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https://en.wikipedia.org/wiki/Jeremy%20Quastel | Jeremy Daniel Quastel , is a Canadian mathematician specializing in probability theory, stochastic processes, partial differential equations. He is currently head of the mathematics department at the University of Toronto. He grew up in Vancouver, British Columbia, and now lives in Toronto, Ontario.
Career
Quastel earned his PhD at Courant Institute of Mathematical Sciences at New York University in 1990; the advisory was S. R. Srinivasa Varadhan. He was a postdoctoral student at the Mathematical Sciences Research Institute in Berkeley, then a faculty member at University of California, Davis for the next six years; returned to Canada in 1998.
Research
Jeremy Quastel is recognized as one of the top probabilists in the world in the fields of hydrodynamic theory, stochastic partial differential equations, and integrable probability. In particular, his research is on the large scale behaviour of interacting particle systems and stochastic partial differential equations.
Awards, distinctions, and recognitions
Fellow of the Royal Society (2021)
CMS Jeffery–Williams Prize (2019)
CRM-Fields-PIMS prize (2018)
Royal Society of Canada Fellow (2016)
Killam Research Fellowship (2013) for his research of stochastic processes and partial differential equations used to describe natural processes of change and evolution
invited speaker at the Current Developments in Mathematics (2011)
invited speaker at the International Congress of Mathematicians in Hyderabad (2010)
Sloan Fellow (1996–98)
Family
Jeremy Quastel is the grandson of biochemist Juda Hirsch Quastel.
Sources
External links
1963 births
Living people
Academic staff of the University of Toronto
Probability theorists
PDE theorists
Scientists from Toronto
Scientists from Vancouver
20th-century Canadian mathematicians
21st-century Canadian mathematicians |
https://en.wikipedia.org/wiki/Quasicircle | In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by and , in the older literature (in German) they were referred to as quasiconformal curves, a terminology which also applied to arcs. In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems.
Definitions
A quasicircle is defined as the image of a circle under a quasiconformal mapping of the extended complex plane. It is called a K-quasicircle if the quasiconformal mapping has dilatation K. The definition of quasicircle generalizes the characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane. In particular a quasicircle is a Jordan curve. The interior of a quasicircle is called a quasidisk.
As shown in , where the older term "quasiconformal curve" is used, if a Jordan curve is the image of a circle under a quasiconformal map in a neighbourhood of the curve, then it is also the image of a circle under a quasiconformal mapping of the extended plane and thus a quasicircle. The same is true for "quasiconformal arcs" which can be defined as quasiconformal images of a circular arc either in an open set or equivalently in the extended plane.
Geometric characterizations
gave a geometric characterization of quasicircles as those Jordan curves for which the absolute value of the cross-ratio of any four points, taken in cyclic order, is bounded below by a positive constant.
Ahlfors also proved that quasicircles can be characterized in terms of a reverse triangle inequality for three points: there should be a constant C such that if two points z1 and z2 are chosen on the curve and z3 lies on the shorter of the resulting arcs, then
This property is also called bounded turning or the arc condition.
For Jordan curves in the extended plane passing through ∞, gave a simpler necessary and sufficient condition to be a quasicircle. There is a constant C > 0 such that if
z1, z2 are any points on the curve and z3 lies on the segment between them, then
These metric characterizations imply that an arc or closed curve is quasiconformal whenever it arises as the image of an interval or the circle under a bi-Lipschitz map f, i.e. satisfying
for positive constants Ci.
Quasicircles and quasisymmetric homeomorphisms
If φ is a quasisymmetric homeomorphism of the circle, then there are conformal maps f of [z| < 1 and g of |z|>1 into disjoint regions such that the complement of the images of f and g is a Jordan curve. The maps f and g extend continuously to the circle |z| = 1 and the sewing equation
holds. The image of the circle is a quasicircle.
Conversely, using the Riemann mapping theorem, the conformal maps f and g uniformizi |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2010–11.
Scottish First Division
Play-offs
Semi-finals
Final
Scottish Second Division
Play-offs
Semi-finals
Final
Scottish Third Division
Final
See also
2010–11 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/Lethargy%20theorem | In mathematics, a lethargy theorem is a statement about the distance of points in a metric space from members of a sequence of subspaces; one application in numerical analysis is to approximation theory, where such theorems quantify the difficulty of approximating general functions by functions of special form, such as polynomials. In more recent work, the convergence of a sequence of operators is studied: these operators generalise the projections of the earlier work.
Bernstein's lethargy theorem
Let be a strictly ascending sequence of finite-dimensional linear subspaces of a Banach space X, and let be a decreasing sequence of real numbers tending to zero. Then there exists a point x in X such that the distance of x to Vi is exactly .
See also
Bernstein's theorem (approximation theory)
References
Preprint
Theorems in approximation theory |
https://en.wikipedia.org/wiki/Positive%20harmonic%20function | In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.
Herglotz-Riesz representation theorem for harmonic functions
A positive function f on the unit disk with f(0) = 1 is harmonic if and only if there is a probability measure μ on the unit circle such that
The formula clearly defines a positive harmonic function with f(0) = 1.
Conversely if f is positive and harmonic and rn increases to 1, define
Then
where
is a probability measure.
By a compactness argument (or equivalently in this case
Helly's selection theorem for Stieltjes integrals), a subsequence of these probability measures has a weak limit which is also a probability measure μ.
Since rn increases to 1, so that fn(z) tends to f(z), the Herglotz formula follows.
Herglotz-Riesz representation theorem for holomorphic functions
A holomorphic function f on the unit disk with f(0) = 1 has positive real part if and only if there is a probability measure μ on the unit circle such that
This follows from the previous theorem because:
the Poisson kernel is the real part of the integrand above
the real part of a holomorphic function is harmonic and determines the holomorphic function up to addition of a scalar
the above formula defines a holomorphic function, the real part of which is given by the previous theorem
Carathéodory's positivity criterion for holomorphic functions
Let
be a holomorphic function on the unit disk. Then f(z) has positive real part on the disk
if and only if
for any complex numbers λ0, λ1, ..., λN, where
for m > 0.
In fact from the Herglotz representation for n > 0
Hence
Conversely, setting λn = zn,
See also
Bochner's theorem
References
Harmonic analysis
Complex analysis
Harmonic functions |
https://en.wikipedia.org/wiki/1971%20S%C3%A3o%20Paulo%20FC%20season | The 1971 football season was São Paulo's 42nd season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 60 (22 Campeonato Paulista, 27 Campeonato Brasileiro, 11 Friendly match)
|-
|Games won || 36 (17 Campeonato Paulista, 10 Campeonato Brasileiro, 8 Friendly match)
|-
|Games drawn || 13 (2 Campeonato Paulista, 10 Campeonato Brasileiro, 1 Friendly match)
|-
|Games lost || 11 (3 Campeonato Paulista, 7 Campeonato Brasileiro, 2 Friendly match)
|-
|Goals scored || 84
|-
|Goals conceded || 50
|-
|Goal difference || +34
|-
|Best result || 4–1 (H) v Portuguesa - Campeonato Paulista - 1971.06.194–1 (H) v Botafogo - Campeonato Brasileiro - 1971.12.15
|-
|Worst result || 0–3 (H) v Grêmio - Campeonato Brasileiro - 1971.08.07
|-
|Most appearances || Gilberto Sorriso (59)
|-
|Top scorer || Toninho Guerreiro (31)
|-
Friendlies
I Festival de Futebol do São Paulo F.C.
Official competitions
Campeonato Paulista
Record
Campeonato Brasileiro
Record
External links
official website
Association football clubs 1971 season
1971
1971 in Brazilian football |
https://en.wikipedia.org/wiki/Boye%20Habekost | Boye Habekost (born 22 September 1968) is a Danish football coach and a former player who is manager at Give Fremad
External links
DBU statistics
BT: Habekost stopper
Boye Habekost ny træner i Give Fremad
1968 births
Living people
Danish men's footballers
Danish football managers
Vejle Boldklub players
Esbjerg fB players
FC Fredericia players
Kolding IF managers
Sportspeople from Fredericia
Men's association football goalkeepers
Footballers from the Region of Southern Denmark |
https://en.wikipedia.org/wiki/H-closed%20space | In mathematics, a Hausdorff space is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.
Examples and equivalent formulations
The unit interval , endowed with the smallest topology which refines the euclidean topology, and contains as an open set is H-closed but not compact.
Every regular Hausdorff H-closed space is compact.
A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.
See also
Compact space
References
K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by Jack Porter and Johannes Vermeer)
Properties of topological spaces
Compactness (mathematics) |
https://en.wikipedia.org/wiki/Tommaso%20Boggio | Tommaso Boggio (22 December 1877 – 25 May 1963) was an Italian mathematician. Boggio worked in mathematical physics, differential geometry, analysis, and financial mathematics. He was an invited speaker in International Congress of Mathematicians 1908 in Rome. He wrote, with Burali-Forti, Meccanica Razionale, published in 1921 by S. Lattes & Compagnia.
Notes
External links
An Italian short biography of Tommaso Boggio at the University of Turin
1877 births
1963 deaths
19th-century Italian mathematicians
20th-century Italian mathematicians
Mathematical analysts
Scientists from Turin
Academic staff of the University of Turin
Academic staff of the University of Genoa |
https://en.wikipedia.org/wiki/Jules%20Gosselet | Jules-Auguste Gosselet (19 April 1832 – 20 March 1916) was a French geologist born in Cambrai, France.
Following unsuccessful studies of pharmacy, and a stint as a mathematics teacher at the Lycée du Quesnoy, he pursued a career in natural history. In 1853 he became a preparateur of geology at the Sorbonne, later obtaining his doctorate with a thesis titled Mémoire sur les terrains primaires de la Belgique, des environs d'Avesnes et du Boulonnais (1860).
He later taught high school physics and chemistry in Bordeaux, afterwards serving as an instructor of natural history at the Faculty of Poitiers. In 1864 he was appointed to the chair of geology at the recently established Faculty of Lille. In 1913 he became a non-resident member of the Academy of Sciences.
Gosselet is remembered for his geological studies of northern France, as demonstrated by the title of one of his better known works: Esquisse géologique du Nord de la France et des contrées voisines (Geological sketches of Nord and neighboring regions), (1880). Other significant writings include a geological treatise on the Ardennes, L'Ardenne (1888), and hydrogeological research on the aquifers of Nord, Leçons sur les nappes aquifères du Nord (1887). Also, he is credited with providing a thorough description of the geological beds in the limestone at Étrœungt (schistes et calcaire d’Etroeungt).
In the field of paleontology, he performed pioneer "zoometric" research of a species of brachiopod known as Spirifer verneuilli (Etudes sur les variations du Spirifer Verneuilli).
Since 1910, the Prix Gosselet is awarded every four years for accomplishments made in the field of applied geology. The mineral gosseletite, a phase of manganian andalusite, is named after him.
References
External links
Drupal Gardens (bibliography)
French geologists
People from Cambrai
1832 births
1916 deaths
Academic staff of the University of Lille Nord de France
Academic staff of the University of Poitiers
Members of the French Academy of Sciences
~ |
https://en.wikipedia.org/wiki/Alexander%20Nabutovsky | Alexander Nabutovsky is a leading Canadian mathematician specializing in differential geometry, geometric calculus of variations and quantitative aspects of topology of manifolds. He is a professor at the University of Toronto Department of Mathematics.
Nabutovsky earned a Ph.D. degree from the Weizmann Institute of Science
in 1993; his advisor was Shmuel Kiro.
He was an invited speaker on "Geometry" at International Congress of Mathematicians, 2010 in Hyderabad.
References
External links
Living people
Canadian mathematicians
Academic staff of the University of Toronto
Geometers
Weizmann Institute of Science alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Johann%20Georg%20B%C3%BCsch | Johann Georg Büsch (January 3, 1728 at Alten-Weding in Hanover – August 5, 1800 in Hamburg) was a German mathematics teacher and writer on statistics and commerce.
Biography
He was educated at Hamburg and Göttingen, and in 1756 was made professor of mathematics in the Hamburg gymnasium, which post he held until his death. Besides suggesting many theoretical improvements in the carrying on of trade by the city, he brought about the establishment of an association for the promotion of art and industry (), and the foundation of a school of trade, instituted in 1767, which became under his direction one of the most noted establishments of its class in the world. For some time before his death Büsch was almost totally blind.
As a mathematics teacher he mentored and helped the young Johann Elert Bode, who later became a famous astronomer.
Works
Besides a history of trade (Geschichte der merkwürdigsten Welthändel, Hamburg, 1781), he wrote voluminously on all subjects connected with commerce and political economy. His collected works were published in 16 volumes at Zwickau in 1813–16, and 8 volumes of selected writings, comprising those on trade alone, at Hamburg, 1824–27.
Notes
References
1728 births
1800 deaths
German statisticians
18th-century German educators
German activists
Writers from Hamburg
German schoolteachers |
https://en.wikipedia.org/wiki/Artin%20conductor | In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function.
Local Artin conductors
Suppose that L is a finite Galois extension of the local field K, with Galois group G. If is a character of G, then the Artin conductor of is the number
where Gi is the i-th ramification group (in lower numbering), of order gi, and χ(Gi) is the average value of on Gi. By a result of Artin, the local conductor is an integer. Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if χ is unramified, then its Artin conductor is zero. Thus if L is unramified over K, then the Artin conductors of all χ are zero.
The wild invariant or Swan conductor of the character is
in other words, the sum of the higher order terms with i > 0.
Global Artin conductors
The global Artin conductor of a representation of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be
where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of to the decomposition group of some prime of L lying over p. Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in L/K.
Artin representation and Artin character
Suppose that L is a finite Galois extension of the local field K, with Galois group G. The Artin character aG of G is the character
and the Artin representation AG is the complex linear representation of G with this character. asked for a direct construction of the Artin representation. showed that the Artin representation can be realized over the local field Ql, for any prime l not equal to the residue characteristic p. showed that it can be realized over the corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field Qp, suggesting that there is no easy way to construct the Artin representation explicitly.
Swan representation
The Swan character swG is given by
where rg is the character of the regular representation and 1 is the character of the trivial representation. The Swan character is the character of a representation of G. showed that there is a unique projective representation of G over the l-adic integers with character the Swan character.
Applications
The Artin conductor appears in the conductor-discriminant formula for the discriminant of a global field.
The optimal level in the Serre modularity conjecture is expressed in terms of the Artin conductor.
The Artin conductor appears in the functional equation of the Artin L-function.
The Artin and Swan representations are used to defined the conductor of an elliptic curve or abelian variety.
Notes
References
Number theory
Representation theory
Zeta and L-functions |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20FK%20Partizan%20season | The 2006–07 season was FK Partizan's 1st season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club played during the 2006–07 season.
Players
Squad information
Squad statistics
Transfers
In
Out
Loan out
Competitions
Overview
Serbian SuperLiga
First stage
Championship round
UEFA Cup
See also
List of FK Partizan seasons
References
External links
Official website
Partizanopedia 2006-2007 (in Serbian)
FK Partizan seasons
Partizan |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20PFC%20CSKA%20Sofia%20season | The 2005–06 season was PFC CSKA Sofia's 58th consecutive season in A Group.
Below is a list of player statistics and all matches (official and friendly) that the club played during the 2005–06 season.
Club
Team kits
The team kits for the 2005–06 season are produced by Uhlsport and sponsored by Vivatel.
Squad
Source:
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Bulgarian Supercup
CSKA lost the game on penalty shoot-out after the extra time.
UEFA Champions League
Second qualifying round
Third qualifying round
UEFA Cup
First round
Group stage
References
External links
CSKA Official Site
CSKA Fan Page with up-to-date information
Bulgarian A Professional Football Group Site
PFC CSKA Sofia seasons
Cska Sofia |
https://en.wikipedia.org/wiki/Serbia%20men%27s%20national%20water%20polo%20team%20statistics |
Serbia men's national water polo team championship results
Olympic Games
World Aquatics Championship
European Championship
FINA World League
FINA Water Polo World Cup
Mediterranean Games
Junior and Youth Results
World Junior Championship
World Youth Championship U-18
European U-19 Championship
European Junior Championship
Player statistics and records
Most appearances
Professional friendly and competitive matches only where Yugoslavia, Serbia and Montenegro and now Serbia were represented.
Top scorers
Professional friendly and competitive matches only where Yugoslavia, Serbia and Montenegro and now Serbia were represented.
Statistics accurate as of matches played 11 August 2017
References
+
National team
Serbia |
https://en.wikipedia.org/wiki/Glossary%20of%20areas%20of%20mathematics | Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, see . The Mathematics Subject Classification is a hierarchical list of areas and subjects of study that has been elaborated by the community of mathematicians. It is used by most publishers for classifying mathematical articles and books.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
See also
Lists of mathematics topics
Outline of mathematics
:Category:Glossaries of mathematics
References
Areas of mathematics
Areas of mathematics
Wikipedia glossaries using description lists |
https://en.wikipedia.org/wiki/1939%E2%80%9340%20Galatasaray%20S.K.%20season | The 1939–40 season was Galatasaray SK's 36th in existence and the club's 28th consecutive season in the Istanbul Football League.
Squad statistics
Competitions
Istanbul Football League
Standings
Matches
Kick-off listed in local time (EEST)
Milli Küme
Classification
Matches
Friendly Matches
References
Atabeyoğlu, Cem. 1453–1991 Türk Spor Tarihi Ansiklopedisi. page(155–159).(1991) An Grafik Basın Sanayi ve Ticaret AŞ
Tekil, Süleyman. Dünden bugüne Galatasaray, (1983), page(88, 123–125, 184). Arset Matbaacılık Kol.Şti.
Futbol vol.2. Galatasaray. Page: 565, 586. Tercüman Spor Ansiklopedisi. (1981)Tercüman Gazetecilik ve Matbaacılık AŞ.
1940 Milli Küme Maçları. Türk Futbol Tarihi vol.1. page(81). (June 1992) Türkiye Futbol Federasyonu Yayınları.
External links
Galatasaray Sports Club Official Website
Turkish Football Federation – Galatasaray A.Ş.
uefa.com – Galatasaray AŞ
Galatasaray S.K. (football) seasons
Turkish football clubs 1939–40 season
1930s in Istanbul
1940s in Istanbul |
https://en.wikipedia.org/wiki/Mohamed%20Ahmed%20Bashir | Mohamed Ahmed Bashir (born 22 July 1983), also known as Muhamed Besha, is a Sudanese football coach and former player. He played as an attacking midfielder .
Career statistics
Scores and results list Sudan's goal tally first.
References
External links
Living people
1987 births
People from Khartoum North
Sudanese men's footballers
Men's association football midfielders
Sudan men's international footballers
Sudan men's A' international footballers
2012 Africa Cup of Nations players
2011 African Nations Championship players
2018 African Nations Championship players
Saudi Pro League players
Al Wehda FC players
Al-Hilal Club (Omdurman) players
Association football coaches
Sudanese expatriate men's footballers
Sudanese expatriate sportspeople in Saudi Arabia
Expatriate men's footballers in Saudi Arabia |
https://en.wikipedia.org/wiki/Marvin%20Knopp | Marvin Isadore Knopp (January 4, 1933 – December 24, 2011) was an American mathematician who worked primarily in number theory. He made
notable contributions to the theory of modular forms.
Life and education
Knopp was born on January 4, 1933, in Chicago, Illinois. He received his PhD under Paul T. Bateman from the University of Illinois in 1958 where he became friends with fellow student Gene Golub.
Over the course of his career, he advised twenty Ph.D. students. He is the father of pianist Seth Knopp, and of Yehudah, Abby, and Elana. Marvin was married to Josephine Zadovsky Knopp for 25 years but the marriage ended in divorce. Knopp died on December 24, 2011, during a vacation in Florida. Marvin found happiness from his children, old movies, great music and numbers. The last 30 years of Knopp's life was shared with Phyllis Zemble. During the 6 years following his death, Zemble organized his papers and books (with the help of Wladimer Pribitkin), his photographs and his mathematical correspondence, which she donated to the American Institute of Mathematics (AIM). On AIM's website, you can find 131 of Knopp's reprints.
Personal life
Knopp was born in Chicago, Illinois in 1933. He was an Ashkenazi Jew.
Career
After receiving his PhD in 1958, Knopp taught at the University of Wisconsin and then, for a few years, at the
University of Illinois Chicago before moving, in 1976, to Temple University where he stayed until his sudden death in 2011.
Knopp was a leading expert in the theory of modular forms and a pioneering figure in the theory of Eichler cohomology, modular integrals and generalized modular forms. He was closely associated with Emil Grosswald. In Jean Dieudonne's influential book A Panorama of Pure Mathematics (Academic Press, 1982),
he is mentioned (p. 95) as one of those who "made substantial contributions" to the theory of modular forms.
Selected publications
Further reading
A set of papers in honor of Grosswald; includes reminiscences, list of PhD students, and a list of papers and books. Temple Tribute
References
External links
American Institute of Mathematics reprints by Marvin Knopp https://aimath.org/cgi-bin/library.cgi?database=reprints;mode=display;BrowseTitle=Knopp, Marvin
20th-century American mathematicians
21st-century American mathematicians
Mathematical analysts
Number theorists
Temple University faculty
University of Illinois alumni
1933 births
2011 deaths
Mathematicians from Pennsylvania
People from Chicago
University of Wisconsin–Madison faculty
University of Illinois Chicago faculty
Mathematicians from Illinois |
https://en.wikipedia.org/wiki/Ghulam%20Dastagir%20Alam | Ghulam Dastagir Alam Qasmi (Urdu: غلام دستگیر عالم قاسمی ; popularly known as G.D. Alam; ), was a Pakistani theoretical physicist and professor of mathematics at the Quaid-e-Azam University. Alam is best known for conceiving and embarking on research on the gas centrifuge during Pakistan's integrated atomic bomb project in the 1970s, and he also conceived the research on charge density, nuclear fission, and gamma-ray bursts throughout his career.
After the atomic bomb project, Alam joined the Department of Mathematics at the Quaid-e-Azam University (QAU) as well as serving as visiting faculty at the Institute of Physics, and co-authored papers on variation calculus and fission isomer. He was one of the notable theoretical physicists at the Pakistan Atomic Energy Commission (PAEC) and QAU. At one point, his fellow theorist, Munir Ahmad Khan, called Alam "the problem solving brain of the PAEC".
Biography
Alam was educated at the Government College University in Lahore where he studied in 1951 and graduated with Bachelor of Science (BSc) in Mathematics in 1955 under the supervision of Abdus Salam–a theoretical physicist. He then went to attend the physics program at the Punjab University where he graduated with Master of Science (MSc) in Physics in 1957, supervised under Dr. Rafi Muhammad– a nuclear physicist.
His thesis titled: The Emission of Electromagnetic Radiations from metals by high energy particles, had contained investigations on electromagnetic radiation emitted from the heavy metals by bombarding the elementary particles.
In 1964, Alam joined the doctoral program in physics at the University College London (UCL) under the Colombo Plan Scholarship, initially joining the doctoral group led by British physicist, John B. Hasted. He learned the course on atomic physics under Harrie Massey and worked on experimental physics under J.B. Hasted. In 1967, he submitted his doctoral thesis supervised by Dr. J.B. Hasted which he defended successfully and graduated with Doctor of Philosophy in theoretical physics. His doctoral thesis, titled: Electron Capture by Multiply Charged Ions, provided scientific investigations on charge-crossing involving potential curve crossing, a concept in quantum mechanics.
In 1967, he published another thesis jointly written by J.B. Hasted and D.K. Bohme on physics of atomic collision and potential energy curves— their work was supported and funded by the United States Department of Defense.
While in the United Kingdom, Alam continued publishing and working on the atomic physics and atomic collisions, collaborating with many other of his British colleagues. However, Alam lost interests in atomic physics and became interested in computer programming and mathematics. In 1970, he published a paper on gamma rays and performed an experiment on isomers, proposing and later proving mathematically that, in the isomer state, the average kinetic energy associated with the decay process of Isomer state is about the same in |
https://en.wikipedia.org/wiki/Hodge%E2%80%93Tate%20module | In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. introduced and named Hodge–Tate structures using the results of on p-divisible groups.
Definition
Suppose that G is the absolute Galois group of a p-adic field K. Then G has a canonical cyclotomic character χ given by its action on the pth power roots of unity. Let C be the completion of the algebraic closure of K. Then a finite-dimensional vector space over C with a semi-linear action of the Galois group G is said to be of Hodge–Tate type if it is generated by the eigenvectors of integral powers of χ.
See also
p-adic Hodge theory
Mumford–Tate group
References
Algebraic geometry
Number theory
Hodge theory |
https://en.wikipedia.org/wiki/Alias%20method | In computing, the alias method is a family of efficient algorithms for sampling from a discrete probability distribution, published in 1974 by A. J. Walker. That is, it returns integer values according to some arbitrary probability distribution . The algorithms typically use or preprocessing time, after which random values can be drawn from the distribution in time.
Operation
Internally, the algorithm consults two tables, a probability table and an alias table (for ). To generate a random outcome, a fair dice is rolled to determine an index into the two tables. Based on the probability stored at that index, a biased coin is then flipped, and the outcome of the flip is used to choose between a result of and .
More concretely, the algorithm operates as follows:
Generate a uniform random variate .
Let and . (This makes uniformly distributed on and uniformly distributed on .)
If , return . This is the biased coin flip.
Otherwise, return .
An alternative formulation of the probability table, proposed by Marsaglia et al. as the "square histogram" method, uses the condition in the third step (where ) instead of computing .
Table generation
The distribution may be padded with additional probabilities to increase to a convenient value, such as a power of two.
To generate the table, first initialize . While doing this, divide the table entries into three categories:
The "overfull" group, where ,
The "underfull" group, where and has not been initialized, and
The "exactly full" group, where or has been initialized.
If , the corresponding value will never be consulted and is unimportant, but a value of is sensible.
As long as not all table entries are exactly full, repeat the following steps:
Arbitrarily choose an overfull entry and an underfull entry . (If one of these exists, the other must, as well.)
Allocate the unused space in entry to outcome , by setting .
Remove the allocated space from entry by changing .
Entry is now exactly full.
Assign entry to the appropriate category based on the new value of .
Each iteration moves at least one entry to the "exactly full" category (and the last moves two), so the procedure is guaranteed to terminate after at most iterations. Each iteration can be done in time, so the table can be set up in time.
Vose points out that floating-point rounding errors may cause the guarantee referred to in step 1 to be violated. If one category empties before the other, the remaining entries may have set to 1 with negligible error. The solution accounting for floating point is sometimes called the Walker-Vose method or the Vose alias method.
The Alias structure is not unique.
As the lookup procedure is slightly faster if (because does not need to be consulted), one goal during table generation is to maximize the sum of the . Doing this optimally turns out to be NP hard, but a greedy algorithm comes reasonably close: rob from the richest and give to the poorest. That is, |
https://en.wikipedia.org/wiki/Cindy%20Watson | Cindy Watson (born 24 March 1978) is a retired tennis player from Australia.
Career statistics
Her highest singles ranking is world No. 131, (achieved on 28 October 2002) and her highest doubles ranking is No. 108 (reached on 8 August 2005). Watson won 13 titles on the ITF Circuit in her career: seven in singles and six in doubles. Watson has taken part in many WTA Tour events.
Biography
Her biggest career highlight is reaching the third round of the 2002 Australian Open. She defeated María José Martínez Sánchez and Emmanuelle Gagliardi in the first and second rounds, respectively, before falling to fourth seed Kim Clijsters, 1–6, 2–6 in the third round.
Watson played on two other Grand Slam tournaments- she fell in the first rounds of the 1999 Australian Open to Mary Pierce and of the 2005 Australian Open to Sania Mirza.
ITF Circuit finals
Singles: 15 (7–8)
Doubles: 14 (6–8)
References
External links
Living people
1978 births
Australian female tennis players
Place of birth missing (living people)
21st-century Australian women |
https://en.wikipedia.org/wiki/Oscillator%20representation | In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself.
The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.
Historical overview
The mathematical formulation of quantum mechanics by Werner Heisenberg and Erwin Schrödinger was originally in terms of unbounded self-adjoint operators on a Hilbert space. The fundamental operators corresponding to position and momentum satisfy the Heisenberg commutation relations. Quadratic polynomials in these operators, which include the harmonic oscillator, are also closed under taking commutators.
A large amount of operator theory was developed in the 1920s and 1930s to provide a rigorous foundation for quantum mechanics. Part of the theory was formulated in terms of unitary groups of operators, largely through the contributions of Hermann Weyl, Marshall Stone and John von Neumann. In turn these results in mathematical physics were subsumed within mathematical analysis, starting with the 1933 lecture notes of Norbert Wiener, who used the heat kernel for the harmonic oscillator to derive the properties of the Fourier transform.
The uniqueness of the Heisenberg commutation relations, as formulated in the Stone–von Neumann theorem, was later interpreted within group representation theory, in particular the theory of induced representations initiated by George Mackey. The quadratic operators were understood in terms of a projective unitary representation of the group SU(1,1) and its Lie algebra. Irving Segal and David Shale generalized this construction to the symplectic group in finite and infinite dimensions—in physics, this is often referred to as bosonic quantization: it is constructed as the symmetric algebra of an infinite-dimensional space. Segal and Shale have also treated the c |
https://en.wikipedia.org/wiki/Nigel%20Cutland | Nigel J. Cutland is Professor of Mathematics at the University of York. His main fields of interest are non-standard analysis, Loeb spaces, and applications in probability and stochastic analysis. He was Editor-in-Chief of Logic and Analysis and Journal of Logic and Analysis.
Books
See also
Influence of non-standard analysis
References
20th-century British mathematicians
21st-century British mathematicians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Victor%20Schlegel | Victor Schlegel (4 March 1843 – 22 November 1905) was a German mathematician. He is remembered for promoting the geometric algebra of Hermann Grassmann and for a method of visualizing polytopes called Schlegel diagrams.
In the nineteenth century there were various expansions of the traditional field of geometry through the innovations of hyperbolic geometry, non-Euclidean geometry and algebraic geometry. Hermann Grassmann was one of the more advanced innovators with his anticipation of linear algebra and multilinear algebra that he called "Extension theory" (Ausdehnungslehre). As recounted by David E. Rowe in 2010:
The most important new convert was Victor Schlegel, Grassmann’s colleague at Stettin Gymnasium from 1866 to 1868. Afterward Schlegel accepted a position as Oberlehrer at the Gymnasium in Waren, a small town in Mecklenburg.
In 1872 Schlegel published the first part of his System der Raumlehre which used Grassmann’s methods to develop plane geometry. Schlegel used his book to put forth Grassmann’s case, arguing that
"Grassmann’s ideas had been neglected because he had not held a university chair."
Continuing his criticism of academics, Schlegel expressed the reactionary view that
Rather than developing a sound basis for their analytic methods, contemporary mathematicians tended to invent new symbolisms on an ad hoc basis, creating a Babel-like cacophony of unintelligible languages.
Schlegel’s attitude was that no basis for scientific method was shown in mathematics, and "neglect of foundations had led to the widely acknowledged lack of interest in mathematics in the schools."
The mathematician Felix Klein addressed Schlegel’s book in a review criticizing him for neglect of cross ratio and failure to contextualize Grassmann in the flow of mathematical developments. Rowe indicates that Klein was most interested in developing his Erlangen program.
In 1875 Schlegel countered with the second part of his System der Raumlehre, answering Klein in the preface. This part developed conic sections, harmonic ranges, projective geometry, and determinants.
Schlegel published a biography of Hermann Grassmann in 1878. Both parts of his textbook, and the biography, are now available at the Internet Archive; see External links section below.
At the Summer meeting of the American Mathematical Society on August 15, 1894, Schlegel presented an essay on the problem of finding the place which is at a minimum total distance from given points.
In 1899 Schlegel became German national secretary for the international Quaternion Society and reported on it in Monatshefte für Mathematik.
References
Bibliography
David E. Rowe (2010) "Debating Grassmann’s Mathematics: Schlegel Versus Klein", Mathematical Intelligencer 32(1):41–8.
Victor Schlegel (1883) "Theorie der homogen zusammengesetzten Raumgebilde", Nova Acta, Ksl. Leop.-Carol. Deutsche Akademie der Naturforscher, Band XLIV, Nr. 4, Druck von E. Blochmann & Sohn in Dresden.
Victor Schlegel (1886) Ueber Pro |
https://en.wikipedia.org/wiki/Kasch%20ring | In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other.
Kasch rings are named in honor of mathematician Friedrich Kasch. Kasch originally called Artinian rings whose proper ideals have nonzero annihilators S-rings. The characterizations below show that Kasch rings generalize S-rings.
Definition
Equivalent definitions will be introduced only for the right-hand version, with the understanding that the left-hand analogues are also true. The Kasch conditions have a few equivalent statements using the concept of annihilators, and this article uses the same notation appearing in the annihilator article.
In addition to the definition given in the introduction, the following properties are equivalent definitions for a ring R to be right Kasch. They appear in :
For every simple right R module M, there is a nonzero module homomorphism from M into R.
The maximal right ideals of R are right annihilators of ring elements, that is, each one is of the form where x is in R.
For any maximal right ideal T of R, .
For any proper right ideal T of R, .
For any maximal right ideal T of R, .
R has no dense right ideals except R itself.
Examples
The content below can be found in references such as , , .
Let R be a semiprimary ring with Jacobson radical J. If R is commutative, or if R/J is a simple ring, then R is right (and left) Kasch. In particular, commutative Artinian rings are right and left Kasch.
For a division ring k, consider a certain subring R of the four-by-four matrix ring with entries from k. The subring R consists of matrices of the following form:
This is a right and left Artinian ring which is right Kasch, but not left Kasch.
Let S be the ring of power series on two noncommuting variables X and Y with coefficients from a field F. Let the ideal A be the ideal generated by the two elements YX and Y2. The quotient ring S/A is a local ring which is right Kasch but not left Kasch.
Suppose R is a ring direct product of infinitely many nonzero rings labeled Ak. The direct sum of the Ak forms a proper ideal of R. It is easily checked that the left and right annihilators of this ideal are zero, and so R is not right or left Kasch.
The two-by-two upper (or lower) triangular matrix ring is not right or left Kasch.
A ring with right socle zero (i.e. ) cannot be right Kasch, since the ring contains no minimal right ideals. So, for example, domains which are not division rings are not right or left Kasch.
References
Algebraic structures
Ring theory |
https://en.wikipedia.org/wiki/Loeb%20space | In mathematics, a Loeb space is a type of measure space introduced by using nonstandard analysis.
Construction
Loeb's construction starts with a finitely additive map from an internal algebra of sets to the nonstandard reals. Define to be given by the standard part of , so that is a finitely additive map from to the extended reals . Even if is a nonstandard -algebra, the algebra need not be an ordinary -algebra as it is not usually closed under countable unions. Instead the algebra has the property that if a set in it is the union of a countable family of elements of , then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as ) from to the extended reals is automatically countably additive. Define to be the -algebra generated by . Then by Carathéodory's extension theorem the measure on extends to a countably additive measure on , called a Loeb measure.
References
External links
Home page of Peter Loeb
Measure theory
Nonstandard analysis |
https://en.wikipedia.org/wiki/Gevrey%20class | In mathematics, the Gevrey classes on a domain , introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions and the space of smooth (infinitely differentiable) functions . In particular, for , the Gevrey class , consists of those smooth functions such that for every compact subset there exists a constant , depending only on , such that
Where denotes the partial derivative of order (see multi-index notation).
When , coincides with the class of analytic functions , but for there are compactly supported functions in the class that are not identically zero (an impossibility in ). It is in this sense that they interpolate between and . The Gevrey classes find application in discussing the smoothness of solutions to certain partial differential equations: Gevrey originally formulated the definition while investigating the homogeneous heat equation, whose solutions are in .
Application
Gevrey functions are used in control engineering for trajectory planning.
A typical example is the function
with
and Gevrey order
See also
Denjoy–Carleman theorem
References
Smooth functions |
https://en.wikipedia.org/wiki/Kidegembye | Kidegembye is a town and ward in Njombe Rural District in the Njombe Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 8,329 people in the ward, from 8,068 in 2012.
References
Wards of Njombe Region |
https://en.wikipedia.org/wiki/Igongolo | Igongolo is a town and ward in Njombe district in the Iringa Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 8,720 people in the ward, from 8,447 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Igosi | Igosi is a town and ward in Wanging'ombe district in the NJOMBE of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 7,437 people in the ward, from 7,204 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Ikondo%2C%20Njombe | Ikondo is a town and ward in Njombe district in the Njombe Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 7,163 people in the ward, from 6,312 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Ikuna | Ikuna (Ikuka) is a town and ward in Njombe Rural District in the Njombe Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 9,474 people in the ward, from 9,178 in 2012.
References
Wards of Njombe Region |
https://en.wikipedia.org/wiki/Iwungilo | Iwungilo is a town and ward in Njombe Urban District in the Njombe Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 8,691 people in the ward, from 8,419 in 2012.
References
Wards of Njombe Region |
https://en.wikipedia.org/wiki/Luduga | Luduga is a village and ward in Wanging'ombe District in the Njombe Region of the Tanzanian Southern Highlands.
In 2016 the Tanzania National Bureau of Statistics report there were 9,587 people in the ward, from 11,947 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Lupembe | Lupembe is a town and ward in Njombe Rural District in the Njombe Region of the Tanzanian Southern Highlands.
In 2016 the Tanzania National Bureau of Statistics report there were 7,958 people in the ward, from 7,709 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Mahongole%2C%20Njombe | Mahongole is a town and ward in Njombe district in the Iringa Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 9,520 people in the ward, from 9,222 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Usuka | Usuka is a town and ward in Njombe District in the Iringa Region of the Tanzanian Southern Highlands. In 2016 the Tanzania National Bureau of Statistics report there were 6,344 people in the ward, from 6,146 in 2012.
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Structural%20synthesis%20of%20programs | Structural synthesis of programs (SSP) is a special form of (automatic) program synthesis that is based on propositional calculus. More precisely, it uses intuitionistic logic for describing the structure of a program in such a detail that the program can be automatically composed from pieces like subroutines or even computer commands. It is assumed that these pieces have been implemented correctly, hence no correctness verification of these pieces is needed. SSP is well suited for automatic composition of services for service-oriented architectures and for synthesis of large simulation programs.
History
Automatic program synthesis began in the artificial intelligence field, with software intended for automatic problem solving. The first program synthesizer was developed by Cordell Green in 1969. At about the same time, mathematicians including R. Constable, Z. Manna, and R. Waldinger explained the possible use of formal logic for automatic program synthesis. Practically applicable program synthesizers appeared considerably later.
The idea of structural synthesis of programs was introduced at a conference on algorithms in modern mathematics and computer science organized by Andrey Ershov and Donald Knuth in 1979. The idea originated from G. Pólya’s well-known book on problem solving. The method for devising a plan for solving a problem in SSP was presented as a formal system. The inference rules of the system were restructured and justified in logic by G. Mints and E. Tyugu in 1982. A programming tool PRIZ that uses SSP was developed in the 1980s.
A recent Integrated development environment that supports SSP is CoCoViLa — a model-based software development platform for implementing domain specific languages and developing large Java programs.
The logic of SSP
Structural synthesis of programs is a method for composing programs from already implemented components (e.g. from computer commands or software object methods) that can be considered as functions. A specification for synthesis is given in intuitionistic propositional logic by writing axioms about the applicability of functions. An axiom about the applicability of a function f is a logical implication
X1 ∧ X2 ∧ ... ∧ Xm → Y1 ∧ Y2 ... Yn,
where X1, X2, ... Xm are preconditions and Y1, Y2, ... Yn are postconditions of the application of the function f. In intuitionistic logic, the function f is called a realization of this formula. A precondition can be a proposition stating that input data exists, e.g. Xi may have the meaning “variable xi has received a value”, but it may denote also some other condition, e.g. that resources needed for using the function f are available, etc. A precondition may also be an implication of the same form as the axiom given above; then it is called a subtask. A subtask denotes a function that must be available as an input when the function f is applied. This function itself must be synthesized in the process of SSP. In this case, realization of t |
https://en.wikipedia.org/wiki/Twentse%20Derby | The Twentse Derby is a Dutch football derby between FC Twente and Heracles Almelo.
Statistics
Results
All time goal scorers
References
FC Twente
Heracles Almelo
Football derbies in the Netherlands |
https://en.wikipedia.org/wiki/Minimal%20ideal | In the branch of abstract algebra known as ring theory, a minimal right ideal of a ring R is a non-zero right ideal which contains no other non-zero right ideal. Likewise, a minimal left ideal is a non-zero left ideal of R containing no other non-zero left ideals of R, and a minimal ideal of R is a non-zero ideal containing no other non-zero two-sided ideal of R .
In other words, minimal right ideals are minimal elements of the partially ordered set (poset) of non-zero right ideals of R ordered by inclusion. The reader is cautioned that outside of this context, some posets of ideals may admit the zero ideal, and so the zero ideal could potentially be a minimal element in that poset. This is the case for the poset of prime ideals of a ring, which may include the zero ideal as a minimal prime ideal.
Definition
The definition of a minimal right ideal N of a ring R is equivalent to the following conditions:
N is non-zero and if K is a right ideal of R with , then either or .
N is a simple right R-module.
Minimal ideals are the dual notion to maximal ideals.
Properties
Many standard facts on minimal ideals can be found in standard texts such as , , , and .
In a ring with unity, maximal right ideals always exist. In contrast, minimal right, left, or two-sided ideals in a ring with unity need not exist.
The right socle of a ring is an important structure defined in terms of the minimal right ideals of R.
Rings for which every right ideal contains a minimal right ideal are exactly the rings with an essential right socle.
Any right Artinian ring or right Kasch ring has a minimal right ideal.
Domains that are not division rings have no minimal right ideals.
In rings with unity, minimal right ideals are necessarily principal right ideals, because for any nonzero x in a minimal right ideal N, the set xR is a nonzero right ideal of R inside N, and so .
Brauer's lemma: Any minimal right ideal N in a ring R satisfies or for some idempotent element e of R .
If N1 and N2 are non-isomorphic minimal right ideals of R, then the product equals {0}.
If N1 and N2 are distinct minimal ideals of a ring R, then
A simple ring with a minimal right ideal is a semisimple ring.
In a semiprime ring, there exists a minimal right ideal if and only if there exists a minimal left ideal .
Generalization
A non-zero submodule N of a right module M is called a minimal submodule if it contains no other non-zero submodules of M. Equivalently, N is a non-zero submodule of M which is a simple module. This can also be extended to bimodules by calling a non-zero sub-bimodule N a minimal sub-bimodule of M if N contains no other non-zero sub-bimodules.
If the module M is taken to be the right R-module RR, then the minimal submodules are exactly the minimal right ideals of R. Likewise, the minimal left ideals of R are precisely the minimal submodules of the left module R R. In the case of two-sided ideals, we see that the minimal ideals of R are exactly the minim |
https://en.wikipedia.org/wiki/Kahe%20Mashariki | Kahe Mashariki is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,210 people in the ward, from 11,384 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20SpVgg%20Greuther%20F%C3%BCrth%20season | The 2011–12 SpVgg Greuther Fürth season started on 15 July against Eintracht Frankfurt in the 2. Bundesliga.
Review and events
Results
Legend
2. Bundesliga
DFB-Pokal
Roster and statistics
Sources
SpVgg Greuther Fürth seasons
Greuther Furth |
https://en.wikipedia.org/wiki/Kibosho%20Magharibi | Kibosho Magharibi is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 21,763 people in the ward, from 20,291 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Clifton%E2%80%93Pohl%20torus | In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.
Definition
Consider the manifold with the metric
Any homothety is an isometry of , in particular including the map:
Let be the subgroup of the isometry group generated by . Then has a proper, discontinuous action on . Hence the quotient which is topologically the torus, is a Lorentz surface that is called the Clifton–Pohl torus. Sometimes, by extension, a surface is called a Clifton–Pohl torus if it is a finite covering of the quotient of by any homothety of ratio different from .
Geodesic incompleteness
It can be verified that the curve
is a geodesic of M that is not complete (since it is not defined at ). Consequently, (hence also ) is geodesically incomplete, despite the fact that is compact. Similarly, the curve
is a null geodesic that is incomplete. In fact, every null geodesic on or is incomplete.
The geodesic incompleteness of the Clifton–Pohl torus is better seen as a direct consequence of the fact that is extendable, i.e. that it can be seen as a subset of a bigger Lorentzian surface. It is a direct consequence of a simple change of coordinates. With
consider
The metric (i.e. the metric expressed in the coordinates ) reads
But this metric extends naturally from to , where
The surface , known as the extended Clifton–Pohl plane, is geodesically complete.
Conjugate points
The Clifton–Pohl tori are also remarkable by the fact that they were the first known non-flat Lorentzian tori with no conjugate points. The extended Clifton–Pohl plane contains a lot of pairs of conjugate points, some of them being in the boundary of i.e. "at infinity" in .
Recall also that, by Hopf–Rinow theorem no such tori exists in the Riemannian setting.
References
Lorentzian manifolds
Metric geometry |
https://en.wikipedia.org/wiki/Kibosho%20Mashariki | Kibosho Mashariki is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,174 people in the ward, from 14,148 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Kindi%20%28Tanzanian%20ward%29 | Kindi is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 22,943 people in the ward, from 21,391 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Kirima%20%28Tanzanian%20ward%29 | Kirima is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,486 people in the ward, from 10,709 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Kirua%20Vunjo%20Kusini | Kirua Vunjo Kusini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 20,784 people in the ward, from 19,378 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Mabogini | Mabogini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania.
In 2016 the Tanzania National Bureau of Statistics report there were 31,095 people in the ward, from 28,992 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20SC%20Preu%C3%9Fen%20M%C3%BCnster%20season | The 2011–12 SC Preußen Münster season started on 23 July against SpVgg Unterhaching in the 3rd Liga.
Review and events
Results
Legend
3rd Liga
Roster and statistics
Sources
SC Preußen Münster seasons
Preussen Munster |
https://en.wikipedia.org/wiki/Marcus%20Sandberg | Marcus Sandberg (born 7 November 1990) is a Swedish footballer who plays for Eliteserien club HamKam as a goalkeeper.
Career statistics
Honours
Club
IFK Göteborg
Svenska Cupen: 2012–13, 2014–15
References
External links
1990 births
Living people
People from Tjörn Municipality
Men's association football goalkeepers
IFK Göteborg players
Swedish men's footballers
Allsvenskan players
Superettan players
Sweden men's under-21 international footballers
Sweden men's youth international footballers
Vålerenga Fotball players
Stabæk Fotball players
Hamarkameratene players
Eliteserien players
Swedish expatriate men's footballers
Expatriate men's footballers in Norway
Swedish expatriate sportspeople in Norway
Footballers from Västra Götaland County |
https://en.wikipedia.org/wiki/Mohammad%20Omar%20Shishani | Mohammad Omar Shishani () (born April 24, 1989) is a Jordanian football player who plays as a striker for Al-Faisaly.
International career statistics
References
External links
kooora.com
1989 births
Living people
Jordanian men's footballers
Jordan men's international footballers
Men's association football forwards
Footballers at the 2010 Asian Games
Footballers from Amman
Al-Faisaly SC players
Shabab Al-Ordon SC players
Al-Hussein SC (Irbid) players
Al-Ramtha SC players
Al-Baqa'a SC players
Al-Ahli SC (Amman) players
Asian Games competitors for Jordan
Jordanian people of Chechen descent |
https://en.wikipedia.org/wiki/Bel%E2%80%93Robinson%20tensor | In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by:
Alternatively,
where is the Weyl tensor. It was introduced by Lluís Bel in 1959. The Bel–Robinson tensor is constructed from the Weyl tensor in a manner analogous to the way the electromagnetic stress–energy tensor is built from the electromagnetic tensor. Like the electromagnetic stress–energy tensor, the Bel–Robinson tensor is totally symmetric and traceless:
In general relativity, there is no unique definition of the local energy of the gravitational field. The Bel–Robinson tensor is a possible definition for local energy, since it can be shown that whenever the Ricci tensor vanishes (i.e. in vacuum), the Bel–Robinson tensor is divergence-free:
References
Tensors in general relativity
Differential geometry |
https://en.wikipedia.org/wiki/Kim%20Sung-ju | Kim Sung-ju (; born 15 November 1990) is a South Korean football player. He is a left-footed play-making midfielder.
Club statistics
References
External links
Kim Sung-ju at Korea Football Association
1990 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's under-23 international footballers
South Korean expatriate men's footballers
Albirex Niigata players
Kataller Toyama players
Seoul E-Land FC players
Gimcheon Sangmu FC players
Ulsan Hyundai FC players
Jeju United FC players
Incheon United FC players
Pohang Steelers players
J1 League players
J2 League players
K League 2 players
K League 1 players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
People from Pohang
Footballers from North Gyeongsang Province |
https://en.wikipedia.org/wiki/Ahmed%20Al-Fahmi | Ahmed Al-Fahmi (; born 12 December 1990) is a Saudi Arabian footballer who plays for Al-Sahel as a goalkeeper.
Club career statistics
References
1990 births
Living people
Sportspeople from Mecca
Saudi Arabian men's footballers
Al Wehda FC players
Al-Qaisumah FC players
Al Qadsiah FC players
Al-Washm Club players
Al-Kawkab FC players
Al-Arabi SC (Saudi Arabia) players
Al-Nairyah Club players
Al-Sahel SC (Saudi Arabia) players
Saudi First Division League players
Saudi Pro League players
Saudi Second Division players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/List%20of%20United%20Sikkim%20FC%20managers | This is a list of United Sikkim Football Club's managers and their records from 2011, when the first professional manager was appointed, to the present day.
Statistics
References
Managers
United Sikkim |
https://en.wikipedia.org/wiki/1970%20S%C3%A3o%20Paulo%20FC%20season | The 1970 football season was São Paulo's 41st season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 61 (18 Campeonato Paulista, 16 Torneio Roberto Gomes Pedrosa, 27 Friendly match)
|-
|Games won || 23 (12 Campeonato Paulista, 3 Torneio Roberto Gomes Pedrosa, 8 Friendly match)
|-
|Games drawn || 20 (3 Campeonato Paulista, 5 Torneio Roberto Gomes Pedrosa, 12 Friendly match)
|-
|Games lost || 18 (3 Campeonato Paulista, 8 Torneio Roberto Gomes Pedrosa, 7 Friendly match)
|-
|Goals scored || 81
|-
|Goals conceded || 69
|-
|Goal difference || +12
|-
|Best result || 8–0 (H) v Mitsubishi - Friendly match - 1970.02.01
|-
|Worst result || 0–4 (A) v Santos - Friendly match - 1970.03.21
|-
|Top scorer || Toninho Guerreiro (17)
|-
Friendlies
Taça São Paulo
Official competitions
Campeonato Paulista
Record
Torneio Roberto Gomes Pedrosa
Record
External links
official website
Association football clubs 1970 season
1970
1970 in Brazilian football |
https://en.wikipedia.org/wiki/Mamba%20Kaskazini | Mamba Kaskazini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,723 people in the ward, from 9,065 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Mamba%20Kusini | Mamba Kusini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,790 people in the ward, from 10,060 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Marangu%20Magharibi | Marangu Magharibi is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 20,352 people in the ward, from 18,976 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Marangu%20Mashariki | Marangu Mashariki is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 25,456 people in the ward, from 23,734 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Mbokomu | Mbokomu is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,665 people in the ward, from 14,606 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Mwika%20Kaskazini | Mwika Kaskazini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 22,713 people in the ward, from 21,177 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Okaoni | Okaoni is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,204 people in the ward, from 10,446 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Old%20Moshi%20Mashariki | Old Moshi Mashariki is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,219 people in the ward, from 9,528 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Uru%20Kaskazini | Uru Kaskazini is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,885 people in the ward, from 11,081 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Uru%20Mashariki | Uru Mashariki is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,853 people in the ward, from 14,781 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Uru%20Shimbwe | Uru Shimbwe is a town and ward in the Moshi Rural district of the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,550 people in the ward, from 6,107 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Uru%20Kusini | Uru South is a ward in the Moshi Rural district in the Kilimanjaro Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 24,565 people in the ward, from 22,904 in 2012.
References
Wards of Kilimanjaro Region |
https://en.wikipedia.org/wiki/Density%20%28polytope%29 | In geometry, the density of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions,
representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.
The same calculation can be performed for any convex polyhedron, even one without symmetries, by choosing any point interior to the polyhedron as its center. For these polyhedra, the density will be 1.
More generally, for any non-self-intersecting (acoptic) polyhedron, the density can be computed as 1 by a similar calculation that chooses a ray from an interior point that only passes through facets of the polyhedron, adds one when this ray passes from the interior to the exterior of the polyhedron, and subtracts one when this ray passes from the exterior to the interior of the polyhedron. However, this assignment of signs to crossings does not generally apply to star polyhedra, as they do not have a well-defined interior and exterior.
Tessellations with overlapping faces can similarly define density as the number of coverings of faces over any given point.
Polygons
The density of a polygon is the number of times that the polygonal boundary winds around its center. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem.
The density of a polygon can also be called its turning number; the sum of the turn angles of all the vertices divided by 360°. This will be an integer for all unicursal paths in a plane.
The density of a compound polygon is the sum of the densities of the component polygons.
Regular star polygons
For a regular star polygon {p/q}, the density is q. It can be visually determined by counting the minimum number of edge crossings of a ray from the center to infinity.
Examples
Polyhedra
A polyhedron and its dual have the same density.
Total curvature
A polyhedron can be considered a surface with Gaussian curvature concentrated at the vertices and defined by an angle defect. The density of a polyhedron is equal to the total curvature (summed over all its vertices) divided by 4π.
For example, a cube has 8 vertices, each with 3 squares, leaving an angle defect of π/2. 8×π/2=4π. So the density of the cube is 1.
Simple polyhedra
The density of a polyhedron with simple faces and vertex figures is half of the Euler Characteristic, χ. If its genus is g, its density is 1-g.
χ = V − E + F = 2D = 2(1-g).
Regular star polyhedra
Arthur Cayley used density as a way to modify Euler's polyhedron formula (V − E + F = 2) to work for t |
https://en.wikipedia.org/wiki/List%20of%20the%20busiest%20airports%20in%20Spain | This is a list of the busiest airports in Spain, including airports in the Balearic Islands and the Canary Islands. Data is compiled from statistics published by Aena, the public body that owns and operates the majority of airports in the country.
At a glance
30 busiest airports in Spain by passenger traffic
2021
The following is a list of the 30 busiest Spanish airports in 2021, from Aena statistics.
2020
The following is a list of the 30 busiest Spanish airports in 2020, from Aena statistics. The large decrease in passenger numbers was caused by travel restrictions imposed in response to the COVID-19 pandemic.
2019
The following is a list of the 30 busiest Spanish airports in 2019, from provisional AENA statistics.
2018
The following is a list of the 30 busiest Spanish airports in 2018, from provisional AENA statistics.
2017
The following is a list of the 30 busiest Spanish airports in 2017, from provisional AENA statistics.
2016
The following is a list of the 30 busiest Spanish airports in 2016, from AENA statistics.
2015
The following is a list of the 30 busiest Spanish airports in 2015, from final AENA statistics.
2014
The following is a list of the 30 busiest Spanish airports in 2014, from AENA statistics.
2013
The following is a list of the 30 busiest Spanish airports in 2013, from AENA statistics.
2012
The following is a list of the 30 busiest Spanish airports in 2012, from AENA statistics.
2011
The following is a list of the 30 busiest Spanish airports in 2011, from AENA statistics.
References
Spain
.x
S
Busiest
it:Aeroporti più trafficati in Europa
tr:Yolcu trafiğine göre Avrupa'nın en kalabalık havalimanları listesi |
https://en.wikipedia.org/wiki/Neat%20submanifold | In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold.
To define this more precisely, first let
be a manifold with boundary, and
be a submanifold of .
Then is said to be a neat submanifold of if it meets the following two conditions:
The boundary of is a subset of the boundary of . That is, .
Each point of has a neighborhood within which 's embedding in is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.
More formally, must be covered by charts of such that where is the dimension For instance, in the category of smooth manifolds, this means that the embedding of must also be smooth.
See also
Local flatness
References
Differential topology |
https://en.wikipedia.org/wiki/Hadamard%20product%20%28matrices%29 | In mathematics, the Hadamard product (also known as the element-wise product, entrywise product or Schur product) is a binary operation that takes in two matrices of the same dimensions and returns a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product. It is attributed to, and named after, either French-Jewish mathematician Jacques Hadamard or German-Jewish mathematician Issai Schur.
The Hadamard product is associative and distributive. Unlike the matrix product, it is also commutative.
Definition
For two matrices and of the same dimension , the Hadamard product (sometimes ) is a matrix of the same dimension as the operands, with elements given by
For matrices of different dimensions ( and , where or ), the Hadamard product is undefined.
For example, the Hadamard product for two arbitrary 2 × 3 matrices is:
Properties
The Hadamard product is commutative (when working with a commutative ring), associative and distributive over addition. That is, if A, B, and C are matrices of the same size, and k is a scalar:
The identity matrix under Hadamard multiplication of two matrices is an matrix where all elements are equal to 1. This is different from the identity matrix under regular matrix multiplication, where only the elements of the main diagonal are equal to 1. Furthermore, a matrix has an inverse under Hadamard multiplication if and only if none of the elements are equal to zero.
For vectors and , and corresponding diagonal matrices and with these vectors as their main diagonals, the following identity holds: where denotes the conjugate transpose of . In particular, using vectors of ones, this shows that the sum of all elements in the Hadamard product is the trace of where superscript T denotes the matrix transpose, that is, . A related result for square and , is that the row-sums of their Hadamard product are the diagonal elements of : Similarly, Furthermore, a Hadamard matrix-vector product can be expressed as: where is the vector formed from the diagonals of matrix .
The Hadamard product is a principal submatrix of the Kronecker product.
The Hadamard product satisfies the rank inequality
If and are positive-definite matrices, then the following inequality involving the Hadamard product holds: where is the th largest eigenvalue of .
If and are diagonal matrices, then
The Hadamard product of two vectors and is the same as matrix multiplication of one vector by the corresponding diagonal matrix of the other vector:
The vector to diagonal matrix operator may be expressed using the Hadamard product as: where is a constant vector with elements and is the identity matrix.
The mixed-product property
where is Kronecker product, assuming has the same dimensions of and with .
where denotes face-splitting product.
where is column-wise Khatri–Rao product.
Schur product theorem
The Hadamard produ |
https://en.wikipedia.org/wiki/Superpartient%20ratio | In mathematics, a superpartient ratio, also called superpartient number or epimeric ratio, is a rational number that is greater than one and is not superparticular. The term has fallen out of use in modern pure mathematics, but continues to be used in music theory and in the historical study of mathematics.
Superpartient ratios were written about by Nicomachus in his treatise Introduction to Arithmetic.
Overview
Mathematically, a superpartient number is a ratio of the form
where a is greater than 1 (a > 1) and is also coprime to n. Ratios of the form are also greater than one and fully reduced, but are called superparticular ratios and are not superpartient.
Etymology
"Superpartient" comes from Greek ἐπιμερής epimeres "containing a whole and a fraction," literally "superpartient".
See also
Mathematics of musical scales
Further reading
Partch, Harry (1979). Genesis of a Music, p.68. .
Rational numbers
Intervals (music) |
https://en.wikipedia.org/wiki/Roman%20Holowinsky | Roman Holowinsky (born July 26, 1979) is an American mathematician known for his work in number theory and, in particular, the theory of modular forms. He is currently an associate professor with tenure at the Ohio State University.
Holowinsky was awarded the SASTRA Ramanujan Prize in 2011 for his contributions to "areas of mathematics influenced by the genius Srinivasa Ramanujan", for proving, with Kannan Soundararajan, an important case of the quantum unique ergodicity (QUE) conjecture. In 2011, Holowinsky was also awarded a Sloan Fellowship.
Holowinsky received a Bachelors in Science Degree from Rutgers University in 2001. Afterwards, he continued his studies at Rutgers and received his PhD in 2006 under the direction of Henryk Iwaniec.
In 2017, he founded the Erdős Institute, a multi-university collaboration focused on helping graduate students, postdocs, and graduate alumni find rewarding jobs in industry.
References
External links
Erdős Institute
1979 births
Living people
Number theorists
Rutgers University alumni
21st-century American mathematicians
Recipients of the SASTRA Ramanujan Prize |
https://en.wikipedia.org/wiki/Newman%20Catholic%20College | Newman Catholic College (formerly Cardinal Hinsley Maths and Computing College) is an all-boys Catholic school, located in the London Borough of Brent. The school has a student age range of 11–19 years old and is Voluntary Aided. Newman College (before the name change) was founded in 1959. Half of the school's students have English as a second language.
Notable staff
Mohamed Mohamud Ibrahim, Deputy Prime Minister of Somalia
Notable alumni
Cyrille Regis (1958–2018) - professional footballer, West Bromwich Albion F.C., Coventry City F.C., England
Bashy/Ashley Thomas (b. 1985) - actor and rapper
Controversies
Bomb threat
On 27 November 2012, students and teachers were evacuated from Newman Catholic College in Harlesden following reports of a bomb in the school's grounds. Police were called to Newman Catholic College in Harlesden Road, Harlesden. Students and staff were evacuated and the surrounding area was cordoned off for public safety while an explosive dog unit was called to search the area. However, nothing was found and the area was reopened at 10:20 am. Police have investigated an allegation of malicious communication.
Departure of staff
Of 50 teachers at the Roman Catholic boys' school in Harlesden, 26 left in 2000. Documents obtained by The Guardian include a letter by union representatives to the chairman of governors, John Fox, alleging that the school environment had become "unsafe for pupils and staff".
References
External links
NCC profile at bbc.co.uk
NCC profile at teachweb.co.uk
NCC profile at foxtons.co.uk
Member of Parliament Sarah Teather visits Newman Catholic College
Department Of Education data for NCC
Ofsted Inspection Data
Brent Council School Contact
1958 establishments in England
Boys' schools in London
Secondary schools in the London Borough of Brent
Educational institutions established in 1958
Voluntary aided schools in London
Catholic secondary schools in the Archdiocese of Westminster |
https://en.wikipedia.org/wiki/Nathalie%20Bock | Nathalie Bock (born 21 October 1988) is a German football midfielder.
Career
Statistics
International career
As an Under-19 international Bock won the 2007 U-19 European Championships, where she scored two goals including Germany's opening goal in the final's extra time.
References
External links
1988 births
Living people
People from Recklinghausen
Footballers from Münster (region)
German women's footballers
Women's association football midfielders
Frauen-Bundesliga players
2. Frauen-Bundesliga players
SG Wattenscheid 09 (women) players
FFC Heike Rheine players
VfL Wolfsburg (women) players |
https://en.wikipedia.org/wiki/1969%20S%C3%A3o%20Paulo%20FC%20season | The 1969 football season was São Paulo's 40th season since club's existence.
Statistics
Scorers
Overall
{|class="wikitable"
|-
|Games played || 65 (29 Campeonato Paulista, 16 Torneio Roberto Gomes Pedrosa, 20 Friendly match)
|-
|Games won || 34 (17 Campeonato Paulista, 5 Torneio Roberto Gomes Pedrosa, 12 Friendly match)
|-
|Games drawn || 10 (3 Campeonato Paulista, 4 Torneio Roberto Gomes Pedrosa, 3 Friendly match)
|-
|Games lost || 21 (9 Campeonato Paulista, 7 Torneio Roberto Gomes Pedrosa, 5 Friendly match)
|-
|Goals scored || 104
|-
|Goals conceded || 86
|-
|Goal difference || +18
|-
|Best result || 4–1 (H) v Guarani - Campeonato Paulista - 1969.02.02 4–1 (H) v São Bento - Campeonato Paulista - 1969.02.26 4–1 (H) v Flamengo - Torneio Roberto Gomes Pedrosa - 1969.11.19
|-
|Worst result || 0–4 (A) v Valencia - Friendly match - 1969.06.290–4 (A) v America-RJ - Torneio Roberto Gomes Pedrosa - 1969.11.26
|-
|Most appearances || Cláudio Deodato (62)
|-
|Top scorer || Zé Roberto (27)
|-
Friendlies
Troféo Bodas de Oro
Friendly tournament played in June 1969 to commemorate the first 50 years of Valencia CF.
Troféo Colombino
Tournoi Mohammed V
Official competitions
Campeonato Paulista
Record
Torneio Roberto Gomes Pedrosa
Record
References
External links
official website
Association football clubs 1969 season
1969
1969 in Brazilian football |
https://en.wikipedia.org/wiki/Trirectangular%20tetrahedron | In geometry, a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles. That vertex is called the right angle of the trirectangular tetrahedron and the face opposite it is called the base. The three edges that meet at the right angle are called the legs and the perpendicular from the right angle to the base is called the altitude of the tetrahedron.
Only the bifurcating graph of the Affine Coxeter group has a Trirectangular tetrahedron fundamental domain.
Metric formulas
If the legs have lengths a, b, c, then the trirectangular tetrahedron has the volume
The altitude h satisfies
The area of the base is given by
De Gua's theorem
If the area of the base is and the areas of the three other (right-angled) faces are , and , then
This is a generalization of the Pythagorean theorem to a tetrahedron.
Integer solution
Perfect body
The area of the base (a,b,c) is always (Gua) an irrational number. Thus a trirectangular tetrahedron with integer edges is never a perfect body. The trirectangular bipyramid (6 faces, 9 edges, 5 vertices) built from these trirectangular tetrahedrons and the related left-handed ones connected on their bases have rational edges, faces and volume, but the inner space-diagonal between the two trirectangular vertices is still irrational. The later one is the double of the altitude of the trirectangular tetrahedron and a rational part of the (proved) irrational space-diagonal of the related Euler-brick (bc, ca, ab).
Integer edges
Trirectangular tetrahedrons with integer legs and sides of the base triangle exist, e.g. (discovered 1719 by Halcke). Here are a few more examples with integer legs and sides.
a b c d e f
240 117 44 125 244 267
275 252 240 348 365 373
480 234 88 250 488 534
550 504 480 696 730 746
693 480 140 500 707 843
720 351 132 375 732 801
720 132 85 157 725 732
792 231 160 281 808 825
825 756 720 1044 1095 1119
960 468 176 500 976 1068
1100 1008 960 1392 1460 1492
1155 1100 1008 1492 1533 1595
1200 585 220 625 1220 1335
1375 1260 1200 1740 1825 1865
1386 960 280 1000 1414 1686
1440 702 264 750 1464 1602
1440 264 170 314 1450 1464
Notice that some of these are multiples of smaller ones. Note also .
Integer faces
Trirectangular tetrahedrons with integer faces and altitude h exist, e.g. without or with coprime .
See also
Irregular tetrahedra
Standard simplex
Euler Brick
References
External links
Polyhedra |
https://en.wikipedia.org/wiki/C.S.%20Mar%C3%ADtimo%20statistics%20and%20records | Club Sport Marítimo has a long history in Portuguese football. Although the club was excluded from participating in the national championships for several years, this article includes statistics and records recorded throughout the club's history.
Statistics
Seasons in all competitions
Key
1D = 1 Division
2D = 2 Division
Pos = Final position
P = Games played
W = Games won
D = Games drawn
L = Games lost
GS = Goals score
GA = Goals against
Pts = Points
CP = Campeonato de Portugal (Championship of Portugal)
TP = Taça de Portugal (Portuguese Cup)
TL = Taça da Liga (Portuguese League Cup)
UEL = UEFA Europa League
» NOTES
Club Sport Marítimo began participation in the national championship in 1973–74.
Club Sport Marítimo began participation in the Taça de Portugal in 1939–40.
The League Cup tournament began in 2007–08.
Last update: 12 June 2023
Top scorers by season
» NOTES
The data used is since the last promotion to the I League in 1985
Last update: 12 June 2023
Top 10 goalscorers
» NOTES
The data used is since the last promotion to the I League in 1985
A player's name in bold denotes active player
Last update 17 June 2023
Top 10 appearances
Competitive, professional matches only. Appearances as substitute (in parentheses) included in total.
» NOTES
The data used is since the last promotion to the I League in 1985
A player's name in bold denotes active player
Last update 18 January 2016
International players
Portugal
» NOTES
Last update: 16 June 2015
Other nations
» NOTES
Last update: 17 October 2023
References
C.S. Marítimo |
https://en.wikipedia.org/wiki/1939%20Albanian%20National%20Championship | Statistics of Albanian National Championship in the 1939 season. This event is still not officially recognized from AFA, but in December 2012 the Albanian sports media have reported that this championship, along with the other two championships of World War II is expected to be recognized soon.
Overview
1939 Albanian National Championship was the 8th season of Albania's annual main competition. It started on 1 July 1939, and ended on 30 September 1939. Eight teams were separated in two groups of 4 teams each, playing two leg matches with a knock-out system. Group A teams were: SK Tirana, Rinia Korçare, Bashkimi Elbasanas and Dragoj. Group B teams were: Vllaznia, Besa, Teuta and Ismail Qemali.
KF Tirana won the championship.
Results
First round
In this round entered all the teams in two groups.
Group A:
Group B:
Semifinals
In this round entered the four winners from the previous round.
Finals
In this round entered the two winners from the previous round.
Teams, scorers and referee:
SK Tirana: Gurashi I; Maluçi, Janku ; Myzeqari, F.Hoxha, Karapici; Kryeziu, Lisi, Korça, Lushta, Plluska.
Vllaznija: Jubani; Hila, Pali; Alibali, Vasija, Koxhja; Shkjezi, L.Hoxha, Boriçi, Radovani, Gjinali.
Goals: Plluska 6', Lisi 13', Kryeziu 32', Kryeziu 40', Vasija 43', L.Hoxha, Lushta, Korça 60’, Boriçi 76', Boriçi 79', Boriçi 80'
Referee: Sinnico.
References
Kategoria Superiore seasons
1
Albania
Albania |
https://en.wikipedia.org/wiki/1940%20Albanian%20National%20Championship | Statistics of Albanian National Championship in the 1940 season. This event is still not officially recognized from AFA, but in December 2012 the Albanian sports media have reported that this championship, along with the other two championships of World War II, is expected to be recognized soon.
Overview
1940 Albanian National Championship was the 9th season of Albania's annual main competition. It started on 3 March 1940, and ended on 26 May 1940. Eight teams were separated in two groups of 4 teams each, playing double round-robin system and only the first team of each group would go into the finals. Group A teams were: Tirana, Shkodra, Elbasani and Durrësi. Group B teams were: Gjirokastra, Berati, Korça and Vlora.
Shkodra won the championship.
Results
First round
In this round entered all the teams in two groups.
Group A
Group A results:
Vllaznia - Elbasani 6-1
Tirana - Teuta 3-1
Teuta - Vllaznia 4-2
Elbasani - Tirana 1-3
Tirana - Vllaznia 2-2
Teuta - Elbasani 3-3
Tirana - Elbasani 4-3
Vllaznia - Teuta 2-1
Vllaznia - Tirana 2-0
Elbasani - Teuta 1-2
Elbasani - Vllaznia 2-7
Teuta - Tirana 1-0
Group B
Group B results:
Luftëtari - Flamurtari 3-3
Tomori - Skënderbeu 0-2
Flamurtari - Skënderbeu 4-0
Tomori - Luftëtari 5-0
Tomori - Flamurtari 1-1
Skënderbeu - Luftëtari 3-0
Luftëtari - Tomori 0-2
Skënderbeu - Flamurtari 4-1
Flamurtari - Tomori 5-3
Luftëtari - Skënderbeu 1-1
Skënderbeu - Tomori 3-0
Flamurtari - Luftëtari 3-0
Finals
In this round entered the two winners from the previous round.
Teams, scorers and referee:
Shkodra: Jubani; Rusi, Fakja; Xharra, Koxhja, Osmani II; Shkjezi, Vasija, Boriçi, S.Gjinali, Z.Berisha.
Korca: Mihallaqi; Dumja, Bitri; P.Saro, Luarasi, Ademi; Dimço, Plluska, Qirinxhi II, Duro, Merolli.
Goals: S.Gjinali 8', Shkjezi 21', Qirinxhi 58' (11-m).
Referee: Enver Kulla.
Shkodra: Jubani; Vasija, Pali; Osmani, Koxhja, Xharra; Rusi, Berisha, Boriçi, S.Gjinali, Shkjezi.
Korca: Mihallaqi; Bitri, Dume; Zeka, Lauarasi, Saro; Dimço, Plluska, Duro, Qirinxhi, Merolli.
Goals: Rusi 11', Boriçi 22', Boriçi 46', Rusi, Rusi, Rusi, Rusi, Rusi, Rusi
Referee: Sabit Çoku.
References
Kategoria Superiore seasons
1
Albania
Albania |
https://en.wikipedia.org/wiki/1942%20Albanian%20National%20Championship | Statistics of Albanian National Championship in the 1942 season. It was the first and only Nationwide Championship ever played, including the participation of Albanian and Kosovan teams at an official football event. This event is still not officially recognized from AFA, but in December 2012 the Albanian sports media have reported that this championship, along with the other two championships of World War II is expected to be recognized soon.
Overview
1942 Albanian National Championship was the 10th season of Albania's annual main competition. It started on 14 June 1942, and ended on 29 June 1942. Ten teams were separated in one group of 4 teams and two groups of 3 teams, playing single round-robin system. Two first teams of the bigger group and only the first team of other two smaller groups would go into the semifinals. North Zone teams were: Shkodra, Prishtina, Peja and Prizreni.
Middle Zone teams were: Tirana, Elbasani and Durrësi. South Zone teams were: Gjirokastra, Berati and Korça.
KF Tirana won the championship.
Results
First round
In this round entered all the teams in three groups.
North Zone
North Zone results:
Prizreni - Peja 2-1
Prishtina - Vllaznia 0-2
Peja - Vllaznia 0-1
Prizreni - Prishtina 5-0
Prizreni - Vllaznia 3-1
Peja - Prishtina 2-2
Middle Zone*
Middle Zone results:
Tirana - Elbasani 5-1
Tirana - Teuta 1-1
Teuta - Elbasani result is yet unknown.
South Zone
South Zone results:
Tomori - Luftëtari 6-0
Tomori - Skënderbeu 2-2
Skënderbeu - Luftëtari 9-2
Semifinals
In this round entered the four winners from the previous round.
Finals
In this round entered the two winners from the previous round.
Note: Regular match ended 1-1 draw, but Shkodra refused to play 2x15 minutes extra time, therefore Italian referee Carone and AFA decided the match in Tirana's favour.
Teams, scorers and referee:
Tirana: Kamba; Peza, Qiri; Visha, Fagu, Kurani; Parapani, Derani, Lisi, Bylyku, Mexhid Dibra.
Shkodra: Nuti; Boshnjaku, Muhamet Dibra; Pelingu, Nd.Pali, Gj.Berisha; Z.Berisha, Shaqiri, Puka, Kavaja, Nehani.
Goals: Bylyku 65', Nd.Pali 90'.
Referee: Michele Carone (Italy).
References
Kategoria Superiore seasons
1
Albania
Albania |
https://en.wikipedia.org/wiki/%CE%A4-additivity | In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces.
A measure or set function on a space whose domain is a sigma-algebra is said to be if for any upward-directed family of nonempty open sets such that its union is in the measure of the union is the supremum of measures of elements of that is,:
See also
References
.
Measure theory |
https://en.wikipedia.org/wiki/Orthocentric%20tetrahedron | In geometry, an orthocentric tetrahedron is a tetrahedron where all three pairs of opposite edges are perpendicular. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.
In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the orthocenter, and it has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid. Hence the orthocenter coincides with the Monge point of the tetrahedron.
Characterizations
All tetrahedra can be inscribed in a parallelepiped. A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi. If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric.
A tetrahedron is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:
In fact, it is enough for only two pairs of opposite edges to satisfy this condition for the tetrahedron to be orthocentric.
Another necessary and sufficient condition for a tetrahedron to be orthocentric is that its three bimedians have equal length.
Volume
The characterization regarding the edges implies that if only four of the six edges of an orthocentric tetrahedron are known, the remaining two can be calculated as long as they are not opposite to each other. Therefore the volume of an orthocentric tetrahedron can be expressed in terms of four edges a, b, c, d. The formula is
where c and d are opposite edges, and .
See also
Disphenoid
Trirectangular tetrahedron
References
Polyhedra |
https://en.wikipedia.org/wiki/Fox%E2%80%93Wright%20function | In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of and :
Upon changing the normalisation
it becomes pFq(z) for A1...p = B1...q = 1.
The Fox–Wright function is a special case of the Fox H-function :
A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution with the pdf on is given as , where denotes the Fox–Wright Psi function.
Wright function
The entire function is often called the Wright function. It is the special case of of the Fox–Wright function. Its series representation is
This function is used extensively in fractional calculus and the stable count distribution. Recall that . Hence, a non-zero with zero is the simplest nontrivial extension of the exponential function in such context.
Three properties were stated in Theorem 1 of Wright (1933) and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955) (p. 212)
Equation (a) is a recurrence formula. (b) and (c) provide two paths to reduce a derivative. And (c) can be derived from (a) and (b).
A special case of (c) is . Replacing with , we have
A special case of (a) is . Replacing with , we have
Two notations, and , were used extensively in the literatures:
M-Wright function
is known as the M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes, generally referred to as time-fractional diffusion processes.
Its properties were surveyed in Mainardi et al (2010).
Through the stable count distribution, is connected to Lévy's stability index .
Its asymptotic expansion of for is
where
See also
Hypergeometric function
Generalized hypergeometric function
Modified half-normal distribution with the pdf on is given as , where denotes the Fox–Wright Psi function.
References
External links
hypergeom on GitLab
Factorial and binomial topics
Hypergeometric functions
Series expansions |
https://en.wikipedia.org/wiki/Quasi-split%20group | In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.
Examples
All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.
showed that all simple algebraic groups over finite fields are quasi-split.
Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups On,n+2, the unitary groups SUn,n and SUn,n+1, and the form of E6 with signature 2.
References
Linear algebraic groups |
https://en.wikipedia.org/wiki/Euclidean%20random%20matrix | Within mathematics, an N×N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f(r, r′) and of N points {ri} randomly distributed in a region V of d-dimensional Euclidean space. The element Aij of the matrix is equal to f(ri, rj): Aij = f(ri, rj).
History
Euclidean random matrices were first introduced in 1999. They studied a special case of functions f that depend only on the distances between the pairs of points: f(r, r′) = f(r - r′) and imposed an additional condition on the diagonal elements Aii,
Aij = f(ri - rj) - u δijΣkf(ri - rk),
motivated by the physical context in which they studied the matrix.
A Euclidean distance matrix is a particular example of Euclidean random matrix with either f(ri - rj) = |ri - rj|2 or f(ri - rj) = |ri - rj|.
For example, in many biological networks, the strength of interaction between two nodes depends on the physical proximity of those nodes. Spatial interactions between nodes can be modelled as a Euclidean random matrix, if nodes are placed randomly in space.
Properties
Because the positions of the points {ri} are random, the matrix elements Aij are random too. Moreover, because the N×N elements are completely determined by only N points and, typically, one is interested in N≫d, strong correlations exist between different elements.
Hermitian Euclidean random matrices
Hermitian Euclidean random matrices appear in various physical contexts, including supercooled liquids, phonons in disordered systems, and waves in random media.
Example 1: Consider the matrix  generated by the function f(r, r′) = sin(k0|r-r′|)/(k0|r-r′|), with k0 = 2π/λ0. This matrix is Hermitian and its eigenvalues Λ are real. For N points distributed randomly in a cube of side L and volume V = L3, one can show that the probability distribution of Λ is approximately given by the Marchenko-Pastur law, if the density of points ρ = N/V obeys ρλ03 ≤ 1 and 2.8N/(k0 L)2 < 1 (see figure).
Non-Hermitian Euclidean random matrices
A theory for the eigenvalue density of large (N≫1) non-Hermitian Euclidean random matrices has been developed and has been applied to study the problem of random laser.
Example 2: Consider the matrix  generated by the function f(r, r′) = exp(ik0|r-r′|)/(k0|r-r′|), with k0 = 2π/λ0 and f(r= r′) = 0. This matrix is not Hermitian and its eigenvalues Λ are complex. The probability distribution of Λ can be found analytically if the density of point ρ = N/V obeys ρλ03 ≤ 1 and 9N/(8k0 R)2 < 1 (see figure).
References
Random matrices
Mathematical physics |
https://en.wikipedia.org/wiki/Smooth%20completion | In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset. Smooth completions exist and are unique over a perfect field.
Examples
An affine form of a hyperelliptic curve may be presented as where and () has distinct roots and has degree at least 5. The Zariski closure of the affine curve in is singular at the unique infinite point added. Nonetheless, the affine curve can be embedded in a unique compact Riemann surface called its smooth completion. The projection of the Riemann surface to is 2-to-1 over the singular point at infinity if has even degree, and 1-to-1 (but ramified) otherwise.
This smooth completion can also be obtained as follows. Project the affine curve to the affine line using the x-coordinate. Embed the affine line into the projective line, then take the normalization of the projective line in the function field of the affine curve.
Applications
A smooth connected curve over an algebraically closed field is called hyperbolic if where g is the genus of the smooth completion and r is the number of added points.
Over an algebraically closed field of characteristic 0, the fundamental group of X is free with generators if r>0.
(Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions O(X) on X is a finitely generated abelian group of rank r -1.
Construction
Suppose the base field is perfect. Any affine curve X is isomorphic to an open subset of an integral projective (hence complete) curve. Taking the normalization (or blowing up the singularities) of the projective curve then gives a smooth completion of X. Their points correspond to the discrete valuations of the function field that are trivial on the base field.
By construction, the smooth completion is a projective curve which contains the given curve as an everywhere dense open subset, and the added new points are smooth. Such a (projective) completion always exists and is unique.
If the base field is not perfect, a smooth completion of a smooth affine curve doesn't always exist. But the above process always produces a regular completion if we start with a regular affine curve (smooth varieties are regular, and the converse is true over perfect fields). A regular completion is unique and, by the valuative criterion of properness, any morphism from the affine curve to a complete algebraic variety extends uniquely to the regular completion.
Generalization
If X is a separated algebraic variety, a theorem of Nagata says that X can be embedded as an open subset of a complete algebraic variety. If X is moreover smooth and the base field has characteristic 0, then by Hironaka's theorem X can even be embedded as an open subset of a complete smooth algebraic variety, with boundary a normal crossing divisor. If X is quasi-projective, the smooth completion can be |
https://en.wikipedia.org/wiki/Hierarchical%20Dirichlet%20process | In statistics and machine learning, the hierarchical Dirichlet process (HDP) is a nonparametric Bayesian approach to clustering grouped data. It uses a Dirichlet process for each group of data, with the Dirichlet processes for all groups sharing a base distribution which is itself drawn from a Dirichlet process. This method allows groups to share statistical strength via sharing of clusters across groups. The base distribution being drawn from a Dirichlet process is important, because draws from a Dirichlet process are atomic probability measures, and the atoms will appear in all group-level Dirichlet processes. Since each atom corresponds to a cluster, clusters are shared across all groups. It was developed by Yee Whye Teh, Michael I. Jordan, Matthew J. Beal and David Blei and published in the Journal of the American Statistical Association in 2006, as a formalization and generalization of the infinite hidden Markov model published in 2002.
Model
This model description is sourced from. The HDP is a model for grouped data. What this means is that the data items come in multiple distinct groups. For example, in a topic model words are organized into documents, with each document formed by a bag (group) of words (data items). Indexing groups by , suppose each group consist of data items .
The HDP is parameterized by a base distribution that governs the a priori distribution over data items, and a number of concentration parameters that govern the a priori number of clusters and amount of sharing across groups. The th group is associated with a random probability measure which has distribution given by a Dirichlet process:
where is the concentration parameter associated with the group, and is the base distribution shared across all groups. In turn, the common base distribution is Dirichlet process distributed:
with concentration parameter and base distribution . Finally, to relate the Dirichlet processes back with the observed data, each data item is associated with a latent parameter :
The first line states that each parameter has a prior distribution given by , while the second line states that each data item has a distribution parameterized by its associated parameter. The resulting model above is called a HDP mixture model, with the HDP referring to the hierarchically linked set of Dirichlet processes, and the mixture model referring to the way the Dirichlet processes are related to the data items.
To understand how the HDP implements a clustering model, and how clusters become shared across groups, recall that draws from a Dirichlet process are atomic probability measures with probability one. This means that the common base distribution has a form which can be written as:
where there are an infinite number of atoms, , assuming that the overall base distribution has infinite support. Each atom is associated with a mass . The masses have to sum to one since is a probability measure. Since is itself the base distr |
https://en.wikipedia.org/wiki/BioNumerics | BioNumerics is a bioinformatics desktop software application that manages microbiological data. It is developed by Applied Maths NV, a bioMérieux company.
History
BioNumerics was first released in 1998. PulseNet, a network run by the Centers for Disease Control and Prevention (CDC), uses BioNumerics to compare pulsed field gel electrophoresis (PFGE) patterns and whole genome sequences from different bacterial strains. CaliciNet, an outbreak surveillance network for noroviruses, is another example of a network which uses BioNumerics to submit norovirus sequences and basic epidemiologic information to a central database.
Features
The basis of BioNumerics is a database consisting of entries. The entries correspond to the individual organisms or samples under study and are characterized by a unique key and by a number of user-defined information fields. Each entry in a database may be characterized by one or more experiments that can be linked easily to the entry. In BioNumerics, experiments are divided in seven classes: fingerprints, spectra, characters, sequences, sequence read sets, trend data and matrices.
Examples of BioNumerics applications are whole genome Multi Locus Sequence Typing (wgMLST), whole genome Single Nucleotide Polymorphisms (wgSNP), genome comparison, identification based on MALDI-TOF Mass Spectrometry, PFGE typing, Amplified Fragment Length Polymorphism (AFLP) typing, sequence-based typing of viruses, antibiotic resistance profiling and functional genotyping.
References
External links
Applied Maths’ official website
Scientific articles using BioNumerics
Bioinformatics software |
https://en.wikipedia.org/wiki/Exponentially%20modified%20Gaussian%20distribution | In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as , where X and Y are independent, X is Gaussian with mean μ and variance σ2, and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.
It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution.
Definition
The probability density function (pdf) of the exponentially modified normal distribution is
where erfc is the complementary error function defined as
This density function is derived via convolution of the normal and exponential probability density functions.
Alternative forms for computation
An alternative but equivalent form of the EMG distribution is used for description of peak shape in chromatography. This is as follows
where
is the amplitude of Gaussian,
is exponent relaxation time, is a variance of exponential probability density function.
This function cannot be calculated for some values of parameters (for example, ) because of arithmetic overflow. Alternative, but equivalent form of writing the function was proposed by Delley:
where is a scaled complementary error function
In the case of this formula arithmetic overflow is also possible, region of overflow is different from the first formula, except for very small τ.
For small τ it is reasonable to use asymptotic form of the second formula:
Decision on formula usage is made on the basis of the parameter :
for z < 0 computation should be made according to the first formula,
for 0 ≤ z ≤ 6.71·107 (in the case of double-precision floating-point format) according to the second formula,
and for z > 6.71·107 according to the third formula.
Mode (position of apex, most probable value) is calculated using derivative of formula 2; the inverse of scaled complementary error function erfcxinv() is used for calculation. Approximate values are also proposed by Kalambet et al. Though the mode is at a value higher than that of the original Gaussian, the apex is always located on the original (unmodified) Gaussian.
Parameter estimation
There are three parameters: the mean of the normal distribution (μ), the standard deviation of the normal distribution (σ) and the exponential decay parameter (τ = 1 / λ). The shape K = τ / σ is also sometimes used to characterise the distribution. Depending on the values of the parameters, the distribution may vary in shape from almost normal to almost exponential.
The parameters of the distribution can be estimated from the sample data with the method of moments as follows:
where m is the sample mean, s is the sample standard deviation, and γ1 is the skewness.
Solving these for the parameters gives:
Recommendations
Ratcliff has suggested that there be at least 100 data points in the sa |
https://en.wikipedia.org/wiki/Eigen%20%28C%2B%2B%20library%29 | Eigen is a high-level C++ library of template headers for linear algebra, matrix and vector operations, geometrical transformations, numerical solvers and related algorithms.
Eigen is open-source software licensed under the Mozilla Public License 2.0 since version 3.1.1. Earlier versions were licensed under the GNU Lesser General Public License. Version 1.0 was released in Dec 2006.
Eigen is implemented using the expression templates metaprogramming technique, meaning it builds expression trees at compile time and generates custom code to evaluate these. Using expression templates and a cost model of floating point operations, the library performs its own loop unrolling and vectorization. Eigen itself can provide BLAS and a subset of LAPACK interfaces.
Release 3.4 (2021) includes many improvements.
See also
List of numerical libraries
Numerical linear algebra
References
C++ libraries
C++ numerical libraries
Free computer libraries
Free software programmed in C++
Numerical analysis software for Linux
Software using the Mozilla license |
https://en.wikipedia.org/wiki/Balanced%20module | In the subfield of abstract algebra known as module theory, a right R module M is called a balanced module (or is said to have the double centralizer property) if every endomorphism of the abelian group M which commutes with all R-endomorphisms of M is given by multiplication by a ring element. Explicitly, for any additive endomorphism f, if fg = gf for every R endomorphism g, then there exists an r in R such that f(x) = xr for all x in M. In the case of non-balanced modules, there will be such an f that is not expressible this way.
In the language of centralizers, a balanced module is one satisfying the conclusion of the double centralizer theorem, that is, the only endomorphisms of the group M commuting with all the R endomorphisms of M are the ones induced by right multiplication by ring elements.
A ring is called balanced if every right R module is balanced. It turns out that being balanced is a left-right symmetric condition on rings, and so there is no need to prefix it with "left" or "right".
The study of balanced modules and rings is an outgrowth of the study of QF-1 rings by C.J. Nesbitt and R. M. Thrall. This study was continued in V. P. Camillo's dissertation, and later it became fully developed. The paper gives a particularly broad view with many examples. In addition to these references, K. Morita and H. Tachikawa have also contributed published and unpublished results. A partial list of authors contributing to the theory of balanced modules and rings can be found in the references.
Examples and properties
Examples
Semisimple rings are balanced.
Every nonzero right ideal over a simple ring is balanced.
Every faithful module over a quasi-Frobenius ring is balanced.
The double centralizer theorem for right Artinian rings states that any simple right R module is balanced.
The paper contains numerous constructions of nonbalanced modules.
It was established in that uniserial rings are balanced. Conversely, a balanced ring which is finitely generated as a module over its center is uniserial.
Among commutative Artinian rings, the balanced rings are exactly the quasi-Frobenius rings.
Properties
Being "balanced" is a categorical property for modules, that is, it is preserved by Morita equivalence. Explicitly, if F(–) is a Morita equivalence from the category of R modules to the category of S modules, and if M is balanced, then F(M) is balanced.
The structure of balanced rings is also completely determined in , and is outlined in .
In view of the last point, the property of being a balanced ring is a Morita invariant property.
The question of which rings have all finitely generated right R modules balanced has already been answered. This condition turns out to be equivalent to the ring R being balanced.
Notes
References
Module theory
Ring theory |
https://en.wikipedia.org/wiki/Last%20geometric%20statement%20of%20Jacobi | In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi, which states:
Every caustic from any point on an ellipsoid other than umbilical points has exactly four cusps.
Numerical experiments had indicated the statement is true before it was proven rigorously in 2004 by Itoh and Kiyohara. It has since been extended to a wider class of surfaces beyond the ellipsoid.
See also
Four-vertex theorem
References
Differential geometry
Algebraic geometry
Conjectures that have been proved |
https://en.wikipedia.org/wiki/Edward%20J.%20McShane | Edward James McShane (May 10, 1904 – June 1, 1989) was an American mathematician noted for his advancements of the calculus of variations, integration theory, stochastic calculus, and exterior ballistics.
His name is associated with the McShane–Whitney extension theorem and McShane integral.
McShane was professor of mathematics at the University of Virginia, president of the American Mathematical Society, president of the Mathematical Association of America, a member of the National Science Board and a member of both the National Academy of Sciences and the American Philosophical Society.
Life and career
McShane was born and raised in New Orleans.
He received his bachelor of engineering and Bachelor of Science degrees from Tulane University in 1925, following with a M.S. degree from Tulane in 1927.
McShane received his Ph.D. in mathematics from the University of Chicago.
He taught at the University of Virginia for 39 years until he retired in 1974. His doctoral students include Victor Klee, Billy James Pettis, and David Lowdenslager, who collaborated with Henry Helson.
McShane died of congestive heart failure at the University of Virginia hospital.
Selected publications
Articles
(over 650 citations)
Books
as translator
References
External links
MAA presidents: Edward James McShane
1904 births
1989 deaths
20th-century American mathematicians
Tulane University alumni
University of Virginia faculty
University of Chicago alumni
Members of the United States National Academy of Sciences
Presidents of the Mathematical Association of America
Presidents of the American Mathematical Society
Members of the American Philosophical Society |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Hertha%20BSC%20season | The 1998–99 Hertha BSC season started on 16 August 1998 against Werder Bremen and ended on 29 May 1999.
Review and events
Results
Legend
Bundesliga
DFB-Pokal
Roster and statistics
Sources:
|}
References
Match reports
Other sources
External links
Hertha BSC
Hertha BSC seasons |
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