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https://en.wikipedia.org/wiki/Functionally%20graded%20element | In materials science and mathematics, functionally graded elements are elements used in finite element analysis. They can be used to describe a functionally graded material.
See also
Graded (mathematics)
Finite element method
Materials science |
https://en.wikipedia.org/wiki/Karambayam | Karambayam is a village in Pattukkottai in the Indian state of Tamil Nadu. The village has Muthu Mariyamman Temple, an Amman temple.
Statistics
External links
Thanjavur
ANNUAL EMPLOYMENT REPORT,THANJAVUR
Villages in Thanjavur district |
https://en.wikipedia.org/wiki/List%20of%20Blackpool%20F.C.%20records%20and%20statistics | This page details Blackpool Football Club's all-time records.
Club records
Results
Largest victory: 10–0 (v. Lanerossi Vicenza, Anglo-Italian Cup on 10 June 1972)
Largest defeat: 1–10 (v. Small Heath, Division Two, on 2 March 1901 and v. Huddersfield Town, Division One, on 13 December 1930)
Consecutive victories: 12 (between 31 March 2007 and 14 August 2007)
Consecutive defeats: 8 (between 26 November 1898 and 7 January 1899 and between 28 November 1964 and 16 January 1965)
League finishing positions
Highest: 2nd in Division One (1956)
Lowest: 21st in Division Four (1983)
Transfer fees
Largest transfer fee paid: £1,250,000 (DJ Campbell, to Leicester City, 2010)
Largest transfer fee received: £6,750,000 (Charlie Adam, from Liverpool, 2011)
Record transfer-fee progression
Paid
Received
Individual records
Players
Most Football League appearances: Jimmy Armfield (569; between 27 December 1954 and 1 May 1971)
Most consecutive League appearances: Georgie Mee (195; between 25 December 1920 and 12 September 1925)
Most goals in total: Jimmy Hampson (252; between 15 October 1927 and 8 January 1938)
Most Football League goals: Jimmy Hampson (248)
Most League goals in one season: Jimmy Hampson (45; in 1929–30)
Most goals in one game: 5 (Jimmy Hampson; v. Reading on 10 November 1928 and Jimmy McIntosh; v. Preston North End on 1 May 1948)
Fastest goal: 11 seconds (Bill Slater; v. Stoke City on 10 December 1949 and James Quinn; v. Bristol City on 12 August 1995
Most capped player: Jimmy Armfield (43; for England)
Managers
Longest-serving manager: Joe Smith (22 years, 9 months; from 1 August 1935 to 30 April 1958)
Personal honours
Ballon d'Or
The following players have won the Ballon d'Or while playing for Blackpool:
Stanley Matthews – 1956
FWA Footballer of the Year
The following players have won the FWA Footballer of the Year award while playing for Blackpool:
Stanley Matthews – 1947–48
Harry Johnston – 1950–51
Attendances
Largest attendance – Pre-2002: 38,098; v. Wolverhampton Wanderers on 17 September 1955.
Largest attendance – 2002 onwards: 16,116 (99.48% of capacity; v. Manchester City on 17 October 2010)
Home gate receipts
Highest: £72,949 (v. Tottenham Hotspur, FA Cup third round, 5 January 1991)
References
Records and Statistics
Blackpool |
https://en.wikipedia.org/wiki/List%20of%20Inter%20Milan%20records%20and%20statistics | Football Club Internazionale Milano is an Italian professional association football club based in Milan that currently plays in the Italian Serie A. They were one of the founding members of Serie A in 1929, and are the only club never to have been relegated from the league. They have also been involved in European football, winning the UEFA Champions League and the UEFA Cup three times each. Inter become the first Italian club to win back-to-back European Cups, achieving the feat in 1964 and 1965.
This list encompasses the major honours won by Inter Milan 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 Inter Milan players on the international stage, and the highest transfer fees paid and received by the club.
Inter has set various records since its founding. In 2010, Inter became the first Italian club to win the treble consisting of Serie A, the Coppa Italia and the UEFA Champions League. Between 2005 and 2010, Inter won five consecutive national championships, a record which was broken by Juventus in the 2016–17 season. Inter has also signed several high-profile players, setting the world record in transfer fees on two occasions with the purchase of Ronaldo in 1997 and Christian Vieri in 1999.
The statistics listed below are updated to 10 June 2023.
Honours
Inter Milan have won 35 domestic trophies, including the league nineteen times, the Coppa Italia nine times and the Supercoppa Italiana seven times. From 2006 to 2010, the club won five successive league titles, equalling the all-time record in that period. Inter has won the Champions League three times; two back-to-back titles in 1964 and 1965, and then another in 2010. The 2010 title completed an unprecedented Italian treble along with the Coppa Italia and the Scudetto. The club has also won three UEFA Cups, two Intercontinental Cups and one FIFA Club World Cup.
National titles
Serie A:
Winners (19): 1909–10, 1919–20, 1929–30, 1937–38, 1939–40, 1952–53, 1953–54, 1962–63, 1964–65, 1965–66, 1970–71, 1979–80, 1988–89, 2005–06, 2006–07, 2007–08, 2008–09, 2009–10, 2020–21
Runners-up (16): 1932–33, 1933–34, 1934–35, 1940–41, 1948–49, 1950–51, 1961–62, 1963–64, 1966–67, 1969–70, 1992–93, 1997–98, 2002–03, 2010–11, 2019–20, 2021–22
Coppa Italia:
Winners (9): 1938–39, 1977–78, 1981–82, 2004–05, 2005–06, 2009–10, 2010–11, 2021–22, 2022–23
Runners-up (6): 1958–59, 1964–65, 1976–77, 1999–2000, 2006–07, 2007–08
Supercoppa Italiana:
Winners (7): 1989, 2005, 2006, 2008, 2010, 2021, 2022
Runners-up (4): 2000, 2007, 2009, 2011
International titles
The following titles include only those which are recognised by UEFA and FIFA.
World-wide titles
Intercontinental Cup:
Winners (2): 1964, 1965
Intercontinental Supercup:
Runners-up (1): 1968
FIFA Club World Cup:
Winners (1): 2010
European titles
Europ |
https://en.wikipedia.org/wiki/Microregions%20in%20Goi%C3%A1s | The state of Goiás is divided into 18 statistical microregions by the Instituto Brasileiro de Geografia e Estatístic (IBGE). These have no administrative function but are used only for statistics. The total area of the state is 341,289 km2 and the population is 5,450,303 as of 2007.
Thirty seven percent of the population—2,032,305—lives in the Goiânia Microregion and 17.6%--960,141—of the population lives in the Entorno do Distrito Federal Microregion. The GDP of the state of Goiás was R$50,536 billion in 2005, the last year data was gathered.
Microregions of Goiás
References
Sepin
IBGE |
https://en.wikipedia.org/wiki/Adams%20Prize%20%28disambiguation%29 | Adams Prize may refer to:
Herbert Baxter Adams Prize, of the American Historical Association
Adams Prize, by the University of Cambridge and St John's College for research in mathematics
Douglas Adams prize, in honor of Douglas Adams, given by St John's College for humorous writing |
https://en.wikipedia.org/wiki/Violin%20plot | A violin plot is a statistical graphic for comparing probability distributions. It is similar to a box plot, with the addition of a rotated kernel density plot on each side.
History
The violin plot was proposed in 1997 by Jerry L. Hintze and Ray D. Nelson as a way to display even more information than box plots, which were created by John Tukey in 1977. The name comes from the plot's alleged resemblance to a violin.
About
Violin plots are similar to box plots, except that they also show the probability density of the data at different values, usually smoothed by a kernel density estimator. A violin plot will include all the data that is in a box plot: a marker for the median of the data; a box or marker indicating the interquartile range; and possibly all sample points, if the number of samples is not too high.
While a box plot shows a summary statistics such as mean/median and interquartile ranges, the violin plot shows the full distribution of the data. The violin plot can be used in multimodal data (more than one peak). In this case a violin plot shows the presence of different peaks, their position and relative amplitude.
Like box plots, violin plots are used to represent comparison of a variable distribution (or sample distribution) across different "categories" (for example, temperature distribution compared between day and night, or distribution of car prices compared across different car makers).
A violin plot can have multiple layers. For instance, the outer shape represents all possible results. The next layer inside might represent the values that occur 95% of the time. The next layer (if it exists) inside might represent the values that occur 50% of the time.
Violin plots are less popular than box plot. Violin plots may be harder to understand for readers not familiar with them. In this case, a more accessible alternative is to plot a series of stacked histograms or kernel density distributions.
References
External links
Vioplot add-in for Stata
Violinplot from a wide-form dataset with the seaborn statistical visualization library based on matplotlib
Statistical charts and diagrams |
https://en.wikipedia.org/wiki/Maryse%20Marpsat | Maryse Marpsat (born 1951) is a French sociologist and statistician whose work employs methods drawn from sociology and statistics but also mathematics. Her major sociological works concern poverty, inequality and homeless situation. She is a civil servant, administrator of the French National Institute of Statistics (INSEE) and a fellow of the CSU, a French research institute specializing in sociological studies in urban societies.
Between 1983 and 1993, as a fellow of two French-British groups, the Cambridge Group for the History of Population and Social Structure and the Centre for Economic Policy Research in Cambridge, she worked to develop the comparisons of the household ways of living in France and England, particularly with Richard Wall and Bruce Penhale, from the City University of London.
Since 2007, Maryse Marpsat has been giving some courses on the topics of her researches. She focuses her last teaching, first, on relationships between statistic tools and their outcomes, and second, on "inequalities", their range, their origins and the different ways of their evaluation. Moreover, since 2007, she has provided on a monthly base some research workshops in her institute of statistics, for example in 2012 going through the feeling of inequalities either for owners of the French minimum wage called "RSA" or for retired people.
Notes
Works
« Combiner les méthodes et les points de vue. De l’enquête statistique au journal intime d’Albert Vanderburg », in Pichon Pascale, SDF, sans-abri, itinérant. Oser la comparaison, Presses de l'Université de Louvain, Louvain, 2009.
(with Pascale Pichon), « La genèse de la recherche en France : composer avec le mouvement associatif, observer et analyser des situations vécues et des rapports sociaux, caractériser et compter », in Pichon Pascale, SDF, sans-abri, itinérant. Oser la comparaison, Presses de l'Université de Louvain, Louvain, 2009.
2008, (with Cécile Brousse and Jean-Marie Firdion), Les sans-domicile, La découverte, Collection Repère,
2008, « L’enquête de l’Insee sur les sans-domicile : quelques éléments historiques », Courrier des statistiques, INSEE, Paris, n° 123, janvier-avril p. 53-64
2006, « La légitimation de l’Art Brut. De la conservation à la consécration », in Gérard Mauger (edited), Droits d’entrée. Modalités et conditions d'accès aux univers artistiques, Maison des Sciences de l’Homme, 89-130, Paris
2006, « Une forme discrète de pauvreté : les personnes logées utilisant les distributions de repas chauds », Économie et Statistique, n°391-392, INSEE, Paris
2004, (with Albert Vanderburg), Le monde d'Albert la Panthère, cybernaute et sans-domicile à Honolulu, Bréal, coll. d'Autre Part, Paris
2004, « Les personnes sans domicile ou mal logées (The homeless or poorly housed) », Travail, Genre et Sociétés, n° 11, pp. 79–93.
2004, (with Martine Quaglia and Nicolas Razafindratsima), « Les sans domicile et les services itinérants », Travaux de l'Observatoire de l'exclusion 2003-2004 |
https://en.wikipedia.org/wiki/Matthew%20Bound | Matthew Terence Bound (born 9 November 1972) is an English former football defender.
Career statistics
Honours
Individual
PFA Team of the Year: 1999–2000 Third Division
References
External links
Since 1888... The Searchable Premiership and Football League Player Database (subscription required)
1972 births
Living people
English men's footballers
Men's association football defenders
Premier League players
Southampton F.C. players
Hull City A.F.C. players
Stockport County F.C. players
Lincoln City F.C. players
Swansea City A.F.C. players
Oxford United F.C. players
Weymouth F.C. players
Eastleigh F.C. players
English Football League players
People from Melksham
Footballers from Wiltshire |
https://en.wikipedia.org/wiki/Lineament | See also Line (geometry)
A lineament is a linear feature in a landscape which is an expression of an underlying geological structure such as a fault. Typically a lineament will appear as a fault-aligned valley, a series of fault or fold-aligned hills, a straight coastline or indeed a combination of these features. Fracture zones, shear zones and igneous intrusions such as dykes can also be expressed as geomorphic lineaments.
Lineaments are often apparent in geological or topographic maps and can appear obvious on aerial or satellite photographs. There are for example, several instances within Great Britain. In Scotland the Great Glen Fault and Highland Boundary Fault give rise to lineaments as does the Malvern Line in western England and the Neath Disturbance in South Wales.
The term 'megalineament' has been used to describe such features on a continental scale. The trace of the San Andreas Fault might be considered an example.
The Trans Brazilian Lineament and the Trans-Saharan Belt, taken together, form perhaps the longest coherent shear zone on the Earth, extending for about 4,000 km.
Lineaments have also been identified on other planets and their moons. Their origins may be radically different from those of terrestrial lineaments due to the differing tectonic processes involved.
References
Geomorphology
Structural geology |
https://en.wikipedia.org/wiki/ABC%40Home | ABC@Home was an educational and non-profit network computing project finding abc-triples related to the abc conjecture in number theory using the Berkeley Open Infrastructure for Network Computing (BOINC) volunteer computing platform.
In March 2011, there were more than 7,300 active participants from 114 countries with a total BOINC credit of more than 2.9 billion, reporting about 10 teraflops (10 trillion operations per second) of processing power.
In 2011, the project met its goal of finding all abc-triples of at most 18 digits. By 2015, the project had found 23.8 million triples in total, and ceased operations soon after.
See also
List of volunteer computing projects
References
External links
The Mathematical Institute of Leiden University
Science in society
Computational number theory
Volunteer computing projects |
https://en.wikipedia.org/wiki/Kohei%20Tanaka%20%28footballer%29 | is a former Japanese football player.
Club statistics
Updated to 5 September 2014.
References
External links
1985 births
Living people
Association football people from Hokkaido
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Kashima Antlers players
Montedio Yamagata players
Vegalta Sendai players
FC Ryukyu players
Men's association football forwards |
https://en.wikipedia.org/wiki/Beltrami%20identity | The Beltrami identity, named after Eugenio Beltrami, is a special case of the Euler–Lagrange equation in the calculus of variations.
The Euler–Lagrange equation serves to extremize action functionals of the form
where and are constants and .
If , then the Euler–Lagrange equation reduces to the Beltrami identity,
where is a constant.
Derivation
By the chain rule, the derivative of is
Because , we write
We have an expression for from the Euler–Lagrange equation,
that we can substitute in the above expression for to obtain
By the product rule, the right side is equivalent to
By integrating both sides and putting both terms on one side, we get the Beltrami identity,
Applications
Solution to the brachistochrone problem
An example of an application of the Beltrami identity is the brachistochrone problem, which involves finding the curve that minimizes the integral
The integrand
does not depend explicitly on the variable of integration , so the Beltrami identity applies,
Substituting for and simplifying,
which can be solved with the result put in the form of parametric equations
with being half the above constant, , and being a variable. These are the parametric equations for a cycloid.
Notes
References
Calculus of variations
Optimal control |
https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard%20inequality | In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [a, b] → R is convex, then the following chain of inequalities hold:
The inequality has been generalized to higher dimensions: if is a bounded, convex domain and is a positive convex function, then
where is a constant depending only on the dimension.
A corollary on Vandermonde-type integrals
Suppose that , and choose distinct values from . Let be convex, and let denote the "integral starting at " operator; that is,
.
Then
Equality holds for all iff is linear, and for all iff is constant, in the sense that
The result follows from induction on .
References
Jacques Hadamard, "Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann", Journal de Mathématiques Pures et Appliquées, volume 58, 1893, pages 171–215.
Zoltán Retkes, "An extension of the Hermite–Hadamard Inequality", Acta Sci. Math. (Szeged), 74 (2008), pages 95–106.
Mihály Bessenyei, "The Hermite–Hadamard Inequality on Simplices", American Mathematical Monthly, volume 115, April 2008, pages 339–345.
Flavia-Corina Mitroi, Eleutherius Symeonidis, "The converse of the Hermite-Hadamard inequality on simplices", Expo. Math. 30 (2012), pp. 389–396. ;
Stefan Steinerberger, The Hermite-Hadamard Inequality in Higher Dimensions, The Journal of Geometric Analysis, 2019.
Inequalities
Theorems involving convexity |
https://en.wikipedia.org/wiki/Steiner%20inellipse | In geometry, the Steiner inellipse, midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inellipse. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.
The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's centroid.
Definition and properties
Definition
An ellipse that is tangent to the sides of a triangle at its midpoints is called the Steiner inellipse of .
Properties:
For an arbitrary triangle with midpoints of its sides the following statements are true:
a) There exists exactly one Steiner inellipse.
b) The center of the Steiner inellipse is the centroid of .
c1) The triangle has the same centroid and the Steiner inellipse of is the Steiner ellipse of the triangle
c2) The Steiner inellipse of a triangle is the scaled Steiner Ellipse with scaling factor 1/2 and the centroid as center. Hence both ellipses have the same eccentricity, are similar.
d) The area of the Steiner inellipse is -times the area of the triangle.
e) The Steiner inellipse has the greatest area of all inellipses of the triangle.
Proof
The proofs of properties a),b),c) are based on the following properties of an affine mapping: 1) any triangle can be considered as an affine image of an equilateral triangle. 2) Midpoints of sides are mapped onto midpoints and centroids on centroids. The center of an ellipse is mapped onto the center of its image.
Hence its suffice to prove properties a),b),c) for an equilateral triangle:
a) To any equilateral triangle there exists an incircle. It touches the sides at its midpoints. There is no other (non-degenerate) conic section with the same properties, because a conic section is determined by 5 points/tangents.
b) By a simple calculation.
c) The circumcircle is mapped by a scaling, with factor 1/2 and the centroid as center, onto the incircle. The eccentricity is an invariant.
d) The ratio of areas is invariant to affine transformations. So the ratio can be calculated for the equilateral triangle.
e) See Inellipse.
Parametric representation and semi-axes
Parametric representation:
Because a Steiner inellipse of a triangle is a scaled Steiner ellipse (factor 1/2, center is centroid) one gets a parametric representation derived from the trigonometric representation of the Steiner ellipse :
The 4 vertices of the Steiner inellipse are
where is the solution of
with
Semi-axes:
With the abbreviations
one gets for the semi-axes (where ):
The linear eccentricity of the Steiner inellipse is
Trilinea |
https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20of%20Cambodia | The National Institute of Statistics is the branch of the Cambodian Ministry of Planning responsible for the collection, processing, and dissemination of official national statistics. It oversees the Social Statistics Department, the Census and Survey Department, the General Statistics Department, and the Economics Statistics Department.
See also
Cabinet of Cambodia
Demographics of Cambodia
List of national and international statistical services
References
http://www.nis.gov.kh/
http://globaledge.msu.edu/countryInsights/country.asp?CountryID=16
External links
Cambodia National Institute of Statistics homepage
Ministry of Planning
Government agencies of Cambodia
Cambodia |
https://en.wikipedia.org/wiki/Hyperbolic%20point | In mathematics, a hyperbolic point is a certain kind of point, one of:
A point in a hyperbolic geometry
A point of negative Gaussian curvature on a smooth surface
A hyperbolic equilibrium point of a dynamical system |
https://en.wikipedia.org/wiki/Oscar%20Johansson%20%28ice%20hockey%29 | Oscar Petter Johansson (born May 11, 1988) is a Swedish professional ice hockey winger, currently playing for Borås HC in the HockeyAllsvenskan.
Career statistics
Regular season and playoffs
References
External links
1988 births
Living people
Borås HC players
IF Sundsvall Hockey players
Swedish ice hockey left wingers
Timrå IK players
People from Karlshamn
Sportspeople from Blekinge County |
https://en.wikipedia.org/wiki/Optical%20granulometry | Optical granulometry is the process of measuring the different grain sizes in a granular material, based on a photograph. Technology has been created to analyze a photograph and create statistics based on what the picture portrays. This information is vital in maintaining machinery in various trades worldwide. Mining companies can use optical granulometry to analyze inactive or moving rock to quantify the size of these fragments. Forestry companies can zero in on wood chip sizes without stopping the production process, and minimize sizing errors.
With more photoanalysis technologies being produced, mining companies have shown an increased interest in these types of systems because of their ability to maintain efficiency throughout the mining process. Companies are saving millions of dollars annually because of this new technology, and are cutting back on maintenance costs on equipment.
In order for optical granulometry to be completely successful, an accurate photo must be taken – under sufficient lighting, and using proper technology – to obtain quantified results. If these requirements are met, an image analysis system can be implemented.
The process
Software uses four basic steps in determining the average size of material:
See the Wikipedia article on Photoanalysis to see how mining, forestry and agricultural companies are using this technology to improve quality control techniques.
See also
Particle-size distribution
Grain size
Granulometry (morphology)
Notes
References
Measurement of Blast Fragmentation: Proceedings of a Workshop Held Parallel With Fragblast-5, Montreal, 26–29 August 1996, by John A. Franklin, Takis Katsabanis, Published by Taylor & Francis, 1996,
External links
Free Web Application for Particle and Grain Analysis
Mining techniques
Image processing
Granulometric analyses |
https://en.wikipedia.org/wiki/Geometry%20Wars%3A%20Retro%20Evolved%202 | Geometry Wars: Retro Evolved 2 is a multidirectional shooter video game created by Activision subsidiary Bizarre Creations, released on Xbox Live Arcade on July 30, 2008 as a sequel to
Geometry Wars: Retro Evolved. It was followed by Geometry Wars 3: Dimensions, a sequel published in 2014 by Lucid Games, which was founded by former members of Bizarre Creations.
Gameplay
The player controls a maneuverable claw-shaped figure that can move and fire independently in any direction. The objective of the game is to score points by destroying a variety of enemy shapes which spawn around the playing field, contact with any enemy results in death and the loss of a life. Bombs destroy all enemies on the playing field but award no points.
Crucial to effective play is the score multiplier, which increases as the player collects "geoms", small green objects dropped by enemies upon destruction. The number of points scored by destroying an enemy depends on the multiplier, which can reach into the thousands.
There are six different game modes available:
Deadline: The player must score as many points as possible with a time limit of three minutes and unlimited lives.
King: The player has only one life and no bombs. Circular safety zones appear sporadically in the playing field; enemies and geoms cannot enter the zones, while the player can only shoot inside them. Zones shrink and disappear after being entered, forcing players to move between zones to survive and collect geoms.
Evolved: Similar in style to Geometry Wars: Retro Evolved, the player is challenged to score as many points as possible with a finite number of lives and bombs, with no time limit. Additional lives are earned at set point breaks.
Pacifism: The player has one life and cannot shoot. To avoid large swarms of slow-moving blue enemies, the player must fly through gates, which explode and destroy nearby enemies upon contact. This mode was inspired by an achievement in the first game called "Pacifist" where the objective was to survive for 60 seconds without firing.
Waves: The player has one life and must avoid and destroy inline waves of orange 'rocket' enemies that pace horizontally and vertically from the edges of the playing field. This mode was introduced as a minigame inside Project Gotham Racing 4.
Sequence: The mode consists of twenty sequential, predetermined levels with 30 seconds allotted per level. Points are awarded for successfully destroying all enemies in each level without death and under the time limit.
Retro Evolved 2 provides local cooperative and competitive multiplayer modes for two to four players simultaneously, and an exclusive "Co-Pilot" mode in which two players control the same ship, with one moving and the other firing. Additionally, the game provides support for worldwide leaderboards in each game mode and, by default, displays the player's ranking against their friends during play.
Development history
In developing the sequel the team struggled with cre |
https://en.wikipedia.org/wiki/Additively%20indecomposable%20ordinal | In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any , we have Additively indecomposable ordinals are also called gamma numbers or additive principal numbers. The class of additively indecomposable ordinals may be denoted , from the German "Hauptzahl". The additively indecomposable ordinals are precisely those ordinals of the form for some ordinal .
From the continuity of addition in its right argument, we get that if and α is additively indecomposable, then
Obviously 1 is additively indecomposable, since No finite ordinal other than is additively indecomposable. Also, is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.
The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by .
The derivative of (which enumerates its fixed points) is written Ordinals of this form (that is, fixed points of ) are called epsilon numbers. The number is therefore the first fixed point of the sequence
Multiplicatively indecomposable
A similar notion can be defined for multiplication. If α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable. 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (also called delta numbers) are those of the form for any ordinal α. Every epsilon number is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable. The delta numbers (other than 2) are the same as the prime ordinals that are limits.
Higher indecomposables
Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of ), and so on. Therefore, is the first ordinal which is -indecomposable for all , where denotes Knuth's up-arrow notation.
See also
Ordinal arithmetic
References
Ordinal numbers |
https://en.wikipedia.org/wiki/Vertex%20angle | In geometry, a vertex is an angle (shape) associated with a vertex of an n-dimensional polytope. In two dimensions it refers to the angle formed by two intersecting lines, such as at a "corner" (vertex) of a polygon. In higher dimensions there can be more than two lines (edges) meeting at a vertex, making a description of the angle shape more complicated.
In three dimensions and three-dimensional polyhedra, a vertex angle is a polyhedral angle or n-hedral angle. It is described by a sequence of n face angles, which are the angles formed by two edges of polyhedron meeting at the vertex, or by a sequence of n dihedral angles, which are the angles between two faces sharing the vertex. The angle may be quantified using a single number by the interior solid angle at the vertex (the spherical excess), which is related to the sum of the dihedral angles, or by the angular defect (or excess) of the vertex, which is related to the sum of the face angles, or other metrics such as the polar sine. The simplest type of polyhedral angle is a trihedral angle or trihedron (bounded by three planes), as found at the vertices of a Parallelepiped or tetrahedron.
For higher-dimensional polytopes, a vertex angle can be quantified using a higher-dimensional solid angle, i.e. by the portion of the n-sphere around the vertex that is interior to the polytope. Face and dihedral angles also generalize to higher dimensions.
The term vertex angle is sometimes used synonymously with face angle, i.e. the angle between two edges meeting at a vertex. It may also refer to the (higher-dimensional) interior solid angle at a vertex.
Properties
A vertex angle in a polygon is often measured on the interior side of the vertex. For any simple n-gon, the sum of the interior angles is π(n − 2) radians or 180(n − 2) degrees.
The face and dihedral angles of a polyhedral angle can be related to each other by interpreting the polyhedral angle as a spherical polygon, whose side lengths are the face angles and whose vertex angles are the dihedral angles; the surface area of the polygon is the solid angle of the vertex (see spherical trigonometry, in particular the spherical law of cosines).
The higher dimensional analogue of a right angle is the vertex angle formed by mutually perpendicular edges, such as at the vertex of a hypercube. In three dimensions such an angle can be found in a trirectangular tetrahedron or a corner reflector.
See also
Complementary angles
Supplementary angles
Dihedral angle
Polar sine
Solid angle
References
Euclidean geometry
Angle |
https://en.wikipedia.org/wiki/Spiral%20of%20Theodorus | In geometry, the spiral of Theodorus (also called square root spiral, Spiral of Einstein, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene.
Construction
The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle is formed, an automedian right triangle with one leg being the hypotenuse of the prior triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second triangle is the square root of 3. The process then repeats; the th triangle in the sequence is a right triangle with the side lengths and 1, and with hypotenuse . For example, the 16th triangle has sides measuring , 1 and hypotenuse of .
History and uses
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus.
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.
Hypotenuse
Each of the triangles' hypotenuses gives the square root of the corresponding natural number, with .
Plato, tutored by Theodorus, questioned why Theodorus stopped at . The reason is commonly believed to be that the hypotenuse belongs to the last triangle that does not overlap the figure.
Overlapping
In 1958, Kaleb Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line, they will never pass through any of the other vertices of the total figure.
Extension
Theodorus stopped his spiral at the triangle with a hypotenuse of . If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
Growth rate
Angle
If is the angle of the th triangle (or spiral segment), then:
Therefore, the growth of the angle of the next triangle is:
The sum of the angles of the first triangles is called the total angle for the th triangle. It grows proportionally to the square root of , with a bounded correction term :
where
().
Radius
The growth of the radius of the spiral at a certain triangle is
Archimedean spiral
The Spiral of Theodorus approximates the Archimedean spiral. Just as the distance between two windings of the Archimedean spiral equals mathematical constant , as the number of spins of the spiral of Theodorus approaches infinity, the distance between two consecutive windings quickly approaches .
The following is a table showing of two windings of the spiral approaching pi:
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to .
Conti |
https://en.wikipedia.org/wiki/Simply%20connected%20at%20infinity | In topology, a branch of mathematics, a topological space X is said to be simply connected at infinity if for any compact subset C of X, there is a compact set D in X containing C so that the induced map
is the zero map. Intuitively, this is the property that loops far away from a small subspace of X can be collapsed, no matter how bad the small subspace is.
The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity. Since this property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3.
However, it is a theorem of John R. Stallings that for , a contractible n-manifold is homeomorphic to Rn precisely when it is simply connected at infinity.
References
Algebraic topology
Properties of topological spaces |
https://en.wikipedia.org/wiki/Uniform%20law | Uniform law may refer to:
Uniform distribution (disambiguation), any of several concepts in mathematics
Uniform Act, a model statute designed to be adopted by many jurisdictions
A body of harmonised laws, see harmonisation of law
Dress code
School uniform rules or regulations |
https://en.wikipedia.org/wiki/Lam%C3%A9%20function | In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials.
The Lamé equation
Lamé's equation is
where A and B are constants, and is the Weierstrass elliptic function. The most important case is when , where is the elliptic sine function, and for an integer n and the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points.
By changing the independent variable to with , Lamé's equation can also be rewritten in algebraic form as
which after a change of variable becomes a special case of Heun's equation.
A more general form of Lamé's equation is the ellipsoidal equation or ellipsoidal wave equation which can be written (observe we now write , not as above)
where is the elliptic modulus of the Jacobian elliptic functions and and are constants. For the equation becomes the Lamé equation with . For the equation reduces to the Mathieu equation
The Weierstrassian form of Lamé's equation is quite unsuitable for calculation (as Arscott also remarks, p. 191). The most suitable form of the equation is that in Jacobian form, as above. The algebraic and trigonometric forms are also cumbersome to use. Lamé equations arise in quantum mechanics as equations of small fluctuations about classical solutions—called periodic instantons, bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials.
Asymptotic expansions
Asymptotic expansions of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of have been obtained by Müller.
The asymptotic expansion obtained by him for the eigenvalues is, with approximately an odd integer (and to be determined more precisely by boundary conditions – see below),
(another (fifth) term not given here has been calculated by Müller, the first three terms have also been obtained by Ince). Observe terms are alternately even and odd in and (as in the corresponding calculations for Mathieu functions, and oblate spheroidal wave functions and prolate spheroidal wave functions). With the following boundary conditions (in which is the quarter period given by a complete elliptic integral)
as well as (the prime meaning derivative)
defining respectively the ellipsoidal wave functions
of periods and for one obtains
Here the upper sign refers to the solutions and the lower to the solutions . Finally expanding about one obtains
In the limit of the Mathieu equation (to which the Lamé equation can be reduced) these expressions reduce to the corresponding expressions of the Ma |
https://en.wikipedia.org/wiki/Economics%20of%20the%20FIFA%20World%20Cup | The FIFA World Cup is said to have a significant impact on the host country's economy.
Statistics
Italy (1990)
United States (1994)
The World Cup in the United States was hosted in a number of cities. In Los Angeles, site of the final, there was a total economic profit of 623 million dollars that went directly into the metropolitan economy. To help one better understand this figure, in comparison of that same year the Super Bowl only accounted for 182 million dollars (Nodell). These figures were calculated over just a one-month period in which these games took place. Just in California, reports from the Pasadena Convention and Visitor's Bureau conclude that 1,700 part-time jobs became available during the preparation and duration of the event (Deady). New York City, San Francisco, and Boston received combined revenue of one billion and forty-five million dollars. The overall increase on hotels and food and beverages was ten and fifteen percent from the previous year. This money spent on hotels and restaurants helps the entire U.S. economy in that many of these hotels and restaurants are chains and corporations. Hence, the money made is spread throughout the corporation and it was found to be used for the opening of new facilities and expansions of the corporation.
In addition to the direct impacts of the 1994 World Cup, there are many indirect impacts as well. In order to host the Cup the United States had to develop a national soccer league, resulting in the formation of Major League Soccer (MLS) in 1996. Construction of new facilities, sponsorship of new teams, and the revenue of the ticket sales all resulted in economic boosts. The newly introduced professional league engendered one of the fastest growing youth sports in the country. Youth soccer took off and the selling of apparel and gear for the sport was a target for private businesses to focus on selling.
France (1998)
South Korea / Japan (2002)
In the 2002 World Cup, several other advantages were discovered when the host was split between Japan and Korea. This was the first time the tournament had been hosted in two countries, with thirty two matches being played in each country with a grand total of sixty four matches. With the three million live spectators ticket sales were 1.2 billion dollars. FIFA promised each country 110 million for hosting and all revenue from their ticket sales.
Each country expanded their 20 soccer facilities with a total investment of 4.7 billion. A host country can also see value in the national exposure with so many people viewing and attending the event.
It was predicted prior to the 2002 Cup that the England team's absence would cost the economy 4.7 billion in lost output or about .3% of their GDP (Gross Domestic Product) were they to win the entire tournament. Should the England team lose in just the first two weeks however, the losses are only expected to reach a total of 1.8 billion.
Germany (2006)
The 2006 World Cup was judged a success co |
https://en.wikipedia.org/wiki/List%20of%20convolutions%20of%20probability%20distributions | In probability theory, the probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively. Many well known distributions have simple convolutions. The following is a list of these convolutions. Each statement is of the form
where are independent random variables, and is the distribution that results from the convolution of . In place of and the names of the corresponding distributions and their parameters have been indicated.
Discrete distributions
Continuous distributions
The following three statements are special cases of the above statement:
where is a random sample from and
Mixed distributions:
See also
Algebra of random variables
Relationships among probability distributions
Infinite divisibility (probability)
Bernoulli distribution
Binomial distribution
Cauchy distribution
Erlang distribution
Exponential distribution
Gamma distribution
Geometric distribution
Hypoexponential distribution
Lévy distribution
Poisson distribution
Stable distribution
Mixture distribution
Sum of normally distributed random variables
References
Sources
Convolutions
Probability distributions, convolutions |
https://en.wikipedia.org/wiki/Carath%C3%A9odory%20conjecture | In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924. Carathéodory did publish a paper on a related subject, but never committed the conjecture into writing. In, John Edensor Littlewood mentions the conjecture and Hamburger's contribution as an example of a mathematical claim that is easy to state but difficult to prove. Dirk Struik describes in the formal analogy of the conjecture with the Four Vertex Theorem for plane curves. Modern references to the conjecture are the problem list of Shing-Tung Yau, the books of Marcel Berger, as well as the books.
The conjecture has had a troubled history with published proofs in the analytic case which contained gaps, and claims of proof in the general smooth case which have not been accepted for publication.
Statement of the conjecture
The conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points. In the sense of the conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of the umbilicus. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.
The case of real analytic surfaces
The invited address of Stefan Cohn-Vossen to the International Congress of Mathematicians of 1928 in Bologna was on the subject and in the 1929 edition of Wilhelm Blaschke's third volume on Differential Geometry he states:
While this book goes into print, Mr. Cohn-Vossen has succeeded in proving that closed real-analytic surfaces do not have umbilic points of index > 2 (invited talk at the ICM in Bologna 1928). This proves the conjecture of Carathéodory for such surfaces, namely that they need to have at least two umbilics.
Here Blaschke's index is twice the usual definition for an index of an umbilic point, and the global conjecture follows by the Poincaré–Hopf index theorem. No paper was submitted by Cohn-Vossen to the proceedings of the International Congress, while in later editions of Blaschke's book the above comments were removed. It is, therefore, reasonable to assume that this work was inconclusive.
For analytic surfaces, an affirmative answer to this conjecture was given in 1940 by Hans Hamburger in a long paper published in three parts. The approach of Hamburger was also via a local index estimate for isolated umbilics, which he had shown to imply the conjecture in his earlier work. In 1943, a shorter proof was proposed by Gerrit Bol, see also, but, in 1959, Tilla Klotz found and corrected a gap in Bol's proof in. Her proof, in turn, was announced to be incomplete in Hanspeter Scherbel's dissertation (no results of that dissertation related to the Carathéodory conjecture were published f |
https://en.wikipedia.org/wiki/Engel%20group | In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the ''n-Engel condition that the repeated commutator [...[[x,y],y], ..., y] with n copies of y is trivial (where [x, y] means xyx−1y−1 or the Lie bracket). It is called an Engel element if it satisfies the Engel condition that it is n-Engel for some n.
A Lie group or Lie algebra is said to satisfy the Engel or n-Engel conditions if every element does. Such groups or algebras are called Engel groups, n-Engel groups, Engel algebras, and n''-Engel algebras.
Every nilpotent group or Lie algebra is Engel. Engel's theorem states that every finite-dimensional Engel algebra is nilpotent. gave examples of non-nilpotent Engel groups and algebras.
Notes
Group theory
Lie algebras |
https://en.wikipedia.org/wiki/Santiago%20Kuhl | Santiago Jorge Kuhl (born 21 October 1981 in Buenos Aires) is an Argentine footballer.
References
Clausura 2007 Statistics at Terra.com.ar
Swiss Challenge League 2007/08 Statistics at Eurosoccer.ch
2003-04 Statistics at LFP.es
FC Locarno profile
1981 births
Living people
Footballers from Buenos Aires
Argentine men's footballers
Men's association football midfielders
Argentinos Juniors footballers
FC Baden players
FC Luzern players
AD Ceuta footballers
CD Leganés players
FC Locarno players
Argentine Primera División players
Argentine expatriate men's footballers
Argentine expatriate sportspeople in Spain
Expatriate men's footballers in Switzerland
Expatriate men's footballers in Spain
Argentine people of German descent |
https://en.wikipedia.org/wiki/Joint%20embedding%20property | In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for all structures A and B in K, there is a structure C in K such that both A and B have embeddings into C.
It is one of the three properties used to define the age of a structure.
A first-order theory has the joint embedding property if the class of its models of has the joint embedding property. A complete theory has the joint embedding property. Conversely a model-complete theory with the joint embedding property is complete.
A similar but different notion to the joint embedding property is the amalgamation property. To see the difference, first consider the class K (or simply the set) containing three models with linear orders, L1 of size one, L2 of size two, and L3 of size three. This class K has the joint embedding property because all three models can be embedded into L3. However, K does not have the amalgamation property. The counterexample for this starts with L1 containing a single element e and extends in two different ways to L3, one in which e is the smallest and the other in which e is the largest. Now any common model with an embedding from these two extensions must be at least of size five so that there are two elements on either side of e.
Now consider the class of algebraically closed fields. This class has the amalgamation property since any two field extensions of a prime field can be embedded into a common field. However, two arbitrary fields cannot be embedded into a common field when the characteristic of the fields differ.
Notes
References
Model theory |
https://en.wikipedia.org/wiki/Buchberger | Buchberger may refer to:
People
Bruno Buchberger (born 1942), professor of computer mathematics at Johannes Kepler University
Hubert Buchberger (born 1951), German violinist, conductor and music university teacher
Kelly Buchberger (born 1966), Canadian professional ice hockey coach and former player
Kerri Buchberger (born 1970), retired female volleyball player from Canada
Michael Buchberger (1874–1961), Roman Catholic priest
Walter Buchberger (1895–1970), Czechoslovak skier of German ethnicity
Other
Buchberger Leite, a gorge near Hohenau in the Lower Bavarian county of Freyung-Grafenau in Bavaria
Buchberger's algorithm, a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order |
https://en.wikipedia.org/wiki/Geometric%20lattice | In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite, or finite and infinite, matroids, and every geometric or matroid lattice comes from a matroid in this way.
Definition
A lattice is a poset in which any two elements and have both a least upper bound, called the join or supremum, denoted by , and a greatest lower bound, called the meet or infimum, denoted by .
The following definitions apply to posets in general, not just lattices, except where otherwise stated.
For a minimal element , there is no element such that .
An element covers another element (written as or ) if and there is no element distinct from both and so that .
A cover of a minimal element is called an atom.
A lattice is atomistic if every element is the supremum of some set of atoms.
A poset is graded when it can be given a rank function mapping its elements to integers, such that whenever , and also whenever .
When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one.
A graded lattice is semimodular if, for every and , its rank function obeys the identity
A matroid lattice is a lattice that is both atomistic and semimodular. A geometric lattice is a finite matroid lattice.
Many authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.
Lattices vs. matroids
The geometric lattices are equivalent to (finite) simple matroids, and the matroid lattices are equivalent to simple matroids without the assumption of finiteness (under an appropriate definition of infinite matroids; there are several such definitions). The correspondence is that the elements of the matroid are the atoms of the lattice and an element x of the lattice corresponds to the flat of the matroid that consists of those elements of the matroid that are atoms
Like a geometric lattice, a matroid is endowed with a rank function, but that function maps a set of matroid elements to a number rather than taking a lattice element as its argument. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and it must be submodular, meaning that it obeys an inequality similar to the one for semimodular ranked lattices:
for sets X and Y of matroid elements.
The maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union. In this way, the flats of a matroid form a matroid lattice, or (if the |
https://en.wikipedia.org/wiki/Ernst%20equation | In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist .
The Ernst equation
The equation reads:
For its Lax pair and other features see e.g. and references therein.
Usage
The Ernst equation is employed in order to produce the exact solutions of the Einstein's equations in the general theory of relativity.
References
Partial differential equations
General relativity
Integrable systems |
https://en.wikipedia.org/wiki/Mart%C3%ADn%20Miguel%20Cort%C3%A9s | Martín Miguel Cortes (born 7 January 1983 in Punta Alta) is an Argentine-born Chilean footballer, who plays for Curicó Unido.
External links
Profile at Defe.com.ar
2004-05 Statistics at Sport.be
1983 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football midfielders
R.A.E.C. Mons (1910) players
Defensores de Belgrano footballers
Paysandu Sport Club players
Ñublense footballers
Curicó Unido footballers
Chilean Primera División players
Primera B de Chile players
Footballers from Buenos Aires Province
Expatriate men's footballers in Belgium
Expatriate men's footballers in Brazil
Expatriate men's footballers in Chile
Naturalized citizens of Chile |
https://en.wikipedia.org/wiki/Modal%20algebra | In algebra and logic, a modal algebra is a structure such that
is a Boolean algebra,
is a unary operation on A satisfying and for all x, y in A.
Modal algebras provide models of propositional modal logics in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics.
Stone's representation theorem can be generalized to the Jónsson–Tarski duality, which ensures that each modal algebra can be represented as the algebra of admissible sets in a modal general frame.
A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying . Magari algebras correspond to provability logic.
See also
Interior algebra
Heyting algebra
References
A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Logic Guides vol. 35, Oxford University Press, 1997.
Algebra
Boolean algebra |
https://en.wikipedia.org/wiki/List%20of%20Super%20League%20records | The top tier of English rugby league was renamed the Super League for the start of the 1996 season. The following page details the records and statistics of the Super League.
League Records
Titles
Most titles:
10 St Helens
Most consecutive title wins:
4 St. Helens (2019, 2020, 2021, 2022)
Most Grand Final appearances:
13 St. Helens
League Leaders Shield
Most League Leaders Shield wins:
8 St. Helens
Most consecutive League Leaders Shield wins:
4 St. Helens (2005, 2006, 2007, 2008)
Biggest League Leaders Shield winning margin:
, 2017; Castleford Tigers (52 points) over Leeds Rhinos (42 points)
Smallest League Leaders Shield winning margin:
0 points and 14 points difference - 2002, St. Helens (+405) over Bradford Bulls (+391). Saint Helens won the League Leaders Shield with a superior points difference.
0 points and 26 points difference - 2015; Leeds Rhinos (+294) over Wigan Warriors (+268). Both teams finished on 41 points, but Leeds won the League Leaders Shield with a superior points difference.
0 points and 60 points difference - 2023; Wigan Warriors (+362) over Catalans Dragons (+302). Both teams finished on 40 points (along with St Helens (+247), but Wigan won the League Leaders Shield with a superior points difference.
Team records
Most points scored in a season (total)
1152, Leeds Rhinos, 2005
Most points scored in a season (points per game)
43.18 (950 pts in 22 games), St. Helens, 1996
Most tries scored in a season
213, Leeds Rhinos, 2005
Most goals scored in a season
204, Bradford Bulls, 2001
Most drop goals scored in a season
11, Halifax, 1999 and Warrington Wolves, 2002
Fewest points conceded in a season (total)
195, St Helens, 2020 (shortened season due to COVID-19 pandemic meant St Helens only played 17 games)
Fewest points conceded in a season (points per game)
9.65 (222 pts in 23 games) Wigan Warriors, 1998
Most points conceded in a season
1210, Leigh Leopards (Previously called 'Centurions'), 2005
Longest undefeated streak and longest winning streak:
21 games, Bradford Bulls, 24 August 1996 to 22 August 1997
Longest winless streak and longest losing streak:
27 games, Halifax, 7 March 2003 to 21 September 2003
Individual records
Games
Most Super League games:
495, James Roby for St. Helens (2004-2023)
Tries
Most Super League tries:
247, Danny Maguire for Leeds Rhinos and Hull KR (2001-2019)
Most tries in a season
40, Denny Solomona, for Castleford Tigers in 2016
Most tries in a game:
7, Bevan French for ( Wigan Warriors v. Hull FC, (15 July 2022)
Fastest try in a game:
7 seconds, Ben Crooks for Hull Kingston Rovers vs Huddersfield Giants (16 April 2021)
Goals
Most goals in a season:
178, Henry Paul ( Bradford Bulls, 2001)
Most goals in a game (inc' drop goals):
14, Henry Paul (for Bradford Bulls v. Salford Red Devils, 25 June 2000)
Most drop goals in a season:
11, Lee Briers ( Warrington Wolves, 2002)
Most drop goals in a game:
5, Lee Briers (for Warrington W |
https://en.wikipedia.org/wiki/Albert%20Fox | Dr. Albert Whiting Fox (29 April 1881 – 29 April 1964) was an American chess master.
Chess career
Born in Boston, he spent a few years in Germany, studying mathematics. By the end of his sojourn in Europe, he won several brilliant games in 1900 and 1901 at Café de la Régence in Paris, and in Antwerp and Heidelberg.
Fox returned to America in 1901. He tied for 10–11th at Cambridge Springs 1904 (won by Frank James Marshall), won Manhattan Chess Club Championship in 1905/06, tied for 2nd–3rd with Marshall, behind Eugene Delmar, at New York 1906, took 3rd at Trenton Falls 1906 (Quadrangular, Emanuel Lasker won), and tied for 7–8th at New York 1916 (Rice tournament, José Raúl Capablanca won).
He played for the Manhattan Chess Club in cable matches against Franklin Chess Club of Philadelphia, and Chicago Chess Club in 1904–1906, and twice in the Anglo-American cable chess matches between Britain and the United States (1907 and 1911).
By 1915, Fox moved to Washington D.C. to "engage in newspaper work" for the Washington Post and shortly after gave up professional chess play. Fox died in Washington, D.C..
References
Further reading
Washington Star, April 30, 1964
Who's Who in Law, 1937, p. 326
Who's Who in the Nation's Capital, 1934–5, p. 338
1881 births
1964 deaths
American chess players
People from Boston
American expatriates in Germany |
https://en.wikipedia.org/wiki/Locally%20finite%20space | In the mathematical field of topology, a locally finite space is a topological space in which every point has a finite neighborhood, that is, an open neighborhood consisting of finitely many elements.
A locally finite space is an Alexandrov space.
A T1 space is locally finite if and only if it is discrete.
References
General topology
Properties of topological spaces |
https://en.wikipedia.org/wiki/Principle%20value | Principle value may refer to:
Principle value (ethics)
Cauchy principal value (mathematics) |
https://en.wikipedia.org/wiki/Skew-symmetric%20graph | In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs.
Skew-symmetric graphs were first introduced under the name of antisymmetrical digraphs by , later as the double covering graphs of polar graphs by , and still later as the double covering graphs of bidirected graphs by . They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding matchings in graphs, in testing whether a still life pattern in Conway's Game of Life may be partitioned into simpler components, in graph drawing, and in the implication graphs used to efficiently solve the 2-satisfiability problem.
Definition
As defined, e.g., by , a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties:
For every vertex v, σ(v) ≠ v,
For every vertex v, σ(σ(v)) = v,
For every edge (u,v), (σ(v),σ(u)) must also be an edge.
One may use the third property to extend σ to an orientation-reversing function on the edges of G.
The transpose graph of G is the graph formed by reversing every edge of G, and σ defines a graph isomorphism from G to its transpose. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a different vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism.
A path or cycle in a skew-symmetric graph is said to be regular if, for each vertex v of the path or cycle, the corresponding vertex σ(v) is not part of the path or cycle.
Examples
Every directed path graph with an even number of vertices is skew-symmetric, via a symmetry that swaps the two ends of the path. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs.
Similarly, a directed cycle graph is skew-symmetric if and only if it has an even number of vertices. In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle.
Polar/switch graphs, double covering graphs, and bidirected graphs
A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph or switch graph, which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew-symmetric graph. This equivalence is the one used by to model p |
https://en.wikipedia.org/wiki/Associated%20prime | In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by and sometimes called the assassin or assassinator of (word play between the notation and the fact that an associated prime is an annihilator).
In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.
Definitions
A nonzero R module N is called a prime module if the annihilator for any nonzero submodule N' of N. For a prime module N, is a prime ideal in R.
An associated prime of an R module M is an ideal of the form where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent: if R is commutative, an associated prime P of M is a prime ideal of the form for a nonzero element m of M or equivalently is isomorphic to a submodule of M.
In a commutative ring R, minimal elements in (with respect to the set-theoretic inclusion) are called isolated primes while the rest of the associated primes (i.e., those properly containing associated primes) are called embedded primes.
A module is called coprimary if xm = 0 for some nonzero m ∈ M implies xnM = 0 for some positive integer n. A nonzero finitely generated module M over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime. A submodule N of M is called P-primary if is coprimary with P. An ideal I is a P-primary ideal if and only if ; thus, the notion is a generalization of a primary ideal.
Properties
Most of these properties and assertions are given in starting on page 86.
If M' ⊆M, then If in addition M' is an essential submodule of M, their associated primes coincide.
It is possible, even for a commutative local ring, that the set of associated primes of a finitely generated module is empty. However, in any ring satisfying the ascending chain condition on ideals (for example, any right or left Noetherian ring) every nonzero module has at least one associated prime.
Any uniform module has either zero or one associated primes, making uniform modules an example of coprimary modules.
For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum If R is an Artinian ring, then this map becomes a bijection.
Matlis' Theorem: For a commutative Noetherian ring R, the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is gi |
https://en.wikipedia.org/wiki/Hans%20Samelson | Hans Samelson (3 March 1916 – 22 September 2005) was a German-American mathematician who worked in differential geometry, topology and the theory of Lie groups and Lie algebras—important in describing the symmetry of analytical structures.
Career and personal life
The eldest of three sons, Samelson was born on 3 March 1916, in Strassburg, Germany (now Strasbourg, France). His brother Klaus later became a mathematician and early computer science pioneer in Germany. His parents—one of Protestant and one of Jewish background—were both pediatricians. He spent most of his youth in Breslau, Silesia, Germany (now Wrocław, Poland), and began his advanced mathematical education there, at the University of Breslau. His family helped him leave Nazi Germany in 1936 for Zurich, Switzerland, where he studied with the geometer Heinz Hopf and received his doctorate in 1940 from the Swiss Federal Institute of Technology.
In 1941, he accepted a position at the Institute for Advanced Study in Princeton and immigrated to the United States; he arrived by ship six months before the United States entered World War II and acquired U.S. citizenship several years later. After leaving Princeton, he held faculty positions at the University of Wyoming (1942–1943), Syracuse University (1943–1946) and the University of Michigan (1946–1960) before coming to Stanford in 1960. He was recognized with the Dean's Award for Distinguished Teaching in 1977. He served as chair of the Mathematics Department from 1979 to 1982.
Though he became emeritus in 1986, he remained professionally active throughout his retirement, publishing articles on both contemporary and historical mathematical topics. One solved an architectural puzzle associated with the construction of the Brunelleschi Dome in Florence, Italy.
He was active in the Palo Alto Friends Meeting (Quakers) during his retirement, serving as treasurer for several years.
See also
Bott–Samelson variety
Publications
Notes on Lie Algebras
Selected Chapters on Geometry (translation by Hans Samelson)
Hans Samelson, renowned mathematician, dead; Nov. 6 memorial service set
External links
1916 births
2005 deaths
20th-century German mathematicians
American people of German-Jewish descent
Jewish emigrants from Nazi Germany to the United States
ETH Zurich alumni
Stanford University Department of Mathematics faculty
University of Michigan faculty
20th-century American mathematicians
Topologists
Differential geometers |
https://en.wikipedia.org/wiki/Supercompact | In mathematics, the term supercompact may refer to:
In set theory, a supercompact cardinal
In topology, a supercompact space. |
https://en.wikipedia.org/wiki/1975%20in%20Japanese%20football | Japanese football in 1975
Japan Soccer League
Division 1
Division 2
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/Gustave%20Choquet | Gustave Choquet (; 1 March 1915 – 14 November 2006) was a French mathematician.
Choquet was born in Solesmes, Nord. His contributions include work in functional analysis, potential theory, topology and measure theory. He is known for creating the Choquet theory, the Choquet integral and the theory of capacities.
He did postgraduate work at the École Normale Supérieure Paris where his advisor was Arnaud Denjoy. He was Professor at the University of Paris (subsequently Paris VI) from 1940 to 1984 and was also Professor at the École Polytechnique from 1960 to 1969. His honours and awards included being a Member of the Académie des Sciences, and an Officier of the Légion d’Honneur.
His students include Haïm Brezis, Gilles Godefroy, Nassif Ghoussoub, Michel L. Lapidus, and Michel Talagrand.
He was married to mathematician and mathematical physicist Yvonne Choquet-Bruhat, with whom he had a son Daniel and a daughter Geneviève. He died in Lyon in 2006.
Bibliography
, available from Gallica. It is an historical account on the development of the theory of capacities by the founder of the theory and one of the main contributors; an English translation of the title reads as: "The birth of capacity theory: reflections on a personal experience".
Choquet, Gustave (1969), Lectures on Analysis, 3 Vols., W.A. Benjamin, Inc., New York.
See also
Capacity of a set
Choquet game
Choquet integral
Choquet theory
Radó–Kneser–Choquet theorem
External links
A biography (in French) by the Académie des Sciences.
A short biography (in French).
A commemorative section (396kb PDF) of the N° 111 Gazette des Mathématiciens (2007) of the Société Mathématique de France.
1915 births
2006 deaths
People from Nord (French department)
20th-century French mathematicians
Academic staff of the University of Paris
École Normale Supérieure alumni
Members of the French Academy of Sciences
Officers of the Legion of Honour |
https://en.wikipedia.org/wiki/Nguyen%20Quoc%20Quan | Dr. Nguyen Quoc Quan (, born November 20, 1953) is a Vietnamese-born American mathematics researcher and human rights activist and a member of the leadership committee of the anti-communist organization Viet Tan. He was detained in April 17, 2012 after arriving at Tan Son Nhat airport in Ho Chi Minh City, Vietnam. On April 28, 2012, Vietnam's state media reported the "pro-democracy activist" has been arrested and accused of organizing "terrorism" activities. Previously, Dr Quan was arrested in Ho Chi Minh City in Vietnam on a trip on November 17, 2007 for preparing pro-democracy flyers. During that first trip, he brought in a Vietnamese translation of the book From Dictatorship to Democracy about nonviolent resistance.
He stood trial in Vietnam on May 13, 2008 on charges of "terrorism" and was sentenced to 6 months in prison. He was eventually released on May 17, 2008 and returned to his home in Elk Grove, California to his wife and two teenage sons. In 2012, he was re-arrested on another trip to Vietnam, and held in prison for 9 months. Following intense US pressure, he was deported on January 30, 2013.
Background
Nguyen Quoc Quan is a former high school math teacher in Kien Giang, Vietnam. He escaped from Vietnam on a fishing boat in 1981, ending up in the United States where he earned a doctorate degree in mathematics from North Carolina State University. He has also practiced software engineering. He is a long-time democracy activist, a devotee of Martin Luther King Jr. and member of senior leadership committee of Viet Tan.
2007 arrest
Nguyen Quoc Quan entered Vietnam on November 15, 2007 on a bicycle through the Cambodian border.
On November 17, 2007 along with two other Viet Tan members (Truong Van Ba, a Hawaiian restaurant owner, and Frenchwoman Nguyen Thi Thanh Van, a contributor to Viet Tan's Radio Chan Troi Moi radio show) he was arrested in the southern suburb of Ho Chi Minh City. At the time, they were leading a "democracy seminar" and preparing pro-democracy pamphlets, when 20 security officers raided the house. Also arrested in the same group were Vietnamese citizens Nguyen The Vu, Nguyen Trong Khiem, Nguyen Viet Trung and Thai journalist Somsak Khunmi.
The two-page pamphlet titled "Non-Violent Struggle: The Approach To Overcome Dictatorship" (translated from Vietnamese) and protesters to "faithfully maintain the discipline of non-violence."
The arrests were not officially confirmed by the Vietnamese government until November 22, 2007. During the press briefing, officials declined to state which laws the detained individuals have broken, nor released any information about Nguyen Quoc Quan, whose whereabouts remained unknown for almost a week.
At the beginning, state-controlled media in Vietnam acknowledged jailing only some, but not all activists. The website of the newspaper Sai Gon Giai Phong originally showed an image of US national Nguyen Quoc Quan wearing prison garb, but hours later replaced it with a manipulated imag |
https://en.wikipedia.org/wiki/Megamaths | Megamaths is a BBC educational television series for primary schools that was originally aired on BBC Two from 16 September 1996 to 4 February 2002. For its first three series, it was set in a castle on top of Table Mountain, populated by the four card suits (Kings, Queens and Jacks/Jackies, and a Joker who looked after children that visited the castle and took part in mathematical challenges). There were two gargoyles at the portcullis of the castle named Gar and Goyle who spoke mostly in rhyme, and an animated dragon called Brimstone who lived in the castle cellar (with his pet kitten, Digit). Each episode featured a song explaining the episode's mathematical content.
The three remaining series, however, were set in a "Superhero School" space station, featuring a trainee superhero named Maths Man who was initially guided by a female tutor, Her Wholeness, in the fifth series, and later by a male tutor, His Wholeness, in the fifth and sixth series. In the fourth series, there were also recurring sketches of a quiz show named Find that Fraction hosted by Colin Cool (played by Simon Davies who co-wrote the second to fourth series with director Neil Ben and had played the King of Diamonds in all four Table Mountain series), and a sports show named Sports Stand hosted by Sue Harker (a spoof of Sue Barker, who was played by Liz Anson) and Harry Fraction (a spoof of Harry Gration, who was also played by Simon Davies), along with a supervillain named The Diddler who Maths Man had to solve mathematical problems caused by when he ventured down to Earth (in the final episode, she was revealed to actually be Her Wholeness in disguise). In the sixth series, the Superhero School gained an on-board computer named VERA (whose initials stood for "Voice-Enhanced Resource Activator", and was voiced by Su Douglas who also played the Queen of Spades in the fourth series) and a character named 2D3D who appeared in his virtual reality glasses (Maths Man now also spoke directly to the audience when he ventured down to Earth calling them his "Maths Team", and His Wholeness set a puzzle for them at the end of each episode). In the seventh and final series, the episodes were shortened from twenty minutes to fifteen, and again featured Maths Man getting sent down to Earth to solve mathematical problems in everyday life.
Episodes
Tables (1996)
The first series, which was co-written by Christopher Lillicrap (who had previously written the first, second and fourth series of the BBC's earlier primary maths show, Numbertime, as well as the El Nombre sketches of its third series), comprised ten episodes focusing on multiplication. Each episode opened and ended with the episode's table being chanted, and the Joker (played by Jenny Hutchinson) introduced it in rhyme while speaking directly to the audience (she would also welcome teams of schoolchildren who came to visit the castle and give them advice as they took part in mathematical challenges). The two gargoyles, Gar (male) |
https://en.wikipedia.org/wiki/Alexandra%20Bellow | Alexandra Bellow (née Bagdasar; previously Ionescu Tulcea; born 30 August 1935) is a Romanian-American mathematician, who has made contributions to the fields of ergodic theory, probability and analysis.
Biography
Bellow was born in Bucharest, Romania, on August 30, 1935, as Alexandra Bagdasar. Her parents were both physicians. Her mother, Florica Bagdasar (née Ciumetti), was a child psychiatrist. Her father, , was a neurosurgeon. She received her M.S. in mathematics from the University of Bucharest in 1957, where she met and married her first husband, mathematician Cassius Ionescu-Tulcea. She accompanied her husband to the United States in 1957 and received her Ph.D. from Yale University in 1959 under the direction of Shizuo Kakutani with thesis Ergodic Theory of Random Series. After receiving her degree, she worked as a research associate at Yale from 1959 until 1961, and as an assistant professor at the University of Pennsylvania from 1962 to 1964. From 1964 until 1967 she was an associate professor at the University of Illinois at Urbana–Champaign. In 1967 she moved to Northwestern University as a Professor of Mathematics. She was at Northwestern until her retirement in 1996, when she became Professor Emeritus.
During her marriage to Cassius Ionescu-Tulcea (1956–1969), she and her husband co-wrote many papers and a research monograph on lifting theory.
Alexandra's second husband was the writer Saul Bellow, who was awarded the Nobel Prize in Literature in 1976, during their marriage (1975–1985). Alexandra features in Bellow's writings; she is portrayed lovingly in his memoir To Jerusalem and Back (1976), and, his novel The Dean's December (1982), more critically, satirically in his last novel, Ravelstein (2000), which was written many years after their divorce. The decade of the nineties was for Alexandra a period of personal and professional fulfillment, brought about by her marriage in 1989 to the mathematician Alberto P. Calderón.
Mathematical work
Some of her early work involved properties and consequences of lifting. Lifting theory, which had started with the pioneering papers of John von Neumann and later Dorothy Maharam, came into its own in the 1960s and 1970s with the work of the Ionescu Tulceas and provided the definitive treatment for the representation theory of linear operators arising in probability, the process of disintegration of measures. Their Ergebnisse monograph from 1969 became a standard reference in this area.
By applying a lifting to a stochastic process, the Ionescu Tulceas obtained a ‘separable’ process; this gives a rapid proof of Joseph Leo Doob's theorem concerning the existence of a separable modification of a stochastic process (also a ‘canonical’ way of obtaining the separable modification). Furthermore, by applying a lifting to a ‘weakly’ measurable function with values in a weakly compact set of a Banach space, one obtains a strongly measurable function; this gives a one line proof of Phillips's clas |
https://en.wikipedia.org/wiki/1965%20in%20Japanese%20football | Japanese football in 1965
Japan Soccer League
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1966%20in%20Japanese%20football |
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1968%20in%20Japanese%20football | Japanese football in 1968
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1967%20in%20Japanese%20football | Japanese football in 1967
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1969%20in%20Japanese%20football | Japanese football in 1969
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1970%20in%20Japanese%20football | Japanese football in 1970
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1973%20in%20Japanese%20football | Japanese football in 1973
Japan Soccer League
Division 1
Division 2
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1974%20in%20Japanese%20football | Japanese football in 1974
Japan Soccer League
Division 1
Division 2
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/1971%20in%20Japanese%20football | Japanese football in 1971
Japan Soccer League
Japanese Regional Leagues
Emperor's Cup
National team
Results
Players statistics
External links
Seasons in Japanese football |
https://en.wikipedia.org/wiki/Robin%20Sibson | Robin Sibson (4 May 1944 – 19 March 2017) was a British mathematician and educator.
He was a fellow of King's College, Cambridge, professor of statistics at the University of Bath and then vice-chancellor of the University of Kent. He was chief executive of the Higher Education Statistics Agency from 2001 until 2009. He was also a member of the Committee for Higher Education and Research (CC-HER), the predecessor to the Steering Committee for Higher Education and Research (CDESR).
He died on 19 March 2017 at the age of 72.
Research
He was the developer of natural neighbour interpolation on discrete sets of points in space.
Selected bibliography
Books
Chapter in books
References
External links
Professor Robin Sibson appointed as new Chief Executive of HESA
1944 births
2017 deaths
20th-century British mathematicians
21st-century British mathematicians
Fellows of King's College, Cambridge
Academics of the University of Bath
Vice-Chancellors of the University of Kent
Alumni of King's College, Cambridge |
https://en.wikipedia.org/wiki/Algebraic%20statistics | Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.
Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics.
The tradition of algebraic statistics
In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes.
Design of experiments
For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs.
Algebraic analysis and abstract statistical inference
Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics.
Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory.
Partially ordered sets and lattices
Partially ordered vector spaces and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues.
Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam.
Recent work using commutative algebra and algebraic geometry
In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties.
Introductory example
Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities
and these numbers satisfy
Conversely, |
https://en.wikipedia.org/wiki/Distance-hereditary%20graph | In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph) is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph.
Distance-hereditary graphs were named and first studied by , although an equivalent class of graphs was already shown to be perfect in 1970 by Olaru and Sachs.
It has been known for some time that the distance-hereditary graphs constitute an intersection class of graphs, but no intersection model was known until one was given by .
Definition and characterization
The original definition of a distance-hereditary graph is a graph such that, if any two vertices and belong to a connected induced subgraph of , then some shortest path connecting and in must be a subgraph of , so that the distance between and in is the same as the distance in .
Distance-hereditary graphs can also be characterized in several other equivalent ways:
They are the graphs in which every induced path is a shortest path, or equivalently the graphs in which every non-shortest path has at least one edge connecting two non-consecutive path vertices.
They are the graphs in which every cycle of length five or more has at least two crossing diagonals.
They are the graphs in which, for every four vertices , , , and , at least two of the three sums of distances , , and are equal to each other.
They are the graphs that do not have as isometric subgraphs any cycle of length five or more, or any of three other graphs: a 5-cycle with one chord, a 5-cycle with two non-crossing chords, and a 6-cycle with a chord connecting opposite vertices.
They are the graphs that can be built up from a single vertex by a sequence of the following three operations, as shown in the illustration:
Add a new pendant vertex connected by a single edge to an existing vertex of the graph.
Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced vertex. The new pair of vertices are called false twins of each other.
Replace any vertex of the graph by a pair of vertices, each of which has as its neighbors the neighbors of the replaced vertex together with the other vertex of the pair. The new pair of vertices are called true twins of each other.
They are the graphs that can be completely decomposed into cliques and stars (complete bipartite graphs ) by a split decomposition. In this decomposition, one finds a partition of the graph into two subsets, such that the edges separating the two subsets form a complete bipartite subgraph, forms two smaller graphs by replacing each of the two sides of the partition by a single vertex, and recursively partitions these two subgraphs.
They are the graphs that have rank-width one, where the rank-width of a graph is defined as the minimum, over all hierarchical partitions of the vertices of the graph, of |
https://en.wikipedia.org/wiki/Mathematics%2C%20Civics%20and%20Sciences%20Charter%20School | The Mathematics, Civics and Sciences Charter School (MCSCS) is a charter school serving students in grades 1–12 in Philadelphia, Pennsylvania, United States. Founded in 1999, the school is located in the Center City neighborhood and had a 100% graduation rate in 2015–2016.
History
The Mathematics, Civics and Sciences Charter School opened in 1999 with over 720 students in grades first through twelfth, adding a grade every year for 4 years.
Governance
The school is led by a Chief Administrative Officer and a principal. A six-member Board of Trustees meets with the CAO and principal regularly.
Campus
The school is located in Center City, in Philadelphia, Pennsylvania.
Extracurricular activities
Extracurricular activities offered at the school include cheerleading, exercise, sewing, public speaking, Mock Trial, and debating and choir.
Athletics
MCSCS has no gym or other athletic facility, so most practice and games take place at a local YMCA near Girard, not owned by the school. The school basketball team is known as the Elephants, named after the school mascot, an African Elephant. The school currently has no other athletic programs.
Curriculum
The school uses a back-to-basics curriculum. Students in grades 5 through 12 are required to select a school-to-college course of study; current selections are Law, Medical, Education, Computer Science, and Accounting.
As of 2015, the school offers AP courses.
Achievements
The Mathematics, Civics and Sciences Charter School was named one of the top 10 schools in Philadelphia, including public and charter schools. Also, every Thursday, the school participates in its self-created Homeless Project, in which the CAO and several students go out and feed and clothe homeless citizens of Philadelphia.
Notable alumni
Mike Watkins (born 1995), basketball player for Hapoel Haifa in the Israeli Basketball Premier League
References
External links
Mathematics, Civics and Sciences Charter School
Educational institutions established in 1999
Public elementary schools in Pennsylvania
Public middle schools in Pennsylvania
Public high schools in Pennsylvania
Charter schools in Pennsylvania
Schools in Philadelphia
1999 establishments in Pennsylvania
Callowhill, Philadelphia |
https://en.wikipedia.org/wiki/Facundo%20Arg%C3%BCello%20%28footballer%29 | Facundo Martin Argüello (born 23 February 1979 in Buenos Aires) is an Argentine footballer who currently plays for Club Atlético Nueva Chicago.
External links
Statistics at FutbolXXI.com
BDFA profile
1979 births
Living people
Footballers from Buenos Aires
Argentine expatriate men's footballers
Argentine men's footballers
Men's association football defenders
Primera Nacional players
Categoría Primera A players
Club Atlético Nueva Chicago footballers
Atlético de Rafaela footballers
Instituto Atlético Central Córdoba footballers
Club Atlético Huracán footballers
Millonarios F.C. players
Hapoel Petah Tikva F.C. players
Expatriate men's footballers in Colombia
Expatriate men's footballers in Israel
Argentine expatriate sportspeople in Israel
Expatriate men's footballers in Romania
Expatriate men's footballers in Malaysia |
https://en.wikipedia.org/wiki/Hans%20Georg%20Bock | Hans Georg Bock (born 9 May 1948) is a German university professor for mathematics and scientific computing.
He has served as managing director of Interdisciplinary Center for Scientific Computing of Heidelberg University from 2005 to 2017.
Before this, he had been vice managing director from 1993 to 2004.
Hans Georg Bock is a member of the European Mathematical Society's committee for developing countries (CDC-EMS) and responsible member for the region of Asia therein.
In appreciation of his merits with respect to Vietnamese-German relations and his role in the establishment of high performance scientific computing in Vietnam, he was awarded the honorary degree of the Vietnamese Academy of Science and Technology in 2000. In 2003, he was awarded the Medal of Merit of the Vietnamese Ministry for Education and Training.
Academic profile
Hans Georg Bock graduated from University of Cologne in 1974 with a diploma thesis in mathematics titled "Numerische Optimierung zustandsbeschränkter parameterabhängiger Prozesse mit linear auftretender Steuerung unter Anwendung der Mehrzielmethode" (Numerical optimization of state-constrained parameter-dependent processes with linearly entering controls by application of the direct multiple shooting method) completed under the supervision of professor Roland Z. Bulirsch.
With his PhD thesis "Randwertproblemmethoden zur Parameteridentifizierung in Systemen nichtlinearer Differentialgleichungen" (Boundary-value problem methods for parameter estimation in systems of nonlinear differential equations) completed under the supervision of Jens Frehse and Roland Z. Bulirsch, he received a Ph.D. in applied mathematics from the University of Bonn in 1986.
After staying in Heidelberg for two years as a visiting professor for numerical mathematics from 1987 to 1988, he accepted a
full professorship at the University of Augsburg. In 1991 Hans Georg Bock accepted a call onto the chair for scientific computing and optimization at Heidelberg University.
Research
Hans Georg Bock authored or co-authored more than 190 scientific publications. In particular, his scientific work comprises advances in the fields of
adaptive discretization and approximate Newton-type methods for large-scale optimization,
simultaneous or one-shot methods for DAE and PDE constrained nonlinear optimization and optimal control problems,
real-time computation of constrained closed-loop control problems subject to DAE and PDE, especially nonlinear model predictive control,
numerical methods for state and parameter estimation, and optimal experimental design for DAE and PDE,
numerical methods for differential algebraic equations (DAE),
nonlinear mixed-integer dynamic optimization,
optimization under uncertainty,
non-standard optimization and optimal control problems such as stability optimization of gait patterns,
computational methods for the cultural heritage, and
applications in aerospace, mechanical and biomechanical engineering, chemical |
https://en.wikipedia.org/wiki/Direct%20multiple%20shooting%20method | In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods.
Single shooting methods
Shooting methods can be used to solve boundary value problems (BVP) like
in which the time points ta and tb are known and we seek
Single shooting methods proceed as follows. Let y(t; t0, y0) denote the solution of the initial value problem (IVP)
Define the function F(p) as the difference between y(tb; p) and the specified boundary value yb: F(p) = y(tb; p) − yb. Then for every solution (ya, yb) of the boundary value problem we have ya=y0 while yb corresponds to a root of F. This root can be solved by any root-finding method given that certain method-dependent prerequisites are satisfied. This often will require initial guesses to ya and yb. Typically, analytic root finding is impossible and iterative methods such as Newton's method are used for this task.
The application of single shooting for the numerical solution of boundary value problems suffers from several drawbacks.
For a given initial value y0 the solution of the IVP obviously must exist on the interval [ta,tb] so that we can evaluate the function F whose root is sought.
For highly nonlinear or unstable ODEs, this requires the initial guess y0 to be extremely close to an actual but unknown solution ya. Initial values that are chosen slightly off the true solution may lead to singularities or breakdown of the ODE solver method. Choosing such solutions is inevitable in an iterative root-finding method, however.
Finite precision numerics may make it impossible at all to find initial values that allow for the solution of the ODE on the whole time interval.
The nonlinearity of the ODE effectively becomes a nonlinearity of F, and requires a root-finding technique capable of solving nonlinear systems. Such methods typically converge slower as nonlinearities become more severe. The boundary value problem solver's performance suffers from this.
Even stable and well-conditioned ODEs may make for unstable and ill-conditioned BVPs. A slight alteration of the initial value guess y0 may generate an extremely large step in the ODEs solution y(tb; ta, y0) and thus in the values of the function F whose root is sought. Non-analytic root-finding methods can seldom cope with this behaviour.
Multiple shooting
A direct multiple shooting method partitions the interval [ta, tb] by introducing additional grid points
The method starts by guessing somehow the values of y at all grid points tk with . Denote these guess |
https://en.wikipedia.org/wiki/Ming-Jun%20Lai | Ming-Jun Lai is an American mathematician, currently a Professor of Mathematics at the University of Georgia. His area of research is splines and their numerical analysis. He has published a text on splines called Splines Functions on Triangulations. He was born in Hangzhou, China.
Lai received a B.Sc. from Hangzhou University and a Ph.D. in mathematics from the Texas A&M University in 1989. His dissertation was entitled "On Construction of Bivariate and Trivariate Vertex Splines on Arbitrary Mixed Grid Partitions" and supervised by Charles K. Chui.
References
Ming-Jun Lai at Math Genealogy Project
Year of birth missing (living people)
Living people
Hangzhou University alumni
Texas A&M University alumni
University of Utah alumni
20th-century American mathematicians
21st-century American mathematicians
Chinese emigrants to the United States
University of Georgia faculty |
https://en.wikipedia.org/wiki/Lee%20Powell%20%28footballer%29 | Lee Powell (born 2 June 1973) is a Welsh football forward, who played for Southampton.
Career statistics
References
External links
Profile
1973 births
Living people
Welsh men's footballers
Men's association football forwards
Premier League players
Southampton F.C. players
Hamilton Academical F.C. players
Yeovil Town F.C. players
Wales men's under-21 international footballers
Scottish Football League players |
https://en.wikipedia.org/wiki/Mean%20%28disambiguation%29 | Mean is a term used in mathematics and statistics.
Mean may also refer to:
Music
Mean (album), a 1987 album by Montrose
"Mean" (song), a 2010 country song by Taylor Swift from Speak Now
"Mean", a song by Pink from Funhouse
Meane, or mean, a vocal music term from 15th and 16th century England
Other uses
Content (measure theory), finitely-additive measures, sometimes called "means"
Ethic mean, a sociology term
Mean (magazine), an American bi-monthly magazine
Meanness, a personal quality
MEAN (solution stack), a free and open-source JavaScript software stack for building dynamic web sites and web applications
A synonym of frugal
See also
Meaning (disambiguation)
Means (disambiguation)
Meen (disambiguation) |
https://en.wikipedia.org/wiki/Miranda%20Municipality%2C%20M%C3%A9rida | Miranda is one of the 23 municipalities (municipios) that makes up the Venezuelan state of Mérida and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 22,879. The town of Timotes is the shire town of the Miranda Municipality. The municipality is one of several in Venezuela named Miranda Municipality after the Venezuelan revolutionary and independence hero Francisco de Miranda.
See also
Timotes
Mérida
Municipalities of Venezuela
References
External links
miranda-merida.gob.ve
Municipalities of Mérida (state) |
https://en.wikipedia.org/wiki/Negative%20value | Negative value may refer to:
Negative predictive value in statistics
Negative ethic or philosophic value
Negative pricing
insolvency |
https://en.wikipedia.org/wiki/Jack%20Fairless | Jack Fairless was manager of the English football club Darlington from 1928 to 1933.
Managerial statistics
External links
Darlington F.C. managers
Year of death missing
Year of birth missing
English football managers
Place of birth missing |
https://en.wikipedia.org/wiki/El%20Kabir%20Pene | El Kabir Pene, born 18 December 1984 in Thiès, Senegal, is a Senegalese basketball player.
Statistics
Height : 1m90
Position : guard
Regular number:
Biography
He plays as a guard for the Senegal national basketball team.
El Kabir Pene participated in the 2006 World Championships in Japan.
Clubs
2003 - 2005 : US Gorée (1st division)
December 2005 - May 2006 : Stade Clermontois (Pro A)
May 2006 - June 2006 : Vichy (Pro B)
2006 - 2007 : Stade Clermontois (Pro A)
2008 - 2009 : ASM Basket Le Puy-en-Velay (France)
Career with the Senegal national team
Senegalese international, El Kabir Pene participated in the African Championships in 2005 and the 2006 World Championships in Japan.
Titles
Silver medal at the 2005 African Championships, (Alger, Algeria)
External links
www.allbasketball.com
World Statistics 2006
His profile at the LNB site
1984 births
Living people
African Games medalists in basketball
African Games silver medalists for Senegal
Competitors at the 2003 All-Africa Games
Senegalese expatriate basketball people in France
Senegalese men's basketball players
Sportspeople from Thiès
2006 FIBA World Championship players
Stade Clermontois BA players
JA Vichy players |
https://en.wikipedia.org/wiki/Cotangent%20complex | In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If is a morphism of geometric or algebraic objects, the corresponding cotangent complex can be thought of as a universal "linearization" of it, which serves to control the deformation theory of . It is constructed as an object in a certain derived category of sheaves on using the methods of homotopical algebra.
Restricted versions of cotangent complexes were first defined in various cases by a number of authors in the early 1960s. In the late 1960s, Michel André and Daniel Quillen independently came up with the correct definition for a morphism of commutative rings, using simplicial methods to make precise the idea of the cotangent complex as given by taking the (non-abelian) left derived functor of Kähler differentials. Luc Illusie then globalized this definition to the general situation of a morphism of ringed topoi, thereby incorporating morphisms of ringed spaces, schemes, and algebraic spaces into the theory.
Motivation
Suppose that and are algebraic varieties and that is a morphism between them. The cotangent complex of is a more universal version of the relative Kähler differentials . The most basic motivation for such an object is the exact sequence of Kähler differentials associated to two morphisms. If is another variety, and if is another morphism, then there is an exact sequence
In some sense, therefore, relative Kähler differentials are a right exact functor. (Literally this is not true, however, because the category of algebraic varieties is not an abelian category, and therefore right-exactness is not defined.) In fact, prior to the definition of the cotangent complex, there were several definitions of functors that might extend the sequence further to the left, such as the Lichtenbaum–Schlessinger functors and imperfection modules. Most of these were motivated by deformation theory.
This sequence is exact on the left if the morphism is smooth. If Ω admitted a first derived functor, then exactness on the left would imply that the connecting homomorphism vanished, and this would certainly be true if the first derived functor of f, whatever it was, vanished. Therefore, a reasonable speculation is that the first derived functor of a smooth morphism vanishes. Furthermore, when any of the functors which extended the sequence of Kähler differentials were applied to a smooth morphism, they too vanished, which suggested that the cotangent complex of a smooth morphism might be equivalent to the Kähler differentials.
Another natural exact sequence related to Kähler differentials is the conormal exact sequence. If f is a closed immersion with ideal sheaf I, then there is an exact sequence
This is an extension of the exact sequence above: There is a new term on the left, the conormal sheaf of f, and the relative differentials ΩX/Y have vanis |
https://en.wikipedia.org/wiki/A%C2%B9%20homotopy%20theory | In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval , which is not an algebraic variety, with the affine line , which is. The theory has seen spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.
Construction
homotopy theory is founded on a category called the homotopy category . Simply put, the homotopy category, or rather the canonical functor , is the universal functor from the category of smooth -schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line becomes contractible. Here is some prechosen base scheme (e.g., the spectrum of the complex numbers ).
This definition in terms of a universal property is not possible without infinity categories. These were not available in the 90's and the original definition passes by way of Quillen's theory of model categories. Another way of seeing the situation is that Morel-Voevodsky's original definition produces a concrete model for (the homotopy category of) the infinity category .
This more concrete construction is sketched below.
Step 0
Choose a base scheme . Classically, is asked to be Noetherian, but many modern authors such as Marc Hoyois work with quasi-compact quasi-separated base schemes. In any case, many important results are only known over a perfect base field, such as the complex numbers, it's perfectly fine to consider only this case.
Step 1
Step 1a: Nisnevich sheaves. Classically, the construction begins with the category of Nisnevich sheaves on the category of smooth schemes over . Heuristically, this should be considered as (and in a precise technical sense is) the universal enlargement of obtained by adjoining all colimits and forcing Nisnevich descent to be satisfied.
Step 1b: simplicial sheaves. In order to more easily perform standard homotopy theoretic procedures such as homotopy colimits and homotopy limits, replaced with the following category of simplicial sheaves.
Let be the simplex category, that is, the category whose objects are the sets
and whose morphisms are order-preserving functions. We let denote the category of functors . That is, is the category of simplicial objects on . Such an object is also called a simplicial sheaf on .
Step 1c: fibre functors. For any smooth -scheme , any point , and any sheaf , let's write for the stalk of the restriction of to the small Nisnevich site of . Explicitly, where the colimit is over factorisations of the canonical inclusion via an étale morphism . The collection is a |
https://en.wikipedia.org/wiki/Digital%20Morse%20theory | In mathematics, digital Morse theory is a digital adaptation of continuum Morse theory for scalar volume data. This is not about the Samuel Morse's Morse code of long and short clicks or tones used in manual electric telegraphy. The term was first promulgated by DB Karron based on the work of JL Cox and DB Karron.
The main utility of a digital Morse theory is that it serves to provide a theoretical basis for isosurfaces (a kind of embedded manifold submanifold) and perpendicular streamlines in a digital context. The intended main application of DMT is in the rapid semiautomatic segmentation objects such as organs and anatomic structures from stacks of medical images such as produced by three-dimensional computed tomography by CT or MRI technology.
DMT Tree
A DMT tree is a digital version of a Reeb graph or contour tree graph, showing the relationship and connectivity of one isovalued defined object to another. Typically, these are nested objects, one inside another, giving a parent-child relationship, or two objects standing alone with a peer relationship.
The essential insight of Morse theory can be given in a little parable.
The fish tank thought experiment
The fish tank thought experiment: Counting islands as the water level changes
The essential insight of continuous Morse theory can be intuited by a thought experiment. Consider a rectangular glass fish tank. Into this tank, we pour a small quantity of sand such that we have two smoothly sloping small hills, one taller than the other. Now, we fill this tank to the brim with water. We now start a count of the number of island objects as we very slowly drain the tank.
Our initial observation is that there are no island features in our tank scene. As the water level drops, we observe the water level just coincident with the peak of the tallest sand hill.
We next observe the behavior of the water at the critical peak of the hill. We see a degenerate point island contour, with zero area, zero perimeter, and infinite curvature. A vanishing small change in the water level and this point contour expand into a tiny island.
We now increment our island object count by +1.
We continue to drain water from the tank.
We next observe the creation of the second island at the peak of the second little hill. We again increment our island object count by +1 to two objects. Our little sea has two island objects in it.
As we continue to slowly lower the water level in our little tank sea.
We now observe the two island contours gradually expand and grow toward each other. As the water level reaches the level of the critical saddle point between the two hills the island contours touch at precisely the saddle point.
We observe that our object count decrements by –1 to give a total island count of one.
The essential feature of this rubric is that we only need to count the peaks and passes to inventory all of the islands in our sea, or objects in our scene. This approach works even as we increase the com |
https://en.wikipedia.org/wiki/Invertible%20knot | In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed. A non-invertible knot is any knot which does not have this property. The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.
There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.
Background
It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963. It is now known almost all knots are non-invertible.
Invertible knots
All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible. The problem can be translated into algebraic terms, but unfortunately there is no known algorithm to solve this algebraic problem.
If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot.
Strongly invertible knots
A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot. By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot. All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible.
Non-invertible knots
The simplest example of a non-invertible knot is the knot 817 (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.
See also
Chiral knot
References
External links
Jablan, Slavik & Sazdanovic, Radmila. Basic graph theory: Non-invertible knot and links , LinKnot.
Explanation with a video, Nrich.Maths.org. |
https://en.wikipedia.org/wiki/Ravindra%20Khattree | Ravindra Khattree (born 1959) is an Indian-American statistician and a distinguished professor of statistics at Oakland University and a co-director of the Center for Data Science and Big Data Analytics at the same university. His contribution to the Fountain–Khattree–Peddada Theorem in Pitman measure of closeness is one of the important results of his work. Khattree is the coauthor of two books and has coedited two volumes. He has served as an associate editor of the Communications in Statistics journal and the editor of the Interstat online journal. He was Chief editor of Journal of Statistics and Applications for more than ten years. He is an elected fellow of the American Statistical Association.
Khattree was born in Uttar Pradesh, India. He attended the Ewing Christian College-Allahabad University and the Indian Statistical Institute. In 1985, he earned a doctorate from the University of Pittsburgh with Calyampudi Radhakrishna Rao as his advisor. He became a faculty member at Oakland University in 1991. He was the biostatistics group leader in the Biomedical Research and Informatics Center and a professor of biostatistics in the College of Human Medicine, Michigan State University during 2005–2006. He worked as a senior research scientist at US National Academy of Sciences with assignment at the Radiation Effects Research Foundation (formerly known as Atomic Bomb Casualty Commission), Hiroshima during 2010–2011.
Prior to joining Oakland University, he had been a faculty member at the North Dakota State University, Case Western Reserve University and also worked at BFGoodrich Chemical Group. He is the paternal grandson of Binda Prasad Khattri.
Recognition
In 2002, Khattree received the Young Researcher Award from the International Indian Statistical Association. Khattree was honored with fellowship in the American Statistical Association in 2003 and became an elected member of the International Statistical Institute in 2004. He is also a recipient of Oakland University Research Excellence Award (2008).
Bibliography
Applied Multivariate Statistics with SAS Software (1995, 1999) with Dayanand N. Naik.
Multivariate Data Reduction and Discrimination with SAS Software (2000) with Dayanand N. Naik.
Handbook of Statistics Volume 22, Statistics in Industry (2002) with C.R. Rao.
Computational Methods in Biomedical Research (2008) with Dayanand N. Naik.
References
External links
Mathematical Genealogy
University of Pittsburgh alumni
Fellows of the American Statistical Association
Indian emigrants to the United States
Oakland University faculty
1959 births
Living people
Elected Members of the International Statistical Institute
American academics of Indian descent |
https://en.wikipedia.org/wiki/Topological%20semigroup | In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous.
Every topological group is a topological semigroup.
See also
References
Topological algebra
Topological groups |
https://en.wikipedia.org/wiki/Paratopological%20group | In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not required to be continuous.
As with topological groups, some authors require the topology to be Hausdorff.
Compact paratopological groups are automatically topological groups.
References
Topological groups |
https://en.wikipedia.org/wiki/MSOR | MSOR can mean:
Marine Special Operations Regiment (United States)
Maths, Stats & OR Network |
https://en.wikipedia.org/wiki/Bonfin%C3%B3polis%20de%20Minas | Bonfinópolis de Minas is a municipality in the north of the Brazilian state of Minas Gerais. The population of the municipality in 2020 by the Brazilian Institute of Geography and Statistics (IBGE) is 5,444 inhabitants in a total area of 1,778 km2. The elevation of the municipal seat is 651. It became a city in 1962.
Bonfinópolis is located in the statistical micro-region of Unaí. The nearest regional centers are Unaí, João Pinheiro, and Paracatu. Bonfinópolis de Minas is connected to Unaí by MG-181 and BR-251. The distance is 141 km. Most of the road is unpaved.
The economy is based on cattle raising (41,000 head in 2006) and the growing of crops such as cotton (400 ha.), beans (3,635 ha.), soybeans (11,500 ha.), corn (6,800 ha. ), and sorghum (600 ha.). In 2006 there were 632 farms with a total agricultural area of 164,517 ha. 32,000 ha. were planted area. There were 1,909 workers related to the producer and 546 persons not related to the producer. There were 327 tractors.
In the town the biggest sectors of employment were commerce and public administration. In 2007 there were 504 automobiles. There was one financial institution.
The school network had 1,208 students in 13 primary schools and 341 students in one middle school. In the health sector there were 3 public health clinics. The nearest hospital was in Unaí, 141 km. away on bad roads. Many patients go as far as Brasília for more serious treatment.
Municipal Human Development Index
MHDI: .754
Ranking in the state: 255 out of 853 municipalities in 2000
Ranking in the country: 1768 out of 5,138 municipalities in 2000
Life expectancy: 72.2
Literacy rate: .83 (See Frigoletto for the complete list for Minas
See also
List of municipalities in Minas Gerais
References
External links
IBGE
Citybrazil
Municipalities in Minas Gerais |
https://en.wikipedia.org/wiki/Standard%20Geographical%20Classification%20code%20%28Canada%29 | The Standard Geographical Classification (SGC) is a system maintained by Statistics Canada for categorizing and enumerating the census geographic units of Canada. Each geographic area receives a unique numeric code ranging from one to seven digits, which extend telescopically to refer to increasingly small areas. This geocode is roughly analogous to the ONS coding system in use in the United Kingdom.
Regions
The SGC code format for regions is , where X is a unique identifier incrementing from east to west, then north.
: Atlantic Canada
: Quebec
: Ontario
: Prairies
: British Columbia
: Northern Canada
Provinces and Territories
The SGC code format for provinces and territories is , where
X is the above regional prefix, and Y is a further identifier incrementing from east to west. Taken as a single digit, each value of Y is unique within the province group, or unique within the territory group.
: Newfoundland and Labrador
: Prince Edward Island
: Nova Scotia
: New Brunswick
: Quebec
: Ontario
: Manitoba
: Saskatchewan
: Alberta
: British Columbia
: Yukon
: Northwest Territories
: Nunavut
Census divisions
The SGC code format for census divisions is , where XX is the above province/territory code, and YY is the census division's code, unique within its own province. Census divisions are generally numbered from east to west. In some locations, a similar policy to American FIPS county codes has been adopted, with even-numbered slots being left vacant for future expansion.
Examples:
: Division No. 4, Newfoundland and Labrador
: Division No. 5, Newfoundland and Labrador
: Kent County, New Brunswick
: Northumberland County, New Brunswick
: York County, New Brunswick
: Les Moulins Regional County Municipality, Quebec
: Territoire équivalent of Laval, Quebec
: Territoire équivalent of Montreal, Quebec
: Roussillon Regional County Municipality, Quebec
: Les Jardins-de-Napierville Regional County Municipality, Quebec
: Leeds and Grenville United Counties, Ontario
: [vacant slot]
: Lanark County, Ontario
: Frontenac Census Division, Ontario
: Division No. 4, Saskatchewan
: Division No. 5, Alberta
: Regional District of East Kootenay, British Columbia
: [vacant slot]
: Regional District of Central Kootenay, British Columbia
: [vacant slot]
: Regional District of Kootenay Boundary, British Columbia
Census subdivisions
The SGC code format for census subdivisions is , where XX is the province/territory code, YY is the census division code, and ZZZ is the census subdivision's code, unique within its own census division. Census subdivisions are again generally numbered from east to west, and the practice has been to leave even-numbered slots vacant for future expansion.
Examples:
: Tyendinaga, Ontario
: Deseronto, Ontario
: [vacant slot]
: Tyendinaga Mohawk Territory, Ontario
: Belleville, Ontario
: [vacant slot]
: Sanikiluaq, Nunavut
: [vacant slot]
: Iqaluit, Nunavut
: [vacant slot]
: Kimmirut, Nunavut
: [vacant slot]
External links
Statistics Cana |
https://en.wikipedia.org/wiki/Double%20%28manifold%29 | In the subject of manifold theory in mathematics, if is a manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where for all .
Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that is non-empty and is compact.
Doubles bound
Given a manifold , the double of is the boundary of . This gives doubles a special role in cobordism.
Examples
The n-sphere is the double of the n-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively. More generally, if is closed, the double of is . Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the Möbius strip is the Klein bottle.
If is a closed, oriented manifold and if is obtained from by removing an open ball, then the connected sum is the double of .
The double of a Mazur manifold is a homotopy 4-sphere.
References
Differential topology
Manifolds |
https://en.wikipedia.org/wiki/Anwar%20Boudjakdji | Anwar Mohamed Boudjakdji (born September 1, 1976) is a retired Algerian football player who played as a midfielder.
National team statistics
Honours
Won the Algerian League once with JS Kabylie in 2006
Won the Arab Champions League once with WA Tlemcen in 1998
Won the Algerian Cup once with WA Tlemcen in 1998
Has 12 caps for the Algerian National Team
References
1976 births
Algerian men's footballers
Living people
JS Kabylie players
Algeria men's international footballers
MC Oran players
WA Tlemcen players
People from Tlemcen
Men's association football midfielders
21st-century Algerian people |
https://en.wikipedia.org/wiki/Ali%20Yaakoub | Ali Hassan Yaakoub (; born 26 January 1985) is a Lebanese footballer who plays as a midfielder for club Mabarra.
Career statistics
International
Scores and results list Lebanon's goal tally first, score column indicates score after each Yaakoub goal.
References
External links
1985 births
Living people
People from Baalbek District
Lebanese men's footballers
Men's association football midfielders
Al Ahed FC players
Nejmeh SC players
Al Mabarra Club players
Shabab Baalbeck SC players
Lebanese Premier League players
Lebanese Second Division players
Lebanon men's youth international footballers
Lebanon men's international footballers |
https://en.wikipedia.org/wiki/Comb%20space | In mathematics, particularly topology, a comb space is a particular subspace of that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation on the comb space.
Formal definition
Consider with its standard topology and let K be the set . The set C defined by:
considered as a subspace of equipped with the subspace topology is known as the comb space. The deleted comb space, D, is defined by:
.
This is the comb space with the line segment deleted.
Topological properties
The comb space and the deleted comb space have some interesting topological properties mostly related to the notion of connectedness.
1. The comb space, C, is path connected and contractible, but not locally contractible, locally path connected, or locally connected.
2. The deleted comb space, D, is connected:
Let E be the comb space without . E is also path connected and the closure of E is the comb space. As E D the closure of E, where E is connected, the deleted comb space is also connected.
3. The deleted comb space is not path connected since there is no path from (0,1) to (0,0):
Suppose there is a path from p = (0, 1) to the point (0, 0) in D. Let f : [0, 1] → D be this path. We shall prove that f −1{p} is both open and closed in [0, 1] contradicting the connectedness of this set. Clearly we have f −1{p} is closed in [0, 1] by the continuity of f. To prove that f −1{p} is open, we proceed as follows: Choose a neighbourhood V (open in R2) about p that doesn’t intersect the x–axis. Suppose x is an arbitrary point in f −1{p}. Clearly, f(x) = p. Then since f −1(V) is open, there is a basis element U containing x such that f(U) is a subset of V. We assert that f(U) = {p} which will mean that U is an open subset of f −1{p} containing x. Since x was arbitrary, f −1{p} will then be open. We know that U is connected since it is a basis element for the order topology on [0, 1]. Therefore, f(U) is connected. Suppose f(U) contains a point s other than p. Then s = (1/n, z) must belong to D. Choose r such that 1/(n + 1) < r < 1/n. Since f(U) does not intersect the x-axis, the sets A = (−∞, r) × and B = (r, +∞) × will form a separation on f(U); contradicting the connectedness of f(U). Therefore, f −1{p} is both open and closed in [0, 1]. This is a contradiction.
4. The comb space is homotopic to a point but does not admit a deformation retract onto a point for every choice of basepoint.
See also
Connected space
Hedgehog space
Infinite broom
List of topologies
Locally connected space
Order topology
Topologist's sine curve
References
Topological spaces
Trees (topology) |
https://en.wikipedia.org/wiki/MuLinux | muLinux is an Italian, English-language lightweight Linux distribution maintained by mathematics and physics professor Michele Andreoli, meant to allow very old and obsolete computers (80386, 80486 and Pentium Pro hardware dating from 1986 through 1998) to be used as basic intranet/Internet servers or text-based workstations with a UNIX-like operating system. It was also designed for quickly turning any 80386 or later computer into a temporary, powerful Linux machine, along with system repair, education, forensic analysis and what the developer called proselytizing. In 2004 reviewer Paul Zimmer wrote, "Although there are several other single-floppy Linux distributions, none can match muLinux's extensive and unique combination of useful features." The last version update was in 2004, when further development of this "linux-on-a-floppy" distribution ended.
Name
The name muLinux comes from the Greek letter mu which is the SI symbol meaning one millionth, harking to the very small size of this OS.
Minimalist design
muLinux was based on the Linux 2.0.36 kernel. Development was frozen in 2004 at version 14r0, with some of the code and packages taken from software releases going back to 1998 (owing only to their smaller sizes). An experimental, unstable version called Lepton had the 2.4 kernel.
muLinux could be both booted or installed to a hard drive on an obsolete machine from floppy disks. A highly functional UNIX-like, network-enabled server with a Unix shell could be had from but one floppy disk. Another floppy disk added workstation functionality and a legacy X Window VGA GUI came with a third floppy. One reviewer noted, "It's not gorgeous, but the whole X subsystem fits onto a single floppy. Egad." muLinux could also be unpacked and installed by a self-executable archive, or extracted directly, onto an old DOS or Windows 9x (umsdos) partition without harming the current OS. If the machine had a floppy disk drive muLinux also would run on an otherwise diskless computer and no CD-ROM drive was needed.
Owing to its minimalist design muLinux was a single-user OS, with all operations performed by the root user. It used the ext2 Linux native file system (rather than the slower Minix file system seen in other single-floppy takes on Linux). The OS was robust when used for text-based tasks along with basic file, light web page or email serving. It could also be adapted as a very tiny, stand-alone embedded system.
muLinux was sometimes installed by Windows users who wanted to learn about the commands and configuration of a Unix-like operating system before taking the step of installing a full Linux distribution or BSD release, although on later computers this could easily be done with any one of many live CD distributions. Since the distribution was always wholly targeted at old hardware and meant to have a tiny footprint, Andreoli warned at the time that muLinux should not be used to evaluate Linux or open source software. The OS came with a lean an |
https://en.wikipedia.org/wiki/Ministry%20of%20Statistics%20and%20Programme%20Implementation | The Ministry of Statistics and Programme Implementation (MoSPI) is a ministry of Government of India concerned with coverage and quality aspects of statistics released. The surveys conducted by the Ministry are based on scientific sampling methods.
History
The Ministry of Statistics and Programme Implementation (MOSPI) came into existence as an Independent Ministry on 15 October 1999 after the merger of the Department of Statistics and the Department of Programme Implementation.
Departments
The Ministry has two departments, one relating to Statistics and the other Programme Implementation. The Statistics department having the National Statistical Office (NSO) which consists of the
(i) Central Statistical Office (CSO).
(ii) Computer center.
(iii) National Sample Survey Office (NSSO).
The Programme Implementation Wing has three Divisions, namely, (i) Twenty Point Programme (ii) Infrastructure Monitoring and Project Monitoring and (iii) Member of Parliament Local Area Development Scheme. Besides these two wings, there is National Statistical Commission created through a Resolution of Government of India (MOSPI) and one autonomous Institute, viz., Indian Statistical Institute declared as an institute of National importance by an Act of Parliament.
The Ministry of Statistics and Programme Implementation attaches considerable importance to coverage and quality aspects of statistics released in the country. The statistics released are based on administrative sources, surveys and censuses conducted by the center and State Governments and non-official sources and studies. The surveys conducted by the Ministry are based on scientific sampling methods. Field data are collected through dedicated field staff. In line with the emphasis on the quality of statistics released by the Ministry, the methodological issues concerning the compilation of national accounts are overseen Committees like Advisory Committee on National Accounts, Standing Committee on Industrial Statistics, Technical Advisory Committee on Price Indices. The Ministry compiles data sets based on current data, after applying standard statistical techniques and extensive scrutiny and supervision.
Responsibilities
National Statistical Office (NSO) is mandated with the following responsibilities:
acts as the nodal agency for planned development of the statistical system in the country, lays down and maintains norms and standards in the field of statistics, involving concepts and definitions, methodology of data collection, processing of data and dissemination of results;
coordinates the statistical work in respect of the Ministries/Departments of the Government of India and State Statistical Bureaus (SSBs), advises the Ministries/Departments of the Government of India on statistical methodology and on statistical analysis of data;
prepares national accounts as well as publishes annual estimates of national product, government and private consumption expenditure, capital formation, savi |
https://en.wikipedia.org/wiki/Duncan%20Lawson | Duncan Austin Lawson is a British mathematician known for work in mathematics education including university-wide mathematics and statistics support.
Early life and education
Lawson attended Bury Grammar School and later obtained a BA and D.Phil. from the University of Oxford.
Career
Lawson worked for British Gas plc before moving to Coventry University in 1987. His research interests are in efficient computational methods for the calculation of thermal radiative heat transfer.
From 2005-10 Lawson with Tony Croft established the sigma Centre for Excellence in University-wide mathematics and statistics support, which was awarded the 2011 Times Higher Education Award for Outstanding Support for Students. The Centre later developed under the National HE STEM Programme into the sigma Network for Excellence in Mathematics and Statistics Support.
Lawson was awarded National Teaching Fellowship in 2005. Lawson also acted as Assistant Chief Executive of the Higher Education Academy from 2012-13.
Lawson was professor and associate dean of the faculty of science and engineering at Coventry University (2002–13), director of the Maths, Stats & OR Network of the Higher Education Academy (2005-9), and chair of the executive group of the More Maths Grads project (2008–10).
From 2013-18, Lawson was pro-vice-chancellor of formative education at Newman University in Birmingham, before returning to Coventry.
Lawson received the Institute of Mathematics and its Applications' Gold Medal in 2016 jointly with Croft for "an outstanding contribution to the improvement of the teaching of mathematics" by establishing sigma.
Lawson was appointed Member of the Order of the British Empire (MBE) in the 2019 Birthday Honours for services to mathematics in higher education.
Lawson is currently Director of the sigma maths support centre at Coventry University.
Footnotes
External links
Year of birth missing (living people)
Living people
Alumni of the University of Oxford
20th-century British mathematicians
21st-century British mathematicians
Academics of Coventry University
Members of the Order of the British Empire |
https://en.wikipedia.org/wiki/More%20Maths%20Grads | More Maths Grads was a three-year project run from 2007 to 2010 by a consortium of British mathematics organisations which aimed to increase the supply of mathematical sciences graduates in England and to widen participation within the mathematical sciences from groups of learners who have not previously been well represented in higher education.
History
The project was launched to address a perceived problem with numbers of students studying mathematics at university - that higher education participation had increased since 2001 but numbers studying mathematical sciences remained almost constant, and had particular focus on encouraging participation from groups of learners who were not well represented in higher education. The project was initially called The Increasing the Supply of Mathematical Science Graduates programme before later being renamed More Maths Grads.
Funding of £3.3M was provided by the Higher Education Funding Council for England under the 'Strategically Important Subjects' initiative.
More Maths Grads was led by the Maths, Stats & OR Network on behalf of a consortium which also included the Institute of Mathematics and its Applications, the London Mathematical Society, the Royal Statistical Society, and HoDoMS, the Heads of Departments of Mathematical Sciences. The project concentrated its activity on three regions: West Midlands, Yorkshire & Humberside and London. It worked in collaboration with Coventry University, University of Leeds, Queen Mary, University of London and Sheffield Hallam University. It was overseen by a steering committee chaired by Duncan Lawson. The project was managed first by Helen Orr and later by Makhan Singh.
Work areas
The More Maths Grads project ran four strands of activity:
Careers theme, producing information about career opportunities with mathematics;
Student theme, focused on enrichment activities;
Teaching theme, professional development for teachers;
HE Curriculum theme, research about the current higher education mathematical sciences curriculum.
The HE Curriculum theme was concerned with curriculum content and also issues around student experience and teaching practice
Legacy
The conclusion of More Maths Grads after three years was marked by a Parliamentary Reception 'Where will maths take you?' on 27 January 2010, hosted by Charles Clarke MP, who claimed the project had made "an impact in improving standards of mathematics education and increasing the number continuing to study mathematics". At the event, project manager Makhan Singh claimed the project had "touched the lives of tens of thousands school students, plus many more members of the wider public" and highlighted the resources and good practice generated by the project, which included the Maths in a Box resource.
The project was followed by the National HE STEM Programme, which built on its work.
References
External links
Project web site
2007 establishments in the United Kingdom
2010 disestablishments in the |
https://en.wikipedia.org/wiki/Subdirect%20product | In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff in 1944 and has proved to be a powerful generalization of the notion of direct product.
Definition
A subdirect product is a subalgebra (in the sense of universal algebra) A of a direct product ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective.
A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A.
An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
Examples
Any distributive lattice L is subdirectly representable as a subalgebra of a direct power of the two-element distributive lattice. This can be viewed as an algebraic formulation of the representability of L as a set of sets closed under the binary operations of union and intersection, via the interpretation of the direct power itself as a power set. In the finite case such a representation is direct (i.e. the whole direct power) if and only if L is a complemented lattice, i.e. a Boolean algebra.
The same holds for any semilattice when "semilattice" is substituted for "distributive lattice" and "subsemilattice" for "sublattice" throughout the preceding example. That is, every semilattice is representable as a subdirect power of the two-element semilattice.
The chain of natural numbers together with infinity, as a Heyting algebra, is subdirectly representable as a subalgebra of the direct product of the finite linearly ordered Heyting algebras. The situation with other Heyting algebras is treated in further detail in the article on subdirect irreducibles.
The group of integers under addition is subdirectly representable by any (necessarily infinite) family of arbitrarily large finite cyclic groups. In this representation, 0 is the sequence of identity elements of the representing groups, 1 is a sequence of generators chosen from the appropriate group, and integer addition and negation are the corresponding group operations in each group applied coordinate-wise. The representation is faithful (no two integers are represented by the same sequence) because of the size requirement, and the projections are onto because every coordinate eventually exhausts its group.
Every vector space over a given field is subdirectly representable by the one-dimensional space over that field, with the finite-dimensional spaces being directly representable in this way. (For vector spaces, as for abelian groups, direct product with finitely many factors is synonymous w |
https://en.wikipedia.org/wiki/Subdirectly%20irreducible%20algebra | In the branch of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect product of "simpler" algebras. Subdirectly irreducible algebras play a somewhat analogous role in algebra to primes in number theory.
Definition
A universal algebra A is said to be subdirectly irreducible when A has more than one element, and when any subdirect representation of A includes (as a factor) an algebra isomorphic to A, with the isomorphism being given by the projection map.
Examples
The two-element chain, as either a Boolean algebra, a Heyting algebra, a lattice, or a semilattice, is subdirectly irreducible. In fact, the two-element chain is the only subdirectly irreducible distributive lattice.
Any finite chain with two or more elements, as a Heyting algebra, is subdirectly irreducible. (This is not the case for chains of three or more elements as either lattices or semilattices, which are subdirectly reducible to the two-element chain. The difference with Heyting algebras is that a → b need not be comparable with a under the lattice order even when b is.)
Any finite cyclic group of order a power of a prime (i.e. any finite p-group) is subdirectly irreducible. (One weakness of the analogy between subdirect irreducibles and prime numbers is that the integers are subdirectly representable by any infinite family of nonisomorphic prime-power cyclic groups, e.g. just those of order a Mersenne prime assuming there are infinitely many.) In fact, an abelian group is subdirectly irreducible if and only if it is isomorphic to a finite p-group or isomorphic to a Prüfer group (an infinite but countable p-group, which is the direct limit of its finite p-subgroups).
A vector space is subdirectly irreducible if and only if it has dimension one.
Properties
The subdirect representation theorem of universal algebra states that every algebra is subdirectly representable by its subdirectly irreducible quotients. An equivalent definition of "subdirect irreducible" therefore is any algebra A that is not subdirectly representable by those of its quotients not isomorphic to A. (This is not quite the same thing as "by its proper quotients" because a proper quotient of A may be isomorphic to A, for example the quotient of the semilattice (Z, ) obtained by identifying just the two elements 3 and 4.)
An immediate corollary is that any variety, as a class closed under homomorphisms, subalgebras, and direct products, is determined by its subdirectly irreducible members, since every algebra A in the variety can be constructed as a subalgebra of a suitable direct product of the subdirectly irreducible quotients of A, all of which belong to the variety because A does. For this reason one often studies not the variety itself but just its subdirect irreducibles.
An algebra A is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient, e |
https://en.wikipedia.org/wiki/L-estimator | In statistics, an L-estimator is an estimator which is a linear combination of order statistics of the measurements (which is also called an L-statistic). This can be as little as a single point, as in the median (of an odd number of values), or as many as all points, as in the mean.
The main benefits of L-estimators are that they are often extremely simple, and often robust statistics: assuming sorted data, they are very easy to calculate and interpret, and are often resistant to outliers. They thus are useful in robust statistics, as descriptive statistics, in statistics education, and when computation is difficult. However, they are inefficient, and in modern times robust statistics M-estimators are preferred, though these are much more difficult computationally. In many circumstances L-estimators are reasonably efficient, and thus adequate for initial estimation.
Examples
A basic example is the median. Given n values , if is odd, the median equals , the -th order statistic; if is even, it is the average of two order statistics: . These are both linear combinations of order statistics, and the median is therefore a simple example of an L-estimator.
A more detailed list of examples includes: with a single point, the maximum, the minimum, or any single order statistic or quantile; with one or two points, the median; with two points, the mid-range, the range, the midsummary (trimmed mid-range, including the midhinge), and the trimmed range (including the interquartile range and interdecile range); with three points, the trimean; with a fixed fraction of the points, the trimmed mean (including interquartile mean) and the Winsorized mean; with all points, the mean.
Note that some of these (such as median, or mid-range) are measures of central tendency, and are used as estimators for a location parameter, such as the mean of a normal distribution, while others (such as range or trimmed range) are measures of statistical dispersion, and are used as estimators of a scale parameter, such as the standard deviation of a normal distribution.
L-estimators can also measure the shape of a distribution, beyond location and scale. For example, the midhinge minus the median is a 3-term L-estimator that measures the skewness, and other differences of midsummaries give measures of asymmetry at different points in the tail.
Sample L-moments are L-estimators for the population L-moment, and have rather complex expressions. L-moments are generally treated separately; see that article for details.
Robustness
L-estimators are often statistically resistant, having a high breakdown point. This is defined as the fraction of the measurements which can be arbitrarily changed without causing the resulting estimate to tend to infinity (i.e., to "break down"). The breakdown point of an L-estimator is given by the closest order statistic to the minimum or maximum: for instance, the median has a breakdown point of 50% (the highest possible), and a n% trimmed or Winso |
https://en.wikipedia.org/wiki/Titchmarsh%20theorem | In mathematics, particularly in the area of Fourier analysis, the Titchmarsh theorem may refer to:
The Titchmarsh convolution theorem
The theorem relating real and imaginary parts of the boundary values of a Hp function in the upper half-plane with the Hilbert transform of an Lp function. See Hilbert transform#Titchmarsh's theorem. |
https://en.wikipedia.org/wiki/Kazuo%20Honma | is a Japanese professional footballer who plays as striker for Samut Prakan of the Thai League 3.
Club career statistics
References
External links
Kazuo Honma Interview (1)
Kazuo Honma Interview (2)
Kazuo Honma Interview (3)
1980 births
Living people
Sportspeople from Yokohama
Japanese men's footballers
Men's association football forwards
Thespakusatsu Gunma players
FK Mačva Šabac players
Tisza Volán SC footballers
Pápai FC footballers
Diósgyőri VTK players
Nyíregyháza Spartacus FC players
BFC Siófok players
Vasas SC players
Ferencvárosi TC footballers
Nemzeti Bajnokság I players
F.C. Chanthabouly players
Japanese expatriate men's footballers
Expatriate men's footballers in Serbia and Montenegro
Expatriate men's footballers in Hungary
Expatriate men's footballers in Laos
Japanese expatriate sportspeople in Serbia and Montenegro
Japanese expatriate sportspeople in Hungary
Japanese expatriate sportspeople in Laos
Japanese expatriate sportspeople in Thailand
Japanese expatriate sportspeople in Bhutan
Expatriate men's footballers in Thailand
Expatriate men's footballers in Bhutan |
https://en.wikipedia.org/wiki/The%20Review%20of%20Economics%20and%20Statistics | The Review of Economics and Statistics is a peer-reviewed academic journal that covers applied economics, with specific relevance to the scope of econometrics. The editors-in-chief are Will Dobbie (Harvard University) and Raymond Fisman (Boston University).
History
The journal, founded initially as The Review of Economic Statistics at Harvard University in 1917, published its official “inaugural volume” in 1919. The journal obtained its current title in 1948.
As the first editor-in-chief, Charles J. Bullock remarked in his Prefatory Statement to the first issue that "the purpose of the Review is to promote the collection, criticism, and interpretation of economic statistics, with a view to making them more accurate and valuable than they are at present for business and scientific purposes."
Editors-in-chief
The following persons are or have been editors-in-chief:
Notable papers
The following papers have been cited most:
References
External links
Econometrics journals
English-language journals
Harvard University academic journals
MIT Press academic journals
Academic journals established in 1919
5 times per year journals |
https://en.wikipedia.org/wiki/Sharp%20map | In differential geometry, the sharp map is the mapping that converts 1-forms into corresponding vectors, given a non-degenerate (0,2)-tensor.
Definition
Let be a manifold and denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field , for example a metric tensor or a symplectic form. The definition
yields a linear map sometimes called the flat map
which is an isomorphism, since is non-degenerate. Its inverse
is called the sharp map.
See also
Flat map
Differential topology
Differential geometry |
https://en.wikipedia.org/wiki/Giuseppe%20Rapisarda | Giuseppe Rapisarda (born 6 September 1985) is a Swiss football player. He currently plays for FC Wohlen.
External links
Statistics at T-Online.de
FC Aarau profile
Swiss Football League profile
FC Zurich stats
1985 births
Living people
Swiss men's footballers
FC Aarau players
FC Zürich players
FC Wohlen players
FC Baden players
FC Chiasso players
Swiss Super League players
Men's association football defenders |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Primera%20Divisi%C3%B3 | Statistics of Primera Divisió for the 1996–97 season.
Overview
It was contested by 12 teams, and Principat won the championship.
League table
Results
References
Primera Divisió seasons
Andorra
1996–97 in Andorran football |
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