated by subtracting off the minimum possible score, n(n + 1)/2 for each group:
UA = 54 − 7(8)/2 = 26
UB = 37 − 6(7)/2 = 16
According to the minimum of these two values is distributed according to a Wilcoxon rank-sum distribution with parameters given by the two group sizes:
Which allows the calculation of a p-value for this test according to the |
https://en.wikipedia.org/wiki/KdV%20hierarchy | In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which contains the Korteweg–de Vries equation.
Details
Let be translation operator defined on real valued functions as . Let be set of all analytic functions that satisfy , i.e. periodic functions of period 1. For each , define an operator
on the space of smooth functions on . We define the Bloch spectrum to be the set of such that there is a nonzero function with and . The KdV hierarchy is a sequence of nonlinear differential operators such that for any we have an analytic function and we define to be and
,
then is independent of .
The KdV hierarchy arises naturally as a statement of Huygens' principle for the D'Alembertian.
Explicit equations for first three terms of hierarchy
The first three partial differential equations of the KdV hierarchy are
where each equation is considered as a PDE for for the respective .
The first equation identifies and as in the original KdV equation. These equations arise as the equations of motion from the (countably) infinite set of independent constants of motion by choosing them in turn to be the Hamiltonian for the system. For , the equations are called higher KdV equations and the variables higher times.
Application to periodic solutions of KdV
One can consider the higher KdVs as a system of overdetermined PDEs for
Then solutions which are independent of higher times above some fixed and with periodic boundary conditions are called finite-gap solutions. Such solutions turn out to correspond to compact Riemann surfaces, which are classified by their genus . For example, gives the constant solution, while corresponds to cnoidal wave solutions.
For , the Riemann surface is a hyperelliptic curve and the solution is given in terms of the theta function. In fact all solutions to the KdV equation with periodic initial data arise from this construction .
See also
Witten's conjecture
Huygens' principle
References
Sources
External links
KdV hierarchy at the Dispersive PDE Wiki.
Partial differential equations
Solitons
Exactly solvable models |
https://en.wikipedia.org/wiki/Aleksandrov%E2%80%93Clark%20measure | In mathematics, Aleksandrov–Clark (AC) measures are specially constructed measures named after the two mathematicians, A. B. Aleksandrov and Douglas Clark, who discovered some of their deepest properties. The measures are also called either Aleksandrov measures, Clark measures, or occasionally spectral measures.
AC measures are used to extract information about self-maps of the unit disc, and have applications in a number of areas of complex analysis, most notably those related to operator theory. Systems of AC measures have also been constructed for higher dimensions, and for the half-plane.
Construction of the measures
The original construction of Clark relates to one-dimensional perturbations of compressed shift operators on subspaces of the Hardy space:
By virtue of Beurling's theorem, any shift-invariant subspace of this space is of the form
where is an inner function. As such, any invariant subspace of the adjoint of the shift is of the form
We now define to be the shift operator compressed to , that is
Clark noticed that all the one-dimensional perturbations of , which were also unitary maps, were of the form
and related each such map to a measure, on the unit circle, via the Spectral theorem. This collection of measures, one for each on the unit circle , is then called the collection of AC measures associated with .
An alternative construction
The collection of measures may also be constructed for any analytic function (that is, not necessarily an inner function). Given an analytic self map, , of the unit disc, , we can construct a collection of functions, , given by
one for each . Each of these functions is positive and harmonic, so by Herglotz' Theorem each is the Poisson integral of some positive measure on . This collection is the set of AC measures associated with . It can be shown that the two definitions coincide for inner functions.
References
Hardy spaces |
https://en.wikipedia.org/wiki/Ra%C3%BAl%20Rojas | Raúl Rojas González (born 1955, in Mexico City) is an emeritus professor of Computer Science and Mathematics at the Free University of Berlin, and a renowned specialist in artificial neural networks. The FU-Fighters, football-playing robots he helped build, were world champions in 2004 and 2005. He is now leading an autonomous car project called Spirit of Berlin.
He and his team were awarded the Wolfgang von Kempelen Prize for his work on Konrad Zuse and the history of computers. Although most of his current research and teaching revolves around artificial intelligence and its applications, he holds academic degrees in mathematics and economics.
In 2009 the Mexican government created the Raúl Rojas González Prize for scientific achievement by Mexican citizens. The first recipient of the prize was Luis Rafael Herrera Estrella, for contributions to plant biotechnology.
He ran for president at the Free University of Berlin in 2010.
History
Rojas was born on June 25, 1955, in Mexico City to an engineer and a teacher. He attended university at the National Polytechnic Institute in Mexico City, where he majored in Mathematics and Physics. He moved to Germany in 1982 as a doctoral student in economics under the guidance of the political economist Elmar Altvater. The resulting dissertation was published under the title "Die Armut der Nationen – Handbuch zur Schuldenkrise von Argentinien bis Zaire" (The poverty of nations – Handbook of debt crisis from Argentina to Zaire). He became a full professor at University of Halle-Wittenberg in 1994, and later moved to the Free University of Berlin, where he remains today in the Informatics department. His wife, Margarita Esponda Argüero, is a professor in the same department.
Prizes
2001: Gründerpreis Multimedia of the German Ministry of Finance and Technology
2002: European Academic Software Award
2004 and 2005: World champions in football robots with the FU-Fighters
2005: Wolfgang von Kempelen Prize for the History of Informatics
2009: Received the Heberto Castillo gold medal for contributions to science from the Mexico City government
2015: Was named Professor of the year by the Association of German Universities
2015: Received the National Prize of Sciences and Arts by the Mexican Government in the category of Technology and Design
Works
Available as a free e-book
References
External links
Homepage of Raúl Rojas at the Free University of Berlin
Curriculum vitae of Raúl Rojas
FU-Fighters football robots
Autonomous car project
1955 births
Living people
National Autonomous University of Mexico alumni
Instituto Politécnico Nacional alumni
Scientists from Mexico City
Mexican scientists
Academic staff of the Free University of Berlin |
https://en.wikipedia.org/wiki/O%C4%BE%C5%A1avka%2C%20Stropkov%20District | Oľšavka (; ) is a village and municipality in Stropkov District in the Prešov Region of north-eastern Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Stropkov District
Šariš |
https://en.wikipedia.org/wiki/L-Seven | L-Seven was an American post-punk band from Detroit, Michigan, United States. The band existed during 1980–1983. Some band members had been formerly active in Detroit punk bands The Blind, Algebra Mothers, and Retro. Anecdotally, they lifted their name from the Rick James album Bustin' Out of L Seven. The band was founded by Michael Smith (Smitt E. Smitty), Dave Rice, Frank Callis, and Charles McEvoy, and then recruited Larissa Stolarchuck to be their lead singer after they realized she was a gifted lyricist and front person. In February 1982, they recorded a self-titled three-song EP at Multi Trac Studios in Redford, Michigan. The EP was released as a 7" titled "L-Seven" by Touch and Go Special Forces in 1982. Although Touch and Go Special Forces was created to issue records of a different nature than the hardcore records that Touch and Go was issuing at the time, L-Seven's record was the only release under the "Special Forces" imprint.
During their brief existence, L-Seven supported many well-known post-punk bands such as The Gun Club, Killing Joke, The Stranglers, Iggy Pop, Bauhaus, U2, and The Birthday Party.
The last of these was one of the inspirations for the Laughing Hyenas, the band Singer Larissa Stolarchuk (under the nom de plume Larissa Strickland) went on to form with former Negative Approach singer John Brannon. In the Hyenas, she switched to playing guitar, relinquishing vocal duties to Brannon.
Stolarchuk died on October 9, 2006. Drummer Kory Clarke fronts the long-running band Warrior Soul. Guitarist Dave Rice would go on to form the bands The Linkletters and Sandy Duncan's Eye and at one point auditioned for British post punk band Public Image Limited. In 2020, with the involvement of Sonic Youth drummer and Michigan native Steve Shelley, a long time fan of the band, Third Man Records released a compilation of unreleased L-Seven Demos and live recordings, which includes live renditions of the Misfits song "London Dungeon" and the Alice Cooper song "You Drive Me Nervous". The website Detroit Punk Archive has a compiled an oral history of the band.
Discography
References
External links
"L-Seven" at the "Touch and Go Records" website
Corey Rusk notes the death of Larissa Stolarchuk
American pop music groups
Musical groups from Detroit
Touch and Go Records artists
Musical groups established in 1980
Musical groups disestablished in 1983
1980 establishments in Michigan |
https://en.wikipedia.org/wiki/Direct%20image%20with%20compact%20support | In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations.
Definition
Let f: X → Y be a continuous mapping of locally compact Hausdorff topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor
f!: Sh(X) → Sh(Y)
that sends a sheaf F on X to the sheaf f!(F) given by the formula
f!(F)(U) := {s ∈ F(f −1(U)) | f|supp(s): supp(s) → U is proper}
for every open subset U of Y. Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines f!(F) as a subsheaf of the direct image sheaf f∗(F), and the functoriality of this construction then follows from basic properties of the support and the definition of sheaves.
The assumption that the spaces be locally compact Hausdorff is imposed in most sources (e.g., Iversen or Kashiwara–Schapira). In slightly greater generality, Olaf Schnürer and Wolfgang Soergel have introduced the notion of a "locally proper" map of spaces and shown that the functor of direct image with compact support remains well-behaved when defined for separated and locally proper continuous maps between arbitrary spaces.
Properties
If f is proper, then f! equals f∗.
If f is an open embedding, then f! identifies with the extension by zero functor.
References
, esp. section VII.1
Sheaf theory
Theory of continuous functions |
https://en.wikipedia.org/wiki/Katz%27s%20back-off%20model | Katz back-off is a generative n-gram language model that estimates the conditional probability of a word given its history in the n-gram. It accomplishes this estimation by backing off through progressively shorter history models under certain conditions. By doing so, the model with the most reliable information about a given history is used to provide the better results.
The model was introduced in 1987 by Slava M. Katz. Prior to that, n-gram language models were constructed by training individual models for different n-gram orders using maximum likelihood estimation and then interpolating them together.
Method
The equation for Katz's back-off model is:
where
C(x) = number of times x appears in training
wi = ith word in the given context
Essentially, this means that if the n-gram has been seen more than k times in training, the conditional probability of a word given its history is proportional to the maximum likelihood estimate of that n-gram. Otherwise, the conditional probability is equal to the back-off conditional probability of the (n − 1)-gram.
The more difficult part is determining the values for k, d and α.
is the least important of the parameters. It is usually chosen to be 0. However, empirical testing may find better values for k.
is typically the amount of discounting found by Good–Turing estimation. In other words, if Good–Turing estimates as , then
To compute , it is useful to first define a quantity β, which is the left-over probability mass for the (n − 1)-gram:
Then the back-off weight, α, is computed as follows:
The above formula only applies if there is data for the "(n − 1)-gram". If not, the algorithm skips n-1 entirely and uses the Katz estimate for n-2. (and so on until an n-gram with data is found)
Discussion
This model generally works well in practice, but fails in some circumstances. For example, suppose that the bigram "a b" and the unigram "c" are very common, but the trigram "a b c" is never seen. Since "a b" and "c" are very common, it may be significant (that is, not due to chance) that "a b c" is never seen. Perhaps it's not allowed by the rules of the grammar. Instead of assigning a more appropriate value of 0, the method will back off to the bigram and estimate P(c | b), which may be too high.
References
Language modeling
Statistical natural language processing |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.