source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Kato%20surface | In mathematics, a Kato surface is a compact complex surface with positive first Betti number that has a global spherical shell. showed that Kato surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite cyclic fundamental group, and are never Kähler manifolds. Examples of Kato surfaces include Inoue-Hirzebruch surfaces and Enoki surfaces. The global spherical shell conjecture claims that all class VII surfaces with positive second Betti number are Kato surfaces.
References
Complex surfaces |
https://en.wikipedia.org/wiki/Enoki%20surface | In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor D with H0(O(D)) ≠ 0 and (D, D) = 0. constructed some examples. They are surfaces of class VII, so are non-Kähler and have Kodaira dimension −∞.
References
Complex surfaces |
https://en.wikipedia.org/wiki/Monica%20Seles%20career%20statistics | This is a list of the main career statistics of former tennis player Monica Seles.
Significant finals
Grand Slam finals
Singles: 13 finals (9 titles, 4 runner-ups)
Year-end championships finals
Singles: 4 finals (3 titles, 1 runner-up)
(i) = Indoor
Tier I finals
Singles: 18 finals (9 titles, 9 runner-ups)
Doubles: 4 finals (3 titles, 1 runner-up)
Career finals
Singles: 85 (53 titles, 32 runner-ups)
Doubles: 9 (6 titles, 3 runner-ups)
Team competition
Finals: 5 (3 titles, 2 runner-up)
Olympic singles bronze medal match
Fed Cup
Wins (3)
Participating (19)
Singles (17)
Doubles (2)
Singles performance timeline
Career Grand Slam tournament seedings
The tournaments won by Seles are in boldface, and advanced into finals by Seles are in italics.
WTA Tour career earnings
Head-to-head statistics
Head-to-head vs. top 10 ranked players
Top 10 wins
No. 1 wins
Longest winning streaks
36-match win streak (1990)
41-match Grand Slam win streak (1991–92)
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Mwaru | Mwaru is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,930 people in the ward, from 11,784 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Muhintiri | Muhintiri is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,761 people in the ward, from 8,896 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Msisi%20%28Singida%20Rural%20ward%29 | Msisi is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,220 people in the ward, from 9,314 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Minyughe | Minyughe is an administrative ward in the Ikungi District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,163 people in the ward, from 18,440 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mgungira | Mgungira is an administrative ward in the Ikungi District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,185 people in the ward, from 6,548 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Maghojoa | Maghojoa is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,857 people in the ward, from 8,983 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Kinyeto | Kinyeto is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,936 people in the ward, from 9,055 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Ikhanoda | Ikhanoda is an administrative ward in the Singida Rural district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,968 people in the ward, from 10,907 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Shri%20K.%20Singh | S. K. Singh was a professor of mathematics from University of Missouri - Kansas City. He received his Ph.D. on the Entire and Meromorphic functions from Aligarh Muslim University in 1953. His advisor was S. M. Shah. Singh was one of the founder fathers and Head of the Department of Mathematics, Karnataka University, Dharwar.
Notes
Aligarh Muslim University
University of Missouri–Kansas City faculty |
https://en.wikipedia.org/wiki/S.%20M.%20Shah | Swarupchand Mohanlal Shah (30 December 1905 – 21 April 1996) was a Distinguished Professor of Mathematics at the University of Kentucky. He received his Ph.D. from University of London in 1942, advised by Edward Titchmarsh who was a Ph.D. student of G. H. Hardy. He was a fellow of the Royal Society of Edinburgh.
Selected publications
"Advanced differential equations with piecewise constant argument deviations", SM Shah, J Wiener, International Journal of Mathematics and Mathematical …, 1983, hindawi.com
"Univalent functions with univalent derivatives", SM Shah, SY Trimble, Bulletin of the American Mathematical Society, 1969, ams.org
"Trigonometric series with quasi-monotone coefficients", SM Shah, Proceedings of the American Mathematical Society, 1962, jstor.org
Notes
1905 births
1996 deaths
University of Kentucky faculty
Alumni of the University of London
Indian expatriates in the United Kingdom
Indian emigrants to the United States |
https://en.wikipedia.org/wiki/Nati%20Azaria | Nati Azaria (; born May 31, 1967) is a former Israeli footballer and now manager.
Azaria was a striker who scored over 100 goals in a career that lasted 15 years.
References
Statistics
External links
1967 births
Living people
Israeli Jews
Israeli men's footballers
Maccabi Netanya F.C. players
Hapoel Kfar Saba F.C. players
Hapoel Tayibe F.C. players
Hapoel Iksal F.C. players
Maccabi Sha'arayim F.C. players
Maccabi Netanya F.C. managers
Footballers from Netanya
Men's association football forwards
Israeli football managers |
https://en.wikipedia.org/wiki/Spherical%20measure | In mathematics — specifically, in geometric measure theory — spherical measure σn is the "natural" Borel measure on the n-sphere Sn. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that σn(Sn) = 1.
Definition of spherical measure
There are several ways to define spherical measure. One way is to use the usual "round" or "arclength" metric ρn on Sn; that is, for points x and y in Sn, ρn(x, y) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of Rn+1). Now construct n-dimensional Hausdorff measure Hn on the metric space (Sn, ρn) and define
One could also have given Sn the metric that it inherits as a subspace of the Euclidean space Rn+1; the same spherical measure results from this choice of metric.
Another method uses Lebesgue measure λn+1 on the ambient Euclidean space Rn+1: for any measurable subset A of Sn, define σn(A) to be the (n + 1)-dimensional volume of the "wedge" in the ball Bn+1 that it subtends at the origin. That is,
where
The fact that all these methods define the same measure on Sn follows from an elegant result of Christensen: all these measures are obviously uniformly distributed on Sn, and any two uniformly distributed Borel regular measures on a separable metric space must be constant (positive) multiples of one another. Since all our candidate σn's have been normalized to be probability measures, they are all the same measure.
Relationship with other measures
The relationship of spherical measure to Hausdorff measure on the sphere and Lebesgue measure on the ambient space has already been discussed.
Spherical measure has a nice relationship to Haar measure on the orthogonal group. Let O(n) denote the orthogonal group acting on Rn and let θn denote its normalized Haar measure (so that θn(O(n)) = 1). The orthogonal group also acts on the sphere Sn−1. Then, for any x ∈ Sn−1 and any A ⊆ Sn−1,
In the case that Sn is a topological group (that is, when n is 0, 1 or 3), spherical measure σn coincides with (normalized) Haar measure on Sn.
Isoperimetric inequality
There is an isoperimetric inequality for the sphere with its usual metric and spherical measure (see Ledoux & Talagrand, chapter 1):
If A ⊆ Sn−1 is any Borel set and B⊆ Sn−1 is a ρn-ball with the same σn-measure as A, then, for any r > 0,
where Ar denotes the "inflation" of A by r, i.e.
In particular, if σn(A) ≥ and n ≥ 2, then
References
(See chapter 1)
(See chapter 3)
Measures (measure theory) |
https://en.wikipedia.org/wiki/Uniformly%20distributed%20measure | In mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends only on its radius and not on its centre. By convention, the measure is also required to be Borel regular, and to take positive and finite values on open balls of finite radius. Thus, if (X, d) is a metric space, a Borel regular measure μ on X is said to be uniformly distributed if
for all points x and y of X and all 0 < r < +∞, where
Christensen's lemma
As it turns out, uniformly distributed measures are very rigid objects. On any "decent" metric space, the uniformly distributed measures form a one-parameter linearly dependent family:
Let μ and ν be uniformly distributed Borel regular measures on a separable metric space (X, d). Then there is a constant c such that μ = cν.
References
(See chapter 3)
Measures (measure theory) |
https://en.wikipedia.org/wiki/Ruled%20variety | In algebraic geometry, a variety over a field k is ruled if it is birational to the product of the projective line with some variety over k. A variety is uniruled if it is covered by a family of rational curves. (More precisely, a variety X is uniruled if there is a variety Y and a dominant rational map Y × P1 – → X which does not factor through the projection to Y.) The concept arose from the ruled surfaces of 19th-century geometry, meaning surfaces in affine space or projective space which are covered by lines. Uniruled varieties can be considered to be relatively simple among all varieties, although there are many of them.
Properties
Every uniruled variety over a field of characteristic zero has Kodaira dimension −∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodaira dimension −∞ over a field of characteristic zero should be uniruled. A related statement is known in all dimensions: Boucksom, Demailly, Păun and Peternell showed that a smooth projective variety X over a field of characteristic zero is uniruled if and only if the canonical bundle of X is not pseudo-effective (that is, not in the closed convex cone spanned by effective divisors in the Néron-Severi group tensored with the real numbers). As a very special case, a smooth hypersurface of degree d in Pn over a field of characteristic zero is uniruled if and only if d ≤ n, by the adjunction formula. (In fact, a smooth hypersurface of degree d ≤ n in Pn is a Fano variety and hence is rationally connected, which is stronger than being uniruled.)
A variety X over an uncountable algebraically closed field k is uniruled if and only if there is a rational curve passing through every k-point of X. By contrast, there are varieties over the algebraic closure k of a finite field which are not uniruled but have a rational curve through every k-point. (The Kummer variety of any non-supersingular abelian surface over p with p odd has these properties.) It is not known whether varieties with these properties exist over the algebraic closure of the rational numbers.
Uniruledness is a geometric property (it is unchanged under field extensions), whereas ruledness is not. For example, the conic x2 + y2 + z2 = 0 in P2 over the real numbers R is uniruled but not ruled. (The associated curve over the complex numbers C is isomorphic to P1 and hence is ruled.) In the positive direction, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled. Smooth cubic 3-folds and smooth quartic 3-folds in P4 over C are uniruled but not ruled.
Positive characteristic
Uniruledness behaves very differently in positive characteristic. In particular, there are uniruled (and even unirational) surfaces of general type: an example is the surface xp+1 + yp+1 + zp+1 + wp+1 = 0 in P3 over p, for any prime number p ≥ 5. So uniruledness does not imply that the Kodaira dimension is −∞ in positive characteristic.
A variety X is separably |
https://en.wikipedia.org/wiki/K.%20S.%20Chandrasekharan | Komaravolu Chandrasekharan (21 November 1920 – 13 April 2017)
was a professor at ETH Zurich and a founding faculty member of School of Mathematics, Tata Institute of Fundamental Research (TIFR). He is known for his work in number theory and summability. He received the Padma Shri, the Shanti Swarup Bhatnagar Award, and the Ramanujan Medal, and he was an honorary fellow of TIFR. He was president of the International Mathematical Union (IMU) from 1971 to 1974.
Biography
Chandrasekharan was born on 21 November 1920 in Machilipatnam, Andhra Pradesh. Chandrasekharan completed his high school from Bapatla village in Guntur from Andhra Pradesh. He completed M.A. in mathematics from the Presidency College, Chennai and a PhD from the Department of Mathematics, University of Madras in 1942, under the supervision of K. Ananda Rau.
When Chandrasekharan was with the Institute for Advanced Study, Princeton, US, Homi Bhabha invited Chandrashekharan to join the School of Mathematics of the Tata Institute of Fundamental Research (TIFR). Chandrashekharan persuaded mathematicians L. Schwarz, C. L. Siegel and others from all over the world to visit TIFR and deliver lectures. In 1965, Chandrasekharan left the Tata Institute of Fundamental Research to join the ETH Zurich, where he retired in 1988.
He was a fellow of the American Mathematical Society.
Selected works
with Salomon Bochner:
with S. Minakshisundaram:
reprinting 2012
Notes
References
– India's who is who
External links
Komaravolu Chandrasekharan in Historical Dictionary of Switzerland (German)
1920 births
2017 deaths
20th-century Indian mathematicians
Telugu people
People from Andhra Pradesh
Recipients of the Padma Shri in literature & education
Indian number theorists
Mathematical analysts
Academic staff of ETH Zurich
Fellows of the American Mathematical Society
University of Madras alumni
21st-century Indian mathematicians
People from Guntur district
Scientists from Andhra Pradesh
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science
Indian emigrants to Switzerland
Presidents of the International Mathematical Union |
https://en.wikipedia.org/wiki/Canonical%20singularity | In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal models. They were introduced by . Terminal singularities are important in the minimal model program because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.
Definition
Suppose that Y is a normal variety such that its canonical class KY is Q-Cartier, and let f:X→Y be a resolution of the singularities of Y.
Then
where the sum is over the irreducible exceptional divisors, and the ai are rational numbers, called the discrepancies.
Then the singularities of Y are called:
terminal if ai > 0 for all i
canonical if ai ≥ 0 for all i
log terminal if ai > −1 for all i
log canonical if ai ≥ −1 for all i.
Properties
The singularities of a projective variety V are canonical if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V has the same plurigenera as any resolution of its singularities. V has canonical singularities if and only if it is a relative canonical model.
The singularities of a projective variety V are terminal if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V the pullback of any section of Vm vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities.
Classification in small dimensions
Two dimensional terminal singularities are smooth.
If a variety has terminal singularities, then its singular points have codimension at least 3, and in particular in dimensions 1 and 2 all terminal singularities are smooth. In 3 dimensions they are isolated and were classified by .
Two dimensional canonical singularities are the same as du Val singularities, and are analytically isomorphic to quotients
of C2 by finite subgroups of SL2(C).
Two dimensional log terminal singularities are analytically isomorphic to quotients
of C2 by finite subgroups of GL2(C).
Two dimensional log canonical singularities have been classified by .
Pairs
More generally one can define these concepts for a pair where is a formal linear combination of prime divisors with rational coefficients such that is -Cartier. The pair is called
terminal if Discrep
canonical if Discrep
klt (Kawamata log terminal) if Discrep and
plt (purely log terminal) if Discrep
lc (log canonical) if Discrep.
References
Singularity theory
Algebraic geometry |
https://en.wikipedia.org/wiki/Andr%C3%A1s%20Frank | András Frank (born 3 June 1949) is a Hungarian mathematician, working in combinatorics, especially in graph theory, and combinatorial optimisation. He is director of the Institute of Mathematics of the Faculty of Sciences of the Eötvös Loránd University, Budapest.
Mathematical work
Using the LLL-algorithm, Frank, and his student, Éva Tardos developed a general method, which could transform some polynomial-time algorithms into strongly polynomial. He solved the problem of finding the minimum number of edges to be added to a given undirected graph so that in the resulting graph the edge-connectivity between any two vertices u and v is at least a predetermined number f(u,v).
Degrees, awards
He received the Candidate of Mathematical Science degree in 1980, advisor: László Lovász, and the Doctor of Mathematical Science degree (1990) from the Hungarian Academy of Sciences. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He was awarded the Tibor Szele Prize of the János Bolyai Mathematical Society in 2002 and the Albert Szent-Györgyi Prize in 2009. In June 2009 the ELTE Mathematical Institute sponsored a workshop in honor of his 60th birthday.
References
External links
1949 births
Living people
Mathematicians from Budapest
University of Szeged alumni
Combinatorialists
Academic staff of Eötvös Loránd University |
https://en.wikipedia.org/wiki/Science%2C%20Technology%2C%20Engineering%20and%20Mathematics%20Network | The Science, Technology, Engineering and Mathematics Network or STEMNET is an educational charity in the United Kingdom that seeks to encourage participation at school and college in science and engineering-related subjects (science, technology, engineering, and mathematics) and (eventually) work.
History
It is based at Woolgate Exchange near Moorgate tube station in London and was established in 1996. The chief executive is Kirsten Bodley. The STEMNET offices are housed within the Engineering Council.
Function
Its chief aim is to interest children in science, technology, engineering and mathematics. Primary school children can start to have an interest in these subjects, leading secondary school pupils to choose science A levels, which will lead to a science career. It supports the After School Science and Engineering Clubs at schools. There are also nine regional Science Learning Centres.
STEM ambassadors
To promote STEM subjects and encourage young people to take up jobs in these areas, STEMNET have around 30,000 ambassadors across the UK. these come from a wide selection of the STEM industries and include TV personalities like Rob Bell.
Funding
STEMNET used to receive funding from the Department for Education and Skills. Since June 2007, it receives funding from the Department for Children, Schools and Families and Department for Innovation, Universities and Skills, since STEMNET sits on the chronological dividing point (age 16) of both of the new departments.
See also
The WISE Campaign
Engineering and Physical Sciences Research Council
National Centre for Excellence in Teaching Mathematics
Association for Science Education
Glossary of areas of mathematics
Glossary of astronomy
Glossary of biology
Glossary of chemistry
Glossary of engineering
Glossary of physics
References
External links
DIUS page
STEM Partnerships (extensive background educational information)
Department for Business, Innovation and Skills
Department for Education
Educational charities based in the United Kingdom
Educational institutions established in 1996
Engineering education in the United Kingdom
Engineering organizations
Mathematics education in the United Kingdom
Mathematics organizations
Organisations based in the London Borough of Camden
Science and technology in the United Kingdom
1996 establishments in the United Kingdom |
https://en.wikipedia.org/wiki/3-fold | In algebraic geometry, a 3-fold or threefold is a 3-dimensional algebraic variety.
The Mori program showed that 3-folds have minimal models.
References |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Real%20Madrid%20CF%20season | The 2009–10 season was Real Madrid Club de Fútbol's 79th season in La Liga. This article shows player statistics and all matches (official and friendly) that the club played during the 2009–10 season.
The newly constructed Second Galácticos of President Pérez looked to reverse the misfortunes of past years. The 2009–10 season, however, was a transitional one as Madrid again finished second in the league, although this time amassing 96 points, the club's record at the time, and went out of the Champions League at the hands of Lyon. The season was marred by Cristiano Ronaldo's injury, that sidelined him for seven weeks, although he still topped the goalscoring charts with 33 goals, and Madrid became the highest scoring team in La Liga, with 102 goals. Real Madrid also had the misfortune to become the runners-up with the highest points total in the history of Europe's top five leagues, until surpassed by Liverpool's 97 points in the 2018–19 Premier League.
Club
Coaching staff
Kit
|
|
|
Other information
Players
Squad information
Players in / out
In
Total spending: €261 million
Out
Squad stats
Goals
Disciplinary record
.
Overall
{|class="wikitable" style="text-align: center;"
|-
!
!Total
! Home
! Away
|-
|align=left| Games played || 48 || 24 || 24
|-
|align=left| Games won || 36 || 21 || 15
|-
|align=left| Games drawn || 5 || 1 || 4
|-
|align=left| Games lost || 7 || 2 || 5
|-
|align=left| Biggest win || 6–0 vs Zaragoza || 6–0 vs Zaragoza || 5–1 vs Tenerife
|-
|align=left| Biggest loss || 0–4 vs Alcorcón || 0–2 vs Barcelona || 0–4 vs Alcorcón
|-
|align=left| Biggest win (League) || 6–0 vs Zaragoza || 6–0 vs Zaragoza || 5–1 vs Tenerife
|-
|align=left| Biggest win (Cup) || 1–0 vs Alcorcón || 1–0 vs Alcorcón || –
|-
|align=left| Biggest win (Europe) || 5–2 vs Zürich || 3–0 vs Marseille || 5–2 vs Zürich
|-
|align=left| Biggest loss (League) || 0–2 vs Barcelona || 0–2 vs Barcelona || 1–2 vs Sevilla
|-
|align=left| Biggest loss (Cup) || 0–4 vs Alcorcón || – || 0–4 vs Alcorcón
|-
|align=left| Biggest loss (Europe) || 2–3 vs Milan || 2–3 vs Milan || 0–1 vs Lyon
|-
|align=left| Clean sheets || 18 || 12 || 6
|-
|align=left| Goals scored || 119 || 68 || 51
|-
|align=left| Goals conceded || 48 || 22 || 26
|-
|align=left| Goal difference || +71 || +46 || +25
|-
|align=left| Average per game || || ||
|-
|align=left| Average per game || || ||
|-
|align=left| Yellow cards || 118 || 46 || 72
|-
|align=left| Red cards || 3 || 2 || 1
|-
|align=left| Most appearances || Iker Casillas (46) || colspan=2|–
|-
|align=left| Most minutes played || Iker Casillas (4326) || colspan=2|–
|-
|align=left| Top scorer || Cristiano Ronaldo (33) || colspan=2|–
|-
|align=left| Top assistor || Guti (10) || colspan=2|–
|-
|align=left| Points || 113/144 (%) || 64/72 (%) || 49/72 (%)
|-
|align=left| Winning rate || % || % || %
|-
Competitions
La Liga
League table
Results by round
Matches
Copa del Rey
Round o |
https://en.wikipedia.org/wiki/Andr%C3%A1s%20S%C3%A1rk%C3%B6zy | András Sárközy (born in Budapest) is a Hungarian mathematician, working in analytic and combinatorial number theory, although his first works were in the fields of geometry and classical analysis. He has the largest number of papers co-authored with Paul Erdős (a total of 62); he has an Erdős number of one. He proved the Furstenberg–Sárközy theorem that every sequence of natural numbers with positive upper density contains two members whose difference is a full square. He was elected a corresponding member (1998), and a full member (2004) of the Hungarian Academy of Sciences. He received the Széchenyi Prize (2010). He is the father of the mathematician Gábor N. Sárközy.
References
Living people
1941 births
Mathematicians from Budapest
Members of the Hungarian Academy of Sciences
Number theorists |
https://en.wikipedia.org/wiki/Albert%20Taylor%20Bledsoe | Albert Taylor Bledsoe (November 9, 1809 – December 8, 1877) was an American Episcopal priest, attorney, professor of mathematics, and officer in the Confederate army and was best known as a staunch defender of slavery and, after the South lost the American Civil War, an architect of the Lost Cause. He was the author of Liberty and Slavery (1856), "the most extensive philosophical treatment of slavery ever produced by a Southern academic", which defended slavery laws as ensuring proper societal order.
Early life and education
Bledsoe was born on November 9, 1809, in Frankfort, Kentucky, the oldest of five children of Moses Owsley Bledsoe and Sophia Childress Taylor (who was a relative of President Zachary Taylor). He was a cadet at the United States Military Academy at West Point from 1825 to 1830, where he was a fellow cadet of Jefferson Davis and Robert E. Lee. After serving two years in the United States Army, he studied law and theology at Kenyon College in Gambier, Ohio, and received his M.A. and LL.M. In 1836. he married Harriet Coxe of Burlington NJ, and they had seven children, four of whom survived childhood.
His daughter was the author Sophia Bledsoe Herrick.
College professor and mathematician
Adjunct Professor of Mathematics and French, Kenyon College, (OH) 1833–1834.
Professor of Mathematics, Miami University (OH), 1834–1835.
Professor of Mathematics and Astronomy, University of Mississippi, 1848–1854.
Professor of Mathematics, University of Virginia, 1854–1861.
Bledsoe in his lectures at the University of Virginia would frequently "interlard his demonstration of some difficult problem in differential or integral calculus—for example, the lemniscata of —with some vigorous remarks in the doctrine of States' rights". His book The Philosophy of Mathematics was one of the earliest American works on mathematics and includes chapters on Descartes, Leibnitz, and Newton. Bledsoe is perhaps best remembered for his treatise An Essay on Liberty and Slavery, which presented an extended proslavery argument. Bledsoe argued that the natural state of humans was in society, not in nature, and that humans in society needed to have restraints on their actions. That is, he argued that liberty was greatest when humans were allowed to exercise only the amount of freedom they were naturally suited to. Some had to be restrained; others were entitled to freedom.
Clergyman
In 1835, Bledsoe became an Episcopal minister and became an assistant to Bishop Smith of Kentucky. He abandoned his clerical career in 1838 because of his opposition to infant baptism. Later in life, he was ordained a Methodist minister in 1871, but he never took charge of a church. He was a strenuous advocate of the doctrine of free will and his views are set forth in his book Examination of Edwards on the Will (1845).
Lawyer
In 1838, Bledsoe moved to Springfield, Illinois, where he was a law partner of Edward D. Baker, and where he practiced law in the same courts as Abraham L |
https://en.wikipedia.org/wiki/Steffi%20Graf%20career%20statistics | This is a list of the main career statistics of professional tennis player Steffi Graf.
Performance timelines
Only results in WTA Tour (incl. Grand Slams) main-draw, Olympic Games and Fed Cup are included in win–loss records.
Singles
Notes:
Only results in WTA Tour (incl. Grand Slams) main-draw, Olympic Games and Fed Cup are included in win–loss records.
Graf retired in August 1999 while ranked world No. 3, She was not included in the official year end ranking.
Doubles
Grand Slam finals
Singles: 31 (22 titles, 9 runner-ups)
Doubles: 4 (1 title, 3 runner-ups)
Year-end championship finals
Singles: 6 (5 titles, 1 runner-up)
Olympic finals
Singles: 2 (1 gold, 1 silver medal)
Graf also won the 1984 demonstration event at the 1984 Los Angeles Games, but this was for players aged 21 or under, and it was not an official Olympic event.
Doubles
Graf and Kohde-Kilsch lost in the semifinals to Jana Novotná and Helena Suková 7–5, 6–3. In 1988, there was no bronze medal match, and both beaten semifinalists received bronze medals.
Category 5 / Tier I finals
Singles: 36 (26 titles, 10 runner-ups)
Doubles: 4 (2 titles, 2 runner-ups)
Career finals
Singles (107 titles, 31 runner-ups)
Doubles (11 titles, 7 runner-ups)
Fed Cup
Wins (2)
Participations (32)
Singles (22)
Doubles (10)
Career Grand Slam tournament seedings
The tournaments won by Graf are in boldface, and advanced into finals by Graf are in italics''.
WTA Tour career earnings
Head-to-head vs. top 10 ranked players
Top 10 wins
Graf has a record against players who were, at the time the match was played, ranked in the top 10.
Double bagel matches (6-0, 6-0)
Awards
1986: Most Improved Player, by the Women's Tennis Association (WTA)
1987 Player of the Year, by the WTA
1987 World Champion, by the International Tennis Federation (ITF)
1988 Player of the Year, by the WTA
1988 World Champion, by the ITF
1988 BBC Overseas Sports Personality of the Year
1989 Player of the Year, by the WTA
1989 World Champion, by the ITF
1989 Female Athlete of the Year, by the Associated Press
1990 Player of the Year, by the WTA
1990 World Champion, by the ITF
1993 Player of the Year, by the WTA
1993 World Champion, by the ITF
1994 Player of the Year, by the WTA
1995 Player of the Year, by the WTA
1995 World Champion, by the ITF
1996 Player of the Year, by the WTA
1996 World Champion, by the ITF
1996 Most Exciting Player of the Year, by the WTA
1998 Most Exciting Player of the Year, by the WTA
1999 Most Exciting Player of the Year, by the WTA
1999 Prince of Asturias Award, one of the most important awards of Spain and named after the heir apparent of Spain, Prince Felipe
1999 Germany Television Award for her outstanding performance as tennis player and her importance to the German public.
1999 Athlete of the Century for the category Female Athlete in Ballsports by a panel of the International Olympic Committee (IOC)
1999 The Greatest Female Tennis Player of the 20th century, by a panel of tennis ex |
https://en.wikipedia.org/wiki/Determinantal%20variety | In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the Segre embedding of a product of two projective spaces.
Definition
Given m and n and r < min(m, n), the determinantal variety Y r is the set of all m × n matrices (over a field k) with rank ≤ r. This is naturally an algebraic variety as the condition that a matrix have rank ≤ r is given by the vanishing of all of its (r + 1) × (r + 1) minors. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j, these minors are polynomials of degree r + 1. The ideal of k[x i,j] generated by these polynomials is a determinantal ideal. Since the equations defining minors are homogeneous, one can consider Y r either as an affine variety in mn-dimensional affine space, or as a projective variety in (mn − 1)-dimensional projective space.
Properties
The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
Assuming that we consider Y r as an affine variety, its dimension is r(m + n − r). One way to see this is as follows: form the product space over where is the Grassmannian of r-planes in an m-dimensional vector space, and consider the subspace , which is a desingularization of (over the open set of matrices with rank exactly r, this map is an isomorphism), and is a vector bundle over which is isomorphic to where is the tautological bundle over the Grassmannian. So since they are birationally equivalent, and since the fiber of has dimension nr.
The above shows that the matrices of rank <r contains the singular locus of , and in fact one has equality. This fact can be verified using that the radical ideal is given by the minors along with the Jacobian criterion for nonsingularity.
The variety Y r naturally has an action of , a product of general linear groups. The problem of determining the syzygies of , when the characteristic of the field is zero, was solved by Alain Lascoux, using the natural action of G.
Related topics
One can "globalize" the notion of determinantal varieties by considering the space of linear maps between two vector bundles on an algebraic variety. Then the determinantal varieties fall into the general study of degeneracy loci. An expression for the cohomology class of these degeneracy loci is given by the Thom-Porteous formula, see (Fulton-Pragacz).
References
Algebraic geometry
Algebraic varieties |
https://en.wikipedia.org/wiki/Du%20Val%20singularity | In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by Patrick du Val and Felix Klein.
The Du Val singularities also appear as quotients of by a finite subgroup of SL2; equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups. The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory.
Classification
The possible Du Val singularities are (up to analytical isomorphism):
See also
Brieskorn–Grothendieck resolution
General elephant conjecture
References
External links
Algebraic surfaces
Singularity theory |
https://en.wikipedia.org/wiki/Alfred%20Brauer | Alfred Theodor Brauer (April 9, 1894 – December 23, 1985) was a German-American mathematician who did work in number theory. He was born in Charlottenburg, and studied at the University of Berlin. As he served Germany in World War I, even being injured in the war, he was able to keep his position longer than many other Jewish academics who had been forced out after Hitler's rise to power. In 1935 he lost his position and in 1938 he tried to leave Germany, but was not able to until the following year. He initially worked in the Northeast, but in 1942 he settled into a position at the University of North Carolina at Chapel Hill. A good deal of his works, and the Alfred T. Brauer library, would be linked to this university. He occasionally taught at Wake Forest University after he retired from Chapel Hill at 70. He died in North Carolina, aged 91.
He was the brother of the mathematician Richard Brauer, who was the founder of modular representation theory.
See also
Brauer chain
Scholz–Brauer conjecture
References
Further reading
External links
20th-century German mathematicians
20th-century American mathematicians
Number theorists
Academic staff of the Humboldt University of Berlin
University of North Carolina at Chapel Hill faculty
Wake Forest University faculty
Jewish American scientists
Scientists from Berlin
1894 births
1985 deaths
Jewish emigrants from Nazi Germany to the United States
German Jewish military personnel of World War I
People from Charlottenburg
People from the Province of Brandenburg
20th-century American Jews |
https://en.wikipedia.org/wiki/Elementary%20cellular%20automaton | In mathematics and computability theory, an elementary cellular automaton is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. There is an elementary cellular automaton (rule 110, defined below) which is capable of universal computation, and as such it is one of the simplest possible models of computation.
The numbering system
There are 8 = 23 possible configurations for a cell and its two immediate neighbors. The rule defining the cellular automaton must specify the resulting state for each of these possibilities so there are 256 = 223 possible elementary cellular automata. Stephen Wolfram proposed a scheme, known as the Wolfram code, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written in order, 111, 110, ..., 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110d=011011102. So rule 110 is defined by the transition rule:
Reflections and complements
Although there are 256 possible rules, many of these are trivially equivalent to each other up to a simple transformation of the underlying geometry. The first such transformation is reflection through a vertical axis and the result of applying this transformation to a given rule is called the mirrored rule. These rules will exhibit the same behavior up to reflection through a vertical axis, and so are equivalent in a computational sense.
For example, if the definition of rule 110 is reflected through a vertical line, the following rule (rule 124) is obtained:
Rules which are the same as their mirrored rule are called amphichiral. Of the 256 elementary cellular automata, 64 are amphichiral.
The second such transformation is to exchange the roles of 0 and 1 in the definition. The result of applying this transformation to a given rule is called the complementary rule.
For example, if this transformation is applied to rule 110, we get the following rule
and, after reordering, we discover that this is rule 137:
There are 16 rules which are the same as their complementary rules.
Finally, the previous two transformations can be applied successively to a rule to obtain the mirrored complementary rule. For example, the mirrored complementary rule of rule 110 is rule 193. There are 16 rules which are the same as their mirrored complementary rules.
Of the 256 elementary cellular automata, there are 88 which are inequivalent under these transformations.
It turns out that reflection and complementation are automorphisms of the monoid of one-dimensional cellular automata, as they both preserve composition.
Single 1 histories
One method used to study these automata is to follow |
https://en.wikipedia.org/wiki/Richard%20A.%20Brualdi | Richard Anthony Brualdi is a professor emeritus of combinatorial mathematics at the University of Wisconsin–Madison.
Brualdi received his Ph.D. from Syracuse University in 1964; his advisor was H. J. Ryser. Brualdi is an Editor-in-Chief of the Electronic Journal of Combinatorics. He has over 200 publications in several mathematical journals. According to current on-line database of Mathematics Genealogy Project, Richard Brualdi has 37 Ph.D. students and 48 academic descendants. The concept of incidence coloring was introduced in 1993 by Brualdi and Massey.
He received the Euler medal from the Institute of Combinatorics and its Applications in 2000. In 2012, he was elected a fellow of the Society for Industrial and Applied Mathematics. That same year, he became an inaugural fellow of the American Mathematical Society.
Books
(with Herbert J. Ryser) Combinatorial Matrix Theory, Cambridge Univ. Press
Richard A. Brualdi, Introductory Combinatorics, Prentice-Hall, Upper Saddle River, N.J.
V. Pless, R. A. Brualdi, and W. C. Huffman, Handbook of Coding Theory, Elsevier Science, New York, 1998
Richard A. Brualdi and Dragos Cvetkovic, A Combinatorial Approach to Matrix Theory and Its Applications, CRC Press, Boca Raton Fla., 2009.
Richard A. Brualdi and Bryan Shader, Matrices of Sign-Solvable Linear Systems, Cambridge Tracts in Mathematics, Vol. 116, Cambridge Univ. Press, 1995.
Richard A. Brualdi, The Mutually Beneficial Relationship Between Graphs and Matrices, American Mathematical Society, CBMS Series, 2012.
Selected articles
with Jeffrey A. Ross:
with J. Csima:
with Bo Lian Liu:
References
External links
Richard Brualdi at University of Wisconsin- Madison website
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Syracuse University alumni
University of Wisconsin–Madison faculty
Fellows of the American Mathematical Society
Living people
1939 births
Fellows of the Society for Industrial and Applied Mathematics
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Hidden%20Markov%20random%20field | In statistics, a hidden Markov random field is a generalization of a hidden Markov model. Instead of having an underlying Markov chain, hidden Markov random fields have an underlying Markov random field.
Suppose that we observe a random variable , where . Hidden Markov random fields assume that the probabilistic nature of is determined by the unobservable Markov random field , .
That is, given the neighbors of is independent of all other (Markov property).
The main difference with a hidden Markov model is that neighborhood is not defined in 1 dimension but within a network, i.e. is allowed to have more than the two neighbors that it would have in a Markov chain. The model is formulated in such a way that given , are independent (conditional independence of the observable variables given the Markov random field).
In the vast majority of the related literature, the number of possible latent states is considered a user-defined constant. However, ideas from nonparametric Bayesian statistics, which allow for data-driven inference of the number of states, have been also recently investigated with success, e.g.
See also
Hidden Markov model
Markov network
Bayesian network
References
Markov networks |
https://en.wikipedia.org/wiki/Boris%20Vuk%C4%8Devi%C4%87 | Boris Vukčević (born 16 March 1990) is a German former professional footballer of Croatian descent who played as a midfielder. Due to the aftermaths of a car accident in 2012 he retired prematurely in 2014.
Club career
He made his debut in the Fußball-Bundesliga on 23 May 2009 for TSG 1899 Hoffenheim in a game against FC Schalke 04.
Personal life
On 28 September 2012, Vukčević was involved in a traffic accident near Bammental, when his Mercedes-Benz C63 AMG Coupé collided with a truck. He underwent an emergency surgery at the University Hospital Heidelberg and was placed in an induced coma. His condition was described as critical. According to a joint press release from the prosecutor's office and the police, the cause of the accident was hypoglycemia. On November 16, 2012, he was reported as no longer being in the coma.
It was not the first time that Vukčević being involved in a car accident due to hypoglycemia. On 18 October 2010, on the state road near Bad Rappenau his car collided with the trailer of a truck after hitting the guard rail several times. On 16 November, Vukcevic awoke from his coma and began communicating with his family.
In April 2014 he made his first public appearance after the car accident when he attended a home of fixture of his club against FC Augsburg. At this occasion he also expressed his desire to play football again. On 1 June 2014 although 1899 Hoffenheim released that his expiring contract wouldn't be extended. However the club would support him finding his way back to a normal life and promised him a new contract when he would be able to play football again.
He retired prematurely at the age of 24.
References
External links
Boris Vukčević at kicker.de
1990 births
Living people
People from Osijek
Croatian emigrants to Germany
Naturalized citizens of Germany
German men's footballers
Germany men's youth international footballers
Germany men's under-21 international footballers
Bundesliga players
Regionalliga players
TSG 1899 Hoffenheim II players
TSG 1899 Hoffenheim players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Stericated%205-simplexes | In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.
Stericated 5-simplex
A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).
Alternate names
Expanded 5-simplex
Stericated hexateron
Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)
Cross-sections
The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.
Coordinates
The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents the positive orthant facet of the stericated 6-orthoplex.
A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:
(1,-1,0,0,0,0)
The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:
Root system
Its 30 vertices represent the root vectors of the simple Lie group A5. It is also the vertex figure of the 5-simplex honeycomb.
Images
Steritruncated 5-simplex
Alternate names
Steritruncated hexateron
Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)
Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
(0,1,1,1,2,3)
This construction exists as one of 64 orthant facets of the steritruncated 6-orthoplex.
Images
Stericantellated 5-simplex
Alternate names
Stericantellated hexateron
Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)
Coordinates
The coordinates can be made in 6-space, as permutations of:
(0,1,1,2,2,3)
This construction exists as one of 64 orthant facets of the stericantellated 6-orthoplex.
Images
Stericantitruncated 5-simplex
Alternate names
Stericantitruncated hexateron
Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)
Coordinates
The coordinates can be made in 6-space, as 360 permutations of:
(0,1,1,2,3,4)
This construction exists as one of 64 orthant facets of the stericantitruncated 6-orthoplex.
Images
Ste |
https://en.wikipedia.org/wiki/Uniform%20honeycombs%20in%20hyperbolic%20space | In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.
Hyperbolic uniform honeycomb families
Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups.
Compact uniform honeycomb families
The nine compact Coxeter groups are listed here with their Coxeter diagrams,
in order of the relative volumes of their fundamental simplex domains.
These 9 families generate a total of 76 unique uniform honeycombs. The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. Two known examples are cited with the {3,5,3} family below. Only two families are related as a mirror-removal halving: [5,31,1] ↔ [5,3,4,1+].
There are just two radical subgroups with non-simplicial domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is [(4,3,4,3*)], represented by Coxeter diagrams an index 6 subgroup with a trigonal trapezohedron fundamental domain ↔ , which can be extended by restoring one mirror as . The other is [4,(3,5)*], index 120 with a dodecahedral fundamental domain.
Paracompact hyperbolic uniform honeycombs
There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity.
Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains (double tetrahedra) as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, with the constraint that at least one node must be ringed across infinite order branches.
[3,5,3] family
There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or
One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps, called a tetrahedrally diminished dodecahedron. Another is constructed with 2 antipodal vertices removed.
The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew polyhedrons: {4,10|3} and {10,4|3}.
[5,3,4] family
There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or .
This family is related to the group [5,31,1] by a half symmetry [5,3,4,1+], or ↔ , when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive ↔ .
[5,3,5] fa |
https://en.wikipedia.org/wiki/Mori%20dream%20space | In algebraic geometry, a Mori dream space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". showed that Mori dream spaces are quotients of affine varieties by torus actions. The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.
Properties
In general, it is difficult to find a non-trivial example of a Mori dream space, as being a Mori Dream Space is equivalent to all (multi-)section rings being finitely generated.
It has been shown that a variety which admits a surjective morphism from a Mori dream space is again a Mori dream space.
See also
spherical variety
References
Algebraic geometry |
https://en.wikipedia.org/wiki/General%20elephant | In algebraic geometry, general elephant is an idiosyncratic name for a general element of the anticanonical system of a variety, introduced by Miles Reid. For 3-folds the general elephant problem (or conjecture) asks whether general elephants have at most du Val singularities; this has been proved in several cases.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Brest%20Airport | Brest Airport (; ) is an airport serving Brest, a city in Belarus.
Statistics
References
External links
Airports in Belarus
Buildings and structures in Brest, Belarus |
https://en.wikipedia.org/wiki/List%20of%20Indian%20Premier%20League%20records%20and%20statistics | The Indian Premier League is a Twenty20 competition in men's cricket. Organised by the Board of Control for Cricket in India (BCCI), the tournament has taken place every year since 2008. Seven teams have won a title since the beginning of the league, with Mumbai Indians and Chennai Super Kings both winning five titles.
Indian batsman and former national team captain Virat Kohli holds the record for the most runs scored and most centuries (seven) made in the league. Chris Gayle holds many other individual batting records including the highest-individual score in a match (175 runs from 66 balls), the most sixes scored (326), the most sixes in a match (17) and the fastest century (from 30 balls). Yuzvendra Chahal holds the record for the most wickets taken (187).
Royal Challengers Bangalore (RCB) has scored the most runs in a match with a score of 263/5 against Pune Warriors India in 2013, the same match in which Gayle hit his record score and set the record for the most sixes in an innings. The second highest score was set by Lucknow Super Giants against Punjab Kings, making 257/5 in 2023. The highest successful run chase in the league's history was by Rajasthan Royals who chased a target of 224 set by Punjab Kings in 2020. RCB also scored the lowest total score, making a score of 49 against Kolkata Knight Riders in 2017.
Listing criteria
In general, the top five are listed in each category, except when there is a tie for the last place among the five, when all the tied record holders are noted.
Listing notation
Team notation
(200/3) indicates that a team scored 200 runs for three wickets and the innings was closed, either due to a successful run chase or if no playing time remained
(200) indicates that a team scored 200 runs and was all out
Batting notation
(100) indicates that a batsman scored 100 runs and was out
(100*) indicates that a batsman scored 100 runs and was not out
Bowling notation
(5/20) indicates that a bowler has captured five wickets while conceding 20 runs
Currently playing
indicates a current cricketer
Start Date
indicates the date the match was played
Team records
By season
Out of the fifteen franchises that have played in the league, two teams have won the competition five times, one team has won twice and four other teams have won once. Mumbai Indians and Chennai Super Kings are the most successful teams in the league's history with five IPL titles, with Kolkata Knight Riders having won two titles. The four teams who have won the tournament once are Rajasthan Royals, Deccan Chargers, Sunrisers Hyderabad and Gujarat Titans. The current champions are Chennai who beat defending champions Gujarat in the 2023 Indian Premier League final to clinch their fifth title.
† Team now defunct
Team wins, losses and draws
Source: CricInfo
Notes:
Tie+W and Tie+L indicates matches tied and then won or lost by super over
The result percentage excludes no results and counts ties (irrespective of a tiebreaker) as ha |
https://en.wikipedia.org/wiki/Fano%20fibration | In algebraic geometry, a Fano fibration or Fano fiber space, named after Gino Fano, is a morphism of varieties whose general fiber is a Fano variety (in other words has ample anticanonical bundle) of positive dimension. The ones arising from extremal contractions in the minimal model program are called Mori fibrations or Mori fiber spaces (for Shigefumi Mori). They appear as standard forms for varieties without a minimal model.
See also
Ample line bundle
Fiber bundle
Fibration
Quasi-fibration
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Supersingular%20prime%20%28algebraic%20number%20theory%29 | In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E is defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field Fp.
Noam Elkies showed that every elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero (if E does not have complex multiplication). conjectured that the number of supersingular primes less than a bound X is within a constant multiple of , using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open.
More generally, if K is any global field—i.e., a finite extension either of Q or of Fp(t)—and A is an abelian variety defined over K, then a supersingular prime for A is a finite place of K such that the reduction of A modulo is a supersingular abelian variety.
References
Classes of prime numbers
Algebraic number theory
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Null%20sign | The null sign (∅) is often used in mathematics for denoting the empty set (however, the variant is more commonly used). The same letter in linguistics represents zero, the lack of an element. It is commonly used in phonology, morphology, and syntax.
Encodings
The symbol ∅ is available at Unicode point U+2205. It can be coded in HTML as and as . It can be coded in LaTeX as .
Similar letters
Similar letters and symbols include the following:
Diameter sign in geometry:
Scandinavian letter Ø: majuscule and minuscule are a part of the alphabet of Scandinavian languages. The minuscule letter is also used in the International Phonetic Alphabet (IPA) to represent close-mid front rounded vowel.
Greek letter Φ: majuscule and minuscule are a part of the Greek alphabet. It sometimes take the form of and is used as a sign in different fields of studies. The is used in the IPA for voiceless bilabial fricative.
Greek letter Θ: majuscule and minuscule are a part of the Greek alphabet. The minuscule is used in the IPA for voiceless dental fricative. The capital letter sometimes are rendered as .
Cyrillic letter Ө: majuscule and minuscule are a part of the Cyrillic script. It is used in the IPA for close-mid central rounded vowel.
Cyrillic letter Ф: majuscule and minuscule are a part of the Cyrillic script. The letter took the place of fita ( and ), a letter of Early Cyrillic alphabet in modern usages.
Use in mathematics
In mathematics, the null sign (∅) denotes the empty set. Note that a null set is not necessarily an empty set. Common notations for the empty set include "{}", "∅", and "". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets (and not related in any way to the Greek letter Φ).
Empty sets are used in set operations. For example:
There are no common elements in the solution; so it should be denoted as:
or
Use in linguistics
In linguistics, the null sign is used to indicate the absence of an element, such as a phoneme or morpheme.
Morphology
The English language was a fusional language, this means the language makes use of inflectional changes to convey grammatical meanings. Although the inflectional complexity of English has been largely reduced in the course of development, the inflectional endings can be seen in earlier forms of English, such as the Early Modern English (abbreviated as EModE).
The verb endings of EModE was summarised in the table below by Roger Lass:
References
Mathematical symbols |
https://en.wikipedia.org/wiki/Probability%20Surveys | Probability Surveys is an open-access electronic journal that is jointly sponsored by the Bernoulli Society and the Institute of Mathematical Statistics. It publishes review articles on topics of interest in probability theory.
Managing Editors
David Aldous (2004–2008)
Geoffrey Grimmett (2009–2011)
Laurent Saloff-Coste (2012–2014)
Ben Hambly (2014– )
Probability journals
Academic journals established in 2004
Institute of Mathematical Statistics academic journals |
https://en.wikipedia.org/wiki/Statistics%20Surveys | Statistics Surveys is an open-access electronic journal, founded in 2007, that is jointly sponsored by the American Statistical Association, the Bernoulli Society, the Institute of Mathematical Statistics and the Statistical Society of Canada. It publishes review articles on topics of interest in statistics. Wendy L. Martinez serves as the coordinating editor.
External links
Official page
Mathematics journals
Statistics journals
Academic journals established in 2007
Institute of Mathematical Statistics academic journals
American Statistical Association academic journals |
https://en.wikipedia.org/wiki/Paulo%20Emilio%20%28footballer%2C%20born%201936%29 | Paulo Emilio Frossard Jorge (3 January 1936 – 17 May 2016) was a Brazilian football manager. He died aged 80 of a brain lymphoma in May 2016.
Managerial statistics
Source:
Honours
Manager
Desportiva
Campeonato Capixaba: 1967
Torneio Início do Espírito Santo: 1967
Taça Cidade de Vitória: 1968
Nacional-AM
Campeonato Amazonense: 1972
Santa Cruz
Campeonato Pernambucano: 1973
Bahia
Campeonato Baiano: 1974
Fluminense
Campeonato Carioca: 1975
Taça Guanabara: 1975
Taça Rio: 1990
Vasco da Gama
Taça Guanabara: 1976
Sporting
Taça de Portugal: 1977-78
Goiás
Campeonato Goiano: 1981
Fortaleza
Campeonato Cearense: 1983
Cerezo Osaka
Japan Football League: 1994
References
External links
1936 births
2016 deaths
Brazilian football managers
Expatriate football managers in Portugal
Expatriate football managers in Saudi Arabia
Expatriate football managers in Japan
Campeonato Brasileiro Série A managers
Campeonato Brasileiro Série B managers
Primeira Liga managers
Saudi Pro League managers
J1 League managers
Associação Portuguesa de Desportos managers
Associação Desportiva Ferroviária Vale do Rio Doce managers
America Football Club (Rio de Janeiro) managers
Nacional Futebol Clube managers
Santa Cruz Futebol Clube managers
Clube do Remo managers
Esporte Clube Bahia managers
Fluminense FC managers
CR Vasco da Gama managers
Guarani FC managers
Sporting CP managers
Goiás Esporte Clube managers
Paysandu Sport Club managers
Botafogo de Futebol e Regatas managers
Clube Náutico Capibaribe managers
Santos FC managers
Fortaleza Esporte Clube managers
São José Esporte Clube managers
Esporte Clube Noroeste managers
Al Hilal SFC managers
Club Athletico Paranaense managers
Cerezo Osaka managers
Brazilian men's footballers
Men's association football defenders |
https://en.wikipedia.org/wiki/Canonical%20model%20%28disambiguation%29 | Canonical model may refer to:
Canonical model, a design pattern used to communicate between different data formats
Canonical ring in mathematics
in modal logic
Relative canonical model in mathematics
See also
Canonical ensemble |
https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20Lempert | László Lempert (4 June 1952, in Budapest) is a Hungarian-American mathematician, working in several complex variables and complex geometry. He proved that the Carathéodory and Kobayashi distances agree on convex domains. He further proved that a compact, strictly pseudoconvex real analytic hypersurface can be embedded into the unit sphere of a Hilbert space.
Life
Lempert graduated from the Eötvös Loránd University in 1975. He was at the Analysis Department of the same university (1977–1988) and is a professor of Purdue University since 1988. He was a visiting research fellow at the Université de Paris VII (1979–1980), visiting lecturer at the Princeton University (1984–1985), and visiting professor at the Eötvös Loránd University (1994–1995).
Degrees, awards
Lempert received the Candidate of the mathematical sciences degree from the Hungarian Academy of Sciences in 1984. He was an invited session speaker at the International Congress of Mathematicians, in Berkeley, California, 1986. He won the Stefan Bergman Prize in 2001. He was elected an external member of the Hungarian Academy of Sciences (2004). In 2012 he became a fellow of the American Mathematical Society.
References
External links
Lempert and Webster receive 2001 Bergman Prize, Notices of the American Mathematical Society, 48(2001), 998–999.
1952 births
Living people
20th-century American mathematicians
20th-century Hungarian mathematicians
21st-century American mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Liz%20Waldner | Liz Waldner is an American poet.
Life
Waldner was raised in small town Mississippi. At 28, she received a B.A. in philosophy and mathematics from St. John's College; she later studied at the Summer Language School in French Middlebury College, and received an M.F.A. from the Iowa Writers' Workshop. Waldner was a Regents Fellow in the Communication Department at the University of California, San Diego.
She is the author of eight poetry collections, most recently Play (Lightful Press) and Trust (winner of the Cleveland State University Poetry Center Open Competition). Her collection, Dark Would (the missing person) (University of Georgia Press), was the winner of the 2002 Contemporary Poetry Series; her collection, Self and Simulacra (2001), won the Beatrice Hawley Award; and her collection, A Point Is That Which Has No Part (2000), received the 1999 Iowa Poetry Prize and the 2000 James Laughlin Award from the Academy of American Poets.
Other honors include grants from the Washington State Professional Development Grant for Artists, Massachusetts Cultural Council Artist Fellowship, the Boomerang Foundation, the Gertrude Stein Award for Innovative Poetry and the Barbara Deming Money for Women Grant. She received fellowships from the Vermont Studio Center, the Djerassi Foundation, Centrum, Hedgebrook, Virginia Center for the Creative Arts, Villa Montalvo, Fundación Valparaiso and the MacDowell Colony.
Waldner's poem "The Ballad of Barding Gaol", along with a selection of others, won the Poetry Society of America's Robert M. Winner Memorial Award, and her poetry has appeared in literary journals and magazines such as Ploughshares, Poetry, The New Yorker, The American Poetry Review, The Journal, Parnassus West, The Cortland Review, Electronic Poetry Review, Colorado Review, Denver Quarterly, New American Writing, Indiana Review, Abacus, and VOLT.
She was an adjunct at Millsaps College in Jackson MS (1988–90) where she used the "Eyes On The Prize" PBS series as a text in her freshman comp course, inviting the college community to regard it as an all-college text; sponsored and served as panelist on the first Environmental Symposium; began with her students a campus recycling program; was advisor for the Rape Awareness office; co-led an NIH symposium on Suffering and Tragedy, gave a paper at the Philosophy Department's Colloquium, and attempted to live on $1000 a course.
Her other teaching positions included Lecturer at Tufts University, the Institute for Language and Thinking at Bard College, Cornell College, Hugo House (Seattle), and the College of Wooster.
Other Awards
2017: Foundation for Contemporary Arts Dorothea Tanning Award
2004: Northern California Book Awards
2001: Beatrice Hawley Award
2000: James Laughlin Award
Published works
Full-length poetry collections
Chapbooks
Works published in periodicals
Ploughshares
References
External links
Academy of American Poets > Poets > Liz Waldner
Video: Liz Waldner Poetry Reading - Pa |
https://en.wikipedia.org/wiki/Ipembe | Ipembe is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,039 people in the ward, from 1,858 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Kindai%2C%20Tanzania | Kindai is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,889 people in the ward, from 12,658 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Majengo%2C%20Singida | Majengo is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,282 people in the ward, from 9,370 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mandewa | Mandewa is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 19,676 people in the ward, from 17,932 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mughanga | Mughanga is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,245 people in the ward, from 2,046 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mungumaji | Mungumaji is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,740 people in the ward, from 4,320 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mwankoko | Mwankoko is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,131 people in the ward, from 10,548 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Unyambwa | Unyambwa is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,206 people in the ward, from 9,301 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Unyamikumbi | Unyamikumbi is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,115 people in the ward, from 12,616 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Utemini | Utemini is an administrative ward in the Singida Urban district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,365 people in the ward, from 11,269 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Aghondi | Aghondi is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,000 people in the ward, from 5,468 in 2012.
References
Wards of Singida Region
Manyoni District |
https://en.wikipedia.org/wiki/Chikola%20%28Manyoni%20ward%29 | Chikola is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,855 people in the ward, from 13,668 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Chikuyu | Chikuyu is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,118 people in the ward, from 6,487 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Heka-Azimio | Heka-Azimio is a village in the administrative ward of Heka in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,692 people in the ward, from 7,921 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Idodyandole | Idodyandole is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,291 people in the ward, from 11,201 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Ipande%20%28Manyoni%20ward%29 | Ipande is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,017 people in the ward, from 10,040 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Isseke | Isseke is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,402 people in the ward, from 12,214 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Makanda%20%28Manyoni%20ward%29 | Makanda is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,718 people in the ward, from 9,768 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Makuru | Makuru is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,029 people in the ward, from 11,874 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Manyoni%20%28Tanzanian%20ward%29 | Manyoni is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 27,986 people in the ward, from 25,505 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Maweni | Maweni is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,639 people in the ward.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mgandu | Mgandu is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,129 people in the ward, from 13,788 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Nkonko | Nkonko is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,378 people in the ward, from 11,281 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Rungwa%20%28Tanzanian%20ward%29 | Rungwa is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 2,424 people in the ward, from 2,209 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Sanjaranda | Sanjaranda is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,687 people in the ward, from 8,828 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Sanza%20%28Tanzanian%20ward%29 | Sanza is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,397 people in the ward, from 10,387 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Sasajila | Sasajila is an administrative ward in the Manyoni District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,836 people in the ward, from 7,141 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Logarithmic%20pair | In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
Definition
A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1.
A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D.
The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X.
A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form
d log(z) = dz/z along components of the divisor given locally by z=0.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Elliptic%20singularity | In algebraic geometry, an elliptic singularity of a surface, introduced by , is a surface singularity such that the arithmetic genus of its local ring is 1.
See also
Rational singularity
References
Algebraic surfaces
Singularity theory |
https://en.wikipedia.org/wiki/Ken%20Stroud | Kenneth Arthur Stroud (; Richmond, Surrey, December, 1908 – Hertfordshire township, February 3, 2000) was a mathematician and Principal Lecturer in Mathematics at Lanchester Polytechnic in Coventry, England. He is most widely known as the author of several mathematics textbooks, especially the very popular Engineering Mathematics.
Education
Stroud held a B.Sc. and a DipEd.
Work
Stroud was an innovator in programmed learning and the identification of precise learning outcomes, and Nigel Steele calls his textbook Engineering Mathematics, based on the programmed learning approach, "one of the most successful mathematics textbooks ever published."
He died in February 2000, aged 91.
Bibliography
Laplace Transforms: Programmes and Problems, Stanley Thornes Ltd, 1973, and 1978, .
Fourier Series and Harmonic Analysis, Nelson Thornes Ltd, 1983, and Stanley Thornes Ltd, 1986, .
Engineering Mathematics, Macmillan, 1970, . 6th ed., (with Dexter J. Booth), Industrial Press, 2007, .
Advanced Engineering Mathematics (with Dexter J. Booth), 5th ed., Industrial Press, 2011, , 4th ed., Palgrave Macmillan, 2003, .
Differential Equations (with Dexter J. Booth), Industrial Press, 2004, .
Vector Analysis (with Dexter J. Booth), Industrial Press, 2005, .
Complex Variables (with Dexter J. Booth), Industrial Press, 2007, .
Linear Algebra (with Dexter J. Booth), Industrial Press, 2008, .
Essential Mathematics for Science and Technology: A Self-Learning Guide (with Dexter J. Booth), Industrial Press, 2009, .
Further Engineering Mathematics : Programmes and Problems, Palgrave Macmillan, 3 October 1986, . 2nd ed., Springer-Verlag, 1 November 1989, . 2nd ed., Palgrave Macmillan, June 1990, . 3rd Revised Edition, Palgrave Macmillan, 27 March 1996, .
Mathematics for engineering technicians, Stanley Thornes Ltd., 1978, . Book 2A : Practical applications, Stanley Thornes Ltd., 1981, .
References
1908 births
2000 deaths
20th-century English mathematicians
People from Richmond, London |
https://en.wikipedia.org/wiki/Cliff%20Joslyn | Cliff Joslyn (born 1963) is an American mathematician, cognitive scientist, and cybernetician. He is currently the Chief Knowledge Scientist and Team Lead for Mathematics of Data Science at the Pacific Northwest National Laboratory in Seattle, Washington, US, and visiting professor of Systems Science at Binghamton University (SUNY).
Biography
Cliff Joslyn studied at Oberlin College and received a BA in Cognitive Science and Mathematics, with High Honors in Cybernetics, in 1985. In 1987 he continued at the State University of New York at Binghamtonm studying under George Klir. In 1989 he received an MS, and in 1994 received a PhD, both in Systems Science, including the thesis "Possibilistic Processes for Complex Systems Modeling".
From 1994 to 1996 he was an NRC Research Associate at NASA Goddard Space Flight Center. From 1996 to 2007 he was team leader at Los Alamos National Laboratory, where he led the Knowledge and Information Systems Science research team in the Modeling, Algorithms and Informatics (CCS-3) Group of the Computer, Computational, and Statistical Sciences Division. Since 2007 he is Chief Scientist for Knowledge Sciences and the Team Leader for Mathematics of Data Science at the Pacific Northwest National Laboratory in Seattle, Washington.
In 2022 Joslyn took a position as a visiting professor of Systems Science in the Systems Science and Industrial Engineering department at Binghamton University.
Joslyn is on the editorial boards of the "International Journal of General Systems" and "Biosemiotics".
In 1991 Joslyn was awarded the Sir Geoffrey Vickers' Award for Best Student Paper from the International Society for the Systems Sciences. In 1997 he received the Distinguished Performance Award: A Large Team Award for IRS Fraud Detection Projects, Los Alamos National Laboratory.
Work
Joslyn's research interests extend from "order theoretical approaches to knowledge discovery and database analysis to include computational semiotics, qualitative modeling, and generalized information theory, with applications in computational biology, infrastructure protection, homeland defense, intelligence analysis, and defense transformation".
Principia Cybernetica
Principia Cybernetica is an international organization in the field of cybernetics and systems science focused on the collaborative development of a "computer-supported evolutionary-systemic philosophy in the context of the transdisciplinary academic fields of Systems Science and Cybernetics".
The organisation was initiated in 1989 by Joslyn, Valentin Turchin of the City College of New York, and Francis Heylighen from the Vrije Universiteit Brussel in Belgium. These three scientists managed the project and worked together in an editorial board, which manages the collection, selection and development of the material, and the implementation of the computer system.
Knowledge and Information Systems Science
At Los Alamos Joslyn and his Knowledge and Information Systems Science |
https://en.wikipedia.org/wiki/Vanishing%20theorem | In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish.
Andreotti–Grauert vanishing theorem
Bogomolov–Sommese vanishing theorem
Grauert–Riemenschneider vanishing theorem
Kawamata–Viehweg vanishing theorem
Kodaira vanishing theorem
Le Potier's vanishing theorem
Mumford vanishing theorem
Nakano vanishing theorem
Ramanujam vanishing theorem
Serre's vanishing theorem |
https://en.wikipedia.org/wiki/Ramanujam%20vanishing%20theorem | In algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to , that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface. The Kawamata–Viehweg vanishing theorem generalizes it.
See also
Mumford vanishing theorem
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Kawamata%E2%80%93Viehweg%20vanishing%20theorem | In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982.
The theorem states that if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K,
then the coherent cohomology groups Hi(L⊗K) vanish for all positive i.
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Mumford%20vanishing%20theorem | In algebraic geometry, the Mumford vanishing theorem proved by Mumford in 1967 states that if L is a semi-ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then
The Mumford vanishing theorem is related to the Ramanujam vanishing theorem, and is generalized by the Kawamata–Viehweg vanishing theorem.
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Uniform%20convergence%20in%20probability | Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory.
The law of large numbers says that, for each single event , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class from one and the same sample. Moreover it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds.
Definitions
For a class of predicates defined on a set and a set of samples , where , the empirical frequency of on is
The theoretical probability of is defined as
The Uniform Convergence Theorem states, roughly, that if is "simple" and we draw samples independently (with replacement) from according to any distribution , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability.
Here "simple" means that the Vapnik–Chervonenkis dimension of the class is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole.
The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis using the concept of growth function.
Uniform convergence theorem
The statement of the uniform convergence theorem is as follows:
If is a set of -valued functions defined on a set and is a probability distribution on then for and a positive integer, we have:
where, for any ,
and . indicates that the probability is taken over consisting of i.i.d. draws from the distribution .
is defined as: For any -valued functions over and ,
And for any natural number , the shattering number is defined as:
From the point of Learning Theory one can consider to be the Concept/Hypothesis class defined over the instance set . Before getting into the details of the proof of the theorem we will state Sauer's Lemma which we will need in our proof.
Sauer–Shelah lemma
The Sauer–Shelah lemma relates the shattering number to the VC Dimension.
Lemma: , where is the VC Dimension of the concept class .
Corollary: .
Proof of uniform convergence theorem
and are the sources of the proof below. Before we get into the details of the proof of the Unif |
https://en.wikipedia.org/wiki/Bakki%20Airport | Bakki Airport is an airport on the southern coast of Iceland, used mainly for short-haul flights to and from the Westman Islands.
Statistics
Passengers and movements
See also
Transport in Iceland
List of airports in Iceland
Notes
References
External links
OurAirports - Bakki
OpenStreetMap - Bakki
Airports in Iceland |
https://en.wikipedia.org/wiki/Mohammad%20Bannout | Mohammad Ali Bannout (محمد علي بنوت; born 17 December 1976, in Beirut, Lebanon), informally referred to as Moe Bannout, is a Lebanese IFBB professional bodybuilder.
Competitive statistics
Age:
Height: 1.78 m
Competitive weight: 108 kg
Off Competitive weight : 120 kg
Competitive history
2002, The Hero of Heroes of Lebanon
2003, The Hero of Heroes of Lebanon
2004, The Hero of Heroes of Lebanon
2005, The Hero of Heroes of Lebanon
2005, Arab Bodybuilding Championship, Jordan, 5th
2006, Arab Bodybuilding Championship, Jordan
2007, IFBB World Amateur Bodybuilding Championships, Light Heavyweight, 3rd
2009, IFBB Ironman Pro Invitational, 7th
2010, IFBB Phoenix Pro, Open, 10th
2014, IFBB Phoenix Pro, Open, 1st
2015, IFBB Mr Olympia, Open, 16th
See also
IFBB Professional League
List of male professional bodybuilders
External links
Gallery at bodybuilding.com
2007 IFBB World Amateur Championships gallery
2007 IFBB World Amateur Championships, list of participants
2009 IFBB Ironman Pro Invitational, list of participants
References
1976 births
Lebanese bodybuilders
Male bodybuilders
Living people
Professional bodybuilders
Sportspeople from Beirut |
https://en.wikipedia.org/wiki/Grauert%E2%80%93Riemenschneider%20vanishing%20theorem | In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold, due to .
Grauert–Riemenschneider conjecture
The Grauert–Riemenschneider conjecture is a conjecture related to the Grauert–Riemenschneider vanishing theorem:
This conjecture was proved by using the Riemann–Roch type theorem (Hirzebruch–Riemann–Roch theorem) and by using Morse theory.
Note
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Audrey%20Terras | Audrey Anne Terras (born September 10, 1942) is an American mathematician who works primarily in number theory. Her research has focused on quantum chaos and on various types of zeta functions.
Early life and education
Audrey Terras was born September 10, 1942, in Washington, D.C.
She received a BS degree in mathematics from the University of Maryland, College Park (UMD) in 1964, and MA and PhD degrees from Yale University in 1966 and 1970 respectively. She was married to fellow UMD alumnus Riho Terras. She stated in a 2008 interview that she chose to study mathematics because "The U.S. government paid me! And not much! It was the time of Sputnik, so we needed to produce more mathematicians, and when I was deciding between Math and History, they weren’t paying me to do history, they were paying me to do math."
Career
Terras joined the University of California, San Diego as an assistant professor in 1972, and became a full professor there in 1983. She retired in 2010, and currently holds the title of Professor Emerita.
As an undergraduate Terras was inspired by her teacher Sigekatu Kuroda to become a number theorist; she was especially interested in the use of analytic techniques to get algebraic results. Today her research interests are in number theory, harmonic analysis on symmetric spaces and finite groups, special functions, algebraic graph theory, zeta functions of graphs, arithmetical quantum chaos, and the Selberg trace formula.
Recognition
Terras was elected a Fellow of the American Association for the Advancement of Science in 1982. She was the Association for Women in Mathematics-
Mathematical Association of America AWM/MAA Falconer Lecturer in 2000, speaking on "Finite Quantum Chaos,"
and the AWM's Noether Lecturer in 2008, speaking on "Fun with Zeta Functions of Graphs". In 2012 she became a fellow of the American Mathematical Society.
She is part of the 2019 class of fellows of the Association for Women in Mathematics.
Selected publications
Article based on her 2000 Falconer lecture.
Draft of a book on zeta functions of graphs.
Notes
Further reading
Interview conducted October 30, 2008.
Terras's "Five Simple Rules for (Academic) Success (or at Least Survival)."
External links
Terras's home page at UCSD
2019 AWM Fellows
AWM Falconers Past Winners
Noether Lectures List
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Number theorists
Fellows of the American Association for the Advancement of Science
Fellows of the American Mathematical Society
Fellows of the Association for Women in Mathematics
University of Maryland, College Park alumni
Yale University alumni
University of California, San Diego faculty
1942 births
Living people
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Empty%20semigroup | In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist on the underlying set of a semigroup being non-empty. One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup. Not excluding the empty semigroup simplifies certain results on semigroups. For example, the result that the intersection of two subsemigroups of a semigroup T is a subsemigroup of T becomes valid even when the intersection is empty.
When a semigroup is defined to have additional structure, the issue may not arise. For example, the definition of a monoid requires an identity element, which rules out the empty semigroup as a monoid.
In category theory, the empty semigroup is always admitted. It is the unique initial object of the category of semigroups.
A semigroup with no elements is an inverse semigroup, since the necessary condition is vacuously satisfied.
See also
Field with one element
Semigroup with one element
Semigroup with two elements
Semigroup with three elements
Special classes of semigroups
References
Algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/Paired%20difference%20test | In statistics, a paired difference test is a type of location test that is used when comparing two sets of paired measurements to assess whether their population means differ. A paired difference test uses additional information about the sample that is not present in an ordinary unpaired testing situation, either to increase the statistical power, or to reduce the effects of confounders.
Specific methods for carrying out paired difference tests are, for normally distributed difference t-test (where the population standard deviation of difference is not known) and the paired Z-test (where the population standard deviation of the difference is known), and for differences that may not be normally distributed the Wilcoxon signed-rank test as well as the paired permutation test.
The most familiar example of a paired difference test occurs when subjects are measured before and after a treatment. Such a "repeated measures" test compares these measurements within subjects, rather than across subjects, and will generally have greater power than an unpaired test. Another example comes from matching cases of a disease with comparable controls.
Use in reducing variance
Paired difference tests for reducing variance are a specific type of blocking. To illustrate the idea, suppose we are assessing the performance of a drug for treating high cholesterol. Under the design of our study, we enroll 100 subjects, and measure each subject's cholesterol level. Then all the subjects are treated with the drug for six months, after which their cholesterol levels are measured again. Our interest is in whether the drug has any effect on mean cholesterol levels, which can be inferred through a comparison of the post-treatment to pre-treatment measurements.
The key issue that motivates the paired difference test is that unless the study has very strict entry criteria, it is likely that the subjects will differ substantially from each other before the treatment begins. Important baseline differences among the subjects may be due to their gender, age, smoking status, activity level, and diet.
There are two natural approaches to analyzing these data:
In an "unpaired analysis", the data are treated as if the study design had actually been to enroll 200 subjects, followed by random assignment of 100 subjects to each of the treatment and control groups. The treatment group in the unpaired design would be viewed as analogous to the post-treatment measurements in the paired design, and the control group would be viewed as analogous to the pre-treatment measurements. We could then calculate the sample means within the treated and untreated groups of subjects, and compare these means to each other.
In a "paired difference analysis", we would first subtract the pre-treatment value from the post-treatment value for each subject, then compare these differences to zero.
If we only consider the means, the paired and unpaired approaches give the same result. To see thi |
https://en.wikipedia.org/wiki/D.%20Raghavarao | Damaraju Raghavarao (1938–2013) was an Indian-born statistician, formerly the Laura H. Carnell professor of statistics and chair of the department of statistics at Temple University in Philadelphia.
Raghavarao is an elected fellow of the Institute of Mathematical Statistics, American Statistical Association, and an elected member of The International Statistical Institute. He has been specialized in combinatorics and applications of experimental designs.
Raghavarao received his M.A. in mathematics from Nagpur University, India in 1957 and earned the gold medal. He earned his Ph.D. in statistics from the University of Mumbai in 1961 for his work in designs of experiments; his Ph.D. advisor was M. C. Chakrabarti. Raghavarao was a professor of statistics at Punjab Agricultural University, University of North Carolina at Chapel Hill, Cornell University, and University of Guelph before joining Temple University.
He died on February 6, 2013.
Books
References
External links
Webpage at Temple university
Brief Resume of Damaraju Raghavarao
Indian statisticians
American statisticians
Fellows of the American Statistical Association
1938 births
2013 deaths
Indian combinatorialists
University of Mumbai alumni
Temple University faculty
Fellows of the Institute of Mathematical Statistics
Elected Members of the International Statistical Institute
Scientists from Andhra Pradesh
20th-century Indian mathematicians |
https://en.wikipedia.org/wiki/Du%20Bois%20singularity | In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by .
gave the following characterisation of Du Bois singularities. Suppose that is a reduced closed subscheme of a smooth scheme .
Take a log resolution of in that is an isomorphism outside , and let be the reduced preimage of in . Then has Du Bois singularities if and only if the induced map is a quasi-isomorphism.
References
Singularity theory
Algebraic geometry |
https://en.wikipedia.org/wiki/Krasner%27s%20lemma | In number theory, more specifically in p-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions.
Statement
Let K be a complete non-archimedean field and let be a separable closure of K. Given an element α in , denote its Galois conjugates by α2, ..., αn. Krasner's lemma states:
if an element β of is such that
then K(α) ⊆ K(β).
Applications
Krasner's lemma can be used to show that -adic completion and separable closure of global fields commute. In other words, given a prime of a global field L, the separable closure of the -adic completion of L equals the -adic completion of the separable closure of L (where is a prime of above ).
Another application is to proving that Cp — the completion of the algebraic closure of Qp — is algebraically closed.
Generalization
Krasner's lemma has the following generalization.
Consider a monic polynomial
of degree n > 1
with coefficients in a Henselian field (K, v) and roots in the
algebraic closure . Let I and J be two disjoint,
non-empty sets with union {1,...,n}. Moreover, consider a
polynomial
with coefficients and roots in . Assume
Then the coefficients of the polynomials
are contained in the field extension of K generated by the
coefficients of g. (The original Krasner's lemma corresponds to the situation where g has degree 1.)
Notes
References
Lemmas in number theory
Field (mathematics) |
https://en.wikipedia.org/wiki/Bingo%20%28British%20version%29 | Bingo is a game of probability in which players mark off numbers on cards as the numbers are drawn randomly by a caller, the winner being the first person to mark off all their numbers. Bingo, also previously known in the UK as Housey-Housey, became increasingly popular across the UK following the Betting and Gaming Act 1960 with more purpose-built bingo halls opened every year until 2005. Since 2005, bingo halls have seen a marked decline in revenues and the closure of many halls. The number of bingo clubs in Britain has dropped from nearly 600 in 2005 to under 400 as of January 2014. These closures are blamed on high taxes, the smoking ban, and the rise in online gambling, amongst other things.
Bingo played in the UK (90-ball bingo) is not to be confused with bingo played in the US (75-ball bingo), as the tickets and the calling are slightly different. In the US, it tends to be much more competitive and intense, whereas in the UK the approach is more relaxed, despite the faster pace of the game.
History
The game itself, not originally called bingo, is thought to have had its roots in Italy in the 16th century, specifically, around 1530. Bingo originates from the Italian lottery, Il Gioco del Lotto d'Italia. The game spread to France from Italy and was known as Le Lotto, played by the French aristocracy. The game is then believed to have migrated to Great Britain and other parts of Europe in the 18th century. Players mark off numbers on a ticket as they are randomly called out in order to achieve a winning combination. The similar Tombola was used in nineteenth-century Germany as an educational tool to teach children multiplication tables, spelling, and even history.
The origins of the modern version of the game and its current name bingo are unclear. Early British slang records bingo as... "A customs officer's term, the triumphal cry employed on a successful search." But it definitely gained its initial popularity with the first modern version of the game appearing at carnivals and fairs in the 1920s and is attributed to Hugh J. Ward, who most probably took the name from pre-existing slang for marketing reasons. The modern bingo card design patent went to Edwin S. Lowe in 1942.
The introduction of the Betting and Gaming Act 1960 on 1 January 1961 saw large cash-prizes legalised and the launch of Mecca Bingo by Mecca Leisure Group, led by Eric Morley, who had a large chain of dancehalls and introduced bingo into 60 of them, including the Lyceum Ballroom. Circuit Management Association, who managed the cinemas and dancehalls of The Rank Organisation, was the other large operator at the time, including hosting bingo at their largest cinema, the Blackpool Odeon.
Description of the game
Bingo is a game of probability in which players mark off numbers on cards as the numbers are drawn randomly by a caller, the winner being the first person to mark off all their numbers.
Bingo ticket
A typical bingo ticket contains 27 spaces, arranged in nine |
https://en.wikipedia.org/wiki/Poisson%20distribution | In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson (; ). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
It plays an important role for discrete-stable distributions.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution with mean 3. The most likely number of calls received are 2 and 3, but 1 and 4 are also likely. There is a small probability of it being as low as zero and a very small probability it could be 10 or even higher.
Another example is the number of decay events that occur from a radioactive source during a defined observation period.
History
The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837). The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus . This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.
In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.
A further practical application of this distribution was made by Ladislaus Bortkiewicz in 1898 when he was given the task of investigating the number of soldiers in the Prussian army killed accidentally by horse kicks; this experiment introduced the Poisson distribution to the field of reliability engineering.
Definitions
Probability mass function
A discrete random variable is said to have a Poisson distribution, with parameter if it has a probability mass function given by:
where
is the number of occurrences ()
is Euler's number ()
is the factorial function.
The positive real number is equal to the expected value of and also to its variance.
The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circu |
https://en.wikipedia.org/wiki/Gieseking%20manifold | In mathematics, the Gieseking manifold is a cusped hyperbolic 3-manifold of finite volume. It is non-orientable and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately . It was discovered by . The volume is called Gieseking constant and has a closed-form,
with Clausen function . Compare to the related Catalan's constant which also manifests as a volume,
The Gieseking manifold can be constructed by removing the vertices from a tetrahedron, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0,1,2 to the face with vertices 3,1,0 in that order. Glue the face 0,2,3 to the face 3,2,1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of David B. A. Epstein and Robert C. Penner. Moreover, the angle made by the faces is . The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
The Gieseking manifold has a double cover homeomorphic to the figure-eight knot complement. The underlying compact manifold has a Klein bottle boundary, and the first homology group of the Gieseking manifold is the integers.
The Gieseking manifold is a fiber bundle over the circle with fiber the once-punctured torus and monodromy given by
The square of this map is Arnold's cat map and this gives another way to see that the Gieseking manifold is double covered by the complement of the figure-eight knot.
See also
List of mathematical constants
References
3-manifolds
Geometric topology
Hyperbolic geometry |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Switzerland | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of Switzerland for statistical purposes. As a member of EFTA Switzerland is included in the NUTS standard, although the standard is developed and regulated by the European Union, an organization that Switzerland does not belong to. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for Switzerland is CH and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation - the local administrative unit (LAU). In Switzerland, the LAUs are districts (LAU-1) and municipalities (LAU-2).
Overall
The three NUTS levels are:
NUTS codes
The NUTS codes are as follows:
Local Administrative Units
Below the NUTS levels, there are two Local Administrative Units (LAU) levels
LAU-1: Districts
LAU-2: Municipalities
Notes and references
See also
Subdivisions of Switzerland
ISO 3166-2 codes of Switzerland
FIPS region codes of Switzerland
Comparison of ISO, FIPS, and NUTS codes of the cantons of Switzerland
List of regions of Switzerland by Human Development Index
External links
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EFTA countries - Statistical regions at level 1
SCHWEIZ/SUISSE/SVIZZERA - Statistical regions at level 2
SCHWEIZ/SUISSE/SVIZZERA - Statistical regions at level 3
Correspondence between the regional levels and the national administrative units
Cantons of Switzerland, Statoids.com
Switzerland
Subdivisions of Switzerland
Regions of Switzerland |
https://en.wikipedia.org/wiki/Spherical%20segment | In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes.
It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.
The surface of the spherical segment (excluding the bases) is called spherical zone.
If the radius of the sphere is called , the radii of the spherical segment bases are and and the height of the segment (the distance from one parallel plane to the other) called , then the volume of the spherical segment is
The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by
See also
Spherical cap
Spherical wedge
Spherical sector
References
External links
Summary of spherical formulas
Spherical geometry |
https://en.wikipedia.org/wiki/Idun%20Reiten | Idun Reiten (born 1 January 1942) is a Norwegian professor of mathematics. She is considered to be one of Norway's greatest mathematicians today.
Career
She took her PhD degree at the University of Illinois in 1971. She was appointed as a professor at the University of Trondheim in 1982, now named the Norwegian University of Science and Technology.
Her research area is representation theory for Artinian algebras, commutative algebra, and homological algebra. Her work with Maurice Auslander now forms the part of the study of Artinian algebras known as Auslander–Reiten theory.
Awards
In 2007, Reiten was awarded the Möbius prize. In 2009 she was awarded Fridtjof Nansen's award for successful researchers, (in the field of mathematics and the natural sciences), and the "Nansen medal for outstanding research.
In 2007, she was elected a foreign member of the Royal Swedish Academy of Sciences. She is also a member of the Norwegian Academy of Science and Letters, the Royal Norwegian Society of Sciences and Letters, and Academia Europaea.
In 2012, she became a fellow of the American Mathematical Society. She was named MSRI Clay Senior Scholar and Simons Professor for 2012-13.
She delivered the Emmy Noether Lecture at the International Congress of Mathematicians (ICM) in 2010 in Hyderabad and was an Invited Speaker at the ICM in 1998 in Berlin.
In 2014, the Norwegian King appointed Reiten as commander of the Order of St. Olav "for her work as a mathematician".
See also
Krull–Schmidt category
References
External links
Publikasjonsliste
Publication List at the Mathematical Reviews
1942 births
Living people
University of Illinois Urbana-Champaign alumni
Academic staff of the Norwegian University of Science and Technology
Norwegian women mathematicians
20th-century Norwegian mathematicians
21st-century Norwegian mathematicians
Algebraists
Fellows of the American Mathematical Society
Members of the Norwegian Academy of Science and Letters
Royal Norwegian Society of Sciences and Letters
Members of the Royal Swedish Academy of Sciences
Members of Academia Europaea
Norwegian women academics
Scientists from Trondheim
20th-century women mathematicians
21st-century women mathematicians
20th-century Norwegian women scientists |
https://en.wikipedia.org/wiki/Morse%E2%80%93Smale%20system | In dynamical systems theory, an area of pure mathematics, a Morse–Smale system is a smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds. Morse–Smale systems are structurally stable and form one of the simplest and best studied classes of smooth dynamical systems. They are named after Marston Morse, the creator of the Morse theory, and Stephen Smale, who emphasized their importance for smooth dynamics and algebraic topology.
Definition
Consider a smooth and complete vector field X defined on a compact differentiable manifold M with dimension n. The flow defined by this vector field is a Morse-Smale system if
X has only a finite number of singular points (i.e. equilibrium points of the flow), and all of them are hyperbolic equilibrium points.
X has only a finite number of periodic orbits, and all of them are hyperbolic periodic orbits.
The limit sets of all orbits of X tends to a singular point or a periodic orbit.
The stable and unstable manifolds of the singular points and periodic orbits intersect transversely. In other words, if is a singular point (or periodic orbit) and (respectively, ) its stable (respectively, unstable) manifold, then implies that the corresponding tangent spaces of the stable and unstable manifold satisfy .
Examples
Any Morse function f on a compact Riemannian manifold M defines a gradient vector field. If one imposes the condition that the unstable and stable manifolds of the critical points intersect transversely, then the gradient vector field and the corresponding smooth flow form a Morse–Smale system. The finite set of critical points of f forms the non-wandering set, which consists entirely of fixed points.
Gradient-like dynamical systems are a particular case of Morse–Smale systems.
For Morse–Smale systems on the 2D-sphere all equilibrium points and periodical orbits are hyperbolic; there are no separatrice loops.
Properties
By Peixoto's theorem, the vector field on a 2D manifold is structurally stable if and only if this field is Morse-Smale.
Consider a Morse-Smale system defined on compact differentiable manifold M with dimension n, and let the index of an equilibrium point (or a periodic orbit) be defined as the dimension of its associated unstable manifold. In Morse-Smale systems, the indices of the equilibrium points (and periodic orbits) are related with the topology of M by the Morse-Smale inequalities. Precisely, define mi as the sum of the number of equilibrium points with index i and the number of periodic orbits with indices i and i + 1, and bi as the i-th Betti number of M. Then the following inequalities are valid:
Notes
References
Dynamical systems |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.