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https://en.wikipedia.org/wiki/2011%20Division%204%20%28lower%20tier%29%20%28Swedish%20football%29 | Statistics of the lower tier of Swedish football Division 4 for the 2011 season. These 5 divisions feed into the Halland and Småland Elit divisions in Division 4
League standings
Halland 2011
Småland Nordvästra 2011
Småland Nordöstra 2011
Småland Sydvästra 2011
Småland Sydöstra 2011
Footnotes
References
Swedish Football Division 4 seasons
6 |
https://en.wikipedia.org/wiki/2002%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 2002 season.
League standings
Norra Norrland 2002
Mellersta Norrland 2002
Södra Norrland 2002
Norra Svealand 2002
Östra Svealand 2002
Västra Svealand 2002
Nordöstra Götaland 2002
Nordvästra Götaland 2002
Mellersta Götaland 2002
Sydöstra Götaland 2002
Sydvästra Götaland 2002
Södra Götaland 2002
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/Mixed%20Poisson%20process | In probability theory, a mixed Poisson process is a special point process that is a generalization of a Poisson process. Mixed Poisson processes are simple example for Cox processes.
Definition
Let be a locally finite measure on and let be a random variable with almost surely.
Then a random measure on is called a mixed Poisson process based on and iff conditionally on is a Poisson process on with intensity measure .
Comment
Mixed Poisson processes are doubly stochastic in the sense that in a first step, the value of the random variable is determined. This value then determines the "second order stochasticity" by increasing or decreasing the original intensity measure .
Properties
Conditional on mixed Poisson processes have the intensity measure and the Laplace transform
.
Sources
Poisson point processes |
https://en.wikipedia.org/wiki/Northwest%20Rankin%20High%20School | {
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Northwest Rankin High School is a suburban public high school located in Flowood, Mississippi, United States. The school serves grades 9-12 and is part of the Rankin County School District. The school's attendance was approximately 1,700 students as of the 2018 campus census.
Plans for the construction for a new high school in between the current campus and the campus of Northwest Rankin Middle School were approved by the Rankin Country School District board. It is expected to be completed by the start of the 2021–2022 school year. Funding for this project was approved by the District's bond issue.
Academics
Northwest Rankin offers Pre-Advanced Placement (Pre-AP) and Advanced Placement (AP) courses in English, Math, Science, and Social Studies. In addition, there are five four-year academic specializations (referred to as academies) offered in addition to several dual enrollment courses with Hinds Community College.
Academies
The academic specializations include the health science nursing academy, engineering academy, convergent media academy, educational leadership academy, and JROTC. All of these academies are four year programs that the student can receive an endorsement in upon graduation.
Classes
In addition to pre-AP courses, several AP courses are also offered, including AP Art, AP US History, AP US Government, AP English Literature, AP English Language, AP Spanish, AP French, AP Physics 1, AP Chemistry, AP Calculus AB, and AP Statistics. Dual credit courses including DC Algebra I, DC Algebra II, DC Biology I, DC Biology II, DC Composition I, DC Composition II, and DC Public Speaking are offered.
Student body
Demographics
As of the 2018-2019 school year, the school had an attendance of 1706 students.
Matriculation statistics
In the typical graduating class, 70% of all students attend a 2- or 4-year college or university, with 20% joining the workforce and 10% joining active US military service. From the 2017 graduating class, the estimated total scholarship award to national colleges and universities amounted to $110 million.
Clubs and organizations
Northwest Rankin hosts several extracurricular activities for its students, covering several areas of academic, leadership, and cultural interests.
Student government
A yearly committee is elected by the student population. The government is divided into executive positions (campus-wide representation) and class positions (Grade 10, 11, or 12 Representation). All positions have one seat available; the noteworthy exception is the co-president position, which has two seats.
Executive positions:
Co-President (2)
Vice President (1)
Secretary (1)
Treasurer (1)
Reporter (1)
Class positions:
Co-President (2)
Vi |
https://en.wikipedia.org/wiki/Walid%20Belhamri | Walid Belhamri (born 19 November 1990 in Bouinan, Algeria) is an Algerian professional footballer who plays for IB Khémis El Khechna.
Statistics
References
External links
1990 births
Living people
Algerian men's footballers
Algerian Ligue 2 players
People from Blida Province
Men's association football midfielders
Olympique de Médéa players
USM Blida players
AS Khroub players
MC El Eulma players
WA Tlemcen players
JS Saoura players
21st-century Algerian people |
https://en.wikipedia.org/wiki/2011%20Swedish%20football%20Division%205%20%28Part%201%29 | Statistics of Swedish football Division 5 for the 2011 season. This is Part 1 which covers Blekinge, Bohuslän, Dalarna, Dalsland, Gestrikland, Gotland, Göteborg, Halland, Hälsingland, Jämtland-Härjedalen, Medelpad, Norrbotten, Skåne and Småland.
See also 2011 Division 5 (Part 2) which covers Stockholm, Södermanland, Uppland, Värmland, Västerbotten, Västergötland, Västmanland, Ångermanland, Örebro Läns and Östergötland.
League standings
Blekinge 2011
Bohuslän 2011
Dalarna Norra 2011
Dalarna Södra 2011
Dalsland 2011
Gotland 2011
Gästrikland 2011
Norrham Withdrew
Göteborg A 2011
Göteborg B 2011
Halland Norra 2011
Halland Södra 2011
Hälsingland 2011
Trönö IK 2 Withdrew
Jämtland/Härjedalen 2011
Medelpad 2011
Norrbotten Norra 2011
Pajala IF Withdrew
Norrbotten Södra 2011
Skåne Nordvästra 2011
Skåne Nordöstra 2011
Skåne Sydvästra 2011
Skåne Sydöstra 2011
Skåne Södra 2011
Skåne Västra 2011
Småland Nordvästra 2011
Småland Nordöstra 2011
Småland Norra 2011
Småland Sydöstra 2011
Småland Södra 2011
Småland Västra 2011
See also
2011 Swedish football Division 5 (Part 2)
Footnotes
References
Swedish Football Division 5 seasons
7 |
https://en.wikipedia.org/wiki/2011%20Swedish%20football%20Division%205%20%28Part%202%29 | Statistics of Swedish football Division 5 for the 2011 season. This is Part 2 which covers Stockholm, Södermanland, Uppland, Värmland, Västerbotten, Västergötland, Västmanland, Ångermanland, Örebro Läns and Östergötland.
See also 2011 Division 5 (Part 1) which covers Blekinge, Bohuslän, Dalarna, Dalsland, Gestrikland, Gotland, Göteborg, Halland, Hälsingland, Jämtland-Härjedalen, Medelpad, Norrbotten, Skåne and Småland.
League standings
Stockholm Mellersta 2011
Stockholm Norra 2011
Stockholm Södra 2011
Södermanland Gul 2011
Irakona International SC Withdrew
Södermanland Svart 2011
Uppland Västra 2011
Uppland Östra 2011
Värmland Västra 2011
Värmland Östra 2011
Västerbotten Norra 2011
Västerbotten Södra 2011
Västergötland Nordvästra 2011
Västergötland Norra 2011
Västergötland Sydvästra 2011
Västergötland Sydöstra 2011
Västergötland Västra 2011
Västergötland Östra 2011
Västmanland 2011
Ångermanland 2011
Örebro Norra 2011
Axbergs IF Withdrew
Örebro Södra 2011
Östergötland Mellersta 2011
Östergötland Västra 2011
Östergötland Östra 2011
See also
2011 Swedish football Division 5 (Part 1)
Footnotes
References
Swedish Football Division 5 seasons
7 |
https://en.wikipedia.org/wiki/Scholz%27s%20reciprocity%20law | In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by and rediscovered by .
Statement
Suppose that p and q are rational primes congruent to 1 mod 4 such that the Legendre symbol (p/q) is 1. Then the ideal (p) factorizes in the ring of integers of Q() as (p)=𝖕𝖕' and similarly (q)=𝖖𝖖' in the ring of integers of Q().
Write εp and εq for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that
[εp/𝖖] = [εq/𝖕]
where [] is the quadratic residue symbol in a quadratic number field.
References
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Mohamed%20Billal%20Rait | Mohamed Billal Rait (born 16 May 1986 in Boufarik, Algeria) is an Algerian professional footballer. He currently plays as a midfielder for the Algerian Ligue 1 club RC Arbaâ.
Statistics
References
External links
1986 births
Living people
Algerian men's footballers
Olympique de Médéa players
RC Arbaâ players
ES Sétif players
Algerian Ligue 2 players
Algerian Ligue Professionnelle 1 players
People from Boufarik
Men's association football midfielders
WA Boufarik players
21st-century Algerian people |
https://en.wikipedia.org/wiki/Mohamed%20Deroukdal | Mohamed Deroukdal (born 2 December 1980) is an Algeria]] professional footballer. He plays as a defender for the Algerian Ligue 2 club Olympique de Médéa.
Statistics
References
1980 births
Living people
Algerian men's footballers
Olympique de Médéa players
Algerian Ligue Professionnelle 1 players
Algerian Ligue 2 players
Paradou AC players
OMR El Annasser players
USM Blida players
RC Kouba players
People from Kouba
IB Khémis El Khechna players
Men's association football defenders
21st-century Algerian people |
https://en.wikipedia.org/wiki/2001%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 2001 season.
League standings
Norra Norrland 2001
Mellersta Norrland 2001
Södra Norrland 2001
Norra Svealand 2001
Östra Svealand 2001
Västra Svealand 2001
Nordöstra Götaland 2001
Nordvästra Götaland 2001
Mellersta Götaland 2001
Sydöstra Götaland 2001
Sydvästra Götaland 2001
Södra Götaland 2001
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/2000%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 2000 season.
League standings
Norra Norrland 2000
Mellersta Norrland 2000
Södra Norrland 2000
Norra Svealand 2000
Östra Svealand 2000
Västra Svealand 2000
Nordöstra Götaland 2000
Nordvästra Götaland 2000
Mellersta Götaland 2000
Sydöstra Götaland 2000
Sydvästra Götaland 2000
Södra Götaland 2000
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/1999%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 1999 season.
League standings
Norra Norrland 1999
Mellersta Norrland 1999
Södra Norrland 1999
Norra Svealand 1999
Östra Svealand 1999
Västra Svealand 1999
Nordöstra Götaland 1999
Nordvästra Götaland 1999
Mellersta Götaland 1999
Sydöstra Götaland 1999
Sydvästra Götaland 1999
Södra Götaland 1999
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/1998%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 1998 season.
League standings
Norra Norrland 1998
Mellersta Norrland 1998
Södra Norrland 1998
Norra Svealand 1998
Östra Svealand 1998
Västra Svealand 1998
Nordöstra Götaland 1998
Nordvästra Götaland 1998
Mellersta Götaland 1998
Sydöstra Götaland 1998
Sydvästra Götaland 1998
Södra Götaland 1998
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/1997%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 1997 season.
League standings
Norra Norrland 1997
Mellersta Norrland 1997
Södra Norrland 1997
Norra Svealand 1997
Östra Svealand 1997
Västra Svealand 1997
Nordöstra Götaland 1997
Nordvästra Götaland 1997
Mellersta Götaland 1997
Sydöstra Götaland 1997
Sydvästra Götaland 1997
Södra Götaland 1997
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/Project%20SEED | Project SEED is a mathematics education program which worked in urban school districts across the United States. Project SEED is a nonprofit organization that worked in partnership with school districts, universities, foundations, and corporations to teach advanced mathematics to elementary and middle school students as a supplement to their regular math instruction. Project SEED also provides professional development for classroom teachers. Founded in 1963 by William F. Johntz, its primary goal is to use mathematics to increase the educational options of low-achieving, at-risk students.
The model is to hire people with a high appreciation and love for mathematics, for example, mathematicians, engineers, and physicists to be trained to teach. They are pre-trained in the program to teach Socratically, that is, only by asking questions of the students, rarely ever making statements, and even more rarely, validating or rejecting any answers given. A unique set of hand/arm signals are taught for use by the students constantly throughout the 45 min. lesson to wave their agreement, disagreement, uncertainty, desire to ask a question, partial agreement or desire to amend, or to signal a high five to each answer given by a student to the instructor's leading question. Lessons were lively, rapid paced at times. The signals allow students to support each other, while giving the instructor a way to gauge who's understood, who hasn't got it yet, and even, who is not paying much attention. Various signals also supported classroom management. The classroom atmosphere is one of utmost respect for the inquiry process and students' participation. No student ever feels put down; when their fellow students respectfully disagreed, one is invited to state their case, and the whole class each individually would use whatever signal indicated whether they agreed or disagreed. Logic, detection of patterns, drawing a picture of the problem, and many more reasoning skills were taught. The curriculum addresses primarily algebra and some calculus -- math topics with which their regular classroom teacher is often not well versed. Changing the expectations of the students' teachers, parents and family after they witnessed the students' mental abilities to understand and articulate many truths of mathematics, elevated their expectations for the students' academic abilities generating a more positive environment for their academic success.
About
Project SEED is primarily a mathematics instruction program delivered to intact classes of elementary and middle school students, many from low-income backgrounds, to better prepare them for high school and college math. SEED Instruction utilized the Socratic method, in which instructors use a question-and-answer approach to guide students to the discovery of mathematical principles.
The SEED instructors are math subject specialists, with degrees in mathematics or math-based sciences, who use a variety of techniques including han |
https://en.wikipedia.org/wiki/Inverted%20bell%20curve | In statistics, an inverted bell curve is a term used loosely or metaphorically to refer to a bimodal distribution that falls to a trough between two peaks, rather than (as in a standard bell curve) rising to a single peak and then falling off on both sides.
References
Continuous distributions
Gaussian function |
https://en.wikipedia.org/wiki/Bhima%20I | {
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Bhima I (r. c. 1022–1064 CE) was a Chaulukya king who ruled parts of present-day Gujarat, India. The early years of his reign saw an invasion from the Ghaznavid ruler Mahmud, who sacked the Somnath temple. Bhima left his capital and took shelter in Kanthkot during this invasion, but after Mahmud's departure, he recovered his power and retained his ancestral territories. He crushed a rebellion by his vassals at Arbuda, and unsuccessfully tried to invade the Naddula Chahamana kingdom. Towards the end of his reign, he formed an alliance with the Kalachuri king Lakshmi-Karna, and played an important role in the downfall of the Paramara king Bhoja.
The earliest of the Dilwara Temples and the Modhera Sun Temple were built during Bhima's reign. The construction of Rani ki vav is attributed to his queen Udayamati.
Early life
Bhima's father Nagaraja was a son of the Chaulukya king Chamunda-raja. Chamunda was succeeded by Nagaraja's brothers, Vallabha-raja and Durlabha-raja, in that order. Both Vallabha and Durlabha died childless. According to the 12th century author Hemachandra, Durlabha was very fond of his nephew Bhima, and appointed Bhima as his successor before his death. Durlabha and Nagaraja died soon after Bhima's ascension to throne.
Military career
Ghaznavid invasion
Early during his reign, Bhīma faced an invasion by Mahmūd of Ghazni, whose plunder of the Somnāth temple has been described in detail by the medieval Muslim historians. According to Ali ibn al-Athīr, Mahmūd started out from Ghazni on 18 October 1025. At Multan, he planned his march in detail and gathered supplies. He left Multan on 26 November, with a large army well-equipped to cross the Thar desert, and reached the Chaulukya capital in December 1025 CE.
According to the Muslim accounts, Bhīma fled his capital Aṇahilapāṭaka (called Nahrwāla by the medieval Muslim historians). He took shelter in Kanthakot, allowing Mahmud to enter the Chaulukya capital unopposed. Mahmūd's sudden invasion, coupled with the lack of any fortifications in Nahrwāla, may have forced Bhima to abandon his capital. Other residents of the city also appear to have evacuated it, as the Muslim historians do not mention any massacre or looting in the Chaulukya capital.
Mahmūd rested at Nahrwāla for a few days, replenished his supplies, and then left for Somnāth. A relatively small force of 20,000 soldiers unsuccessfully tried to chec |
https://en.wikipedia.org/wiki/Omar%20Ibrahim%20Hammad | Omar Ibrahim Hammad is a Sudanese footballer who plays . He plays as a right winger.
Club career statistics
Statistics accurate as of 21 August 2012
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
External links
Player profile - QSL.com.qa
References
1986 births
Living people
Sudanese men's footballers
Sudanese expatriate men's footballers
Sudanese expatriate sportspeople in Qatar
Al Ahli SC (Doha) players
Al-Arabi SC (Qatar) players
Al-Sailiya SC players
Umm Salal SC players
Men's association football wingers |
https://en.wikipedia.org/wiki/Lee-Jen%20Wei | Lee-Jen Wei () is a Taiwanese-American professor of biostatistics at Harvard University.
Career
He was graduated from Fu Jen Catholic University's Mathematics Department in 1970. He obtained his PhD from the University of Wisconsin–Madison in 1975. He has been a tenured Professor of Biostatistics at Harvard University since 1991 and was the co-director of the Bioinformatics Core at the Harvard School of Public Health from 2003 to 2007. From 2003 to 2004, he served as the acting chair of the Department of Biostatistics at Harvard University. Under his supervision, the department successfully converted the doctor of science degree program in biostatistics (a professional degree) to a conventional (art and sciences) Ph.D. program at the Harvard Graduate School. This was an important accomplishment since the department had tried this conversion for more than 20 years without success.
Early career
Before Harvard University, he was a tenured Professor of Biostatistics and Statistics at the University of South Carolina, University of Wisconsin–Madison, the University of Michigan, and the George Washington University from 1982 to 1991. He was named Cancer Expert by The National Cancer Institute in 1980.
Research and contributions
Wei has developed and published a number of novel quantitative methods for analyzing data from experimental and observational studies. Specifically, he has published many papers on monitoring drug and device safety and related topics. The resulting procedures have been utilized for various drug and device regulatory evaluations involving safety issues. His extensive experience in quantitative science for making inferences about the drug and device safety is readily applicable to the general industry product safety issues.
Wei has also served on numerous Data Safety Monitoring Boards for experimental studies for the drug industry. And has moreover been intimately involved in designing, monitoring and analyzing various kinds of studies in assessing postmarketing surveillance data to identify signals of safety concerns.
Wei's scholarly writings include over 130 articles in peer-reviewed academic journals. He is responsible for developing numerous novel statistical methods for practitioners. Many of these methods have been included in the most commonly used statistical packages such as SAS, S-plus, and R. He has additionally served on the editorial boards of a number of statistical journals and am an elected Fellow of the American Statistical Association and Institute of Mathematical Statistics.
Awards and recognition
In 1986 he was elected as a Fellow of the American Statistical Association.
In 1999 he was awarded the Outstanding Alumni Award of Fu Jen Catholic University.
Professor Wei was named “Statistician of the Year” in 2007 by the Boston Chapter of the American Statistical Association.
The American Statistical Association gave him the Wilks Memorial Award in 2009 "for statistical methods used in clinical trials.", |
https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Faber%E2%80%93Krahn%20inequality | In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eigenvalue of the Laplace operator on a bounded domain in , . It states that the first Dirichlet eigenvalue is no less than the corresponding Dirichlet eigenvalue of a Euclidean ball having the same volume. Furthermore, the inequality is rigid in the sense that if the first Dirichlet eigenvalue is equal to that of the corresponding ball, then the domain must actually be a ball. In the case of , the inequality essentially states that among all drums of equal area, the circular drum (uniquely) has the lowest voice.
More generally, the Faber–Krahn inequality holds in any Riemannian manifold in which the isoperimetric inequality holds. In particular, according to Cartan–Hadamard conjecture, it should hold in all simply connected manifolds of nonpositive curvature.
See also
Hearing the shape of a drum
References
Elliptic partial differential equations
Riemannian geometry
Spectral theory |
https://en.wikipedia.org/wiki/Denis%20Creissels | Denis Creissels (born 18 September 1943) is a French professor of linguistics at the University of Lyon.
After studying mathematics and Russian, he has taught general linguistics at the University of Grenoble from 1970 to 1996, and at the University of Lyon from 1996 to 2008. He is now professor emeritus at the University of Lyon, and member of the research team Dynamique du Langage.
He is specialised in languages of Africa and the Caucasus. He has done extensive research on Hungarian, Tswana, Malinke and Akhvakh.
His 1995 and 2006 books on syntax are widely used as textbooks in linguistic typology and syntax in French-speaking universities.
He has taught as invited professor at the Summer School on Linguistic Typology in Leipzig in 2010.
Monographs
2009 Le malinké de Kita. Cologne : Rüdiger Köppe.Rüdiger Köppe Verlag
2006 Syntaxe générale, une introduction typologique. Paris : Hermès. (2 vol., 412 p. & 334 p.)
1997 (en collaboration avec A.M. Chebanne et H.W. Nkhwa) Tonal morphology of the Setswana verb. Munich : LINCOM Europa. 227 p.
1995 Eléments de syntaxe générale. Presses Universitaires de France. 332 p.
1994 Aperçu sur les structures phonologiques des langues négro-africaines, 2ème édition entièrement revue et complétée. Grenoble : Editions Littéraires et Linguistiques de l’Université de Grenoble. 320 p.
1991 Description des langues négro-africaines et théorie syntaxique. Grenoble : Editions Littéraires et Linguistiques de l’Université de Grenoble, 466 p.
1989 Aperçu sur les structures phonologiques des langues négro-africaines, 1ère édition. Grenoble : Editions Littéraires et Linguistiques de l’Université de Grenoble. 288 p.
1983 Eléments de grammaire de la langue mandinka. Grenoble : Publications de l’Université des Langues et Lettres de Grenoble. 223 p.
1979 (en collaboration avec N. Kouadio) Les tons du baoulé (parler de la région de Toumodi). Abidjan : Institut de Linguistique Appliquée. 123 p.
1977 (en collaboration avec N. Kouadio) Description phonologique et grammaticale d’un parler baoulé. Abidjan : Institut de Linguistique Appliquée. 642 p.
References
External links
http://www.ddl.ish-lyon.cnrs.fr/annuaires/index.asp?Langue=FR&Page=Denis%20CREISSELS
http://deniscreissels.fr/
1943 births
Living people
Linguists from France
Linguists of Bantu languages
Linguists of Mande languages
Linguists of Northeast Caucasian languages |
https://en.wikipedia.org/wiki/Geometrography | In the mathematical field of geometry, geometrography is the study of geometrical constructions. The concepts and methods of geometrography were first expounded by Émile Lemoine (1840–1912), a French civil engineer and a mathematician, in a meeting of the French Association for the Advancement of the Sciences held at Oran in 1888.
Lemoine later expanded his ideas in another memoir read at the Pau meeting of the same Association held in 1892.
It is well known in elementary geometry that certain geometrical constructions are simpler than certain others. But in many case it turns out that the apparent simplicity of a construction does not consist in the practical execution of the construction, but in the brevity of the statement of what has to be done. Can then any objective criterion be laid down by which an estimate may be formed of the relative simplicity of several different constructions for attaining the same end? Lemoine developed the ideas of geometrography to answer this question. The question of the ubiquity of a construction is also raised. Whether or not a construction, regardless of simplicity, can be applied in all or most conditions, or just in the special cases, is an important consideration.
Basic ideas
In developing the ideas of geometrography, Lemoine restricted himself to Euclidean constructions using rulers and compasses alone. According to the analysis of Lemoine, all such constructions can be executed, as a sequence of operations selected form a fixed set of five elementary operations. The five elementary operations identified by Lemoine are the following:
Elementary operations in a geometrical construction
In a geometrical construction the fact that an operation X is to be done n times is denoted by the expression nX. The operation of placing a ruler in
coincidence with two points is indicated by 2R1. The operation of putting one point of the compasses on a determinate point and the other point of the compasses
on another determinate point is 2C1.
Every geometrical construction can be represented by an expression of the following form
l1R1 + l2R2 + m1C1 + m2C2 + m3C3.
Here the coefficients l1, etc. denote the number of times any
particular operation is performed.
Coefficient of simplicity
The number l1 + l2 + m1 +m2 + m3 is called the coefficient of simplicity, or the simplicity of the construction. It denotes the total number of operations.
Coefficient of exactitude
The number l1 + m1 + m2 is
called the coefficient of exactitude, or the exactitude of the construction; it denotes the number of preparatory operations, on which the exactitude of the construction depends.
Examples
Lemoine applied his scheme to analyze more than sixty problems in elementary geometry.
The construction of a triangle given the three vertices can be represented by the expression 4R1 + 3R2.
A certain construction of the regular heptadecagon involving the Carlyle circles can be represented by the expression 8R1 + 4R2 + 22C1 + 11C3 a |
https://en.wikipedia.org/wiki/QUEL%20query%20languages | QUEL is a relational database query language, based on tuple relational calculus, with some similarities to SQL. It was created as a part of the Ingres DBMS effort at University of California, Berkeley, based on Codd's earlier suggested but not implemented Data Sub-Language ALPHA. QUEL was used for a short time in most products based on the freely available Ingres source code, most notably in an implementation called POSTQUEL supported by POSTGRES. As Oracle and DB2 gained market share in the early 1980s, most companies then supporting QUEL moved to SQL instead. QUEL continues to be available as a part of the Ingres DBMS, although no QUEL-specific language enhancements have been added for many years.
Usage
QUEL statements are always defined by tuple variables, which can be used to limit queries or return result sets. Consider this example, taken from one of the first original Ingres papers:
range of E is EMPLOYEE
retrieve into W
(COMP = E.Salary / (E.Age - 18))
where E.Name = "Jones"
Here E is a tuple variable that ranges over the EMPLOYEE relation, and all tuples in that relation are found which satisfy the qualification `E.Name = "Jones"`. The result of the query is a new relation W, which has a single domain COMP that has been calculated for each qualifying tuple. Additional queries can then be made against the relation W.
An equivalent SQL statement is:
create table W as
select (E.salary / (E.age - 18)) as COMP
from employee as E
where E.name = 'Jones'
In this example, the relation is being stored in a new table W. This is not a direct analog of the QUEL version; relations in QUEL are more similar to temporary tables seen in most modern SQL implementations.
Here is a sample of a simple session that creates a table, inserts a row into it, and then retrieves and modifies the data inside it and finally deletes the row that was added (assuming that name is a unique field).
Another feature of QUEL was a built-in system for moving records en-masse into and out of the system. Consider this command:
copy student(name=c0, comma=d1, age=c0, comma=d1, sex=c0, comma=d1, address=c0, nl=d1)
into "/student.txt"
which creates a comma-delimited file of all the records in the student table. The d1 indicates a delimiter, as opposed to a data type. Changing the to a reverses the process. Similar commands are available in many SQL systems, but usually as external tools, as opposed to being internal to the SQL language. This makes them unavailable to stored procedures.
QUEL has an extremely powerful aggregation capability. Aggregates can be nested, and different aggregates can have independent by-lists and/or restriction clauses. For example:
retrieve (a=count(y.i by y.d where y.str = "ii*" or y.str = "foo"), b=max(count(y.i by y.d)))
This example illustrates one of the arguably less desirable quirks of QUEL, namely that all string comparisons are potentially pattern matches. matches all values starting with . In contrast, SQL uses only for exac |
https://en.wikipedia.org/wiki/Mohammad%20Shatnawi | Mohammad Shatnawi (; born 17 August 1985) is a Jordanian professional footballer who plays as a goalkeeper for Jordanian club Sahab.
Career statistics
International
Honours
Club
Al-Hussein
Jordan FA Shield: 2005
Al-Faisaly
Jordan Premier League: 2011–12
Jordan FA Cup: 2011–12, 2014–15
Jordan FA Shield: 2011
Jordan Super Cup: 2012, 2015
References
External links
Goal.com
Living people
1985 births
Jordan men's international footballers
Al-Faisaly SC players
Al-Ramtha SC players
Al-Hussein SC (Irbid) players
Jordanian men's footballers
Men's association football goalkeepers
Mansheyat Bani Hasan SC players
Footballers at the 2006 Asian Games
Asian Games competitors for Jordan
Al-Salt SC players
Jordanian Pro League players
Sahab SC players
Ma'an SC players |
https://en.wikipedia.org/wiki/Rational%20reciprocity%20law | In number theory, a rational reciprocity law is a reciprocity law involving residue symbols that are related by a factor of +1 or –1 rather than a general root of unity.
As an example, there are rational biquadratic and octic reciprocity laws. Define the symbol (x|p)k to be +1 if x is a k-th power modulo the prime p and -1 otherwise.
Let p and q be distinct primes congruent to 1 modulo 4, such that (p|q)2 = (q|p)2 = +1. Let p = a2 + b2 and q = A2 + B2 with aA odd. Then
If in addition p and q are congruent to 1 modulo 8, let p = c2 + 2d2 and q = C2 + 2D2. Then
References
Algebraic number theory |
https://en.wikipedia.org/wiki/1996%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 1996 season.
League standings
Norra Norrland 1996
Mellersta Norrland 1996
Södra Norrland 1996
Norra Svealand 1996
Östra Svealand 1996
Västra Svealand 1996
Nordöstra Götaland 1996
Nordvästra Götaland 1996
Mellersta Götaland 1996
Sydöstra Götaland 1996
Sydvästra Götaland 1996
Södra Götaland 1996
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/Quality%20%26%20Quantity | Quality & Quantity is an interdisciplinary double-blind peer-reviewed academic journal dealing with methodological issues in the fields of economics, psychology and sociology, mathematics, and statistics. The journal is published by Springer Science+Business Media.
References
External links
Sociology journals
Mathematical and statistical psychology journals
Economics journals
English-language journals
Research methods journals
Springer Science+Business Media academic journals
Bimonthly journals |
https://en.wikipedia.org/wiki/1995%20Swedish%20Football%20Division%203 | Statistics of Swedish football Division 3 for the 1995 season.
League standings
Norra Norrland 1995
Mellersta Norrland 1995
Södra Norrland 1995
Norra Svealand 1995
Östra Svealand 1995
Västra Svealand 1995
Nordöstra Götaland 1995
Nordvästra Götaland 1995
Mellersta Götaland 1995
Sydöstra Götaland 1995
Sydvästra Götaland 1995
Södra Götaland 1995
Footnotes
References
Swedish Football Division 3 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/Convex%20space | In mathematics, a convex space (or barycentric algebra) is a space in which it is possible to take convex combinations of any sets of points.
Formal Definition
A convex space can be defined as a set equipped with a binary convex combination operation for each satisfying:
(for )
From this, it is possible to define an n-ary convex combination operation, parametrised by an n-tuple , where .
Examples
Any real affine space is a convex space. More generally, any convex subset of a real affine space is a convex space.
History
Convex spaces have been independently invented many times and given different names, dating back at least to Stone (1949). They were also studied by Neumann (1970) and Świrszcz (1974), among others.
References
Convex geometry
Algebraic structures |
https://en.wikipedia.org/wiki/Riemann%20%28surname%29 | Riemann is a German surname. Notable people with this surname include the following:
Bernhard Riemann (1826–1866), German mathematician, originator of Riemannian geometry
Fritz Riemann (1859–1932), German chess master
Fritz Riemann (psychologist) (1902–1979), German psychoanalyst
Hugo Riemann (1849–1919), German musicologist
Johannes Riemann (1888–1959), German actor
Katja Riemann (born 1963), German actress
Leopold Reimann (1892-1917), German flying ace
Manuel Riemann (born 1988), German football (soccer) player
Paula Riemann (born 1993), German actress
Solomon Riemann (died c. 1873), Jewish traveller
See also
Reimann, a similar surname
List of topics named after Bernhard Riemann
Riemann (crater), a lunar crater
German-language surnames |
https://en.wikipedia.org/wiki/Cofree%20coalgebra | In algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space over a field exists, though it is more complicated than one might expect by analogy with the free algebra.
Definition
If V is a vector space over a field F, then the cofree coalgebra C (V), of V, is a coalgebra together with a linear map C (V) → V, such that any linear map from a coalgebra X to V factors through a coalgebra homomorphism from X to C (V). In other words, the functor C is right adjoint to the forgetful functor from coalgebras to vector spaces.
The cofree coalgebra of a vector space always exists, and is unique up to canonical isomorphism.
Cofree cocommutative coalgebras are defined in a similar way, and can be constructed as the largest cocommutative coalgebra in the cofree coalgebra.
Construction
C (V) may be constructed as a completion of the tensor coalgebra T(V) of V. For k ∈ N = {0, 1, 2, ...}, let TkV denote the k-fold tensor power of V:
with T0V = F, and T1V = V. Then T(V) is the direct sum of all TkV:
In addition to the graded algebra structure given by the tensor product isomorphisms TjV ⊗ TkV → Tj+kV for j, k ∈ N, T(V) has a graded coalgebra structure Δ : T(V) → T(V) ⊠ T(V) defined by extending
by linearity to all of T(V).
Here, the tensor product symbol ⊠ is used to indicate the tensor product used to define a coalgebra; it must not be confused with the tensor product ⊗, which is used to define the bilinear multiplication operator of the tensor algebra. The two act in different spaces, on different objects. Additional discussion of this point can be found in the tensor algebra article.
The sum above makes use of a short-hand trick, defining to be the unit in the field . For example, this short-hand trick gives, for the case of in the above sum, the result that
for . Similarly, for and , one gets
Note that there is no need to ever write as this is just plain-old scalar multiplication in the algebra; that is, one trivially has that
With the usual product this coproduct does not make T(V) into a bialgebra, but is instead dual to the algebra structure on T(V∗), where V∗ denotes the dual vector space of linear maps V → F. It can be turned into a bialgebra with the product where (i,j) denotes the binomial coefficient . This bialgebra is known as the divided power Hopf algebra. The product is dual to the coalgebra structure on T(V∗) which makes the tensor algebra a bialgebra.
Here an element of T(V) defines a linear form on T(V∗) using the nondegenerate pairings
induced by evaluation, and the duality between the coproduct on T(V) and the product on T(V∗) means that
This duality extends to a nondegenerate pairing
where
is the direct product of the tensor powers of V. (The direct sum T(V) is the subspace of the direct product for which only finitely many components are nonzero.) However, the coproduct Δ on T(V) only extends to a linear map
wi |
https://en.wikipedia.org/wiki/Gassmann%20triple | In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a group G together with two faithful actions on sets X and Y, such that X and Y are not isomorphic as G-sets but every element of G has the same number of fixed points on X and Y. They were introduced by Fritz Gassmann in 1926.
Applications
Gassmann triples have been used to construct examples of pairs of mathematical objects with the same invariants that are not isomorphic, including arithmetically equivalent number fields and isospectral graphs and isospectral Riemannian manifolds.
Examples
The simple group G = SL3(F2) of order 168 acts on the projective plane of order 2, and the actions on the 7 points and 7 lines give a Gassmann triple.
References
Permutation groups |
https://en.wikipedia.org/wiki/Shahrul%20Azhar%20Ture | Shahrul Azhar bin Ture (born 15 September 1985) is a Malaysian footballer who plays as a midfielder for the Malaysia Super League club PKNS.
Career statistics
Club
References
1985 births
Living people
Malaysian men's footballers
Malaysia Super League players
Sabah F.C. (Malaysia) players
Selangor F.C. II players
Footballers from Sabah
Men's association football midfielders |
https://en.wikipedia.org/wiki/Jack%20van%20Wijk | Jarke J. (Jack) van Wijk (born 1959) is a Dutch computer scientist, a professor in the Department of Mathematics and Computer Science at the Eindhoven University of Technology, and an expert in information visualization.
Biography
Van Wijk received his M.S. from the Delft University of Technology in 1982. His master's thesis, on simulation of traffic collisions, led him to become interested in computer visualization, and he remained at Delft for his doctoral studies, completing a Ph.D. in 1986 under the supervision of Dennis J. McConalogue. He worked at the Energy Research Centre of the Netherlands from 1988 until 1998, when he joined the Eindhoven faculty; he was promoted to a full professorship in 2001.
As well as being a faculty member at TU Eindhoven, van Wijk is the vice president for scientific affairs at MagnaView, a Dutch information visualization company.
Research
In information visualization, Van Wijk is known for his research in texture synthesis, treemaps, and flow visualization. His work on map projection won the 2009 Henry Johns Award of the British Cartographic Society for best cartographic journal article.
He has twice been program co-chair for IEEE Visualization, and once for IEEE InfoVis. In 2007, he received an IEEE Technical Achievement Award for his visualization research.
In graph drawing, van Wijk has worked on the visualization of small-world networks and on the depiction of abstract trees as biological trees. He has also conducted user studies that showed that the standard depiction of directed edges in graph drawings using arrowheads is less effective at conveying the directionality of the edges to readers than other conventions such as tapering. He was one of two invited speakers at the 19th International Symposium on Graph Drawing in 2011, and the capstone speaker at the IEEE Visualization 2013.
References
External links
Home page at TU Eindhoven
1959 births
Living people
Dutch computer scientists
Graph drawing people
Information visualization experts
Delft University of Technology alumni
Academic staff of the Eindhoven University of Technology |
https://en.wikipedia.org/wiki/P%C3%A9ter%20Komj%C3%A1th | Péter Komjáth (born 8 April 1953) is a Hungarian mathematician, working in set theory, especially combinatorial set theory. Komjáth is a professor at the Faculty of Sciences of the Eötvös Loránd University. He is currently a visiting faculty member at Emory University in the department of Mathematics and Computer Science.
Komjáth won a gold medal at the International Mathematical Olympiad in 1971. His Ph.D. advisor at Eötvös was András Hajnal, and he has two joint papers with Paul Erdős. He received the Paul Erdős Prize in 1990. He is a member of the Hungarian Academy of Sciences.
Selected publications
Komjáth, Péter and Vilmos Totik: Problems and Theorems in Classical Set Theory, Springer-Verlag, Berlin, 2006.
.
.
.
References
External links
Hungarian Academy of Sciences Webpage
Eötvös University Webpage
Rutgers University Webpage
1953 births
Living people
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Set theorists
Mathematicians from Budapest
Eötvös Loránd University alumni
Academic staff of Eötvös Loránd University
International Mathematical Olympiad participants |
https://en.wikipedia.org/wiki/2011%20Mongolian%20Premier%20League | Statistics of Niislel Lig in the 2011 season. The title was won by FC Ulaanbaatar which was their first ever title.
League standings
In the first round of the competition, all the teams played each other twice. The top four teams advanced to the semi-finals.
Play-offs
Semi-finals
Final
Super Cup
The 2011 Super Cup was played on 1 October 2011 between the league champion FC Ulaanbaatar and the cup winner Erchim. The winner would qualify for the 2012 AFC President's Cup.
Erchim won 2–1 and became the first Mongolian representative in the AFC President's Cup.
References
External links
FIFA.com
Soccerway.com
Mongolia Premier League seasons
Mongolia
Mongolia
football |
https://en.wikipedia.org/wiki/Knut%20Hartwig | Knut Hartwig (born 13 November 1969) is retired a German football midfielder. Hartwig portrayed Fritz Walter in the movie The Miracle of Bern.
Career statistics
Personal life
Knut Hartwig is the father of Luis Hartwig.
References
External links
1969 births
Living people
German men's footballers
VfL Bochum II players
Wuppertaler SV players
SC Preußen Münster players
Rot-Weiss Essen players
2. Bundesliga players
Footballers from Münster
Men's association football midfielders |
https://en.wikipedia.org/wiki/Walter%20Zastrau | Walter Zastrau (born 30 May 1935) is a German retired professional footballer who played as a defender.
Career statistics
References
External links
Living people
1935 births
German men's footballers
Men's association football defenders
Germany men's international footballers
Germany men's B international footballers
Rot-Weiss Essen players
FC Schalke 04 players
VfL Bochum players |
https://en.wikipedia.org/wiki/Parallel%20postulate | In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.
This postulate does not specifically talk about parallel lines; it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates.
Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate.
The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate does not hold is known as a non-Euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the modern equivalent of the first four postulates) is known as absolute geometry (or sometimes "neutral geometry").
Equivalent properties
Probably the best-known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
This axiom by itself is not logically equivalent to the Euclidean parallel postulate since there are geometries in which one is true and the other is not. However, in the presence of the remaining axioms which give Euclidean geometry, each of these can be used to prove the other, so they are equivalent in the context of absolute geometry.
Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. These equivalent statements include:
There is at most one line that can be drawn parallel to another given one through an external point. (Playfair's axiom)
The sum of the angles in every triangle is 180° (triangle postulate).
There exists a triangle whose angles add up to 180°.
The sum of the angles is the same for every triangle.
There exists a pair of similar, but not congruent, triangles.
Every triangle can be circumscribed.
If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.
There exists a quadrilateral in which all angles are right angles, that is, a rectangle.
There exists a pair of straight lines that are at constant distance from each other.
Two lines that a |
https://en.wikipedia.org/wiki/Playfair%27s%20axiom | In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.
It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line L and a point P not on L, as follows:
Construct a perpendicular: Using the axioms and previously established theorems, you can construct a line perpendicular to line L that passes through P.
Construct another perpendicular: A second perpendicular line is drawn to the first one, starting from point P.
Parallel Line: This second perpendicular line will be parallel to L by the definition of parallel lines (i.e the alternate interior angles are congruent as per the 4th axiom).
The statement is often written with the phrase, "there is one and only one parallel". In Euclid's Elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used.
This axiom is used not only in Euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. In the affine geometry setting, the stronger form of Playfair's axiom (where "at most one" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as Euclid's parallel axiom, even though it was not Euclid's version of the axiom.
History
Proclus (410–485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31).
In 1785 William Ludlam expressed the parallel axiom as follows:
Two straight lines, meeting at a point, are not both parallel to a third line.
This brief expression of Euclidean parallelism was adopted by Playfair in his textbook Elements of Geometry (1795) that was republished often. He wrote
Two straight lines which intersect one another cannot be both parallel to the same straight line.
Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point.
In 1883 Arthur Cayley was president of the British Association and expressed this opinion in his address to the Association:
My own view is that Euclid's Twelfth Axiom in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, which is the representation lying at the bottom of all external experience.
When David Hilbert wrote his book, Foundations of Geometry (1899), providing |
https://en.wikipedia.org/wiki/Hyperoperation | In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).
After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) and can be written as using n − 2 arrows in Knuth's up-arrow notation.
Each hyperoperation may be understood recursively in terms of the previous one by:
It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:
This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes's number and googolplexplex (e.g. is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3).
This recursion rule is common to many variants of hyperoperations.
Definition
Definition, most common
The hyperoperation sequence is the sequence of binary operations , defined recursively as follows:
(Note that for n = 0, the binary operation essentially reduces to a unary operation (successor function) by ignoring the first argument.)
For n = 0, 1, 2, 3, this definition reproduces the basic arithmetic operations of successor (which is a unary operation), addition, multiplication, and exponentiation, respectively, as
The operations for n ≥ 3 can be written in Knuth's up-arrow notation.
So what will be the next operation after exponentiation? We defined multiplication so that and defined exponentiation so that so it seems logical to define the next operation, tetration, so that with a tower of three 'a'. Analogously, the pentation of (a, 3) will be tetration(a, tetration(a, a)), with three "a" in it.
Knuth's notation could be extended to negative indices ≥ −2 in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
The hyperoperations can thus be seen as an answer to the question "what's next" in the sequence: successor, addition, multiplication, exponentiation, and so on. Noting that
the relationship between basic arithmetic operations is illustrated, allowing the higher operations to be defined naturally as above. The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term; so a is the base, b is the exponent (or hyperexponent), and n is the rank (or grade), and moreover, is read as "the bth n-ation of a", e.g. is read as "the 9th tetration of 7", and is read as "the 789th 123-a |
https://en.wikipedia.org/wiki/Kunita%E2%80%93Watanabe%20inequality | In stochastic calculus, the Kunita–Watanabe inequality is a generalization of the Cauchy–Schwarz inequality to integrals of stochastic processes.
It was first obtained by Hiroshi Kunita and Shinzo Watanabe and plays a fundamental role in their extension of Ito's stochastic integral to square-integrable martingales.
Statement of the theorem
Let M, N be continuous local martingales and H, K measurable processes. Then
where the angled brackets indicates the quadratic variation and quadratic covariation operators. The integrals are understood in the Lebesgue–Stieltjes sense.
References
Probability theorems
Probabilistic inequalities |
https://en.wikipedia.org/wiki/Dallas%20Mavericks%20all-time%20roster%20and%20statistics%20leaders | The Dallas Mavericks are an American professional basketball team based in Dallas. They play in the Southwest Division of the Western Conference in the National Basketball Association (NBA). The team joined the NBA in 1980 as an expansion team and won their first NBA championship in 2011. The Mavericks have played their home games at the American Airlines Center since 2001. Their principal owner is Mark Cuban. Their current staff includes Nico Harrison as general manager and Jason Kidd as head coach.
Dirk Nowitzki, who played his entire NBA career with the Mavericks starting in 1998, is the franchise's longest-serving player. He played more games, played more minutes, scored more points, and recorded more rebounds than any other Maverick. He also leads the franchise in field goals made, three-pointers made, and free throws made. His achievements include the Most Valuable Player Award in 2007, Finals Most Valuable Player Award in 2011, 14 All-Star Game selections, and 12 consecutive All-NBA Team selections.
Other prominent Mavericks include Steve Nash, who was selected to two All-NBA Teams and two All-Star Games. He and Luka Dončić are the only other Mavericks who have been selected to the All-NBA Team. Rolando Blackman and Mark Aguirre were selected to four and three All-Star games, respectively, and Dončić has been selected to three as of 2022. Five other players, James Donaldson, Michael Finley, Chris Gatling, Josh Howard, and Jason Kidd, were selected to the All-Star Game at least once during their Mavericks careers. Three Mavericks have won the NBA Sixth Man of the Year Award: Roy Tarpley in 1988, Antawn Jamison in 2004, and Jason Terry in 2009. Ten players were selected to the All-Rookie Team, including Kidd, who won the Rookie of the Year Award in 1995, and Dončić, who earned the distinction in 2019. Three players, Adrian Dantley, Alex English, and Dennis Rodman, have been inducted to the Basketball Hall of Fame, although all of them spent most of their careers elsewhere, and each spent less than two seasons with the Mavericks. Guard Derek Harper, who played 12 seasons with the Mavericks during two separate stints, is the franchise leader in assists and steals. Before being passed by Nowitzki, the blocked shots category was led by center Shawn Bradley, who once led the league in blocks.
The Mavericks have four retired jersey numbers: the number 12 jersey worn by Derek Harper, the number 15 jersey worn by Brad Davis, the number 22 jersey worn by Rolando Blackman, and the number 41 jersey worn by Dirk Nowitzki. Davis, who played 12 seasons with the Mavericks until his retirement in 1992, had his number 15 jersey retired by the franchise in November 1992. Blackman, who played 11 seasons with the Mavericks after being selected by the team in the 1981 draft, had his number 22 jersey retired in March 2000. Harper, who played parts of 12 seasons in two stints with the Mavericks starting with the 1983 draft, had his number 12 jersey retired in J |
https://en.wikipedia.org/wiki/James%20Colliander | James Ellis Colliander (born 22 June 1967) is an American-Canadian mathematician. He is currently Professor of Mathematics at University of British Columbia and served as Director of the Pacific Institute for the Mathematical Sciences (PIMS) during 2016-2021. He was born in El Paso, Texas, and lived there until age 8 and then moved to Hastings, Minnesota. He graduated from Macalester College in 1989. He worked for two years at the United States Naval Research Laboratory on fiber optic sensors and then went to graduate school to study mathematics. He received his PhD from the University of Illinois at Urbana–Champaign in 1997 and was advised by Jean Bourgain. Colliander was a postdoctoral fellow at the University of California, Berkeley and spent semesters at the University of Chicago, the Institute for Advanced Study and the Mathematical Sciences Research Institute.
He is also an award-winning teacher.
Research
Colliander's research mostly addresses dynamical aspects of solutions of Hamiltonian partial differential equations, especially non-linear Schrödinger equation.
Colliander is a collaborator with Markus Keel, Gigliola Staffilani, Hideo Takaoka, and Terence Tao, forming a group known as the "I-team". The name of this group has been said to come from a mollification operator used in the team's method of almost conserved quantities, or as an abbreviation for "interaction", referring both to the teamwork of the group and to the interactions of light waves with each other. The group's work was featured in the 2006 Fields Medal citations for group member Tao.
Organization creation
Colliander is co-founder of the education technology company called Crowdmark.
Colliander, with colleagues from PIMS, created Syzygy, a project that provides interactive computing for students and teachers at universities across Canada. Syzygy operates on infrastructure provided by Compute Canada.
Colliander, with colleagues from PIMS and Cybera, created Callysto, a project designed to improve computational thinking for students and teachers in grades 5-12.
Colliander is co-founder of the International Interactive Computing Collaboration (2i2c).
Major publications
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness for Schrödinger equations with derivative. SIAM J. Math. Anal. 33 (2001), no. 3, 649–669.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. A refined global well-posedness result for Schrödinger equations with derivative. SIAM J. Math. Anal. 34 (2002), no. 1, 64–86.
Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation. Math. Res. Lett. 9 (2002), no. 5-6, 659–682.
Christ, Michael; Colliander, James; Tao, Terence. Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125 (2003), no. 6, 1235–1293.
Colliander, J.; Keel, M.; Staffilani, G.; Ta |
https://en.wikipedia.org/wiki/Hermite%27s%20cotangent%20identity | In mathematics, Hermite's cotangent identity is a trigonometric identity discovered by Charles Hermite. Suppose a1, ..., an are complex numbers, no two of which differ by an integer multiple of . Let
(in particular, A1,1, being an empty product, is 1). Then
The simplest non-trivial example is the case n = 2:
Notes and references
Trigonometry |
https://en.wikipedia.org/wiki/Biggest%20little%20polygon | In geometry, the biggest little polygon for a number n is the n-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one n-gons. One non-unique solution when n = 4 is a square, and the solution is a regular polygon when n is an odd number, but the solution is irregular otherwise.
Quadrilaterals
For n = 4, the area of an arbitrary quadrilateral is given by the formula S = pq sin(θ)/2 where p and q are the two diagonals of the quadrilateral and θ is either of the angles they form with each other. In order for the diameter to be at most 1, both p and q must themselves be at most 1. Therefore, the quadrilateral has largest area when the three factors in the area formula are individually maximized, with p = q = 1 and sin(θ) = 1. The condition that p = q means that the quadrilateral is an equidiagonal quadrilateral (its diagonals have equal length), and the condition that sin(θ) = 1 means that it is an orthodiagonal quadrilateral (its diagonals cross at right angles). The quadrilaterals of this type include the square with unit-length diagonals, which has area 1/2. However, infinitely many other orthodiagonal and equidiagonal quadrilaterals also have diameter 1 and have the same area as the square, so in this case the solution is not unique.
Odd numbers of sides
For odd values of n, it was shown by Karl Reinhardt in 1922 that a regular polygon has largest area among all diameter-one polygons.
Even numbers of sides
In the case n = 6, the unique optimal polygon is not regular. The solution to this case was published in 1975 by Ronald Graham, answering a question posed in 1956 by Hanfried Lenz; it takes the form of an irregular equidiagonal pentagon with an obtuse isosceles triangle attached to one of its sides, with the distance from the apex of the triangle to the opposite pentagon vertex equal to the diagonals of the pentagon. Its area is 0.674981.... , a number that satisfies the equation
4096 x10 +8192x9 − 3008x8 − 30848x7 + 21056x6 + 146496x5 − 221360x4 + 1232x3 + 144464x2 − 78488x + 11993 = 0.
Graham conjectured that the optimal solution for the general case of even values of n consists in the same way of an equidiagonal (n − 1)-gon with an isosceles triangle attached to one of its sides, its apex at unit distance from the opposite (n − 1)-gon vertex. In the case n = 8 this was verified by a computer calculation by Audet et al.
Graham's proof that his hexagon is optimal, and the computer proof of the n = 8 case, both involved a case analysis of all possible n-vertex thrackles with straight edges.
The full conjecture of Graham, characterizing the solution to the biggest little polygon problem for all even values of n, was proven in 2007 by Foster and Szabo.
See also
Hansen's small octagon
Reinhardt polygon, the polygons maximizing perimeter for their diameter, maximizing width for their diameter, and maximizing width for their perimeter
|
https://en.wikipedia.org/wiki/2009%20Montedio%20Yamagata%20season | 2009 Montedio Yamagata season
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Montedio Yamagata
Montedio Yamagata seasons |
https://en.wikipedia.org/wiki/2009%20Kashima%20Antlers%20season | 2009 Kashima Antlers season
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Kashima Antlers
Kashima Antlers seasons |
https://en.wikipedia.org/wiki/2009%20Omiya%20Ardija%20season | 2009 Omiya Ardija season
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Omiya Ardija
Omiya Ardija seasons |
https://en.wikipedia.org/wiki/2009%20JEF%20United%20Chiba%20season | 2009 JEF United Ichihara Chiba season
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JEF United Ichihara Chiba
JEF United Chiba seasons |
https://en.wikipedia.org/wiki/2009%20Kashiwa%20Reysol%20season | 2009 Kashiwa Reysol season
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Kashiwa Reysol
Kashiwa Reysol seasons |
https://en.wikipedia.org/wiki/2009%20FC%20Tokyo%20season | The 2009 FC Tokyo season was the team's 11th as a member of J.League Division 1.
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Tokyo
2009 |
https://en.wikipedia.org/wiki/2009%20Kawasaki%20Frontale%20season | 2009 Kawasaki Frontale season
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Kawasaki Frontale
Kawasaki Frontale seasons |
https://en.wikipedia.org/wiki/2009%20Yokohama%20F.%20Marinos%20season | 2009 Yokohama F. Marinos season
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Yokohama F. Marinos
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https://en.wikipedia.org/wiki/2009%20J%C3%BAbilo%20Iwata%20season | 2009 Júbilo Iwata season
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Jubilo Iwata
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https://en.wikipedia.org/wiki/2009%20Kyoto%20Sanga%20FC%20season | 2009 Kyoto Sanga F.C. season
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Kyoto Sanga F.C.
Kyoto Sanga FC seasons |
https://en.wikipedia.org/wiki/2009%20Gamba%20Osaka%20season | 2009 Gamba Osaka season
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Gamba Osaka
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https://en.wikipedia.org/wiki/2009%20Vissel%20Kobe%20season | 2009 Vissel Kobe season
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Vissel Kobe
Vissel Kobe seasons |
https://en.wikipedia.org/wiki/2009%20Sanfrecce%20Hiroshima%20season | 2009 Sanfrecce Hiroshima season
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Sanfrecce Hiroshima
Sanfrecce Hiroshima seasons |
https://en.wikipedia.org/wiki/2009%20Oita%20Trinita%20season | 2009 Oita Trinita season
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Oita Trinita seasons |
https://en.wikipedia.org/wiki/2009%20Consadole%20Sapporo%20season | 2009 Consadole Sapporo season
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Consadole Sapporo
Hokkaido Consadole Sapporo seasons |
https://en.wikipedia.org/wiki/2009%20Vegalta%20Sendai%20season | 2009 Vegalta Sendai season.
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Vegalta Sendai
Vegalta Sendai seasons |
https://en.wikipedia.org/wiki/2009%20Mito%20HollyHock%20season | 2009 Mito HollyHock season
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Mito HollyHock
Mito HollyHock seasons |
https://en.wikipedia.org/wiki/2009%20Thespa%20Kusatsu%20season | 2009 Thespa Kusatsu season
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Thespa Kusatsu
Thespakusatsu Gunma seasons |
https://en.wikipedia.org/wiki/2009%20Tokyo%20Verdy%20season | 2009 Tokyo Verdy season
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Tokyo Verdy
Tokyo Verdy seasons |
https://en.wikipedia.org/wiki/2009%20Yokohama%20FC%20season | 2009 Yokohama FC season
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Yokohama FC
Yokohama FC seasons |
https://en.wikipedia.org/wiki/2009%20Shonan%20Bellmare%20season | 2009 Shonan Bellmare season
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Shonan Bellmare
Shonan Bellmare seasons |
https://en.wikipedia.org/wiki/2009%20Ventforet%20Kofu%20season | 2009 Ventforet Kofu season
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J. League official site
Statistical information for the 2009 league.
Ventforet Kofu
Ventforet Kofu seasons |
https://en.wikipedia.org/wiki/2009%20Kataller%20Toyama%20season | 2009 Kataller Toyama season
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Kataller Toyama
Kataller Toyama seasons |
https://en.wikipedia.org/wiki/2009%20FC%20Gifu%20season | 2009 F.C. Gifu season
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Gifu
2009 |
https://en.wikipedia.org/wiki/2009%20Cerezo%20Osaka%20season | 2009 Cerezo Osaka season
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Cerezo Osaka
Cerezo Osaka seasons |
https://en.wikipedia.org/wiki/2009%20Fagiano%20Okayama%20season | 2009 Fagiano Okayama season
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League table
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Fagiano Okayama
Fagiano Okayama seasons |
https://en.wikipedia.org/wiki/2009%20Tokushima%20Vortis%20season | 2009 Tokushima Vortis season
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Tokushima Vortis
Tokushima Vortis seasons |
https://en.wikipedia.org/wiki/2009%20Ehime%20FC%20season | 2009 Ehime F.C. season
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Ehime F.C.
Ehime FC seasons |
https://en.wikipedia.org/wiki/2009%20Avispa%20Fukuoka%20season | 2009 Avispa Fukuoka season
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Avispa Fukuoka
Avispa Fukuoka seasons |
https://en.wikipedia.org/wiki/2009%20Sagan%20Tosu%20season | 2009 Sagan Tosu season
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Sagan Tosu
Sagan Tosu seasons |
https://en.wikipedia.org/wiki/2009%20Roasso%20Kumamoto%20season | 2009 Roasso Kumamoto season
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Roasso Kumamoto
Roasso Kumamoto seasons |
https://en.wikipedia.org/wiki/Energy%20in%20Slovakia | Primary energy use in Slovakia was 194 TWh and 36 TWh per million inhabitants in 2009.
Statistics
Energy plan
Slovakia has a plan to get renewable sources of energy up to 19.2% by 2030.
Energy types
From 2024, following the completion of two new nuclear reactors, Slovakia will return to being a net exporter of electricity.
Fossil fuels
Oil
Slovnaft is the largest oil refinery in Slovakia.
In 2022 Slovakia sought to reduce its reliance on oil from Russia.
Natural gas
Slovenský plynárenský priemysel (Slovak Gas Industry) is the main natural gas supplier in Slovakia.
In 2022 Slovakia sought to reduce its reliance on natural gas from Russia who was supplying 81% in 2020.
Coal
Two coal power stations operate in Slovakia, with one at Nováky, scheduled to close in 2023.
Nuclear Energy
Five operating reactors in two power plants Bohunice Nuclear Power Plant with two reactors dating from the 1980's and three at Mochovce Nuclear Power Plant), with two from the 1990's and the 3rd being commissioned in January 2023. Total electricity generation from nuclear in 2020 was 15.4 TWh. One additional reactor is near completion at Mochovce.
Renewable energy
Renewable energy includes wind, solar, biomass and geothermal energy sources.
Wind power
In the end of 2022 wind power capacity in Slovakia 3MW which has not changed since 2010. In the National Energy and Climate Plan the Government plans to build 500MW of wind by 2030.
Solar power
In 2019 Slovakia had 472 MW of installed solar power capacity. It is expected to rise to 750 MW by 2030.
Biomass
Biomass provides around 4% of electricity generation capacity.
Hydro power
There is hydropower potential in Vah and Orava rivers (before Stary Hrad, and after Kralovianski Meander, Oravka tunnel), with power plants over 30MW as extremely profitable (for low cost/installed MW).
Climate change
Emissions of carbon dioxide in total, per capita in 2007 were 6.8 tons CO2 compared to EU 27 average 7.9 tons . Emission change 2007/1990 (%) was -35.1%. In Europe in 2007 the Slovak emissions of carbon dioxide per capita (6.8 tons CO2) were higher than in Hungary 5.4, Sweden 5.1, Portugal 5.2 or Switzerland 5.6 and lower than in Czech Republic 11.8, Luxembourg 22.4, Finland 12.2, Netherlands 11.1, Germany 9.7 or Ireland 10.1
1990 emissions were 74 Mt eq. The Kyoto protocol target is reduction of 6 Mt (-8%).
See also
2009 Handlová mine blast
List of power stations in Slovakia
Renewable energy by country
References |
https://en.wikipedia.org/wiki/Kummer%27s%20theorem | In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, .
Statement
Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation of the binomial coefficient is equal to the number of carries when m is added to n − m in base p.
An equivalent formation of the theorem is as follows:
Write the base- expansion of the integer as , and define to be the sum of the base- digits. Then
The theorem can be proved by writing as and using Legendre's formula.
Examples
To compute the largest power of 2 dividing the binomial coefficient write and in base as and . Carrying out the addition in base 2 requires three carries:
{| cellpadding=5 style="border:none"
| || 1 || 1 || 1 || || ||
|-
| || || || 1 || 1 2
|-
| + || || 1 || 1 || 1 2
|-
| style='border-top: 1px solid' |
| style='border-top: 1px solid' | 1
| style='border-top: 1px solid' | 0
| style='border-top: 1px solid' | 1
| style='border-top: 1px solid' | 0 2
|}
Therefore the largest power of 2 that divides is 3.
Alternatively, the form involving sums of digits can be used. The sums of digits of 3, 7, and 10 in base 2 are , , and respectively. Then
Multinomial coefficient generalization
Kummer's theorem can be generalized to multinomial coefficients as follows:
See also
Lucas's theorem
References
Theorems in number theory |
https://en.wikipedia.org/wiki/2010%20Vegalta%20Sendai%20season | 2010 Vegalta Sendai season.
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Vegalta Sendai
Vegalta Sendai seasons |
https://en.wikipedia.org/wiki/2010%20Montedio%20Yamagata%20season | 2010 Montedio Yamagata season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Montedio Yamagata
Montedio Yamagata seasons |
https://en.wikipedia.org/wiki/2010%20Kashima%20Antlers%20season | 2010 Kashima Antlers season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Kashima Antlers
Kashima Antlers seasons |
https://en.wikipedia.org/wiki/2010%20Urawa%20Red%20Diamonds%20season | 2010 Urawa Red Diamonds season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Urawa Red Diamonds
Urawa Red Diamonds seasons |
https://en.wikipedia.org/wiki/2010%20Omiya%20Ardija%20season | 2010 Omiya Ardija season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Omiya Ardija
Omiya Ardija seasons |
https://en.wikipedia.org/wiki/2010%20FC%20Tokyo%20season | The 2010 FC Tokyo season was the team's 12th as a member of J.League Division 1.
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Tokyo
2010 |
https://en.wikipedia.org/wiki/2010%20Kawasaki%20Frontale%20season | The 2010 Kawasaki Frontale season was their sixth consecutive season in J.League 1, the top division of football in Japan.
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Kawasaki Frontale
Kawasaki Frontale seasons |
https://en.wikipedia.org/wiki/2010%20Yokohama%20F.%20Marinos%20season | 2010 Yokohama F. Marinos season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J.League official site
Yokohama F. Marinos
Yokohama F. Marinos seasons |
https://en.wikipedia.org/wiki/2010%20J%C3%BAbilo%20Iwata%20season | 2010 Júbilo Iwata season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Jubilo Iwata
Júbilo Iwata seasons |
https://en.wikipedia.org/wiki/2010%20Kyoto%20Sanga%20FC%20season | The 2010 Kyoto Sanga F.C. season was the 11th season of the club in J. League Division 1.
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Kyoto Sanga F.C.
Kyoto Sanga FC seasons |
https://en.wikipedia.org/wiki/2010%20Cerezo%20Osaka%20season | 2010 Cerezo Osaka season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Cerezo Osaka
Cerezo Osaka seasons |
https://en.wikipedia.org/wiki/2010%20Vissel%20Kobe%20season | 2010 Vissel Kobe season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Vissel Kobe
Vissel Kobe seasons |
https://en.wikipedia.org/wiki/2010%20Sanfrecce%20Hiroshima%20season | 2010 Sanfrecce Hiroshima season
Competitions
J. League 1
Emperor's Cup
J. League Cup
Player statistics
Other pages
J. League official site
Sanfrecce Hiroshima
Sanfrecce Hiroshima seasons |
https://en.wikipedia.org/wiki/2010%20Consadole%20Sapporo%20season | 2010 Consadole Sapporo season
Competitions
Player statistics
Other pages
J. League official site
Consadole Sapporo
Hokkaido Consadole Sapporo seasons |
https://en.wikipedia.org/wiki/2010%20Mito%20HollyHock%20season | 2010 Mito HollyHock season
Competitions
Player statistics
Other pages
J. League official site
Mito HollyHock
Mito HollyHock seasons |
https://en.wikipedia.org/wiki/2010%20Thespa%20Kusatsu%20season | 2010 Thespa Kusatsu season
Competitions
Player statistics
Other pages
J. League official site
Thespa Kusatsu
Thespakusatsu Gunma seasons |
https://en.wikipedia.org/wiki/2010%20JEF%20United%20Chiba%20season | 2010 JEF United Ichihara Chiba season
Competitions
Player statistics
Other pages
J. League official site
JEF United Ichihara Chiba
JEF United Chiba seasons |
https://en.wikipedia.org/wiki/2010%20Kashiwa%20Reysol%20season | 2010 Kashiwa Reysol season
Competitions
Player statistics
Other pages
J. League official site
Kashiwa Reysol
Kashiwa Reysol seasons |
https://en.wikipedia.org/wiki/2010%20Tokyo%20Verdy%20season | 2010 Tokyo Verdy season
Competitions
Player statistics
Other pages
J. League official site
Tokyo Verdy
Tokyo Verdy seasons |
https://en.wikipedia.org/wiki/2010%20Yokohama%20FC%20season | 2010 Yokohama FC season
Competitions
Player statistics
Other pages
J. League official site
Yokohama FC
Yokohama FC seasons |
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