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https://en.wikipedia.org/wiki/Stufe%20%28algebra%29 | In field theory, a branch of mathematics, the Stufe (/ʃtuːfə/; German: level) s(F) of a field F is the least number of squares that sum to −1. If −1 cannot be written as a sum of squares, s(F) = . In this case, F is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.
Powers of 2
If then for some natural number .
Proof: Let be chosen such that . Let . Then there are elements such that
Both and are sums of squares, and , since otherwise , contrary to the assumption on .
According to the theory of Pfister forms, the product is itself a sum of squares, that is, for some . But since , we also have , and hence
and thus .
Positive characteristic
Any field with positive characteristic has .
Proof: Let . It suffices to prove the claim for .
If then , so .
If consider the set of squares. is a subgroup of index in the cyclic group with elements. Thus contains exactly elements, and so does .
Since only has elements in total, and cannot be disjoint, that is, there are with and thus .
Properties
The Stufe s(F) is related to the Pythagoras number p(F) by p(F) ≤ s(F) + 1. If F is not formally real then s(F) ≤ p(F) ≤ s(F) + 1. The additive order of the form (1), and hence the exponent of the Witt group of F is equal to 2s(F).
Examples
The Stufe of a quadratically closed field is 1.
The Stufe of an algebraic number field is ∞, 1, 2 or 4 (Siegel's theorem). Examples are Q, Q(√−1), Q(√−2) and Q(√−7).
The Stufe of a finite field GF(q) is 1 if q ≡ 1 mod 4 and 2 if q ≡ 3 mod 4.
The Stufe of a local field of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field Q2 is 4.
Notes
References
Further reading
Field (mathematics) |
https://en.wikipedia.org/wiki/David%20Minda | Carl David Minda is an American mathematician, the Charles Phelps Taft Professor of Mathematics at the University of Cincinnati.
Minda did his undergraduate studies at the University of Cincinnati, earning a bachelor's degree in 1965 and a masters in 1966. He then earned his Ph.D. in 1970 from the University of California, San Diego, under the supervision of Burton Rodin. He taught at the University of Minnesota, and then returned to the Cincinnati faculty. He was given the Taft Professorship in 1999. His research falls within the branch of mathematics known as Complex Analysis. His research interests include
structure of hyperbolic metric, Riemann surfaces, and geometric Schwarz-Pick lemma.
In 2001, Minda won the University of Cincinnati's Dolly Cohen Award for Excellence in Teaching, and in 2002, he won the distinguished teaching award of the Ohio section of the Mathematical Association of America.
References
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
University of Cincinnati alumni
University of California, San Diego alumni
University of Minnesota faculty
University of Cincinnati faculty |
https://en.wikipedia.org/wiki/Kriszti%C3%A1n%20Pogacsics | Krisztián Pogacsics (; born 17 October 1985) is a Hungarian Slovene football player who currently plays for Puskás Akadémia FC.
Club statistics
Updated to games played as of 27 June 2020.
External links
1985 births
Living people
Sportspeople from Zalaegerszeg
Footballers from Zala County
Hungarian men's footballers
Men's association football goalkeepers
Zalaegerszegi TE players
FC Bihor Oradea (1958) players
Panionios F.C. players
AFC Săgeata Năvodari players
Győri ETO FC players
Puskás Akadémia FC players
Balmazújvárosi FC players
Kaposvári Rákóczi FC players
Nemzeti Bajnokság I players
Liga I players
Super League Greece players
Hungarian people of Slovenian descent
Hungarian expatriate men's footballers
Expatriate men's footballers in Romania
Expatriate men's footballers in Greece
Hungarian expatriate sportspeople in Romania
Hungarian expatriate sportspeople in Greece |
https://en.wikipedia.org/wiki/Exercise%20%28mathematics%29 | A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers. Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions.
Usually instructors prepare students with worked examples: the exercise is stated, then a model answer is provided. Often several worked examples are demonstrated before students are prepared to attempt exercises on their own. Some texts, such as those in Schaum's Outlines, focus on worked examples rather than theoretical treatment of a mathematical topic.
Overview
In primary school students start with single digit arithmetic exercises. Later most exercises involve at least two digits. A common exercise in elementary algebra calls for factorization of polynomials. Another exercise is completing the square in a quadratic polynomial. An artificially produced word problem is a genre of exercise intended to keep mathematics relevant. Stephen Leacock described this type:
The student of arithmetic who has mastered the first four rules of his art and successfully striven with sums and fractions finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance.
A distinction between an exercise and a mathematical problem was made by Alan H. Schoenfeld:
Students must master the relevant subject matter, and exercises are appropriate for that. But if rote exercises are the only kinds of problems that students see in their classes, we are doing the students a grave disservice.
He advocated setting challenges:
By "real problems" ... I mean mathematical tasks that pose an honest challenge to the student and that the student needs to work at in order to obtain a solution.
A similar sentiment was expressed by Marvin Bittinger when he prepared the second edition of his textbook:
In response to comments from users, the authors have added exercises that require something of the student other than an understanding of the immediate objectives of the lesson at hand, yet are not necessarily highly challenging.
The zone of proximal development for each student, or cohort of students, sets exercises at a level of difficulty that challenges but does not frustrate them.
Some comments in the preface of a calculus textbook show the cen |
https://en.wikipedia.org/wiki/National%20Mathematics%20Year | In India and in Nigeria the year 2012 CE was celebrated National Mathematics Year. In India, the National Mathematics Year was a tribute to the mathematical genius Srinivasa Ramanujan who was born on 22 December 1887 and whose 125th birthday falls on 22 December 2012. In Nigeria, the year 2012 was observed as National Mathematics Year as part of the federal government's effort to promote and popularize the study of mathematics.
National Mathematics Year in India
The decision to designate the year 2012 CE as National Mathematics Year was announced by Dr Manmohan Singh, Prime Minister of India
, during the inaugural ceremony of the celebrations to mark the 125th birth anniversary of Srinivasa Ramanujan held at the Madras University Centenary Auditorium on 26 February 2012. The Prime Minister also announced that December 22 would be celebrated as National Mathematics Day from 2012 onwards.
An Organising Committee with Professor M.S. Raghunathan, President of the Ramanujan Mathematical Society as chair, and Professor Dinesh Singh, Secretary of the Ramanujan Mathematical Society as secretary, has been formed to formulate and implement programmes and projects as part of the observance of the National Mathematics Year. A National Committee with Minister for
Kapil Sibal as the chair supervises the activities of the Organising Committee.
National Mathematics Year in Nigeria
In Nigeria, the various activities planned as part of the celebration of National Mathematics Year would be centred on the theme Mathematics: The Key to Transformation. The events were inaugurated on 1 March 2012 at a function in Musa Yar’adua Dome, Abuja. Thirteen projects of national importance are planned as part of the celebrations.
References
Science and technology in India
Indian mathematics
Srinivasa Ramanujan
Observances in India
2012 in India |
https://en.wikipedia.org/wiki/Wirtinger%20sextic | In mathematics, the Wirtinger plane sextic curve, studied by Wirtinger, is a degree 6 genus 4 plane curve with double points at the 6 vertices of a complete quadrilateral.
References
Sextic curves |
https://en.wikipedia.org/wiki/Table%20of%20congruences | In mathematics, a congruence is an equivalence relation on the integers. The following sections list important or interesting prime-related congruences.
Table of congruences characterizing special primes
Other prime-related congruences
There are other prime-related congruences that provide necessary and sufficient conditions on the primality of certain subsequences of the natural numbers.
Many of these alternate statements characterizing primality are related to Wilson's theorem, or are restatements of this classical result given in terms of other
special variants of generalized factorial functions. For instance, new variants of Wilson's theorem stated in terms of the
hyperfactorials, subfactorials, and superfactorials are given in.
Variants of Wilson's theorem
For integers , we have the following form of Wilson's theorem:
If is odd, we have that
Clement's theorem concerning the twin primes
Clement's congruence-based theorem characterizes the twin primes pairs of the form through the following conditions:
P. A. Clement's original 1949 paper provides a
proof of this interesting elementary number theoretic criteria for twin primality based on Wilson's theorem.
Another characterization given in Lin and Zhipeng's article provides that
Characterizations of prime tuples and clusters
The prime pairs of the form for some include the special cases of the cousin primes (when ) and the sexy primes (when ). We have elementary congruence-based characterizations of the primality of such pairs, proved for instance in the article. Examples of congruences characterizing these prime pairs include
and the alternate characterization when is odd such that given by
Still other congruence-based characterizations of the primality of triples, and more general prime clusters (or prime tuples) exist and are typically proved starting from Wilson's theorem (see, for example, Section 3.3 in ).
References
Congruences
Modular arithmetic |
https://en.wikipedia.org/wiki/Equivalent%20territory | An equivalent territory (), formally known as territory equivalent to a regional county municipality (), is a territorial unit used by Statistics Canada and the Institut de la statistique du Québec.
Quebec is divided into 87 regional county municipalities (RCMs), equivalent to counties in other jurisdictions. However, the RCMs do not cover the entire territory, since major cities are outside any RCM (). To ensure complete territorial coverage for certain purposes, such as the census, the equivalent territories are defined.
Most equivalent territories correspond to certain urban agglomerations; the others are Jamésie, Eeyou Istchee, and Kativik, which comprise the Nord-du-Québec region.
Equivalent territories by region
See also
List of regional county municipalities and equivalent territories in Quebec
External links
Local government in Quebec |
https://en.wikipedia.org/wiki/La%20Romaine%2C%20Quebec%20%28unconstituted%20locality%29 | La Romaine is an unconstituted locality (as defined by Statistics Canada in the Canada 2011 Census) within the municipality of Côte-Nord-du-Golfe-du-Saint-Laurent in the Côte-Nord region of Quebec, Canada. Its population in the 2011 census was 96.
It is directly adjacent to the much larger (in population) Indian reserve of the same name.
In 2017, Mrs Danielle Collard acts as the municipal secretary.
Education
Commission scolaire du Littoral operates the Marie-Sarah School for adults. Its school program for children was suspended in 2014. It was formerly a francophone school.
References
Communities in Côte-Nord
Unconstituted localities in Quebec
Road-inaccessible communities of Quebec |
https://en.wikipedia.org/wiki/Rotunda%20%28geometry%29 | In geometry, a rotunda is any member of a family of dihedral-symmetric polyhedra. They are similar to a cupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. The pentagonal rotunda is a Johnson solid.
Other forms can be generated with dihedral symmetry and distorted equilateral pentagons.
Examples
Star-rotunda
See also
Birotunda
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Johnson solids |
https://en.wikipedia.org/wiki/Asoka%20Handagama | Asoka Handagama (born December 1962) is a Sri Lankan filmmaker. He obtained his primary and secondary schooling in a provincial school and went on to study mathematics at the University of Kelaniya where he was awarded a first class honours degree. He obtained his MSc in Development Economics at Warwick University in 1995. He is also an Assistant Governor of the Sri Lankan Central Bank.
Theatre
Asoka Handagama's entry to filmmaking was via the theatre and television. His maiden theatrical effort, Bhoomika (film), was to address the seedling emerging ethnic crisis in the Island. The play won the National Youth award for best direction in 1985. His second stage play Thunder, was placed second runner up in the Best script in 1987 State drama Festival. The country was a real killing field when he directed his third, and most controversial play, Magatha. The play, with its radical theatrical form, bravely questioned the existing judicial system of the country. Magatha was shown almost all parts of the country, not only in the theatres, in the paddy fields and work places as well. Among all the controversies, the play won the Best Original Script and Best Director award in 1989, State Drama Festival. The script was published in 2011, and won the State Literary Award for Best Drama Script.
Television series
Asoka Handagama's exercises in the field of TV art were unique. Dunhidda Addara is a clear landmark in the history of so-called tele-dramas in Sri Lanka. It won all nine main awards including the Best Script and Direction, at the OCIC awards in 1994. Diyaketa Pahana, his third TV work, added a new dimension to the traditional tele-feature series. Synthetic Sihina explored a way to have a post-modern political discussion in the form of a serious episodic tele-play. He exploited the short spell of "ceasefire" (2003–2006) observed by Government forces and the Liberation Tigers of Tamil Ealam (LTTE) to shoot his next tele-feature series, Take This Road in Jaffna, the Northern capital of Sri Lanka to create a dialogue on the root causes of the ongoing war. East is Calling, the tele-feature series was on the same theme set in a tsunami rehabilitation camp.
Films
Chanda Kinnarie was his debut effort in cinema. Breaking the rules of so-called realism, this film clearly indicated the formation of a cinematic language consisting of hyper-realistic images. The film won the Award for Most Promising Director at the Critics' awards in 1994. It was also awarded Best Film, Best Director and Best Screenplay at the 1998 OCIC awards. Asoka Handagama's academic background in mathematics and development economics has stood him in good stead as an artist. Certainly it has helped him tackle the technical intricacies of film, television and the theatre, and to use these forms to maximum creative effect. More importantly this background has enabled him to sort out his priorities as a creative artiste who is conscious of the joys, sorrow and contradictions of daily life |
https://en.wikipedia.org/wiki/Ramanujan%20Mathematical%20Society | Ramanujan Mathematical Society is an Indian organisation of persons formed with the aim of "promoting mathematics at all levels". The Society was founded in 1985 and registered in Tiruchirappalli, Tamil Nadu, India. Professor G. Shankaranarayanan was the first President, Professor R. Balakrishnan the first Secretary and Professor E. Sampathkumar the first Academic Secretary. The initial impetus for the formation of the Society
was the deeply felt need of a new mathematical journal and the necessity of an organisation to launch and nourish the journal.
Publications
The publications of Ramanujan Mathematical Society include the following:
Mathematics Newsletter: A journal catering to the needs of students, research scholars, and teachers. The Newsletter was launched in the year 1991 with Professor R Balakrishnan as Chief Editor. Currently, Professor S Ponnusamy of IIT Madras is the Chief Editor.
Journal of the Ramanujan Mathematical Society : The Journal was started in 1986 with Professor K S Padmanabhan as Editor-in-Chief. Initially, it was a biannual Journal. Now it has four issues per year. The present Editor-in-Chief is Professor R Parimala of Emory University, Atlanta, United States and the Managing Editor is Professor E Sampathkumar of University of Mysore.
Little Mathematical Treasures: This is envisaged as a series of books addressed to mathematically mature readers and to bright students. So far only one book has been published under this series: "Adventures in Iteration" by Dr Shilesh A Shirali.
RMS Lecture Notes Series in Mathematics: This is a series consisting of monographs and proceedings of conferences.
Endowment Lectures
The Society organises the following endowment lectures every year.
Professor W H Abdi Memorial Lecture: The lectures were started in the year 2000 and are sponsored by Department of Mathematics, Cochin University of Science and Technology, of which Professor Wazir Hasan Abdi (1922–1999) was the Head during the period 1977 – 1982.
Professor C S Venkataraman Memorial Lectures: The lectures, started in 1996, are sponsored by Dr C S Venkataraman Memorial Trust, Thrissur, Kerala State.
Professor M N Gopalan Endowment Lectures: The lectures, started in 2000, are sponsored by Professor M N Gopalan, Mysore.
Prof J N Kapur Endowment Lectures: The lectures, started in 2002, are sponsored by Professor J N Kapur, New Delhi.
New members are taken in based on their achievements and capabilities.
Executive committee
References
Srinivasa Ramanujan
Mathematical societies
Indian mathematics
Organisations based in Tiruchirappalli
Science and technology in Tamil Nadu
1985 establishments in Tamil Nadu
Scientific organizations established in 1985 |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20PFC%20Levski%20Sofia%20season | The 2012–13 season is Levski Sofia's 91st season in the First League. This article shows player statistics and all matches (official and friendly) that the club has played during the 2012–13 season.
Transfers
Summer transfers
In:
Out:
See List of Bulgarian football transfers summer 2012
Winter transfers
In:
Out:
See List of Bulgarian football transfers winter 2013
Squad
Statistics
Goalscorers
Cards
Pre-season and friendlies
Summer
Winter
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
UEFA Europa League
Second qualifying round
References
PFC Levski Sofia seasons
Levski Sofia |
https://en.wikipedia.org/wiki/1958%20Liga%20Espa%C3%B1ola%20de%20Baloncesto | The 1958 season was the 2nd season of the Liga Española de Baloncesto. Real Madrid won their title.
Teams
Venues and locations
League table
Relegation playoffs
|}
Individual statistics
Points
External links
ACB.com
linguasport
1957
1958 in basketball
1957–58 in Spanish basketball |
https://en.wikipedia.org/wiki/National%20Mathematics%20Day%20%28India%29 | The 2012 Indian stamp featured Srinivasa Ramanujan.
The Indian government declared 22 December to be National Mathematics Day. It was introduced by Prime Minister Manmohan Singh on 26 December 2011 at Madras University, to mark the 125th birth anniversary of the Indian mathematician Srinivasa Ramanujan. On this occasion Prime minister Manmohan Singh also announced that 2012 would be celebrated as the National Mathematics Year.
Since then, India's National Mathematics Day is celebrated on 22 December every year with numerous educational events held at schools and universities throughout the country. In 2017, the day's significance was enhanced by the opening of the Ramanujan Math Park in Kuppam, in Chittoor, Andhra Pradesh. National Mathematics Day is celebrated in all schools and universities throughout the country.
References
Science and technology in India
Indian mathematics
Observances in India |
https://en.wikipedia.org/wiki/Committee%20for%20the%20Coordination%20of%20Statistical%20Activities | The Committee for the Coordination of Statistical Activities (CCSA) is composed of international and supranational organisations, whose mandate includes the provision of statistics. The CCSA promotes inter-agency coordination and cooperation on statistical programmes and consistency in statistical practices and development. As a forum of committed members, the CCSA fosters good practices in the statistical activities of international and supranational organisations, in accordance with the principles governing international statistical activities. The members of the CCSA contribute actively to the development of a coordinated global statistical system producing and disseminating high-quality statistics.
The CCSA Secretariat and CCSA website are hosted by the United Nations Statistics Division (UNSD). The Committee meets twice a year and is represented by the top level of the statistical services of its member organisations.
Membership as of November 2014
African Development Bank (AfDB)
Arab Institute for Training and Research in Statistics (AITRS)
Asian Development Bank (ADB)
Bank for International Settlements (BIS)
Caribbean Community (CARICOM)
European Central Bank (ECB)
Food and Agriculture Organization of the United Nations (FAO)
Inter-American Development Bank (IDB)
International Atomic Energy Agency (IAEA)
International Civil Aviation Organization (ICAO)
International Labour Organization (ILO)
International Monetary Fund (IMF)
International Telecommunication Union (ITU)
Interstate Statistical Committee of the Commonwealth of Independent States (CISSTAT)
Office for the Coordination of Humanitarian Affairs of the Secretariat (UNOCHA)
Office of the United Nations High Commissioner for Human Rights (OHCHR)
Organization for Economic Cooperation and Development (OECD)
PARIS21 (P21)
Statistical Centre for the Cooperation Council for the Arab Countries of the Gulf (GCC-Stat)
Statistical, Economic and Social Research and Training Centre for Islamic Countries (SESRIC)
Statistical Office of the European Union (EUROSTAT)
The Economic and Statistical Observatory of Sub-Saharan Africa (Afristat)
United Nations Children’s Fund (UNICEF)
United Nations Conference on Trade and Development (UNCTAD)
United Nations Development Programme (UNDP)
United Nations Economic Commission for Africa (UNECA)
United Nations Economic Commission for Europe (UNECE)
United Nations Economic Commission for Latin America and the Caribbean (UNECLAC)
United Nations Economic and Social Commission for Asia and the Pacific (UNESCAP)
United Nations Economic and Social Commission for Western Asia (UNESCWA)
United Nations Educational, Scientific and Cultural Organization Institute for Statistics (UNESCO-IS)
United Nations Entity for Gender Equality and the Empowerment of Women (UN-Women)
United Nations Environment Programme (UNEP)
United Nations High Commissioner for Refugees (UNHCR)
United Nations Human Settlements Programme (UN-Habitat)
United Nations Industrial Development Organ |
https://en.wikipedia.org/wiki/Dara%20%C3%93%20Briain%3A%20School%20of%20Hard%20Sums | Dara Ó Briain: School of Hard Sums is a British comedy game show about the subject of mathematics and it is based on the Japanese show Comaneci University Mathematics. The programme is broadcast on Dave and is presented by Dara Ó Briain and stars Marcus du Sautoy. Each episode is themed and Ó Briain, along with a guest or guests, attempts to solve various conundrums set by du Sautoy. At the end of each episode, Ó Briain sets homework questions for viewers; the answers can be checked on the show's website.
Transmissions
Episodes
A pilot episode was produced with the title Dara O'Briain's University of Practical Mathematics, but this was never broadcast.
The coloured backgrounds denote the result of each of the shows:
Series 1
Series 2
Series 3
Reaction
Readers of UKGameshows.com named it on a two way tie, the seventh best new game show of 2012 in their "Hall of fame" poll.
References
External links
2010s British comedy television series
2012 British television series debuts
2014 British television series endings
Dave (TV channel) original programming
English-language television shows |
https://en.wikipedia.org/wiki/Haji%20Ahmadov | Haji Ahmadov (, born on 23 November 1993 in Baku) is an Azerbaijani football defender who plays for Zagatala.
Career
Ahmadov left Qarabağ at the end of the 2014–15 season.
Career statistics
Achievements
Baku
Azerbaijan Cup (1): 2009–10
Qarabağ
Azerbaijan Premier League (2): 2013–14, 2014–15
Azerbaijan Cup (1): 2014–15
References
External links
1993 births
Living people
Azerbaijani men's footballers
Footballers from Baku
FC Baku players
Sumgayit FK players
Qarabağ FK players
Zira FK players
Azerbaijan Premier League players
Men's association football defenders
Azerbaijan men's youth international footballers
Azerbaijan men's under-21 international footballers
Azerbaijan men's international footballers |
https://en.wikipedia.org/wiki/Miriam%20G.%20Sherin | Miriam G. Sherin is a professor in the School of Education and Social Policy and the Learning Sciences Department at Northwestern University. Her areas of research include mathematics teaching and learning, teacher cognition, and teacher education. Sherin has published articles in Journal of Teacher Education, Teaching and Teacher Education, and Journal of Mathematics Teacher Education. Her most recent book, Mathematics Teacher Noticing: Seeing Through Teachers' Eyes, was publish in 2011 by Taylor & Francis. Since 2018 she has been associate provost for undergraduate education at Northwestern University. She is the sister of sociology scholar Adam Gamoran, currently president of the William T. Grant Foundation.
Career
Sherin taught math and science at Lincoln Junior High School (1987–88) in Oceanside, CA and Lincoln Middle School (1988–89) in Vista, CA. She then spent 6 years substitute teaching in Berkeley Unified School District (1989–92) in Berkeley, CA and West Contra Costa Unified School District (1993–95) in Richmond, CA.
In 1996, Sherin began her post-doctoral fellowship at the Stanford Graduate School of Education. After completing her research at Stanford University, Sherin began as an assistant professor in the Northwestern University School of Education and Social Policy in 1997. She is currently a professor at Northwestern University and director of the undergraduate education program.
Education
After attending public schools in Palatine, Illinois, Sherin received a BA in mathematics from University of Chicago in 1985. She went on to receive a master's degree in mathematics from University of California, San Diego in 1987. In 1996, she received a PhD in Science and Mathematics Education from University of California, Berkeley. Her thesis was titled The nature and dynamics of teachers' content knowledge.
Awards and honors
In 1996 Sherin received a postdoctoral fellowship from the James S. McDonnell Foundation to examine the demands that mathematics reform places on teachers' knowledge.
In 2001 she received a postdoctoral fellowship from the National Academy of Education and the Spencer Foundation to examine how video clubs can support the development of teachers' professional vision.
Sherin was also awarded a five-year Early Career Grant from the National Science Foundation to study the ways that video can support teacher learning.
In April 2003, Sherin received the Kappa Delta Pi/American Educational Research Association Division K Award for early career achievements in research on teaching and teacher education.
Selected publications
Brantlinger, A., Sherin, M. G., & Linsenmeier, K. (2011). Discussing discussion: A video club in the service of math teachers' National Board preparation. Teachers and Teaching, 17(1), 5 – 33.
Sherin, M. G., Linsenmeier, K. L., (2010). Principals' views of mathematics teacher learning. Journal of Mathematics Education Leadership. 20-32
van Es, E. A. & Sherin, M. G. (2010). The influen |
https://en.wikipedia.org/wiki/Jonckheere%27s%20trend%20test | In statistics, the Jonckheere trend test (sometimes called the Jonckheere–Terpstra test) is a test for an ordered alternative hypothesis within an independent samples (between-participants) design. It is similar to the Kruskal–Wallis test in that the null hypothesis is that several independent samples are from the same population. However, with the Kruskal–Wallis test there is no a priori ordering of the populations from which the samples are drawn. When there is an a priori ordering, the Jonckheere test has more statistical power than the Kruskal–Wallis test. The test was developed by Aimable Robert Jonckheere, who was a psychologist and statistician at University College London.
The null and alternative hypotheses can be conveniently expressed in terms of population medians for k populations (where k > 2). Letting θi be the population median for the ith population, the null hypothesis is:
The alternative hypothesis is that the population medians have an a priori ordering e.g.:
≤ ≤ ≤
with at least one strict inequality.
Procedure
The test can be seen as a special case of Maurice Kendall’s more general method of rank correlation and makes use of the Kendall's S statistic. This can be computed in one of two ways:
The ‘direct counting’ method
Arrange the samples in the predicted order
For each score in turn, count how many scores in the samples to the right are larger than the score in question. This is P.
For each score in turn, count how many scores in the samples to the right are smaller than the score in question. This is Q.
S = P – Q
The ‘nautical’ method
Cast the data into an ordered contingency table, with the levels of the independent variable increasing from left to right, and values of the dependent variable increasing from top to bottom.
For each entry in the table, count all other entries that lie to the ‘South East’ of the particular entry. This is P.
For each entry in the table, count all other entries that lie to the ‘South West’ of the particular entry. This is Q.
S = P – Q
Note that there will always be ties in the independent variable (individuals are ‘tied’ in the sense that they are in the same group) but there may or may not be ties in the dependent variable. If there are no ties – or the ties occur within a particular sample (which does not affect the value of the test statistic) – exact tables of S are available; for example, Jonckheere provided selected tables for values of k from 3 to 6 and equal samples sizes (m) from 2 to 5. Leach presented critical values of S for k = 3 with sample sizes ranging from 2,2,1 to 5,5,5.
Normal approximation to S
The standard normal distribution can be used to approximate the distribution of S under the null hypothesis for cases in which exact tables are not available. The mean of the distribution of S will always be zero, and assuming that there are no ties scores between the values in two (or more) different samples the variance is given by
Where n is the total number of sc |
https://en.wikipedia.org/wiki/Sobel%20test | In statistics, the Sobel test is a method of testing the significance of a mediation effect. The test is based on the work of Michael E. Sobel, a statistics professor at Columbia University in New York, NY, and is an application of the delta method. In mediation, the relationship between the independent variable and the dependent variable is hypothesized to be an indirect effect that exists due to the influence of a third variable (the mediator). As a result when the mediator is included in a regression analysis model with the independent variable, the effect of the independent variable is reduced and the effect of the mediator remains significant. The Sobel test is basically a specialized t test that provides a method to determine whether the reduction in the effect of the independent variable, after including the mediator in the model, is a significant reduction and therefore whether the mediation effect is statistically significant.
Theoretical basis
When evaluating a mediation effect three different regression models are examined:Model 1: YO = γ1 + τXI + ε1
Model 2: XM = γ2 + αXI + ε2
Model 3: YO = γ3 + τXI + βXM + ε3
In these models YO is the dependent variable, XI is the independent variable and XM is the mediator. The parameters γ1, γ2, and γ3 represent the intercepts for each model, while ε1, ε2, and ε3 represent the error term for each equation. τ denotes the relationship between the independent variable and the dependent variable in model 1, while τ denotes that same relationship in model 3 after controlling for the effect of the mediator. The terms αXI and βXM represent the relationship between the independent variable and the mediator, and the mediator and the dependent variable after controlling for the independent variable, respectively.
Product of coefficients
From these models, the mediation effect is calculated as (τ – τ). This represents the change in the magnitude of the effect that the independent variable has on the dependent variable after controlling for the mediator. From examination of these equations it can be determined that (αβ) = (τ – τ). The α term represents the magnitude of the relationship between the independent variable and the mediator. The β term represents the magnitude of the relationship between the mediator and dependent variable after controlling for the effect of the independent variable. Therefore (αβ) represents the product of these two terms. In essence this is the amount of variance in the dependent variable that is accounted for by the independent variable through the mechanism of the mediator. This is the indirect effect, and the (αβ) term has been termed the product of coefficients.
Venn diagram approach
Another way of thinking about the product of coefficients is to examine the figure below. Each circle represents the variance of each of the variables. Where the circles overlap represents variance the circles have in common and thus the effect of one variable on the second variable. For |
https://en.wikipedia.org/wiki/List%20of%20highest-grossing%20films%20in%20Malaysia | The statistics on international films' Box Office in Malaysia has started in 2008. Box Office Mojo is the only website that provides the box office numbers for international films released in Malaysia. However, this does not include the numbers for local films. For top local films gross, please view Cinema of Malaysia. Box Office - Yahoo! Malaysia and Cinema Online Malaysia are two current websites that show the ranking of films weekly inclusive of local films, but not providing any box office number. Golden Screen Cinemas (GSC) also provide only the ranking of both local and international films weekly, based on the popularity at its own cinema. The ranking can be accessed at the right bottom corner of GSC's website.
Highest-grossing local films in Malaysia
Below is the list of top 30 highest-grossing local films ever in Malaysia.
List of grossing local films
, Mat Kilau: Kebangkitan Pahlawan directed by Syamsul Yusof is currently the highest-grossing local film (and overall) of all time in Malaysia.
Highest-grossing international films in Malaysia
Below is the list of top 50 highest-grossing international films ever in Malaysia.
Note: All grosses are based on the final week of the film in local theatres and taken in estimation of local currency, as different rates between USD and Ringgit of Malaysia applied every week.
List of grossing international films
Transformers: Dark of the Moon grossed RM34,661,020 but due to the exchange rates, United International Pictures claimed that the film banked in a total of RM37,252,441.
^^ Note 2: Including the total gross of the re-release of the special edition.
Note 3: Films that are currently running in theatres nationwide.
Highest opening weekend films in history
Below is the list of the 30 biggest opening weekend of all time in Malaysia. As of 4 August 2019, Avengers: Endgame has the biggest opening weekend of all time in Malaysia, grossing RM 41,835,999 in the opening weekend alone and surpassed RM 10,000,000 on the opening day itself
As of 31 May 2023
^ Note 1: Furious 7 grossed RM20,769,458 on its opening weekend from Thursday to Sunday. However, if include sneak previews from 9pm onward on Wednesday, it opened to a total of RM26,623,534.
^^ Note 2: Avengers: Age of Ultron grossed RM17,876,018 on its opening weekend from Thursday to Sunday. If include sneak previews from Wednesday midnight onward, it opened to RM19,351,790.
^^^ Note 3: Iron Man 3 was considered to have higher opening weekend as it was a 3-day opening weekend compared to Transformers: Dark of the Moon 4-day opening weekend. The list ranks the films according to the total gross of opening weekend regardless of total days of opening weekend.
List of highest-grossing Indian films
2.0 holds the record for the biggest opening day (MYR 2,730,000) and the biggest opening weekend of any Indian movie in Malaysia.
Highest-grossing animated films in Malaysia
Below is the list of highest-grossing animated films ever in Malaysia ( |
https://en.wikipedia.org/wiki/Vincent%27s%20theorem | In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients.
Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them.
Sign variation
Let c0, c1, c2, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold:
If r = l+1 the numbers cl and cr have opposite signs.
If r ≥ l+2 the numbers cl+1, ..., cr−1 are all zero and the numbers cl and cr have opposite signs.
This is called a sign variation or sign change between the numbers cl and cr.
When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients.
Two versions of this theorem are presented: the continued fractions version due to Vincent, and the bisection version due to Alesina and Galuzzi.
Vincent's theorem: Continued fractions version (1834 and 1836)
If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
where are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:
Moreover, if infinitely many numbers satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.
The above statement is an exact translation of the theorem found in Vincent's original papers; however, the following remarks are needed for a clearer understanding:
If denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all either has no sign variation or it has one sign variation. In the latter case has a single positive real root for all .
The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming , we can only obtain a root of the original equation that is > 1. To obtain an arbitrary positive root we need to assume that .
Negative roots are obtained by replacing x by −x, in which case the negative roots become positive.
Vincent's theorem: Bisection version (Alesina and Galuzzi 200 |
https://en.wikipedia.org/wiki/Chapman%E2%80%93Kolmogorov%20equation | In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was derived independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov. CKE prominently used in recent Variational Bayesian methods.
Mathematical description
Suppose that { fi } is an indexed collection of random variables, that is, a stochastic process. Let
be the joint probability density function of the values of the random variables f1 to fn. Then, the Chapman–Kolmogorov equation is
i.e. a straightforward marginalization over the nuisance variable.
(Note that nothing yet has been assumed about the temporal (or any other) ordering of the random variables—the above equation applies equally to the marginalization of any of them.)
Application to time-dilated Markov chains
When the stochastic process under consideration is Markovian, the Chapman–Kolmogorov equation is equivalent to an identity on transition densities. In the Markov chain setting, one assumes that i1 < ... < in. Then, because of the Markov property,
where the conditional probability is the transition probability between the times . So, the Chapman–Kolmogorov equation takes the form
Informally, this says that the probability of going from state 1 to state 3 can be found from the probabilities of going from 1 to an intermediate state 2 and then from 2 to 3, by adding up over all the possible intermediate states 2.
When the probability distribution on the state space of a Markov chain is discrete and the Markov chain is homogeneous, the Chapman–Kolmogorov equations can be expressed in terms of (possibly infinite-dimensional) matrix multiplication, thus:
where P(t) is the transition matrix of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps.
As a corollary, it follows that to calculate the transition matrix of jump t, it is sufficient to raise the transition matrix of jump one to the power of t, that is
The differential form of the Chapman–Kolmogorov equation is known as a master equation.
See also
Fokker–Planck equation (also known as Kolmogorov forward equation)
Kolmogorov backward equation
Examples of Markov chains
Further reading
External links
Equations
Markov processes
Stochastic calculus |
https://en.wikipedia.org/wiki/Moderated%20mediation | In statistics, moderation and mediation can occur together in the same model. Moderated mediation, also known as conditional indirect effects, occurs when the treatment effect of an independent variable A on an outcome variable C via a mediator variable B differs depending on levels of a moderator variable D. Specifically, either the effect of A on B, and/or the effect of B on C depends on the level of D.
Langfred (2004) model
Langfred (2004) was the first to provide a comprehensive treatment of the question of how to conceptualize moderated mediation, classify different types of moderated mediation models, and to develop the logic and methodology for the statistical analysis of such models using multiple regression.
Because there was no established procedure to analyze models with moderated mediation, Langfred (2004) first describes the different types of moderated mediation models that might exist, noting that there are two primary forms of moderated mediation. Type 1, in which the moderator operates on the relationship between the independent variable and the mediator, and Type 2, in which the moderator operates on the relationship between the mediator and the dependent variable. Langfred reviews the existing perspectives on moderated mediation (James and Brett, 1984), and notes that an accepted statistical approach already exists for Type 1 moderated mediation, as demonstrated by Korsgaard, Brodt, and Whitener (2002). Type 2 moderation, however, is more statistically difficult, so Langfred reviews three different possible approaches for the analysis, and ultimately recommends one of them as the correct technique.
Langfred (2004) is often overlooked because the academic paper itself is not about statistical methodology. Rather, because the model in the paper involved moderated mediation, a very large appendix was included, in which the definitions and procedures for the regression analysis were developed.
Muller, Judd, & Yzerbyt (2005)
Muller, Judd, and Yzerbyt (2005) provided additional clarity and definition of moderated mediation. The following regression equations are fundamental to their model of moderated mediation, where A = independent variable, C = outcome variable, B = mediator variable, and D = moderator variable.
C = β40 + β41A + β42D + β43AD + ε4
This equation assesses moderation of the overall treatment effect of A on C.
B = β50 + β51A + β52D + β53AD + ε5
This equation assesses moderation of the treatment effect of A on the mediator B.
C = β60 + β61A + β62D + β63AD + β64B + β65BD + ε6
This equation assesses moderation of the effect of the mediator B on C, as well as moderation of the residual treatment effect of A on C.
This fundamental equality exists among these equations:
β43 – β63 = β64β53 + β65β51
In order to have moderated mediation, there must be an overall treatment effect of A on the outcome variable C (β41), which does not depend on the moderator (β43 = 0). In addition, the treatment effect of A o |
https://en.wikipedia.org/wiki/Gaussian%20moat | In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. More colorfully, if one imagines the Gaussian primes to be stepping stones in a sea of complex numbers, the question is whether one can walk from the origin to infinity with steps of bounded size, without getting wet. The problem was first posed in 1962 by Basil Gordon (although it has sometimes been erroneously attributed to Paul Erdős) and it remains unsolved.
With the usual prime numbers, such a sequence is impossible: the prime number theorem implies that there are arbitrarily large gaps in the sequence of prime numbers, and there is also an elementary direct proof: for any n, the n − 1 consecutive numbers n! + 2, n! + 3, ..., n! + n are all composite.
The problem of finding a path between two Gaussian primes that minimizes the maximum hop size is an instance of the minimax path problem, and the hop size of an optimal path is equal to the width of the widest moat between the two primes, where a moat may be defined by a partition of the primes into two subsets and its width is the distance between the closest pair that has one element in each subset. Thus, the Gaussian moat problem may be phrased in a different but equivalent form: is there a finite bound on the widths of the moats that have finitely many primes on the side of the origin?
Computational searches have shown that the origin is separated from infinity by a moat of width 6.
It is known that, for any positive number k, there exist Gaussian primes whose nearest neighbor is at distance k or larger. In fact, these numbers may be constrained to be on the real axis. For instance, the number 20785207 is surrounded by a moat of width 17. Thus, there definitely exist moats of arbitrarily large width, but these moats do not necessarily separate the origin from infinity.
References
Further reading
External links
Prime numbers
Unsolved problems in number theory
Complex numbers |
https://en.wikipedia.org/wiki/David%20Christiani | David Christiani (25 December 1610 – 13 February 1688) was a German mathematician, philosopher and Lutheran theologian. He became an ordinary professor of mathematics at the University of Marburg in 1643, ordinary professor of theology at the University of Giessen in 1681, and rector of the University of Giessen in 1686.
References
1610 births
1688 deaths
People from Gryfice
People from Pomerania
17th-century German mathematicians
German philosophers
German Lutheran theologians
17th-century German Protestant theologians
17th-century German philosophers
German male non-fiction writers
17th-century German writers
17th-century German male writers |
https://en.wikipedia.org/wiki/Hans%20Smees | Hans Smees (born 3 August 1970) is a Dutch motorcycle racer. He won the Dutch 250cc Championship in 2005 and 2006.
Career statistics
Grand Prix motorcycle racing
By season
Races by year
(key)
References
External links
Profile on MotoGP.com
Living people
1970 births
Dutch motorcycle racers
250cc World Championship riders
21st-century Dutch people |
https://en.wikipedia.org/wiki/CDF-based%20nonparametric%20confidence%20interval | In statistics, cumulative distribution function (CDF)-based nonparametric confidence intervals are a general class of confidence intervals around statistical functionals of a distribution. To calculate these confidence intervals, all that is required is an
independently and identically distributed (iid) sample from the distribution and known bounds on the support of the distribution. The latter requirement simply means that all the nonzero probability mass of the distribution must be contained in some known interval .
Intuition
The intuition behind the CDF-based approach is that bounds on the CDF of a distribution can be translated into bounds on statistical functionals of that distribution. Given an upper and lower bound on the CDF, the approach involves finding the CDFs within the bounds that maximize and minimize the statistical functional of interest.
Properties of the bounds
Unlike approaches that make asymptotic assumptions, including bootstrap approaches and those that rely on the central limit theorem, CDF-based bounds are valid for finite sample sizes. And unlike bounds based on inequalities such as Hoeffding's and McDiarmid's inequalities, CDF-based bounds use properties of the entire sample and thus often produce significantly tighter bounds.
CDF bounds
When producing bounds on the CDF, we must differentiate between pointwise and simultaneous bands.
Pointwise band
A pointwise CDF bound is one which only guarantees their Coverage probability of percent on any individual point of the empirical CDF. Because of the relaxed guarantees, these intervals can be much smaller.
One method of generating them is based on the Binomial distribution. Considering a single point of a CDF of value , then the empirical distribution at that point will be distributed proportional to the binomial distribution with and set equal to the number of samples in the empirical distribution. Thus, any of the methods available for generating a Binomial proportion confidence interval can be used to generate a CDF bound as well.
Simultaneous Band
CDF-based confidence intervals require a probabilistic bound on the CDF of the distribution from which the sample were generated. A variety of methods exist for generating confidence intervals for the CDF of a distribution, , given an i.i.d. sample drawn from the distribution. These methods are all based on the empirical distribution function (empirical CDF). Given an i.i.d. sample of size n, , the empirical CDF is defined to be
where is the indicator of event A. The Dvoretzky–Kiefer–Wolfowitz inequality, whose tight constant was determined by Massart, places a confidence interval around the Kolmogorov–Smirnov statistic between the CDF and the empirical CDF. Given an i.i.d. sample of size n from , the bound states
This can be viewed as a confidence envelope that runs parallel to, and is equally above and below, the empirical CDF.
The equally spaced confidence interval around the empirical CDF a |
https://en.wikipedia.org/wiki/Hesse%27s%20theorem | In geometry, Hesse's theorem, named for Otto Hesse, states that if two pairs of opposite vertices of a quadrilateral are conjugate with respect to some conic, then so is the third pair. A quadrilateral with this property is called a Hesse quadrilateral.
References
Theorems in projective geometry |
https://en.wikipedia.org/wiki/Jannik%20Stevens | Jannik Stevens (born 21 July 1992) is a German footballer who plays as a defender for SV Straelen.
Career statistics
References
External links
1992 births
Living people
German men's footballers
Men's association football defenders
2. Bundesliga players
Regionalliga players
VfL Bochum players
VfL Bochum II players
Alemannia Aachen players
SV 19 Straelen players |
https://en.wikipedia.org/wiki/David%20E.%20Muller | David Eugene Muller (November 2, 1924 – April 27, 2008) was an American mathematician and computer scientist. He was a professor of mathematics and computer science at the University of Illinois (1953–92), after which he became an emeritus professor, and was an adjunct professor of mathematics at the New Mexico State University (1995–2008). Muller received his BS in 1947 and his PhD in 1951 in physics from Caltech; an honorary PhD was conferred by the University of Paris in 1989. He was the inventor of the Muller C-element (or Muller C-gate), a device used to implement asynchronous circuitry in electronic computers. He also co-invented the Reed–Muller codes. He discovered the codes, and Irving S. Reed proposed the majority logic decoding for the first time. Furthermore, he invented Muller automata, an automaton model for infinite words. In geometric group theory Muller is known for the Muller–Schupp theorem, joint with Paul Schupp, characterizing finitely generated virtually free groups as finitely generated groups with context-free word problem.
Family
David E. Muller was the son of Hermann Joseph Muller and Jessie Jacobs Muller Offermann (formerly Jesse Marie Jacobs). He was born in Austin, Texas, when his parents taught at The University of Texas. His mother was one of the first women to receive a Ph.D. in mathematics in the United States, and he credited her with inspiring his early interest in mathematics. She lost her position as an instructor in pure mathematics at Texas because she became pregnant, and according to Hermann Joseph Muller's biographer, "her colleagues felt that a mother could not give full attention to classroom duties and remain a good mother." As a child he was with his parents in Berlin and Leningrad in 1933–34. His family was dissolved in the Soviet Union. He returned to Austin with his mother in July 1934. His mother obtained a divorce in Texas in the summer of 1935. Sometime between October 1935 and January 1936, Jessie Muller married Carlos Alberto Offermann, who had been working in Muller's laboratory and was on a visit to Austin from the Soviet Union at that time. Hermann Joseph Muller left the Soviet Union in 1937 after the start of Stalin's political persecutions. After a brief stay in Madrid and Paris, in September 1937, Hermann moved to Edinburgh, where he married Dorothea Kantorowicz in May 1939. They had a daughter, Helen Juliette. Hermann Joseph Muller received the Nobel Prize in Physiology or Medicine in 1946.
David E. Muller died in 2008 in Las Cruces, New Mexico. He is survived by his children, Chandra L. Muller and Kenneth J. Muller. His half-sister, Helen J. Muller, is a professor emerita at the University of New Mexico. He was predeceased by his wife Alice Mimi Muller, who died in Urbana, Illinois, in 1989, and divorced (posthumously) in 2009 from his second wife, Denise Impens Muller, in Las Cruces, New Mexico.
See also
Muller C-element
Reed–Muller code
Reed–Muller expansion
Muller's method (an e |
https://en.wikipedia.org/wiki/Tensors%20in%20curvilinear%20coordinates | Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics.
Vector and tensor algebra in three-dimensional curvilinear coordinates
Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna. Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, Naghdi, Simmonds, Green and Zerna, Basar and Weichert, and Ciarlet.
Coordinate transformations
Consider two coordinate systems with coordinate variables and , which we shall represent in short as just and respectively and always assume our index runs from 1 through 3. We shall assume that these coordinates systems are embedded in the three-dimensional euclidean space. Coordinates and may be used to explain each other, because as we move along the coordinate line in one coordinate system we can use the other to describe our position. In this way Coordinates and are functions of each other
for
which can be written as
for
These three equations together are also called a coordinate transformation from to .Let us denote this transformation by . We will therefore represent the transformation from the coordinate system with coordinate variables to the coordinate system with coordinates as:
Similarly we can represent as a function of as follows:
for
similarly we can write the free equations more compactly as
for
These three equations together are also called a coordinate transformation from to . Let us denote this transformation by . We will represent the transformation from the coordinate system with coordinate variables to the coordinate system with coordinates as:
If the transformation is bijective then we call the image of the transformation,namely , a set of admissible coordinates for . If is linear the coordinate system will be called an affine coordinate system ,otherwise is called a curvilinear coordinate system
The Jacobian
As we now see that the Coordinates and are functions of each other, we can take the derivative of the coordinate variable with respect to the coordinate variable
consider
for , these derivatives can be arranged in a matrix, say ,in which is the element in the -th row and -th column
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates
Let (b1, b2, b3) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal. However, they are required to be linearly independent. Then a vector v can be expressed as
|
https://en.wikipedia.org/wiki/Ambit%20field | In mathematics, an ambit field is a d-dimensional random field describing the stochastic properties of a given system. The input is in general a d-dimensional vector (e.g. d-dimensional space or (1-dimensional) time and (d − 1)-dimensional space) assigning a real value to each of the points in the field. In its most general form, the ambit field, , is defined by a constant plus a stochastic integral, where the integration is done with respect to a Lévy basis, plus a smooth term given by an ordinary Lebesgue integral. The integrations are done over so-called ambit sets, which is used to model the sphere of influence (hence the name, ambit, Latin for "sphere of influence" or "boundary") which affect a given point.
The use and development of ambit fields is motivated by the need of flexible stochastic models to describe turbulence and the evolution of electricity prices for use in e.g. risk management and derivative pricing. It was pioneered by Ole E. Barndorff-Nielsen and Jürgen Schmiegel to model turbulence and tumour growth.
Note, that this article will use notation that includes time as a dimension, i.e. we consider (d − 1)-dimensional space together with 1-dimensional time. The theory and notation easily carries over to d-dimensional space (either including time herin or in a setting involving no time at all).
Intuition and motivation
In stochastic analysis, the usual way to model a random process, or field, is done by specifying the dynamics of the process through a stochastic (partial) differential equation (SPDE). It is known, that solutions of (partial) differential equations can in some cases be given as an integral of a Green's function convolved with another function – if the differential equation is stochastic, i.e. contaminated by random noise (e.g. white noise) the corresponding solution would be a stochastic integral of the Green's function. This fact motivates the reason for modelling the field of interest directly through a stochastic integral, taking a similar form as a solution through a Green's Function, instead of first specifying a SPDE and then trying to find a solution to this. This provides a very flexible and general framework for modelling a variety of phenomena.
Definition
A tempo-spatial ambit field, , is a random field in space-time taking values in . Let be ambit sets in deterministic kernel functions, a stochastic function, a stochastic field (called the energy dissipation field in turbulence and volatility in finance) and a Lévy basis. Now, the ambit field is
Ambit sets
In the above, the ambit sets and describe the sphere of influence for a given point in space-time. I.e. at a given point, the sets and are the points in space-time which affect the value of the ambit field at . When time is considered as one of the dimensions, the sets are often taken to only include time-coordinates which are at or prior to the current time, t, so as to preserve causality of the field (i.e. a given point in space-t |
https://en.wikipedia.org/wiki/Martin%20Csirszki | Martin Csirszki (born 7 January 1995 in Miskolc) is a Hungarian football player who currently plays for Putnok FC.
Club statistics
Updated to games played as of 18 November 2014.
Honours
Diósgyőr
Hungarian League Cup (1): 2013–14
References
DVTK website
HLSZ
Nemzetisport
1995 births
Living people
Footballers from Miskolc
Hungarian men's footballers
Men's association football midfielders
Diósgyőri VTK players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Kriszti%C3%A1n%20P%C3%B3ti | Krisztián Póti (born 28 May 1988 in Budapest) is a Hungarian football player who currently plays for Erzsébeti SMTK.
Club statistics
Updated to games played as of 17 April 2018.
References
Player profile at HLSZ
1988 births
Living people
People from Mátészalka
Hungarian men's footballers
Men's association football defenders
Tököl KSK footballers
Jászberényi SE footballers
Bőcs KSC footballers
Hajdúböszörményi TE footballers
MTK Budapest FC players
Kecskeméti TE players
Nyíregyháza Spartacus FC players
Szigetszentmiklósi TK footballers
Aqvital FC Csákvár players
Soproni VSE players
Balmazújvárosi FC players
Monori SE players
Ceglédi VSE footballers
Erzsébeti Spartacus MTK LE footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Footballers from Szabolcs-Szatmár-Bereg County |
https://en.wikipedia.org/wiki/Eleven-point%20conic | In geometry, an eleven-point conic is a conic associated to four points and a line, containing 11 special points.
References
. Reprinted in 2010.
External links
Eleven-point conic
Projective geometry
Conic sections |
https://en.wikipedia.org/wiki/1958%E2%80%9359%20Liga%20Espa%C3%B1ola%20de%20Baloncesto | The 1958–59 season was the third season of the Liga Española de Baloncesto. Barcelona won their title.
Teams
Venues and locations
League table
Relegation playoffs
|}
Individual statistics
Points
External links
ACB.com
linguasport
Liga Española de Baloncesto (1957–1983) seasons
1958–59 in Spanish basketball |
https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Ganea%20theorem | In mathematics, particularly in homological algebra and algebraic topology, the Eilenberg–Ganea theorem states for every finitely generated group G with certain conditions on its cohomological dimension (namely ), one can construct an aspherical CW complex X of dimension n whose fundamental group is G. The theorem is named after Polish mathematician Samuel Eilenberg and Romanian mathematician Tudor Ganea. The theorem was first published in a short paper in 1957 in the Annals of Mathematics.
Definitions
Group cohomology: Let be a group and let be the corresponding Eilenberg−MacLane space. Then we have the following singular chain complex which is a free resolution of over the group ring (where is a trivial -module):
where is the universal cover of and is the free abelian group generated by the singular -chains on . The group cohomology of the group with coefficient in a -module is the cohomology of this chain complex with coefficients in , and is denoted by .
Cohomological dimension: A group has cohomological dimension with coefficients in (denoted by ) if
Fact: If has a projective resolution of length at most , i.e., as trivial module has a projective resolution of length at most if and only if for all -modules and for all .
Therefore, we have an alternative definition of cohomological dimension as follows,
The cohomological dimension of G with coefficient in is the smallest n (possibly infinity) such that G has a projective resolution of length n, i.e., has a projective resolution of length n as a trivial module.
Eilenberg−Ganea theorem
Let be a finitely presented group and be an integer. Suppose the cohomological dimension of with coefficients in is at most , i.e., . Then there exists an -dimensional aspherical CW complex such that the fundamental group of is , i.e., .
Converse
Converse of this theorem is an consequence of cellular homology, and the fact that every free module is projective.
Theorem: Let X be an aspherical n-dimensional CW complex with π1(X) = G, then cdZ(G) ≤ n.
Related results and conjectures
For n = 1 the result is one of the consequences of Stallings theorem about ends of groups.
Theorem: Every finitely generated group of cohomological dimension one is free.
For the statement is known as the Eilenberg–Ganea conjecture.
Eilenberg−Ganea Conjecture: If a group G has cohomological dimension 2 then there is a 2-dimensional aspherical CW complex X with .
It is known that given a group G with , there exists a 3-dimensional aspherical CW complex X with .
See also
Eilenberg–Ganea conjecture
Group cohomology
Cohomological dimension
Stallings theorem about ends of groups
References
.
Kenneth S. Brown, Cohomology of groups, Corrected reprint of the 1982 original, Graduate Texts in Mathematics, 87, Springer-Verlag, New York, 1994. .
Homological algebra
Theorems in algebraic topology |
https://en.wikipedia.org/wiki/Johannes%20Scheubel | Johannes Scheubel (18 August 1494 – 20 February 1570) was a German mathematician. His books include De Numeris et Diversis Rationibus (1545) and Algebrae Compendiosa (1551).
References
1494 births
1570 deaths
16th-century German mathematicians
16th-century German writers
16th-century German male writers |
https://en.wikipedia.org/wiki/Matrix%20completion | Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry represents the rating of movie by customer , if customer has watched movie and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. Another example is the document-term matrix: The frequencies of words used in a collection of documents can be represented as a matrix, where each entry corresponds to the number of times the associated term appears in the indicated document.
Without any restrictions on the number of degrees of freedom in the completed matrix this problem is underdetermined since the hidden entries could be assigned arbitrary values. Thus we require some assumption on the matrix to create a well-posed problem, such as assuming it has maximal determinant, is positive definite, or is low-rank.
For example, one may assume the matrix has low-rank structure, and then seek to find the lowest rank matrix or, if the rank of the completed matrix is known, a matrix of rank that matches the known entries. The illustration shows that a partially revealed rank-1 matrix (on the left) can be completed with zero-error (on the right) since all the rows with missing entries should be the same as the third row. In the case of the Netflix problem the ratings matrix is expected to be low-rank since user preferences can often be described by a few factors, such as the movie genre and time of release. Other applications include computer vision, where missing pixels in images need to be reconstructed, detecting the global positioning of sensors in a network from partial distance information, and multiclass learning. The matrix completion problem is in general NP-hard, but under additional assumptions there are efficient algorithms that achieve exact reconstruction with high probability.
In statistical learning point of view, the matrix completion problem is an application of matrix regularization which is a generalization of vector regularization. For example, in the low-rank matrix completion problem one may apply the regularization penalty taking the form of a nuclear norm
Low rank matrix completion
One of the variants of the matrix completion problem is to find the lowest rank matrix which matches the matrix , which we wish to recover, for all entries in the set of observed entries. The mathematical formulation of this problem is as follows:
Candès and Recht proved that with assumptions on the sampling of the observed entries and sufficiently many sampled entries this problem has a unique solution with high probability.
An equivalent formulation, given that the matrix to be recovered is known to be of rank , is |
https://en.wikipedia.org/wiki/Layered%20queueing%20network | In queueing theory, a discipline within the mathematical theory of probability, a layered queueing network (or rendezvous network) is a queueing network model where the service time for each job at each service node is given by the response time of a queueing network (and those service times in turn may also be determined by further nested networks). Resources can be nested and queues form along the nodes of the nesting structure. The nesting structure thus defines "layers" within the queueing model.
Layered queueing has applications in a wide range of distributed systems which involve different master/slave, replicated services and client-server components, allowing each local node to be represented by a specific queue, then orchestrating the evaluation of these queues.
For large population of jobs, a fluid limit has been shown in PEPA to be a give good approximation of performance measures.
External links
Tutorial Introduction to Layered Modeling of Software Performance by Murray Woodside, Carleton University
References
Distributed computing
Queueing theory
Network performance |
https://en.wikipedia.org/wiki/Egyptian%20Sign%20Language | Egyptian Sign Language is a sign language used by members of the deaf community in Egypt.
Sign language
Although there are no official statistics on the number of deaf people or the number of people who use Egyptian Sign Language as their primary language,
Gallaudet University's library resources website quotes a 1999 estimate of 2 million hearing impaired children, while a 2007 study by the World Health Organization places the prevalence of hearing loss in Egypt at 16.02% across all age groups. Egyptian Sign Language is not formally recognized by the government.
Linguistically, Egyptian Sign Language is not related to other sign languages of the Arab World, such as Jordanian Sign Language, Palestinian Sign Language, or Libyan Sign Language. Attempts at unification, creating an "Arabic Sign Language", have failed, as the unified form would be an entirely new language.
See also
Deafness in Egypt
References
External links
Egyptian Sign Puddle – Dictionary of Egyptian Sign Language (under construction)
YouTube ESL Dictionary – Animated Dictionary of Egyptian Sign Language (under construction)
The Deaf Unit – Private School in Cairo, associated with The Nardine Association NGO – جمعية الناردين
Arab sign languages
Languages of Egypt |
https://en.wikipedia.org/wiki/Pseudo%20great%20rhombicuboctahedron | In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboctahedron (a uniform polyhedron), but is not a uniform polyhedron (due to not being isogonal), and has a smaller symmetry group. It can be obtained from the great rhombicuboctahedron by taking a square face and the 8 faces with a common vertex to it (forming a crossed square cupola) and rotating them by an angle of . It is related to the nonconvex great rhombicuboctahedron in the same way that the pseudo rhombicuboctahedron is related to the rhombicuboctahedron.
Related polyhedra
The pseudo-great rhombicuboctahedron may also be termed an elongated crossed square gyrobicupola, in analogy to the name of the elongated square gyrobicupola.
References
. Reprinted in .
External links
George Hart - Pseudo Rhombicuboctahedra
Pseudo-uniform polyhedra |
https://en.wikipedia.org/wiki/McMullen%20problem | The McMullen problem is an open problem in discrete geometry named after Peter McMullen.
Statement
In 1972, David G. Larman wrote about the following problem:
Larman credited the problem to a private communication by Peter McMullen.
Equivalent formulations
Gale transform
Using the Gale transform, this problem can be reformulated as:
The numbers of the original formulation of the McMullen problem and of the Gale transform formulation are connected by the relationships
Partition into nearly-disjoint hulls
Also, by simple geometric observation, it can be reformulated as:
The relation between and is
Projective duality
The equivalent projective dual statement to the McMullen problem is to determine the largest number such that every set of hyperplanes in general position in d-dimensional real projective space form an arrangement of hyperplanes in which one of the cells is bounded by all of the hyperplanes.
Results
This problem is still open. However, the bounds of are in the following results:
David Larman proved in 1972 that
Michel Las Vergnas proved in 1986 that
Jorge Luis Ramírez Alfonsín proved in 2001 that
The conjecture of this problem is that . This has been proven for .
References
Discrete geometry
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/List%20of%20United%20States%20Davis%20Cup%20team%20representatives | This is a list of tennis players who have represented the United States Davis Cup team in an official Davis Cup match. The United States has taken part in the Davis Cup since 1901. Statistics correct as of 18 September 2017.
Players
References
Lists of Davis Cup tennis players
Davis |
https://en.wikipedia.org/wiki/Log-linear%20analysis | Log-linear analysis is a technique used in statistics to examine the relationship between more than two categorical variables. The technique is used for both hypothesis testing and model building. In both these uses, models are tested to find the most parsimonious (i.e., least complex) model that best accounts for the variance in the observed frequencies. (A Pearson's chi-square test could be used instead of log-linear analysis, but that technique only allows for two of the variables to be compared at a time.)
Fitting criterion
Log-linear analysis uses a likelihood ratio statistic that has an approximate chi-square distribution when the sample size is large:
where
natural logarithm;
observed frequency in cellij (i = row and j = column);
expected frequency in cellij.
the deviance for the model.
Assumptions
There are three assumptions in log-linear analysis:
1. The observations are independent and random;
2. Observed frequencies are normally distributed about expected frequencies over repeated samples. This is a good approximation if both (a) the expected frequencies are greater than or equal to 5 for 80% or more of the categories and (b) all expected frequencies are greater than 1. Violations to this assumption result in a large reduction in power. Suggested solutions to this violation are: delete a variable, combine levels of one variable (e.g., put males and females together), or collect more data.
3. The logarithm of the expected value of the response variable is a linear combination of the explanatory variables. This assumption is so fundamental that it is rarely mentioned, but like most linearity assumptions, it is rarely exact and often simply made to obtain a tractable model.
Additionally, data should always be categorical. Continuous data can first be converted to categorical data, with some loss of information. With both continuous and categorical data, it would be best to use logistic regression. (Any data that is analysed with log-linear analysis can also be analysed with logistic regression. The technique chosen depends on the research questions.)
Variables
In log-linear analysis there is no clear distinction between what variables are the independent or dependent variables. The variables are treated the same. However, often the theoretical background of the variables will lead the variables to be interpreted as either the independent or dependent variables.
Models
The goal of log-linear analysis is to determine which model components are necessary to retain in order to best account for the data. Model components are the number of main effects and interactions in the model. For example, if we examine the relationship between three variables—variable A, variable B, and variable C—there are seven model components in the saturated model. The three main effects (A, B, C), the three two-way interactions (AB, AC, BC), and the one three-way interaction (ABC) gives the seven model components.
The log-linear models can be tho |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Scunthorpe%20United%20F.C.%20season | The 2012–13 season is Scunthorpe United F.C.'s second consecutive in Third Tier of the Football League.
League One data
Standings
Results summary
Result round by round
Squad
Statistics
|-
|colspan="14"|Players currently on loan:
|-
|colspan="14"|Players featured for club who have left:
|}
Goalscoring record
Disciplinary record
Transfers
In
Loans in
Out
Loans out
Contracts
Fixtures & Results
Pre-season
League One
FA Cup
League Cup
League Trophy
Overall summary
Summary
Score overview
References
2012–13
2012–13 Football League One by team |
https://en.wikipedia.org/wiki/Harcourt%27s%20theorem | Harcourt's theorem is a formula in geometry for the area of a triangle, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle.
The theorem is named after J. Harcourt, an Irish professor.
Statement
Let a triangle be given with vertices A, B, and C, opposite sides of lengths a, b, and c, area K, and a line that is tangent to the triangle's incircle at any point on that circle. Denote the signed perpendicular distances of the vertices from the line as a ', b ', and c ', with a distance being negative if and only if the vertex is on the opposite side of the line from the incenter. Then
Degenerate case
If the tangent line contains one of the sides of the triangle, then two of the distances are zero and the formula collapses to the familiar formula that twice the area of a triangle is a base (the coinciding triangle side) times the altitude from that base.
Extension
If the line is instead tangent to the excircle opposite, say, vertex A of the triangle, then
Dual property
If rather than a', b', c' referring to distances from a vertex to an arbitrary incircle tangent line, they refer instead to distances from a sideline to an arbitrary point, then the equation
remains true.
References
Theorems about triangles and circles |
https://en.wikipedia.org/wiki/Analytic%20subgroup%20theorem | In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s. It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences.
Statement
If is a commutative algebraic group defined over an algebraic number field and is a Lie subgroup of with Lie algebra defined over the number field then does not contain any non-zero algebraic point of unless contains a proper algebraic subgroup.
One of the central new ingredients of the proof was the theory of multiplicity estimates of group varieties developed by David Masser and Gisbert Wüstholz in special cases and established by Wüstholz in the general case which was necessary for the proof of the analytic subgroup theorem.
Consequences
One of the spectacular consequences of the analytic subgroup theorem was the Isogeny Theorem published by Masser and Wüstholz. A direct consequence is the Tate conjecture for abelian varieties which Gerd Faltings had proved with totally different methods which has many applications in modern arithmetic geometry.
Using the multiplicity estimates for group varieties Wüstholz succeeded to get the final expected form for lower bound for linear forms in logarithms. This was put into an effective form in a joint work of him with Alan Baker which marks the current state of art. Besides the multiplicity estimates a further new ingredient was a very sophisticated use of geometry of numbers to obtain very sharp lower bounds.
See also
Algebraic curve
Citations
References
Transcendental numbers |
https://en.wikipedia.org/wiki/Seymour%20Lipschutz | Seymour Saul Lipschutz (born 1931 died March 2018) was an author of technical books on pure mathematics and probability, including a collection of Schaum's Outlines.
Lipschutz received his Ph.D. in 1960 from New York University's Courant Institute . He received his BA and MA degrees in Mathematics at Brooklyn College. He was a mathematics professor at Temple University, and before that on the faculty at the Polytechnic Institute of Brooklyn.
Bibliography
Schaum's Outline of Discrete Mathematics
Schaum's Outline of Probability
Schaum's Outline of Finite Mathematics
Schaum's Outline of Linear Algebra
Schaum's Outline of Beginning Linear Algebra
Schaum's Outline of Set Theory
Schaum's Outline of General Topology
Schaum's Outline of Data Structures
Schaum's Outline of Differential Geometry
References
1931 births
2018 deaths
20th-century American mathematicians
21st-century American mathematicians
Courant Institute of Mathematical Sciences alumni
Temple University faculty
Polytechnic Institute of New York University faculty
Brooklyn College alumni |
https://en.wikipedia.org/wiki/William%20Galbraith%20%28mathematician%29 | Rev William Galbraith (1786 – 27 October 1850) was a Scottish mathematician. He taught mathematics and nautical astronomy in Edinburgh, and took an interest in surveying work, becoming an advocate of the extension of the work of triangulating Great Britain.
Early life
He was born at Greenlaw, Berwickshire. Initially he was a schoolmaster. His pupil
William Rutherford walked long distances to attend his school at Eccles. Subsequently, he moved to Edinburgh, and graduated A.M. at the University of Edinburgh in 1821.
Surveyor
During the 1830s Galbraith became interested in the surveying problems of Scotland. In 1831 he pointed out that Arthur's Seat had a strongly magnetic peak. In 1837 he pointed out the impact of anomalies in measurement, work that received recognition; it was topical because of the 1836 geological map of Scotland by John MacCulloch, with which critics had found fault on topographical as well as geological grounds. A paper on the locations of places on the River Clyde was recognised in 1837 by a gold medal, from the Society for the Encouragement of the Useful Arts for Scotland.
Galbraith followed with detailed Remarks on the Geographical Position of some Points on the West Coast of Scotland (1838). Having made some accurate surveys of his own, he lobbied for further attention from the national survey.
Later life
About 1832 Galbraith was licensed a minister by the presbytery of Dunse. He married Eleanor Gale in 1833.
Galbraith was buried with his wife in the north-east section of the Grange Cemetery in Edinburgh.
Works
Galbraith's major works combined textbook material with mathematical tables:
Mathematical and Astronomical Tables (1827): review.
Trigonometrical Surveying, Levelling, and Railway Engineering (1842)
He edited John Ainslie's 1812 treatise on land surveying (1849), and with William Rutherford revised John Bonnycastle's Algebra.
Notes
External links
Online Books page
1786 births
1850 deaths
Scottish mathematicians
19th-century Scottish clergy
Alumni of the University of Edinburgh
19th-century Scottish mathematicians |
https://en.wikipedia.org/wiki/Double%20pushout%20graph%20rewriting | In computer science, double pushout graph rewriting (or DPO graph rewriting) refers to a mathematical framework for graph rewriting. It was introduced as one of the first algebraic approaches to graph rewriting in the article "Graph-grammars: An algebraic approach" (1973). It has since been generalized to allow rewriting structures which are not graphs, and to handle negative application conditions, among other extensions.
Definition
A DPO graph transformation system (or graph grammar) consists of a finite graph, which is the starting state, and a finite or countable set of labeled spans in the category of finite graphs and graph homomorphisms, which serve as derivation rules. The rule spans are generally taken to be composed of monomorphisms, but the details can vary.
Rewriting is performed in two steps: deletion and addition.
After a match from the left hand side to is fixed, nodes and edges that are not in the right hand side are deleted. The right hand side is then glued in.
Gluing graphs is in fact a pushout construction in the category of graphs, and the deletion is the same as finding a pushout complement, hence the name.
Uses
Double pushout graph rewriting allows the specification of graph transformations by specifying a pattern of fixed size and composition to be found and replaced, where part of the pattern can be preserved. The application of a rule is potentially non-deterministic: several distinct matches can be possible. These can be non-overlapping, or share only preserved items, thus showing a kind of concurrency known as parallel independence, or they may be incompatible, in which case either the applications can sometimes be executed sequentially, or one can even preclude the other.
It can be used as a language for software design and programming (usually a variant working on richer structures than graphs is chosen). Termination for DPO graph rewriting is undecidable because the Post correspondence problem can be reduced to it.
DPO graph rewriting can be viewed as a generalization of Petri nets.
Generalization
Axioms have been sought to describe categories in which DPO rewriting will work. One possibility is the notion of an adhesive category, which also enjoys many closure properties. Related notions are HLR systems, quasi-adhesive categories and -adhesive categories, adhesive HLR categories.
The concepts of adhesive category and HLR system are related (an adhesive category with coproducts is a HLR system).
Hypergraph, typed graph and attributed graph rewriting, for example, can be handled because they can be cast as adhesive HLR systems.
Notes
Graph algorithms
Graph rewriting |
https://en.wikipedia.org/wiki/Napoleon%20points | In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.
The name "Napoleon points" has also been applied to a different pair of triangle centers, better known as the isodynamic points.
Definition of the points
First Napoleon point
Let be any given plane triangle. On the sides of the triangle, construct outwardly drawn equilateral triangles respectively. Let the centroids of these triangles be respectively. Then the lines are concurrent. The point of concurrence is the first Napoleon point, or the outer Napoleon point, of the triangle .
The triangle is called the outer Napoleon triangle of . Napoleon's theorem asserts that this triangle is an equilateral triangle.
In Clark Kimberling's Encyclopedia of Triangle Centers, the first Napoleon point is denoted by X(17).
The trilinear coordinates of :
The barycentric coordinates of :
Second Napoleon point
Let be any given plane triangle. On the sides of the triangle, construct inwardly drawn equilateral triangles respectively. Let the centroids of these triangles be respectively. Then the lines are concurrent. The point of concurrence is the second Napoleon point, or the inner Napoleon point, of .
The triangle is called the inner Napoleon triangle of . Napoleon's theorem asserts that this triangle is an equilateral triangle.
In Clark Kimberling's Encyclopedia of Triangle Centers, the second Napoleon point is denoted by X(18).
The trilinear coordinates of :
The barycentric coordinates of :
Two points closely related to the Napoleon points are the Fermat-Torricelli points (ETC's X(13) and X(14)). If instead of constructing lines joining the equilateral triangles' centroids to the respective vertices one now constructs lines joining the equilateral triangles' apices to the respective vertices of the triangle, the three lines so constructed are again concurrent. The points of concurrence are called the Fermat-Torricelli points, sometimes denoted and . The intersection of the Fermat line (i.e., that line joining the two Fermat-Torricelli points) and the Napoleon line (i.e., that line joining the two Napoleon points) is the triangle's symmedian point (ETC's X(6)).
Generalizations
The results regarding the existence of the Napoleon points can be generalized in different ways. In defining the Napoleon points we begin with equilateral triangles drawn on the sides of and then consider the centers of these triangles. These centers can be thought as the vertices of isosceles triangles erected on the sides of triangle ABC with the base angles equal to /6 (30 degrees). The generalizations seek to determine other triangles th |
https://en.wikipedia.org/wiki/List%20of%20Hertha%20BSC%20managers | This is list details of Hertha BSC coaches and their statistics, trophies and other records.
Tenure, wins, draws, losses and winning percentage
Combined records
Trophies won
References
Managers
Lists of football managers in Germany by club |
https://en.wikipedia.org/wiki/Cho%20Young-hoon | Cho Young-hoon (; born 13 April 1989) is a South Korean footballer who plays as a defender for FC Anyang in the K League 2.
Club career statistics
External links
1989 births
Living people
Men's association football defenders
South Korean men's footballers
Daegu FC players
FC Anyang players
K League 1 players
K League 2 players
Dongguk University alumni |
https://en.wikipedia.org/wiki/Karl%20Reinhardt%20%28mathematician%29 | Karl August Reinhardt (27 January 1895 – 27 April 1941) was a German mathematician whose research concerned geometry, including polygons and tessellations. He solved one of the parts of Hilbert's eighteenth problem, and is the namesake of the Reinhardt polygons.
Life
Reinhardt was born on January 27, 1895, in Frankfurt, the descendant of farming stock. One of his childhood friends was mathematician Wilhelm Süss. After studying at the gymnasium there, he became a student at the University of Marburg in 1913 before his studies were interrupted by World War I. During the war, he became a soldier, a high school teacher, and an assistant to mathematician David Hilbert at the University of Göttingen.
Reinhardt completed his Ph.D. at Goethe University Frankfurt in 1918. His dissertation, Über die Zerlegung der Ebene in Polygone, concerned tessellations of the plane, and was supervised by Ludwig Bieberbach. He began working as a secondary school teacher while working on his habilitation with Bieberbach, which he completed in 1921; titled Über Abbildungen durch analytische Funktionen zweier Veränderlicher, it concerned functional analysis.
Bieberbach moved to Berlin in 1921, taking Süss as an assistant. They left Reinhardt in Frankfurt, working two jobs as a high school teacher and junior faculty at the university. In 1924, Reinhardt moved to the University of Greifswald as an extraordinary professor, under the leadership of Johann Radon; this gave him an income sufficient to support himself without a second job, and afforded him more time for research. He became an ordinary professor at Greifswald in 1928.
He remained in Greifswald for the rest of his career, "with an outstanding research record and a reputation as a fine, thoughtful teacher". However, despite his now-comfortable position, his health was poor, and he died in Berlin on April 27, 1941, aged 46.
Contributions
In his doctoral dissertation, Reinhardt discovered the five tile-transitive pentagon tilings. In a 1922 paper, Extremale Polygone gegebenen Durchmessers, he solved the odd case of the biggest little polygon problem, and found the Reinhardt polygons, equilateral polygons inscribed in Reuleaux polygons that solve several related optimization problems.
He had long been interested in Hilbert's eighteenth problem, a shared interest with Bieberbach, who in 1911 had solved a part of the problem asking for the classification of space groups. A second part of the problem asked for a tessellation of Euclidean space by a tile that is not the fundamental region of any group. In a 1928 paper, Zur Zerlegung der euklidischen Räume in kongeuente Polytope Reinhardt solved this part by finding an example of such a tessellation. In a later development, Heinrich Heesch showed in 1935 that tilings with this property exist even in the two-dimensional Euclidean plane.
Another of his works, Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurve |
https://en.wikipedia.org/wiki/Eboardmuseum | The Eboardmuseum, founded in 1987 by the musician, mathematics teacher and engineer Gert Prix, is a collection of electronic keyboard instruments.
It very quickly outgrew the original venue and, in 2007, the collection was moved into a hall at the fair area in the centre of Klagenfurt am Wörthersee in Austria and is now considered to be the largest museum of its kind worldwide. In 2010, the Eboardmuseum was honored with the Austrian "Museumsguetesiegel" seal of quality.
At the invitation of Google, the Eboardmuseum has been participating in the Music, Makers & Machines project as part of Google Arts & Culture since March 2021. As part of this project, it presents a small selection of its exhibits, accompanied by video examples.
The Museum
In an area of about 2,000 m2 the Eboardmuseum has about 2,000 exhibits.
Focusing on electronic keyboards the Eboardmuseum covers the entire history of these instruments from a 1935 Hammond model A to an up-to-date Moog Voyager.
Among the exhibits there are numerous preliminary models and unique items such as a Hohner Clavinet, Rhodes Piano, Mellotron as well as original instruments from international stars like Keith Emerson (Emerson, Lake & Palmer), Geoff Downes (Asia), Rick Wakeman (Yes), Peter Wolf (Frank Zappa), Tangerine Dream, Ken Hensley (Uriah Heep), Dave Greenslade (Colosseum), Eddie Hardin (Spencer Davis Group), King Crimson, George Duke, Ray Charles, Opus and Grateful Dead.
Unlike other musical instrument museums the Eboardmuseum not only presents its exhibits in guided tours, but also allows visitors to play the instruments. Professionals and music enthusiasts from all around the world make use of this opportunity.
The Eboardmuseum library contains literature about electronic music, focusing on keyboards, pop music and pop culture.
Guided tours in the Eboardmuseum target everybody including non-musicians and children. They aim to offer an entertaining and family-friendly trip into the world of music.
Events
The Eboardmuseum hosts, on average, 50 live concerts a year. The integrated event area in the Eboardmuseum is furnished with sofas, offering a living room experience. Due to its unique location and quirky atmosphere, it constantly attracts big names from the world of music. In spite of its a relatively small auditorium, musicians such as Carl Palmer (Emerson, Lake & Palmer, Asia), Ian Paice (Deep Purple), Peter Ratzenbeck, Brian Auger, Wolfgang Ambros, Alex Ligertwood (Santana), Ken Hensley, Hans Theessink, Barbara Dennerlein, Nick Simper and Don Airey (Deep Purple) and Waterloo & Robinson have all performed on the Eboardmuseum stage. The previous and future programs are documented on the Eboardmuseum website.
Service area
In open workshops visitors can watch the museum technicians at work and have an insight into the inner workings of the instruments.
See also
List of music museums
References
External links
Eboardmuseum website
Buildings and structures in Klagenfurt
Museu |
https://en.wikipedia.org/wiki/Algebraic%20semantics | Algebraic semantics may refer to:
Algebraic semantics (computer science)
Algebraic semantics (mathematical logic) |
https://en.wikipedia.org/wiki/Algebraic%20semantics%20%28computer%20science%29 | In computer science, algebraic semantics is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program specifications in a formal manner.
Syntax
The syntax of an algebraic specification is formulated in two steps: (1) defining a formal signature of data types and operation symbols, and (2) interpreting the signature through sets and functions.
Definition of a signature
The signature of an algebraic specification defines its formal syntax. The word "signature" is used like the concept of "key signature" in musical notation.
A signature consists of a set of data types, known as sorts, together with a family of sets, each set containing operation symbols (or simply symbols) that relate the sorts.
We use to denote the set of operation symbols relating the sorts to the sort .
For example, for the signature of integer stacks, we define two sorts, namely, and , and the following family of operation symbols:
where denotes the empty string.
Set-theoretic interpretation of signature
An algebra interprets the sorts and operation symbols as sets and functions.
Each sort is interpreted as a set , which is called the carrier of of sort , and each symbol in is mapped to a function , which is called an operation of .
With respect to the signature of integer stacks, we interpret the sort as the set of integers, and interpret the sort as the set of integer stacks. We further interpret the family of operation symbols as the following functions:
Semantics
Semantics refers to the meaning or behavior. An algebraic specification provides both the meaning and behavior of the object in question.
Equational axioms
The semantics of an algebraic specifications is defined by axioms in the form of conditional equations.
With respect to the signature of integer stacks, we have the following axioms:
For any and ,
where "" indicates "not found".
Mathematical semantics
The mathematical semantics (also known as denotational semantics) of a specification refers to its mathematical meaning.
The mathematical semantics of an algebraic specification is the class of all algebras that satisfy the specification.
In particular, the classic approach by Goguen et al. takes the initial algebra (unique up to isomorphism) as the "most representative" model of the algebraic specification.
Operational semantics
The operational semantics of a specification means how to interpret it as a sequence of computational steps.
We define a ground term as an algebraic expression without variables. The operational semantics of an algebraic specification refers to how ground terms can be transformed using the given equational axioms as left-to-right rewrite rules, until such terms reach their normal forms, where no more rewriting is possible.
Consider the axioms for integer stacks. Let "" denote "rewrites to".
Canonical property
An algebraic specification is said to be confluent (also known as Church-Rosser) if the rewriting of any |
https://en.wikipedia.org/wiki/Spieker%20center | In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in the shape of . The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.
Location
The following result can be used to locate the Spieker center of any triangle.
The Spieker center of triangle is the incenter of the medial triangle of .
That is, the Spieker center of is the center of the circle inscribed in the medial triangle of . This circle is known as the Spieker circle.
The Spieker center is also located at the intersection of the three cleavers of triangle . A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. Each cleaver contains the center of mass of the boundary of , so the three cleavers meet at the Spieker center.
To see that the incenter of the medial triangle coincides with the intersection point of the cleavers, consider a homogeneous wireframe in the shape of triangle consisting of three wires in the form of line segments having lengths . The wire frame has the same center of mass as a system of three particles of masses placed at the midpoints of the sides . The centre of mass of the particles at and is the point which divides the segment in the ratio . The line is the internal bisector of . The centre of mass of the three particle system thus lies on the internal bisector of . Similar arguments show that the center mass of the three particle system lies on the internal bisectors of and also. It follows that the center of mass of the wire frame is the point of concurrence of the internal bisectors of the angles of the triangle , which is the incenter of the medial triangle .
Properties
Let be the Spieker center of triangle .
The trilinear coordinates of are
The barycentric coordinates of are
is the radical center of the three excircles.
is the cleavance center of triangle
is collinear with the incenter (), the centroid (), and the Nagel point () of triangle . Moreover,
Thus on a suitably scaled and positioned number line, , , , and .
lies on the Kiepert hyperbola. is the point of concurrence of the lines where are similar, isosceles and similarly situated triangles constructed on the sides of triangle as bases, having the common base angle
References
Triangle centers |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Manitoba | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Manitoba had 148 designated places, an increase from 135 in 2016. Designated place types in Manitoba include 9 dissolved municipalities, 44 local urban districts, 46 northern communities, and 48 unincorporated urban centres. In 2021, the 148 designated places had a cumulative population of 89,803 and an average population of . Manitoba's largest designated place is Oakbank with a population of 5,041.
List
See also
List of census agglomerations in Manitoba
List of population centres in Manitoba
Notes
References
Designated |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20New%20Brunswick | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, New Brunswick had 161 designated places, an increase from 157 in 2016. Designated place types in New Brunswick include 8 former local governments, 152 local service districts and a single retired population centre. In 2021, the 161 designated places had a cumulative population of 93,925 and an average population of . New Brunswick's largest designated place is Tracadie with a population of 5,349.
List
See also
List of census agglomerations in Atlantic Canada
List of population centres in New Brunswick
Notes
References
Lists of populated places in New Brunswick
Local government in New Brunswick |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Newfoundland%20and%20Labrador | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Newfoundland and Labrador had 207 designated places, an increase from 199 in 2016. Among these designated places are 5 retired population centres. In 2021, the 207 designated places had a cumulative population of 44,012 and an average population of . Newfoundland and Labrador's largest designated place is Goulds with a population of 4,441.
List
See also
List of census agglomerations in Atlantic Canada
List of communities in Newfoundland and Labrador
List of local service districts in Newfoundland and Labrador
List of municipalities in Newfoundland and Labrador
List of population centres in Newfoundland and Labrador
Notes
References
Designated |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Nova%20Scotia | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Nova Scotia had 70 designated places, an increase from 68 in 2016. Designated place types in Nova Scotia include 66 class IV areas and 4 retired population centres. In 2021, the 70 designated places had a cumulative population of 44,090 and an average population of . Nova Scotia's largest designated place is Bible Hill with a population of 5,076.
List
See also
List of census agglomerations in Atlantic Canada
List of population centres in Nova Scotia
Notes
References
Designated |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Ontario | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Ontario had 135 designated places, an increase from 129 in 2016. Designated place types in Ontario include 45 dissolved municipalities, 44 local service boards, 37 municipal defined places, and 9 dissolved population centres. In 2021, the 135 designated places had a cumulative population of 74,105 and an average population of . Ontario's largest designated place is Breslau with a population of 5,053.
List
See also
List of census agglomerations in Ontario
List of population centres in Ontario
Notes
References
Designated |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Quebec | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Quebec had 120 designated places, an increase from 117 in 2016. Designated place types in Quebec include 14 retired population centres, 94 dissolved municipalities (municipalité dissoute), and 12 unconstituted localities (localité non constituée). In 2021, the 120 designated places had a cumulative population of 80,697 and an average population of . Quebec's largest designated place is Sainte-Agathe-des-Monts with a population of 6,740.
List
See also
List of census agglomerations in Quebec
List of population centres in Quebec
List of unconstituted localities in Quebec
Municipal history of Quebec
References
Lists of populated places in Quebec |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Saskatchewan | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places, so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, Saskatchewan had 198 designated places, an increase from 193 in 2016. Designated place types in Saskatchewan include 2 cluster subdivisions, 40 dissolved municipalities, 9 northern settlements, 143 organized hamlets, 2 resort subdivisions, and 2 retired population centre. In 2021, the 198 designated places had a cumulative population of 11,858, and an average population of . Saskatchewan's largest designated place is Gravelbourg with a population of 986.
List
See also
List of census agglomerations in Saskatchewan
List of cities in Saskatchewan
List of communities in Saskatchewan
List of ghost towns in Saskatchewan
List of hamlets in Saskatchewan
List of Indian reserves in Saskatchewan
List of municipalities in Saskatchewan
List of population centres in Saskatchewan
List of resort villages in Saskatchewan
List of rural municipalities in Saskatchewan
List of towns in Saskatchewan
List of villages in Saskatchewan
Notes
References
Designated |
https://en.wikipedia.org/wiki/Theodor%20Spieker | Theodor Spieker (8 August 1823 – 9 April 1913) was a German mathematician, a teacher at a gymnasium in Potsdam.
Spieker's geometry textbook (Verlag von August Stein, Potsdam, 1862) was republished in many editions. A copy of this textbook was given to Albert Einstein by his tutor when Einstein was twelve, and quickly led Einstein to become interested in higher mathematics.
Spieker is the namesake of the Spieker circle of a triangle (the circle inscribed in its medial triangle) and the Spieker center (the center of the Spieker circle).
References
1823 births
1913 deaths
19th-century German mathematicians |
https://en.wikipedia.org/wiki/Catalan%20pseudoprime | In mathematics, a Catalan pseudoprime is an odd composite number n satisfying the congruence
where Cm denotes the m-th Catalan number. The congruence also holds for every odd prime number n that justifies the name pseudoprimes for composite numbers n satisfying it.
Properties
The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then p2 is a Catalan pseudoprime.
References
Catalan pseudoprimes. Research in Scientific Computing in Undergraduate Education.
Pseudoprimes |
https://en.wikipedia.org/wiki/Michael%20Esser | Michael Esser (born 22 November 1987) is a German professional footballer who plays as a goalkeeper for Bundesliga club VfL Bochum.
Career statistics
References
External links
1987 births
Living people
People from Castrop-Rauxel
Footballers from Münster (region)
German men's footballers
Men's association football goalkeepers
Bundesliga players
2. Bundesliga players
Regionalliga players
Landesliga players
Austrian Football Bundesliga players
VfL Bochum players
VfL Bochum II players
SK Sturm Graz players
SV Darmstadt 98 players
Hannover 96 players
TSG 1899 Hoffenheim players
German expatriate men's footballers
German expatriate sportspeople in Austria
Expatriate men's footballers in Austria |
https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%20theorem | In symplectic topology and dynamical systems, Poincaré–Birkhoff theorem (also known as Poincaré–Birkhoff fixed point theorem and Poincaré's last geometric theorem) states that every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points.
History
The Poincaré–Birkhoff theorem was discovered by Henri Poincaré, who published it in a 1912 paper titled "Sur un théorème de géométrie", and proved it for some special cases. The general case was proved by George D. Birkhoff in his 1913 paper titled "Proof of Poincaré's geometric theorem".
References
Further reading
M. Brown; W. D. Neumann. "Proof of the Poincaré-Birkhoff fixed-point theorem". Michigan Math. J. Vol. 24, 1977, p. 21–31.
P. Le Calvez; J. Wang. "Some remarks on the Poincaré–Birkhoff theorem". Proc. Amer. Math. Soc. Vol. 138, No.2, 2010, p. 703–715.
J. Franks. "Generalizations of the Poincaré-Birkhoff Theorem", Annals of Mathematics, Second Series, Vol. 128, No. 1 (Jul., 1988), pp. 139–151.
Symplectic topology
Dynamical systems
Fixed-point theorems |
https://en.wikipedia.org/wiki/1960%E2%80%9361%20Liga%20Espa%C3%B1ola%20de%20Baloncesto | The 1960–61 season was the 5th season of the Liga Española de Baloncesto. R. Madrid won their title.
Teams and venues
League table
Relegation playoffs
|}
Individual statistics
Points
References
ACB.com
linguasport
Liga Española de Baloncesto (1957–1983) seasons
1960–61 in Spanish basketball |
https://en.wikipedia.org/wiki/Square%20antiprismatic%20molecular%20geometry | In chemistry, the square antiprismatic molecular geometry describes the shape of compounds where eight atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a square antiprism. This shape has D4d symmetry and is one of the three common shapes for octacoordinate transition metal complexes, along with the dodecahedron and the bicapped trigonal prism.
Like with other high coordination numbers, eight-coordinate compounds are often distorted from idealized geometries, as illustrated by the structure of Na3TaF8. In this case, with the small Na+ ions, lattice forces are strong. With the diatomic cation NO+, the lattice forces are weaker, such as in (NO)2XeF8, which crystallizes with a more idealized square antiprismatic geometry.
Examples
Square prismatic geometry and cubic geometry
Square prismatic geometry (D4h) is much less common compared to the square antiprism. An example of a molecular species with square prismatic geometry (a slightly flattened cube) is octafluoroprotactinate(V), [PaF8]3–, as found in its sodium salt, Na3PaF8. While local cubic 8-coordination is common in ionic lattices (e.g., Ca2+ in CaF2), and some 8-coordinate actinide complexes are approximately cubic, there are no reported examples of rigorously cubic 8-coordinate molecular species. A number of other rare geometries for 8-coordination are also known.
References
Stereochemistry
Molecular geometry |
https://en.wikipedia.org/wiki/Capped%20square%20antiprismatic%20molecular%20geometry | In chemistry, the capped square antiprismatic molecular geometry describes the shape of compounds where nine atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a gyroelongated square pyramid.
The gyroelongated square pyramid is a square pyramid with a square antiprism connected to the square base. In this respect, it can be seen as a "capped" square antiprism (a square antiprism with a pyramid erected on one of the square faces).
It is very similar to the tricapped trigonal prismatic molecular geometry, and there is some dispute over the specific geometry exhibited by certain molecules.
Examples
is sometimes described as having a capped square antiprismatic geometry, although its geometry is most often described as tricapped trigonal prismatic.
, a lanthanum(III) complex with a La–La bond.
References
Stereochemistry
Molecular geometry |
https://en.wikipedia.org/wiki/List%20of%201.%20FC%20N%C3%BCrnberg%20managers | This is list details of 1. FC Nürnberg managers and their statistics, trophies and other records.
Managers, tenure, wins, draws, losses and winning percentage
References
Managers
Lists of football managers in Germany by club |
https://en.wikipedia.org/wiki/STEAM%20fields | STEAM fields are the areas of science, technology, engineering, the arts, and mathematics. STEAM is designed to integrate STEM subjects with arts subjects into various relevant education disciplines. These programs aim to teach students innovation, to think critically, and to use engineering or technology in imaginative designs or creative approaches to real-world problems while building on students' mathematics and science base. STEAM programs add arts to STEM curriculum by drawing on reasoning and design principles, and encouraging creative solutions.
STEAM in children's media
Sesame Streets 43rd season continues to focus on STEM but finds ways to integrate art. They state: "This helps make learning STEM concepts relevant and enticing to young children by highlighting how artists use STEM knowledge to enhance their art or solve problems. It also provides context for the importance of STEM knowledge in careers in the arts (e.g. musician, painter, sculptor, and dancer)."
MGA Entertainment created a S.T.E.A.M. based franchise Project Mc2.
Other uses of the STEAM acronym
Other meanings of the "A" that have been promoted include agriculture, architecture, and applied mathematics.
The Rhode Island School of Design has a STEM to STEAM program and at one point maintained an interactive map that showed global STEAM initiatives. Relevant organizations were able to add themselves to the map, though it is no longer available at the location stated in press releases. John Maeda, (2008 to 2013 president of Rhode Island School of Design) has been a champion in bringing the initiative to the political forums of educational policy.
Some programs offer STEAM from a base focus like mathematics and science.
SteamHead is a non-profit organization that promotes innovation and accessibility in education, focusing on STEAM fields.
As part of a $1.5 million Department of Education grant, Wolf Trap's Institute of Education trains and places teaching artists in preschool and kindergarten classrooms. The artists collaborate with the teachers to integrate math and science into the arts.
American Lisa La Bonte, CEO of the Arab Youth Venture Foundation based in the United Arab Emirates, uses the STEAM acronym, but her work does not include arts integration. Starting in 2007, La Bonte created and ran high-profile free public STEAM programs having added an A for "inspired STEM", with the A standing for Aeronautics, Aviation, Astronomy, Aerospace, Ad Astra! and using all things "air and space" as a hook for youth to embark on greater experimentation, studies, and careers in the region's burgeoning space-related industries. One of AYVF's best-known programs, "STEAM@TheMall", served over 200,000 in its first two years at the most popular shopping malls and provided free weekend activity stations such as Mars robotics, science experiments, SkyLab portable planetarium, art/design, and creative writing. In 2008, Sharjah Sheikha Maisa kicked off the "Design booth for youth |
https://en.wikipedia.org/wiki/Ren%C3%A9%20Schoof | René Schoof (born 8 May 1955 in Den Helder) is a mathematician from the Netherlands who works in number theory, arithemtic geometry, and coding theory.
He received his PhD in 1985 from the University of Amsterdam with Hendrik Lenstra (Elliptic Curves and Class Groups). He is now a professor at the University Tor Vergata in Rome.
In 1985, Schoof discovered an algorithm which enabled him to count points on elliptic curves over finite fields in polynomial time. This was important for the use of elliptic curves in cryptography, and represented a theoretical breakthrough, as it was the first deterministic polynomial time algorithm for counting points on elliptic curves. The algorithms known before (e.g. the baby-step giant-step algorithm) were of exponential running time. His algorithm was improved by A. O. L. Atkin (1992) and Noam Elkies (1990).
He obtained the best known result extending Deligne's Theorem for finite flat group schemes to the non commutative setting, over certain local Artinian rings. His interests range throughout Algebraic Number Theory, Arakelov theory, Iwasawa theory, problems related to existence and classification of Abelian varieties over the rationals with bad reduction in one prime only, and algorithms.
In the past, René has also worked with Rubik's cubes by creating a common strategy in speedsolving used to set many world records known as F2L Pairs, in which the solver creates four 2-piece "pairs" with one edge and corner piece which are each "inserted" into F2L slots in the CFOP method to finish the first two layers of a 3x3x3 Rubik's cube. This strategy is also used for all cubes of higher order (4x4x4 and up) in the Reduction, Yau, and Hoya methods if CFOP is used for their 3x3x3 stages.
He also wrote a book on Catalan's conjecture.
See also
Schoof's algorithm
Schoof–Elkies–Atkin algorithm
External links
Homepage
Some publications
Counting points of elliptic curves over finite fields, Journal des Théories des Nombres de Bordeaux, No. 7, 1995, 219–254, pdf
With Gerard van der Geer, Ben Moonen (editors): Number fields and function fields – two parallel worlds, Birkhäuser 2005
Finite flat group schemes over Artin rings, Compositio Mathematica, v. 128 (2001), 115
Catalan's Conjecture, Universitext, Springer, 2008
References
1955 births
Living people
20th-century Dutch mathematicians
21st-century Dutch mathematicians
Number theorists
University of Amsterdam alumni
People from Den Helder
Academic staff of the University of Rome Tor Vergata |
https://en.wikipedia.org/wiki/Pentagonal%20planar%20molecular%20geometry | In chemistry, the pentagonal planar molecular geometry describes the shape of compounds where five atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a pentagon.
Examples
The only two pentagonal planar species known are the isoelectronic (nine valence electrons) ions and . Both are derived from the pentagonal bipyramid with two lone pairs occupying the apical positions and the five fluorine atoms all equatorial.
References
Stereochemistry
Molecular geometry |
https://en.wikipedia.org/wiki/Kantowski%E2%80%93Sachs%20metric | In general relativity the Kantowski-Sachs metric (named after Ronald Kantowski and Rainer K. Sachs) describes a homogeneous but anisotropic universe whose spatial section has the topology of . The metric is:
The isometry group of this spacetime is . Remarkably, the isometry group does not act simply transitively on spacetime, nor does it possess a subgroup with simple transitive action.
See also
Bianchi classification
Dust solution
Notes
General relativity
Physical cosmology |
https://en.wikipedia.org/wiki/France%20Kri%C5%BEani%C4%8D | France Križanič (3 March 1928 – 17 January 2002) was a Slovene mathematician, author of numerous books and textbooks on mathematics. He was professor of mathematical analysis at the Faculty of Mathematics and Physics of the University of Ljubljana.
Križanič won the Levstik Award twice, in 1951 for his book Kratkočasna matematika (Maths for Fun) and in 1960 for Križem po matematiki and Elektronski aritmetični računalniki (Criss Cross Across Maths and Electronic Calculators).
Published works
Kratkočasna matematika (Maths for Fun), 1951
Križem kražem po matematiki (Criss Cross Across Maths), 1960
Elektronski aritmetični računalniki (Electronic Calculators), 1960
Vektorji, matrike, tenzorji (Vectors, Matrices, Tensors), 1962
Aritmetika, algebra in analiza – I.del (Arithmetics, Algebra and Analysis – Part I.), 1963
Aritmetika, algebra in analiza – II.del (Arithmetics, Algebra and Analysis – Part II.), 1964
Aritmetika, algebra in analiza – III.del (Arithmetics, Algebra and Analysis – Part III.), 1964
Aritmetika, algebra in analiza – IV.del (Arithmetics, Algebra and Analysis – Part IV.), 1964
Operatorski račun in integralske transformacije (Operational Calculus and Integral Transforms), 1965
Vektorska in tenzorska analiza (Vector and Tensor Analysis), 1966
Linearna algebra in linearna analiza (Linear Algebra and Linear Analysis), 1969
Funkcije več kompleksnih spremenljivk (Functions of Complex Numbers), 1971
Dinamični sistemi (Dynamical Systems), 1972
Topološke grupe (Topological Groups), 1974
Navadne diferencialne enačbe in variacijski račun (Ordinary Differential Equations and Variational Calculus), 1974
Liejeve grupe (Lie groups), 1976
Liejeve algebra (Lie Algebra), 1978
Linearna algebra (Linear Algebra), 1978
Matematika – prvo berilo (Mathematics – First Year Textbook), 1981
Matematika – drugo berilo (Mathematics – Second Year Textbook), 1981
Matematika – tretje berilo (Mathematics – Third Year Textbook), 1983
Matematika – četrto berilo (Mathematics – Fourth Year Textbook), 1985
Linearna analiza na grupah (Linear Analysis on Groups), 1982
Nihalo, prostor in delci (The pendulum, Space and Particles), 1982
Temelji realne matematične analize (The Foundation of Real Mathematical Analysis), 1990
Linearna algebra in linearna analiza (Linear Algebra and Linear Analysis), 1993
Vektorska in tenzorska analiza – abeceda globalne analize (Vector and Tensor Analysis – the Alphabet of Global Analysis), 1996
Splošno in posebno : (nakladanja in otepanja) (General and Specific Stuff : Collection of Notes), published posthumously 2003
References
1928 births
2002 deaths
20th-century Slovenian mathematicians
People from Maribor
Levstik Award laureates
University of Ljubljana alumni
Academic staff of the University of Ljubljana
Yugoslav mathematicians |
https://en.wikipedia.org/wiki/Discrete%20least%20squares%20meshless%20method | In mathematics the discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries.
Description
While most of the existing meshless methods need background cells for numerical integration, DLSM did not require a numerical integration procedure due to the use of the discrete least squares method to discretize the governing differential equation. A Moving least squares (MLS) approximation method is used to construct the shape function, making the approach a fully least squares-based approach.
Arzani and Afshar developed the DLSM method in 2006 for the solution of Poisson's equation. Firoozjaee and Afshar proposed the collocated discrete least squares meshless (CDLSM) method to solve elliptic partial differential equations, and studied the effect of the collocation points on the convergence and accuracy of the method. The method can be considered as an extension the earlier method of DLSM by the introduction of a set of collocation points for the calculation of the least squares functional.
CDLSM was later used by Naisipour et al. to solve elasticity problems regarding the irregular distribution of nodal points. Afshar and Lashckarbolok used the CDLSM method for the adaptive simulation of hyperbolic problems. A simple a posteriori error indicator based on the value of the least squares functional and a node moving strategy was used and tested on 1-D hyperbolic problems. Shobeyri and Afshar simulated free surface problems using the DLSM method.
The method was then extended for adaptive simulation of two-dimensional shocked hyperbolic problems by Afshar and Firoozjaee. Also, adaptive node-moving refinement and multi-stage node enrichment adaptive refinement are formulated in the DLSM for the solution of elasticity problems.
Amani, Afshar and Naisipour. proposed mixed discrete least squares meshless (MDLSM) formulation for solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet type, which is easily incorporated via a penalty method. Because this is a least squares based algorithm of the MDLSM method, the proposed method does not need to be satisfied by the Ladyzhenskaya–Babuška–Brezzi (LBB) condition.
Notes
References
H. Arzani, M.H. Afshar, Solving Poisson's equations by the discrete least square meshless method, WIT Transactions on Modelling and Simulation 42 (2006) 23–31.
M. H. Afshar, M. Lashckarbolok, Collocated discre |
https://en.wikipedia.org/wiki/Hermite%20reciprocity | In mathematics, Hermite's law of reciprocity, introduced by , states that the degree m covariants of a binary form of degree n correspond to the degree n covariants of a binary form of degree m. In terms of representation theory it states that the representations Sm Sn C2 and Sn Sm C2 of GL2 are isomorphic.
References
Invariant theory |
https://en.wikipedia.org/wiki/Statistics%20Without%20Borders | Statistics Without Borders (SWB) is an organization of volunteers that provide pro bono statistical consulting and assistance to organizations or governments to help deal with health issues. SWB is sponsored by the American Statistical Association. Their goal is to help international health initiatives and projects be delivered more effectively through better use of statistics.
Some of the past and present projects include design and analysis about a survey about a public health intervention project in Sierra Leone, and another in Haiti, a survey of the impact of the economic impact of an earthquake in Haiti, reviewing food security survey projects for the Food and Nutritional Technical Assistance II (FANTA-2) project at the Academy for Educational Development, and helping to prepare a proposal to survey households in Mexico about their use of bottled water.
Projects
References
External links
Statistics Without Borders homepage
Statistical societies
Organizations established in 2008 |
https://en.wikipedia.org/wiki/Petr%E2%80%93Douglas%E2%80%93Neumann%20theorem | In geometry, the Petr–Douglas–Neumann theorem (or the PDN-theorem) is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as Petr–Douglas–Neumann theorem, or as the PDN-theorem for short, is due to Stephen B Gray. This theorem has also been called Douglas's theorem, the Douglas–Neumann theorem, the Napoleon–Douglas–Neumann theorem and Petr's theorem.
The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.
Statement of the theorem
The Petr–Douglas–Neumann theorem asserts the following.
If isosceles triangles with apex angles 2kπ/n are erected on the sides of an arbitrary n-gon A0, and if this process is repeated with the n-gon formed by the free apices of the triangles, but with a different value of k, and so on until all values 1 ≤ k ≤ n − 2 have been used (in arbitrary order), then a regular n-gon An−2 is formed whose centroid coincides with the centroid of A0.
Specialisation to triangles
In the case of triangles, the value of n is 3 and that of n − 2 is 1. Hence there is only one possible value for k, namely 1. The specialisation of the theorem to triangles asserts that the triangle A1 is a regular 3-gon, that is, an equilateral triangle.
A1 is formed by the apices of the isosceles triangles with apex angle 2π/3 erected over the sides of the triangle A0. The vertices of A1 are the centers of equilateral triangles erected over the sides of triangle A0. Thus the specialisation of the PDN theorem to a triangle can be formulated as follows:
If equilateral triangles are erected over the sides of any triangle, then the triangle formed by the centers of the three equilateral triangles is equilateral.
The last statement is the assertion of the Napoleon's theorem.
Specialisation to quadrilaterals
In the case of quadrilaterals, the value of n is 4 and that of n − 2 is 2. There are two possible values for k, namely 1 and 2, and so two possible apex angles, namely:
(2×1×π)/4 = π/2 = 90° ( corresponding to k = 1 )
(2×2×π)/4 = π = 180° ( corresponding to k = 2 ).
According to the PDN-theorem the quadrilateral A2 is a regular 4-gon, that is, a square. The two-stage process yielding the square A2 can be carried out in two different ways. (The apex Z of an isosceles triangle with apex angle π erected over a line segment XY is the midpoint of the line segment XY.)
Construct A1 using apex angle π/2 and then A2 with apex angle π.
In this case the vertices of A1 are the free apices of isosceles triangles with apex angles π/2 erecte |
https://en.wikipedia.org/wiki/2012%20President%27s%20Cup%20%28Maldives%29%20final | The 2012 President's Cup (Maldives) Final was the 62nd Final of the Maldives President's Cup.
Route to the final
Match
Details
Statistics
See also
2012 President's Cup (Maldives)
References
President's Cup (Maldives) finals
Pres |
https://en.wikipedia.org/wiki/Multipartition | In number theory and combinatorics, a multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn a partition. The concept is also found in the theory of Lie algebras.
r-component multipartitions
An r-component multipartition of an integer n is an r-tuple of partitions λ(1),...,λ(r) where each λ(i) is a partition of some ai and the ai sum to n. The number of r-component multipartitions of n is denoted Pr(n). Congruences for the function Pr(n) have been studied by A. O. L. Atkin.
References
Number theory
Combinatorics |
https://en.wikipedia.org/wiki/Ternary%20cubic | In mathematics, a ternary cubic form is a homogeneous degree 3 polynomial in three variables.
Invariant theory
The ternary cubic is one of the few cases of a form of degree greater than 2 in more than 2 variables whose ring of invariants was calculated explicitly in the 19th century.
The ring of invariants
The algebra of invariants of a ternary cubic under SL3(C) is a polynomial algebra generated by two invariants S and T of degrees 4 and 6, called Aronhold invariants. The invariants are rather complicated when written as polynomials in the coefficients of the ternary cubic, and are given explicitly in
The ring of covariants
The ring of covariants is given as follows.
The identity covariant U of a ternary cubic has degree 1 and order 3.
The Hessian H is a covariant of ternary cubics of degree 3 and order 3.
There is a covariant G of ternary cubics of degree 8 and order 6 that vanishes on points x lying on the
Salmon conic of the polar of x with respect to the curve and its Hessian curve.
The Brioschi covariant J is the Jacobian of U, G, and H of degree 12, order 9.
The algebra of covariants of a ternary cubic is generated over the ring of invariants by U, G, H, and J, with a relation that the square of J is a polynomial in the other generators.
The ring of contravariants
The Clebsch transfer of the discriminant of a binary cubic is a contravariant F of ternary cubics of degree 4 and class 6, giving the dual cubic of a cubic curve.
The Cayleyan P of a ternary cubic is a contravariant of degree 3 and class 3.
The quippian Q of a ternary cubic is a contravariant of degree 5 and class 3.
The Hermite contravariant Π is another contravariant of ternary cubics of degree 12 and class 9.
The ring of contravariants is generated over the ring of invariants by F, P, Q, and Π, with a relation that Π2 is a polynomial in the other generators.
The ring of concomitants
and described the ring of concomitants, giving 34 generators.
The Clebsch transfer of the Hessian of a binary cubic is a concomitant of degree 2, order 2, and class 2.
The Clebsch transfer of the Jacobian of the identity covariant and the Hessian of a binary cubic is a concomitant of ternary cubics of degree 3, class 3, and order 3
See also
Ternary quartic
Invariants of a binary form
References
Invariant theory |
https://en.wikipedia.org/wiki/Roland%20Weitzenb%C3%B6ck | Roland Weitzenböck (26 May 1885 – 24 July 1955) was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection. He was appointed professor of mathematics at the University of Amsterdam in 1923 at the initiative of Brouwer, after Hermann Weyl had turned down Brouwer’s offer.
Biography
Roland Weitzenböck was born in Kremsmünster, Austria-Hungary. He studied from 1902 to 1904 at the Imperial and Royal Technical Military Academy (now HTL Vienna) and was a captain in the Austrian army. He then studied at the University of Vienna, where he graduated in 1910 with the dissertation Zum System von 3 Strahlenkomplexen im 4-dimensionalen Raum (The system of 3-rays complexes in 4-dimensional space). After further studies at Bonn and Göttingen, he became professor at the University of Graz in 1912. After Army service in World War I, he became Professor of Mathematics at the Karl-Ferdinand University in Prague in 1918.
In 1923 Weitzenböck took a position of professor of mathematics at the University of Amsterdam, where he stayed until 1945. He settled in Blaricum, where he became a fully accepted member of the community. He was a man of few words, without observable political views. Appearances are often, however, deceptive, and in this case the solid imperturbable exterior hid a considerable amount of frustration resulting from the disastrous course of the First World War. As so many German and Austrian ex-service men, Weitzenböck became a hard-core revanchist, and an implacable enemy of France. But whereas Brouwer actively campaigned for the rehabilitation of German scientists, Weitzenböck refrained from political activity. However, after the ‘Anschluss’ of Austria in 1938, he started to vent his approval of Hitler’s policies in private conversations. Weitzenböck was elected member of the Royal Netherlands Academy of Arts and Sciences (KNAW) in May 1924, but suspended in May 1945 because of his attitude during the war. Weitzenböck had been a member of the National Socialist Movement in the Netherlands.
In 1923 Weitzenböck published a modern monograph on the theory of invariants on manifolds that included tensor calculus. In the Preface of this monograph one can read an offensive acrostic. One finds that the first letter of the first word in the first 21 sentences spell out:
NIEDER MIT DEN FRANZOSEN (down with the French).
He also published papers on torsion. In fact, in his paper "Differential Invariants in Einstein’s Theory of Tele-parallelism" Weitzenböck had given a supposedly complete bibliography of papers on torsion without mentioning Élie Cartan. Weitzenböck died in Zelhem, Netherlands in 1955. His doctoral students include G. F. C. Griss, Daniel Rutherford and Max Euwe.
Publications
Neuere Arbeiten zur algebraischen Invariantentheorie. Differentialinvarianten. Enzyklopädie der mathematischen Wissenschaften, III, Bd.3, Teubner 1921
Differentialinvarianten in der Einsteinschen Theorie des Fernpara |
https://en.wikipedia.org/wiki/Ternary%20quartic | In mathematics, a ternary quartic form is a degree 4 homogeneous polynomial in three variables.
Hilbert's theorem
showed that a positive semi-definite ternary quartic form over the reals can be written as a sum of three squares of quadratic forms.
Invariant theory
The ring of invariants is generated by 7 algebraically independent invariants of degrees 3, 6, 9, 12, 15, 18, 27 (discriminant) , together with 6 more invariants of degrees 9, 12, 15, 18, 21, 21, as conjectured by . discussed the invariants of order up to about 15.
The Salmon invariant is a degree 60 invariant vanishing on ternary quartics with an
inflection bitangent.
Catalecticant
The catalecticant of a ternary quartic is the resultant of its 6 second partial derivatives. It vanishes when the ternary quartic can be written as a sum of five 4th powers of linear forms.
See also
Ternary cubic
Invariants of a binary form
References
.
External links
Invariants of the ternary quartic
Invariant theory |
https://en.wikipedia.org/wiki/Lemoine%27s%20problem | In geometry, Lemoine's problem is a straightedge and compass construction problem posed by French mathematician Émile Lemoine in 1868:
Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle.
The problem was published as Question 864 in (Series 2, Volume 7 (1868), p 191). The chief interest in the problem is that a discussion of the solution of the problem by Ludwig Kiepert published in (series 2, Volume 8 (1869), pp 40–42) contained a description of a hyperbola which is now known as the Kiepert hyperbola.
Ludwig Kiepert's solution
Kiepert establishes the validity of his construction by proving a few lemmas.
Problem
Let be the vertices of the equilateral triangles placed on the sides of a triangle Given construct .
Lemma 1
If on the three sides of an arbitrary triangle one describes equilateral triangles then the line segments are equal, they concur in a point , and the angles they form one another are equal to 60°.
Lemma 2
If on one makes the same construction as that on there will have three equilateral triangles three equal line segments which will also concur at the point .
Lemma 3
are respectively the midpoints of
Solution
Describe on the segments the equilateral triangles respectively.
The midpoints of are, respectively, the vertices of the required triangle.
Other solutions
Several other people in addition to Kiepert submitted their solutions during 1868–9, including Messrs Williere (at Arlon), Brocard, Claverie (Lycee de Clermont), Joffre (Lycee Charlemagne), Racine (Lycee de Poitiers), Augier (Lycee de Caen), V. Niebylowski, and L. Henri Lorrez.
Kiepert's solution was more complete than the others.
References
Triangle problems |
https://en.wikipedia.org/wiki/Protein%20fold%20class | In molecular biology, protein fold classes are broad categories of protein tertiary structure topology. They describe groups of proteins that share similar amino acid and secondary structure proportions. Each class contains multiple, independent protein superfamilies (i.e. are not necessarily evolutionarily related to one another).
Generally recognised classes
Four large classes of protein that are generally agreed upon by the two main structure classification databases (SCOP and CATH).
all-α
All-α proteins are a class of structural domains in which the secondary structure is composed entirely of α-helices, with the possible exception of a few isolated β-sheets on the periphery.
Common examples include the bromodomain, the globin fold and the homeodomain fold.
all-β
All-β proteins are a class of structural domains in which the secondary structure is composed entirely of β-sheets, with the possible exception of a few isolated α-helices on the periphery.
Common examples include the SH3 domain, the beta-propeller domain, the immunoglobulin fold and B3 DNA binding domain.
α+β
α+β proteins are a class of structural domains in which the secondary structure is composed of α-helices and β-strands that occur separately along the backbone. The β-strands are therefore mostly antiparallel.
Common examples include the ferredoxin fold, ribonuclease A, and the SH2 domain.
α/β
α/β proteins are a class of structural domains in which the secondary structure is composed of alternating α-helices and β-strands along the backbone. The β-strands are therefore mostly parallel.
Common examples include the flavodoxin fold, the TIM barrel and leucine-rich-repeat (LRR) proteins such as ribonuclease inhibitor.
Additional classes
Membrane proteins
Membrane proteins interact with biological membranes either by inserting into it, or being tethered via a covalently attached lipid. They are one of the common types of protein along with soluble globular proteins, fibrous proteins, and disordered proteins. They are targets of over 50% of all modern medicinal drugs. It is estimated that 20–30% of all genes in most genomes encode membrane proteins.
Intrinsically disordered proteins
Intrinsically disordered proteins lack a fixed or ordered three-dimensional structure. IDPs cover a spectrum of states from fully unstructured to partially structured and include random coils, (pre-)molten globules, and large multi-domain proteins connected by flexible linkers. They constitute one of the main types of protein (alongside globular, fibrous and membrane proteins).
Coiled coil proteins
Coiled coil proteins form long, insoluble fibers involved in the extracellular matrix. There are many scleroprotein superfamilies including keratin, collagen, elastin, and fibroin. The roles of such proteins include protection and support, forming connective tissue, tendons, bone matrices, and muscle fiber.
Small proteins
Small proteins typically have a tertiary structure that is maintain |
https://en.wikipedia.org/wiki/Quaternary%20cubic | In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.
Invariants
and studied the ring of invariants of a quaternary cubic, which is a ring generated by invariants of
degrees 8, 16, 24, 32, 40, 100. The generators of degrees 8, 16, 24, 32, 40 generate a polynomial ring. The generator of degree 100 is a skew invariant, whose square is a polynomial in the other generators given explicitly by Salmon. Salmon also gave an explicit formula for the discriminant as a polynomial in the generators, though pointed out that the formula has a widely copied misprint in it.
Sylvester pentahedron
A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cube roots of unity. This was conjectured by Sylvester in 1851, and proven 10 years later by Clebsch. The union of the 5 planes where these 5 linear forms vanish is called the Sylvester pentahedron.
See also
Ternary cubic
Ternary quartic
Invariants of a binary form
References
Invariant theory
Algebraic surfaces |
https://en.wikipedia.org/wiki/Jun%20Shimanuki | is a former Japanese football player.
Club statistics
References
External links
J. League (#24)
1988 births
Living people
Association football people from Tokyo
Japanese men's footballers
J2 League players
Mito HollyHock players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Song%20Han-ki | Song Han-Ki (, born August 7, 1988) is a South Korean football player.
Club statistics
External links
1988 births
Living people
Men's association football defenders
South Korean men's footballers
South Korean expatriate men's footballers
J2 League players
K League 2 players
Korea National League players
Shonan Bellmare players
Kamatamare Sanuki players
Goyang Zaicro FC players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan |
https://en.wikipedia.org/wiki/Yusuke%20Tanahashi | is a Japanese football player. He plays for Vonds Ichihara.
Club statistics
References
External links
1987 births
Living people
Hannan University alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
Japan Football League players
Kataller Toyama players
FC Ryukyu players
Vonds Ichihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takamasa%20Sakai | is a former Japanese football player.
Club statistics
References
External links
J. League (#26)
1988 births
Living people
Kochi University alumni
Association football people from Hiroshima Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Kataller Toyama players
Kamatamare Sanuki players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Extinction%20probability | Extinction probability is the chance of an inherited trait becoming extinct as a function of time t. If t = ∞ this may be the complement of the chance of becoming a universal trait.
Statistical genetics
Stochastic processes
Population models |
https://en.wikipedia.org/wiki/Adilson%20%28footballer%2C%20born%201987%29 | Adilson dos Santos Souza (born 18 February 1987 in Barra do Rocha), simply known as Adilson, is a Brazilian footballer who plays as a forward.
Statistics
References
External links
1987 births
Living people
Footballers from Bahia
Brazilian men's footballers
Men's association football forwards
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Sport Club do Recife players
Grêmio Barueri Futebol players
Ipatinga Futebol Clube players
Esporte Clube XV de Novembro (Piracicaba) players
Mogi Mirim Esporte Clube players
Esporte Clube Noroeste players
Sport Club Corinthians Paulista players
Santa Cruz Futebol Clube players
Associação Portuguesa de Desportos players
Yangon United F.C. players
Marília Atlético Clube players
Clube Atlético Juventus players
Ituano FC players
Lemense Futebol Clube players
Rio Claro Futebol Clube players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Myanmar
Expatriate men's footballers in Myanmar |
https://en.wikipedia.org/wiki/National%20Museum%20of%20Mathematics | The National Museum of Mathematics or MoMath is a museum in Manhattan, New York City dedicated to mathematics.
Opened on December 15, 2012, it is the only museum in North America dedicated to mathematics and features over thirty interactive exhibits. The mission of the museum is to "enhance public understanding and perception of mathematics". The museum is known for a special tricycle with square wheels, which operates smoothly on a catenary surface.
History
In 2006 the Goudreau Museum on Long Island, at the time the only museum in the United States dedicated to mathematics, closed its doors. In response, a group led by founder and former CEO Glen Whitney met to explore the opening of a new museum. They received a charter from the New York State Department of Education in 2009, and raised over 22 million dollars in under four years.
With this funding, a space was leased in the Goddard Building at 11-13 East 26th Street, located in the Madison Square North Historic District. Despite some opposition to the architectural plans within the local community, permission for construction was granted by the New York City Landmarks Preservation Commission and the Department of Buildings.
The current board chair is John Overdeck, co-chairman of Two Sigma Investments.
Programs
Math Midway is a traveling exhibition of math-based interactive displays. The exhibits include a square-wheeled tricycle that travels smoothly over an undulating cycloidal track; the Ring of Fire, which uses lasers to intersect three-dimensional objects with a two-dimensional plane to uncover interesting shapes; and an "organ function grinder" which allows users to create their own mathematical functions and see the results. After making its debut at the World Science Festival in 2009, Math Midway traveled the country, reaching more than a half million visitors. The Midway's schedule included stops in New York, Pennsylvania, Texas, California, New Jersey, Ohio, Maryland, Florida, Indiana, and Oregon. In 2016, the Math Midway exhibit was sold to the Science Centre Singapore.
Math Midway 2 Go (MM2GO) is a spinoff of Math Midway. MM2GO includes six of the most popular Math Midway Exhibits. MM2GO began traveling to science festivals, schools, community centers, and libraries in the autumn of 2012.
Math Encounters is a monthly speaker series presented by the Museum of Math and the Simons Foundation. The lectures initially took place at Baruch College in Manhattan on the first Wednesday of each month, but moved to MoMath's visitor center at 11 East 26th Street in March, 2013. Every month a different mathematician is invited to deliver a lecture. Lecturers have included Google's Director of Research Peter Norvig, journalist Paul Hoffman, and computer scientist Craig Kaplan. Examples of topics are "The Geometry of Origami", "The Patterns of Juggling", and "Mathematical Morsels from The Simpsons and Futurama". The lectures are meant to be accessible and engaging for high school stude |
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