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https://en.wikipedia.org/wiki/1969%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1969 season.
Overview
It was contested by 11 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1970%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1970 season.
Overview
It was contested by 11 teams, and Nacional won the championship.
League standings
Playoff
Champions
Relegation group
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1971%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1971 season.
Overview
It was contested by 12 teams, and Nacional won the championship.
League standings
Playoff
Champions
Relegation group
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1972%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1972 season.
Overview
It was contested by 12 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1973%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1973 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1974%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1974 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1975%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1975 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1976%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1976 season.
Overview
It was contested by 12 teams, and Defensor won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1977%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1977 season.
Overview
It was contested by 12 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1978%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1978 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/1979%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1979 season.
Overview
It was contested by 13 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1980%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1980 season.
Overview
It was contested by 14 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1981%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1981 season.
Overview
It was contested by 15 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1982%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1982 season.
Overview
It was contested by 14 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1983%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1983 season.
Overview
It was contested by 13 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1984%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1984 season.
Overview
It was contested by 13 teams, and Central Español won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1985%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1985 season.
Overview
It was contested by 13 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1987%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1987 season.
Overview
It was contested by 13 teams, and Defensor won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1989%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1989 season.
Overview
It was contested by 13 teams, and Progreso won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1
Uru |
https://en.wikipedia.org/wiki/1990%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1990 season. It was contested by 14 teams, and Bella Vista won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1990 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1991%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1991 season.
Overview
It was contested by 14 teams, and Defensor Sporting won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1991 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1992%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1992 season.
Overview
It was contested by 13 teams, and Nacional won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1992 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1993%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1993 season.
Overview
It was contested by 13 teams, and Peñarol won the championship.
League standings
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1993 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1994%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1994 season.
Overview
It was contested by 13 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Peñarol 1-1 ; 1-1 ; 2-1 Defensor Sporting
Peñarol won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1994 in Uruguayan football |
https://en.wikipedia.org/wiki/1995%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1995 season.
Overview
It was contested by 13 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Peñarol 1-0 ; 1-2 ; 3-1 Nacional
Peñarol won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1995 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1996%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1996 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Peñarol 1-0 ; 1-1 Nacional
Peñarol won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1996 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1997%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1997 season.
Overview
It was contested by 12 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Semifinal
Peñarol 3-2 Nacional
Final
Peñarol 1-0 ; 3-0 Defensor Sporting
Peñarol won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
1997 in Uruguayan football
Uru |
https://en.wikipedia.org/wiki/1998%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1998 season.
Overview
It was contested by 12 teams, and Nacional won the championship.
Apertura
Clausura
Overall
References
Uruguay – List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1998 in Uruguayan football |
https://en.wikipedia.org/wiki/1999%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 1999 season.
Overview
It was contested by 15 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Peñarol 1–1 ; 1–1 ; 2–1 Nacional
Peñarol won the championship.
References
Uruguay – List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1999 in Uruguayan football |
https://en.wikipedia.org/wiki/2000%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 2000 season.
Overview
It was contested by 18 teams, and Nacional won the championship.
Apertura
Clausura
Overall
Playoff
Nacional 1-0 ; 1-1 Peñarol
Nacional won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
2000 in Uruguayan football |
https://en.wikipedia.org/wiki/2002%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 2002 season.
Overview
It was contested by 18 teams, and Nacional won the championship.
Classification
Group A
Group B
Group C
Overall
Champions
Apertura
Clausura
Playoff
Danubio 1-2 ; 1-2 Nacional
Nacional won the championship.
Relegation group
Group A
Group B
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
1 |
https://en.wikipedia.org/wiki/2003%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 2003 season.
Overview
It was contested by 18 teams, and Peñarol won the championship.
Apertura
Clausura
Overall
Playoff
Peñarol 1-0 Nacional
Peñarol won the championship.
References
Uruguay - List of final tables (RSSSF)
Uruguayan Primera División seasons
Uru
2003 in Uruguayan football |
https://en.wikipedia.org/wiki/2004%20Uruguayan%20Primera%20Divisi%C3%B3n | Statistics of Primera División Uruguaya for the 2004 season.
Overview
It was contested by 18 teams, and Danubio won the championship.
League standings
Champions
Apertura
Clausura
Championship playoff
Nacional and Danubio qualified to the championship playoffs as the Apertura and Clausura winners, respectively
First leg
Second leg
Danubio F.C. became champions by winning the qualifier and the annual table.
Relegation group
References
Uruguay 2004 (RSSSF)
Uruguayan Primera División seasons
Uru
2004 in Uruguayan football |
https://en.wikipedia.org/wiki/Paco%20Lagerstrom | Paco Axel Lagerstrom (February 24, 1914 – February 16, 1989) was an applied mathematician and aeronautical engineer. He was trained formally in mathematics, but worked for much of his career in aeronautical applications. He was known for work in applying the method of asymptotic expansion to fluid mechanics problems. Several of his works have become classics, including "Matched Asymptotic Expansions: Ideas And Techniques".
Biography
He was born on February 24, 1914, in Oskarshamn, Sweden.
Lagerstrom earned bachelor's and master's degrees, in 1935 and 1939 respectively, at the University of Stockholm. He then came to America as a graduate student at Princeton University, earning a PhD in 1942 in mathematics under Salomon Bochner with a dissertation entitled "Measure and Integral in Partially Ordered Spaces". During this time, Lagerstrom was also a mathematics instructor.
He left Princeton in 1944 to work briefly at Bell Aircraft in Niagara Falls, New York until 1945, after which he worked for a similarly brief period at Douglas Aircraft in Santa Monica. While he had already published significant results in pure mathematics, he was, by this time, firmly interested in its applications to fluid dynamic and aerodynamic problems. In 1946, Lagerstrom was recruited by Hans Liepmann to the Guggenheim Aeronautical Laboratory at Caltech. He was later promoted to Professor of Aeronautics in 1952 and Professor of Applied Mathematics in 1967, having departed only briefly to the University of Paris in 1960-1961 as visiting professor on a Guggenheim Fellowship.
He died on February 16, 1989.
Publications
His book "Laminar flow theory", initially published in 1964 in the Theory of Laminar flows, edited by F.K.Moore, is still considered as the standard textbook for fluid mechanics.
References
External links
20th-century American mathematicians
Fluid dynamicists
Swedish mathematicians
20th-century American engineers
1989 deaths
1914 births
Swedish emigrants to the United States |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Cyprus | The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard used for referencing the subdivisions of Cyprus for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code assigned to Cyprus is CY, and Eurostat has established a hierarchical structure consisting of three levels. However, Cyprus does not have subdivisions covered by the NUTS levels, as its population is small enough to be covered within a single level.
In addition to the NUTS levels, there are further levels of geographic organisation known as the local administrative units (LAU). In Cyprus, LAU 1 corresponds to districts, and LAU 2 represents municipalities. These divisions help in the administration and organization of statistical data at a local level.
Overall and NUTS codes
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Cyprus can be downloaded here: ''
See also
Subdivisions of Cyprus
ISO 3166-2 codes of Cyprus
FIPS region codes of Cyprus
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
KYPROS / KIBRIS - NUTS level 2
KYPROS / KIBRIS - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Cyprus
Nuts |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Luxembourg | In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Luxembourg (LU), the three levels are:
NUTS codes
LU0 Luxembourg
LU00 Luxembourg
LU000 Luxembourg
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Luxembourg can be found in this spread sheet of all 28 EU member states LAU codes here, current as of 2018:
See also
Subdivisions of Luxembourg
ISO 3166-2 codes of Luxembourg
FIPS region codes of Luxembourg
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
LUXEMBOURG - NUTS level 2
LUXEMBOURG - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Luxembourg
Nuts |
https://en.wikipedia.org/wiki/Filipinos%20in%20Pakistan | Filipinos in Pakistan () consist of migrants from the Philippines. In 2008, there were an estimated 1,500 Filipinos in Pakistan according to the statistics of the Philippine government. Many Filipinos came to Pakistan for work and those who later married Pakistani men are now holding Pakistani citizenship. Pakistan comparatively has experience good immigration rate from Philippines despite security issues.
Migration history
Many Filipino people entered Pakistan's commercial center Karachi illegally with fake passports and false identity cards as early in 1990s along with hundreds of Nepali, Bengals, Sri Lankans and Indians however later deported. As of April 2010, there are 546 registered Filipino living in Pakistan who were eligible to vote in 2010 Philippine presidential election as per Philippines Foreign Affairs ministry.
Employment
Many Filipinos in Pakistan are domestic workers, including the housemaids of high government officials and rich Pakistanis. There are some three Filipino maids at house of former Pakistani Prime Minister Yousaf Raza Gillani and many Filipinos working as chefs in Japanese restaurants in Karachi and Islamabad.
A small number of Filipinos studying Islam in the country is reported by the Philippines Embassy in Islamabad while thousands of Muslim students from various Southeast Asian countries including Philippines illegally studying in the Pakistani Madrasahs. Some Filipinos are also nurses in Pakistan.
Relations with Pakistani society
In 2007, following a state of emergency declared by Pakistani President Pervez Musharraf, about 200 Filipinos gathered in Islamabad on the advice of Ambassador Jimmy Yambao to call for protest. However, there have no direct threats to safety of Filipinos in Pakistan reported.
See also
Pakistan–Philippines relations
References
External links
Article on Filipinos in Pakistan
Human trafficking: FIA opens probe into Filipinas smuggling
From Philippines to Pakistan: Maids can be brought in for only a few thousand rupees
Pakistan
Pakistan
Ethnic groups in Pakistan
Immigration to Pakistan |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Liechtenstein | As a member of the EFTA, Liechtenstein (LI) is included in the Nomenclature of Territorial Units for Statistics (NUTS). The three NUTS levels all correspond to the country itself:
NUTS-1: LI0 Liechtenstein
NUTS-2: LI00 Liechtenstein
NUTS-3: LI000 Liechtenstein
Below the NUTS levels, there are two LAU levels (LAU-1: electoral districts; LAU-2: municipalities).
See also
Subdivisions of Liechtenstein
Electoral District of Oberland
Electoral District of Unterland
ISO 3166-2 codes of Liechtenstein
FIPS region codes of Liechtenstein
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EFTA countries - Statistical regions at level 1
LIECHTENSTEIN - Statistical regions at level 2
LIECHTENSTEIN - Statistical regions at level 3
Correspondence between the regional levels and the national administrative units
Communes of Liechtenstein, Statoids.com
Liechtenstein
Subdivisions of Liechtenstein |
https://en.wikipedia.org/wiki/Norman%20Mason%20%28canoeist%29 | Norman Mason (born 23 July 1952) is a maths teacher in Leicester at Wyggeston and Queen Elizabeth I College, and was a British sprint canoer who competed in the mid-1970s. He was eliminated in the repechages of the K-2 1000 m event at the 1976 Summer Olympics in Montreal.
References
Sports-Reference.com profile
1952 births
Canoeists at the 1976 Summer Olympics
Living people
Olympic canoeists for Great Britain
British male canoeists
21st-century English educators |
https://en.wikipedia.org/wiki/Variation%20of%20information | In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings (partitions of elements). It is closely related to mutual information; indeed, it is a simple linear expression involving the mutual information. Unlike the mutual information, however, the variation of information is a true metric, in that it obeys the triangle inequality.
Definition
Suppose we have two partitions and of a set into disjoint subsets, namely and .
Let:
and
Then the variation of information between the two partitions is:
.
This is equivalent to the shared information distance between the random variables i and j with respect to the uniform probability measure on defined by for .
Explicit information content
We can rewrite this definition in terms that explicitly highlight the information content of this metric.
The set of all partitions of a set form a compact Lattice where the partial order induces two operations, the meet and the join , where the maximum is the partition with only one block, i.e., all elements grouped together, and the minimum is , the partition consisting of all elements as singletons. The meet of two partitions and is easy to understand as that partition formed by all pair intersections of one block of, , of and one, , of . It then follows that and .
Let's define the entropy of a partition as
,
where . Clearly, and . The entropy of a partition is a monotonous function on the lattice of partitions in the sense that .
Then the VI distance between and is given by
.
The difference is a pseudo-metric as doesn't necessarily imply that . From the definition of , it is .
If in the Hasse diagram we draw an edge from every partition to the maximum and assign it a weight equal to the VI distance between the given partition and , we can interpret the VI distance as basically an average of differences of edge weights to the maximum
.
For as defined above, it holds that the joint information of two partitions coincides with the entropy of the meet
and we also have that coincides with the conditional entropy of the meet (intersection) relative to .
Identities
The variation of information satisfies
,
where is the entropy of , and is mutual information between and with respect to the uniform probability measure on . This can be rewritten as
,
where is the joint entropy of and , or
,
where and are the respective conditional entropies.
The variation of information can also be bounded, either in terms of the number of elements:
,
Or with respect to a maximum number of clusters, :
References
Further reading
External links
Partanalyzer includes a C++ implementation of VI and other metrics and indices for analyzing partitions and clusterings
C++ implementation with MATLAB mex files
Entropy and information
Summary statistics for contingency tables
Clustering criteria |
https://en.wikipedia.org/wiki/Urquhart%20graph | In computational geometry, the Urquhart graph of a set of points in the plane, named after Roderick B. Urquhart, is obtained by removing the longest edge from each triangle in the Delaunay triangulation.
The Urquhart graph was described by , who suggested that removing the longest edge from each Delaunay triangle would be a fast way of constructing the relative neighborhood graph (the graph connecting pairs of points and when there does not exist any third point that is closer to both and than they are to each other). Since Delaunay triangulations can be constructed in time , the same time bound holds for the Urquhart graph as well. Although it was later shown that the Urquhart graph is not exactly the same as the relative neighborhood graph, it can be used as a good approximation to it. The problem of constructing relative neighborhood graphs in time, left open by the mismatch between the Urquhart graph and the relative neighborhood graph, was solved by .
Like the relative neighborhood graph, the Urquhart graph of a set of points in general position contains the Euclidean minimum spanning tree of its points, from which it follows that it is a connected graph.
References
Computational geometry
Geometric graphs |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Portugal | The Nomenclature of Territorial Units for Statistics (NUTS) is developed by Eurostat, and employed in both Portugal and the entire European Union for statistical purposes. The NUTS branch extends from NUTS1, NUTS2 and NUTS3 regions, with the complementary LAU (Local Administrative Units) sub-categorization being used to differentiate the local areas, of trans-national importance.
Developed by Eurostat and implemented in 1998, the Nomenclature of Territorial Units for Statistics (NUTS) regions, which comprises three levels of the Portuguese territory, are instrumental in European Union's Structural Fund delivery mechanisms. The standard was developed by the European Union and extensively used by national governments, Eurostat and other EU bodies for statistical and policy matters. Until 4 November 2002, the Sistema Estatístico Nacional (SEN) used a NUTS codification system that was distinct from the Eurostat system. With the enactment of Decree Law 244/2002 (5 November 2002), published in the Diário da República, this system was abandoned in order to harmonize the national system with that of Eurostat.
Subdivisions
The NUTS system subdivides the nation into three levels: NUTS I, NUTS II and NUTS III. In some European partners, as is the case with Portugal, a complementary hierarchy, respectively LAU I and LAU II (posteriorly referred to as NUTS IV and NUTS V) is employed. The LAU, or Local Administrative Units, in the Portuguese context pertains to the 308 municipalities (LAU I) and 3092 civil parishes (LAU II) respectively. In the broadest sense, the NUTS hierarchy, while they may follow some of the borders (municipal or parish) diverge in their delineation.
Changes NUTS 2-3 (1986—2013)
NUTS I
The first and broadest subdivision of Portugal is between continental Portugal and the two autonomous regions of the Azores and Madeira.
NUTS II
Although the districts are still the most socially relevant subdivision, their function is being phased in favour of locally oriented regional units, and regions are growing in importance. Portugal is divided into five regions, administered by the Commissions for Coordination and Regional Development () in continental Portugal, plus the two autonomous regions that are their own NUTS II regions.
NUTS III
The seven regions of Portugal are likewise subdivided into 25 subregions () that, from 2015, represent the 2 metropolitan areas, the 21 intermunicipal communities and the 2 autonomous regions. Therefore, since the 2013 revision (enforced in 2015), the Portuguese subregions have a statutory and administrative relevance.
The two autonomous regions () in the Atlantic, correspond to their own NUTS I, II and III categories.
NUTS Codes
{| class="wikitable"
! width="60px" |Code
! width="200px" |NUTS 1
! width="60px" |Code
! width="200px" |NUTS 2
! width="60px" |Code
! width="250px" |NUTS 3
|-
| rowspan="23" align="center" |PT1
| rowspan="23" align="center" |Continente
| rowspan="8" align="center" |PT11
| rowsp |
https://en.wikipedia.org/wiki/Donald%20Firesmith | Donald G. Firesmith (born June 14, 1952) is an American software engineer, consultant, and trainer at the Software Engineering Institute.
Biography
Firesmith received his B.A. in Mathematics and German from Linfield College in 1975 and his M.A. in Mathematics from Arizona State University in 1977. He also studied one year at Ludwig Maximilian University of Munich.
Firesmith started working in the computer business as a software developer in 1979 and has been quality engineer, configuration manager, and data manager for Computer Science Corporation in the US, Germany, and Switzerland. From 1984 to 1988 he was an OO methodologist at Magnavox Electronic Systems Corporation. And from 1988 to 1995 he was President of Advanced Software Technology Specialists, a small consulting and training company.
From 1994 to 1997 he has been an acquisition editor and editor and chief of Reference Books at SIGS Books. And further he was a Senior advisory software engineer at StorageTek, where he worked as a technical leader, requirements engineer, and software architect, Chief architect Lante Corporation, which specialized in producing eMarketplaces, and Chief architect for the North American Business Unit of Cambridge Technology Partners. From 2003 through 2020, he was a Principal Engineer at the Software Engineering Institute where he works in the Client Technical Solutions Software Solutions Division helping the United States Government acquire software-intensive systems. Since retiring in 2020, he has been a full-time novelist.
Firesmith was named a Distinguished Engineer by the Association for Computing Machinery in 2015.
Work
Method Engineering (ME) and Open Process Environment and Notation (OPEN)
Firesmith is a co-founder with Brian Henderson-Sellers and Ian Graham of the international OPEN Consortium. Firesmith was the principal developer of the OPEN Modeling Language. Firesmith is the founder of the OPEN Process Framework Repository Organization and the developer of its large repository of free, open-source, reusable method components.
The OPEN approach to software development is founded on situational method engineering (SME). This is a means by which a software development team can construct a method and process that is appropriate for their own particular situation or circumstances. Fragments of methods, conformant with an international software engineering metamodel standard such as ISO/IEC 24744 and stored in a repository, are individually selected and the method composed from these method fragments. The SME approach is based on research by many groups worldwide – results from a recent conference are published.
Method Framework for Engineering System Architectures (MFESA)
Firesmith is the primary developer of the Method Framework for Engineering System Architectures (MFESA). This framework consists of the following:
Ontology defining the key concepts of system architecture engineering and their relationships
Metamodel defining the foundati |
https://en.wikipedia.org/wiki/Public%20estate%20in%20the%20United%20Kingdom | The public estate in the United Kingdom is the collection of all government-owned real property assets in the United Kingdom. The Office for National Statistics estimated in 2008 that the public estate has a book value of £380 billion, which is about £6,000 for every UK resident. Of this, approximately £240 billion is held by local government, while the rest—£130 billion—is held by the central government and public corporations.
In 2007, the Office of Government Commerce estimated that the government's office portfolio was worth £30 billion, and cost £6 billion annually to run.
In the mid-1990s, government real estate management functions were passed from the Property Services Agency to individual departments. The National Audit Office stated in 2006 that the motive behind this change was that it brought greater clarity and accountability, although it also diminished economies of scale and synergies between departments. The Property Services Agency was succeeded by Property Holdings, which in 1996, was succeeded by the Property Advisors to the Civil Estate.
The Office of Government Commerce (OGC) has no authority to direct other government departments. Rather, it issues guidelines and has an advisory role.
Central government bodies adhere to the Civil Estate Co-ordination Protocol (CECP).
Government departments are accountable for managing and using their own property portfolio.
Notes
References
Real estate in the United Kingdom |
https://en.wikipedia.org/wiki/Martin%20Aigner | Martin Aigner (28 February 1942 – 11 October 2023) was an Austrian mathematician and professor at Freie Universität Berlin from 1974 with interests in combinatorial mathematics and graph theory.
Biography
Martin Aigner was born on 28 February 1942. He received his Ph.D from the University of Vienna. His book Proofs from THE BOOK (co-written with Günter M. Ziegler) has been translated into 12 languages.
Aigner died on 11 October 2023, at the age of 81.
Awards
Aigner was a recipient of a 1996 Lester R. Ford Award from the Mathematical Association of America for his expository article Turán's Graph Theorem. In 2018, Aigner received the Leroy P. Steele Prize for Mathematical Exposition (jointly with Günter M. Ziegler).
Bibliography
Combinatorial Theory (1997 reprint: , 1979: ; )
(with Günter M. Ziegler) Proofs from THE BOOK
A Course in Enumeration, 2007,
Mathematics Everywhere. Martin Aigner (Author, Editor), Ehrhard Behrends (Editor), 2010
Alles Mathematik: Von Pythagoras zum CD-player, by Martin Aigner, Ehrhard Behrends, 2008,
Combinatorial search. Teubner, Stuttgart 1988,
Graphentheorie. Eine Entwicklung aus dem 4-Farben-Problem. Teubner, Stuttgart 1984,
Diskrete Mathematik. Mit über 500 Übungsaufgaben.
Vieweg, Braunschweig/Wiesbaden 1993, ,
corrected edition 12006,
References
1942 births
2023 deaths
Scientists from Linz
Austrian Roman Catholics
Austrian mathematicians
Academic staff of the Free University of Berlin
Cartellverband members
University of Vienna alumni
Members of the Austrian Academy of Sciences
Corresponding Members of the Austrian Academy of Sciences |
https://en.wikipedia.org/wiki/Expenditure%20and%20Food%20Survey | The Expenditure and Food Survey is now the Living Costs and Food Survey.
The Expenditure and Food Survey (EFS) is a survey conducted by the Office for National Statistics (ONS) and the Department for Environment, Food and Rural Affairs (DEFRA) which collects data about private household expenditure and food consumption in Great Britain.
History
From 1957 until 2001, there were two different surveys conducted each year: the Family Expenditure Survey (FES) and the National Food Survey (NFS). These two surveys were combined in the Expenditure and Food Survey (EFS) which completely replaced the former series. The survey is conducted by the Office for National Statistics (ONS), the DEFRA sponsors the food data.
The design of the survey is based on that of the FES but there have been some changes, notably with the introduction of the European Standard Classification of Individual Consumption by Purpose (COICOP). Another change has occurred with the use of a new processing software, SPSS, which has affected the structure of the datasets.
Additionally, the EFS has changed from a financial year based system to a calendar year based one in 2006; from January 2008, the EFS became known as the Living Costs and Food (LCF) module of the Integrated Household Survey (IHS).
Methodology and scope
The survey is made up of a comprehensive household questionnaire, an individual questionnaire for each adult over 16 years of age, a personal expenditure diary kept by each person over a period of two weeks as well as a simplified diary kept by children aged seven to 15.
Survey results
The data collected in the EFS is multi-purpose but it is primarily used for the Retail Prices Index, National Accounts estimates of household expenditure, the analysis of the effect of taxes and benefits and trends in nutrition.
Data access
Data from the Living Costs and Food Survey (2008 to present), as well as the Expenditure and Food Survey (2001 to 2007) can be downloaded for research and teaching use from the UK Data Service website.
References
External links
ONS website retrieved 8 November 2013
DEFRA website retrieved 8 November 2013
Demographics of the United Kingdom
Expenditure
Household surveys
Office for National Statistics
Publications established in 2001 |
https://en.wikipedia.org/wiki/Opinions%20%28Omnibus%29%20Survey | The Omnibus Survey, now called the Opinions Survey, is a survey conducted monthly by the Office for National Statistics (ONS) in Great Britain in order to collect information for different governmental departments as well as non-profit organisations in the academic and voluntary sector.
History
The Opinions (Omnibus) Survey was first set up in 1990 by the Office for National Statistics as a source of reliable, quick and cost-effective information on topics which are generally too brief to be addressed in a separate survey.
Since 2004, the survey consists of two different parts: the core questions concerning demographic information which are the same every month and non-core questions which cover different topics. Particular topics can be added for one month or longer and there are now over 300 questions modules. These modules cover issues such as contraception, tobacco consumption, changes to family income, unused medicines, internet access, etc.
Due to increased demand for the data collected, priority has been given to government departments and agencies relevant for policy-making purposes, though the data is also still available to non-profit organisations. In January 2008, the Opinions (Omnibus) Survey became part of the Integrated Household Survey (IHS) and also changed its name to the Opinions Survey.
Methodology and scope
The survey is a repeated, cross-sectional study which is conducted 12 times a year by face-to-face interviews, using a sample of about 1,800 adults per survey. Each survey period lasts 14 weeks overall, including a four-week-long testing period and five weeks of evaluation at the end. The core questions are to be answered by all household member aged 15 and older whereas the varying module questions are only answered by one selected person.
Survey results
The Opinions (Omnibus) survey allows the user to obtain results quickly and it can measure the efficacy of publicity campaigns. It can also give an indicator for public awareness of new policies and the survey is useful for question testing and piloting.
Re-using the data
Registered users can obtain Omnibus data from the Economic and Social Data Service (ESDS) website.
References
External links
Economic and Social Data Service (ESDS) website
Office for National Statistics (ONS) website
Demographics of the United Kingdom
Publications established in 1990
Office for National Statistics |
https://en.wikipedia.org/wiki/Annual%20Population%20Survey | The Annual Population Survey (APS) is a combined statistical survey of households in Great Britain which is conducted quarterly by the Office for National Statistics (ONS). It combines results from the Labour Force Survey (LFS) and the English, Welsh and Scottish Labour Force Survey boosts which are funded by the Department for Education and Skills (DfES), the Department for Work and Pensions (DWP), the Welsh Government and the Scottish Government.
History
APS data was first published in July 2005, containing data collected between January and December 2004. Since then, APS data has been published quarterly but with each dataset relating to a whole year. Between January 2004 and December 2005, an additional sample boost, the APS boost, was introduced but then discontinued in 2006 due to a lack of funding. In 2007, the APS and the LFS data were reweighted by the ONS, now using population estimates for 2007–2008.
The key feature of the APS is its emphasis on relatively small geographic areas, providing information on selected social and socio-economic variables between the 10-yearly censuses. Some of the topics included in the APS are, for example, education, health, employment and ethnicity.
Methodology and scope
The APS is published quarterly with each dataset containing 12 months of data. For each dataset, the sample size is approximately 170,000 households and 360,000 individuals.
Re-using the data
Users can obtain Annual Population Survey data from the UK Data Service website; certain data can also be accessed at the NOMIS website. The Labour Force Survey data service provides tables using APS data.
References
External links
Economic and Social Data Service (ESDS) website
NOMIS, official labour market statistics website
Office for National Statistics (ONS) website
Censuses in the United Kingdom
Demographics of the United Kingdom
Household surveys
Office for National Statistics
Publications established in 2005 |
https://en.wikipedia.org/wiki/Global%20Development%20Finance | The World Bank’s Global Development Finance, External Debt of Developing Countries (GDF) is the sole repository for statistics on the external debt of developing countries on a loan-by-loan basis. This edition of GDF presents reported or estimated data on the total external debt of all low-and middle-income countries in both electronic and print formats. Data are shown for 128 individual countries that report to the World Bank's Debtor reporting System (DRS). GDF includes over 200 time series indicators from 1970 to 2009, for most reporting countries.
Methodology and scope
GDF focuses on financial flows, trends in external debt, major economic aggregates, key debt ratios, average terms of new commitments, currency composition of long-term debt, debt restructuring, scheduled debt service projections, and other major financial indicators for developing countries.
The online database is updated twice a year, once in January and another in April, coinciding with the World Development Indicators database update. The GDF publication is released once a year in the month of January.
Accessing the data
The World Bank’s Open Data site provides access to the GDF database free of charge to all users. A selection of GDF data is featured at data.worldbank.org. Users can browse the data by Country, Indicators, Topics, and Data Catalog. GDF is listed in the catalog and can be accessed directly via dataBank.
The Economic and Social Data Service (ESDS) International provides the macro-economic datasets free of charge for members of UK higher and further education institutions. In order to access the data, users have to be registered which can be done here.
References
External links
World Bank, Global Development Finance, latest edition, retrieved March 6, 2011
World Bank Open Data, retrieved March 6, 2011
About ESDS International, retrieved September 14, 2009
See also
The World Bank
World Development Indicators
Africa Development Indicators
ESDS International
Economic data
World Bank |
https://en.wikipedia.org/wiki/World%20Development%20Indicators | World Development Indicators (WDI) is the World Bank’s premier compilation of international statistics on global development. Drawing from officially recognized sources and including national, regional, and global estimates, the WDI provides access to approximately 1,600 indicators for 217 economies, with some time series extending back more than 50 years. The database helps users find information related to development, both current and historical. The topics covered in the WDI range from poverty, health, and demographics to GDP, trade, and the environment.
The World Development Indicators website provides access to data as well as information about data coverage, curation, and methodologies, and allows users to discover what type of indicators are available, how they are collected, and how they can be visualized to analyze development trends.
Data sources
A share of the indicators in WDI come from World Bank Group surveys and data collection efforts, but the majority are based on data originally collected, compiled and published by other sources, including other international organizations such as UN specialized agencies (sometimes in cooperation with the World Bank), national statistical offices, organizations with a specific research or monitoring focus, the private sector, and academic studies.
Accessing the data
The World Bank’s Open Data site provides access to the WDI database free of charge to all users. Users can browse the data by Country, Indicators, Topics, and via the Data Catalog. The WDI database can be accessed directly via DataBank, a query tool where users can select series, economies, and time periods, and do bulk downloads in Excel or CSV, or via API. In addition, data can be programmatically accessed using Stata, R, and Python modules.
WDI and the Sustainable Development Goals
World Development Indicators takes a comprehensive view of the world and currently includes many of the official SDG indicators as well as other data that are relevant to SDGs. For example, in addition to official indicators on the income/consumption growth of the bottom 40 percent of the population relative to the average as per SDG target 10.1, the WDI presents indicators like the Gini index or income shares by decile or quintile that are relevant for SDG goal 10 on inequality. SDG related indicators can also be explored in the SDG dashboard, which uses WDI data.
References
External links
World Development Indicators
World Bank Open Data
DataBank
WDI and the Sustainable Development Goals
The Atlas of Sustainable Development Goals 2018
Google - public data
Demographics indicators
World Bank
Year of introduction missing |
https://en.wikipedia.org/wiki/Morphism%20of%20algebraic%20varieties | In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function.
A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the concepts of rational and birational maps are widely used as well; they are partial functions that are defined locally by rational fractions instead of polynomials.
An algebraic variety has naturally the structure of a locally ringed space; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces.
Definition
If X and Y are closed subvarieties of and (so they are affine varieties), then a regular map is the restriction of a polynomial map . Explicitly, it has the form:
where the s are in the coordinate ring of X:
where I is the ideal defining X (note: two polynomials f and g define the same function on X if and only if f − g is in I). The image f(X) lies in Y, and hence satisfies the defining equations of Y. That is, a regular map is the same as the restriction of a polynomial map whose components satisfy the defining equations of .
More generally, a map f:X→Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of f(x) such that f(U) ⊂ V and the restricted function f:U→V is regular as a function on some affine charts of U and V. Then f is called regular, if it is regular at all points of X.
Note: It is not immediately obvious that the two definitions coincide: if X and Y are affine varieties, then a map f:X→Y is regular in the first sense if and only if it is so in the second sense. Also, it is not immediately clear whether regularity depends on a choice of affine charts (it does not.) This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an (abstract) algebraic variety is defined to be a particular kind of a locally ringed space. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces.
The composition of regular maps is again regular; thus, algebraic varieties form the category of algebraic varieties where the morphisms are the regular maps.
Regular maps between affine varieties correspond contravariantly in one-to-one to algebra homomorphisms between the coordinate rings: if f:X→Y is a morphism of affine varieties, then it defines the algebra homomorphism
where are the coordinate rings of X and Y; it is well-defined since is a polynomial in elements of . Conversely, if is an algebra homomorphism, then it induces the morphism
given by: writing
where are the images of 's. Note as well as In particular, f is an isomorphism |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Slovenia | In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Slovenia (SI), the three levels are:
NUTS codes
SI0 Slovenia
SI03 Eastern Slovenia (Vzhodna Slovenija)
SI031 Mura Statistical Region (pomurska statistična regija)
SI032 Drava Statistical Region (podravska statistična regija)
SI033 Carinthia Statistical Region (koroška statistična regija)
SI034 Savinja Statistical Region (savinjska statistična regija)
SI035 Central Sava Statistical Region (zasavska statistična regija)
SI036 Lower Sava Statistical Region (spodnjeposavska statistična regija)
SI037 Southeast Slovenia Statistical Region (jugovzhodna Slovenija)
SI038 Littoral–Inner Carniola Statistical Region (primorsko-notranjska statistična regija)
SI04 Western Slovenia (Zahodna Slovenija)
SI041 Central Slovenia Statistical Region (osrednjeslovenska statistična regija)
SI042 Upper Carniola Statistical Region (gorenjska statistična regija)
SI043 Gorizia Statistical Region (goriška statistična regija)
SI044 Coastal–Karst Statistical Region (obalno-kraška statistična regija)
In the 2003 version, the codes were as follows:
SI0 Slovenia
SI00 Slovenia
SI001 Mura Statistical Region (pomurska statistična regija)
SI002 Drava Statistical Region (podravska statistična regija)
SI003 Carinthia Statistical Region (koroška statistična regija)
SI004 Savinja Statistical Region (savinjska statistična regija)
SI005 Central Sava Statistical Region (zasavska statistična regija)
SI006 Lower Sava Statistical Region (spodnjeposavska statistična regija)
SI009 Upper Carniola Statistical Region (gorenjska statistična regija)
SI00A Inner Carniola–Karst Statistical Region (notranjsko-kraška statistična regija)
SI00B Gorizia Statistical Region (goriška statistična regija)
SI00C Coastal–Karst Statistical Region (obalno-kraška statistična regija)
SI00D Southeast Slovenia Statistical Region (jugovzhodna Slovenija)
SI00E Central Slovenia Statistical Region (osrednjeslovenska statistična regija)
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Slovenia can be downloaded here:
See also
Subdivisions of Slovenia
ISO 3166-2 codes of Slovenia
FIPS region codes of Slovenia
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
SLOVENIJA - NUTS level 2
SLOVENIJA - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Regions of Slovenia, Statoids.com
Eurostat, Regions in the European Union, Nomenclature of territorial units for statistics NUTS 2013/EU-28, ISSN 2363-197X
Slovenia
Nuts |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Slovakia | In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Slovakia (SK), the three levels are:
NUTS codes
SK0 Slovakia
SK01 Bratislava Region
SK010 Bratislava Region
SK02 Western Slovakia (Západné Slovensko)
SK021 Trnava Region
SK022 Trenčín Region
SK023 Nitra Region
SK03 Central Slovakia (Stredné Slovensko)
SK031 Žilina Region
SK032 Banská Bystrica Region
SK04 Eastern Slovakia (Východné Slovensko)
SK041 Prešov Region
SK042 Košice Region
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Slovakia can be downloaded here:
See also
Subdivisions of Slovakia
ISO 3166-2 codes of Slovakia
FIPS region codes of Slovakia
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
SLOVENSKÁ REPUBLIKA - NUTS level 2
SLOVENSKÁ REPUBLIKA - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Regions of Slovakia, Statoids.com
Slovakia
Nuts |
https://en.wikipedia.org/wiki/Truncated%205-simplexes | In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.
Truncated 5-simplex
The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).
Alternate names
Truncated hexateron (Acronym: tix) (Jonathan Bowers)
Coordinates
The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.
Images
Bitruncated 5-simplex
Alternate names
Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)
Coordinates
The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.
Images
Related uniform 5-polytopes
The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3x3o3o3o - tix, o3x3x3o3o - bittix
External links
Polytopes of Various Dimensions, Jonathan Bowers
Truncated uniform polytera (tix), Jonathan Bowers
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Michael%20Somos | Michael Somos is an American mathematician, who was a visiting scholar in the Georgetown University Mathematics and Statistics department for four years and is a visiting scholar at Catholic University of America. In the late eighties he proposed a conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers. Somos' quadratic recurrence constant is also named after him.
Notes
References
Michael Somos and Robert Haas, "A Linked Pair of Sequences Implies the Primes Are Infinite", The American Mathematical Monthly, volume 110, number 6 (June – July, 2003), pp. 539–540
External links
Michael Somos's homepage
Somos sequence in mathworld
The Troublemaker Number. Numberphile video on the Somos sequences
Living people
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Littlewood%E2%80%93Paley%20theory | In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about L2 functions to Lp functions for 1 < p < ∞. It is typically used as a substitute for orthogonality arguments which only apply to Lp functions when p = 2. One implementation involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley g-function to compare it with its Poisson integral. The 1-variable case was originated by and developed further by Polish mathematicians A. Zygmund and J. Marcinkiewicz in the 1930s using complex function theory . E. M. Stein later extended the theory to higher dimensions using real variable techniques.
The dyadic decomposition of a function
Littlewood–Paley theory uses a decomposition of a function f into a sum of functions fρ with localized frequencies. There are several ways to construct such a decomposition; a typical method is as follows.
If f(x) is a function on R, and ρ is a measurable set (in the frequency space) with characteristic function , then fρ is defined via its Fourier transform
.
Informally, fρ is the piece of f whose frequencies lie in ρ.
If Δ is a collection of measurable sets which (up to measure 0) are disjoint and have union on the real line, then a well behaved function f can be written as a sum of functions fρ for ρ ∈ Δ.
When Δ consists of the sets of the form
for k an integer, this gives a so-called "dyadic decomposition" of f : Σρ fρ.
There are many variations of this construction; for example, the characteristic function of a set used in the definition of fρ can be replaced by a smoother function.
A key estimate of Littlewood–Paley theory is the Littlewood–Paley theorem, which bounds the size of the functions fρ in terms of the size of f. There are many versions of this theorem corresponding to the different ways of decomposing f. A typical estimate is to bound the Lp norm of (Σρ |fρ|2)1/2 by a multiple of the Lp norm of f.
In higher dimensions it is possible to generalize this construction by replacing intervals with rectangles with sides parallel to the coordinate axes. Unfortunately these are rather special sets, which limits the applications to higher dimensions.
The Littlewood–Paley g function
The g function is a non-linear operator on Lp(Rn) that can be used to control the Lp norm of a function f in terms of its Poisson integral.
The Poisson integral u(x,y) of f is defined for y > 0 by
where the Poisson kernel P on the upper half space
is given by
The Littlewood–Paley g function g(f) is defined by
A basic property of g is that it approximately preserves norms. More precisely, for 1 < p < ∞, the ratio of the Lp norms of f and g(f) is bounded above and below by fixed positive constants depending on n and p but not on f.
Applications
One early application of Littlewood–Paley theory was the proof that if Sn are the partial sums of the Fourier series of a p |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Croatia | Croatia (HR) is included in the Nomenclature of Territorial Units for Statistics (NUTS) of the European Union. The NUTS of Croatia were defined during the Accession of Croatia to the European Union, codified by the Croatian Bureau of Statistics in early 2007. The regions were revised twice, first in 2012, and then in 2021.
The three NUTS levels are:
NUTS-1: Croatia
NUTS-2: 4 regions (non-administrative)
NUTS-3: 21 counties (administrative)
The NUTS codes are as follows:
Below the NUTS levels, there are two LAU levels:
LAU-1: none (same as NUTS-3)
LAU-2: Cities and municipalities (Gradovi i općine)
See also
Subdivisions of Croatia
ISO 3166-2 codes of Croatia
FIPS region codes of Croatia
References
Croatia
Subdivisions of Croatia |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20North%20Macedonia | As a candidate country of the European Union, North Macedonia (MK) is included in the Nomenclature of Territorial Units for Statistics (NUTS). The three NUTS levels are:
NUTS-1: MK0 North Macedonia
NUTS-2: MK00 North Macedonia
NUTS-3: 8 Statistical regions
MK001 Vardarski
MK002 Istočen
MK003 Jugozapaden
MK004 Jugoistočen
MK005 Pelagoniski
MK006 Pološki
MK007 Severoistočen
MK008 Skopski
Below the NUTS levels, there are two LAU levels (LAU-1: municipalities; LAU-2: settlements).
See also
ISO 3166-2 codes of North Macedonia
FIPS region codes of North Macedonia
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of CC (Candidate countries) - Statistical regions at level 1
Macedonia - Statistical regions at level 2
Macedonia - Statistical regions at level 3
Correspondence between the regional levels and the national administrative units
Municipalities of Macedonia, Statoids.com
Macedonia
Subdivisions of North Macedonia |
https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Turkey | As a candidate country of the European Union, Turkey (TR) is included in the Nomenclature of Territorial Units for Statistics (NUTS). Defined in 2002 in agreement between Eurostat and the Turkish authorities, Turkey's NUTS classifications are officially termed statistical regions, as Turkey is not a member of the EU and Eurostat only defines NUTS for member states. The three NUTS levels are:
NUTS-1: 12 Regions
NUTS-2: 26 Subregions
NUTS-3: 81 Provinces
Below the NUTS levels, there are two LAU levels:
LAU-1: 923 districts
LAU-2: 37,675 municipalities
The NUTS codes are as follows:
See also
FIPS region codes of Turkey
ISO 3166-2 codes of Turkey
Subdivisions of Turkey
List of Turkish regions by Human Development Index
References
External links
Overview of the NUTS 2013 Classification
Statistics Illustrated. Interactive map of NUTS levels.
Regional Statistics Illustrated. Interactive map of statistics indicators by NUTS levels.
Provinces of Turkey, Statoids.com
Turkey
Statistical subregions of Turkey
Administrative divisions of Turkey
Turkey–European Union relations |
https://en.wikipedia.org/wiki/Precision%20%28statistics%29 | In statistics, the precision matrix or concentration matrix is the matrix inverse of the covariance matrix or dispersion matrix, .
For univariate distributions, the precision matrix degenerates into a scalar precision, defined as the reciprocal of the variance, .
Other summary statistics of statistical dispersion also called precision (or imprecision)
include the reciprocal of the standard deviation, ;
the standard deviation itself and the relative standard deviation;
as well as the standard error and the confidence interval (or its half-width, the margin of error).
Usage
One particular use of the precision matrix is in the context of Bayesian analysis of the multivariate normal distribution: for example, Bernardo & Smith prefer to parameterise the multivariate normal distribution in terms of the precision matrix, rather than the covariance matrix, because of certain simplifications that then arise. For instance, if both the prior and the likelihood have Gaussian form, and the precision matrix of both of these exist (because their covariance matrix is full rank and thus invertible), then the precision matrix of the posterior will simply be the sum of the precision matrices of the prior and the likelihood.
As the inverse of a Hermitian matrix, the precision matrix of real-valued random variables, if it exists, is positive definite and symmetrical.
Another reason the precision matrix may be useful is that if two dimensions and of a multivariate normal are conditionally independent, then the and elements of the precision matrix are . This means that precision matrices tend to be sparse when many of the dimensions are conditionally independent, which can lead to computational efficiencies when working with them. It also means that precision matrices are closely related to the idea of partial correlation.
The precision matrix plays a central role in generalized least squares, compared to ordinary least squares, where is the identity matrix, and to weighted least squares, where is diagonal (the weight matrix).
Etymology
The term precision in this sense ("mensura praecisionis observationum") first appeared in the works of Gauss (1809) "Theoria motus corporum coelestium in sectionibus conicis solem ambientium" (page 212). Gauss's definition differs from the modern one by a factor of . He writes, for the density function of a normal distribution with precision (reciprocal of standard deviation),
where (see: Exponentiation#History of the notation).
Later Whittaker & Robinson (1924) "Calculus of observations" called this quantity the modulus (of precision), but this term has dropped out of use.
References
Statistical deviation and dispersion
Bayesian statistics |
https://en.wikipedia.org/wiki/Smallpeice%20Trust | The Smallpeice Trust is a British charity that provides programmes to promote engineering careers to young people aged 10 to 18 through residential courses, Science, Technology, Engineering and Maths (STEM) days, STEM clubs and STEM teacher training days.
Cosby Smallpeice (died 1977), a pioneering engineer and inventor of the Smallpeice lathe, founded the trust, following the market flotation of his company Martonair. Dr Smallpeice invested his energy and part of his personal fortune to set up the Trust to ensure that industry could benefit from his proven design and engineering philosophies “Simplicity in design, economy in production”.
The trust is governed by eminent non-executive trustees and members from a range of engineering, industry, educational and professional bodies.
In the academic year ended July 2011, The Smallpeice Trust reached out to 17,495 young people through 35 different subsidised 3 to 5 day residential courses in a range of engineering disciplines, 1-day in-school STEM days and STEM clubs. The trust has also trained 1,280 teachers to enhance their delivery of STEM in the classroom through its teacher training days.
The trust maintains a strong interface with industry, education and professional bodies that help to support, promote and develop the courses. Through these relationships the trust can provide tailored or specialised courses.
The trust awards most of its annual Arkwright Engineering Scholarships supporting students in A Level or Standard Grade standard/vocational studies at its office in Leamington Spa, England.
References
External links
Twitter feed/posts
Facebook page
Educational charities based in the United Kingdom
Engineering education in the United Kingdom
Funding bodies in the United Kingdom |
https://en.wikipedia.org/wiki/B%C3%BCchi%27s%20problem | In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such that every sequence of M or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (x + i)2, i = 1, 2, ..., M,... for some integer x. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer M such that, for all integers x and a, the quantity (x + n)2 + a cannot be a square for more than M consecutive values of n, unless a = 0?
Statement of Büchi's problem
Büchi's problem can be stated in the following way: Does there exist a positive integer M such that the system of equations
has only solutions satisfying
Since the first difference of the sequence is the sequence , the second difference of is
Therefore, the above system of equations is equivalent to the single equation
where the unknown is the sequence .
Examples
Observe that for any integer x we have
Hence the equation has solutions, called trivial Büchi sequences of length three, such that and . For example, the sequences (2, 3, 4) and (2, −3, 4) are trivial Büchi sequences. A nontrivial Büchi sequence of length three is given for example by the sequence (0, 7, 10), as it satisfies 102 − 2·72 + 02 = 2, while 02, 72 and 102 are not consecutive squares.
Replacing x by x + 1 in equation , we obtain . Hence the system of equations
has trivial Büchi solutions of length 4, namely the one satisfying for n = 0, 1, 2, 3. In 1983, D. Hensley showed that there are infinitely many nontrivial Büchi sequences of length four. It is not known whether there exist any non-trivial Büchi sequence of length five (Indeed, Büchi asked originally the question only for M = 5.).
Original motivation
A positive answer to Büchi's problem would imply, using the negative answer to Hilbert's tenth problem by Yuri Matiyasevich, that there is no algorithm to decide whether a system of diagonal quadratic forms with integer coefficients represents an integer tuple. Indeed, Büchi observed that squaring, therefore multiplication, would be existentially definable in the integers over the first-order language having two symbols of constant for 0 and 1, a symbol of function for the sum, and a symbol of relation P to express that an integer is a square.
Some results
Paul Vojta proved in 1999 that a positive answer to Büchi's Problem would follow from a positive answer to a weak version of the Bombieri–Lang conjecture. In the same article, he proves that the analogue of Büchi's Problem for the field of meromorphic functions over the complex numbers has a positive answer. Positive answers to analogues of Büchi's Problem in various other rings of functions have been obtained since then (in the case of rings of functions, one adds the hypothesis that not all xn are constant |
https://en.wikipedia.org/wiki/FIFA%20U-20%20World%20Cup%20records%20and%20statistics | This is a list of records and statistics of the FIFA U-20 World Cup.
Debut of national teams
Overall team records
In this ranking 3 points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored.
.
Former countries
Comprehensive team results by tournament
Legend
— Champions
— Runners-up
— Third place
— Fourth place
QF — Quarter-finals (since 1979; round of 16)
R2 — Round 2 (since 1997; round of 16)
R1 — Round 1 (group stage)
— Did not qualify
— Did not enter / Withdrew / Banned
— Country did not exist or national team was inactive
— Hosts
Q — Qualified for upcoming tournament
For each tournament, the number of teams in each finals tournament are shown (in parentheses).
Teams that have finished in the top four
1 = includes results representing Yugoslavia
2 = includes results representing Soviet Union
3 = includes results representing West Germany
Results of defending champions
Results of host nations
Result of confederation
AFC
CAF
CONCACAF
CONMEBOL
OFC
UEFA
Awards
At the end of each FIFA U-20 World Cup tournament, several awards are presented to the players and teams which have distinguished themselves in various aspects of the game.
There are four awards:
the Golden Ball (commercially termed "adidas Golden Ball") for best player assigned by members of the media;
the Golden Boot (commercially termed "adidas Golden Boot" for best scorer;
the Golden Glove Award (commercially termed "adidas Golden Glove" for the best goalkeeper assigned since 2009 FIFA U-20 World Cup;
the FIFA Fair Play Trophy for the team that advanced to the second round with the best record of fair play;
Golden Ball
The Adidas Golden Ball award is awarded to the player who plays the most outstanding football during the tournament. It is selected by the media poll. Since the 2007 tournament, those who finish as runners-up in the vote receive the Silver Ball and Bronze Ball awards as the second and third most outstanding players in the tournament respectively.
Golden Boot
The Golden Boot (known commercially as the Adidas Golden Shoe) is awarded to the top goalscorer of the tournament. If more than one players are equal by same goals, the players will be selected based by the most assists made and, if still tied, less playing minutes recorded during the tournament.
Golden Glove
The Golden Glove is awarded to the best goalkeeper of the tournament.
FIFA Fair Play Award
FIFA Fair Play Award is given to the team who has the best fair play record during the tournament with the criteria set by FIFA Fair Play Committee.
Records and statistics
Most World Cup appearances 19, , as of 2023
Most consecutive finals tournaments 16, (1981–2011)
Most tournament wins (player) 2, three players:
Fernando Brassard (; 198 |
https://en.wikipedia.org/wiki/Circular%20surface | In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ : I × S1 → R3, where I ⊂ R is an open interval and S1 is the unit circle, defined by
where γ, u, v : I → R3 and r : I → R>0, when Moreover, it is usually assumed that u · u = v · v = 1 and u · v = 0, where dot denotes the canonical scalar product on R3, i.e. u and v are unit length and mutually perpendicular. The map γ : I → R3 is called the base curve for the circular surface and the two maps u, v : I → R3 are called the direction frame for the circular surface. For a fixed t0 ∈ I the image of ƒ(t0, θ) is called a generating circle of the circular surface.
Circular surfaces are an analogue of ruled surfaces. In the case of circular surfaces the generators are circles; called the generating circles. In the case of ruled surface the generators are straight lines; called rulings.
References
Surfaces |
https://en.wikipedia.org/wiki/Cantellated%205-simplexes | In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.
There are unique 4 degrees of cantellation for the 5-simplex, including truncations.
Cantellated 5-simplex
The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).
Alternate names
Cantellated hexateron
Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)
Coordinates
The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.
Images
Bicantellated 5-simplex
Alternate names
Bicantellated hexateron
Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)
Coordinates
The coordinates can be made in 6-space, as 90 permutations of:
(0,0,1,1,2,2)
This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex.
Images
Cantitruncated 5-simplex
Alternate names
Cantitruncated hexateron
Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)
Coordinates
The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.
Images
Bicantitruncated 5-simplex
Alternate names
Bicantitruncated hexateron
Great birhombated dodecateron (Acronym: gibrid) (Jonathan Bowers)
Coordinates
The coordinates can be made in 6-space, as 180 permutations of:
(0,0,1,2,3,3)
This construction exists as one of 64 orthant facets of the bicantitruncated 6-orthoplex.
Images
Related uniform 5-polytopes
The cantellated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honey |
https://en.wikipedia.org/wiki/Georges-Andre%20Machia | Georges-Andre Machia Malock (born 26 March 1988) is a Cameroonian former football player who plays as a striker.
Career statistics in Hong Kong
As of 24 September 2009
References
External links
Player Information on tswpegasus.com
1988 births
Living people
Cameroonian men's footballers
Dhivehi Premier League players
Hong Kong First Division League players
Indonesian Premier Division players
Georges-Andre Machia
Expatriate men's footballers in Hong Kong
Expatriate men's footballers in Indonesia
Expatriate men's footballers in the Maldives
Expatriate men's footballers in Thailand
Cameroonian expatriate sportspeople in Hong Kong
Cameroonian expatriate sportspeople in Indonesia
Cameroonian expatriate sportspeople in the Maldives
Cameroonian expatriate sportspeople in Thailand
Men's association football forwards
Victory Sports Club players
Persis Solo players
Hong Kong Pegasus FC players
PSLS Lhokseumawe players
Persik Kediri players
Georges-Andre Machia
Georges-Andre Machia |
https://en.wikipedia.org/wiki/Algebra%20%28singer%29 | Algebra Felicia Blessett (born April 9, 1976), usually known as Algebra Blessett or just Algebra, is an American contemporary R&B singer.
Early life
Blessett's mother was a gospel singer and bass player, and Blessett grew up to the sounds of soul music, gospel and R&B. Like many R&B singers, she sang in a gospel choir when she was in school. However, she was not passionate about the experience, and decided to do it only because she was not good at sports, but still wanted to stay after school with her friends.
Music career
Blessett started doing background vocals, among others for R&B artists Monica and Bilal. This earned her a contract with Rowdy Records in Atlanta. She has toured with Anthony Hamilton, and collaborated with India.Arie. At a later age she learned to play the guitar, and started to do her own gigs in the Atlanta club scene. She writes her own songs.
Blessett released her first single, "U Do It For Me", on the Kedar Entertainment label in 2006. She released her first album, Purpose, in 2008.
In 2014, her sophomore effort Recovery was released.
Charts
Blessett's debut album Purpose was on the US Billboard R&B chart for 14 weeks, and reached No. 56. It also landed No. 37 on Heatseekers Albums.
Her second album, Recovery, was her debut on the Billboard 200, charting No. 149. It also entered No. 2 on Heatseekers Albums and No. 23 on Top R&B/Hip-Hop Albums.
Discography
Albums
Singles
2006: "U Do It For Me"
2008: "Run and Hide"
2012: "Black Gold" – credited as 'Esperanza Spalding with Algebra Blessett'
2013: "Nobody But You"
References
External links
2 Algebra pictures, Vibe Magazine Gallery, retrieved September 26, 2009
Algebra on NeoSoulVille, retrieved September 26, 2009
Algebra Blessett on Myspace
1976 births
Living people
Singers from Atlanta
Songwriters from Georgia (U.S. state)
American contemporary R&B singers
21st-century American women singers
21st-century American singers
American neo soul singers |
https://en.wikipedia.org/wiki/Sourav%20Chatterjee | Sourav Chatterjee (born November 1979) is an Indian mathematician, specializing in mathematical statistics and probability theory. Chatterjee is credited with work on the study of fluctuations in random structures, concentration and super-concentration inequalities, Poisson and other non-normal limits, first-passage percolation, Stein's method and spin glasses. He has received a Sloan Fellowship in mathematics, Tweedie Award, Rollo Davidson Prize, Doeblin Prize, Loève Prize, and Infosys Prize in mathematical sciences. He was an invited speaker at the International Congress of Mathematicians in 2014.
Career
Chatterjee received a Bachelor and Master of Statistics from Indian Statistical Institute, Kolkata, and a Ph.D. from Stanford University in 2005, where he worked under the supervision of Persi Diaconis. Chatterjee joined University of California, Berkeley, as a visiting assistant professor, then received a tenure-track Assistant Professor position in 2006. In July 2009 he became an Associate Professor of Statistics and Mathematics at University of California, Berkeley. Then in September 2009, Chatterjee became an associate professor of mathematics at the Courant Institute of Mathematical Sciences, New York University. He spent the academic year 2012–2013 as a visiting associate professor of mathematics and statistics at Stanford University. Since autumn 2013 he has joined the faculty of Stanford University as a full professor with joint appointments in the departments of Mathematics and Statistics.
He has served as an associate editor of Annals of Probability, Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, "Probability Theory and Related Fields". He currently serves as an editor of Communications in Mathematical Physics.
Awards and honours
2008 Tweedie New Researcher Award, from the Institute of Mathematical Statistics.
Sloan Research Fellowship in Mathematics, 2007–2009
Rollo Davidson Prize 2010
IMS Medallion Lecture, 2012
Inaugural Wolfgang Doeblin Prize in Probability, 2012
Loève Prize 2013
ICM Invited talk, 2014
Infosys Prize 2020
Fellow of the Royal Society 2023
References
External links
Sourav Chatterjee's homepage (Department of Statistics - Stanford University)
1979 births
21st-century Bengalis
Bengali mathematicians
Living people
Stanford University alumni
University of California, Berkeley College of Letters and Science faculty
Courant Institute of Mathematical Sciences faculty
Probability theorists
Indian Statistical Institute alumni
21st-century Indian mathematicians
Sloan Research Fellows
Fellows of the Institute of Mathematical Statistics
Scientists from Kolkata
21st-century Indian scientists
Fellows of the Royal Society |
https://en.wikipedia.org/wiki/Jim%20Davies%20%28computer%20scientist%29 | Jim Davies is Professor of Software Engineering and current Director of the Software Engineering Programme at the University of Oxford, England.
Biography
Jim Davies studied mathematics at New College, Oxford, joining the Oxford University Computing Laboratory (now the Oxford University Department of Computer Science) in 1986 for a Masters' and Doctorate. After working as a researcher and lecturer in computer science, at Oxford, Reading, and Royal Holloway, University of London, he became a lecturer in software engineering at Oxford in 1995. He has led the Software Engineering Programme since 2000, and was made Professor of Software Engineering in 2006.
Davies is an expert in formal methods, including Communicating Sequential Processes (CSP) and the Z notation.
Books
Jim Davies, Specification and Proof in Real Time CSP. Cambridge University Press, 1993. .
Jim Woodcock and Jim Davies, Using Z: Specification, Refinement, and Proof. Prentice-Hall International Series in Computer Science, 1996. .
Jim Davies, Bill Roscoe, and Jim Woodcock, Millennial Perspectives in Computer Science: Proceedings of the 1999 Oxford-Microsoft Symposium in Honour of Sir Tony Hoare. Palgrave Macmillan, Cornerstones of Computing, 2000. .
References
Year of birth missing (living people)
Living people
Alumni of New College, Oxford
Academics of Royal Holloway, University of London
Academics of the University of Reading
Members of the Department of Computer Science, University of Oxford
Fellows of Kellogg College, Oxford
English computer scientists
Formal methods people
Computer science writers
British textbook writers |
https://en.wikipedia.org/wiki/Algebraic%20sentence | In mathematical logic, an algebraic sentence is one that can be stated using only equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.
Saying that a sentence is algebraic is a stronger condition than saying it is elementary.
Related
Algebraic theory
Algebraic definition
Algebraic expression
Mathematical logic |
https://en.wikipedia.org/wiki/Algebraic%20theory | Informally in mathematical logic, an algebraic theory is a theory that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.
The notion is very close to the notion of algebraic structure, which, arguably, may be just a synonym.
Saying that a theory is algebraic is a stronger condition than saying it is elementary.
Informal interpretation
An algebraic theory consists of a collection of n-ary functional terms with additional rules (axioms).
For example, the theory of groups is an algebraic theory because it has three functional terms: a binary operation a × b, a nullary operation 1 (neutral element), and a unary operation x ↦ x−1 with the rules of associativity, neutrality and inverses respectively. Other examples include:
the theory of semigroups
the theory of lattices
the theory of rings
This is opposed to geometric theory which involves partial functions (or binary relationships) or existential quantors − see e.g. Euclidean geometry where the existence of points or lines is postulated.
Category-based model-theoretical interpretation
An algebraic theory T is a category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n-tuple of morphisms:
proji: n → 1, i = 1, ..., n
This allows interpreting n as a cartesian product of n copies of 1.
Example: Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1, ..., Xn with integer coefficients and with substitution as composition. In this case proji is the same as Xi. This theory T is called the theory of commutative rings.
In an algebraic theory, any morphism n → m can be described as m morphisms of signature n → 1. These latter morphisms are called n-ary operations of the theory.
If E is a category with finite products, the full subcategory Alg(T, E) of the category of functors [T, E] consisting of those functors that preserve finite products is called the category of T-models or T-algebras.
Note that for the case of operation 2 → 1, the appropriate algebra A will define a morphism
A(2) ≈ A(1) × A(1) → A(1)
See also
Algebraic sentence
Algebraic definition
References
Lawvere, F. W., 1963, Functorial Semantics of Algebraic Theories, Proceedings of the National Academy of Sciences 50, No. 5 (November 1963), 869-872
Adámek, J., Rosický, J., Vitale, E. M., Algebraic Theories. A Categorical Introduction To General Algebra
Kock, A., Reyes, G., Doctrines in categorical logic, in Handbook of Mathematical Logic, ed. J. Barwise, North Holland 1977
Mathematical logic |
https://en.wikipedia.org/wiki/Elementary%20sentence | In mathematical logic, an elementary sentence is one that is stated using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory.
Saying that a sentence is elementary is a weaker condition than saying it is algebraic.
Related
Elementary theory
Elementary definition
References
Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.
Mathematical logic |
https://en.wikipedia.org/wiki/Elementary%20theory | In mathematical logic, an elementary theory is a theory that involves axioms using only finitary first-order logic, without reference to set theory or using any axioms which have consistency strength equal to set theory.
Saying that a theory is elementary is a weaker condition than saying it is algebraic.
Examples
Examples of elementary theories include:
The theory of groups
The theory of finite groups
The theory of abelian groups
The theory of fields
The theory of finite fields
The theory of real closed fields
Axiomization of Euclidean geometry
Related
Elementary sentence
Elementary definition
Elementary theory of the reals
References
Mac Lane and Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, page 4.
Mathematical logic |
https://en.wikipedia.org/wiki/APJ | APJ may refer to:
The Astrophysical Journal
Apelin receptor
Absolute probability judgement
Asia-Pacific
APJ Abdul Kalam (1931 - 2015), former President of India
Peach Aviation ICAO Airlines Code |
https://en.wikipedia.org/wiki/Direct%20method%20in%20the%20calculus%20of%20variations | In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.
The method
The calculus of variations deals with functionals , where is some function space and . The main interest of the subject is to find minimizers for such functionals, that is, functions such that:
The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand.
The functional must be bounded from below to have a minimizer. This means
This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence in such that
The direct method may be broken into the following steps
Take a minimizing sequence for .
Show that admits some subsequence , that converges to a with respect to a topology on .
Show that is sequentially lower semi-continuous with respect to the topology .
To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions.
The function is sequentially lower-semicontinuous if
for any convergent sequence in .
The conclusions follows from
,
in other words
.
Details
Banach spaces
The direct method may often be applied with success when the space is a subset of a separable reflexive Banach space . In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence in has a subsequence that converges to some in with respect to the weak topology. If is sequentially closed in , so that is in , the direct method may be applied to a functional by showing
is bounded from below,
any minimizing sequence for is bounded, and
is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence it holds that .
The second part is usually accomplished by showing that admits some growth condition. An example is
for some , and .
A functional with this property is sometimes called coercive. Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method. See below for some theorems for a general class of functionals.
Sobolev spaces
The typical functional in the calculus of variations is an integral of the form
where is a subset of and is a real-valued function on . The argument of is a differentiable function , and its Jacobian is identified with a -vector.
When deriving the Euler–Lagrange equation, the common approach is to assume has a boundary and let the d |
https://en.wikipedia.org/wiki/Haran%27s%20diamond%20theorem | In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian.
Statement of the diamond theorem
Let K be a Hilbertian field and L a separable extension of K. Assume there exist two Galois extensions
N and M of K such that L is contained in the compositum NM, but is contained in neither N nor M. Then L is Hilbertian.
The name of the theorem comes from the pictured diagram of fields, and was coined by Jarden.
Some corollaries
Weissauer's theorem
This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led Haran to his diamond theorem.
Weissauer's theorem
Let K be a Hilbertian field, N a Galois extension of K, and L a finite proper extension of N. Then L is Hilbertian.
Proof using the diamond theorem
If L is finite over K, it is Hilbertian; hence we assume that L/K is infinite. Let x be a primitive element for L/N, i.e., L = N(x).
Let M be the Galois closure of K(x). Then all the assumptions of the diamond theorem are satisfied, hence L is Hilbertian.
Haran–Jarden condition
Another, preceding to the diamond theorem, sufficient permanence condition was given by Haran–Jarden:
Theorem.
Let K be a Hilbertian field and N, M two Galois extensions of K. Assume that neither contains the other. Then their compositum NM is Hilbertian.
This theorem has a very nice consequence: Since the field of rational numbers, Q is Hilbertian (Hilbert's irreducibility theorem), we get that the algebraic closure of Q is not the compositum of two proper Galois extensions.
References
.
.
Galois theory
Theorems in algebra
Number theory |
https://en.wikipedia.org/wiki/Dead%20Man%27s%20Flats | Dead Man's Flats is a former Hamlet, now a Neighbourhood in Canmore, Alberta within the Municipal District of Bighorn No. 8. Statistics Canada also recognizes it as a designated place under the name of Pigeon Mountain. It is located within Alberta's Rockies at Highway 1 exit 98, approximately southeast of Canmore and west of Calgary.
History
A variety of explanations account for the origin of the hamlet's name. One explanation associates it with a murder which took place in 1904 at a dairy farm situated on the flats of the Bow River. Francois Marret stood trial in Calgary for killing his brother Jean, whose body he had disposed of in the Bow River, but the jury acquitted him by reason of insanity. Another account states that two or three First Nations people who were illegally trapping beaver noticed a warden approaching in the distance. Knowing that they did not have time to flee without being spotted, they smeared themselves with beaver blood and pretended to be dead. The warden, fooled by their deception, ran for help. Meanwhile, the trappers took their beaver pelts and escaped. This account is regarded as dubious; for example, no known description of this incident appears in the official wardens' reports.
In 1954, the Calgary Herald wrote that it was "named only 10 to 12 years ago after a man was found shot in a cabin in the area." However, the phrase "Dead Man's flat" (lower-case "f" without the plural "s" at the end) is used in the August 25th, 1924 edition of the Calgary Herald. In an article that describes some recent events in Canmore it is stated that "A party of Canmore boys...returned last week from a seven days' outing at Dead Man's flat." They went on the outing for the purpose of fishing.
From 1974 to 1985 the hamlet was officially called Pigeon Mountain Service Centre, but it changed its name to Dead Man's Flats in 1985 to encourage tourism. The new name had been unofficially used to designate the hamlet for several decades prior.
Before the Trans-Canada highway was constructed through the area in the 1950s, it was sparsely populated Crown land; among the only structures in the area were a corral and a camper's cabin. Proximity to the new national highway spurred the hamlet's development as a commercial service centre and rest stop for travelers and truck drivers. Businesses currently operating include motels, a Husky truck stop with a 24-hour diner, a Shell, gas station with a U-Haul Neighborhood Dealer, and the one98eight restaurant. Recently the area has been the subject of proposed developments which would involve the construction of a new residential neighbourhood and a light industrial park.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Dead Man’s Flats had a population of 377 living in 128 of its 162 total private dwellings, a change of from its 2016 population of 125. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Popu |
https://en.wikipedia.org/wiki/Tibor%20Heffler | Tibor Heffler (born 17 May 1987) is a Hungarian football player. He has a brother, Norbert Heffler, who plays for Gyirmót II.
Honours
Paksi SE
Hungarian Second Division: Winner 2006
Club statistics
Updated to games played as of 9 August 2020.
References
HLSZ
UEFA Official Website
1987 births
Sportspeople from Dunaújváros
Footballers from Fejér County
Hungarian people of German descent
Living people
Hungarian men's footballers
Men's association football midfielders
Men's association football defenders
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Hungary men's international footballers
Paksi FC players
Fehérvár FC players
Puskás Akadémia FC players
Ceglédi VSE footballers
Budapest Honvéd FC players
FC Ajka players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Mathe | Mathe or Mathé or Máthé may refer to:
Mathematics
Given name
Mathé Altéry (born 1927), French soprano singer
Surname
Annanias Mathe (c. 1976–2016), Mozambique criminal
Antoine Félix Mathé (1808–1882), French politician
Édouard Mathé (1886–1934), French silent film actor
Félix Mathé (1834–1911), French politician
Gábor Máthé (footballer) (born 1985), Hungarian football player
Gábor Máthé (lawyer) (born 1941), Hungarian lawyer
Gábor Máthé (tennis) (born 1985), Hungarian male tennis player
Georges Mathé (1922–2010), French oncologist and immunologist
Henri Mathé (1837–1907), French politician
Ketty Mathé (born 1988), French judoka
Lew Mathe (1915–1986), American bridge player
Pierre Mathé (1882–1956), French farmer and politician
Vasily Mathe (1856–1917), Russian artist and engraver
Zsuzsa Mathe (born 1964), Hungarian artist
Films
Mathe Haditu Kogile, Kannada language film
''Mathe Mungaru, Kannada language film |
https://en.wikipedia.org/wiki/Andrew%20Tooke | Andrew Tooke (1673–1732) was an English scholar, headmaster of Charterhouse School, Gresham Professor of Geometry, Fellow of the Royal Society and translator of Tooke's Pantheon, a standard textbook for a century on Greek mythology.
Life
He was second son of Benjamin Tooke, stationer of London, and received his education in the Charterhouse school. He was admitted a scholar of Clare Hall, Cambridge, in 1690, took the degree of B.A. in 1693, and commenced M.A. in 1697.
In 1695 he had become usher in the Charterhouse school, and on 5 July 1704 he was elected professor of geometry in Gresham College in succession to Robert Hooke. On 30 November 1704 he was chosen a fellow of the Royal Society, which held its meetings in his chambers, until they left the college in 1710.
He was chosen master of the Charterhouse on 17 July 1728 in the place of Thomas Walker. He had taken deacon's orders and sometimes preached, but devoted himself principally to education. On 26 June 1729 he resigned his professorship in Gresham College. He died on 20 January 1732, and was buried in the chapel of the Charterhouse, where a monument was erected to his memory. In May 1729 he had married the widow of Dr. Henry Levett, physician to the Charterhouse.
Works
His works are:
'The Pantheon, representing the Fabulous Histories of the Heathen Gods and most Illustrious Heroes', translated from the 'Pantheum Mithicum' of the Jesuit father François Antoine Pomey and illustrated with copperplates, London 1698; 7th edit., London, 1717, 35th edit. London, 1824, 8vo.
'Synopsis Graecae Linguae', London, 1711.
'The Whole Duty of Man, according to the Law of Nature', translated from the Latin Samuel von Pufendorf, 4th edit. London, 1716.
'Institutiones Christianae', London, 1718, a translation of the 'Christian Institutes', by Francis Gastrell.
An edition of Ovid's 'Fasti', London, 1720.
An edition of William Walker's Treatise of English Particles, London, 1720.
'Copy of the last Will and Testament of Sir Thomas Gresham . . . with some Accounts concerning Gresham College, taken from the last Edition of Stow's "Survey of London"' (anon.), London, 1724 (some of these accounts were originally written by him).
Epistles distinguished by the letters A. Z. in the English edition of Pliny's 'Epistles', 11 vols. London, 1724.
References
Attribution
1673 births
1732 deaths
English translators
Fellows of the Royal Society
Headmasters of Charterhouse School |
https://en.wikipedia.org/wiki/Graded%20manifold | In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces.
Graded manifolds
A graded manifold of dimension is defined as a locally ringed space where is an -dimensional smooth manifold and is a -sheaf of Grassmann algebras of rank where is the sheaf of smooth real functions on . The sheaf is called the structure sheaf of the graded manifold , and the manifold is said to be the body of . Sections of the sheaf are called graded functions on a graded manifold . They make up a graded commutative -ring called the structure ring of . The well-known Batchelor theorem and Serre–Swan theorem characterize graded manifolds as follows.
Serre–Swan theorem for graded manifolds
Let be a graded manifold. There exists a vector bundle with an -dimensional typical fiber such that the structure sheaf of is isomorphic to the structure sheaf of sections of the exterior product of , whose typical fibre is the Grassmann algebra .
Let be a smooth manifold. A graded commutative -algebra is isomorphic to the structure ring of a graded manifold with a body if and only if it is the exterior algebra of some projective -module of finite rank.
Graded functions
Note that above mentioned Batchelor's isomorphism fails to be canonical, but it often is fixed from the beginning. In this case, every trivialization chart of the vector bundle yields a splitting domain of a graded manifold , where is the fiber basis for . Graded functions on such a chart are -valued functions
,
where are smooth real functions on and are odd generating elements of the Grassmann algebra .
Graded vector fields
Given a graded manifold , graded derivations of the structure ring of graded functions are called graded vector fields on . They constitute a real Lie superalgebra with respect to the superbracket
,
where denotes the Grassmann parity of . Graded vector fields locally read
.
They act on graded functions by the rule
.
Graded exterior forms
The -dual of the module graded vector fields is called the module of graded exterior one-forms . Graded exterior one-forms locally read so that the duality (interior) product
between and takes the form
.
Provided with the graded exterior product
,
graded one-forms generate the graded exterior algebra of graded exterior forms on a graded manifold. They obey the relation
,
where denotes the form degree of . The graded exterior algebra is a graded differential algebra with respect to the graded exterior differential
,
where the graded derivations , are graded commutative with the graded forms and . There are
the familiar relations
.
Graded differential |
https://en.wikipedia.org/wiki/Locally%20decodable%20code | A locally decodable code (LDC) is an error-correcting code that allows a single bit of the original message to be decoded with high probability by only examining (or querying) a small number of bits of a possibly corrupted codeword.
This property could be useful, say, in a context where information is being transmitted over a noisy channel, and only a small subset of the data is required at a particular time and there is no need to decode the entire message at once. Note that locally decodable codes are not a subset of locally testable codes, though there is some overlap between the two.
Codewords are generated from the original message using an algorithm that introduces a certain amount of redundancy into the codeword; thus, the codeword is always longer than the original message. This redundancy is distributed across the codeword and allows the original message to be recovered with good probability even in the presence of errors. The more redundant the codeword, the more resilient it is against errors, and the fewer queries required to recover a bit of the original message.
Overview
More formally, a -locally decodable code encodes an -bit message to an -bit codeword such that any bit of the message can be recovered with probability by using a randomized decoding algorithm that queries only bits of the codeword , even if up to locations of the codeword have been corrupted.
Furthermore, a perfectly smooth local decoder is a decoder such that, in addition to always generating the correct output given access to an uncorrupted codeword, for every and the query to recover the bit is uniform over .
(The notation denotes the set ). Informally, this means that the set of queries required to decode any given bit are uniformly distributed over the codeword.
Local list decoders are another interesting subset of local decoders. List decoding is useful when a codeword is corrupted in more than places, where is the minimum Hamming distance between two codewords. In this case, it is no longer possible to identify exactly which original message has been encoded, since there could be multiple codewords within distance of the corrupted codeword. However, given a radius , it is possible to identify the set of messages that encode to codewords that are within of the corrupted codeword. An upper bound on the size of the set of messages can be determined by and .
Locally decodable codes can also be concatenated, where a message is encoded first using one scheme, and the resulting codeword is encoded again using a different scheme. (Note that, in this context, concatenation is the term used by scholars to refer to what is usually called composition; see ). This might be useful if, for example, the first code has some desirable properties with respect to rate, but it has some undesirable property, such as producing a codeword over a non-binary alphabet. The second code can then transform the result of the first encoding over a non-binary alpha |
https://en.wikipedia.org/wiki/Tam%C3%A1s%20B%C3%A1l%C3%B3 | Tamás Báló (born 12 January 1984) is a Hungarian football player who plays for Dunaújváros.
Honours
Paksi SE
Hungarian Second Division: Winner 2006
Club statistics
Updated to games played as of 11 March 2020.
References
Paksi FC Official Website
HLSZ
1984 births
Living people
People from Kalocsa
Hungarian men's footballers
Men's association football defenders
Dunaújváros FC players
Paksi FC players
Dunaújváros PASE players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Footballers from Bács-Kiskun County |
https://en.wikipedia.org/wiki/Zsolt%20G%C3%A9vay | Zsolt Gévay (born 19 November 1987) is a Hungarian professional footballer who plays for Paks, as a defender.
Club statistics
Updated to games played as of 15 May 2021.
References
Paksi FC Official Website
HLSZ
1984 births
Living people
Sportspeople from Dunaújváros
Footballers from Fejér County
Hungarian men's footballers
Men's association football defenders
Fehérvár FC players
Makói FC footballers
Paksi FC players
Gyirmót FC Győr players
Mezőkövesdi SE footballers
Nemzeti Bajnokság I players |
https://en.wikipedia.org/wiki/Localization%20theorem | In mathematics, particularly in integral calculus, the localization theorem allows, under certain conditions, to infer the nullity of a function given only information about its continuity and the value of its integral.
Let be a real-valued function defined on some open interval Ω of the real line that is continuous in Ω. Let D be an arbitrary subinterval contained in Ω. The theorem states the following implication:
A simple proof is as follows: if there were a point x0 within Ω for which , then the continuity of would require the existence of a neighborhood of x0 in which the value of was nonzero, and in particular of the same sign than in x0. Since such a neighborhood N, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of over N would evaluate to a nonzero value. However, since x0 is part of the open set Ω, all neighborhoods of x0 smaller than the distance of x0 to the frontier of Ω are included within it, and so the integral of over them must evaluate to zero. Having reached the contradiction that must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no x0 in Ω for which .
The theorem is easily generalized to multivariate functions, replacing intervals with the more general concept of connected open sets, that is, domains, and the original function with some , with the constraints of continuity and nullity of its integral over any subdomain . The proof is completely analogous to the single variable case, and concludes with the impossibility of finding a point such that .
Example
An example of the use of this theorem in physics is the law of conservation of mass for fluids, which states that the mass of any fluid volume must not change:
Applying the Reynolds transport theorem, one can change the reference to an arbitrary (non-fluid) control volume Vc. Further assuming that the density function is continuous (i.e. that our fluid is monophasic and thermodynamically metastable) and that Vc is not moving relative to the chosen system of reference, the equation becomes:
As the equation holds for any such control volume, the localization theorem applies, rendering the common partial differential equation for the conservation of mass in monophase fluids:
Integral calculus |
https://en.wikipedia.org/wiki/Grassland%2C%20Alberta | Grassland is a hamlet in northern Alberta, Canada within Athabasca County. It is on Highway 63, northeast of Edmonton.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Grassland had a population of 46 living in 20 of its 42 total private dwellings, a change of from its 2016 population of 68. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Grassland had a population of 68 living in 26 of its 52 total private dwellings, a change of from its 2011 population of 94. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Athabasca County
Hamlets in Alberta
Designated places in Alberta |
https://en.wikipedia.org/wiki/Conjugate%20diameters | In geometry, two diameters of a conic section are said to be conjugate if each chord parallel to one diameter is bisected by the other diameter. For example, two diameters of a circle are conjugate if and only if they are perpendicular.
Of ellipse
For an ellipse, two diameters are conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram (skewed compared to a bounding rectangle). In his manuscript De motu corporum in gyrum, and in the 'Principia', Isaac Newton cites as a lemma proved by previous authors that all (bounding) parallelograms for a given ellipse have the same area.
It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any bounding parallelogram. For example, in proposition 14 of Book VIII of his Collection, Pappus of Alexandria gives a method for constructing the axes of an ellipse from a given pair of conjugate diameters. Another method is using Rytz's construction, which takes advantage of the Thales' theorem for finding the directions and lengths of the major and minor axes of an ellipse regardless of its rotation or shearing.
Of hyperbola
Similar to the elliptic case, diameters of a hyperbola are conjugate when each bisects all chords parallel to the other. In this case both the hyperbola and its conjugate are sources for the chords and diameters.
In the case of a rectangular hyperbola, its conjugate is the reflection across an asymptote. A diameter of one hyperbola is conjugate to its reflection in the asymptote, which is a diameter of the other hyperbola. As perpendicularity is the relation of conjugate diameters of a circle, so hyperbolic orthogonality is the relation of conjugate diameters of rectangular hyperbolas.
The placement of tie rods reinforcing a square assembly of girders is guided by the relation of conjugate diameters in a book on analytic geometry.
Conjugate diameters of hyperbolas are also useful for stating the principle of relativity in the modern physics of spacetime. The concept of relativity is first introduced in a plane consisting of a single dimension in space, the second dimension being time. In such a plane, one hyperbola corresponds to events a constant space-like interval from the origin event, the other hyperbola corresponds to events a constant time-like interval from it. The principle of relativity can be formulated "Any pair of conjugate diameters of conjugate hyperbolas can be taken for the axes of space and time". This interpretation of relativity was enunciated by E. T. Whittaker in 1910.
In projective geometry
Every line in projective geometry contains a point at infinity, also called a figurative point. The ellipse, parabola, and hyperbola are viewed as conics in projective geometry, and each conic determines a relation of pole and polar between points a |
https://en.wikipedia.org/wiki/Mohamed%20Fawzi%20%28footballer%29 | Mohammed Fawzi Johar Farag Abdulla (; born 22 February 1990) is an Emirati association football player who plays as a right back. He appeared at the 2012 Summer Olympics.
Career statistics
Club
1Continental competitions include the AFC Champions League
2Other tournaments include the UAE President Cup and Etisalat Emirates Cup
National team
As of 27 September 2009
Honours
United Arab Emirates
Gulf Cup of Nations: 2013
AFC Asian Cup third-place: 2015
AFC U-19 Championship: 2008
References
External links
1990 births
Living people
Emirati men's footballers
Ittihad Kalba FC players
Shabab Al Ahli Club players
Baniyas Club players
Al Ain FC players
Al Jazira Club players
Al-Nasr SC (Dubai) players
Footballers at the 2012 Summer Olympics
Olympic footballers for the United Arab Emirates
UAE First Division League players
UAE Pro League players
Asian Games medalists in football
Footballers at the 2010 Asian Games
Asian Games silver medalists for the United Arab Emirates
Men's association football defenders
Medalists at the 2010 Asian Games
United Arab Emirates men's youth international footballers |
https://en.wikipedia.org/wiki/Supergeometry | Supergeometry is differential geometry of modules over graded commutative algebras, supermanifolds and graded manifolds. Supergeometry is part and parcel of many classical and quantum field theories involving odd fields, e.g., SUSY field theory, BRST theory, or supergravity.
Supergeometry is formulated in terms of -graded modules and sheaves over -graded commutative algebras (supercommutative algebras). In particular, superconnections are defined as Koszul connections on these modules and sheaves. However, supergeometry is not particular noncommutative geometry because of a different definition of a graded derivation.
Graded manifolds and supermanifolds also are phrased in terms of sheaves of graded commutative algebras. Graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces. There are different types of supermanifolds. These are smooth supermanifolds (-, -, -supermanifolds), -supermanifolds, and DeWitt supermanifolds. In particular, supervector bundles and principal superbundles are considered in the category of -supermanifolds. Definitions of principal superbundles and principal superconnections straightforwardly follow that of smooth principal bundles and principal connections. Principal graded bundles also are considered in the category of graded manifolds.
There is a different class of Quillen–Ne'eman superbundles and superconnections. These superconnections have been applied to computing the Chern character in K-theory, noncommutative geometry, and BRST formalism.
See also
Supermanifold
Graded manifold
Supersymmetry
Connection (algebraic framework)
Supermetric
References
.
.
.
External links
G. Sardanashvily, Lectures on supergeometry, .
Supersymmetry |
https://en.wikipedia.org/wiki/Schoonschip | Schoonschip was one of the first computer algebra systems, developed in 1963 by Martinus J. G. Veltman, for use in particle physics.
"Schoonschip" refers to the Dutch expression "schoon schip maken": to make a clean sweep, to clean/clear things up (literally: to make the ship clean). The name was chosen "among others to annoy everybody, who could not speak Dutch".
Veltman initially developed the program to compute the quadrupole moment of the W boson, the computation of which involved "a monstrous expression involving in the order of 50,000 terms in intermediate stages"
The initial version, dating to December 1963, ran on an IBM 7094 mainframe. In 1966 it was ported to the CDC 6600 mainframe, and later to most of the rest of Control Data's CDC line. In 1983 it was ported to the Motorola 68000 microprocessor, allowing its use on a number of 68000-based systems running variants of Unix.
FORM can be regarded, in a sense, as the successor to Schoonschip.
Contacts with Veltman about Schoonschip have been important for Stephen Wolfram in building Mathematica.
See also
Comparison of computer algebra systems
References
External links
Documentation
Schoonschip program files, documentation, and examples
Further reading
Close, Frank (2011) The Infinity Puzzle. Oxford University Press. Describes the historical context of and rationale for 'Schoonschip' (Chapter 11: "And Now I Introduce Mr 't Hooft")
Computer algebra systems
Computer science in the Netherlands
Information technology in the Netherlands |
https://en.wikipedia.org/wiki/Census%20of%20Floridablanca | The census of Floridablanca is considered the first Spanish census of population prepared on modern statistics techniques. It was a census document produced in Spain under the direction of the count of Floridablanca, minister of Charles III, between 1785 and 1787.
References
Censuses in Spain |
https://en.wikipedia.org/wiki/Cuboid%20%28disambiguation%29 | Cuboid may refer to:
Cuboid, in geometry a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube
Cuboid (computer vision), a feature used for behavior recognition in video
Cuboid (video game), a puzzle game for the PlayStation Network
Cuboid bone, one of seven tarsal bones in the human foot
Cuboid syndrome, a medical condition of the human foot |
https://en.wikipedia.org/wiki/Max%20Sabbatani | Massimiliano Sabbatani (born 4 August 1975 in Forlì, Italy) is a former Grand Prix motorcycle road racer. 125 cc European Champion in 1998.
Career statistics
Grand Prix motorcycle racing
Races by year
(key) (Races in bold indicate pole position, races in italics indicate fastest lap)
External links
Max Sabbatani – Motogp.com
1975 births
Living people
Italian motorcycle racers
125cc World Championship riders
250cc World Championship riders
Sportspeople from Forlì |
https://en.wikipedia.org/wiki/Susan%20J.%20M.%20Bauman | Susan J. M. Bauman is an attorney and former politician. She was the first woman to be elected Mayor of Madison, Wisconsin. Bauman worked as an 8th grade mathematics teacher in the Madison Public School system for eight years, and became President of the teachers' union, Madison Teachers, Incorporated (MTI). Bauman, along with MTI Executive Director John Matthews, led a two-week strike commencing January 5, 1976. Two years later, Bauman left teaching and pursued a Juris Doctor degree, graduating from the University of Wisconsin Law School in 1981. After serving for twelve years as an Alderperson on the Madison City Council, Bauman was elected Mayor on April 1, 1997 to fill the unexpired term of Paul Soglin, who resigned to run for Congress. Two years later, on April 6, 1999, Bauman was elected to a full four-year term, having defeated Eugene Parks. In 2003, Bauman sought re-election, but failed to place among the top two candidates in the primary election, and was therefore not on the ballot for the general election. She was succeeded by Dave Cieslewicz.
On April 18, 2003, shortly after her electoral defeat, Wisconsin Governor Jim Doyle announced the appointment of Bauman to serve as a Commissioner on the Wisconsin Employment Relations Commission. Bauman served from June 1, 2003 until May 20, 2011.
References
External links
Living people
Mayors of Madison, Wisconsin
Women mayors of places in Wisconsin
Wisconsin city council members
21st-century American women politicians
21st-century American politicians
University of Wisconsin Law School alumni
Women city councillors in Wisconsin
Year of birth missing (living people)
Schoolteachers from Wisconsin
American women educators
20th-century American women politicians
20th-century American politicians |
https://en.wikipedia.org/wiki/Joussard | Joussard is a hamlet in northern Alberta within Big Lakes County. It is north of Highway 2, approximately west of Slave Lake.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Joussard had a population of 334 living in 162 of its 232 total private dwellings, a change of from its 2016 population of 257. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Joussard had a population of 223 living in 100 of its 175 total private dwellings, a change of from its 2011 population of 181. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Hamlets in Alberta
Designated places in Alberta
Big Lakes County |
https://en.wikipedia.org/wiki/Meanook | Meanook is a hamlet in northern Alberta, Canada within Athabasca County. It is east of Highway 2, north of Edmonton.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Meanook had a population of 35 living in 12 of its 14 total private dwellings, a change of from its 2016 population of 30. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Meanook had a population of 30 living in 13 of its 15 total private dwellings, a change of from its 2011 population of 25. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
Meanook Magnetic Observatory
References
Athabasca County
Hamlets in Alberta
Designated places in Alberta |
https://en.wikipedia.org/wiki/Generalized%20gamma%20distribution | The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many distributions commonly used for parametric models in survival analysis (such as the exponential distribution, the Weibull distribution and the gamma distribution) are special cases of the generalized gamma, it is sometimes used to determine which parametric model is appropriate for a given set of data. Another example is the half-normal distribution.
Characteristics
The generalized gamma distribution has two shape parameters, and , and a scale parameter, . For non-negative x from a generalized gamma distribution, the probability density function is
where denotes the gamma function.
The cumulative distribution function is
where denotes the lower incomplete gamma function,
and denotes the regularized lower incomplete gamma function.
The quantile function can be found by noting that where is the cumulative distribution function of the gamma distribution with parameters and . The quantile function is then given by inverting using known relations about inverse of composite functions, yielding:
with being the quantile function for a gamma distribution with .
Related distributions
If then the generalized gamma distribution becomes the Weibull distribution.
If the generalised gamma becomes the gamma distribution.
If then it becomes the exponential distribution.
If and then it becomes the Nakagami distribution.
If and then it becomes a half-normal distribution.
Alternative parameterisations of this distribution are sometimes used; for example with the substitution α = d/p. In addition, a shift parameter can be added, so the domain of x starts at some value other than zero. If the restrictions on the signs of a, d and p are also lifted (but α = d/p remains positive), this gives a distribution called the Amoroso distribution, after the Italian mathematician and economist Luigi Amoroso who described it in 1925.
Moments
If X has a generalized gamma distribution as above, then
Properties
Denote GG(a,d,p) as the generalized gamma distribution of parameters a, d, p.
Then, given and two positive real numbers, if , then
and
.
Kullback-Leibler divergence
If and are the probability density functions of two generalized gamma distributions, then their Kullback-Leibler divergence is given by
where is the digamma function.
Software implementation
In the R programming language, there are a few packages that include functions for fitting and generating generalized gamma distributions. The gamlss package in R allows for fitting and generating many different distribution families including generalized gamma (family=GG). Other options in R, implemented in the package flexsurv, include the function dgengamma, with parameterization: , , , and in the package ggamma with parametrisation: , |
https://en.wikipedia.org/wiki/John%20P.%20Burgess | John Patton Burgess (born 5 June 1948) is an American philosopher. He is John N. Woodhull Professor of Philosophy at Princeton University where he specializes in logic and philosophy of mathematics.
Education and career
Burgess received his Ph.D. from the University of California, Berkeley's Group in Logic and Methodology of Science. His interests include logic, philosophy of mathematics and metaethics. He is the author of numerous articles on logic and philosophy of mathematics. In 2012, he was elected a Fellow of the American Academy of Arts and Sciences. He is the brother of Barbara Burgess.
Selected publications
1997. A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics (with Gideon Rosen), Oxford University Press.
2005. Fixing Frege, Princeton University Press.
2007. Computability and Logic (with George Boolos and Richard C. Jeffrey), Cambridge University Press.
2008. Mathematics, Models, and Modality: Selected Philosophical Essays, Cambridge University Press.
2009. Philosophical Logic, Princeton University Press.
2011. Truth (with Alexis Burgess), Princeton University Press.
2013. Saul Kripke: Puzzles and Mysteries .
2015. Rigor and Structure, Oxford University Press.
References
External links
Home page
John Burgess Video "The Necessity of Origin and the Origin of Necessity", Second Annual Saul Kripke Lecture, The CUNY Graduate Center, November 13th, 2012
American logicians
Princeton University faculty
Living people
Fellows of the American Academy of Arts and Sciences
1948 births |
https://en.wikipedia.org/wiki/Perryvale | Perryvale is a hamlet in northern Alberta, Canada within Athabasca County It is east of Highway 2, north of Edmonton.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Perryvale had a population of 10 living in 5 of its 6 total private dwellings, a change of from its 2016 population of 20. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Perryvale had a population of 20 living in 6 of its 6 total private dwellings, a change of from its 2011 population of 10. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Athabasca County
Hamlets in Alberta
Designated places in Alberta |
https://en.wikipedia.org/wiki/Castelnuovo%E2%80%93Mumford%20regularity | In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space is the smallest integer r such that it is r-regular, meaning that
whenever . The regularity of a subscheme is defined to be the regularity of its sheaf of ideals. The regularity controls when the Hilbert function of the sheaf becomes a polynomial; more precisely dim is a polynomial in m when m is at least the regularity. The concept of r-regularity was introduced by , who attributed the following results to :
An r-regular sheaf is s-regular for any .
If a coherent sheaf is r-regular then is generated by its global sections.
Graded modules
A related idea exists in commutative algebra. Suppose is a polynomial ring over a field k and M is a finitely generated graded R-module. Suppose M has a minimal graded free resolution
and let be the maximum of the degrees of the generators of . If r is an integer such that for all j, then M is said to be r-regular. The regularity of M is the smallest such r.
These two notions of regularity coincide when F is a coherent sheaf such that contains no closed points. Then the graded module
is finitely generated and has the same regularity as F.
See also
Hilbert scheme
Quot scheme
References
Algebraic geometry |
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