source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Petrie%20polygon | In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides (but no three) belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.
For every regular polytope there exists an orthogonal projection onto a plane such that one Petrie polygon becomes a regular polygon with the remainder of the projection interior to it. The plane in question is the Coxeter plane of the symmetry group of the polygon, and the number of sides, , is the Coxeter number of the Coxeter group. These polygons and projected graphs are useful in visualizing symmetric structure of the higher-dimensional regular polytopes.
Petrie polygons can be defined more generally for any embedded graph. They form the faces of another embedding of the same graph, usually on a different surface, called the Petrie dual.
History
John Flinders Petrie (1907–1972) was the son of Egyptologists Hilda and Flinders Petrie. He was born in 1907 and as a schoolboy showed remarkable promise of mathematical ability. In periods of intense concentration he could answer questions about complicated four-dimensional objects by visualizing them.
He first noted the importance of the regular skew polygons which appear on the surface of regular polyhedra and higher polytopes. Coxeter explained in 1937 how he and Petrie began to expand the classical subject of regular polyhedra:
One day in 1926, J. F. Petrie told me with much excitement that he had discovered two new regular polyhedral; infinite but free of false vertices. When my incredulity had begun to subside, he described them to me: one consisting of squares, six at each vertex, and one consisting of hexagons, four at each vertex.
In 1938 Petrie collaborated with Coxeter, Patrick du Val, and H. T. Flather to produce The Fifty-Nine Icosahedra for publication.
Realizing the geometric facility of the skew polygons used by Petrie, Coxeter named them after his friend when he wrote Regular Polytopes.
The idea of Petrie polygons was later extended to semiregular polytopes.
The Petrie polygons of the regular polyhedra
The regular duals, {p,q} and {q,p}, are contained within the same projected Petrie polygon.
In the images of dual compounds on the right it can be seen that their Petrie polygons have rectangular intersections in the points where the edges touch the common midsphere.
The Petrie polygons of the Kepler–Poinsot polyhedra are hexagons {6} and decagrams {10/3}.
Infinite regular skew polygons (apeirogon) can also be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, and 60 degrees of their square, hexagon and triangular faces respectively.
Infinite regular skew polygons also exist as Petrie polygons of the regular hyperboli |
https://en.wikipedia.org/wiki/Soar%20Valley%20College | Soar Valley College is an 11-16 coeducational secondary school located in Leicester, Leicestershire, England.
It was designated a Maths and Computing specialist college in September 2004.
As part of the Building Schools for the Future initiative, the old building was demolished in 2009 and turned into playing fields, and a new building was built on the previous playing fields at a cost of approximately £21.5 million. The school hosted the Special Olympics netball games in 2009 on their newly built netball courts, known as the "Soar Valley Netball Centre" on the other side of the campus from the new building.
It became co-educational in 2016 and has one acre of land. The school has a lanyard system with different colours to show the different years.
Previously a community school administered by Leicester City Council, in June 2023 Soar Valley College converted to academy status. The school is now sponsored by the Aspire Learning Partnership.
Notable former pupils
Parminder Nagra, actress
Rakhee Thakrar, actress
References
External links
Secondary schools in Leicester
Academies in Leicester |
https://en.wikipedia.org/wiki/Ruhollah%20Bigdeli | Ruhollah Bigdeli (, born 21 March 1984) is an Iranian football striker of Mes Sarcheshmeh.
Club career
Club career statistics
Assist Goals
External links
1984 births
Living people
Iranian men's footballers
Persian Gulf Pro League players
Esteghlal Ahvaz F.C. players
Sanat Mes Kerman F.C. players
Foolad F.C. players
Men's association football forwards
Siah Jamegan F.C. players
Mes Sarcheshme players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Ali%20Azari%20Karki | Ali Azari (, born 2 June 1985) is an Iranian football left midfielder of Damash Iranian F.C. He was a member of Iran national under-23 football team.
Club career
Club career statistics
Assist Goals
External links
Profile at Iranproleague.net
1985 births
Living people
Iranian men's footballers
Steel Azin F.C. players
Gahar Zagros F.C. players
Men's association football forwards
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Covariance%20%28disambiguation%29 | In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
Covariance matrix, a matrix of covariances between a number of variables
Covariance or cross-covariance between two random variables or data sets
Autocovariance, the covariance of a signal with a time-shifted version of itself
Covariance function, a function giving the covariance of a random field with itself at two locations
Algebra and geometry
A covariant (invariant theory) is a bihomogeneous polynomial in and the coefficients of some homogeneous form in that is invariant under some group of linear transformations.
Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis
Covariant transformation, a rule that describes how certain physical entities change under a change of coordinate system
Covariance and contravariance of functors, properties of functors
General covariance or simply covariance (inaccurate but common usage in quantum mechanics), the invariance of the mathematical form of physical laws under arbitrary differential coordinate transformations (including Lorentz transformations), strictly meaning invariance
Lorentz covariance, a property of space-time that follows from the special theory of relativity
Poincaré covariance, a related property
Eddy covariance, an atmospheric flux measurement technique
in category theory, covariant functor, the same as a functor (the term is used to be in contrast to the term contravariant functor).
Computer science
Covariance and contravariance (computer science), a system of class relations in computer science
See also
Covariance and correlation
Covariance and contravariance (disambiguation) |
https://en.wikipedia.org/wiki/Mark%20J.%20Ablowitz | Mark Jay Ablowitz (born June 5, 1945, New York) is a professor in the department of Applied Mathematics at the University of Colorado at Boulder, Colorado. He was born in New York City.
Education
Ablowitz received his Bachelor of Science degree in mechanical engineering from University of Rochester, and completed his Ph.D. in Mathematics under the supervision of David Benney at Massachusetts Institute of Technology in 1971.
Career and research
Ablowitz was an assistant professor of Mathematics at Clarkson University during 1971–1975 and an associate professor during 1975–1976. He visited the Program in Applied Mathematics founded by Ahmed Cemal Eringen at Princeton University during 1977–1978. He was a professor of Mathematics at Clarkson during 1976-1985 where he became the Chairman of the Department of Mathematics and Computer Science in 1979. On July 1, 1985, he was appointed as the Dean of Science of Clarkson University and served there until he joined to the department of Applied Mathematics (APPM) at University of Colorado Boulder on June 30, 1989.
Awards and honors
Sloan Fellowship, 1975–1977.
Clarkson Graham Research Award, 1976.
John Simon Guggenheim Foundation Fellowship, 1984.
SIAM Fellow, 2011.
National Academy of Sciences Symposium on Soliton Theory Kiev, USSR 1979.
Fellow of the American Mathematical Society, 2012.
Publications
Solitons and the Inverse Scattering Transform, M.J. Ablowitz and H. Segur, (SIAM Studies in Applied Mathematics) 1981
Topics in Soliton Theory and Exactly Solvable Nonlinear Equations, Eds. M.J. Ablowitz, B. Fuchssteiner and M. D. Kruskal, (World Scientific) 1987
Solitons, Nonlinear Evolution Equations and Inverse Scattering, M.J. Ablowitz and P.A. Clarkson, (London Mathematical Society Lecture Notes Series, 516 pages, (Cambridge University Press, Cambridge, UK, 1991)
Complex Variables: Introduction and Applications, Mark J. Ablowitz and A. S. Fokas, (Cambridge University Press, Cambridge, UK, 1997)
Nonlinear Physics: Theory and Experiment. II, M.J. Ablowitz, M. Boiti, F. Pempinelli and B. Prinari, (World Scientific 2003)
Discrete and Continuous Nonlinear Schrödinger Systems, Mark J. Ablowitz, B. Prinari and D. Trubatch, 258 (Cambridge University Press, Cambridge, UK, 2004)
Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Mark J. Ablowitz, (Cambridge University Press, Cambridge, UK, 2011)
References
Living people
University of Rochester alumni
Massachusetts Institute of Technology School of Science alumni
Clarkson University faculty
University of Colorado Boulder faculty
20th-century American mathematicians
21st-century American mathematicians
1945 births
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics |
https://en.wikipedia.org/wiki/Samuel%20Oppenheim | Samuel Oppenheim (19 November 1857 in Braunsberg – 15 August 1928 in Vienna) was an Austrian astronomer.
In 1875 Oppenheim began to study mathematics, physics and astronomy in Vienna. He took his Staatsexamen in 1880. From 1881–1887 he worked at the Observatory of Vienna and from 1888–1896 at the Kuffner observatory in Vienna. He attained the Doctorate in 1884 and the Habilitation in 1910 for theoretical astronomy. After working as a teacher in Prague, he was Professor ordinarius for astronomy at the University of Vienna.
Oppenheim's field of research was mainly in the field of celestial mechanics (for example he wrote works on comets, Gravitation, Precession, Kinematics and statistics of stars etc.). He was co-editor of the Astronomy section of Enzyklopädie der mathematischen Wissenschaften.
References
Publications
1857 births
1928 deaths
19th-century Austrian astronomers
20th-century Austrian astronomers
Astronomers from Austria-Hungary |
https://en.wikipedia.org/wiki/Conductor%20%28ring%20theory%29 | In ring theory, a branch of mathematics, the conductor is a measurement of how far apart a commutative ring and an extension ring are. Most often, the larger ring is a domain integrally closed in its field of fractions, and then the conductor measures the failure of the smaller ring to be integrally closed.
The conductor is of great importance in the study of non-maximal orders in the ring of integers of an algebraic number field. One interpretation of the conductor is that it measures the failure of unique factorization into prime ideals.
Definition
Let A and B be commutative rings, and assume . The conductor of A in B is the ideal
Here is viewed as a quotient of A-modules, and denotes the annihilator. More concretely, the conductor is the set
Because the conductor is defined as an annihilator, it is an ideal of A.
If B is an integral domain, then the conductor may be rewritten as
where is considered as a subset of the fraction field of B. That is, if a is non-zero and in the conductor, then every element of B may be written as a fraction whose numerator is in A and whose denominator is a. Therefore the non-zero elements of the conductor are those that suffice as common denominators when writing elements of B as quotients of elements of A.
Suppose R is a ring containing B. For example, R might equal B, or B might be a domain and R its field of fractions. Then, because , the conductor is also equal to
Elementary properties
The conductor is the whole ring A if and only if it contains and, therefore, if and only if . Otherwise, the conductor is a proper ideal of A.
If the index is finite, then , so . In this case, the conductor is non-zero. This applies in particular when B is the ring of integers in an algebraic number field and A is an order (a subring for which is finite).
The conductor is also an ideal of B, because, for any in and any in , . In fact, an ideal J of B is contained in A if and only if J is contained in the conductor. Indeed, for such a J, , so by definition J is contained in . Conversely, the conductor is an ideal of A, so any ideal contained in it is contained in A. This fact implies that is the largest ideal of A which is also an ideal of B. (It can happen that there are ideals of A contained in the conductor which are not ideals of B.)
Suppose that S is a multiplicative subset of A. Then
with equality in the case that B is a finitely generated A-module.
Conductors of Dedekind domains
Some of the most important applications of the conductor arise when B is a Dedekind domain and is finite. For example, B can be the ring of integers of a number field and A a non-maximal order. Or, B can be the affine coordinate ring of a smooth projective curve over a finite field and A the affine coordinate ring of a singular model. The ring A does not have unique factorization into prime ideals, and the failure of unique factorization is measured by the conductor .
Ideals coprime to the conductor |
https://en.wikipedia.org/wiki/United%20States%20Census%20of%20Agriculture | The Census of Agriculture is a census conducted every five years by the U.S. Department of Agriculture’s National Agricultural Statistics Service (NASS) that provides the only source of uniform, comprehensive agricultural data for every county in the United States.
Overview
The census is a complete count of U.S. farms and ranches and the people who operate them. The Census looks at land use and ownership, operator characteristics, production practices, income and expenditures and many other areas. This picture, when compared to earlier censuses, helps to measure trends and new developments in the agricultural sector of the nation’s economy.
Title 7 of the United States Code requires all those who receive a census report form to respond – even if they did not operate a farm or ranch during the census year. The same law protects the confidentiality of all census respondents. NASS uses the information only for statistical purposes and publishes data only in tabulated totals. The report cannot be used for purposes of taxation, investigation or regulation. The privacy of individual census records is also protected from disclosure through the Freedom of Information Act.
For census purposes, a farm is defined as a place from which $1,000 or more of agricultural products were produced and sold, or normally would have been sold, during the census year. This farm definition has changed nine times throughout history and the current definition has been in effect since 1974.
History
The first Census of Agriculture was taken in 1840 as part of the sixth decennial population census. The census remained a part of the decennial census through 1950, with separate mid-decade Censuses of Agriculture taken in 1925, 1935 and 1945. As time passed, census years were adjusted until the reference year coincided with the economic censuses covering other sectors of the nation’s economy. Currently, the Census of Agriculture is conducted for years ending in 2 and 7.
The 1997 Census of Agriculture has historical significance because it was the first conducted by NASS after the 1997 Appropriations Act shifted responsibility of the Census of Agriculture from the U.S. Census Bureau, which is part of the U.S. Department of Commerce, to USDA.
Uses of census data
Census data is used by all those who serve farmers and rural communities – federal, state and local governments, agribusinesses, trade associations and many others. For instance:
Companies and cooperatives use the information to determine the locations of facilities that will serve agricultural producers.
Community planners use the information to target needed services to rural residents.
USDA uses the information to ensure that local service centers are staffed at appropriate levels.
Legislators use the information when shaping farm policies and programs.
Farmers and ranchers use Census data to help make informed decisions about the future of their own operations.
External links
Census of Agriculture — official webs |
https://en.wikipedia.org/wiki/Selberg%20integral | In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
Selberg's integral formula
When , we have
Selberg's formula implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto.
Aomoto's integral formula
Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,
A proof is found in Chapter 8 of .
Mehta's integral
When ,
It is a corollary of Selberg, by setting , and change of variables with , then taking .
This was conjectured by , who were unaware of Selberg's earlier work.
It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
Macdonald's integral
conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.
The product is over the roots r of the roots system and the numbers dj are the degrees of the generators of the ring of invariants of the reflection group.
gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.
References
Special functions |
https://en.wikipedia.org/wiki/Peter%20D.%20Jarvis | Peter D. Jarvis is an Australian physicist notable for his work on applications of group theory to physical problems, particularly supersymmetry in the genetic code. He has also applied classical invariant theory to problems of quantum physics (entanglement measures for mixed state systems), and also to phylogenetic reconstruction (entanglement measures, including distance measures, for taxonomic pattern frequencies).
Education
Jarvis obtained his BSc and MSc from the University of Adelaide. He also has a PhD from Imperial College, London, where he studied under Robert Delbourgo, for a thesis entitled Noise Voltages Produced by Flux Motion in Superconductors.
Career
Jarvis works at the School of Mathematics and Physics, at the University of Tasmania. His main focus is on algebraic structures in mathematical physics and their applications, especially combinatorial Hopf algebras in integrable systems and quantum field theory.
See also
Quantum Aspects of Life
Notes
External links
Jarvis' math genealogy
homepage
Living people
People from Tasmania
Alumni of Imperial College London
Australian physicists
Academic staff of the University of Tasmania
Quantum physicists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Partial%20residual%20plot | In applied statistics, a partial residual plot is a graphical technique that attempts to show the relationship between a given independent variable and the response variable given that other independent variables are also in the model.
Background
When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated. Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model.
Definition
Partial residual plots are formed as
where
Residuals = residuals from the full model,
= regression coefficient from the i-th independent variable in the full model,
Xi = the i-th independent variable.
Partial residual plots are widely discussed in the regression diagnostics literature (e.g., see the References section below). Although they can often be useful, they can also fail to indicate the proper relationship. In particular, if Xi is highly correlated with any of the other independent variables, the variance indicated by the partial residual plot can be much less than the actual variance. These issues are discussed in more detail in the references given below.
CCPR plot
The CCPR (component and component-plus-residual) plot is a refinement of the partial residual plot, adding
This is the "component" part of the plot and is intended to show where the "fitted line" would lie.
See also
Partial regression plot
Partial leverage plot
Variance inflation factors for a multi-linear fit.
References
External links
Partial Residual Plot
Statistical charts and diagrams
Regression diagnostics |
https://en.wikipedia.org/wiki/Plane | Plane(s) most often refers to:
Aero- or airplane, a powered, fixed-wing aircraft
Plane (geometry), a flat, 2-dimensional surface
Plane (mathematics), generalizations of a geometrical plane
Plane or planes may also refer to:
Biology
Plane (tree) or Platanus, wetland native plant
Planes (genus), marsh crabs in Grapsidae
Bindahara phocides, the plane butterfly of Asia
Maritime transport
Planing (boat), where weight is predominantly supported by hydrodynamic lift
Plane (wherry), a Norfolk canal boat, in use 1931–1949
Music
"Planes", a 1976 song by Colin Blunstone
"Planes (Experimental Aircraft)", a 1989 song by Jefferson Airplane from Jefferson Airplane
"Planez", originally "Planes", a 2015 song by Jeremih
"The Plane", a 1987 song on the Empire of the Sun soundtrack
"The Plane", a 1997 song by Kinito Méndez
Other entertainment
Plane (Dungeons & Dragons), any fictional realm of the D&D roleplaying game's multiverse
Planes (film), a 2013 animation
Planes: Fire & Rescue, a 2014 sequel
Plane (film), a 2023 film
Plane (Magic: The Gathering), any fictional realm of the multiverse the card game is set in
Places
Plane (Han Pijesak), Republika Srpska, Bosnia and Herzegovina
Plane, Tuzla, Bosnia and Herzegovina
Plane (river), eastern Germany
Plane Island, off Cape Farina, Tunisia
Pláně, Czech Republic
Planès, France
Planes, Alicante, Spain
Religion
Plane (esotericism), a state, level or region of reality
Planē (mythology), an ancient Greek goddess
Technology
Plane (tool), a tool for shaping wood
Plane (Unicode), in the Universal Coded Character Set, a continuous group of 216 code points
Plane, part of a telecommunications network structure
See also
Plain (disambiguation)
Planar (disambiguation)
Plano (disambiguation)
Plane sailing, an approximate method of navigation |
https://en.wikipedia.org/wiki/List%20of%20Oldham%20Athletic%20A.F.C.%20records%20and%20statistics | This article is a list of statistics and records relating to Oldham Athletic Football Club. Oldham Athletic are an English football club based on Oldham. The club was founded in 1895 as Pine Villa Football Club before renaming in 1899. The club joined the Football League in 1907 though did not win a league title until 1952. Oldham Athletic currently play in Football League two, the third tier of English football. The club plays their home games at Boundary Park, having moved from Sheepfoot Lane in 1905.
This list encompasses all honours won by Oldham Athletic and any club records, by their managers and players.
All statistics are correct as of the end of Oldham Athletic's 2016-17 season.
Honours
League
Football League Second Division: (1)
1990–91
Football League Third Division: (1)
1973–74
Football League Third Division North: (1)
1952–53
Ford Sporting League: (1)
1970–71
Cup
Lancashire Senior Cup: (3)
1907–08, 1966–67, 2005–06
Managerial records
First manager: David Ashworth
Longest serving manager by time: Jimmy Frizzell – 12 years, 102 days between 1970 and 1982
Longest serving manager by games: Joe Royle – 602 competitive games as manager
Most victories: 225 victories by Joe Royle, 1982–1994
Players
League Appearances
525, Ian Wood between 1966 and 1980
Goals
Season: 33, Tom Davis, Second Division, 1936–37
League Career: 141, Roger Palmer, between 1980 and 1994
Overall Career: 156, Roger Palmer, between 1980 and 1994
International Caps
25, Gunnar Halle for Norway, between 1991 and 1996
Transfer fees
Received: £1,700,000 for Earl Barrett, Aston Villa (February 1992)
Paid: £750,000 for Ian Olney, Aston Villa (June 1992)
Club records
Victories
Record win: 11–0 v Southport, Fourth Division, 26 December 1962
Record league win: 11–0 v Southport, Fourth Division, 26 December 1962
Record FA Cup win: 10–1 v Lytham, first round, 28 November 1925
Record League Cup win: 7–0 v Scarborough, 25 October 1989
Losses
Record league loss: 9
0–9 v Hull City, Third Division North, 5 April 1958
4–13 v Tranmere Rovers, Third Division North, 26 December 1935
Attendance
47,671 v Sheffield Wednesday, FA Cup Fourth Round, 25 January 1930
League Attendance
45,120 v Blackpool, Second Division, 21 April 1930
Gate receipts
£138,680 v Manchester United, Premier League, 29 December 1993
Points
Most points in a season:
Two points for a win: 61 in 42 matches, Third Division, 1973–74
Three points for a win: 88 in 46 matches, Second Division, 1990–91
Streaks
Longest winning run: 10 (12 January 1974 – 16 March 1974)
Longest losing run: 6 (5 February 2005 – 26 February 2005)
Longest unbeaten run: 17 (25 August 1990 – 10 November 1990)
Longest run without a win: 17 (9 September 1920 – 25 December 1920)
Season-by-season performances
External links
Facts from OldhamAthletic.co.uk
Oldham Athletic all time records from Soccerbase
Oldham Athletic
Records And Statistics |
https://en.wikipedia.org/wiki/1972%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1972 season.
Overview
Sixteen (16) teams competed for the championships, and Zarya Voroshilovgrad won the championship.
League standings
Results
Top scorers
14 goals
Oleg Blokhin (Dynamo Kyiv)
13 goals
Oganes Zanazanyan (Ararat)
12 goals
Gennadi Khromchenkov (Zenit)
Yuri Smirnov (Torpedo Moscow)
11 goals
Viktor Kolotov (Dynamo Kyiv)
10 goals
Anatoliy Banishevskiy (Neftchi)
Vladimir Onischenko (Zorya)
Aleksei Yeskov (SKA Rostov-on-Don)
9 goals
Arkady Andreasyan (Ararat)
Vladimir Muntyan (Dynamo Kyiv)
Pavel Sadyrin (Zenit)
Anatoli Vasilyev (Dinamo Minsk)
Yuri Yeliseyev (Zorya)
References
Soviet Union - List of final tables (RSSSF)
1969
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1973%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1973 season.
Overview
It was contested by 16 teams, and Ararat Yerevan won the championship.
League standings
Results
Results in brackets indicate the results from penalty shoot-outs whenever games were drawn.
Top scorers
18 goals
Oleg Blokhin (Dynamo Kyiv)
16 goals
Anatoli Kozhemyakin (Dynamo Moscow)
13 goals
Arkady Andreasyan (Ararat)
12 goals
Berador Abduraimov (Pakhtakor)
Aleksandr Piskaryov (Spartak Moscow)
11 goals
Mikhail Bulgakov (Spartak Moscow)
Givi Nodia (Dinamo Tbilisi)
10 goals
Eduard Markarov (Ararat)
Vitali Starukhin (Shakhtar)
9 goals
Vladimir Dorofeyev (CSKA)
Viktor Kuznetsov (Zorya)
References
Soviet Union - List of final tables (RSSSF)
1969
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1974%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1974 season.
Overview
It was contested by 16 teams, and Dynamo Kyiv won the championship.
League standings
Results
Top scorers
20 goals
Oleg Blokhin (Dynamo Kyiv)
16 goals
Anatoli Ionkin (Kairat)
Vadim Pavlenko (Dynamo Moscow)
13 goals
Vladimir Makarov (Chornomorets Odessa)
12 goals
Vadim Nikonov (Torpedo Moscow)
11 goals
Mikhail An (Pakhtakor)
Anatoly Baidachny (Dynamo Moscow)
Vladimir Onischenko (Dynamo Kyiv)
Vitali Starukhin (Shakhtar)
10 goals
Vladimir Danilyuk (Karpaty)
Aleksandr Piskaryov (Spartak Moscow)
References
Soviet Union - List of final tables (RSSSF)
1969
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1977%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1977 season.
Overview
It was contested by 16 teams, and Dynamo Kyiv won the championship.
League standings
Results
Top scorers
17 goals
Oleg Blokhin (Dynamo Kyiv)
14 goals
David Kipiani (Dinamo Tbilisi)
12 goals
Yuri Chesnokov (CSKA Moscow)
10 goals
Andrei Yakubik (Dynamo Moscow)
9 goals
Yuri Reznik (Shakhtar)
Nikolai Smolnikov (Neftchi)
Vitali Starukhin (Shakhtar)
8 goals
Revaz Chelebadze (Dinamo Tbilisi)
Vladimir Klementyev (Zenit)
7 goals
Yuri Dubrovny (Karpaty)
Vladimir Kazachyonok (Dynamo Moscow)
Boris Kopeikin (CSKA Moscow)
Khoren Hovhannisyan (Ararat)
Vyacheslav Semyonov (Zorya Voroshylovhrad)
References
Soviet Union - List of final tables (RSSSF)
1969
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1978%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1978 season.
Overview
It was contested by 16 teams, and Dinamo Tbilisi won the championship.
Introduction of draw limit, a number of games tied during a season.
League standings
Results
Top scorers
19 goals
Georgi Yartsev (Spartak Moscow)
15 goals
Ramaz Shengelia (Dinamo Tbilisi)
13 goals
Oleg Blokhin (Dynamo Kyiv)
11 goals
Nikolai Latysh (Shakhtar)
10 goals
Vladimir Klementyev (Zenit)
9 goals
Nikolai Kolesov (Dynamo Moscow)
8 goals
Aleksei Belenkov (CSKA Moscow)
Vakhtang Koridze (Dinamo Tbilisi)
Viktor Kuznetsov (Zorya Voroshylovhrad)
7 goals
Konstantin Bakanov (Pakhtakor)
Anatoliy Banishevskiy (Neftchi)
Yuri Chesnokov (CSKA Moscow)
Vladimir Fyodorov (Pakhtakor)
Yevgeni Khrabrostin (Torpedo Moscow)
David Kipiani (Dinamo Tbilisi)
Aleksandr Maksimenkov (Dynamo Moscow)
Vladimir Onischenko (Dynamo Kyiv)
Vladimir Ploskina (Chornomorets)
Andrei Redkous (Zenit)
Yuri Reznik (Shakhtar)
Vitaliy Shevchenko (Chornomorets)
Vitali Starukhin (Shakhtar)
References
Soviet Union - List of final tables (RSSSF)
1969
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1979%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1979 season.
Overview
1979 Top League was composed of 18 teams, the championship was won by Spartak Moscow.
On 11 August 1979, a mid-air collision killed virtually the entire FC Pakhtakor Tashkent team. The Top League ordered all the other teams to make three players available for a draft to restock the team, and Pakhtakor was given exemption from relegation for the next three years.
League standings
Results
Top scorers
26 goals
Vitali Starukhin (Shakhtar)
17 goals
Sergey Andreyev (SKA Rostov-on-Don)
Oleg Blokhin (Dynamo Kyiv)
Khoren Hovhannisyan (Ararat)
Valeriy Petrakov (Lokomotiv Moscow)
16 goals
Yuri Chesnokov (CSKA Moscow)
Vladimir Kazachyonok (Zenit)
14 goals
Aleksandr Prokopenko (Dinamo Minsk)
Georgi Yartsev (Spartak Moscow)
Nikolai Vasilyev (Torpedo Moscow)
References
Soviet Union - List of final tables (RSSSF)
Soviet Top League seasons
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1980%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1980 season.
Overview
It was contested by 18 teams, and Dynamo Kyiv won the championship.
League standings
Results
Top scorers
20 goals
Sergey Andreyev (SKA Rostov-on-Don)
19 goals
Oleg Blokhin (Dynamo Kyiv)
17 goals
Ramaz Shengelia (Dinamo Tbilisi)
14 goals
Yuriy Horyachev (Chornomorets)
Aleksandr Tarkhanov (CSKA Moscow)
12 goals
Vladimir Kazachyonok (Zenit)
Valeriy Petrakov (Lokomotiv Moscow)
11 goals
Revaz Chelebadze (Dinamo Tbilisi)
10 goals
Andranik Khachatryan (Ararat)
Pyotr Vasilevsky (Dinamo Minsk)
References
Soviet Union - List of final tables (RSSSF)
Soviet Top League seasons
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1981%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1981 season.
Overview
It was contested by 18 teams, and Dynamo Kyiv won the championship.
League standings
Results
Top scorers
23 goals
Ramaz Shengelia (Dinamo Tbilisi)
21 goals
Yuri Gavrilov (Spartak Moscow)
19 goals
Oleg Blokhin (Dynamo Kyiv)
16 goals
Vladimir Kazachyonok (Zenit)
15 goals
Valery Gazzaev (Dynamo Moscow)
14 goals
Khoren Hovhannisyan (Ararat)
Pyotr Vasilevsky (Dinamo Minsk)
13 goals
Andrei Yakubik (Pakhtakor)
12 goals
Viktor Grachyov (Shakhtar)
Aleksandr Pogorelov (Dnipro)
References
Soviet Union - List of final tables (RSSSF)
1981. Higher League. (1981. Высшая лига.) Luhansk Our Futbol portal.
44th USSR Championship, 1981 Higher League (44-й чемпионат СССР, 1981г. Высшая лига). Wildstat website.
Soviet Top League seasons
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1982%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1982 season.
Overview
It was contested by 18 teams, and Dinamo Minsk won the championship.
The rules stated that a team could only have a maximum of 10 draws; all points from matches draw since the eleventh wouldn't be counted.
League standings
Results
Top scorers
23 goals
Andrei Yakubik (Pakhtakor)
18 goals
Merab Megreladze (Torpedo Kutaisi)
16 goals
Ramaz Shengelia (Dinamo Tbilisi)
Aleksandr Tarkhanov (CSKA)
13 goals
Igor Gurinovich (Dinamo Minsk)
12 goals
Boris Chukhlov (Zenit)
Valery Gazzaev (Dynamo Moscow)
Khoren Hovhannisyan (Ararat)
Andrei Redkous (Torpedo Moscow)
Mykhaylo Sokolovsky (Shakhtar)
Medal squads
(league appearances and goals listed in brackets)
Number of teams by union republic
References
Soviet Union - List of final tables (RSSSF)
Soviet Top League seasons
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1983%20Soviet%20Top%20League | Statistics of Soviet Top League for the 1983 season.
Teams
Promoted teams
Zhalgiris Vilnius – champion (returning after 21 seasons)
Nistru Kishinev – 2nd place (returning after nine seasons)
Location
League standings
Results
Top scorers
18 goals
Yuri Gavrilov (Spartak Moscow)
17 goals
Igor Gurinovich (Dinamo Minsk)
15 goals
Volodymyr Fink (Chornomorets)
Khoren Hovhannisyan (Ararat)
Mykhaylo Sokolovsky (Shakhtar)
Andrei Yakubik (Pakhtakor)
14 goals
Sigitas Jakubauskas (Žalgiris)
13 goals
Viktor Kolyadko (CSKA Moscow)
Oleh Taran (Dnipro)
11 goals
Valery Gazzaev (Dynamo Moscow)
Valeriy Petrakov (Torpedo Moscow)
Igor Ponomaryov (Neftchi)
Ramaz Shengelia (Dinamo Tbilisi)
Aleksandr Tarkhanov (CSKA Moscow)
Vadym Yevtushenko (Dynamo Kyiv)
Medal squads
(league appearances and goals listed in brackets)
Number of teams by union republic
References
Soviet Union - List of final tables (RSSSF)
Soviet Top League seasons
1
Soviet
Soviet |
https://en.wikipedia.org/wiki/1992%20Russian%20Top%20League | Statistics of Russian Top League in season 1992.
Overview
Twenty clubs of the former Soviet competition took place in this season. The league was combined out of six clubs of the Soviet Top League, 11 - Soviet First League, and the rest out of the promoted from the Buffer League (Center and East). FC Spartak Moscow won the championship.
The composition of groups may seem kind of uneven with four Top League clubs in Group A and two — in Group B. However the seeding was done upon the completion of the previous Soviet season with Rotor being conditionally promoted to the top level.
First stage
Group A
Table
Results
Group B
Table
Results
Final stage
The results of games played in the first stage were counted in the final stage.
By political agreement with UEFA and Ukraine, Russia inherited the access right of Soviet Union to the European competitions, while Ukraine obtained part of the rights of disbanded East Germany.
Championship Round
Tournament for places 1 to 8
Table
Results
Relegation Round
Tournament for places 9 to 20
Table
Results
Top scorers
Gasimov was the official top scorer as Matveyev and Garin did not play in the Championship Round.
20 goals
Yuri Matveyev (Uralmash)
16 goals
Oleg Garin (Okean)
Vali Gasimov (Dynamo Moscow)
13 goals
Vladimir Kulik (Zenit)
Kirill Rybakov (Asmaral)
12 goals
Dmitri Radchenko (Spartak Moscow)
Nazim Suleymanov (Spartak Vladikavkaz)
10 goals
Rustyam Fakhrutdinov (Krylya Sovetov)
Aleksandr Grishin (CSKA Moscow)
Gennadi Grishin (Torpedo Moscow)
Igor Lediakhov (Spartak Moscow)
Oleg Veretennikov (Rotor)
Medal squads
(league appearances and goals listed in brackets)
See also
1992 Russian First League
1992 Russian Second League
References
Russia - List of final tables (RSSSF)
Russian Premier League seasons
1
Russia
Russia |
https://en.wikipedia.org/wiki/1993%20Russian%20Top%20League | Statistics of Russian Top League in season 1993.
Teams
18 teams are played in the 1993 season. After the 1992 season, Zenit St.Petersburg, Fakel Voronezh, Kuban Krasnodar, Shinnik Yaroslavl and Dinamo-Gazovik were relegated to the 1993 Russian First League. They were replaced by Zhemchuzhina-Sochi, winners of the 1992 Russian First League.
Venues
Personnel and kits
Managerial changes
League standings
Results
Promotion tournament
FC Rostselmash and FC Asmaral were relegated. FC Krylia Sovetov, FC Luch and FC Okean played in a promotion tournament against the winners of the three zones of the 1993 Russian First League, in which three spots in 1994 Russian Top League were contested. FC Krylia Sovetov kept their spot and FC Luch and FC Okean were relegated.
Top scorers
21 goals
Victor Panchenko (KAMAZ)
19 goals
Oleg Veretennikov (Rotor)
18 goals
Vladimir Beschastnykh (Spartak Moscow)
16 goals
Igor Simutenkov (Dynamo Moscow)
14 goals
Mikhail Markhel (Spartak Vladikavkaz)
Nikolai Pisarev (Spartak Moscow)
Nazim Suleymanov (Spartak Vladikavkaz)
13 goals
Gocha Gogrichiani (Zhemchuzhina)
Valeri Karpin (Spartak Moscow)
/Vladimir Niederhaus (Rotor)
Medal squads
(league appearances and goals listed in brackets)
References
Russia - List of final tables (RSSSF)
Russian Premier League seasons
1
Russia
Russia |
https://en.wikipedia.org/wiki/1994%20Russian%20Top%20League | Statistics of Russian Top League in the 1994 season.
Overview
16 teams participated, and FC Spartak Moscow won the championship.
League standings
Results
Top scorers
21 goals
Igor Simutenkov (Dynamo Moscow)
20 goals
Oleg Garin (Lokomotiv Moscow)
12 goals
Oleg Veretennikov (Rotor)
10 goals
Vladimir Beschastnykh (Spartak Moscow)
9 goals
Vladimir Filimonov (Zhemchuzhina)
Yuri Matveyev (Uralmash)
/ Vladimir Niederhaus (Rotor)
Andrei Tikhonov (Spartak Moscow)
8 goals
Andrei Afanasyev (Torpedo Moscow)
Timur Bogatyryov (Zhemchuzhina)
Dmitri Cheryshev (Dynamo Moscow)
Yuri Kalitvintsev (Lokomotiv Nizhny Novgorod)
Aleksandr Smirnov (Dynamo Moscow)
Medal squads
References
Russia - List of final tables (RSSSF)
Russian Premier League seasons
1
Russia
Russia |
https://en.wikipedia.org/wiki/1995%20Russian%20Top%20League | Statistics of Russian Top League in season 1995.
Overview
16 teams participated, and Spartak-Alania Vladikavkaz won the championship.
League standings
Results
Top scorers
25 goals
Oleg Veretennikov (Rotor)
18 goals
Aleksandr Maslov (Rostselmash)
16 goals
Valeri Shmarov (Spartak Moscow)
14 goals
/Vladimir Niederhaus (Rotor)
13 goals
Oleg Garin (Lokomotiv Moscow)
12 goals
Mikhail Kavelashvili (Spartak-Alania)
11 goals
Yevgeni Kharlachyov (Lokomotiv Moscow)
Oleg Teryokhin (Dynamo Moscow)
10 goals
Garnik Avalyan (Krylya Sovetov)
Timur Bogatyryov (Zhemchuzhina)
Dmitri Karsakov (CSKA Moscow)
Sergei Natalushko (Tekstilshchik)
Mirjalol Qosimov (Spartak-Alania)
Bakhva Tedeyev (Spartak-Alania)
Medal squads
See also
1995 Russian First League
1995 Russian Second League
1995 Russian Third League
References
Russia - List of final tables (RSSSF)
Russian Premier League seasons
1
Russia
Russia |
https://en.wikipedia.org/wiki/Rail%20transport%20in%20Mongolia | Rail transport is an important means of travel in the landlocked country of Mongolia, which has relatively few paved roads. According to official statistics, rail transport carried 93% of Mongolian freight and 43% of passenger turnover in 2007. The Mongolian rail system employs 12,500 people. The national operator is UBTZ (Ulaanbataar Railway, ), traditionally also known as Mongolian Railway (MTZ, ). This can be a source of confusion, since MTZ is a distinct company established in 2008. The Mongolian Railway College is located in Ulaanbaatar.
Routes
The Trans-Mongolian Railway connects the Trans-Siberian Railway from Ulan Ude in Russia to Erenhot and Beijing in China through the capital Ulaanbaatar. The Mongolian section of this line runs for . The Trans-Mongolian Railway runs through Mongolia on Russian gauge track, changing to standard gauge track after entering China. There are several spur lines: to the copper combine in Erdenet, to coal mines in Sharyngol, Nalaikh and Baganuur, to the fluorspar mine in Bor-Öndör, to the former Soviet military base and refinery at Züünbayan. Another line links Züünbayan with Khangi on the Chinese border.
A separate railway line exists in the east of the country between Choibalsan and the Trans-Siberian at Borzya; however, that line is closed to passengers beyond the Mongolian town of Chuluunkhoroot. This line used to have a spur line to the uranium mine at Mardai, however this spur line was torn up and sold in the late 1990s/ early 2000s.
For domestic transport, daily trains run from Ulaanbaatar to Darkhan, Sukhbaatar, and Erdenet, as well as Zamyn-Üüd, Choir and Sainshand. Mongolia uses the (Russian gauge) with a total system length of .
The Mongolian Railway (MongolRail) is slated to cover by year 2025. The coverage track distance will get increased by . Mongolian railways transported 20.5 million tons of freight in 2013, which is close to the system's full capacity.
Transporting transit cargo between Russia and China is an important source of revenue for the country's railway system; in addition to this, railways are used to transport domestic coal to power plants.
Proposed lines
A 2010 Mongolian plan proposed of new track, primarily linking Dalanzadgad and Choibalsan, to be built in three stages:
the first stage, totaling and linking Dalanzadgad–Tavan Tolgoi mine–Tsagaan Suvarga mine–Züünbayan (), Sainshand–Baruun-Urt (), Baruun-Urt–Khööt mine (), and Khööt–Choibalsan ();
the second stage, totaling and connecting the first stage with the Chinese border, linking Nariin Sukhait mine–Shivee Khüren (), Tavan Tolgoi–Gashuun Sukhait (), Khööt–Tamsagbulag–Nömrög (), and Khööt–Bichigt (); and
the third stage, totaling ) and not described in detail, but including a connection with Tsagaannuur on the Russian border and a line from Ulaanbaatar to Kharkhorin.
In 2012, a line linking Erdenet–Mörön–Ovoot mine–Arts Suuri on the Russian border () was approved. In 2014, it was announced that the planned |
https://en.wikipedia.org/wiki/Partial%20regression%20plot | In applied statistics, a partial regression plot attempts to show the effect of adding another variable to a model that already has one or more independent variables. Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots.
When performing a linear regression with a single independent variable, a scatter plot of the response variable against the independent variable provides a good indication of the nature of the relationship. If there is more than one independent variable, things become more complicated. Although it can still be useful to generate scatter plots of the response variable against each of the independent variables, this does not take into account the effect of the other independent variables in the model.
Calculation
Partial regression plots are formed by:
Computing the residuals of regressing the response variable against the independent variables but omitting Xi
Computing the residuals from regressing Xi against the remaining independent variables
Plotting the residuals from (1) against the residuals from (2).
Velleman and Welsch
express this mathematically as:
where
Y•[i] = residuals from regressing Y (the response variable) against all the independent variables except Xi
Xi•[i] = residuals from regressing Xi against the remaining independent variables.
Properties
Velleman and Welsch list the following useful properties for this plot:
The least squares linear fit to this plot has the slope and intercept zero.
The residuals from the least squares linear fit to this plot are identical to the residuals from the least squares fit of the original model (Y against all the independent variables including Xi).
The influences of individual data values on the estimation of a coefficient are easy to see in this plot.
It is easy to see many kinds of failures of the model or violations of the underlying assumptions (nonlinearity, heteroscedasticity, unusual patterns). .
Partial regression plots are related to, but distinct from, partial residual plots. Partial regression plots are most commonly used to identify data points with high leverage and influential data points that might not have high leverage. Partial residual plots are most commonly used to identify the nature of the relationship between Y and Xi (given the effect of the other independent variables in the model). Note that since the simple correlation between the two sets of residuals plotted is equal to the partial correlation between the response variable and Xi, partial regression plots will show the correct strength of the linear relationship between the response variable and Xi. This is not true for partial residual plots. On the other hand, for the partial regression plot, the x-axis is not Xi. This limits its usefulness in determining the need for a transformation (which is the primary purpose of the partial residual plot).
See also
Partial residual plot
Partial leverage plot
Variance inflatio |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Real%20Madrid%20CF%20season | The 2001–02 season was Real Madrid CF's 71st season in La Liga. This article lists all matches that the club played in the 2001–02 season, and also shows statistics of the club's players. Although German home appliance giant Teka appeared as a shirt sponsor earlier in the season, Realmadrid.com replaced it as the primary shirt sponsor later in 2001, and there was no shirt sponsor for the second half of the season. The club introduced new grey and black third kits as well.
Summary
Real Madrid endured its worst domestic league performance under Vicente del Bosque's management, finishing only third in the league standings (with 66 points), as well as losing the Copa del Rey final at the Bernabéu to unheralded Deportivo La Coruña, despite the club's world record signing of Zinedine Zidane from Juventus. On a brighter note, del Bosque delivered La Novena's UEFA Champions League title as a consolation prize, following a 2–1 victory against Bayer Leverkusen in the final thanks to Zidane's volley goal.
First-team squad
Transfers
In
Total spending: €72,000,000
Out
Total income: €0
Results
Friendlies
La Liga
League table
Results by round
Matches
Results summary
Copa del Rey
Round of 64
Round of 32
Round of 16
Quarter-finals
Semi-finals
Final
Supercopa de España
Champions League
First group stage
Group A
Second group stage
Group C
Quarter-finals
Semi-finals
Final
FIFA Club World Championship
As winners of the 1999–2000 UEFA Champions League, Real Madrid was one of the 12 teams that were invited to the 2001 FIFA Club World Championship, which was scheduled to be hosted in Spain from 28 July to 12 August 2001. However, the tournament was cancelled, primarily due to the collapse of ISL, which was the marketing partner of FIFA at the time.
Since the fixtures were already released prior to the tournament's cancellation, it is known that Real Madrid would have played its group stage matches at the Bernabéu.
Group stage
Statistics
Appearances and goals
|-
! colspan=14 style=background:#dcdcdc; text-align:center| Players transferred out during the season
Reference:
References
External links
Official site
Real Madrid team page
Real Madrid (Spain) profile
UEFA Champions League
Web Oficial de la Liga de Fútbol Profesional
FIFA
Real Madrid
Real Madrid CF seasons
UEFA Champions League-winning seasons |
https://en.wikipedia.org/wiki/List%20of%20busiest%20airports%20in%20Africa | This is a list of the busiest airports in Africa, ranked by total passengers per year, which includes arrival, departure and transit passengers.
Evolution in graph
2022 statistics
2021 statistics
2020 statistics
2019 statistics
2018 statistics
2017 statistics
2016 statistics
2014 statistics
2013 statistics
2012 statistics
2011 statistics
2010 statistics
2009 statistics
2008 statistics
Gallery
See also
List of eponyms of airports
List of the busiest airports in the Middle East
References
Africa
Aviation in Africa
Busiest |
https://en.wikipedia.org/wiki/Bricard%20octahedron | In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897. The overall shape of one of these polyhedron may change in a continuous motion, without any changes to the lengths of its edges nor to the shapes of its faces.
These octahedra were the first flexible polyhedra to be discovered.
The Bricard octahedra have six vertices, twelve edges, and eight triangular faces, connected in the same way as a regular octahedron. Unlike the regular octahedron, the Bricard octahedra are all non-convex self-crossing polyhedra. By Cauchy's rigidity theorem, a flexible polyhedron must be non-convex, but there exist other flexible polyhedra without self-crossings. Avoiding self-crossings requires more vertices (at least nine) than the six vertices of the Bricard octahedra.
In his publication describing these octahedra, Bricard completely classified the flexible octahedra. His work in this area was later the subject of lectures by Henri Lebesgue at the Collège de France.
Construction
A Bricard octahedron may be formed from three pairs of points, each symmetric around a common axis of 180° rotational symmetry, with no plane containing all six points. These points form the vertices of the octahedron. The triangular faces of the octahedron have one point from each of the three symmetric pairs. For each pair, there are two ways of choosing one point from the pair, so there are eight triangular faces altogether. The edges of the octahedron are the sides of these triangles, and include one point from each of two symmetric pairs. There are 12 edges, which form the octahedral graph .
As an example, the six points (0,0,±1), (0,±1,0), and (±1,0,0) form the vertices of a regular octahedron, with each point opposite in the octahedron to its negation, but this is not flexible. Instead, these same six points can be paired up differently to form a Bricard octahedron, with a diagonal axis of symmetry. If this axis is chosen as the line through the origin and the point (0,1,1), then the three symmetric pairs of points for this axis are
(0,0,1)—(0,1,0), (0,0,−1)—(0,−1,0), and (1,0,0)–(−1,0,0). The resulting Bricard octahedron resembles one of the extreme configurations of the second animation, which has an equatorial antiparallelogram.
As a linkage
It is also possible to think of the Bricard octahedron as a mechanical linkage consisting of the twelve edges, connected by flexible joints at the vertices, without the faces. Omitting the faces eliminates the self-crossings for many (but not all) positions of these octahedra. The resulting kinematic chain has one degree of freedom of motion, the same as the polyhedron from which it is derived.
Explanation
The quadrilaterals formed by the edges between the points in any two symmetric pairs of points can be thought of as equators of the octahedron. These equators have the property (by their symmetry) that opposite pairs of quadrilateral sides have equal length. Every quadrila |
https://en.wikipedia.org/wiki/EuroCup%20Basketball%20individual%20statistics | EuroCup Basketball individual statistics are the individual stats leaders of the European-wide 2nd-tier level league, the EuroCup. The EuroCup is the European-wide league that is one tier level below the top-tier level EuroLeague.
EuroCup Basketball season by season individual stats leaders
Points per game
2002–03 / Jamie Arnold (KRKA Novo Mesto): 20.25 (in 16 games)
2003–04 Rasheed Brokenborough (Superfund Bulls Kapfenberg): 26.55 (in 9 games)
2004–05 Todor Stoykov (Lukoil Academic Sofia): 23.92 (in 12 games)
2005–06 Horace Jenkins (Hapoel Jerusalem): 20.44 (in 16 games)
2006–07 / Milan Gurović (Crvena Zvezda Belgrade): 25.86 (in 14 games)
2007–08 / De'Teri Mayes (Allianz Swans Gmunden): 21.1 (in 10 games)
2008–09 Khalid El-Amin (BC Azovmash): 17.91 (in 11 games)
2009–10 / Darius Washington (Galatasaray Café Crown): 21.64 (in 11 games)
2010–11 / Jaycee Carroll (CB Gran Canaria): 19.00 (in 12 games)
2011–12 Ramel Curry (BC Donetsk): 16.43 (in 14 games)
2012–13 / Walter Hodge (Stelmet Zielona Góra): 21.17 (in 12 games)
2013–14 Errick McCollum (Panionios): 20.19 (in 16 games)
2014–15 Randy Culpepper (Krasny Oktyabr Volgograd): 19.15 (in 13 games)
2015–16 Keith Langford (UNICS Kazan): 19.69 (in 16 games)
2016–17 Alexey Shved (Khimki): 22.14 (in 14 games)
Rebounds per game
2002–03 K'zell Wesson (Cholet Basket): 12.7 (in 10 games)
2003–04 Geert Hammink (RheinEnergie Cologne): 11.5 (in 12 games)
2004–05 Chris Ensminger (GHP Bamberg): 10.67 (in 9 games)
2005–06 Mario Austin (Hapoel Jerusalem): 9.44 (in 16 games)
2006–07 / Tariq Kirksay (SLUC Nancy): 9.58 (in 12 games)
2007–08 Virgil Carutasu (CSU Asesoft Ploiesti): 10 (in 9 games)
2008–09 Deyan Ivanov (KK Zadar): 8.56 (in 9 games)
2009–10 James Augustine (CB Gran Canaria): 7.43 (in 14 games)
2010–11 Maciej Lampe (UNICS Kazan): 8.06 (in 16 games)
2011–12 / Jeremiah Massey (PBC Lokomotiv-Kuban): 8.36 (in 14 games)
2012–13 John Bryant (Ratiopharm Ulm): 9.00 (in 13 games)
2013–14 Vladimir Golubović (Aykon TED Ankara): 10.10 (in 20 games)
2014–15 Sharrod Ford (Paris Levallois): 8.85 (in 20 games)
2015–16 Adrien Moerman (Banvit Bandirma): 8.56 (in 18 games)
2016–17 Drew Gordon (Lietuvos rytas Vilnius): 9.57 (in 14 games)
Assists per game
2002–03 Richard "Scooter" Barry (Cholet Basket): 5.3 (in 10 games)
2003–04 Ivan Tomas (KK Zagreb): 4.78 (in 9 games)
2004–05 Damir Mulaomerović (PAOK Thessaloniki): 7.78 (in 9 games)
2005–06 Lamont Jones (Lukoil Academic Sofia): 6.3 (in 10 games)
2006–07 Mark Dickel (Anwil Wloclawek): 5.67 (in 9 games)
2007–08 / Omar Cook (Crvena Zvezda Belgrade): 6.14 (in 14 games)
2008–09 Khalid El-Amin (BC Azovmash): 5.27 (in 11 games)
2009–10 Marko Popović (UNICS Kazan): 4.75 (in 12 games)
2010–11 / Dontaye Draper (KK Cedevita Zagreb): 6.21 (in 14 games)
2011–12 DaShaun Wood (Alba Berlin): 5.42 (in 12 games)
2012–13 Nick Calathes (PBC Lokomotiv-Kuban): 6.65 (in 17 games)
2013–14 Marko Marinović (Radnički Kragujevac): 8.56 (in 16 games)
2014–15 Mike |
https://en.wikipedia.org/wiki/Ultralimit | In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces . The concept captures the limiting behavior of finite configurations in the spaces employing an ultrafilter to bypass the need for repeatedly consideration of subsequences to ensure convergence. Ultralimits generalize Gromov Hausdorff convergence in metric spaces.
Ultrafilters
An Ultrafilter, denoted as ω, on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset X of , contains either X or . An Ultrafilter on is non-principal if it contains no finite set.
Limit of a sequence of points with respect to an Ultrafilter
In the following, ω is a non-principal Ultrafilter on .
If is a sequence of points in a metric space (X,d) and x∈ X, then the point x is called the ω-limit of xn, denoted as .
If for every there are:
It is observed that,
If an ω-limit of a sequence of points exists, it is unique.
If in the standard sense, . (For this property to hold, it is crucial that the Ultrafilter should be non-principal.)
A fundamental fact states that, if (X,d) is compact and ω is a non-principal Ultrafilter on , the ω-limit of any sequence of points in X exists (and is necessarily unique).
In particular, any bounded sequence of real numbers has a well-defined ω-limit in , as closed intervals are compact.
Ultralimit of metric spaces with specified base-points
Let ω be a non-principal Ultrafilter on . Let (Xn ,dn) be a sequence of metric spaces with specified base-points pn ∈ Xn.
Suppose that a sequence , where xn ∈ Xn, is admissible. If the sequence of real numbers (dn(xn ,pn))n is bounded, that is, if there exists a positive real number C such that , then denote the set of all admissible sequences by .
It follows from the triangle inequality that for any two admissible sequences and the sequence (dn(xn,yn))n is bounded and hence there exists an ω-limit . One can define a relation on the set of all admissible sequences as follows. For , there is whenever This helps to show that is an equivalence relation on
The ultralimit with respect to ω of the sequence (Xn,dn, pn) is a metric space defined as follows.
Written as a set, .
For two -equivalence classes of admissible sequences and , there is
This shows that is well-defined and that it is a metric on the set .
Denote .
On base points in the case of uniformly bounded spaces
Suppose that (Xn ,dn) is a sequence of metric spaces of uniformly bounded diameter, that is, there exists a real number C > 0 such that diam(Xn) ≤ C for every . Then for any choice pn of base-points in Xn every sequence is admissible. Therefore, in this situation the choice of base-points does not have to be specified when defining an ultralimit, and the ultralimit depends only on (Xn,dn) and on ω but does not depend on |
https://en.wikipedia.org/wiki/Corps%20de%20l%27INSEE | The Corps of INSEE (Corps de l'INSEE) is a technical Grand corps de l'Etat with the National Institute of Statistics and Economic Studies (INSEE).
Service
Its members are public servants known as Administrateurs de l'INSEE. Most of them work for INSEE or in the French Ministry of the Economy.
Education
People entering the Corps are educated at the École nationale de la statistique et de l'administration économique−ENSAE. Most of them are from the École polytechnique and are known as X-INSEE. The rest come from the École Normale Supérieure, the regular curriculum of the ENSAE (École nationale de la statistique et de l'administration économique), ENSAI or internal promotion.
References
Institut national de la statistique et des études économiques
INSEE
Demographics of France
National statistical services |
https://en.wikipedia.org/wiki/Oleksandr%20Chyzhov | Oleksandr Oleksandrovych Chyzhov (; born 10 August 1986) is a Ukrainian retired footballer who played as a defender.
Career statistics
Honours
Club
Shakhtar Donetsk
Ukrainian Premier League (3) : 2009–10, 2010–11, 2011–12
Ukrainian Cup (2) : 2010–11, 2011–12
Ukrainian Super Cup (2) : 2008, 2010
UEFA Cup: 2008–09
References
External links
Profile on Official Shakhtar website
Ukrainian Premier League 2007–08 statistics
1986 births
Living people
Footballers from Poltava
Ukrainian men's footballers
Ukraine men's under-21 international footballers
Men's association football defenders
FC Shakhtar Donetsk players
FC Vorskla Poltava players
FC Vorskla-2 Poltava players
FC Sevastopol players
FC Mariupol players
FC Okzhetpes players
SC Poltava players
Ukrainian Premier League players
Ukrainian Second League players
Ukrainian Amateur Football Championship players
Kazakhstan Premier League players
Ukrainian expatriate men's footballers
Expatriate men's footballers in Kazakhstan
Ukrainian expatriate sportspeople in Kazakhstan
Ukrainian football managers
FC Shakhtar Donetsk non-playing staff |
https://en.wikipedia.org/wiki/Ahmad%20Khaziravi | Ahmad Khaziravi (, July 23, 1989 in Abadan, Iran) is an Iranian football player currently playing for Bargh Shiraz F.C. in Azadegan League.
Club career
Club career statistics
Club career
Khaziravi made his first senior team appearance for Esteghlal F.C. on August 5, 2008, against F.C. Aboumoslem
Iran's Premier Football League
Winner: 1
2008–09 with Esteghlal
External links
Profile at Iranproleague.net
Iranian men's footballers
Esteghlal F.C. players
PAS Tehran F.C. players
Bargh Shiraz F.C. players
Footballers from Abadan, Iran
Living people
F.C. Iranjavan Bushehr players
1989 births
Men's association football forwards |
https://en.wikipedia.org/wiki/Ashkan%20Namdari | Ashkan Namdari is an Iranian retired football player and current coach. He played for Esteghlal and Aboomoslem in IPL as a goalkeeper.
Club career
Club career statistics
Last update 12 May 2010
Honours
Esteghlal
Persian Gulf Pro League: 2008–09
External links
Persian League Profile
Iranian men's footballers
Men's association football defenders
F.C. Aboomoslem players
Bargh Shiraz F.C. players
Esteghlal F.C. players
Living people
1977 births
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Fuchs%27%20theorem | In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form
has a solution expressible by a generalised Frobenius series when , and are analytic at or is a regular singular point. That is, any solution to this second-order differential equation can be written as
for some positive real s, or
for some positive real r, where y0 is a solution of the first kind.
Its radius of convergence is at least as large as the minimum of the radii of convergence of , and .
See also
Frobenius method
References
.
.
Differential equations
Theorems in analysis |
https://en.wikipedia.org/wiki/Robbins%27%20problem | In probability theory, Robbins' problem of optimal stopping, named after Herbert Robbins, is sometimes referred to as the fourth secretary problem or the problem of minimizing the expected rank with full information.Let X1, ... , Xn be independent, identically distributed random variables, uniform on [0, 1]. We observe the Xk's sequentially and must stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value?The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not only on simpler sufficient statistics of these. Only bounds are known for the limiting value v as n goes to infinity, namely 1.908 < v < 2.329. It is known that there is some room to improve the lower bound by further computations for a truncated
version of the problem. It is still not known how to improve on the upper bound which stems from the subclass of memoryless threshold rules.
Chow-Robbins game
Another optimal stopping problem bearing Robbins' name is the Chow-Robbins game:Given an infinite sequence of IID random variables with distribution , how to decide when to stop, in order to maximize the sample average where is the stopping time?
The probability of eventually stopping must be 1 (that is, you are not allowed to keep sampling and never stop).For any distribution with finite second moment, there exists an optimal strategy, defined by a sequence of numbers . The strategy is to keep sampling until .
Optimal strategy for very large n
If has finite second moment, then after subtracting the mean and dividing by the standard deviation, we get a distribution with mean zero and variance one. Consequently it suffices to study the case of with mean zero and variance one.
With this, , where is the solution to the equationwhich can be proved by solving the same problem with continuous time, with a Wiener process. At the limit of , the discrete time problem becomes the same as the continuous time problem.
This was proved independently by.
When the game is a fair coin toss game, with heads being +1 and tails being -1, then there is a sharper resultwhere is the Riemann zeta function.
Optimal strategy for small n
When n is small, the asymptotic bound does not apply, and finding the value of is much more difficult. Even the simplest case, where are fair coin tosses, is not fully solved.
For the fair coin toss, a strategy is a binary decision: after tosses, with k heads and (n-k) tails, should one continue or should one stop? Since 1D random walk is recurrent, starting at any , the probability of eventually having more heads than tails is 1. So, if , one should always continue. However, if , it is tricky to decide whether to stop or continue.
found an exact solution fo |
https://en.wikipedia.org/wiki/Essential%20manifold | In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.
Definition
A closed manifold M is called essential if its fundamental class [M] defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding Eilenberg–MacLane space K(, 1), via the natural homomorphism
where n is the dimension of M. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.
Examples
All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere S2.
Real projective space RPn is essential since the inclusion
is injective in homology, where
is the Eilenberg–MacLane space of the finite cyclic group of order 2.
All compact aspherical manifolds are essential (since being aspherical means the manifold itself is already a K(, 1))
In particular all compact hyperbolic manifolds are essential.
All lens spaces are essential.
Properties
The connected sum of essential manifolds is essential.
Any manifold which admits a map of nonzero degree to an essential manifold is itself essential.
References
See also
Gromov's systolic inequality for essential manifolds
Systolic geometry
Algebraic topology
Riemannian geometry
Differential geometry
Systolic geometry
Manifolds |
https://en.wikipedia.org/wiki/Steve%20Brooks%20%28statistician%29 | Stephen Peter "Steve" Brooks is Executive Director of Select Statistical Services Ltd, a statistical research consultancy company based in Exeter, and former professor of statistics at the Statistical Laboratory of the University of Cambridge.
He received a degree in mathematics from Bristol University in 1991, and a master's degree in statistics from the University of Kent. He received his PhD at Cambridge; his supervisor was Gareth Roberts. Post-graduation he then returned to Bristol as a lecturer in the Statistics Group and then Senior Lecturer at the University of Surrey. In 2000 Brooks returned to Cambridge first as a fellow of King's College, Cambridge. and then of Wolfson College.
He is a specialist in Markov chain Monte Carlo and applied statistical methods.
He is one of the founding directors of the National Centre for Statistical Ecology which was set up in 2005.
He left Cambridge in 2006 to become Director of Research for ATASS Sports and is now executive director of Select Statistical Services Ltd a statistical consultancy firm based in Exeter and the Director of the Exeter Initiative for Statistics and its Applications
Career
1989–1991 Undergraduate, University of Bristol
1991–1992 Graduate Student, University of Kent
1992–1993 Research Associate, University of Kent
1993–1996 Research Student, University of Cambridge
1996–1999 Lecturer, University of Bristol
1999–2000 Senior Lecturer, University of Surrey
2000–2002 Lecturer, University of Cambridge
2002–2005 Reader, University of Cambridge
2005–2008 Professor, University of Cambridge
2006–2011 Director of Research ATASS
2011-Executive Director Select Statistical Services Ltd
Degrees and Qualifications
1991 BSc Mathematics, Bristol
1992 MSc Statistics, Kent
1996 PhD, Cambridge
1999 Chartered Statistician
2011 Chartered Scientist
Honours and awards
2005 Royal Statistical Society's Guy medal in Bronze
2004 Philip Leverhulme Prize
1999 Royal Statistical Society's Research prize
Books
Handbook of Markov Chain Monte Carlo edited by Steve Brooks, Andrew Gelman, Galin Jones and Xiao-Li Meng; Chapman and Hall/CRC, 2011
Bayesian Analysis for Population Ecology by Ruth King, Olivier Gimenez, Byron Morgan and Steve Brooks; Chapman and Hall/CRC, 2009
References
External links
Select Statistical Consulting Home Page
ATASS Ltd Home Page
Living people
English statisticians
Alumni of the University of Bristol
Alumni of the University of Kent
Alumni of the University of Cambridge
Fellows of King's College, Cambridge
Fellows of Wolfson College, Cambridge
Cambridge mathematicians
1970 births |
https://en.wikipedia.org/wiki/Uniform%202%20k1%20polytope | {{DISPLAYTITLE:Uniform 2 k1 polytope}}
In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3k,1}.
Family members
The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.
Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, {31,n-2,1}.
The sequence ends with k=6 (n=10), as an infinite hyperbolic tessellation of 9-space.
The complete family of 2k1 polytope polytopes are:
5-cell: 201, (5 tetrahedra cells)
Pentacross: 211, (32 5-cell (201) facets)
221, (72 5-simplex and 27 5-orthoplex (211) facets)
231, (576 6-simplex and 56 221 facets)
241, (17280 7-simplex and 240 231 facets)
251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)
261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets)
Elements
See also
k21 polytope family
1k2 polytope family
References
Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910.
Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988
External links
PolyGloss v0.05: Gosset figures (Gossetoctotope)
Polytopes |
https://en.wikipedia.org/wiki/Uniform%201%20k2%20polytope | {{DISPLAYTITLE:Uniform 1 k2 polytope}}
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3k,2}.
Family members
The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.
Each polytope is constructed from 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {31,n-2,2} polytope is a birectified n-simplex, t2{3n}.
The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space.
The complete family of 1k2 polytope polytopes are:
5-cell: 102, (5 tetrahedral cells)
112 polytope, (16 5-cell, and 10 16-cell facets)
122 polytope, (54 demipenteract facets)
132 polytope, (56 122 and 126 demihexeract facets)
142 polytope, (240 132 and 2160 demihepteract facets)
152 honeycomb, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets)
162 honeycomb, tessellates hyperbolic 9-space (∞ 152 and ∞ demienneract facets)
Elements
See also
k21 polytope family
2k1 polytope family
References
Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910.
Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988
External links
PolyGloss v0.05: Gosset figures (Gossetododecatope)
Multi-dimensional geometry
Polytopes |
https://en.wikipedia.org/wiki/Ted%20Kaczynski | Theodore John Kaczynski ( ; May 22, 1942 – June 10, 2023), also known as the Unabomber ( ), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a primitive lifestyle.
Between 1978 and 1995, Kaczynski murdered three individuals and injured 23 others in a nationwide mail bombing campaign against people he believed to be advancing modern technology and the destruction of the natural environment. He authored Industrial Society and Its Future, a 35,000-word manifesto and social critique opposing industrialization, rejecting leftism, and advocating a nature-centered form of anarchism.
In 1971, Kaczynski moved to a remote cabin without electricity or running water near Lincoln, Montana, where he lived as a recluse while learning survival skills to become self-sufficient. After witnessing the destruction of the wilderness surrounding his cabin, he concluded that living in nature was becoming impossible and resolved to fight industrialization and its destruction of nature through terrorism. In 1979, Kaczynski became the subject of what was, by the time of his arrest in 1996, the longest and most expensive investigation in the history of the Federal Bureau of Investigation (FBI). The FBI used the case identifier UNABOM (University and Airline Bomber) before his identity was known, resulting in the media naming him the "Unabomber".
In 1995, Kaczynski sent a letter to The New York Times promising to "desist from terrorism" if the Times or The Washington Post published his manifesto, in which he argued that his bombings were extreme but necessary in attracting attention to the erosion of human freedom and dignity by modern technologies. The FBI and U.S. Attorney General Janet Reno pushed for the publication of the essay, which appeared in The Washington Post in September 1995. Upon reading it, Kaczynski's brother, David, recognized the prose style and reported his suspicions to the FBI. After his arrest in 1996, Kaczynski—maintaining that he was sane—tried and failed to dismiss his court-appointed lawyers because they wished him to plead insanity to avoid the death penalty. He pleaded guilty to all charges in 1998 and was sentenced to eight consecutive life terms in prison without the possibility of parole. In June 2023, Kaczynski died by suicide in prison.
Early life
Childhood
Theodore John Kaczynski was born in Chicago on May 22, 1942, to working-class parents Wanda Theresa (née Dombek) and Theodore Richard Kaczynski, a sausage maker. The two were Polish Americans who were raised as Roman Catholics but later became atheists. They married on April 11, 1939.
From first to fourth grade (ages six to nine), Kaczynski attended Sherman Elementary School in Chicago, where administrators described him as healthy and well-adjusted. In 1952, three years after his brother David was born, the family moved to suburban Evergreen Park, Illinois, and Ted transferred to Evergreen Pa |
https://en.wikipedia.org/wiki/Uniform%20coloring | In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following different uniform color patterns.
A uniform coloring can be specified by listing the different colors with indices around a vertex figure.
n-uniform figures
In addition, an n-uniform coloring is a property of a uniform figure which has n types vertex figure, that are collectively vertex transitive.
Archimedean coloring
A related term is Archimedean color requires one vertex figure coloring repeated in a periodic arrangement. A more general term are k-Archimedean colorings which count k distinctly colored vertex figures.
For example, this Archimedean coloring (left) of a triangular tiling has two colors, but requires 4 unique colors by symmetry positions and become a 2-uniform coloring (right):
References
Uniform and Archimedean colorings, pp. 102–107
External links
Uniform Tessellations on the Euclid plane
Tessellations of the Plane
David Bailey's World of Tessellations
k-uniform tilings
n-uniform tilings
Uniform tilings
Polyhedra |
https://en.wikipedia.org/wiki/List%20of%20Bolton%20Wanderers%20F.C.%20records%20and%20statistics | Bolton Wanderers F.C. is an English professional association football club based in Horwich, Bolton. The club was founded as Christ Church F.C. in 1874, making them one of the oldest football clubs in England, and turned professional in 1877, before joining the Football League as founder members in 1888. Bolton Wanderers currently play in English Football League, the third tier of English football. They were relegated from the top tier (where they had been since 2001) in 2012 but in their time as a professional club have played in all four professional English leagues.
This list encompasses the major honours won by Bolton Wanderers and records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Bolton Wanderers players on the international stage, and the highest transfer fees paid and received by the club. The club's attendance records, both at The Reebok Stadium, their home since 1997, and Burnden Park, their home between 1895 and 1997, are also included in the list.
The club have won the FA Cup four times, but not since 1958, and have spent the majority of their history in the top flight of English football. Bolton also hold the record for the most years in the top flight of English football without winning the title; 73 years in total. The club's record appearance maker is Eddie Hopkinson, who made 578 appearances between his debut in 1952 and retirement in 1970, and the club's record goalscorer is Nat Lofthouse, who scored 285 goals in 503 games between 1946 and 1960.
All stats accurate as of end of 2018-19 season.
Honours and achievements
League
Second Division / Championship (level 2)
Champions: 1908–09, 1977–78, 1996–97
2nd place promotion: 1899–1900, 1904–05, 1910–11, 1934–35
Play-off winners: 1995, 2001
Third Division / League One (level 3)
Champions: 1972–73
2nd place promotion: 1992–93, 2016–17
Fourth Division / League Two (level 4)
3rd place promotion: 1987–88, 2020–21
Cup
FA Cup
Winners: 1922–23, 1925–26, 1928–29, 1957–58
Runners-up: 1893–94, 1903–04, 1952–53
Football League Cup
Runners-up: 1994–95, 2003–04
FA Charity Shield
Winners: 1958
Football League Trophy
Winners: 1988–89, 2022–23
Runners-up: 1985–86
Football League War Cup
Winners: 1945
Reserves and others
Premier League Asia Trophy Winners (1) – 2005
Peace Cup Runners up (1) – 2007
Carlsberg Cup Winners (1) – 2010
Lancashire Senior Cup (12) – 1886, 1891, 1912, 1922, 1925, 1927, 1932, 1934, 1939 (shared with Preston North End), 1948, 1989, 1991
Central League (2) – 1955, 1995
Premier Reserve League North (1) – 2006–07
Manchester Senior Cup (3) 1922, 1963, 2015
Professional Development League North (1) – 2017–18
Players
All current players are in bold
Appearances
Youngest first-team player: Ray Parry 15 years 267 days (v. Wolves, 13 October 1951).
Oldest first- |
https://en.wikipedia.org/wiki/Sinusoidal%20spiral | In algebraic geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
where is a nonzero constant and is a rational number other than 0. With a rotation about the origin, this can also be written
The term "spiral" is a misnomer, because they are not actually spirals, and often have a flower-like shape. Many well known curves are sinusoidal spirals including:
Rectangular hyperbola ()
Line ()
Parabola ()
Tschirnhausen cubic ()
Cayley's sextet ()
Cardioid ()
Circle ()
Lemniscate of Bernoulli ()
The curves were first studied by Colin Maclaurin.
Equations
Differentiating
and eliminating a produces a differential equation for r and θ:
.
Then
which implies that the polar tangential angle is
and so the tangential angle is
.
(The sign here is positive if r and cos nθ have the same sign and negative otherwise.)
The unit tangent vector,
,
has length one, so comparing the magnitude of the vectors on each side of the above equation gives
.
In particular, the length of a single loop when is:
The curvature is given by
.
Properties
The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a rectangular hyperbola.
The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals.
One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.
When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.
References
Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Spiral" p. 213–214
"Sinusoidal spiral" at www.2dcurves.com
"Sinusoidal Spirals" at The MacTutor History of Mathematics
Plane curves
Algebraic curves |
https://en.wikipedia.org/wiki/Lima%C3%A7on%20trisectrix | In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the Cycloid of Ceva, Quadratrix of Hippias, Trisectrix of Maclaurin, and Tschirnhausen cubic. The limaçon trisectrix a special case of a sectrix of Maclaurin.
Specification and loop structure
The limaçon trisectrix specified as a polar equation is
.
The constant may be positive or negative. The two curves with constants and are reflections of each other across the line . The period of is given the period of the sinusoid .
The limaçon trisectrix is composed of two loops.
The outer loop is defined when on the polar angle interval , and is symmetric about the polar axis. The point furthest from the pole on the outer loop has the coordinates .
The inner loop is defined when on the polar angle interval , and is symmetric about the polar axis. The point furthest from the pole on the inner loop has the coordinates , and on the polar axis, is one-third of the distance from the pole compared to the furthest point of the outer loop.
The outer and inner loops intersect at the pole.
The curve can be specified in Cartesian coordinates as
,
and parametric equations
,
.
Relationship with rose curves
In polar coordinates, the shape of is the same as that of the rose . Corresponding points of the rose are a distance to the left of the limaçon's points when , and to the right when .
As a rose, the curve has the structure of a single petal with two loops that is inscribed in the circle and is symmetric about the polar axis.
The inverse of this rose is a trisectrix since the inverse has the same shape as the trisectrix of Maclaurin.
Relationship with the sectrix of Maclaurin
See the article Sectrix of Maclaurin on the limaçon as an instance of the sectrix.
Trisection properties
The outer and inner loops of the limaçon trisectrix have angle trisection properties. Theoretically, an angle may be trisected using a method with either property, though practical considerations may limit use.
Outer loop trisectrix property
The construction of the outer loop of reveals its angle trisection properties. The outer loop exists on the interval . Here, we examine the trisectrix property of the portion of the outer loop above the polar axis, i.e., defined on the interval .
First, note that polar equation is a circle with radius , center on the polar axis, and has a diameter that is tangent to the line at the pole . Denote the diameter containing the pole as , where is at .
Second, consider any chord of the circle with the polar angle . Since is a right triangle, . The corresponding point on the outer loop has coordinates , where .
Given this construction, it is shown that and two other ang |
https://en.wikipedia.org/wiki/Sectrix%20of%20Maclaurin | In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them.
Equations in polar coordinates
We are given two lines rotating about two poles and . By translation and rotation we may assume and . At time , the line rotating about has angle and the line rotating about has angle , where , , and are constants. Eliminate to get
where and . We assume is rational, otherwise the curve is not algebraic and is dense in the plane. Let be the point of intersection of the two lines and let be the angle at , so . If is the distance from to then, by the law of sines,
so
is the equation in polar coordinates.
The case and where is an integer greater than 2 gives arachnida or araneidan curves
The case and where is an integer greater than 1 gives alternate forms of arachnida or araneidan curves
A similar derivation to that above gives
as the polar equation (in and ) if the origin is shifted to the right by . Note that this is the earlier equation with a change of parameters; this to be expected from the fact that two poles are interchangeable in the construction of the curve.
Equations in the complex plane, rectangular coordinates and orthogonal trajectories
Let where and are integers and the fraction is in lowest terms. In the notation of the previous section, we have
or
.
If then , so the equation becomes
or
. This can also be written
from which it is relatively simple to derive the Cartesian equation given m and n. The function
is analytic so the orthogonal trajectories of the family are the curves , or
Parametric equations
Let where and are integers, and let where is a parameter. Then converting the polar equation above to parametric equations produces
.
Applying the angle addition rule for sine produces
.
So if the origin is shifted to the right by a/2 then the parametric equations are
.
These are the equations for Plateau curves when , or
.
Inversive triplets
The inverse with respect to the circle with radius a and center at the origin of
is
.
This is another curve in the family. The inverse with respect to the other pole produces yet another curve in the same family and the two inverses are in turn inverses of each other. Therefore each curve in the family is a member of a triple, each of which belongs to the family and is an inverse of the other two. The values of q in t |
https://en.wikipedia.org/wiki/J.%20Murdoch%20Ritchie | Joseph Murdoch Ritchie (June 10, 1925 – July 9, 2008) was a Scottish born American biophysicist and a professor at Yale University.
Early life and education
Ritchie studied mathematics and physics at the University of Aberdeen, then did his doctorate at University College, London in biophysics in 1952.
Career
He joined the faculty at Yale in pharmacology in 1968, and later served as chairman of the department and as director of the division of biological sciences (1975–1978). He retired in 2003.
He was elected a Fellow of the Royal Society in 1976. According to his nomination citation "Ritchie's early work was concerned with the factors affecting the onset and duration of the active state in striated muscle, and with other aspects of the dynamics of muscular contraction. In 1954 he turned his attention to the properties of mammalian non-myelinated nerve fibres, and since then has made many distinguished contributions to our knowledge not only of some of the physiological functions served by such fibres, but also of the mechanism of conduction in them. In particular, he has been responsible for definitive studies of the mode of action of acetylcholine and local anaesthetics, of the ionic movements during nervous activity, of the temperature changes during the nervous impulse, of oxidative and glucose metabolism, and of the electrogenic sodium extrusion that underlies post-tetanic hyperpolarization. His most recent work on the specific and non-specific binding of tetrodotoxin has provided new information about the density of sodium channels in various types of nerve."
Ritchie is known for asking the Central Intelligence Agency in 1975 to share its supply of saxitoxin (which were used in suicide pills) with scientists for research and his work in neuroscience. He was the co-author of numerous scientific and technical books and articles.
Personal life
He was married to Brenda Bigland–Ritchie, a physiologist. They had a son, Alasdair Ritchie, a biologist, and a daughter, Jocelyn Ritchie, a neuropsychologist.
References
1925 births
2008 deaths
Alumni of the University of Aberdeen
Alumni of University College London
American biophysicists
American pharmacologists
Scottish emigrants to the United States
Scottish biophysicists
Scottish pharmacologists
Yale University faculty
Fellows of the Royal Society |
https://en.wikipedia.org/wiki/S-unit | In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-units.
Definition
Let K be a number field with ring of integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to positive or negative powers). For the ring of rational integers Z one may take S to be a finite set of prime numbers and define an S-unit to be a rational number whose numerator and denominator are divisible only by the primes in S.
Properties
The S-units form a multiplicative group containing the units of R.
Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.
S-unit equation
The S-unit equation is a Diophantine equation
u + v = 1
with u and v restricted to being S-units of K (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero). The number of solutions of this equation is finite and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendental number theory. A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yn = f(x).
A computational solver for S-unit equation is available in the software SageMath.
References
Chap. V.
Further reading
Algebraic number theory |
https://en.wikipedia.org/wiki/List%20of%20Melbourne%20Victory%20FC%20records%20and%20statistics | Melbourne Victory Football Club is an Australian professional association football club based at the Melbourne Rectangular Stadium. The club was formed in 2004.
The list encompasses the honours won by Melbourne Victory, records set by the club, their managers and their players. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records the highest transfer fees paid and received by the club. Attendance records at Olympic Park, Docklands Stadium and Melbourne Rectangular Stadium, the club's present home, are also included.
Melbourne Victory have won 7 top-flight titles, and have two Australia Cups. The club's record appearance maker is Leigh Broxham, who has currently made 444 appearances from 2005 to the present day. Archie Thompson is Melbourne Victory's record goalscorer, scoring 97 goals in total.
All figures are correct as of 26 January 2023
Honours and achievements
Domestic
A-League Men Premiership
Winners (3): 2006–07, 2008–09, 2014–15
Runners-up (3): 2009–10, 2016–17, 2021–22
A-League Men Championship
Winners (4): 2007, 2009, 2015, 2018
Runners-up (2): 2010, 2017
Australia Cup
Winners (2): 2015, 2021
A-League Pre-Season Challenge Cup
Winners (1): 2008
Player records
Appearances
Most A-League Men appearances: Leigh Broxham, 372
Most national cup appearances: Leigh Broxham, 27
Most Asian appearances: Leigh Broxham, 47
Youngest first-team player: Birkan Kirdar, 16 years, 70 days (against Shanghai SIPG, AFC Champions League group stage, 18 April 2018)
Oldest first-team player: Kevin Muscat, 37 years, 277 days (against Tianjin TEDA, AFC Champions League group stage, 20 April 2011)
Most consecutive appearances: Rodrigo Vargas, 54 (from 16 November 2007 to 18 October 2009)
Most appearances
a. Includes the A-League Pre-Season Challenge Cup and Australia Cup
b. Includes goals and appearances (including those as a substitute) in the 2005 Australian Club World Championship Qualifying Tournament.
Goalscorers
Most league goals in a season: Besart Berisha, 21 goals in the A-League, 2016–17
Youngest goalscorer: Christopher Cristaldo, 18 years, 67 days (against Perth Glory, A-League, 23 March 2013)
Oldest goalscorer: Kevin Muscat, 37 years, 256 days (against Tianjin TEDA, AFC Champions League group stage, 20 April 2011)
Most consecutive goalscoring appearances: Besart Berisha, 6 (from 26 April 2015 to 22 September 2015)
Top goalscorers
Competitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored.
a. Includes the A-League Pre-Season Challenge Cup and Australia Cup
b. Includes goals and appearances (including those as a substitute) in the 2005 Australian Club World Championship Qualifying Tournament.
Award winners
Joe Marston Medal
Johnny Warren Medal
Transfers
Record transfer fees received
Managerial records
First full-time manager: Ernie Merrick managed Melbourne Victory from December |
https://en.wikipedia.org/wiki/Uniform%20tilings%20in%20hyperbolic%20plane | In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.
Uniform tilings can be identified by their vertex configuration, a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents the heptagonal tiling which has 3 heptagons around each vertex. It is also regular since all the polygons are the same size, so it can also be given the Schläfli symbol {7,3}.
Uniform tilings may be regular (if also face- and edge-transitive), quasi-regular (if edge-transitive but not face-transitive) or semi-regular (if neither edge- nor face-transitive). For right triangles (p q 2), there are two regular tilings, represented by Schläfli symbol {p,q} and {q,p}.
Wythoff construction
There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where + + < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic triangle group.
Each symmetry family contains 7 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors. An 8th represents an alternation operation, deleting alternate vertices from the highest form with all mirrors active.
Families with r = 2 contain regular hyperbolic tilings, defined by a Coxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], ....
Hyperbolic families with r = 3 or higher are given by (p q r) and include (4 3 3), (5 3 3), (6 3 3) ... (4 4 3), (5 4 3), ... (4 4 4)....
Hyperbolic triangles (p q r) define compact uniform hyperbolic tilings. In the limit any of p, q or r can be replaced by ∞ which defines a paracompact hyperbolic triangle and creates uniform tilings with either infinite faces (called apeirogons) that converge to a single ideal point, or infinite vertex figure with infinitely many edges diverging from the same ideal point.
More symmetry families can be constructed from fundamental domains that are not triangles.
Selected families of uniform tilings are shown below (using the Poincaré disk model for the hyperbolic plane). Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.
Each uniform tiling generates a dual uniform tiling, with many of them also given below.
Right triangle domains
There are infinitely many ( |
https://en.wikipedia.org/wiki/Height%20%28disambiguation%29 | Height is the measurement of vertical distance.
Height may also refer to:
Mathematics and computer science
Height (abelian group), an invariant that captures the divisibility properties of an element
Height (ring theory), a measurement in commutative algebra
Height (triangle) or altitude
Height function, a function that quantifies the complexity of mathematical objects
Height of a field, exponent of torsion in the Witt group
Height, the logarithm of the first nonzero term in the formal power series
Tree height, length of the longest root-to-leaf path in a tree data structure
Music
Height (musician), Baltimore hip hop artist
Height (album), an album by John Nolan
People
Amy Height (c. 1866–1913), African-American music hall entertainer in the UK
Bob Height, 19th century African-American blackface minstrel performer
Dorothy Height (1912–2010), civil rights activist
See also
The Heights (disambiguation), including "Heights"
Human height
Tree height measurement
Reduced height of a field, the Pythagoras number |
https://en.wikipedia.org/wiki/FIBA%20EuroChallenge%20individual%20statistics | This article contains the individual statistics of players in the FIBA EuroChallenge competition. The FIBA EuroChallenge was the 3rd-tier level European-wide professional basketball league. The league is now defunct.
Statistical leaders
Points
Rebounds
Assists
Steals
Blocks
Individual highs
Statistical top 10s
2003–04 FIBA Europe League
Points per game:
Duane Woodward (EKA AEL Limassol): 21.3
Dametri Hill (Skonto Riga): 20.2
Ashante Johnson (ECM Nymburk): 19.5
Nestoras Kommatos (Aris Thessaloniki): 18.8
Andris Biedriņš (Skonto Riga): 18.6
George Zidek (ECM Nymburk): 18
Nebojša Bogavac (Hemofarm Vršac): 17.7
Nenad Čanak (NIS Vojvodina Novi Sad): 17.6
Steve Goodrich (BC Kyiv): 17.2
Troy Coleman (Kalev Tallinn): 16.8
Milan Gurović (NIS Vojvodina Novi Sad): 16.76
Jason Sasser (GHP Bamberg): 16.71
Assists per game:
Stevin Smith (Strauss Iscar Nahariya): 6.8
Randolph Childress (SLUC Nancy): 5.4
Maurice Whitfield (ECM Nymburk): 5.4
Petr Samoylenko (UNICS Kazan): 5.1
Vidas Ginevičius (Alita Alytus): 4.5
Roderick Blakney (Maroussi Telestet Athens): 4.4
Vassilis Spanoulis (Maroussi Telestet Athens): 4.28
Zakhar Pashutin (Ural Great Perm): 4.2
John Celestand (BC Kyiv): 4.0
Armands Šķēle (Anwil Włocławek): 4
Denis Mujagic (ECM Nymburk): 3.857
Dror Hajaj (Hapoel Tel Aviv): 3.85
Rebounds per game:
Chris Ensminger (GHP Bamberg): 12.5
Christophe Beghin (Telindus Oostende): 8.3
Andris Biedriņš (Skonto Riga): 8.18
Kšyštof Lavrinovič (Ural Great Perm): 8.13
JoJo Garcia (SLUC Nancy): 7.9
Steven Goodrich (BC Kyiv): 7.3
Stanislav Balashov (BC Kyiv): 7.2
Andrej Botichev (Azovmash Mariupol): 7.07
Eric Campbell (Strauss Iscar Nahariya): 7.05
Joey Beard (Telindus Oostende): 6.8
Grigorij Khizhnyak (GS Peristeri Athens): 6.5
Gennadiy Kuznyetsov (MBC Odessa): 6.3
Steals per game:
Vidas Ginevičius (Alita Alytus): 2.9
Denis Mujagic (ECM Nymburk): 2.7
Stevin Smith (Strauss Iscar Nahariya): 2.58
Bekir Yarangüme (Turk Telekom Ankara): 2.57
Duane Woodward (EKA AEL Limassol): 2.4
Roderick Blakney (Maroussi Telestet Athens): 2.3
Blocks per game:
Grigorij Khizhnyak (GS Peristeri Athens): 2.5
Vincent Jones (Ural Great Perm): 2.2
Andris Biedriņš (Skonto Riga): 1.8
Kšyštof Lavrinovič (Ural Great Perm): 1.7
Janar Talts (Kalev Tallinn): 1.53
Denis Ershov (Khimki Moscow): 1.5
2004–05 FIBA Europe League
Points per game:
Alvin Young (Bnei Hasharon): 22.6
Art Long (Azovmash Mariupol): 21.3
Shammond Williams (UNICS Kazan): 21.1
Christian Dalmau (Hapoel Galil Elyon): 20.41
Khalid El-Amin (Beşiktaş Istanbul): 20.4
Sam Hoskin (Ural Great Perm): 20.2
Aleksandar Zečević (RBC Verviers-Pepinster): 20
Kelly McCarty (Dynamo St.Petersburg): 19.4
Damir Mršić (Fenerbahçe Istanbul): 19.36
Kelvin Gibbs (Hapoel Tel Aviv): 19.33
Lior Eliyahu (Hapoel Galil Elyon): 19.0
Sharon Shason (Ural Great Perm): 19.0
Assists per game:
Khalid El-Amin (Beşiktaş Istanbul): 7.1
Maurice Whitfield (CEZ Nymburk): 6.1
Ed Cota (Dynamo St.Peters |
https://en.wikipedia.org/wiki/Walid%20Salah%20Abdel-Latif | Walid Salah Abdel Latif (born 11 November 1977 in Mansoura) is a retired Egyptian football player and a Zamalek international forward.
Career statistics
International goals
Honours
Egypt
African Cup of Nations: Winner 1998
Zamalek
Egyptian League Title (2000/2001 & 2002/2003 & 2003/2004)
Egyptian Cup Title (2001/2002)
Egyptian Super Cup (2000/2001 & 2001/2002)
African Champions League title (2002)
African Super Cup title (2002)
Arab Club Championship title (2003)
Egyptian Saudi Super Cup title (2003)
References
External links
1977 births
Living people
Egyptian men's footballers
Egypt men's international footballers
1998 African Cup of Nations players
Men's association football forwards
Zamalek SC players
People from Mansoura, Egypt
Egyptian Premier League players |
https://en.wikipedia.org/wiki/Official%20statistics | Official statistics are statistics published by government agencies or other public bodies such as international organizations as a public good. They provide quantitative or qualitative information on all major areas of citizens' lives, such as economic and social development, living conditions, health, education, and the environment.
During the 15th and 16th centuries, statistics were a method for counting and listing populations and State resources. The term statistics comes from the Neo-Latin statisticum collegium (council of state) and refers to science of the state. According to the Organisation for Economic Co-operation and Development (OECD), official statistics are statistics disseminated by the national statistical system, excepting those that are explicitly not to be official".
Governmental agencies at all levels, including municipal, county, and state administrations, may generate and disseminate official statistics. This broader possibility is accommodated by later definitions. For example:
Official statistics result from the collection and processing of data into statistical information by a government institution or international organization. They are then disseminated to help users develop their knowledge about a particular topic or geographical area, make comparisons between countries or understand changes over time. Official statistics make information on economic and social development accessible to the public, allowing the impact of government policies to be assessed, thus improving accountability.
Aim
Official statistics provide a picture of a country or different phenomena through data, and images such as graph and maps. Statistical information covers different subject areas (economic, demographic, social etc.). It provides basic information for decision making, evaluations and assessments at different levels.
The goal of statistical organizations is to produce relevant, objective and accurate statistics to keep users well informed and assist good policy and decision-making.
Various categories
The Fundamental Principles of Official Statistics were adopted in 1992 by the United Nations Economic Commission for Europe, and subsequently endorsed as a global standard by the United Nations Statistical Commission. According to the first Principle "Official statistics provide an indispensable element in the information system of a democratic society, serving the government, the economy and the public with data about the economic, demographic, social and environmental situation".
The categorization of the domains of official statistics has been further developed in the Classification of Statistical Activities, endorsed by the Conference of European Statisticians and various other bodies.
Most common indicators used in official statistics
Statistical indicators provide an overview of the social, demographic and economic structure of the society. Moreover, these indicators facilitate comparisons between countries and regions. |
https://en.wikipedia.org/wiki/Ali%20Karimi%20%28footballer%2C%20born%201982%29 | Ali Karimi (; born August 30, 1982) is an Iranian footballer who plays as a striker.
Club career
He played most of his career for his hometown teams Tractor and Shahrdari.
Club career statistics
References
1982 births
Living people
Footballers from Tabriz
Iranian men's footballers
Saipa F.C. players
PAS Tehran F.C. players
Tractor S.C. players
Shahrdari Tabriz F.C. players
Azadegan League players
Men's association football forwards |
https://en.wikipedia.org/wiki/Giro%20d%27Italia%20records%20and%20statistics | Since the first Giro d'Italia in 1909, there have been 2,074 stages. This number includes half-stages, prologues, and a small number of stages cancelled mid-race or immediately before the start. This number is up to date after the 2023 Giro. Since 1931, the race leader following each stage has been awarded the pink jersey ().
Although the leader of the classification after each stage gets a pink jersey, he is not considered the winner of the pink jersey, only the wearer. Only after the final stage is complete, the wearer of the pink jersey is considered the winner of the pink jersey, and thereby the winner of the Giro d'Italia. In 2020, British rider Tao Geoghegan Hart became the first cyclist to win the overall pink jersey, having never worn it during the race itself.
In this article first-place-classifications before 1931 are also counted as if a pink jersey was awarded. Nonetheless, the number of pink jerseys awarded is not equal to the number of stages. In the 1912 Giro d'Italia, the race was contested by teams, so no individual cyclist is counted in this statistic. Sometimes more cyclists were leading the classification (1925 after stages 2 and 3, 1929 after stage 2, 1936 after stage 6, 1938 after stages 2 and 3, 1957 after stage 18 and 1973 after the prologue). On the other hand, jerseys were not awarded in between any of the 51 pairs of half-stages that took place during the history of the Giro. Thus, as of 2023, 2,025 pink jerseys have been awarded in the Giro d'Italia to 285 different riders.
Individual records
Key:
In previous Giri d'Italia, sometimes a stage was split in two. On such occasions, only the cyclist leading at the end of the day is counted. The "Maglia Rosa" column gives the number of days that the cyclist wore the pink jersey, the "Giro wins" column gives the number of days that the cyclist won the pink jersey. The next four columns indicate the number of times the rider won the points classification, the King of the Mountains classification, and the young rider competition, and the years in which the pink jersey was worn, with bold years indicating an overall Giro win. For example: Eddy Merckx has spent 78 stages as leader of the race, won the general classification five times; won the points classification two times, won the mountains classification one time, and never won the young rider classification. He wore the pink jersey in the 1968, 1970, 1972, 1973, and 1974 editions of the race (which he all won) as well as 1969 (which he did not win).
After Alberto Contador was stripped from his victory in the 2011 Giro d'Italia, Michele Scarponi became the new winner.
Ranked by most days in the Maglia Rosa, updated until after Stage 20 of the 2023 Giro d'Italia.
Per country
The pink jersey has been awarded to 28 different countries since 1903. In the table below, "Jerseys" indicates the number of pink jerseys that were given to cyclists of each country. "Giro wins" stands for the number of Giro wins by cyclists of tha |
https://en.wikipedia.org/wiki/Shapiro%27s%20lemma | In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus relates the group cohomology with respect to a group to the cohomology with respect to a subgroup. Shapiro's lemma is named after Arnold S. Shapiro, who proved it in 1961; however, Beno Eckmann had discovered it earlier, in 1953.
Statement for rings
Let R → S be a ring homomorphism, so that S becomes a left and right R-module. Let M be a left S-module and N a left R-module. By restriction of scalars, M is also a left R-module.
If S is projective as a right R-module, then:
If S is projective as a left R-module, then:
See . The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see .
Statement for group rings
When H is a subgroup of finite index in G, then the group ring R[G] is finitely generated projective as a left and right R[H] module, so the previous theorem applies in a simple way. Let M be a finite-dimensional representation of G and N a finite-dimensional representation of H. In this case, the module S ⊗R N is called the induced representation of N from H to G, and RM is called the restricted representation of M from G to H. One has that:
When n = 0, this is called Frobenius reciprocity for completely reducible modules, and Nakayama reciprocity in general. See , which also contains these higher versions of the Mackey decomposition.
Statement for group cohomology
Specializing M to be the trivial module produces the familiar Shapiro's lemma. Let H be a subgroup of G and N a representation of H. For NG the induced representation of N from H to G using the tensor product, and for H the group homology:
H(G, NG) = H(H, N)
Similarly, for NG the co-induced representation of N from H to G using the Hom functor, and for H the group cohomology:
H(G, NG) = H(H, N)
When H has finite index in G, then the induced and coinduced representations coincide and the lemma is valid for both homology and cohomology.
See .
See also
Change of rings
Notes
References
.
, page 59
Homological algebra
Representation theory
Lemmas in algebra |
https://en.wikipedia.org/wiki/Supernatural%20number | In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory.
A supernatural number is a formal product:
where runs over all prime numbers, and each is zero, a natural number or infinity. Sometimes is used instead of . If no and there are only a finite number of non-zero then we recover the positive integers. Slightly less intuitively, if all are , we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide "infinitely often," by taking that prime's corresponding exponent to be the symbol .
There is no natural way to add supernatural numbers, but they can be multiplied, with . Similarly, the notion of divisibility extends to the supernaturals with if for all . The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining
and
.
With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number.
We can also extend the usual -adic order functions to supernatural numbers by defining for each .
Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.
Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
See also
Profinite integer
References
External links
Planet Math: Supernatural number
Number theory
Infinity |
https://en.wikipedia.org/wiki/Fake%204-ball | In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the use of Casson handles and so does not work in the smooth category.
References
Alexandru Scorpan, The Wild World of 4-Manifolds, American Mathematical Society,
4-manifolds
Geometric topology |
https://en.wikipedia.org/wiki/Koichi%20Sato%20%28footballer%29 | is a Japanese football player who plays as a forward for Veertien Mie.
Club statistics
Updated to 23 February 2020.
References
External links
Profile at Zweigen Kanazawa
Profile at V-Varen Nagasaki
Profile at FC Gifu
1986 births
Living people
Yokkaichi University alumni
Association football people from Mie Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
FC Gifu players
V-Varen Nagasaki players
Zweigen Kanazawa players
Ventforet Kofu players
Veertien Mie players
Men's association football forwards |
https://en.wikipedia.org/wiki/Decagonal%20bipyramid | In geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If a decagonal bipyramid is to be face-transitive, all faces must be isosceles triangles. It is an icosahedron, but not the regular one.
Images
It can be drawn as a tiling on a sphere, and represents the fundamental domains of [5,2], *5.2.2 symmetry.
See also
External links
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
VRML models <10>
Conway Notation for Polyhedra Try: dP10
Polyhedra |
https://en.wikipedia.org/wiki/Bass%E2%80%93Serre%20theory | Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.
History
Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in Trees, Serre's 1977 monograph (developed in collaboration with Hyman Bass) on the subject. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. Subsequent work of Bass contributed substantially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subject.
Mathematically, Bass–Serre theory builds on exploiting and generalizing the properties of two older group-theoretic constructions: free product with amalgamation and HNN extension. However, unlike the traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language of covering theory and fundamental groups. Graphs of groups, which are the basic objects of Bass–Serre theory, can be viewed as one-dimensional versions of orbifolds.
Apart from Serre's book, the basic treatment of Bass–Serre theory is available in the article of Bass, the article of G. Peter Scott and C. T. C. Wall and the books of Allen Hatcher, Gilbert Baumslag, Warren Dicks and Martin Dunwoody and Daniel E. Cohen.
Basic set-up
Graphs in the sense of Serre
Serre's formalism of graphs is slightly different from the standard formalism from graph theory. Here a graph A consists of a vertex set V, an edge set E, an edge reversal map such that ≠ e and for every e in E, and an initial vertex map . Thus in A every edge e comes equipped with its formal inverse . The vertex o(e) is called the origin or the initial vertex of e and the vertex o() is called the terminus of e and is denoted t(e). Both loop-edges (that is, edges e such that o(e) = t(e)) and multiple edges are allowed. An orientation on A is a partition of E into the union of two disjoint subsets E+ and E− so that for every edge e exactly one of the edges from the pair e, belongs to E+ and the other belongs to E−.
Graphs of groups
A graph of groups A consists of the following data:
A connected graph A;
An assignment of a vertex group Av to every vertex v of A.
An assignment of an edge group Ae to every edge e of A so that we have for every e ∈ E.
Boundary monomorphisms for all edges e of A, so that each is an injective group homomorphism.
For every the map is a |
https://en.wikipedia.org/wiki/%C3%89ric%20Sitruk | Éric Sitruk (born 14 January 1978, in Bondy) is a French football striker who last played for Stade Brestois 29.
Statistics
Updated 2 May 2010
References
1978 births
Living people
French men's footballers
Stade Lavallois players
FC Rouen players
En Avant Guingamp players
Stade Brestois 29 players
Ligue 2 players
Entente SSG players
FC Versailles 78 players
Men's association football forwards |
https://en.wikipedia.org/wiki/Population%20structure | Population structure may refer to many aspectsof population ecology:
Population structure (genetics), also called population stratification
Population pyramid
Age class structure
F-statistics
Population density
Population distribution
Population dynamics
Population genetics
Population growth
Population size
See also
Demography
Population model
:Category:Population |
https://en.wikipedia.org/wiki/Rotation%20operator | Rotation operator may refer to:
An operator that specifies a rotation (mathematics)
Three-dimensional rotation operator
Rot (operator) aka Curl, a differential operator in mathematics
Rotation operator (quantum mechanics) |
https://en.wikipedia.org/wiki/Vuelta%20a%20Espa%C3%B1a%20records%20and%20statistics | The Vuelta a España is an important cycling race (one of the Grand Tours). The first Vuelta a España was in 1935. As the Vuelta a España is a stage race, a classification based on times is calculated after every stage. The cyclist with the lowest time after a stage is the leader of the general classification after that stage. A jersey is given to that cyclist, which the cyclist wears during the next stage. From 1999 to 2009, this jersey had a golden color and was named the golden jersey. From 2010 onwards, the leader's jersey was red. It has also been black and white at different points in the Vuelta's history.
Although the leader of the classification after a stage gets the leader's jersey, he is not considered the winner of that jersey, only the wearer. Only after the final stage, the wearer of the leader's jersey is considered the winner, and therefore the winner of the Vuelta a España.
Since the first Vuelta a España in 1935, there have been 1,491 stages, up to and including the Stage 12 of the 2021 Vuelta a España The race leader following each stage has been awarded a leader's jersey.
Although the number of stages is 1,491, there have been 1,492 leader's jerseys awarded, because after the first stage of the 1948 Vuelta a España, Bernardo Ruiz and Julián Berrendero shared the lead and both received the leader's jersey. As of 2021, 1,492 leader's jerseys have been awarded in the Vuelta a España to 226 different riders.
Jerseys per rider
Key:
In previous Vueltas a España, sometimes a stage was split in two. On such occasions, only the cyclist leading at the end of the day is counted. The "Leader's jerseys" column gives the number of days that the cyclist wore the leader's jersey, the "Vuelta wins" column gives the number of times that the cyclist won the Vuelta. The next three columns indicate the number of times the rider won the points classification, the mountains classification, and the years in which the rider lead the general classification, with bold years indicating an overall Vuelta win.
For example: Alex Zülle has spent 48 days in the leader's jersey, and won the overall classification two times. He wore the leader's jersey in the Vueltas of 1993, 1996, 1997 and 2000, of which he won the 1996 and 1997 Vueltas.
Jerseys per country
The leader's jersey has been awarded to 23 different countries since 1935. In the table below, "Jerseys" indicates the number of leader's jerseys that were given to cyclists of each country. "Vuelta wins" stands for the number of Vuelta wins by cyclists of that country, "Points" for the number of times the points classification was won by a cyclist of that country and "KoM" for the number of times the mountains classification was won by a cyclist of that country. "Combo'" shows the winners of the combination classification, This classification is calculated by adding the numeral ranks of each cyclist in the general, points, and mountains classifications (a rider must have a score in all).
The "Most r |
https://en.wikipedia.org/wiki/Franz%20Thomas%20Bruss | Franz Thomas Bruss is Emeritus Professor of Mathematics at the Université Libre de Bruxelles, where he had been director of "Mathématiques Générales" and co-director of the probability chair, and where he continues his research as invited professor.
His main research activities in mathematics are in the field of probability:
1/e-law of best choice
Odds algorithm of optimal stopping
Galton–Watson processes
Resource Dependent Branching Processes
Borel–Cantelli lemma
Robbins' problem (of optimal stopping)
Pascal processes
BRS-inequality
Life
Thomas Bruss studied mathematics at the Universities Saarbrücken, Cambridge and Sheffield. In 1977 he obtained the Dr. rer. nat at Saarbrücken with his thesis (Sufficient Conditions for the Extinction of Modified Branching Processes) under Professor Gerd Schmidt, and the legal Dr. en sciences of Belgium one year later. After a scientific career at the University of Namur he moved to the United States and taught at the University of California at Santa Barbara, University of Arizona, Tucson, and then University of California at Los Angeles. In 1990 he returned to Europe as professor of mathematics at Vesalius College, Vrije Universiteit Brussel. In 1993 he was appointed chair of Mathématiques Générales and Probability at the Université Libre de Bruxelles, where he has stayed since then. He held visiting positions at the University of Strathclyde, Glasgow, University of Zaire, University of Antwerp, Purdue University, and repeatedly at the Université Catholique de Louvain.
Bruss is fellow of the Alexander von Humboldt Foundation, fellow of the Institute of Mathematical Statistics, elected member of the Tönissteiner Kreis e.V., Germany, and member of the
International Statistical Institute. In 2004 he received the Jacques Deruyts Prize (period 2000–2004) for distinguished contributions to mathematics from the Belgian Academy of Science Académie Royale de Belgique. In 2011, Thomas Bruss was honoured Commandeur de Order of Leopold of Belgium. Under his presidency
(2017-2019) the Belgian Statistical Society has received royal favour and become the Royal Statistical Society of Belgium (in French: Société Royale Belge de Statistique - in Dutch: Koninklijke Belgische Vereniging voor Statistiek.)
See also
Bruss–Duerinckx theorem
Odds algorithm (Bruss Strategie)
Robbins' problem
BRS-inequality
Royal Statistical Society of Belgium
Sources
Belgian Statistical Society, http://www.sbs-bvs.be/
External links
Thomas Bruss’ Homepage at Département de Mathématique of the Université libre de Bruxelles
His publication list on the Université libre de Bruxelles platform
1949 births
Living people
20th-century German mathematicians
21st-century German mathematicians
Probability theorists
Academic staff of the Université libre de Bruxelles |
https://en.wikipedia.org/wiki/Juggler%20sequence | In number theory, a juggler sequence is an integer sequence that starts with a positive integer a0, with each subsequent term in the sequence defined by the recurrence relation:
Background
Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.
For example, the juggler sequence starting with a0 = 3 is
If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verified for initial terms up to 106, but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems".
For a given initial term n, one defines l(n) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h(n) to be the maximum value in the juggler sequence starting at n. For small values of n we have:
{| class="wikitable"
|-
! n
! Juggler sequence
! l(n)
! h(n)
|-
| 2
| 2, 1
| align="center" | 1
| align="center" | 2
|-
| 3
| 3, 5, 11, 36, 6, 2, 1
| align="center" | 6
| align="center" | 36
|-
| 4
| 4, 2, 1
| align="center" | 2
| align="center" | 4
|-
| 5
| 5, 11, 36, 6, 2, 1
| align="center" | 5
| align="center" | 36
|-
| 6
| 6, 2, 1
| align="center" | 2
| align="center" | 6
|-
| 7
| 7, 18, 4, 2, 1
| align="center" | 4
| align="center" | 18
|-
| 8
| 8, 2, 1
| align="center" | 2
| align="center" | 8
|-
| 9
| 9, 27, 140, 11, 36, 6, 2, 1
| align="center" | 7
| align="center" | 140
|-
| 10
| 10, 3, 5, 11, 36, 6, 2, 1
| align="center" | 7
| align="center" | 36
|}
Juggler sequences can reach very large values before descending to 1. For example, the juggler sequence starting at a0 = 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at a0 = 48443 reaches a maximum value at a60 with 972,463 digits, before reaching 1 at a157.
See also
Arithmetic dynamics
Collatz conjecture
Recurrence relation
References
External links
Juggler sequence (A094683) at the On-Line Encyclopedia of Integer Sequences. See also:
Number of steps needed for juggler sequence (A094683) started at n to reach 1.
n sets a new record for number of iterations to reach 1 in the juggler sequence problem.
Number of steps where the Juggler sequence reaches a new record.
Smallest number which requires n iterations to reach 1 in the juggler sequence problem.
Starting values that produce a larger juggler number than smaller starting values.
Juggler sequence calculator at Collatz Conjecture Calculation Center
Juggler Number pages by Harry J. Smith
Arithmetic dynamics
Integer sequences
Recurrence relations
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Carl%20Leo%20Stearns | Carl Leo Stearns (1892-November 28, 1972) was an American astronomer.
After graduating from Wesleyan University in 1917 with high honors in general scholarship and special honors in mathematics, Stearns received his PhD from Yale University. He became an instructor in mathematics and astronomy at Wesleyan in 1919. He became an assistant professor in 1920, then an associate professor in 1942 and a full professor in 1944. He served as chairman of the astronomy department at Wesleyan, then in 1960 he was named as emeritus Fisk professor of astronomy. From 1960–71, after serving as an assistant, he became director of the Van Vleck Observatory; the second to hold that position.
During his career he computed more than 200 stellar trigonometric parallaxes. In 1927 he discovered the comet 1927 IV (comet Stearns, 1927d). It was observed for 4 years after perihelion until 1931 at a record distance of 11.5 AU. It is difficult to explain its activity based on water-dominated ice.
2035 Stearns, a Mars-crossing asteroid was named after him, as is the crater Stearns on the far side of the Moon.
References
Wesleyan University alumni
1892 births
1972 deaths
American astronomers
Wesleyan University faculty |
https://en.wikipedia.org/wiki/Shorewood-Troy%20Public%20Library | The Shorewood-Troy Public Library serves the village of Shorewood, Illinois and its surrounding areas. The library is near the intersection of U.S. Route 52 and Illinois Route 59.
Library statistics
FY2021 Information
Population served: 19,235
Circulation: 171,930
History
On Monday, November 17, 1975, the Shorewood-Troy Township Library opened its doors to residents of Shorewood. The original building was a storefront in the Shorewood Plaza on US Route 52, containing over 2,000 books, magazines, cassettes and records. Originally part of the Burr Oak Library System, residents had been using the Burr Oak bookmobile as their library service since 1972.
Originally labeled as a “demonstration library”, the storefront library operated under federal grants for several years. In May 1976, the Shorewood-Troy Library District was formed, after a referendum passed establishing a library board and new tax rates.
By 1980, the community of Shorewood was outgrowing its small storefront library, so the search for a new site began. In June 1984, a library construction grant of $250,000 was awarded to the district. Land was donated just north of the Shorewood Plaza by George and William Michas and Chris Dragatsis for the new building. The official ground-breaking of the new facility was August 16, 1984. The Shorewood-Troy Public Library opened to the public the following year. In 1992–93, a lower level of the library was completed to house the Youth Services Department and a meeting room.
Notes
External links
Shorewood-Troy Public Library website
Library buildings completed in 1985
Buildings and structures in Will County, Illinois
Education in Will County, Illinois
Library districts in Illinois
Public libraries in Illinois
Shorewood, Illinois |
https://en.wikipedia.org/wiki/Great%20dodecicosacron | In geometry, the great dodecicosacron (or great dipteral trisicosahedron) is the dual of the great dodecicosahedron (U63). It has 60 intersecting bow-tie-shaped faces.
Proportions
Each face has two angles of and two angles of . The diagonals of each antiparallelogram intersect at an angle of . The dihedral angle equals . The ratio between the lengths of the long edges and the short ones equals , which is the golden ratio. Part of each face lies inside the solid, hence is invisible in solid models.
References
External links
Uniform polyhedra and duals
Dual uniform polyhedra |
https://en.wikipedia.org/wiki/Cotes%27s%20spiral | In physics and in the mathematics of plane curves, a Cotes's spiral (also written Cotes' spiral and Cotes spiral) is one of a family of spirals classified by Roger Cotes.
Description
Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals).
The shape of spirals in the family depends on the parameters. The curves in polar coordinates, (r, θ), r > 0 are defined by one of the following five equations:
A > 0, k > 0 and ε are arbitrary real number constants. A determines the size, k determines the shape, and ε determines the angular position of the spiral.
Cotes referred to the different forms as "cases". The equations of the curves above correspond respectively to his 5 cases.
The Diagram shows representative examples of the different curves. The centre is marked by ‘O’ and the radius from O to the curve is shown when θ is zero. The value of ε is zero unless shown.
The first and third forms are Poinsot's spirals; the second is the equiangular spiral; the fourth is the hyperbolic spiral (more correctly called by its alternative name: the "Reciprocal Spiral" since it has no connection with the hyperbola, or the hyperbolic functions which feature in the Poinsot's spirals); the fifth is the epispiral.
For more information about their properties, reference should be made to the individual curves.
Classical mechanics
Cotes's spirals appear in classical mechanics, as the family of solutions for the motion of a particle moving under an inverse-cube central force. Consider a central force
where μ is the strength of attraction. Consider a particle moving under the influence of the central force, and let h be its specific angular momentum, then the particle moves along a Cotes's spiral, with the constant k of the spiral given by
when μ < h2 (cosine form of the spiral), or
when μ > h2, Poinsot form of the spiral. When μ = h2, the particle follows a hyperbolic spiral. The derivation can be found in the references.
History
In the Harmonia Mensurarum (1722), Roger Cotes analysed a number of spirals and other curves, such as the Lituus. He described the possible trajectories of a particle in an inverse-cube central force field, which are the Cotes's spirals. The analysis is based on the method in the Principia Book 1, Proposition 42, where the path of a body is determined under an arbitrary central force, initial speed, and direction.
Depending on the initial speed and direction he determines that there are 5 different "cases" (excluding the trivial ones, the circle and straight line through the centre).
He notes that of the 5, "the first and the last are described by Newton, by means of the quadrature (i.e. integration) of the hyperbola and the ellipse". |
https://en.wikipedia.org/wiki/List%20of%20gridiron%20football%20quarterbacks%20passing%20statistics | This is a list of gridiron football quarterbacks passing statistics for quarterbacks that have played outdoor professional football in North America. Below is a listing of the combined professional football league leaders for passing yards, passing touchdowns, passing completions, and passing attempts.
Because indoor football is played on a much shorter field and heavily favors offensive scoring, its records are not included in the main list, but are noted in a separate addendum below; likewise, as is standard for statistical record-keeping, exhibition games, all-star games (such as the Pro Bowl) and preseason contests are not counted.
During the NFL season, Tampa Bay Buccaneers quarterback Tom Brady passed Drew Brees to become the all-time passing yards leader in professional football league history. Brees had surpassed Anthony Calvillo for the record in the previous season, while Calvillo had surpassed Damon Allen for the record in the CFL season. Allen broke the previous record held by Warren Moon in . Moon had held the record since when he surpassed the record set by Ron Lancaster, who had surpassed Johnny Unitas during the CFL season. Unitas established the record set forth in this listing, while Aaron Rodgers is the leading active QB chasing Brady.
In the NFL season, Brady surpassed Brees for the all-time passing touchdowns record; Brees had surpassed Peyton Manning for the record the season before. Manning had held the record since the NFL season when he surpassed Brett Favre's record. Prior to that, Brett Favre held the record since the NFL season when he surpassed Warren Moon's record. Moon had held the record since when he surpassed the record set by Fran Tarkenton. Tarkenton had surpassed Johnny Unitas during the NFL season. Ron Lancaster overtook Tarkenton's lead in the touchdown pass category during the CFL season. However, Tarkenton retook the lead during the NFL season. Unitas established the record set forth in this listing.
Brady has held the record for pass completions since passing Brees during the 2021 NFL season; Brees surpassed Brett Favre during the season. Favre had held the previous record since the season when he surpassed Warren Moon's record. Moon had held the record since 1992 when he surpassed the record set by Fran Tarkenton. Tarkenton had surpassed Johnny Unitas during the NFL season. Unitas established the record set forth in this listing.
Brady has held the record for pass attempts since the 2020 NFL season when he surpassed Brees, who had surpassed Favre earlier in the same season. Prior to being surpassed by both Brady and Brees during the 2020 NFL season, Favre had held the record since , when he surpassed Warren Moon. Moon had held the record since when he surpassed the record set by Fran Tarkenton. Tarkenton had surpassed Johnny Unitas during the 1975 NFL season. Unitas established the record set forth in this listing.
Brady, Moon and Unitas are the only gridiron quarterbacks to have he |
https://en.wikipedia.org/wiki/Decidable%20sublanguages%20of%20set%20theory | In mathematical logic, various sublanguages of set theory are decidable. These include:
Sets with Monotone, Additive, and Multiplicative Functions.
Sets with restricted quantifiers.
References
Proof theory
Logic in computer science
Model theory |
https://en.wikipedia.org/wiki/Kurosh%20subgroup%20theorem | In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russian mathematician, in 1934. Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of its intersections with the conjugates of the factors of the original free product.
History and generalizations
After the original 1934 proof of Kurosh, there were many subsequent proofs of the Kurosh subgroup theorem, including proofs of Harold W. Kuhn (1952), Saunders Mac Lane (1958) and others. The theorem was also generalized for describing subgroups of amalgamated free products and HNN extensions. Other generalizations include considering subgroups of free pro-finite products and a version of the Kurosh subgroup theorem for topological groups.
In modern terms, the Kurosh subgroup theorem is a straightforward corollary of the basic structural results of Bass–Serre theory about groups acting on trees.
Statement of the theorem
Let be the free product of groups A and B and let be a subgroup of G. Then there exist a family of subgroups , a family of subgroups , families and of elements of G, and a subset such that
This means that X freely generates a subgroup of G isomorphic to the free group F(X) with free basis X and that, moreover, giAigi−1, fjBjfj−1 and X generate H in G as a free product of the above form.
There is a generalization of this to the case of free products with arbitrarily many factors. Its formulation is:
If H is a subgroup of ∗i∈IGi = G, then
where X ⊆ G and J is some index set and gj ∈ G and each Hj is a subgroup of some Gi.
Proof using Bass–Serre theory
The Kurosh subgroup theorem easily follows from the basic structural results in Bass–Serre theory, as explained, for example in the book of Cohen (1987):
Let G = A∗B and consider G as the fundamental group of a graph of groups Y consisting of a single non-loop edge with the vertex groups A and B and with the trivial edge group. Let X be the Bass–Serre universal covering tree for the graph of groups Y. Since H ≤ G also acts on X, consider the quotient graph of groups Z for the action of H on X. The vertex groups of Z are subgroups of G-stabilizers of vertices of X, that is, they are conjugate in G to subgroups of A and B. The edge groups of Z are trivial since the G-stabilizers of edges of X were trivial. By the fundamental theorem of Bass–Serre theory, H is canonically isomorphic to the fundamental group of the graph of groups Z. Since the edge groups of Z are trivial, it follows that H is equal to the free product of the vertex groups of Z and the free group F(X) which is the fundamental group (in the standard topological sense) of the underlying graph Z of Z. This implies the conclusion of the Kurosh subgroup theorem.
Extension
The result extends to the case that G is the amalgamated product along a common subgroup C, |
https://en.wikipedia.org/wiki/Stellation%20diagram | In geometry, a stellation diagram or stellation pattern is a two-dimensional diagram in the plane of some face of a polyhedron, showing lines where other face planes intersect with this one. The lines cause 2D space to be divided up into regions. Regions not intersected by any further lines are called elementary regions. Usually unbounded regions are excluded from the diagram, along with any portions of the lines extending to infinity. Each elementary region represents a top face of one cell, and a bottom face of another.
A collection of these diagrams, one for each face type, can be used to represent any stellation of the polyhedron, by shading the regions which should appear in that stellation.
A stellation diagram exists for every face of a given polyhedron. In face transitive polyhedra, symmetry can be used to require all faces have the same diagram shading. Semiregular polyhedra like the Archimedean solids will have different stellation diagrams for different kinds of faces.
See also
List of Wenninger polyhedron models
The fifty nine icosahedra
References
M Wenninger, Polyhedron models; Cambridge University Press, 1st Edn (1983), Ppbk (2003).
(1st Edn University of Toronto (1938))
External links
Stellation diagram
Polyhedra Stellations Applet Vladimir Bulatov, 1998
http://bulatov.org/polyhedra/stellation/index.html Polyhedra Stellation (VRML)
http://bulatov.org/polyhedra/icosahedron/index_vrml.html 59 stellations of icosahedron
http://www.queenhill.demon.co.uk/polyhedra/FacetingDiagrams/FacetingDiags.htm facetting diagrams
http://fortran.orpheusweb.co.uk/Poly/Ex/dodstl.htm Stellating the Dodecahedron
http://www.queenhill.demon.co.uk/polyhedra/icosa/stelfacet/StelFacet.htm Towards stellating the icosahedron and faceting the dodecahedron
http://www.mathconsult.ch/showroom/icosahedra/index.html 59 stellations of the icosahedron
http://www.uwgb.edu/dutchs/symmetry/stellate.htm Stellations of Polyhedra
http://www.uwgb.edu/dutchs/symmetry/stelicos.htm Coxeter's Classification and Notation
http://www.georgehart.com/virtual-polyhedra/stellations-icosahedron-index.html |
https://en.wikipedia.org/wiki/Shang-Hua%20Teng | Shang-Hua Teng (; born 1964) is a Chinese-American computer scientist. He is the Seeley G. Mudd Professor of Computer Science and Mathematics at the University of Southern California. Previously, he was the chairman of the Computer Science Department at the Viterbi School of Engineering of the University of Southern California.
Biography
Teng was born in China in 1964. His father, Dr. Teng Zhanhong, was a professor of civil engineering at the Taiyuan University of Technology. His mother, Li Guixin, was an administrator at the same university.
Teng graduated with BA in electrical engineering and BS in computer science, both from Shanghai Jiao Tong University in 1985. He obtained MS in computer science from the University of Southern California in 1988. Teng holds a Ph.D. in computer science from Carnegie Mellon University (in 1991).
Prior to joining USC in 2009, Teng was a professor at Boston University. He has also taught at MIT, the University of Minnesota, and the University of Illinois at Urbana-Champaign. He has worked at Xerox PARC, NASA Ames Research Center, Intel Corporation, IBM Almaden Research Center, Akamai Technologies, Microsoft Research Redmond, Microsoft Research New England and Microsoft Research Asia.
Recognition
In 2008 Teng was awarded the Gödel Prize for his joint work on smoothed analysis of algorithms with Daniel Spielman. They went to win the prize again in 2015 for their contribution on "nearly-linear-time Laplacian solvers". In 2009, he received the Fulkerson Prize given by the American Mathematical Society and the Mathematical Programming Society.
Teng is a Fellow of the Association for Computing Machinery (ACM) as well as an Alfred P. Sloan Research Fellow. He was named a SIAM Fellow in the 2021 class of fellows, "for contributions to scalable algorithm design, mesh generation, and algorithmic game theory, and for pioneering smoothed analysis of linear programming".
Personal life
In 2003, Teng married Diana Irene Williams, then a Ph.D. student of history at Harvard University.
References
External links
Shang-Hua Teng's personal homepage at USC
1964 births
Living people
American computer scientists
Chinese computer scientists
Boston University faculty
Carnegie Mellon University alumni
Chinese emigrants to the United States
Gödel Prize laureates
IBM employees
Intel people
Researchers in geometric algorithms
Fellows of the Association for Computing Machinery
Fellows of the Society for Industrial and Applied Mathematics
Massachusetts Institute of Technology faculty
Microsoft Research people
Shanghai Jiao Tong University alumni
Sloan Research Fellows
University of Illinois Urbana-Champaign faculty
University of Minnesota faculty
USC Viterbi School of Engineering alumni
University of Southern California faculty
Scientists at PARC (company)
Simons Investigator |
https://en.wikipedia.org/wiki/Rostislav%20Grigorchuk | Rostislav Ivanovich Grigorchuk (; b. February 23, 1953) is a mathematician working in different areas of mathematics including group theory, dynamical systems, geometry and computer science. He holds the rank of Distinguished Professor in the Mathematics Department of Texas A&M University. Grigorchuk is particularly well known for having constructed, in a 1984 paper, the first example of a finitely generated group of intermediate growth, thus answering an important problem posed by John Milnor in 1968. This group is now known as the Grigorchuk group and it is one of the important objects studied in geometric group theory, particularly in the study of branch groups, automaton groups and iterated monodromy groups. Grigorchuk is one of the pioneers of asymptotic group theory as well as of the theory of dynamically defined groups. He introduced the notion of branch groups and developed the foundations of the related theory. Grigorchuk, together with his collaborators and students, initiated the theory of groups generated by finite Mealy type automata, interpreted them as groups of fractal type, developed the theory of groups acting on rooted trees, and found numerous applications of these groups in various fields of mathematics including functional analysis, topology, spectral graph theory, dynamical systems and ergodic theory.
Biographical data
Grigorchuk was born on February 23, 1953, in Ternopil Oblast, now Ukraine (in 1953 part of the USSR).
He received his undergraduate degree in 1975 from Moscow State University.
He obtained a PhD (Candidate of Science) in Mathematics in 1978, also from Moscow State University, where his thesis advisor was Anatoly M. Stepin. Grigorchuk received a habilitation (Doctor of Science) degree in Mathematics in 1985 at the Steklov Institute of Mathematics in Moscow. During the 1980s and 1990s, Rostislav Grigorchuk held positions at the Moscow State University of Transportation, and subsequently at the Steklov Institute of Mathematics and Moscow State University. In 2002 Grigorchuk joined the faculty of Texas A&M University as a Professor of Mathematics, and he was promoted to the rank of Distinguished Professor in 2008.
Rostislav Grigorchuk gave an invited address at the 1990 International Congress of Mathematicians in Kyoto an AMS Invited Address at the March 2004 meeting of the American Mathematical Society in Athens, Ohio and a plenary talk at the 2004 Winter Meeting of the Canadian Mathematical Society.
Grigorchuk is the Editor-in-Chief of the journal "Groups, Geometry and Dynamics", published by the European Mathematical Society, and is or was a member of the editorial boards of the journals "Mathematical Notes", "International Journal of Algebra and Computation", "Journal of Modern Dynamics", "Geometriae Dedicata", "Ukrainian Mathematical Journal", "Algebra and Discrete Mathematics", "Carpathian Mathematical Publications", "Bukovinian Mathematical Journal", and "Matematychni Studii".
Mathematical cont |
https://en.wikipedia.org/wiki/Gilton | Gilton Ribeiro or simply Gilton (born March 25, 1989) is a Brazilian football defender who last played for Guarani.
Club statistics
References
External links
1989 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Cruzeiro Esporte Clube players
Clube Atlético Juventus players
Joinville Esporte Clube players
Cerezo Osaka players
Albirex Niigata players
Kashima Antlers players
Ventforet Kofu players
Paraná Clube players
Paysandu Sport Club players
Expatriate men's footballers in Japan
J1 League players
J2 League players
Cuiabá Esporte Clube players
Brusque Futebol Clube players
Men's association football defenders |
https://en.wikipedia.org/wiki/Jo%C3%A3o%20Sales | Joáo Francisco de Sales or simply Sales (born June 9, 1986) is a Brazilian striker, who plays for CA Linense.
Club statistics
References
External links
1986 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Bulgaria
Expatriate men's footballers in the Czech Republic
J2 League players
Czech First League players
First Professional Football League (Bulgaria) players
Associação Desportiva São Caetano players
FC Slovan Liberec players
Ventforet Kofu players
Vegalta Sendai players
Yokohama FC players
Clube Atlético Bragantino players
Associação Desportiva Recreativa e Cultural Icasa players
PFC Beroe Stara Zagora players
Mogi Mirim Esporte Clube players
Clube Atlético Linense players
Vila Nova Futebol Clube players
Men's association football forwards |
https://en.wikipedia.org/wiki/Maranh%C3%A3o%20%28footballer%2C%20born%201984%29 | Luis Carlos dos Santos Martins, or simply Maranhão (born June 19, 1984 in São Luís, Maranhão), is a Brazilian striker. He currently plays for North Bangkok University.
Club statistics
Honours
Ulsan Hyundai
AFC Champions League: 2012
References
External links
Guardian's Stats Centre
1984 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in South Korea
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in South Korea
J1 League players
J2 League players
K League 1 players
K League 2 players
Ventforet Kofu players
Tokyo Verdy players
Ulsan Hyundai FC players
Jeju United FC players
Gangwon FC players
Comercial Futebol Clube (Ribeirão Preto) players
Marília Atlético Clube players
Men's association football forwards |
https://en.wikipedia.org/wiki/Chan%20Cham%20Hei | Chan Cham Hei (, born 17 June 1991 in Hong Kong) is a former Hong Kong professional football player who plays as a left-back and is currently a free agent.
Career statistics
As of 15 December 2012
Honours
Eastern
Hong Kong First Division: 2014–15
Hong Kong FA Cup: 2014-15
Hong Kong League Cup: 2014-15
Kitchee
Hong Kong First Division: 2013-14
South China
Hong Kong First Division: 2012-13
External links
Hong Kong men's footballers
1991 births
Living people
Metro Gallery FC players
Hong Kong Premier League players
Hong Kong First Division League players
South China AA players
Kitchee SC players
Happy Valley AA players
Southern District FC players
Hong Kong Sapling players
Hong Kong Rangers FC players
Men's association football fullbacks |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20FK%20Vojvodina%20season | The 2008–09 season was FK Vojvodina's 3rd season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club played during the 2008–09 season.
Players
Squad information
Squad statistics
Matches
Serbian SuperLiga
Serbian Cup
UEFA Cup
External links
Official website
FK Vojvodina seasons
Vojvodina |
https://en.wikipedia.org/wiki/Ali%20Salmani | Ali Salmani (, born May 10, 1979) is an Iranian footballer.
Club career
Club Career Statistics
Last Update 19 October 2010
Assist Goals
External links
Profile at Iranproleague.net
Iranian men's footballers
Persepolis F.C. players
Steel Azin F.C. players
Paykan F.C. players
Pegah F.C. players
Saba Qom F.C. players
1979 births
Living people
Homa F.C. players
Shahid Ghandi Yazd F.C. players
Shahin Bushehr F.C. players
F.C. Aboomoslem players
Men's association football midfielders
Footballers from Tehran |
https://en.wikipedia.org/wiki/Bolyai%20Prize | The International János Bolyai Prize of Mathematics is an international prize founded by the Hungarian Academy of Sciences. The prize is named after János Bolyai and is awarded every five years to mathematicians for monographs with important new results in the preceding 10 years.
Medalists
1905 – Henri Poincaré
1910 – David Hilbert
2000 – Saharon Shelah for his Cardinal Arithmetic, Oxford University Press, 1994.
2005 – Mikhail Gromov for his Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser, 1999.
2010 – Yuri I. Manin for his Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, American Mathematical Society, 1999.
2015 – Barry Simon for his Orthogonal Polynomials on the Unit Circle, American Mathematical Society, 2005.
2020 - Terence Tao for his Nonlinear Dispersive Equations: Local and Global Analysis, American Mathematical Society, 2006.
See also
List of mathematics awards
References
Mathematics awards
Lists of award winners
Awards of the Hungarian Academy of Sciences
Awards established in 1902 |
https://en.wikipedia.org/wiki/Apache%20Mahout | Apache Mahout is a project of the Apache Software Foundation to produce free implementations of distributed or otherwise scalable machine learning algorithms focused primarily on linear algebra. In the past, many of the implementations use the Apache Hadoop platform, however today it is primarily focused on Apache Spark. Mahout also provides Java/Scala libraries for common math operations (focused on linear algebra and statistics) and primitive Java collections. Mahout is a work in progress; a number of algorithms have been implemented.
Features
Samsara
Apache Mahout-Samsara refers to a Scala domain specific language (DSL) that allows users to use R-Like syntax as opposed to traditional Scala-like syntax. This allows user to express algorithms concisely and clearly.
val G = B %*% B.t - C - C.t + (ksi dot ksi) * (s_q cross s_q)
Backend Agnostic
Apache Mahout's code abstracts the domain specific language from the engine where the code is run. While active development is done with the Apache Spark engine, users are free to implement any engine they choose- and Apache Flink have been implemented in the past and examples exist in the code base.
GPU/CPU accelerators
The JVM has notoriously slow computation. To improve speed, “native solvers” were added which move in-core, and by extension, distributed BLAS operations out of the JVM, offloading to off-heap or GPU memory for processing via multiple CPUs and/or CPU cores, or GPUs when built against the ViennaCL library. . ViennaCL is a highly optimized C++ library with BLAS operations implemented in OpenMP, and OpenCL. As of release 14.1, the OpenMP build considered to be stable, leaving the OpenCL build is still in its experimental POC phase.
Recommenders
Apache Mahout features implementations of Alternating Least Squares, Co-Occurrence, and Correlated Co-Occurrence, a unique-to-Mahout recommender algorithm that extends co-occurrence to be used on multiple dimensions of data.
History
Transition from Map Reduce to Apache Spark
While Mahout's core algorithms for clustering, classification and batch based collaborative filtering were implemented on top of Apache Hadoop using the map/reduce paradigm, it did not restrict contributions to Hadoop-based implementations. Contributions that run on a single node or on a non-Hadoop cluster were also welcomed. For example, the 'Taste' collaborative-filtering recommender component of Mahout was originally a separate project and can run stand-alone without Hadoop.
Starting with the release 0.10.0, the project shifted its focus to building a backend-independent programming environment, code named "Samsara". The environment consists of an algebraic backend-independent optimizer and an algebraic Scala DSL unifying in-memory and distributed algebraic operators. Supported algebraic platforms are Apache Spark, , and Apache Flink. Support for MapReduce algorithms started being gradually phased out in 2014.
Release History
Developers
Apache Mahout is de |
https://en.wikipedia.org/wiki/Grigorchuk%20group | In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.
History and significance
The growth of a finitely generated group measures the asymptotics, as of the size of an n-ball in the Cayley graph of the group (that is, the number of elements of G that can be expressed as words of length at most n in the generating set of G). The study of growth rates of finitely generated groups goes back to the 1950s and is motivated in part by the notion of volume entropy (that is, the growth rate of the volume of balls) in the universal covering space of a compact Riemannian manifold in differential geometry. It is obvious that the growth rate of a finitely generated group is at most exponential and it was also understood early on that finitely generated nilpotent groups have polynomial growth. In 1968 John Milnor posed a question about the existence of a finitely generated group of intermediate growth, that is, faster than any polynomial function and slower than any exponential function. An important result in the subject is Gromov's theorem on groups of polynomial growth, obtained by Gromov in 1981, which shows that a finitely generated group has polynomial growth if and only if this group has a nilpotent subgroup of finite index. Prior to Grigorchuk's work, there were many results establishing growth dichotomy (that is, that the growth is always either polynomial or exponential) for various classes of finitely generated groups, such as linear groups, solvable groups, etc.
Grigorchuk's group G was constructed in a 1980 paper of Rostislav Grigorchuk, where he proved that this group is infinite, periodic and residually finite. In a subsequent 1984 paper Grigorchuk proved that this group has intermediate growth (this result was announced by Grigorchuk in 1983). More precisely, he proved that G has growth b(n) that is faster than but slower than where . The upper bound was later improved by Laurent Bartholdi to
A lower bound of was proved by Yurii Leonov. The precise asymptotics of the growth of G is still unknown. It is conjectured that the limit
exists but even this remained a major open problem. This problem was resolved in 2020 by Erschler and Zheng. They show that the limit equals .
Grigorchuk's group was also the first example of a group that is |
https://en.wikipedia.org/wiki/Robin%20Farquharson | Reginald Robin Farquharson (3 October 1930 – 1 April 1973) was an academic whose interest in mathematics and politics led him to work on game theory. He wrote an influential analysis of voting systems in his doctoral thesis, later published as Theory of Voting.
Farquharson diagnosed himself as suffering from bipolar disorder (manic depression), and episodes of mania made it difficult for him to obtain a permanent university position and also resulted in him losing commercial employment. In later years, he dropped out of mainstream society, and became a prominent counter-cultural figure in late-1960s London. Farquharson wrote an account of his unconventional life in his 1968 book, Drop Out!, in which he described a week of being homeless in London.
In 1973 he died from burns associated with an arson, for which two persons were convicted of unlawful killing.
Education
Robin Farquharson was educated at Michaelhouse, Natal, South Africa, 1944–46. He earned a B.A. in South Africa from Rhodes University College, Grahamstown (1947–50). Subsequently studying at Brasenose and Nuffield Colleges, University of Oxford (1950–53), he obtained a second-class B.A. honours PPE degree. For his B.A. 1953–54 (?), his studies at this time were overseen by David Butler of Nuffield College, Oxford University. His D.Phil. was awarded in June 1958 from Nuffield College for his thesis entitled "An Approach to a Pure Theory of Voting Procedures".
He was given a Research Fellowship at Churchill College, Cambridge in 1964. He also studied at the Sorbonne in Paris.
While an undergraduate at Oxford, Farquharson was a contemporary of John Searle, Rupert Murdoch, and Sir Michael Dummett.
Research on voting
Farquharson wrote a monograph on the analysis of voting procedures and several papers, including a notable paper with Michael Dummett that conjectured the Gibbard–Satterthwaite theorem.
Strategic voting
Farquharson published influential articles on the theory of voting: in particular, in an article with Michael Dummett, he conjectured that deterministic voting rules with more than three issues faced endemic strategic voting. The Dummett–Farquharson conjecture was proved by Allan Gibbard, a philosopher and former student of Kenneth J. Arrow and John Rawls, and by Mark A. Satterthwaite, an economist.
After the establishment of the Farquarson-Dummett conjecture by Gibbard and Sattherthwaite, Michael Dummett contributed three proofs of the Gibbard–Satterthwaite theorem in his monograph on voting.
Theory of Voting
In the field of political game theory, Farquharson's main contribution was his exposition of the Condorcet paradox regarding the sincerity of voters. The problem was initially raised by Pliny the Younger and then picked up again in the political pamphlets of Reverend Charles Lutwidge Dodgson (Lewis Carroll), who was a significant influence on Farquharson.
Theory of Voting was originally Farquharson's doctoral thesis but was deemed to be of such a high quality i |
https://en.wikipedia.org/wiki/Grushko%20theorem | In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann.
Statement of the theorem
Let A and B be finitely generated groups and let A∗B be the free product of A and B. Then
rank(A∗B) = rank(A) + rank(B).
It is obvious that rank(A∗B) ≤ rank(A) + rank(B) since if X is a finite generating set of A and Y is a finite generating set of B then X∪Y is a generating set for A∗B and that |X ∪ Y| ≤ |X| + |Y|. The opposite inequality, rank(A∗B) ≥ rank(A) + rank(B), requires proof.
Grushko, but not Neumann, proved a more precise version of Grushko's theorem in terms of Nielsen equivalence. It states that if M = (g1, g2, ..., gn) is an n-tuple of elements of G = A∗B such that M generates G, <g1, g2, ..., gn> = G, then M is Nielsen equivalent in G to an n-tuple of the form
M = (a1, ..., ak, b1, ..., bn−k) where {a1, ..., ak}⊆A is a generating set for A and where {b1, ..., bn−k}⊆B is a generating set for B. In particular, rank(A) ≤ k, rank(B) ≤ n − k and rank(A) + rank(B) ≤ k + (n − k) = n. If one takes M to be the minimal generating tuple for G, that is, with n = rank(G), this implies that rank(A) + rank(B) ≤ rank(G). Since the opposite inequality, rank(G) ≤ rank(A) + rank(B), is obvious, it follows that rank(G)=rank(A) + rank(B), as required.
History and generalizations
After the original proofs of Grushko (1940) and Neumann(1943), there were many subsequent alternative proofs, simplifications and generalizations of Grushko's theorem. A close version of Grushko's original proof is given in the 1955 book of Kurosh.
Like the original proofs, Lyndon's proof (1965) relied on length-functions considerations but with substantial simplifications. A 1965 paper of Stallings
gave a greatly simplified topological proof of Grushko's theorem.
A 1970 paper of Zieschang gave a Nielsen equivalence version of Grushko's theorem (stated above) and provided some generalizations of Grushko's theorem for amalgamated free products. Scott (1974) gave another topological proof of Grushko's theorem, inspired by the methods of 3-manifold topology Imrich (1984) gave a version of Grushko's theorem for free products with infinitely many factors.
A 1976 paper of Chiswell gave a relatively straightforward proof of Grushko's theorem, modelled on Stallings' 1965 proof, that used the techniques of Bass–Serre theory. The argument directly inspired the machinery of foldings for group actions on trees and for graphs of groups and Dicks' even more straightforward proof of Grushko's theorem (see, for example,
John R. Stallings. "Foldings of G-trees." Arboreal group theory (Berkeley, California, 1988), pp. 355–368, Mathematical Sciences Research Institute |
https://en.wikipedia.org/wiki/Nikos%20Paragios | Nikos Paragios (, born at 1972) is a distinguished professor of Applied mathematics at CentraleSupélec, the school of engineering of the Paris-Saclay_University and founder, president and chief executive officer of TheraPanacea, an information technology company targeting precision medicine in oncology, neurology and beyond through holistic treatment pathways optimization.
Prior to that, he was senior fellow at the Institut Universitaire de France and affiliated scientific leader at Inria (2007-2017), served as the editor in chief of the Computer Vision and Image Understanding Journal (2012-2022) of Elsevier Publishing House, and has held permanent positions at Siemens Corporate Technology, École des ponts ParisTech as well as visiting positions at Rutgers University, Yale University and University of Houston.
He holds a D.Sc. degree in electrical and computer engineering (2005) from Université Côte d'Azur, a PhD in electrical and computer engineering (2000) from Inria and the University of Nice Sophia Antipolis and a MSc/BSc in computer science (1996/1994) from the University of Crete.
Work
medical imaging, computer vision, artificial intelligence and machine learning
Awards
European Research Council Fellow for his contributions to continuous and discrete inference in computer vision, 2011-2016
IEEE Fellow for his contributions to continuous and discrete inference in computer vision, 2011
Bodossaki Foundation Scientific Award in applied and engineering sciences, 2008
Francois Erbsmann Prize (with Ben Glocker), Information Processing in Medical Imaging (IPMI), 2007
TR35 MIT Technology Review Award, 2006
ERCIM Cor Baayen Award Award (honorable mention), 2000
Books
"Geometric Level Set Methods in Imaging, Vision and Graphics" (with Stanley Osher) published in 2003 by Springer
"Handbook of Mathematical Models in Computer Vision" (with Yunmei Chen and Olivier Faugeras) published in 2005 by Springer
"Handbook of Biomedical Imaging: Methodologies and Clinical Research" (with James Duncan and Nicholas Ayache) published by Springer in 2015.
External links
Nikos Paragios Homepage
Google Scholar Profile
Living people
Greek computer scientists
Computer vision researchers
University of Crete alumni
Fellow Members of the IEEE
Year of birth missing (living people)
People from Rhodes |
https://en.wikipedia.org/wiki/Algebra | Algebra () () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics.
Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory.
The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist.
Etymology
The word algebra comes from the from the title of the early 9th century book ʿIlm al-jabr wa l-muqābala "The Science of Restoring and Balancing" by the Persian mathematician and astronomer al-Khwarizmi. In his work, the term al-jabr referred to the operation of moving a term from one side of an equation to the other, المقابلة al-muqābala "balancing" referred to adding equal terms to both sides. Shortened to just algeber or algebra in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the 16th century.
Different meanings of "algebra"
The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.
As a single word without an article, "algebra" names a broad part of mathematics.
As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
With a qualifier, there is the same distinction:
Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary an |
https://en.wikipedia.org/wiki/Trigonometry | Trigonometry () is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.
Trigonometry is known for its many identities. These
trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.
History
Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.
In the 3rd century BC, Hellenistic mathematicians such as Euclid and Archimedes studied the properties of chords and inscribed angles in circles, and they proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. In 140 BC, Hipparchus (from Nicaea, Asia Minor) gave the first tables of chords, analogous to modern tables of sine values, and used them to solve problems in trigonometry and spherical trigonometry. In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. (The value we call sin(θ) can be found by looking up the chord length for twice the angle of interest (2θ) in Ptolemy's table, and then dividing that value by two.) Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds.
The modern definition of the sine is first attested in the Surya Siddhanta, and its properties were further documented in the 5th century (AD) by Indian mathematician and astronomer Aryabhata. These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. In 830 AD, Persian mathematician Habash al-Hasib al-Marwazi produced the first table of cotangents. By the 10th century AD, in the work of Persian mathematician Abū al-Wafā' al-Būzjānī, all six trigonometric functions were used. Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places |
https://en.wikipedia.org/wiki/Korean%20Uruguayans | Korean Uruguayans, numbering 130 individuals, formed the 19th-largest Korean community in Latin America as of 2005, according to the statistics of South Korea's Ministry of Foreign Affairs and Trade.
Migration history
The first Korean immigrants to Uruguay were ten families, totalling 45 persons, admitted in March 1975 to work in the agricultural sector. However, most of them later emigrated to Argentina and Paraguay. Since 1980, a total of 140 Koreans have naturalised as Uruguayan citizens, although many are not resident in the country any longer. The population is transient and fluctuates in size; the peak months of Korean presence in Uruguay are June, July, November and December.
Employment
Most Koreans live in and around Montevideo, where some work as fishermen, while others are involved in the textile industry. The fishermen are almost all from Busan; they earn between US$1,000 and US$1,500 a month, while the engineers on their boats receive US$3,000-4,000 and the captain may make as much as US$6,000. The fishermen often work in dangerous conditions and face language barriers. In February 2007, three Korean fishermen were killed in an explosion on board a fishing boat, along with their Vietnamese colleague. There are several Korean-run restaurants and noraebang (karaoke bars) in Montevideo.
As of 2013, there are 15 South Korean citizens registered in the Uruguayan social security.
Religion
South Korean missionaries of the Church of the Brethren, a Protestant denomination, have been evangelising among Korean fishermen in Uruguay for almost 20 years. One of their earliest converts from among the fishermen, Simon Lee, eventually left the fishing industry to devote himself to religious work; in 2004, he and ten others established a Korean church in Montevideo, which also aimed to serve fishermen from other Asian countries as well.
Notable people
Giovanna Yun (18 July 1992), a female footballer who plays for Uruguayan club Peñarol and the Uruguay national team. She has South Korean heritage from her father's side.
See also
South Korea-Uruguay relations
References
Asian Uruguayan
Ethnic groups in Uruguay
Uruguay
Uruguay
Immigration to Uruguay |
https://en.wikipedia.org/wiki/Rank%20of%20a%20group | In the mathematical subject of group theory, the rank of a group G, denoted rank(G), can refer to the smallest cardinality of a generating set for G, that is
If G is a finitely generated group, then the rank of G is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for p-groups, the rank of the group P is the dimension of the vector space P/Φ(P), where Φ(P) is the Frattini subgroup.
The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group G is the maximum of the ranks of its subgroups:
Sometimes the subgroup rank is restricted to abelian subgroups.
Known facts and examples
For a nontrivial group G, we have rank(G) = 1 if and only if G is a cyclic group. The trivial group T has rank(T) = 0, since the minimal generating set of T is the empty set.
For a free abelian group we have
If X is a set and G = F(X) is the free group with free basis X then rank(G) = |X|.
If a group H is a homomorphic image (or a quotient group) of a group G then rank(H) ≤ rank(G).
If G is a finite non-abelian simple group (e.g. G = An, the alternating group, for n > 4) then rank(G) = 2. This fact is a consequence of the Classification of finite simple groups.
If G is a finitely generated group and Φ(G) ≤ G is the Frattini subgroup of G (which is always normal in G so that the quotient group G/Φ(G) is defined) then rank(G) = rank(G/Φ(G)).
If G is the fundamental group of a closed (that is compact and without boundary) connected 3-manifold M then rank(G)≤g(M), where g(M) is the Heegaard genus of M.
If H,K ≤ F(X) are finitely generated subgroups of a free group F(X) such that the intersection is nontrivial, then L is finitely generated and
rank(L) − 1 ≤ 2(rank(K) − 1)(rank(H) − 1).
This result is due to Hanna Neumann. The Hanna Neumann conjecture states that in fact one always has rank(L) − 1 ≤ (rank(K) − 1)(rank(H) − 1). The Hanna Neumann conjecture has recently been solved by Igor Mineyev and announced independently by Joel Friedman.
According to the classic Grushko theorem, rank behaves additively with respect to taking free products, that is, for any groups A and B we have
rank(AB) = rank(A) + rank(B).
If is a one-relator group such that r is not a primitive element in the free group F(x1,..., xn), that is, r does not belong to a free basis of F(x1,..., xn), then rank(G) = n.
The rank problem
There is an algorithmic problem studied in group theory, known as the rank problem. The problem asks, for a particular class of finitely presented groups if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank p |
https://en.wikipedia.org/wiki/William%20J.%20Milne%20%28educator%29 | William James Milne (1843–1914) was an American educator, academic administrator, and author. He was known for heading two teachers' colleges in New York State, and writing numerous mathematics textbooks.
William J. Milne was born in Scotland in 1843. He was the eldest of six children of Charles and Jean Black Milne. His father brought the family from Scotland to Monroe county NY about 1852 where they resided for a short time then removing to Holley Orleans county NY. Milne worked his own way through school and college and was graduated at the University of Rochester in 1868. Milne was a member of the Delphic Society while a student at Rochester.
In 1871, Milne became principal of what had been planned four years before as Wadsworth Normal and Training School, and officially opened it in Geneseo, N.Y. as the Geneseo Normal and Training School. While at Geneseo, Milne was instrumental in the founding of the Delphic Society (today the Delphic Fraternity.) Milne was on the faculty of what is today the State University of New York at Brockport before his principalship at Geneseo.
He held his Ph.D. by October 1874, when he was ordained an elder of the town's Central Presbyterian Church. He also held an LL.D. degree in 1880, when, in March, that church and the First Presbyterian Church of Geneseo Village united to form the Presbyterian Church of Geneseo Village. He and the other elders of the two churches became the 12 elders of the united church, and in September he was elected as one of its 6 trustees, and also became one of three superintendents of its Sunday school. He continued as an elder and trustee at least into 1887. In 1889 Milne was succeeded as head of Geneseo Normal and Training School by his brother John M. Milne.
In 1889 Milne took the presidency of New York State Normal School at Albany, overseeing development of its mission, as reflected in its name changes to New York State Normal College in 1890, and to New York State College for Teachers in 1914. He died later that year on September 4 at Bethlehem, New Hampshire of a heart ailment. He was buried in Albany, NY.
The institution's early American practice-teaching school was named The Milne School after him, and after the school's closing in 1977, the building has continued, in its new roles, to be called Milne Hall. He was also memorialized—jointly with his brother John M. Milne, who had followed him as president at Geneseo—by the naming of the Milne Library, in 1966, as part of the institution that by then had the name State University College at Geneseo.
Publications
Milne was the author of an extensive mathematics curriculum, with multiple editions, including the following texts and ancillary materials:
The practical arithmetic on the inductive plan including oral and written exercises: Inductive Series (1878)
Basic methods of teaching (1882)
High school algebra: embracing a complete course for high schools and academies (c. 1892)
Standard Arithmetic: embracing a com |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.