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https://en.wikipedia.org/wiki/Chikola%20%28Dodoma%20Rural%20ward%29 | Chikola is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,855 people in the ward, from 13,668 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Chinugulu | Chinugulu is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 5,669 people in the ward, from 5,216 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Chipanga | Chipanga is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,492 people in the ward, from 9,654 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Fufu%20%28Tanzanian%20ward%29 | Fufu (Tanzanian ward) is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 3,770 people in the ward, from 3,469 in 2012.
See also
Dodoma Region
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Ordered%20semigroup | In mathematics, an ordered semigroup is a semigroup (S,•) together with a partial order ≤ that is compatible with the semigroup operation, meaning that x ≤ y implies z•x ≤ z•y and x•z ≤ y•z for all x, y, z in S.
An ordered monoid and an ordered group are, respectively, a monoid or a group that are endowed with a partial order that makes them ordered semigroups. The terms posemigroup, pogroup and pomonoid are sometimes used, where "po" is an abbreviation for "partially ordered".
The positive integers, the nonnegative integers and the integers form respectively a posemigroup, a pomonoid, and a pogroup under addition and the natural ordering.
Every semigroup can be considered as a posemigroup endowed with the trivial (discrete) partial order "=".
A morphism or homomorphism of posemigroups is a semigroup homomorphism that preserves the order (equivalently, that is monotonically increasing).
Category-theoretic interpretation
A pomonoid can be considered as a monoidal category that is both skeletal and thin, with an object of for each element of , a unique morphism from to if and only if , the tensor product being given by , and the unit by .
References
T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, , chap. 11.
Ordered algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/Ibugule | Ibugule is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,745 people in the ward, from 8,046 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Majeleko | Majeleko is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,484 people in the ward, from 6,886 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Makanda%20%28Bahi%20District%29 | Makanda is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,582 people in the ward, from 7,896 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Makang%27wa | Makang'wa is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,937 people in the ward, from 20,337 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Manchali%20%28Tanzanian%20ward%29 | Manchali is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,395 people in the ward, from 10,485 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Manda%20%28Tanzanian%20ward%29 | Manda is an administrative ward in Ludewa District in the Njombe Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,443 people in the ward, from 4,304 in 2012.
Manda is a small town along Lake Nyasa (Nyasa means Lake), in Ludewa District. The small town of Manda was known as Wiedhafen during the time of German East Africa. Germans promoted commerce and economic growth. Lake Nyasa in Tanzania also known as Lake Malawi in Malawi, and Lago Niassa in Mozambique is one of the truly wonderful places in the world.
The lake itself is the 5th largest in the world by volume, the second largest in Tanzania and Africa, and has been in existence for millions of years (estimates range from about 8mya). In just one lake it contains over a thousand species of fishes. Lake Nyasa is between long, and about wide at its widest point. The lake is at its deepest point, located in a major depression in the north-central part. The largest river flowing into it is the Ruhuhu River in Manda, and there is an outlet at its southern end, the Shire River, a tributary that flows into the very large Zambezi River in Mozambique.
The majority of the species are in the family cichlinae, popular with aquarists, but there are also lungfish, elephant-noses, mastacemblid eels and catfish. They are often very colourful and have interesting behaviours - making for fascinating fish-watching. The great diversity in the lake means that most species have unique adaptations and occupy specialised ecological roles in the lake. Some scrape algae from the rocks, some filter sand, some eat snails, some live in empty snail shells, some eat plants, some eat fish, some eat scales, some mimic dead fish to allow them to catch unwary prey. Most are maternal mouth breeders, which mean that the females carry the eggs and young (!) around in their mouths. All of this in a lake that offers warm, clear water, with white sands and clean hot rocks to sunbathe on. It is surrounded by vast plains and the impressive Livingstone Mountains. Manda has experience a ton of historical legacies, ranging from being the host of naval battle between the British and the German.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Manzase | Manzase is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,482 people in the ward, from 11,485 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Membe%2C%20Tanzania | Membe is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,484 people in the ward.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mlowa%20Bwawani | Mlowa Bwawani is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,815 people in the ward, from 9,031 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mpalanga | Mpalanga is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,116 people in the ward, from 10,228 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mpamantwa | Mpamantwa is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,111 people in the ward, from 12,984 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mpwayungu | Mpwayungu is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,731 people in the ward, from 12,634 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Msanga | Msanga is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,502 people in the ward, from 9,663 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Msisi%20%28Bahi%29 | Msisi is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,876 people in the ward, from 11,847 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mtitaa | Mtitaa is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,351 people in the ward, from 8,604 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Muungano | Muungano is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,678 people in the ward, from 10,745 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mvumi%20Mission | Mvumi Mission, also Mvumi Misheni in kiswahili, is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 17,948 people in the ward, from 16,514 in 2012.
References
External links
Mvumi Mission
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mwitikira | Mwitikira is an administrative ward in the Bahi District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,863 people in the ward, from 7,235 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Nghambaku | Nghambaku is an administrative ward in the Chamwino District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 7,976 people in the ward, from 7,339 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/WISE%20Campaign | The WISE Campaign (Women into Science and Engineering) is a United Kingdom-based organization that encourages women and girls to value and pursue science, technology, engineering and maths-related courses in school or college and to move on into related careers and progress. Its mission statement aims to facilitate understanding of these disciplines among women and girls and the opportunities which they present at a professional level. It is operated by UKRC trading as WISE (company number 07533934).
Formation
The campaign began on 17 January 1984, headed by The Baroness Platt of Writtle, a qualified mechanical engineer, at which time women made up 7% of graduate engineers and 3% of professional engineers in the UK. It was a collaboration between the Engineering Council and the Equal Opportunities Commission, originally viewed as a one-year campaign "Women into Science and Engineering" WISE'84.
Activities
One of WISE's main objectives is to listen to students and women qualified or working in these sectors, and understand and voice their opinions to academic institutions, policy-makers and employers. It then works creatively with delivery agencies and others, offering models, tools and approaches to support them in challenging traditional approaches, so as to demonstrate equitable involvement. WISE combats gender stereotypes to get more girls and women involved in careers where female participation was once considered near impossible.
WISE operates throughout the UK, with specialist committees in Wales, Northern Ireland and Scotland. Volunteers, from industry and relevant organisations, attend the various WISE committee meetings, and undertake projects with WISE.
In 2011 the UKRC - an organisation specialising in gender equality in science, engineering and technology - became part of WISE. Trudy Norris-Grey, the Chair of UKRC since 2007 then became Chair of WISE. WISE counts The Princess Royal, Dame Julia Higgins, Kate Bellingham and Joanna Kennedy as its patrons. The Founding Chair and Patron The Baroness Platt of Writtle died on 1 February 2015, aged 91.
Young Professionals' Board
The WISE Campaign has an advisory Board to the main Board called the WISE Young Professionals' Board, formerly the WISE Young Women's Board, with a mandate to act as a sounding board to the WISE Campaign and promote the visibility of young women in STEM.
Notable members and former members include:
Jess Wade was a Young Professionals' Board member 2015-2018 and is a campaigner for Women in STEM and promoting early career researchers.
Structure
It is headquartered at Leeds College of Building, though has been based at the UKRC (UK Resource Centre for Women in Science, Engineering, and Technology) in Bradford.
References
External links
of WISE Campaign
UK Parliament Business, Innovation and Skills Committee: Written evidence submitted by the Women into Science and Engineering (WISE) Campaign 11 Oct 2012
Ingenia March 2010, issue 42 pp 48–50 "Engaging |
https://en.wikipedia.org/wiki/Mahendranagar%2C%20Dhanusha | {
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}Mahendranagar is a town in Chhireshwarnath Municipality of Dhanusa District in the Janakpur Zone of south-eastern Nepal. The formerly Village Development Committee was converted into municipality merging along with existing VDCs Ramdaiya, Sakhuwa Mahendranagar, Hariharpur and Digambarpur on 18 May 2014. At the time of the 1991 Nepal census it had a population of 10,209 persons living in 1916 individual households. Mahendranagar acts as bridge between Dhalkebar and Janakpur. Basically it is popular for its largest cattle market in the Nepal. It is assumed that 65% of cattle for e.g. buffalo, goats in Kathmandu valley are brought from here. The town is named after late king Mahendra.
Banks and financial institutions
Mahendranagar has branches of commercial banks and Co-operative Organization.
Kumari Bank Limited
Rastriya Banijya Bank Limited
Sahayogi Bikash Bank Limited
Machhapuchchhre Bank
Century Bank Limited
Prabhu Bank
NIC Asia Bank
Many Co-operative organization are based in Mahendranagar. Mahalakshi, Laxman rekha, Khusiyali, jaiSi baba, Upyogi Co-operative organization provide Loan and Saving Facilities. They concentrate loan to poor people without any deposit.
Education
Different private schools are opened here for quality education. Most of them are residential English and Nepali medium schools. D R S K Secondary English Boarding, Saraswati English Boarding School. Gangotri Secondary English Boarding School, New Parijat Educational Academy, Gyan Mandir Boarding, D.R.S.K Academy and Pragati Shishu Sadan are also famous school of Chhireswor Municipality. Chhireswor Janta Higher Secondary School is the oldest Educational institute of Chhireshwor Municipality. Here is also Government School, which provides free education facilities. This school also Provide Dress to all the Students.
There are two Colleges in Chhireswornath Municipality. Chhireswor Janta Bahumukhi Campus is the oldest campus of chhireswarnath Municipality. Gangotri Campus is another Campus at Chhireswornath Municipality which is recently commenced.
Agrochemical Suppliers
Durga Beej Bhandar
Durga Beej Bhandar (Nepali: दुर्गा बीज भण्डार) is a well-known agrochemical supplier in Nepal since 1993. It is retail as well as wholesale. It provides insecticides, fungicides, herbicides, pesticides, and many other chemicals that are used for the treatment of crops. Seeds are the most essential component of agriculture. It also provides seasonal as well as hybrid seeds.
The primary business purpose of Durga Beej Bhandar is bringing quality products from India, China & from other countries and distribute the equivalent products in different parts of Nepal through legitimate channels. Durga Beej Bhandar has created num |
https://en.wikipedia.org/wiki/Main%20Building%20%28Statistics%20Canada%29 | The Main Building, or Main Statistics Canada Building, is a four-storey federal government office building located in the Tunney's Pasture area of Ottawa, Ontario, Canada. It is connected on the north side by the Jean Talon Building and on the south side by the R. H. Coats Building. Most of its 1,700 occupants work for Statistics Canada. Other tenants include Health Canada, Public Works and Government Services Canada, and SNC-Lavalin.
The building was completed in 1952 to accommodate the growing staff of the Dominion Bureau of Statistics. At that time the Main Building was only one of two buildings in Tunney’s Pasture - the other being a heating plant.
In recent years, the building has experienced structural and environmental problems such as water leakage and power failures.
Notes
References
Government buildings completed in 1952
Office buildings completed in 1952
Federal government buildings in Ottawa
Modernist architecture in Canada |
https://en.wikipedia.org/wiki/Burau%20representation | In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau during the 1930s. The Burau representation has two common and near-equivalent formulations, the reduced and unreduced Burau representations.
Definition
Consider the braid group to be the mapping class group of a disc with marked points . The homology group is free abelian of rank . Moreover, the invariant subspace of (under the action of ) is primitive and infinite cyclic. Let be the projection onto this invariant subspace. Then there is a covering space corresponding to this projection map. Much like in the construction of the Alexander polynomial, consider as a module over the group-ring of covering transformations , which is isomorphic to the ring of Laurent polynomials . As a -module, is free of rank . By the basic theory of covering spaces, acts on , and this representation is called the reduced Burau representation.
The unreduced Burau representation has a similar definition, namely one replaces with its (real, oriented) blow-up at the marked points. Then instead of considering one considers the relative homology where is the part of the boundary of corresponding to the blow-up operation together with one point on the disc's boundary. denotes the lift of to . As a -module this is free of rank .
By the homology long exact sequence of a pair, the Burau representations fit into a short exact sequence
where (resp. ) is the reduced (resp. unreduced) Burau -module and is the complement to the diagonal subspace, in other words:
and acts on by the permutation representation.
Explicit matrices
Let denote the standard generators of the braid group . Then the unreduced Burau representation may be given explicitly by mapping
for , where denotes the identity matrix. Likewise, for the reduced Burau representation is given by
while for , it maps
Bowling alley interpretation
Vaughan Jones gave the following interpretation of the unreduced Burau representation of positive braids for in – i.e. for braids that are words in the standard braid group generators containing no inverses – which follows immediately from the above explicit description:
Given a positive braid on strands, interpret it as a bowling alley with intertwining lanes. Now throw a bowling ball down one of the lanes and assume that at every crossing where its path crosses over another lane, it falls down with probability and continues along the lower lane. Then the 'th entry of the unreduced Burau representation of is the probability that a ball thrown into the 'th lane ends up in the 'th lane.
Relation to the Alexander polynomial
If a knot is the closure of a braid in , then, up to multiplication by a unit in , the Alexander polynomial of is given by
where is the reduced Burau representation of the braid .
For example, if in , one finds by using the explicit matrices above th |
https://en.wikipedia.org/wiki/Uniform%20tree | In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group G=Aut(X) of the tree, which is a locally compact topological group, is unimodular and G\X is finite. Also equivalent is the existence of a uniform X-lattice in G.
Sources
Trees (graph theory) |
https://en.wikipedia.org/wiki/Shmuel%20Weinberger | The mathematician Shmuel Aaron Weinberger (born February 20, 1963) is an American topologist. He completed a PhD in mathematics in 1982 at New York University under the direction of Sylvain Cappell. Weinberger was, from 1994 to 1996, the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, and he is currently the Andrew MacLeish Professor of Mathematics and chair of the Mathematics department at the University of Chicago.
His research interests include geometric topology, differential geometry, geometric group theory, and, in recent years, applications of topology in other disciplines. He has written a book on topologically stratified spaces and a book on the application of mathematical logic to geometry.
He has given the Porter lectures at Rice University (2000), the Jankowski memorial lecture of the Polish Academy of Sciences (2000), the Zabrodsky Memorial lecture at Hebrew University (2001), the Cairns lectures at University of Illinois at Urbana-Champaign (2002), the Marker lectures in Mathematics at Penn State University (2003), the Lewis Lectures at Rutgers University (2004), the Blumenthal Lectures at Tel Aviv University (2005), the Hardy Lectures of the London Mathematical Society (2008), the William Benter Lecture at the City University of Hong Kong (2010), the Clifford Lectures at Tulane University (2012), the Minerva Lectures at Princeton University (2017) and the Abraham Robinson lecture at Yale (2019). In addition he has given invited lectures at the International Congress of Mathematicians in Zürich (1994), a mini-symposium at the European Congress of Mathematics (2008), the American Mathematical Society (1989), the Canadian Mathematical Society (2006), the Association for Symbolic Logic (2001), and Pembroke Pines Charter High School (2021).
In 2012, he was elected to the first class of AMS fellows. In 2013, he was elected as a fellow of AAAS.
Selected publications
Books
Research articles
References
External links
Shmuel Weinberger's homepage
20th-century American mathematicians
Topologists
University of Chicago faculty
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
Jewish American scientists
Fellows of the American Mathematical Society
1963 births
Living people
Courant Institute of Mathematical Sciences alumni
21st-century American mathematicians
21st-century American Jews |
https://en.wikipedia.org/wiki/RExcel | RExcel is an add-in for Microsoft Excel. It allows access to the statistics package R from within Excel.
The main features are:
data transfer (matrices and data frames) between R and Excel in both directions;
running R code from Excel ranges;
writing macros calling R to perform calculations;
calling R functions from cell formulas, using Excel's auto-update mechanism to trigger recalculation by R;
using Excel as a GUI for R.
RExcel works on Microsoft Windows (XP, Vista , or 7), with Excel 2003, 2007,
2010, and 2013.
It uses the statconnDCOM server and for certain configurations the rcom package to access R from within Excel.
The RExcelInstaller package was removed from CRAN due to FOSS license restrictions.
References
Bibliography
Baier T., Neuwirth E., De Meo M: Creating and Deploying an Application with (R)Excel and R R Journal 3/2, December 2011
Baier T. and Neuwirth. E. Excel :: COM :: R. Computational Statistics 22 (2007)
Heiberger R. and Neuwirth E.: R Through Excel, Springer Verlag 2009.
Neuwirth, E.: R meets the Workplace - Embedding R into Excel and making it more accessible. Paper presented at the UseR 2008, Dortmund
Narasimhan, B.: Disseminating Statistical Methodology and Results via R and Excel: Two Examples. Paper presented at the Interface 2007, Philadelphia
Baier, T., Heiberger, R., Neuwirth, E., Schinagl, K., & Grossmann, W.: Using R for teaching statistics to nonmajors: Comparing experiences of two different approaches. Paper presented at the UseR 2006, Vienna.
Konnert, A.: LabNetAnalysis - An instrument for the analysis of data from laboratory networks based on RExcel Paper presented at the UseR 2006, Vienna.
External links
RExcelInstaller at CRAN
RExcel's website has a master installer RandFriendsSetup which installs R, many R packages, RExcel, and the infrastructure needed to run RExcel (rscproxy, room, the statconnDCOM server)
R (programming language)
Microsoft Office-related statistical software |
https://en.wikipedia.org/wiki/List%20of%20Esteghlal%20F.C.%20records%20and%20statistics | Esteghlal Football Club (, Bashgah-e Futbal-e Esteqlâl), commonly known as Esteghlal (, meaning 'The Independence'), is an Iranian football club based in capital Tehran, that competes in the Persian Gulf Pro League. The club was founded in 1945 as Docharkheh Savaran (; meaning 'The Cyclists') and previously known as Taj (; meaning 'The Crown') between 1949 and 1979.
This page details Esteghlal Football Club records and statistics.
Honours
Esteghlal is the most proud team of Iran with 38 official championship titles in provincial, national and continental cups.
Domestic
League
Iran League
Winners (9): 1970–71, 1974–75, 1989–90, 1997–98, 2000–01, 2005–06, 2008–09, 2012–13, 2021–22
Runners-up (10): 1973–74, 1991–92, 1994–95, 1998–99, 1999–2000, 2001–02, 2003–04, 2010–11, 2016–17, 2019–20
Cups
Hazfi Cup (record)
Winners (7): 1976–77, 1995–96, 1999–2000, 2001–02, 2007–08, 2011–12, 2017–18
Runners-up (6): 1989–90, 1998–99, 2003–04, 2015–16, 2019–20, 2020–21
Super Cup
Winners (1): 2022
Runners-up (1): 2018
Provincial (High Level)
Tehran League (record)
Winners (13): 1949–50, 1952–53, 1956–57, 1957–58, 1959–60, 1960–61, 1962–63, 1968–69, 1970–1971, 1972–73, 1983–84, 1985–86, 1991–92
Runners-up (7): 1946–47, 1951–52, 1958–59, 1969–70, 1982–83, 1989–90, 1990–91
Tehran Hazfi Cup
Winners (4): 1946–47, 1950–51, 1958–59, 1960–61
Runners-up (3): 1945–46, 1957–58, 1969–70
Tehran Super Cup (shared record)
Winners (1): 1994
Continental
AFC Champions League (Iran record)
Winners (2): 1970, 1990–91
Runners-up (2): 1991, 1998–99
Third place (3): 1971, 2001–02, 2013
Doubles and Treble
Esteghlal has achieved the Double on 5 occasions in its history:
Iran League and Tehran League
1957–58 Season
1970–71 Season
Tehran League and Tehran Hazfi Cup
1958–59 Season
1960–61 Season
AFC Champions League and Tehran League
1990–91 Season
Esteghlal has achieved the Treble on 1 occasions in its history:
AFC Champions League and Iran League and Tehran League
1970–71 Season
Minor Tournaments
International
DCM Trophy
Winners (4): 1969, 1970, 1971, 1989
Bordoloi Trophy
Winners (1): 1989
Qatar Independence Cup
Winners (1): 1991
Turkmenistan President's Cup
Winners (1): 1998
Caspian International Cup
Winners (1): 1998
Domestic
Taj Cup
Winner (1): 1958
Doosti Cup
Winners (1): 1972
Ettehad Cup
Winners (1): 1973
Basij Festival
Winner (1): 1992
Iran Third Division
Winner (1): 1993
Kish Quartet Competition Cup
Winners (1): 1998
Iranian Football League Cup
Winners (1): 2002
Solh va Doosti Cup
Winners (1): 2005
Rankings
The club is currently ranked 160 in the world by IFFHS.
Top Ten Asian's clubs of the 20th Century
Esteghlal was placed 3rd in IFFHS continental Clubs of the 20th Century:
Statistics
Official Matchs
Most goals scored in a match:
18 – 0 (1 time) (Iran record)
13 – 0 (1 time)
13 – 1 (1 time)
11 – 0 (2 time)
11 – 1 (1 time)
10 – 0 (4 time)
10 – 1 (1 time)
Player with a most goal in a single match:
Ali Jabbari wi |
https://en.wikipedia.org/wiki/Petar%20Trifonov | Petar Trifonov (: born 14 March 1984, in Pleven) is a Bulgarian footballer who last played as a defender for Svetkavitsa.
External links
2007-08 Statistics, 2006-07 Statistics & 2005-06 Statistics at PFL.bg
1984 births
Living people
Bulgarian men's footballers
First Professional Football League (Bulgaria) players
OFC Belasitsa Petrich players
FC Dunav Ruse players
PFK Svetkavitsa 1922 players
Men's association football defenders
Footballers from Pleven |
https://en.wikipedia.org/wiki/Ars%20Mathematica%20Contemporanea | Ars Mathematica Contemporanea is a quarterly peer-reviewed scientific journal covering discrete mathematics in connection with other branches of mathematics. It is published by the University of Primorska together with the Society of Mathematicians, Physicists and Astronomers of Slovenia, the Institute of Mathematics, Physics, and Mechanics, and the Slovenian Discrete and Applied Mathematics Society. It is a platinum open access journal, with articles published under the Creative Commons Attribution 4.0 license.
Abstracting and indexing
The journal is indexed by:
Current Contents/Physical, Chemical & Earth Sciences
Mathematical Reviews
Science Citation Index Expanded
Scopus
zbMATH
According to the Journal Citation Reports, the journal has a 2018 impact factor of 0.910.
See also
List of academic journals published in Slovenia
References
External links
Combinatorics journals
Open access journals
Academic journals established in 2008
English-language journals
Academic journals published in Slovenia
University of Primorska
Academic journals of Slovenia |
https://en.wikipedia.org/wiki/Maximum-minimums%20identity | In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2n − 1 non-empty subsets of S.
Let S = {x1, x2, ..., xn}. The identity states that
or conversely
For a probabilistic proof, see the reference.
See also
Inclusion–exclusion principle
References
Mathematical identities |
https://en.wikipedia.org/wiki/Center%20for%20Women%20in%20Mathematics | The Center for Women in Mathematics, a part of the Smith College Department of Mathematics and Statistics, is an American educational program founded in 2007 to increase the involvement of women in mathematics. The Center aims for students to engage in coursework and research in a mathematical environment that actively supports women.
Junior Program
The Junior Program is designed for undergraduate women who wish to spend a year or a semester studying mathematics at a women's college. Financial aid funding is provided by the National Science Foundation.
Post-Baccalaureate Program
The Post-Baccalaureate Program is geared towards women with bachelor's degrees who didn't major in mathematics as undergraduates or whose major was light. The post-baccalaureate program is funded through grants from Smith College and the National Science Foundation and students receive tuition waivers and living stipends.
Students of both programs are able to take classes not only at Smith College, but also at any other of the Five Colleges - Amherst, Mt. Holyoke and Hampshire Colleges and UMass Amherst, the last of which also offers graduate-level courses.
WIMIN Conference
Each year the Center hosts the Women in Mathematics in New England (WIMIN) Conference. The conference features two plenary lectures given by prominent female mathematicians: the Dorothy Wrinch Lecture in Biomathematics, and the Alice Dickinson Lecture in Mathematics. It also features short talks by undergraduate and graduate students (of any gender), and a panel intended for students considering graduate studies.
Past Plenary Speakers
Notes
Sources
Center for Women in Mathematics Brochure
External links
Center for Women in Mathematics at Smith College
Women in Mathematics in New England
Smith College website
Smith College
Women and education
United States educational programs
Women in mathematics |
https://en.wikipedia.org/wiki/Total%20curvature | In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
The total curvature of a closed curve is always an integer multiple of 2, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point. This map is similar to the Gauss map for surfaces.
Comparison to surfaces
This relationship between a local geometric invariant, the curvature, and a global topological invariant, the index, is characteristic of results in higher-dimensional Riemannian geometry such as the Gauss–Bonnet theorem.
Invariance
According to the Whitney–Graustein theorem, the total curvature is invariant under a regular homotopy of a curve: it is the degree of the Gauss map. However, it is not invariant under homotopy: passing through a kink (cusp) changes the turning number by 1.
By contrast, winding number about a point is invariant under homotopies that do not pass through the point, and changes by 1 if one passes through the point.
Generalizations
A finite generalization is that the exterior angles of a triangle, or more generally any simple polygon, add up to 360° = 2 radians, corresponding to a turning number of 1. More generally, polygonal chains that do not go back on themselves (no 180° angles) have well-defined total curvature, interpreting the curvature as point masses at the angles.
The total absolute curvature of a curve is defined in almost the same way as the total curvature, but using the absolute value of the curvature instead of the signed curvature.
It is 2 for convex curves in the plane, and larger for non-convex curves. It can also be generalized to curves in higher dimensional spaces by flattening out the tangent developable to into a plane, and computing the total curvature of the resulting curve. That is, the total curvature of a curve in -dimensional space is
where is last Frenet curvature (the torsion of the curve) and is the signum function.
The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2 for the unknot, but by the Fáry–Milnor theorem it is at least 4 for any other knot.
References
Further reading
(translated by Bruce Hunt)
Curves
Curvature (mathematics) |
https://en.wikipedia.org/wiki/Criticism%20of%20nonstandard%20analysis | Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Introduction
The evaluation of nonstandard analysis in the literature has varied greatly. Paul Halmos described it as a technical special development in mathematical logic. Terence Tao summed up the advantage of the hyperreal framework by noting that it
The nature of the criticisms is not directly related to the logical status of the results proved using nonstandard analysis. In terms of conventional mathematical foundations in classical logic, such results are quite acceptable although usually strongly dependent on choice. Abraham Robinson's nonstandard analysis does not need any axioms beyond Zermelo–Fraenkel set theory (ZFC) (as shown explicitly by Wilhelmus Luxemburg's ultrapower construction of the hyperreals), while its variant by Edward Nelson, known as internal set theory, is similarly a conservative extension of ZFC. It provides an assurance that the newness of nonstandard analysis is entirely as a strategy of proof, not in range of results. Further, model theoretic nonstandard analysis, for example based on superstructures, which is now a commonly used approach, does not need any new set-theoretic axioms beyond those of ZFC.
Controversy has existed on issues of mathematical pedagogy. Also nonstandard analysis as developed is not the only candidate to fulfill the aims of a theory of infinitesimals (see Smooth infinitesimal analysis). Philip J. Davis wrote, in a book review of Left Back: A Century of Failed School Reforms by Diane Ravitch:
There was the nonstandard analysis movement for teaching elementary calculus. Its stock rose a bit before the movement collapsed from inner complexity and scant necessity.
Nonstandard calculus in the classroom has been analysed in the study by K. Sullivan of schools in the Chicago area, as reflected in secondary literature at Influence of nonstandard analysis. Sullivan showed that students following the nonstandard analysis course were better able to interpret the sense of the mathematical formalism of calculus than a control group following a standard syllabus. This was also noted by Artigue (1994), page 172; Chihara (2007); and Dauben (1988).
Bishop's criticism
In the view of Errett Bishop, classical mathematics, which includes Robinson's approach to nonstandard analysis, was nonconstructive and therefore deficient in numerical meaning . Bishop was particularly concerned about the use of nonstandard analysis in teaching as he discussed in his essay "Crisis in mathematics" . Specifically, after discussing Hilbert's formalist program he wrote:
A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather that it has met with some degree of success, whether at the expense of giving significantly less meaningful proofs I do not know. My interest in non-standard analysis i |
https://en.wikipedia.org/wiki/Dimensional%20operator | In mathematics, specifically set theory, a dimensional operator on a set E is a function from the subsets of E to the subsets of E.
Definition
If the power set of E is denoted P(E) then a dimensional operator on E is a map
that satisfies the following properties for S,T ∈ P(E):
S ⊆ d(S);
d(S) = d(d(S)) (d is idempotent);
if S ⊆ T then d(S) ⊆ d(T);
if Ω is the set of finite subsets of S then d(S) = ∪A∈Ωd(A);
if x ∈ E and y ∈ d(S ∪ {x}) \ d(S), then x ∈ d(S ∪ {y}).
The final property is known as the exchange axiom.
Examples
For any set E the identity map on P(E) is a dimensional operator.
The map which takes any subset S of E to E itself is a dimensional operator on E.
References
Set theory |
https://en.wikipedia.org/wiki/Descendant%20subgroup | In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predecessor.
The series may be infinite. If the series is finite, then the subgroup is subnormal.
See also
Ascendant subgroup
References
Subgroup properties |
https://en.wikipedia.org/wiki/Fathom%3A%20Dynamic%20Data%20Software | Fathom Dynamic Data Software is software for learning and teaching statistics, at the high school and introductory college level.
Reviews
Technology & Learning Award of Excellence
MacWorld 2005 Review
EHO Review
Statistical software |
https://en.wikipedia.org/wiki/Ernest%20Vinberg | Ernest Borisovich Vinberg (; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg's algorithm and the Koecher–Vinberg theorem.
He was a recipient of the 1997 Humboldt Prize. He was on the executive committee of the Moscow Mathematical Society. In 1983, he was an Invited Speaker with a talk on Discrete reflection groups in Lobachevsky spaces at the International Congress of Mathematicians in Warsaw. In 2010, he was elected an International Honorary Member of the American Academy of Arts and Sciences.
Ernest Vinberg died from pneumonia caused by COVID-19 on 12 May 2020.
Selected publications
editor and co-author: (contains Construction of the exceptional simple Lie algebras)
with A. L. Onishchik: 2012 pbk edition
with V. V. Gorbatsevich, A. L. Onishchik:
(ed.) (contains: Vinberg et alia: Geometry of spaces of constant curvature, Discrete groups of motions of spaces of constant curvature)
References
External links
Humboldt Research Award
Ernest Borisovich Vinberg, Moscow Mathematical Journal
1937 births
2020 deaths
20th-century Russian mathematicians
20th-century Russian non-fiction writers
21st-century Russian mathematicians
21st-century Russian non-fiction writers
Deaths from the COVID-19 pandemic in Russia
Fellows of the American Academy of Arts and Sciences
Group theorists
Moscow State University alumni
Academic staff of Moscow State University
Russian editors
Russian Jews
Russian male writers
Soviet Jews
Soviet mathematicians
Textbook writers
Mathematicians from Moscow |
https://en.wikipedia.org/wiki/Joseph%20Jean%20Baptiste%20Neuberg | Joseph Jean Baptiste Neuberg (30 October 1840 – 22 March 1926) was a Luxembourger mathematician who worked primarily in geometry.
Biography
Neuberg was born on 30 October 1840 in Luxembourg City, Luxembourg. He first studied at a local school, the Athénée de Luxembourg, then progressed to Ghent University, studying at the École normale des Sciences of the science faculty. After graduation, Neuberg taught at several institutions. Between 1862 and 1865, he taught at the École Normale de Nivelle. For the next sixteen years, he taught at the Athénée Royal d'Arlon, though he also taught at the École Normale at Bruges from 1868 onwards.
Neuberg switched from his previous two schools to the Athénée Royal de Liège in 1878. He became an extraordinary professor in the university in the same city in 1884, and was promoted to ordinary professor in 1887. He held this latter position until his retirement in 1910. A year after his retirement, he was elected president of the Belgian Royal Academy, which he had joined earlier, in 1866, after taking Belgian nationality despite his origins.
The professor died on 22 March 1926 in Liège, Belgium, and was commemorated in the Bulletin of the American Mathematical Society.
Contributions
Neuberg worked mainly in geometry, particularly the geometry of the triangle, The Neuberg cubic, a curve defined from a triangle, is named after him, and passes through the isodynamic points of a triangle which he discovered and published in 1885.
Neuberg was also involved in a number of mathematical journals. With Eugène Catalan and Paul Mansion, he founded the journal Nouvelle correspondance mathématique. This journal was founded to honour the earlier journal Correspondance mathématique et physique, which had been edited by Lambert Quetelet and Jean Garnier. Correspondance was published until 1880; after this, Catalan advised Mansion and Neuberg to continue publication of a new journal. They followed his advice, creating Mathesis in 1881, which is perhaps Neuberg's best-known journal.
Several mathematical societies included Neuberg: the Institute of Science of Luxembourg, the Royal Society of Science of Liège, Mathematical Society of Amsterdam, and the Belgian Royal Academy noted in the biography above.
References
External links
A generalization of Neuberg's theorem and the Simson-Wallace line at Dynamic Geometry Sketches, an interactive dynamic geometry sketch
Luxembourgian educators
Belgian mathematicians
19th-century mathematicians
Alumni of the Athénée de Luxembourg
1840 births
1926 deaths
People from Luxembourg City |
https://en.wikipedia.org/wiki/Horikawa%20surface | In mathematics, a Horikawa surface is one of the surfaces of general type introduced by Horikawa.
These are surfaces with q = 0 and pg = c12/2 + 2 or c12/2 + 3/2 (which implies that they are more or less on the Noether line edge of the region of possible values of the Chern numbers).
They are all simply connected, and Horikawa gave a detailed description of them.
References
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Jeremy%20Colman | Jeremy Colman (born April 1948) is a former Auditor General for Wales.
He was born in London and was educated at The John Lyon School, followed by Peterhouse, Cambridge, where he read Mathematics, and Imperial College, London, where he studied for an MSc DIC in Management Science.
His early career was in the civil service, including appointment as Private Secretary to successive holders of the post of Head of the Home Civil Service, and in the Treasury, where he played a leading role in the privatisation of British Airways and of the British Airports Authority. In 1988 he moved to the private sector, first as a Director of a major investment bank (County NatWest), and later as a partner in Price Waterhouse, based in Prague as Head of Corporate Finance.
He joined the National Audit Office (NAO) in 1993, where for 12 years he was responsible for Private Finance Initiatives and Public-private partnerships, before being appointed as the first head of the new Wales Audit Office on 1 April 2005 for an eight-year term of office. He lives in Dinas Powys.
Colman resigned from his post on 3 February 2010, after an internal investigation found child pornography on his work computer. South Wales Police began an investigation, and on 8 February arrested Colman on charges described as "possessing indecent images". Colman pleaded guilty to fifteen separate offences and, in November 2010, he was sentenced to eight months in jail.
References
Living people
Civil servants from London
Alumni of Peterhouse, Cambridge
Alumni of Imperial College London
Government of Wales
People educated at The John Lyon School
Prisoners and detainees of England and Wales
British auditors
Welsh people convicted of child pornography offences
1948 births |
https://en.wikipedia.org/wiki/Roger%20Wolcott%20Richardson | Roger Wolcott Richardson (30 May 1930 – 15 June 1993) was a mathematician noted for his work in representation theory and geometry. He was born in Baton Rouge, Louisiana, and educated at Louisiana State University, Harvard University and University of Michigan, Ann Arbor where he obtained a Ph.D. in 1958 under the supervision of Hans Samelson.
After a postdoc appointment at Princeton University, he accepted a faculty position at the University of Washington in Seattle. He emigrated to the United Kingdom in 1970, taking up a chair at Durham University. In 1978 he moved to the Australian National University in Canberra, where he stayed as faculty until his death.
Richardson's best known result states that if P is a parabolic subgroup of a reductive group, then P has a dense orbit on its nilradical, i.e., one whose closure is the whole space. This orbit is now universally known as the Richardson orbit.
Publications
See also
Prehomogeneous vector space
External links
Mathematical Reviews analysis
References
1930 births
1993 deaths
20th-century American mathematicians
Algebraists
Louisiana State University alumni
Harvard University alumni
University of Michigan alumni
Australian mathematicians
Fellows of the Australian Academy of Science
People from Baton Rouge, Louisiana
Mathematicians from Louisiana
University of Washington faculty
Princeton University people
Academics of Durham University
Academic staff of the Australian National University |
https://en.wikipedia.org/wiki/Mixing%20time | Mixing time may refer to:
Blend time, the time to achieve a predefined level of homogeneity of a flow tracer in a mixing vessel
Mixing (mathematics), an abstract concept originating from physics used to attempt to describe the irreversible thermodynamic process of mixing
Markov chain mixing time, the time to achieve a level of homogeneity in the probability distribution of a state in a Markov process |
https://en.wikipedia.org/wiki/Regular%20skew%20polyhedron | In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
Infinite regular skew polyhedra that span 3-space or higher are called regular skew apeirohedra.
History
According to Coxeter, in 1926 John Flinders Petrie generalized the concept of regular skew polygons (nonplanar polygons) to regular skew polyhedra.
Coxeter offered a modified Schläfli symbol for these figures, with implying the vertex figure, -gons around a vertex, and -gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes.
The regular skew polyhedra, represented by , follow this equation:
A first set , repeats the five convex Platonic solids, and one nonconvex Kepler–Poinsot solid:
{| class=wikitable
!
!Faces
!Edges
!Vertices
!
!Polyhedron
!Symmetryorder
|- BGCOLOR="#e0f0e0" align=center
| {3,3| 3} = {3,3} || 4||6||4 || 0|| Tetrahedron||12
|- BGCOLOR="#f0e0e0" align=center
| {3,4| 4} = {3,4} ||8||12||6 || 0|| Octahedron||24
|- BGCOLOR="#e0e0f0" align=center
| {4,3| 4} = {4,3} ||6||12||8 || 0|| Cube||24
|- BGCOLOR="#f0e0e0" align=center
| {3,5| 5} = {3,5} ||20||30||12 || 0||Icosahedron||60
|- BGCOLOR="#e0e0f0" align=center
| {5,3| 5} = {5,3} ||12||30||20 || 0|| Dodecahedron||60
|- BGCOLOR="#e0f0e0" align=center
| {5,5| 3} = {5,5/2} ||12||30||12 || 4|| Great dodecahedron||60
|}
Finite regular skew polyhedra
Coxeter also enumerated the a larger set of finite regular polyhedra in his paper "regular skew polyhedra in three and four dimensions, and their topological analogues".
Just like the infinite skew polyhedra represent manifold surfaces between the cells of the convex uniform honeycombs, the finite forms all represent manifold surfaces within the cells of the uniform 4-polytopes.
Polyhedra of the form {2p, 2q | r} are related to Coxeter group symmetry of [(p,r,q,r)], which reduces to the linear [r,p,r] when q is 2. Coxeter gives these symmetry as [[(p,r,q,r)]+] which he says is isomorphic to his abstract group (2p,2q|2,r). The related honeycomb has the extended symmetry [[(p,r,q,r)]].
{2p,4|r} is represented by the {2p} faces of the bitruncated {r,p,r} uniform 4-polytope, and {4,2p|r} is represented by square faces of the runcinated {r,p,r}.
{4,4|n} produces a n-n duoprism, and specifically {4,4|4} fits inside of a {4}x{4} tesseract.
A final set is based on Coxeter's further extended form {q1,m|q2,q3...} or with q2 unspecified: {l, m |, q}. These can also be represented a regular finite map or {l, m}2q, and group Gl,m,q.
Higher dimensions
Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew ic |
https://en.wikipedia.org/wiki/Mu%20Sigma%20Rho | Mu Sigma Rho () is the US national statistics honor society.
History
Founded in 1968 at Iowa State University, Mu Sigma Rho seeks to promote and encourage scholarly activity in statistics, and to recognize outstanding achievement among students and faculty thereof. Its activities include outreach and professional service.
The Society publishes an occasional newsletter, the Mu Sigma Rhover.
See also
Kappa Mu Epsilon, (mathematics)
Mu Alpha Theta, (mathematics, high school)
Pi Mu Epsilon, (mathematics)
External links
Mu Sigma Rho and the College Bowl
References
Student organizations established in 1968
Honor societies
Statistical organizations in the United States
1968 establishments in Iowa |
https://en.wikipedia.org/wiki/Ove%20Molin | Ove Molin (born May 27, 1971) is a retired Swedish professional ice hockey player who spent most of his career with Brynäs IF in the Swedish Elite League.
Career statistics
References
External links
1971 births
Swedish ice hockey right wingers
AIK IF players
Brynäs IF players
HIFK (ice hockey) players
Swedish expatriate ice hockey players in Finland
Living people |
https://en.wikipedia.org/wiki/Stallings%20theorem%20about%20ends%20of%20groups | In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group has more than one end if and only if the group admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group has more than one end if and only if admits a nontrivial (that is, without a global fixed point) action on a simplicial tree with finite edge-stabilizers and without edge-inversions.
The theorem was proved by John R. Stallings, first in the torsion-free case (1968) and then in the general case (1971).
Ends of graphs
Let be a connected graph where the degree of every vertex is finite. One can view as a topological space by giving it the natural structure of a one-dimensional cell complex. Then the ends of are the ends of this topological space. A more explicit definition of the number of ends of a graph is presented below for completeness.
Let be a non-negative integer. The graph is said to satisfy if for every finite collection of edges of the graph has at most infinite connected components. By definition, if and if for every the statement is false. Thus if is the smallest nonnegative integer such that . If there does not exist an integer such that , put . The number is called the number of ends of .
Informally, is the number of "connected components at infinity" of . If , then for any finite set of edges of there exists a finite set of edges of with such that has exactly infinite connected components. If , then for any finite set of edges of and for any integer there exists a finite set of edges of with such that has at least infinite connected components.
Ends of groups
Let be a finitely generated group. Let be a finite generating set of and let be the Cayley graph of with respect to . The number of ends of is defined as . A basic fact in the theory of ends of groups says that does not depend on the choice of a finite generating set of , so that is well-defined.
Basic facts and examples
For a finitely generated group we have if and only if is finite.
For the infinite cyclic group we have
For the free abelian group of rank two we have
For a free group where we have .
Freudenthal-Hopf theorems
Hans Freudenthal and independently Heinz Hopf established in the 1940s the following two facts:
For any finitely generated group we have .
For any finitely generated group we have if and only if is virtually infinite cyclic (that is, contains an infinite cyclic subgroup of finite index).
Charles T. C. Wall proved in 1967 the following complementary fact:
A group is virtually infinite cyclic if and only if it has a finite normal subgroup such that is either infinite cyclic or infinite dihedral.
Cuts and almost invariant sets
Let be a finitely generated group, be a finite generating set of and let be the Cayle |
https://en.wikipedia.org/wiki/Radek%20Philipp | Radek Philipp (born February 12, 1977) is a Czech professional ice hockey player with the HC Sparta Praha team in the Czech Extraliga.
Career statistics
References
External links
1977 births
Avtomobilist Yekaterinburg players
Czech ice hockey defencemen
Espoo Blues players
Czech expatriate ice hockey players in Russia
HC Havířov players
HC Sparta Praha players
HC Vítkovice players
Living people
Lahti Pelicans players
Ice hockey people from Ostrava
HC Košice players
Salavat Yulaev Ufa players
Czech expatriate ice hockey players in Slovakia
Czech expatriate ice hockey players in Finland
Czech expatriate ice hockey players in Sweden |
https://en.wikipedia.org/wiki/Baer%E2%80%93Specker%20group | In mathematics, in the field of group theory, the Baer–Specker group, or Specker group, named after Reinhold Baer and Ernst Specker, is an example of an infinite Abelian group which is a building block in the structure theory of such groups.
Definition
The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z. It can equivalently be described as the additive group of formal power series with integer coefficients.
Properties
Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.
The group of homomorphisms from the Baer–Specker group to a free abelian group of finite rank is a free abelian group of countable rank. This provides another proof that the group is not free.
See also
Slender group
Notes
References
.
.
.
.
Cornelius, E. F., Jr. (2009), "Endomorphisms and product bases of the Baer-Specker group", Int'l J Math and Math Sciences, 2009, article 396475, https://www.hindawi.com/journals/ijmms/
External links
Stefan Schröer, Baer's Result: The Infinite Product of the Integers Has No Basis
Abelian group theory |
https://en.wikipedia.org/wiki/Essential%20subgroup | In mathematics, especially in the area of algebra studying the theory of abelian groups, an essential subgroup is a subgroup that determines much of the structure of its containing group. The concept was generalized to essential submodules.
Definition
A subgroup of a (typically abelian) group is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".
References
Subgroup properties
Abelian group theory |
https://en.wikipedia.org/wiki/Homotopy%20fiber | In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.
Construction
The homotopy fiber has a simple description for a continuous map . If we replace by a fibration, then the homotopy fiber is simply the fiber of the replacement fibration. We recall this construction of replacing a map by a fibration:
Given such a map, we can replace it with a fibration by defining the mapping path space to be the set of pairs where and (for ) a path such that . We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology). Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at . Then deformation retracts to this subspace by contracting the paths.
The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiberwhich can be defined as the set of all with and a path such that and for some fixed basepoint . A consequence of this definition is that if two points of are in the same path connected component, then their homotopy fibers are homotopy equivalent.
As a homotopy limit
Another way to construct the homotopy fiber of a map is to consider the homotopy limitpg 21 of the diagramthis is because computing the homotopy limit amounts to finding the pullback of the diagramwhere the vertical map is the source and target map of a path , soThis means the homotopy limit is in the collection of mapswhich is exactly the homotopy fiber as defined above.
If and can be connected by a path in , then the diagrams
and are homotopy equivalent to the diagram and thus the homotopy fibers of and are isomorphic in . Therefore we often speak about the homotopy fiber of a map without specifying a base point.
Properties
Homotopy fiber of a fibration
In the special case that the original map was a fibration with fiber , then the homotopy equivalence given above will be a map of fibrations over . This will induce a morphism of their long exact sequences of homotopy groups, from which (by applying the Five Lemma, as is done in the Puppe sequence) one can see that the map is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.
Duality with mapping cone
The homotopy fiber is dual to the mapping cone, much a |
https://en.wikipedia.org/wiki/List%20of%20Cleveland%20Force%20%281978%E2%80%931988%29%20players | Players for the original Cleveland Force (1978–88) of the Major Soccer League:
Regular season statistics only
A
Luis Alberto, M (1980–82) 47 games, 20 goals, 31 assists
Craig Allen, F (1982–88) 254 games, 275 goals, 180 assists
Gary Allison, G (1979–80) 25 games, 4-10 record, 1 assist, 6.34 GAA
Ian Anderson, D (1980–82) 48 games, 34 goals, 36 assists
Desmond Armstrong, D (1986–88) 93 games, 14 goals, 26 assists
Ruben Astigarraga, F (1979–82) 41 games, 39 goals, 41 assists
Mohammad Attiah, F (1978–81) 37 games, 14 goals, 13 assists
B
Mike Barca, G (1978–79) 10 games, 3-4 record, 4.77 GAA
Mike Barry, M (1979–82) 68 games, 33 goals, 30 assists
Chris Bennett, F (1979–80) 25 games, 6 goals, 6 assists
Clyde Best, F (1979–80) 30 games, 33 goals, 16 assists
Brian Bliss, D (1987–88) 51 games, 4 goals, 4 assists
Rich Brands, G (1978–79) 13 games, 1-8 record, 7.99 GAA
John Brooks, D (1979–80) 9 games, 1 goal, 0 assists
Cliff Brown, G (1979–81) 61 games, 24-24 record, 2 assists, 5.17 GAA
Brian Budd, F (1978–79) 19 games, 25 goals, 4 assists
C
Marine Cano, G (1980–81) 18 games, 5-4 record, 4.92 GAA
Peter Carr, D (1981–82) 14 games, 1 goal, 0 assists
Caesar Cervin, F (1978–79) 19 games, 8 goals, 10 assists
Andy Chapman, F (1984–86) 59 games, 34 goals, 25 assists
Fadi Choujaa, F (1982-1982) 2 games, 0 goals, 1 assists
Chris Chueden, M (1985–87) 44 games, 24 goals, 12 assists
Lou Cioffi, G (1981–83) 17 games, 4-10 record, 1 goal, 5.50 GAA
Prosper Cohen, M-F (1980–82) 60 games, 30 goals, 24 assists
Tom Condric, F (1981–82) 38 games, 3 goals, 7 assists
Charlie Cooke, F (1981–82) 19 games, 4 goals, 0 assists
Brooks Cryder, D (1979–80) 32 games, 6 goals, 3 assists
Everald Cummings, F (1978–79) 6 games, 1 goal, 0 assists
D
Benny Dargle, D (1983–88) 247 games, 38 goals, 40 assists
Vic Davidson, F (1982–84) 100 games, 79 goals, 56 assists
Trevor Dawkins, D (1980–84) 168 games, 16 goals, 23 assists
Pasquale de Luca, D (1985–88) 110 games, 19 goals, 23 assists
Carlos DeVenutto, F (1979–80) 1 game, 0 goals, 1 assist
George Dewsnip, F (1980–82) 54 games, 18 goals, 26 assists
Kyle Dietrich, G (1983–85) 5 games, 4-1 record, 4.71 GAA
Gino DiFlorio, F (1984–88) 128 games, 56 goals, 47 assists
Tony Douglas (1978–79) 4 games, 0 goals, 1 assist
Paul Dueker, G (1978–79) 1 game, 0-1 record, 14.00 GAA
E
Mike England, D (1979–80) 11 games, 0 goals, 1 assist
Gino Epifani, G (1987–88) 2 games, 0-0 record, 27.27 GAA
Pat Ercoli, F (1985–86) 29 games, 6 goals, 4 assists
Bobby Joe Esposito, F (1987–88) 37 games, 10 goals, 4 assists
Gary Evans, M (1979–80) 8 games, 0 goals, 2 assists
F
Gordon Fearnley, F (1978–79) 4 games, 1 goal, 2 assists
Drew Ferguson, D (1983–84) 44 games, 20 goals, 13 assists
George Fernandez, D (1983–85) 6 games, 0 goals, 1 assist
Ivair Ferreira, F (1980–81) 29 games, 16 goals, 4 assists
Pat Fidelia, F (1979–80) 29 games, 10 goals, 9 assists
Trevor Franklin, D (1981–82) 26 games, 2 goals, 6 assists
Keith Furphy, |
https://en.wikipedia.org/wiki/Takeshi%20Terada | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
Hannan University alumni
Association football people from Ibaraki Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Thespakusatsu Gunma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Kei%20Omoto | is a Japanese football player who plays for Ococias Kyoto AC.
Club statistics
Updated to 23 February 2018.
1Includes Playoffs J2/J3.
References
External links
Profile at Blaublitz Akita
1984 births
Living people
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Yokohama F. Marinos players
Thespakusatsu Gunma players
Mito HollyHock players
Tochigi SC players
Blaublitz Akita players
Men's association football defenders |
https://en.wikipedia.org/wiki/Wataru%20Yamazaki | is a former Japanese football player. He first played for Nihon University in 2003 and later transferred to Thespa Kusatsu in 2010 after 7 years.
Club statistics
References
External links
1980 births
Living people
Nihon University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Thespakusatsu Gunma players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Multiplicity%20%28mathematics%29 | In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity".
If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct".
Multiplicity of a prime factor
In prime factorization, the multiplicity of a prime factor is its -adic valuation. For example, the prime factorization of the integer is
the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . Thus, has four prime factors allowing for multiplicities, but only three distinct prime factors.
Multiplicity of a root of a polynomial
Let be a field and be a polynomial in one variable with coefficients in . An element is a root of multiplicity of if there is a polynomial such that and . If , then a is called a simple root. If , then is called a multiple root.
For instance, the polynomial has 1 and −4 as roots, and can be written as . This means that 1 is a root of multiplicity 2, and −4 is a simple root (of multiplicity 1). The multiplicity of a root is the number of occurrences of this root in the complete factorization of the polynomial, by means of the fundamental theorem of algebra.
If is a root of multiplicity of a polynomial, then it is a root of multiplicity of the derivative of that polynomial, unless the characteristic of the underlying field is a divisor of , in which case is a root of multiplicity at least of the derivative.
The discriminant of a polynomial is zero if and only if the polynomial has a multiple root.
Behavior of a polynomial function near a multiple root
The graph of a polynomial function f touches the x-axis at the real roots of the polynomial. The graph is tangent to it at the multiple roots of f and not tangent at the simple roots. The graph crosses the x-axis at roots of odd multiplicity and does not cross it at roots of even multiplicity.
A non-zero polynomial function is everywhere non-negative if and only if all its roots have even multiplicity and there exists an such that .
Multiplicity of a solution of a nonlinear system of equations
For an equation with a single variable solution , the multiplicity is if
and
In other words, the differential functional , defined as the derivative of a function at , vanishes at for up to . Those differential functionals span a vector space, called the Macaulay dual space at , and its dimension is the multiplicity of as a zero of .
Let be a system of equations of varia |
https://en.wikipedia.org/wiki/E8%20polytope | {{DISPLAYTITLE:E8 polytope}}
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively.
These polytopes can be visualized as symmetric orthographic projections in Coxeter planes of the E8 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 255 polytopes can be made in the E8, E7, E6, D7, D6, D5, D4, D3, A7, A5 Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry, and E6, E7, E8 have [12], [18], [30] symmetry respectively. In addition there are two other degrees of fundamental invariants, order [20] and [24] for the E8 group that represent Coxeter planes.
11 of these 255 polytopes are each shown in 14 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
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]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Notes
8-polytopes
Polytope |
https://en.wikipedia.org/wiki/Like%20terms | In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients.
Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients.
More generally, when some variable are considered as parameters, like terms are defined similarly, but "numerical factors" must be replaced by "factors depending only on the parameters".
For example, when considering a quadratic equation, one considers often the expression
where and are the roots of the equation and may be considered as parameters. Then, expanding the above product and regrouping the like terms gives
Generalization
In this discussion, a "term" will refer to a string of numbers being multiplied or divided (that division is simply multiplication by a reciprocal) together. Terms are within the same expression and are combined by either addition or subtraction. For example, take the expression:
There are two terms in this expression. Notice that the two terms have a common factor, that is, both terms have an . This means that the common factor variable can be factored out, resulting in
If the expression in parentheses may be calculated, that is, if the variables in the expression in the parentheses are known numbers, then it is simpler to write the calculation . and juxtapose that new number with the remaining unknown number. Terms combined in an expression with a common, unknown factor (or multiple unknown factors) are called like terms.
Examples
Example
To provide an example for above, let and have numerical values, so that their sum may be calculated. For ease of calculation, let and . The original expression becomes
which may be factored into
or, equally,
.
This demonstrates that
The known values assigned to the unlike part of two or more terms are called coefficients. As this example shows, when like terms exist in an expression, they may be combined by adding or subtracting (whatever the expression indicates) the coefficients, and maintaining the common factor of both terms. Such combination is called combining like terms, and it is an important tool used for solving equations.
Simplifying an expression
Take the expression, which is to be simplified:
The first step to grouping like terms in this expression is to get rid of the parentheses. Do this by distributing (multiplying) each number in front of a set of parentheses to each term in that set of parentheses:
The like terms in this expression are the terms that can be grouped together by having exactly the same set of unknown factors. Here, the sets of unknown factors are and . By the rule in the first example, all terms with the same set of unknown factors, that is, all like terms, may be combined by adding or subtracting their coefficients, while maintaining the unknown factors. Thus, the expression becomes
The expression is considered simplified when all |
https://en.wikipedia.org/wiki/14%20%28number%29 | 14 (fourteen) is a natural number following 13 and preceding 15.
In relation to the word "four" (4), 14 is spelled "fourteen".
In mathematics
Fourteen is the seventh composite number. It is specifically, the third distinct Semiprime, it also being the 3rd of the form (2.q) , where q is a higher prime.
It has an aliquot sum of 8, within an aliquot sequence of two composite numbers (14,8,7,1,0) to the Prime in the 7-aliquot tree.
14 is the first member of the first cluster of two discrete semiprimes (14, 15) the next such cluster is (21, 22).
It is the lowest even for which the equation has no solution, making it the first even nontotient.
A set of real numbers to which it is applied closure and complement operations in any possible sequence generates 14 distinct sets. This holds even if the reals are replaced by a more general topological space; see Kuratowski's closure-complement problem.
14 is the third stella octangula number, and the second square pyramidal number.
14 is also the fourth Companion Pell number, and the fifth Catalan number.
According to the Shapiro inequality, 14 is the least number such that there exist , , , where:
with and
There are fourteen polygons that can fill a plane-vertex tiling, where five polygons tile the plane uniformly, and nine others only tile the plane alongside irregular polygons.
Several distinguished polyhedra in three dimensions contain fourteen faces or vertices as facets:
The cuboctahedron, one of two quasiregular polyhedra, has 14 faces and is the only uniform polyhedron with radial equilateral symmetry.
The rhombic dodecahedron, dual to the cuboctahedron, contains 14 vertices and is the only Catalan solid that can tessellate space.
The truncated octahedron contains 14 faces, is the permutohedron of order four, and the only Archimedean solid to tessellate space.
The dodecagonal prism, which is the largest prism that can tessellate space alongside other uniform prisms, has 14 faces.
The Szilassi polyhedron and its dual, the Császár polyhedron, are the simplest toroidal polyhedra; they have 14 vertices and 14 triangular faces, respectively.
Steffen's polyhedron, the simplest flexible polyhedron without self-crossings, has 14 triangular faces.
The regular tetrahedron, the simplest uniform polyhedron and Platonic solid, is made up of a total of 14 elements: 4 edges, 6 vertices, and 4 faces.
Szilassi's polyhedron and the tetrahedron are the only two known polyhedra where each face shares an edge with each other face, while Császár's polyhedron and the tetrahedron are the only two known polyhedra with a continuous manifold boundary that do not contain any diagonals.
Two tetrahedra that are joined by a common edge whose four adjacent and opposite faces are replaced with two specific seven-faced crinkles will create a new flexible polyhedron, with a total of 14 possible clashes where faces can meet.pp.10-11,14 This is the second simplest known triangular flexible polyhedron, af |
https://en.wikipedia.org/wiki/Estate%20Khmaladze | Estate V. Khmaladze (, born October 20, 1944, in Tbilisi, Georgia) is a Georgian statistician. He is best known for his contribution of Khmaladze transformation in statistics.
Biography
Estate Khmaladze was born October 20, 1944, Tbilisi, Georgia. He graduated from Tbilisi State University in 1966, where the first three years he was studying physics. He finished his PhD in 1971 at V. A. Steklov Mathematical Institute, Moscow, under supervision of L. N. Bolshev, who was head of department of mathematical statistics at Steklov after N. V. Smirnov. From 1972 until 1990, his work was, mostly, split between the Steklov Institute of Mathematics in Moscow, and the A. Razmadze Mathematical Institute at Tbilisi State University.
From 1990 to 1999, he was appointed head of department of probability theory and mathematical statistics of A. Razmadze Mathematical Institute of the Georgian National Academy of Sciences.
In 1996, Khmaladze moved with his family from Tbilisi, Georgia, to Sydney, Australia, and from there to Wellington, New Zealand, where in 2002 he was appointed Professor of Statistics after retirement of his predecessor, David Vere-Jones.
A characteristic feature of Khmaladze's work is the search of connection between distant analytical topics. For example, in Khmaladze (1993), the connections between the theory of spatial martingales and Volterra operators with goodness of fit problems of statistics was demonstrated, and in Khmaladze(2007), the infinitesimal theory for set-valued functions was extended to help with problems of spatial statistics and image analysis.
However, the majority of his mathematical research centers around empirical processes and distribution-free methods of testing statistical hypotheses.
A considerable amount of Khmaladze's work is in applications of statistics, in the fields of cito-genetics, physiology, demography and insurance, statistical analysis of texts, various problems in economics and finance. His current applied interests are focused on statistical theory of diversity and Zipf's law.
References
External links
Estate Khmaladze at Victoria University of Wellington
R:Function to compute Khmaladze Transformation
Professional Profile
Mathematicians from Georgia (country)
Soviet mathematicians
Tbilisi State University alumni
Living people
1944 births
Statisticians from Georgia (country)
Academic staff of Victoria University of Wellington
Mathematical statisticians
People from Tbilisi |
https://en.wikipedia.org/wiki/Dejan%20Rusi%C4%8D | Dejan Rusič (born 5 December 1982) is a Slovenian former footballer who played as a striker.
He was capped four times for the Slovenia national team between 2006 in 2008.
Career statistics
International
Statistics accurate as of match played 6 February 2008
References
External links
1982 births
Living people
People from Brežice
Slovenian men's footballers
Men's association football forwards
NK Krško players
NK Celje players
Slovenian expatriate men's footballers
Slovenian PrvaLiga players
FC Politehnica Timișoara players
PFC Spartak Nalchik players
Expatriate men's footballers in Romania
Slovenian expatriate sportspeople in Romania
Expatriate men's footballers in Russia
Slovenian expatriate sportspeople in Russia
Liga I players
Russian Premier League players
Al Taawoun FC players
Expatriate men's footballers in Saudi Arabia
Khazar Lankaran FK players
Expatriate men's footballers in Azerbaijan
Slovenian expatriate sportspeople in Azerbaijan
Slovenia men's international footballers |
https://en.wikipedia.org/wiki/E7%20polytope | {{DISPLAYTITLE:E7 polytope}}
In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. The three simplest forms are the 321, 231, and 132 polytopes, composed of 56, 126, and 576 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the E7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 127 polytopes can be made in the E7, E6, D6, D5, D4, D3, A6, A5, A4, A3, A2 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 and E7 have 12, 18 symmetry respectively.
For 10 of 127 polytopes (7 single rings, and 3 truncations), they are shown in these 9 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
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, Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(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]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
7-polytopes |
https://en.wikipedia.org/wiki/2%2021%20polytope | {{DISPLAYTITLE:2 21 polytope}}
In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.
Its Coxeter symbol is 221, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221.
The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122.
These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
2_21 polytope
The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube.
For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The Schläfli graph is the 1-skeleton of this polytope.
Alternate names
E. L. Elte named it V27 (for its 27 vertices) in his 1912 listing of semiregular polytopes.
Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)
Coordinates
The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope:
(-2, 0, 0, 0,-2, 0, 0, 0),
( 0,-2, 0, 0,-2, 0, 0, 0),
( 0, 0,-2, 0,-2, 0, 0, 0),
( 0, 0, 0,-2,-2, 0, 0, 0),
( 0, 0, 0, 0,-2, 0, 0,-2),
( 0, 0, 0, 0, 0,-2,-2, 0)
( 2, 0, 0, 0,-2, 0, 0, 0),
( 0, 2, 0, 0,-2, 0, 0, 0),
( 0, 0, 2, 0,-2, 0, 0, 0),
( 0, 0, 0, 2,-2, 0, 0, 0),
( 0, 0, 0, 0,-2, 0, 0, 2)
(-1,-1,-1,-1,-1,-1,-1,-1),
(-1,-1,-1, 1,-1,-1,-1, 1),
(-1,-1, 1,-1,-1,-1,-1, 1),
(-1,-1, 1, 1,-1,-1,-1,-1),
(-1, 1,-1,-1,-1,-1,-1, 1),
(-1, 1,-1, 1,-1,-1,-1,-1),
(-1, 1, 1,-1,-1,-1,-1,-1),
( 1,-1,-1,-1,-1,-1,-1, 1),
( 1,-1, 1,-1,-1,-1,-1,-1),
( 1,-1,-1, 1,-1,-1,-1,-1),
( 1, 1,-1,-1,-1,-1,-1,-1),
(-1, 1, 1, 1,-1,-1,-1, 1),
( 1,-1, 1, 1,-1,-1,-1, 1),
( 1, 1,-1, 1,-1,-1,-1, 1),
( 1, 1, 1,-1,-1,-1,-1, 1),
( 1, 1, 1, 1,-1,-1,-1,-1)
Construction
Its construction is based on the E6 group.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 5-simplex, .
Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), .
Every simplex facet touches a 5-orthoplex facet, while alte |
https://en.wikipedia.org/wiki/Lawrence%E2%80%93Krammer%20representation | In mathematics the Lawrence–Krammer representation is a representation of the braid groups. It fits into a family of representations called the Lawrence representations. The first Lawrence representation is the Burau representation and the second is the Lawrence–Krammer representation.
The Lawrence–Krammer representation is named after Ruth Lawrence and Daan Krammer.
Definition
Consider the braid group to be the mapping class group of a disc with n marked points, . The Lawrence–Krammer representation is defined as the action of on the homology of a certain covering space of the configuration space . Specifically, the first integral homology group of is isomorphic to , and the subgroup of invariant under the action of is primitive, free abelian, and of rank 2. Generators for this invariant subgroup are denoted by .
The covering space of corresponding to the kernel of the projection map
is called the Lawrence–Krammer cover and is denoted . Diffeomorphisms of act on , thus also on , moreover they lift uniquely to diffeomorphisms of which restrict to the identity on the co-dimension two boundary stratum (where both points are on the boundary circle). The action of on
thought of as a
-module,
is the Lawrence–Krammer representation. The group is known to be a free -module, of rank .
Matrices
Using Bigelow's conventions for the Lawrence–Krammer representation, generators for the group are denoted for . Letting denote the standard Artin generators of the braid group, we obtain the expression:
Faithfulness
Stephen Bigelow and Daan Krammer have given independent proofs that the Lawrence–Krammer representation is faithful.
Geometry
The Lawrence–Krammer representation preserves a non-degenerate sesquilinear form which is known to be negative-definite Hermitian provided are specialized to suitable unit complex numbers (q near 1 and t near i). Thus the braid group is a subgroup of the unitary group of square matrices of size . Recently it has been shown that the image of the Lawrence–Krammer representation is a dense subgroup of the unitary group in this case.
The sesquilinear form has the explicit description:
References
Further reading
Braid groups
Representation theory |
https://en.wikipedia.org/wiki/3%2021%20polytope | {{DISPLAYTITLE:3 21 polytope}}
In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.
Its Coxeter symbol is 321, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.
The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132.
These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
321 polytope
In 7-dimensional geometry, the 321 polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplexes.
For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The 1-skeleton of the 321 polytope is the Gosset graph.
This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram: .
Alternate names
It is also called the Hess polytope for Edmund Hess who first discovered it.
It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure.
E. L. Elte named it V56 (for its 56 vertices) in his 1912 listing of semiregular polytopes.
H.S.M. Coxeter called it 321 due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)
Coordinates
The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:
± (-3, -3, 1, 1, 1, 1, 1, 1)
Construction
Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex, .
Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 311, .
Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.
The vertex figure is determined by removing the ringed node and |
https://en.wikipedia.org/wiki/4%2021%20polytope | {{DISPLAYTITLE:4 21 polytope}}
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.
Its Coxeter symbol is 421, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 4-node sequences, .
The rectified 421 is constructed by points at the mid-edges of the 421. The birectified 421 is constructed by points at the triangle face centers of the 421. The trirectified 421 is constructed by points at the tetrahedral centers of the 421.
These polytopes are part of a family of 255 = 28 − 1 convex uniform 8-polytopes, made of uniform 7-polytope facets and vertex figures, defined by all permutations of one or more rings in this Coxeter-Dynkin diagram: .
421 polytope
The 421 polytope has 17,280 7-simplex and 2,160 7-orthoplex facets, and 240 vertices. Its vertex figure is the 321 polytope. As its vertices represent the root vectors of the simple Lie group E8, this polytope is sometimes referred to as the E8 root polytope.
The vertices of this polytope can also be obtained by taking the 240 integral octonions of norm 1. Because the octonions are a nonassociative normed division algebra, these 240 points have a multiplication operation making them not into a group but rather a loop, in fact a Moufang loop.
For visualization this 8-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a regular triacontagon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
Alternate names
This polytope was discovered by Thorold Gosset, who described it in his 1900 paper as an 8-ic semi-regular figure. It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
E. L. Elte named it V240 (for its 240 vertices) in his 1912 listing of semiregular polytopes.
H.S.M. Coxeter called it 421 because its Coxeter-Dynkin diagram has three branches of length 4, 2, and 1, with a single node on the terminal node of the 4 branch.
Dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton (Acronym Fy) - 2160-17280 facetted polyzetton (Jonathan Bowers)
Coordinates
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The 240 vertices of the 421 polytope can be constructed in two sets: 112 (22×8C2) with coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (27) with coordinates obtained from by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be a multiple of 4).
Each vertex has 56 nearest neighbors; for example, the nearest neighbors |
https://en.wikipedia.org/wiki/Edwin%20E.%20Moise | Edwin Evariste Moise (; December 22, 1918 – December 18, 1998)
was an American mathematician and mathematics education reformer. After his retirement from mathematics he became a literary critic of 19th-century English poetry and had several notes published in that field.
Early life and education
Edwin E. Moise was born December 22, 1918, in New Orleans, Louisiana.
He graduated from Tulane University in 1940. He worked as a cryptanalyst and Japanese translator for the Office of the Chief of Naval Operations during World War II.
He received his Ph.D. degree in mathematics from the University of Texas in 1947. His dissertation was titled "An indecomposable continuum which is homeomorphic to each of its nondegenerate subcontinua," a topic in continuum theory, and was written under the direction of Robert Lee Moore. In his dissertation Moise coined the term pseudo-arc.
Career
Moise taught at the University of Michigan from 1947 to 1960. He was James B. Conant Professor of education and mathematics at Harvard University from 1960 to 1971. He held a Distinguished Professorship at Queens College, City University of New York from 1971 to 1987.
Moise started working on the topology of 3-manifolds while at the University of Michigan. During 1949–1951 he held an appointment at the Institute for Advanced Study during which he proved Moise's theorem that every 3-manifold can be triangulated in an essentially unique way.
Moise joined the School Mathematics Study Group when it started in 1958, as a member of the geometry writing team. The team produced several course outlines and sample pages for a 10th grade geometry course, and then Moise and Floyd L. Downs wrote a geometry textbook, based on the team's approach, that was published in 1964. The textbook used metric postulates instead of Euclid's postulates, a controversial approach supported by some mathematicians such as Saunders Mac Lane but opposed by others such as Alexander Wittenberg and Morris Kline.
Moise was a president of the Mathematical Association of America, a vice-president of the American Mathematical Society, a Fellow of the American Academy of Arts and Sciences, and was on the executive committee of the International Commission on Mathematical Instruction.
Moise retired from Queens College in 1987 and started a second career studying 19th century English poetry. He had six short notes of literary criticism published.
In the middle and late 1960s, Moise was among the few members of the senior faculty at Harvard University who strongly and publicly opposed the Vietnam War.
Moise died in New York City on December 18, 1998, aged 79.
See also
Moise's theorem
Selected publications
References
External links
MAA presidents: Edwin Evariste Moise
1918 births
1998 deaths
20th-century American mathematicians
Queens College, City University of New York faculty
Geometers
Harvard University Department of Mathematics faculty
Harvard University faculty
Institute for Advanced Study visiting |
https://en.wikipedia.org/wiki/E6%20polytope | {{DISPLAYTITLE:E6 polytope}}
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the E6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 39 polytopes can be made in the E6, D5, D4, D2, A5, A4, A3 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 has 12 symmetry.
Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E6 symmetry. The vertices and edges drawn with vertices colored by the number of overlapping vertices in each projective position.
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, Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(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]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
6-polytopes |
https://en.wikipedia.org/wiki/Lubom%C3%ADr%20Pokluda | Lubomír Pokluda (born 17 March 1958 in Vojkovice) is a Czech former footballer.
References
League statistics
1958 births
Living people
Czech men's footballers
Czechoslovak men's footballers
Czechoslovakia men's international footballers
Footballers at the 1980 Summer Olympics
Olympic footballers for Czechoslovakia
Olympic gold medalists for Czechoslovakia
AC Sparta Prague players
FK Teplice players
FK Hvězda Cheb players
FK Inter Bratislava players
Lierse S.K. players
Olympic medalists in football
Czechoslovak expatriate men's footballers
Expatriate men's footballers in Belgium
Czechoslovak expatriate sportspeople in Belgium
Medalists at the 1980 Summer Olympics
Men's association football midfielders
People from Frýdek-Místek District
Footballers from the Moravian-Silesian Region |
https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert%20controversy | In a controversy over the foundations of mathematics, in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. The debate concerned fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. Much of the controversy took place while both were involved with Mathematische Annalen, the leading mathematical journal of the time, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board. In 1920, Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen.
Background
The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s.
In his biography of Kurt Gödel, John W. Dawson, Jr observed that "partisans of three principal philosophical positions took part in the debate" – the logicists (Gottlob Frege and Bertrand Russell), the formalists (David Hilbert and his school of collaborators), and the constructivists (Henri Poincaré and Hermann Weyl); within this constructivist school was the radical self-named "intuitionist" L. E. J. Brouwer.
Brief history of Brouwer and intuitionism
Brouwer founded the mathematical philosophy of intuitionism as a challenge to the then-prevailing formalism of David Hilbert and his collaborators, Paul Bernays, Wilhelm Ackermann, John von Neumann and others. As a variety of constructive mathematics, intuitionism is a philosophy of the foundations of mathematics which rejects the law of excluded middle in mathematical reasoning.
After completing his dissertation, Brouwer decided not to share his philosophy until he had established his career. By 1910, he had published a number of important papers, in particular the fixed-point theorem. Hilbert admired Brouwer and helped him receive a regular academic appointment in 1912 at the University of Amsterdam. Then, Brouwer decided to return to intuitionism.
In the later 1920s, Brouwer became involved in a public controversy with Hilbert over editorial policy at Mathematische Annalen, at that time a leading learned journal. He became relatively isolated; the development of intuitionism at its source was taken up by his student Arend Heyting.
Origins of disagreement
The nature of Hilbert's proof of the Hilbert basis theorem from 1888 was controversial. Although Leopold Kronecker, a constructivist, had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" – in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object".
Brouwer was not convinced and in particular objected to the use of the law of excluded middle over infinite sets. Hilbert would respond: "Taking the Principle of the Excluded Mid |
https://en.wikipedia.org/wiki/C.%20B.%20Collins | Christopher Barry Collins is a cosmologist who has written many papers with Stephen Hawking. He is a professor emeritus of applied mathematics at the University of Waterloo.
Collins earned his Ph.D. in 1972 from the University of Cambridge under the supervision of F. Gerard Friedlander.
Among his works with Hawking is a 1973 paper that uses the anthropic principle to provide a solution to the flatness problem.
Selected publications
.
.
.
.
.
See also
List of University of Waterloo people
References
Living people
Canadian cosmologists
20th-century British mathematicians
20th-century Canadian mathematicians
Year of birth missing (living people)
Place of birth missing (living people)
Alumni of the University of Cambridge
Academic staff of the University of Waterloo |
https://en.wikipedia.org/wiki/Stefan%20Bemstr%C3%B6m | Stefan Bemström (born 1 March 1972) is a Swedish retired professional ice hockey who mostly played with the Södertälje SK team in the Swedish Elitserien league.
Career statistics
External links
Bemström retires (Swedish)
References
1972 births
Swedish ice hockey defencemen
Södertälje SK players
Timrå IK players
Leksands IF players
Living people
Ice hockey people from Södertälje |
https://en.wikipedia.org/wiki/Double%20tangent%20bundle | In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space TM of the tangent bundle of a smooth manifold M
. A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle.
Secondary vector bundle structure and canonical flip
Since is a vector bundle in its own right, its tangent bundle has the secondary vector bundle structure where is the push-forward of the canonical projection
In the following we denote
and apply the associated coordinate system
on TM. Then the fibre of the secondary vector bundle structure at X∈TxM takes the form
The double tangent bundle is a double vector bundle.
The canonical flip is a smooth involution j:TTM→TTM that exchanges these vector space structures
in the sense that it is a vector bundle isomorphism between and In the associated coordinates on TM it reads as
The canonical flip has the property that for any f: R2 → M,
where s and t are coordinates of the standard basis of R 2. Note that both partial derivatives are functions from R2 to TTM.
This property can, in fact, be used to give an intrinsic definition of the canonical flip. Indeed, there is a submersion
p: J20 (R2,M) → TTM given by
where p can be defined in the space of two-jets at zero because only depends on f up to order two at zero. We consider the application:
where α(s,t)= (t,s). Then J is compatible with the projection p and induces the canonical flip on the quotient TTM.
Canonical tensor fields on the tangent bundle
As for any vector bundle, the tangent spaces of the fibres TxM of the tangent bundle can be identified with the fibres TxM themselves. Formally this is achieved through the vertical lift, which is a natural vector space isomorphism
defined as
The vertical lift can also be seen as a natural vector bundle isomorphism
from the pullback bundle of over onto the vertical tangent bundle
The vertical lift lets us define the canonical vector field
which is smooth in the slit tangent bundle TM\0. The canonical vector field can be also defined as the infinitesimal generator of the Lie-group action
Unlike the canonical vector field, which can be defined for any vector bundle, the canonical endomorphism
is special to the tangent bundle. The canonical endomorphism J satisfies
and it is also known as the tangent structure for the following reason. If (E,p,M) is any vector bundle
with the canonical vector field V and a (1,1)-tensor field J that satisfies the properties listed above, with VE in place of VTM, then the vector bundle (E,p,M) is isomorphic to the tangent |
https://en.wikipedia.org/wiki/Phi%20Beta%20Kappa%20Award%20in%20Science | The Phi Beta Kappa Award in Science is given annually by the Phi Beta Kappa Society to authors of significant books in the fields of science and mathematics. The award was first given in 1959 to anthropologist Loren Eiseley.
Award winners
Source: Phi Beta Kappa Society
2022 - Chanda Prescod-Weinstein
See also
List of general science and technology awards
Ralph Waldo Emerson Award
References
American non-fiction literary awards |
https://en.wikipedia.org/wiki/Lloyd%27s%20sign | Lloyd's sign is a sign of renal calculus or pyelonephritis when pain is elicited by deep percussion in the back between the 12th rib and the spine. It is closely related to costovertebral angle tenderness in that the area of percussion is the same. However, Lloyd's sign is defined as positive costovertebral angle tenderness along with the absence of tenderness with normal pressure.
References
Nephrology
Abdominal pain |
https://en.wikipedia.org/wiki/Spark%20%28mathematics%29 | In mathematics, more specifically in linear algebra, the spark of a matrix is the smallest integer such that there exists a set of columns in which are linearly dependent. If all the columns are linearly independent, is usually defined to be 1 more than the number of rows. The concept of matrix spark finds applications in error-correction codes, compressive sensing, and matroid theory, and provides a simple criterion for maximal sparsity of solutions to a system of linear equations.
The spark of a matrix is NP-hard to compute.
Definition
Formally, the spark of a matrix is defined as follows:
where is a nonzero vector and denotes its number of nonzero coefficients ( is also referred to as the size of the support of a vector). Equivalently, the spark of a matrix is the size of its smallest circuit (a subset of column indices such that has a nonzero solution, but every subset of it does not).
If all the columns are linearly independent, is usually defined to be (if has m rows).
By contrast, the rank of a matrix is the largest number such that some set of columns of is linearly independent.
Example
Consider the following matrix .
The spark of this matrix equals 3 because:
There is no set of 1 column of which are linearly dependent.
There is no set of 2 columns of which are linearly dependent.
But there is a set of 3 columns of which are linearly dependent. The first three columns are linearly dependent because .
Properties
If , the following simple properties hold for the spark of a matrix :
(If the spark equals , then the matrix has full rank.)
if and only if the matrix has a zero column.
.
Criterion for uniqueness of sparse solutions
The spark yields a simple criterion for uniqueness of sparse solutions of linear equation systems.
Given a linear equation system . If this system has a solution that satisfies , then this solution is the sparsest possible solution. Here denotes the number of nonzero entries of the vector .
Lower bound in terms of dictionary coherence
If the columns of the matrix are normalized to unit norm, we can lower bound its spark in terms of its dictionary coherence:
Here, the dictionary coherence is defined as the maximum correlation between any two columns:
.
Applications
The minimum distance of a linear code equals the spark of its parity-check matrix.
The concept of the spark is also of use in the theory of compressive sensing, where requirements on the spark of the measurement matrix are used to ensure stability and consistency of various estimation techniques. It is also known in matroid theory as the girth of the vector matroid associated with the columns of the matrix. The spark of a matrix is NP-hard to compute.
References
Signal processing
Matrix theory |
https://en.wikipedia.org/wiki/Poloidal%E2%80%93toroidal%20decomposition | In vector calculus, a topic in pure and applied mathematics, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition. It is often used in the spherical coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids.
Definition
For a three-dimensional vector field F with zero divergence
this F can be expressed as the sum of a toroidal field T and poloidal vector field P
where r is a radial vector in spherical coordinates (r, θ, φ). The toroidal field is obtained from a scalar field, Ψ(r, θ, φ), as the following curl,
and the poloidal field is derived from another scalar field Φ(r, θ, φ), as a twice-iterated curl,
This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal, known as Chandrasekhar–Kendall function.
Geometry
A toroidal vector field is tangential to spheres around the origin,
while the curl of a poloidal field is tangential to those spheres
The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields Ψ and Φ vanishes on every sphere of radius r.
Cartesian decomposition
A poloidal–toroidal decomposition also exists in Cartesian coordinates, but a mean-field flow has to be included in this case. For example, every solenoidal vector field can be written as
where denote the unit vectors in the coordinate directions.
See also
Toroidal and poloidal
Chandrasekhar–Kendall function
Notes
References
Hydrodynamic and hydromagnetic stability, Chandrasekhar, Subrahmanyan; International Series of Monographs on Physics, Oxford: Clarendon, 1961, p. 622.
Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the boussinesq-equations, Schmitt, B. J. and von Wahl, W; in The Navier–Stokes Equations II — Theory and Numerical Methods, pp. 291–305; Lecture Notes in Mathematics, Springer Berlin/ Heidelberg, Vol. 1530/ 1992.
Anelastic Magnetohydrodynamic Equations for Modeling Solar and Stellar Convection Zones, Lantz, S. R. and Fan, Y.; The Astrophysical Journal Supplement Series, Volume 121, Issue 1, Mar. 1999, pp. 247–264.
Plane poloidal-toroidal decomposition of doubly periodic vector fields: Part 1. Fields with divergence and Part 2. Stokes equations. G. D. McBain. ANZIAM J. 47 (2005)
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.
.
Vector calculus |
https://en.wikipedia.org/wiki/Hyperbolic%20plane%20%28disambiguation%29 | In mathematics, the term hyperbolic plane may refer to:
A two-dimensional plane in hyperbolic geometry
A quadratic space known as the hyperbolic plane (quadratic forms) |
https://en.wikipedia.org/wiki/Khmaladze%20transformation | In statistics, the Khmaladze transformation is a mathematical tool used in constructing convenient goodness of fit tests for hypothetical distribution functions. More precisely, suppose are i.i.d., possibly multi-dimensional, random observations generated from an unknown probability distribution. A classical problem in statistics is to decide how well a given hypothetical distribution function , or a given hypothetical parametric family of distribution functions , fits the set of observations. The Khmaladze transformation allows us to construct goodness of fit tests with desirable properties. It is named after Estate V. Khmaladze.
Consider the sequence of empirical distribution functions based on a sequence of i.i.d random variables, , as n increases. Suppose is the hypothetical distribution function of each . To test whether the choice of is correct or not, statisticians use the normalized difference,
This , as a random process in , is called the empirical process. Various functionals of are used as test statistics. The change of the variable , transforms to the so-called uniform empirical process . The latter is an empirical processes based on independent random variables , which are uniformly distributed on if the s do indeed have distribution function .
This fact was discovered and first utilized by Kolmogorov (1933), Wald and Wolfowitz (1936) and Smirnov (1937) and, especially after Doob (1949) and Anderson and Darling (1952), it led to the standard rule to choose test statistics based on . That is, test statistics are defined (which possibly depend on the being tested) in such a way that there exists another statistic derived from the uniform empirical process, such that . Examples are
and
For all such functionals, their null distribution (under the hypothetical ) does not depend on , and can be calculated once and then used to test any .
However, it is only rarely that one needs to test a simple hypothesis, when a fixed as a hypothesis is given. Much more often, one needs to verify parametric hypotheses where the hypothetical , depends on some parameters , which the hypothesis does not specify and which have to be estimated from the sample itself.
Although the estimators , most commonly converge to true value of , it was discovered that the parametric, or estimated, empirical process
differs significantly from and that the transformed process , has a distribution for which the limit distribution, as , is dependent on the parametric form of and on the particular estimator and, in general, within one parametric family, on the value of .
From mid-1950s to the late-1980s, much work was done to clarify the situation and understand the nature of the process .
In 1981, and then 1987 and 1993, Khmaladze suggested to replace the parametric empirical process by its martingale part only.
where is the compensator of . Then the following properties of were established:
Although the form of , and therefor |
https://en.wikipedia.org/wiki/K%C3%B6z%C3%A9piskolai%20Matematikai%20%C3%A9s%20Fizikai%20Lapok | Középiskolai Matematikai és Fizikai Lapok [Mathematical and Physical Journal for Secondary Schools] (KöMaL) is a Hungarian mathematics and physics journal for high school students. It was founded by Dániel Arany, a high school teacher from Győr, Hungary and has been continually published since 1894.
KöMaL has been organizing various renowned correspondence competitions for high school students, making a major contribution to Hungarian high school education. Winners of the competition include many leading Hungarian scientists and mathematicians. Since the early 1970s, all of the problems in the KöMaL journal have been translated into English; published solutions, however, are not typically translated.
In addition to problems in mathematics, physics and more recently, informatics, the journal contains articles on those subjects. A 100-year archive of issues is provided online.
The journal's problem section and correspondence competition has been a source of inspiration for the United States of America Mathematical Talent Search.
References
External links
KöMaL homepage
What is KöMaL?
Mathematics journals |
https://en.wikipedia.org/wiki/Triple%20system | In algebra, a triple system (or ternar) is a vector space V over a field F together with a F-trilinear map
The most important examples are Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949 to study subspaces of associative algebras closed under triple commutators [[u, v], w] and triple anticommutators {u, {v, w}}. In particular, any Lie algebra defines a Lie triple system and any Jordan algebra defines a Jordan triple system. They are important in the theories of symmetric spaces, particularly Hermitian symmetric spaces and their generalizations (symmetric R-spaces and their noncompact duals).
Lie triple systems
A triple system is said to be a Lie triple system if the trilinear map, denoted , satisfies the following identities:
The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator, while the third identity means that the linear map Lu,v: V → V, defined by Lu,v(w) = [u, v, w], is a derivation of the triple product. The identity also shows that the space k = span {Lu,v : u, v ∈ V} is closed under commutator bracket, hence a Lie algebra.
Writing m in place of V, it follows that
can be made into a -graded Lie algebra, the standard embedding of m, with bracket
The decomposition of g is clearly a symmetric decomposition for this Lie bracket, and hence if G is a connected Lie group with Lie algebra g and K is a subgroup with Lie algebra k, then G/K is a symmetric space.
Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket [[u, v], w] makes m into a Lie triple system.
Jordan triple systems
A triple system is said to be a Jordan triple system if the trilinear map, denoted {.,.,.}, satisfies the following identities:
The first identity abstracts the symmetry of the triple anticommutator, while the second identity means that if Lu,v:V→V is defined by Lu,v(y) = {u, v, y} then
so that the space of linear maps span {Lu,v:u,v ∈ V} is closed under commutator bracket, and hence is a Lie algebra g0.
Any Jordan triple system is a Lie triple system with respect to the product
A Jordan triple system is said to be positive definite (resp. nondegenerate) if the bilinear form on V defined by the trace of Lu,v is positive definite (resp. nondegenerate). In either case, there is an identification of V with its dual space, and a corresponding involution on g0. They induce an involution of
which in the positive definite case is a Cartan involution. The corresponding symmetric space is a symmetric R-space. It has a noncompact dual given by replacing the Cartan involution by its composite with the involution equal to +1 on g0 and −1 on V and V*. A special case of this construction arises when g0 preserves a complex structure on V. In this case we obtain dual Hermitian symmetric spaces of compact and noncompact type (the latter being bounded symmetric domains).
Jordan pair
A Jordan pair |
https://en.wikipedia.org/wiki/Kamel%20Ouejdide | Kamel Ouejdide (born 1 May 1980) is a French-Moroccan former professional footballer who played as a striker.
Career statistics
References
External links
1980 births
Living people
Footballers from Rabat
Men's association football forwards
Moroccan men's footballers
Ligue 2 players
Championnat National players
Süper Lig players
Challenger Pro League players
Stade Malherbe Caen players
Red Star F.C. players
Göztepe S.K. footballers
Fortuna Düsseldorf players
ASOA Valence players
FC Sète 34 players
Clermont Foot players
AS Cannes players
Racing de Ferrol footballers
R.F.C. Seraing (1922) players
RWS Bruxelles players
Moroccan expatriate men's footballers
Moroccan expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium
Moroccan expatriate sportspeople in Germany
Expatriate men's footballers in Germany
Moroccan expatriate sportspeople in Turkey
Expatriate men's footballers in Turkey
Francs Borains players |
https://en.wikipedia.org/wiki/Askey%E2%80%93Gasper%20inequality | In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture.
Statement
It states that if , , and then
where
is a Jacobi polynomial.
The case when can also be written as
In this form, with a non-negative integer, the inequality was used by Louis de Branges in his proof of the Bieberbach conjecture.
Proof
gave a short proof of this inequality, by combining the identity
with the Clausen inequality.
Generalizations
give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.
See also
Turán's inequalities
References
Inequalities
Special functions
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Ammann%E2%80%93Beenker%20tiling | In geometry, an Ammann–Beenker tiling is a nonperiodic tiling which can be generated either by an aperiodic set of prototiles as done by Robert Ammann in the 1970s, or by the cut-and-project method as done independently by F. P. M. Beenker.
They are one of the five sets of tilings discovered by Ammann and described in Tilings and Patterns.
The Ammann–Beenker tilings have many properties similar to the more famous Penrose tilings:
They are nonperiodic, which means that they lack any translational symmetry.
Their non-periodicity is implied by their hierarchical structure: the tilings are substitution tilings arising from substitution rules for growing larger and larger patches. This substitution structure also implies that:
Any finite region (patch) in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. Thus, the infinite tilings all look similar to one another, if one looks only at finite patches.
They are quasicrystalline: implemented as a physical structure an Ammann–Beenker tiling will produce Bragg diffraction; the diffractogram reveals both the underlying eightfold symmetry and the long-range order. This order reflects the fact that the tilings are organized, not through translational symmetry, but rather through a process sometimes called "deflation" or "inflation."
All of this infinite global structure is forced through local matching rules on a pair of tiles, among the very simplest aperiodic sets of tiles ever found, Ammann's A5 set.
Various methods to describe the tilings have been proposed: matching rules, substitutions, cut and project schemes and coverings. In 1987 Wang, Chen and Kuo announced the discovery of a quasicrystal with octagonal symmetry.
Description of the tiles
Amman's A and B tiles in his pair A5 a 45-135-degree rhombus and a 45-45-90 degree triangle, decorated with matching rules that allowed only certain arrangements in each region, forcing the non-periodic, hierarchical, and quasiperiodic structures of each of the infinite number of individual Ammann–Beenker tilings.
An alternate set of tiles, also discovered by Ammann, and labelled "Ammann 4" in Grünbaum and Shephard, consists of two nonconvex right-angle-edged pieces. One consists of two squares overlapping on a smaller square, while the other consists of a large square attached to a smaller square. The diagrams below show the pieces and a portion of the tilings.
This is the substitution rule for the alternate tileset.
The relationship between the two tilesets.
In addition to the edge arrows in the usual tileset, the matching rules for both tilesets can be expressed by drawing pieces of large arrows at the vertices, and requiring them to piece together into full arrows.
Katz has studied the additional tilings allowed by dropping the vertex constraints and imposing only the requirement that the edge arrows match. Since this requirement is itself preserved by the substitution rules, any new tiling has an infinite seq |
https://en.wikipedia.org/wiki/Danijel%20Mihajlovi%C4%87 | Danijel Mihajlović (; born 2 June 1985) is a Serbian footballer.
Statistics
Statistics accurate as of 4 May 2015
Honours
Red Star
Serbian Cup: 2012
Jagodina
Serbian Cup: 2013
References
External links
Profile and stats at Srbijafudbal
Danijel Mihajlović Stats at Utakmica.rs
1985 births
Living people
Serbian men's footballers
FK Jagodina players
Serbian SuperLiga players
Men's association football defenders
Red Star Belgrade footballers
People from Varvarin
Sportspeople from Rasina District
FK Temnić players |
https://en.wikipedia.org/wiki/PRIMUS%20%28journal%29 | PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies is a peer-reviewed academic journal covering the teaching of undergraduate mathematics, established in 1991. The journal has been published by Taylor & Francis since March 2007. It is abstracted and indexed in Cambridge Scientific Abstracts, MathEduc, PsycINFO, and Zentralblatt MATH.
PRIMUS is an affiliated journal of the Mathematical Association of America, so all MAA members have access to PRIMUS.
Editorial Team
PRIMUS was started by founding editor-in-chief Brian Winkel in 1991 to address the lack of venues for tertiary mathematics educators to share their pedagogical work. In 2011, Jo Ellis-Monaghan became the second editor-in-chief, with Matt Boelkins serving as associate editor. In 2017, Ellis-Monaghan and Boelkins became co-editors-in-chief.
Currently, Matt Boelkins serves as editor in chief, Kathy Weld as associate editor, Brian P Katz serves as associate and communications editor, and Rachel Schwell as managing editor.
References
External links
Community/editorial website
Academic journals established in 1991
English-language journals
Mathematics education journals |
https://en.wikipedia.org/wiki/Hanna%20Neumann%20conjecture | In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957.
In 2011, a strengthened version of the conjecture (see below) was proved independently by Joel Friedman
and by Igor Mineyev.
In 2017, a third proof of the Strengthened Hanna Neumann conjecture, based on homological arguments inspired by pro-p-group considerations, was published by Andrei Jaikin-Zapirain.
History
The subject of the conjecture was originally motivated by a 1954 theorem of Howson who proved that the intersection of any two finitely generated subgroups of a free group is always finitely generated, that is, has finite rank. In this paper Howson proved that if H and K are subgroups of a free group F(X) of finite ranks n ≥ 1 and m ≥ 1 then the rank s of H ∩ K satisfies:
s − 1 ≤ 2mn − m − n.
In a 1956 paper Hanna Neumann improved this bound by showing that :
s − 1 ≤ 2mn − 2m − n.
In a 1957 addendum, Hanna Neumann further improved this bound to show that under the above assumptions
s − 1 ≤ 2(m − 1)(n − 1).
She also conjectured that the factor of 2 in the above inequality is not necessary and that one always has
s − 1 ≤ (m − 1)(n − 1).
This statement became known as the Hanna Neumann conjecture.
Formal statement
Let H, K ≤ F(X) be two nontrivial finitely generated subgroups of a free group F(X) and let L = H ∩ K be the intersection of H and K. The conjecture says that in this case
rank(L) − 1 ≤ (rank(H) − 1)(rank(K) − 1).
Here for a group G the quantity rank(G) is the rank of G, that is, the smallest size of a generating set for G.
Every subgroup of a free group is known to be free itself and the rank of a free group is equal to the size of any free basis of that free group.
Strengthened Hanna Neumann conjecture
If H, K ≤ G are two subgroups of a group G and if a, b ∈ G define the same double coset HaK = HbK then the subgroups H ∩ aKa−1 and H ∩ bKb−1 are conjugate in G and thus have the same rank. It is known that if H, K ≤ F(X) are finitely generated subgroups of a finitely generated free group F(X) then there exist at most finitely many double coset classes HaK in F(X) such that H ∩ aKa−1 ≠ {1}. Suppose that at least one such double coset exists and let a1,...,an be all the distinct representatives of such double cosets. The strengthened Hanna Neumann conjecture, formulated by her son Walter Neumann (1990), states that in this situation
The strengthened Hanna Neumann conjecture was proved in 2011 by Joel Friedman.
Shortly after, another proof was given by Igor Mineyev.
Partial results and other generalizations
In 1971 Burns improved Hanna Neumann's 1957 bound and proved that under the same assumptions as in Hanna Neumann's paper one has
s ≤ 2mn − 3m − 2n + 4.
In a 1990 paper, Walter Neumann formulated the strengthened Hanna Neumann conjecture (see statement above).
Tardos (1992) establish |
https://en.wikipedia.org/wiki/Lebedev%E2%80%93Milin%20inequality | In mathematics, the Lebedev–Milin inequality is any of several inequalities for the coefficients of the exponential of a power series, found by and . It was used in the proof of the Bieberbach conjecture, as it shows that the Milin conjecture implies the Robertson conjecture.
They state that if
for complex numbers and , and is a positive integer, then
See also exponential formula (on exponentiation of power series).
References
.
(Translation of the 1971 Russian edition, edited by P. L. Duren).
Inequalities |
https://en.wikipedia.org/wiki/Ganit | Ganit may refer to:
Mathematics in Sanskrit
Gallium nitrate
Hənifə, Azerbaijan |
https://en.wikipedia.org/wiki/Zlil%20Sela | Zlil Sela is an Israeli mathematician working in the area of geometric group theory.
He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.
Biographical data
Sela received his Ph.D. in 1991 from the Hebrew University of Jerusalem, where his doctoral advisor was Eliyahu Rips.
Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University in New York. While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation.
Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing. He gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic,
and he delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society and the 2005 Tarski Lectures at the University of California at Berkeley.
He was also awarded the 2003 Erdős Prize from the Israel Mathematical Union.
Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic for his work on the Tarski conjecture and on discovering and developing new connections between model theory and geometric group theory.
Mathematical contributions
Sela's early important work was his solution in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper. The machinery of the canonical representatives allowed Rips and Sela to prove algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free groups, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani to the case of relatively hyperbolic groups and played a key role in the solution of the isomorphism problem for toral relatively hyperbolic groups.
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups, motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by oth |
https://en.wikipedia.org/wiki/Transformation%20semigroup | In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group.
A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal.
An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set.
In automata theory, some authors use the term transformation semigroup to refer to a semigroup acting faithfully on a set of "states" different from the semigroup's base set. There is a correspondence between the two notions.
Transformation semigroups and monoids
A transformation semigroup is a pair (X,S), where X is a set and S is a semigroup of transformations of X. Here a transformation of X is just a function from a subset of X to X, not necessarily invertible, and therefore S is simply a set of transformations of X which is closed under composition of functions. The set of all partial functions on a given base set, X, forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X), typically denoted by .
If S includes the identity transformation of X, then it is called a transformation monoid. Obviously any transformation semigroup S determines a transformation monoid M by taking the union of S with the identity transformation. A transformation monoid whose elements are invertible is a permutation group.
The set of all transformations of X is a transformation monoid called the full transformation monoid (or semigroup) of X. It is also called the symmetric semigroup of X and is denoted by TX. Thus a transformation semigroup (or monoid) is just a subsemigroup (or submonoid) of the full transformation monoid of X.
If (X,S) is a transformation semigroup then X can be made into a semigroup action of S by evaluation:
This is a monoid action if S is a transformation monoid.
The characteristic feature of transformation semigroups, as actions, is that they are faithful, i.e., if
then s = t. Conversely if a semigroup S acts on a set X by T(s,x) = s • x then we can define, for s ∈ S, a transformation Ts of X by
The map sending s to Ts is injective if and only if (X, T) is faithful, in which case the image of this map is a transformation semigroup isomorphic to S.
Cayley representation
In group theory, Cayley's theorem asserts that any group G is isomorphic to a subgroup of the symmetric group of G (regarded as a set), so that G is a permutation group. This theorem generalizes straightforwardly to monoids: any monoid M is a transformation monoid of its underlying set, via the action given by left (or right) multiplication. T |
https://en.wikipedia.org/wiki/Architectural%20geometry | Architectural geometry is an area of research which combines applied geometry and architecture, which looks at the design, analysis and manufacture processes. It lies at the core of architectural design and strongly challenges contemporary practice, the so-called architectural practice of the digital age.
Architectural geometry is influenced by following fields: differential geometry, topology, fractal geometry, and cellular automata.
Topics include:
freeform curves and surfaces creation
developable surfaces
discretisation
generative design
digital prototyping and manufacturing
See also
Geometric design
Computer-aided architectural design
Mathematics and architecture
Fractal geometry
Blobitecture
References
External links
Theory
Charles Jencks: The New Paradigm in Architecture
Institutions
Geometric Modeling and Industrial Geometry
Städelschule Architecture Class
SIAL - The Spatial Information Architecture Laboratory
Companies
Evolute Research and Consulting
Events
Smart Geometry
Advances in Architectural Geometry,( Conference Proceedings, 80MB)
Resource collections
Geometry in Action: Architecture
Tools
K3DSurf — A program to visualize and manipulate Mathematical models in three, four, five and six dimensions. K3DSurf supports Parametric equations and Isosurfaces
JavaView — a 3D geometry viewer and a mathematical visualization software.
Generative Components — Generative design software that captures and exploits the critical relationships between design intent and geometry.
ParaCloud GEM— A software for components population based on points of interest, with no requirement for scripting.
Grasshopper— a graphical algorithm editor tightly integrated with Rhino's 3-D modeling tools.
Computer-aided design
Computer-aided design software
ar:جيوميترية العمارة |
https://en.wikipedia.org/wiki/Architecture%20of%20San%20Francisco | The architecture of San Francisco is not so much known for defining a particular architectural style; rather, with its interesting and challenging variations in geography and topology and tumultuous history, San Francisco is known worldwide for its particularly eclectic mix of Victorian and modern architecture. Bay windows were identified as a defining characteristic of San Francisco architecture in a 2012 study that had a machine learning algorithm examine a random sample of 25,000 photos of cities from Google Street View.
Icons of San Francisco architecture include the Golden Gate Bridge, Alcatraz Island, Coit Tower, the Palace of Fine Arts, Lombard Street, Alamo Square, Fort Point, the Transamerica Pyramid, and Chinatown. Included below are summaries of the historical significance of some of these great San Franciscan architectural achievements.
Fort Point
At the foot of the Golden Gate bridge is Fort Point, built to protect the Bay from naval attacks. Designed to allow cannons to hit enemy ships at water level, Fort Point is the only one of its kind in the west. It was originally constructed under the leadership of Spanish Lieutenant Jose Joaquin Moraga. Moraga had been sent up from Monterey, about 100 miles south of San Francisco, to build fortifications in the San Francisco Bay in order to secure Spanish control over the whole area. In 1792, he built a Presidio style fort, which had sufficient fortifications, but was clearly not strong enough to truly act as the main defense of the harbor.
In 1796, Moraga redesigned the fort and gave it the name, Castillo de San Joaquín. Unfortunately, the fort was built from adobe brick and constructed on sand, meaning that the adobe cracked when one of the cannons was fired and that each winter the Fort would be heavily damaged from the weather. After the Mexican War of Independence in 1821, the Spanish were forced to abandon all of posts in California, including the Castillo. Once the fort was abandoned, the Mexican government attempted to maintain it and others of its kind, but the young government simply did not have the resources to do so. All of the forts in California were abandoned by 1835.
After California was granted statehood in 1850, the fort still sat unused until 1856, when the United States government allotted $500,000 (about $14 million in today's money) to install fortifications in California. The people in charge of the fortifications decided to rebuild The Castillo de San Joaquín, but using much more modern materials and building techniques. The fort was built on a granite foundation with masonry walls 12 feet thick. The fort was loaded with weapons and renamed Fort Point. At the time, the fort was incredibly impressive and served as the main fortification protecting the San Francisco bay. It was also the most powerful heavily fortified fort on the west coast when it was built. However, after the civil war, masonry forts of its type were rendered technologically obsolete, leaving F |
https://en.wikipedia.org/wiki/T.%20Tony%20Cai | Tianwen Tony Cai (; born March, 1967) is a Chinese statistician. He is the Daniel H. Silberberg Professor of Statistics and Vice Dean at the Wharton School of the University of Pennsylvania. He is also professor of Applied Math & Computational Science Graduate Group, and associate scholar at the Department of Biostatistics, Epidemiology & Informatics, Perelman School of Medicine, University of Pennsylvania. In 2008 Tony Cai was awarded the COPSS Presidents' Award.
Early life and education
Cai was born in Rui'an, Wenzhou, Zhejiang, China. In 1986, he graduated from the Department of Mathematics, Hangzhou University (previous and current Department of Mathematics, Zhejiang University), at 18 years old. In 1989, he then received an M.Sc. from Shanghai Jiao Tong University and in 1996, earned a PhD from Cornell University
Career
Tony Cai was appointed the Dorothy Silberberg Professor at the Wharton School of the University of Pennsylvania from July 1, 2007 to July 31, 2018 and has been the Daniel H. Silberberg Professor at Wharton since August 1, 2018. Cai has been the Vice Dean of the Wharton School since August 1, 2017.
Tony Cai's research focuses on high-dimensional statistics, statistical machine learning, large-scale inference, nonparametric function estimation, functional data analysis, and statistical decision theory, and applications to genomics, compressed sensing, chemical identification, medical imaging, and financial engineering.
Tony Cai was elected a Fellow of the Institute of Mathematical Statistics in 2006.
In 2008, he received the COPSS Presidents’ Award from the Committee of Presidents of Statistical Societies.
In 2009, Tony Cai was named the Medallion Lecturer at the Institute of Mathematical Statistics.
In 2017, he was elected to the presidency of International Chinese Statistical Association (ICSA).
Additional affiliations and memberships
Tony Cai was a co-editor of the leading statistics journal, the Annals of Statistics from 2010 to 2012. He has also served on editorial boards of several other journals, including Journal of the American Statistical Association (JASA), Journal of the Royal Statistical Society Series B (JRSSB), Statistics Surveys, and Statistica Sinica.
Honors and awards
Fellow, Institute of Mathematical Statistics, 2006.
COPSS Presidents' Award 2008, Committee of Presidents of Statistical Societies.
Personal life
Cai has four brothers and one sister. His sister Tianxi Cai is the John Rock Professor of Population and Translational Data Sciences in the Department of Biostatistics at the Harvard T.H. Chan School of Public Health. She is also a professor at Harvard Medical School. Topics in her research include biomarkers, personalized medicine, survival analysis, and health informatics. His brother Tianwu Michael Cai, majored in physics (PhD, Rochester) is a vice-president of Goldman Sachs. Tony Cai has two children, a son and daughter.
External links
T. Tony Cai's homepage at the Wharton Schoo |
https://en.wikipedia.org/wiki/Abstract%20and%20Applied%20Analysis | Abstract and Applied Analysis is a peer-reviewed mathematics journal covering the fields of abstract and applied analysis and traditional forms of analysis such as linear and nonlinear ordinary and partial differential equations, optimization theory, and control theory.
It is published by Hindawi Publishing Corporation. It was established by Athanassios G. Kartsatos (University of South Florida) in 1996, who was editor-in-chief until 2005. Martin Bohner (Missouri S&T) was editor-in-chief from 2006 until 2011 when the journal converted to a model shared by all Hindawi journals of not having an editor-in-chief, with editorial decisions made by editorial board members.
The journal has faced delisting from the Journal Citation Reports (thus not receive an impact factor), for anomalous citation patterns.
References
External links
Mathematics journals
Hindawi Publishing Corporation academic journals
Monthly journals
English-language journals
Academic journals established in 1996 |
https://en.wikipedia.org/wiki/Laurence%20Baxter | Laurence Alan Baxter (28 February 1954, in London – 8 November 1996, in Long Island) was professor of statistics at the State University of New York at Stony Brook.
Early life
Baxter was born at the Bearstead Jewish Maternity Hospital, Stoke Newington. His family lived in Ilford, Essex. He was educated at University College London (UCL).
Career
Baxter's first job (1975–1977) was at an insurance company. He then went to the Central Electricity Generating Board, where he researched ways to predict the available generator capacity given the incidence of breakdowns and the average time required for generator repairs. This work was accepted by UCL for a Ph.D. in 1980.
Baxter was then offered a temporary post as a lecturer at the University of Delaware. The following year, he moved to the State University of New York at Stony Brook (SUNY) where he was granted tenure about ten years before his death.
Leitmann (1997) reports that "Baxter was internationally renowned for his work in applied probability
and reliability theory" and that he "published over 45 papers and did extensive consulting in this area". He further argues that the "results of his work on separately maintained components have been incorporated into a widely used AT&T Bell Laboratories software package for calculating various characteristics of system availability". Leitmann also notes that Baxter provided extensions to several classic theories in reliability theory naming these continuum structure functions (CSFs). Also that he researched air pollutions impact on mortality.
Baxter conceived the idea for and was editor-in-chief of the book series Stochastic Modeling, published by Chapman and Hall from 1993. He was an editorial board member for Applied Probability Newsletter, Bulletin of the Institute of Mathematical Statistics, the Journal of Mathematical Analysis and its Applications, Naval Research Logistics and the International Journal of Operations and Quantitative Management.
SUNY established the annual Laurence Baxter Memorial Lecture, which is now given each April at Stony Brook.
Notes and references
1954 births
1996 deaths
Jewish scientists
English statisticians
20th-century English mathematicians
People educated at Ilford County High School
Alumni of University College London
People from Ilford
Stony Brook University faculty
English Jews |
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