source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Ibrahim%20Al-Ghanim
|
Ibrahim Al-Ghanim (born June 27, 1983) is a retired Qatari footballer. He played as a defender.
Al-Ghanim was also a member of the Qatar national football team.
Club career statistics
Statistics accurate as of 21 August 2011
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
International goals
References
External links
FIFA.com profile
Goalzz.com profile
1983 births
Living people
Qatari men's footballers
Qatar men's international footballers
Al-Arabi SC (Qatar) players
2007 AFC Asian Cup players
2011 AFC Asian Cup players
Qatar Stars League players
Al-Gharafa SC players
Asian Games medalists in football
Footballers at the 2002 Asian Games
Footballers at the 2006 Asian Games
Asian Games gold medalists for Qatar
Men's association football defenders
Medalists at the 2006 Asian Games
|
https://en.wikipedia.org/wiki/Normality%20test
|
In statistics, normality tests are used to determine if a data set is well-modeled by a normal distribution and to compute how likely it is for a random variable underlying the data set to be normally distributed.
More precisely, the tests are a form of model selection, and can be interpreted several ways, depending on one's interpretations of probability:
In descriptive statistics terms, one measures a goodness of fit of a normal model to the data – if the fit is poor then the data are not well modeled in that respect by a normal distribution, without making a judgment on any underlying variable.
In frequentist statistics statistical hypothesis testing, data are tested against the null hypothesis that it is normally distributed.
In Bayesian statistics, one does not "test normality" per se, but rather computes the likelihood that the data come from a normal distribution with given parameters μ,σ (for all μ,σ), and compares that with the likelihood that the data come from other distributions under consideration, most simply using a Bayes factor (giving the relative likelihood of seeing the data given different models), or more finely taking a prior distribution on possible models and parameters and computing a posterior distribution given the computed likelihoods.
A normality test is used to determine whether sample data has been drawn from a normally distributed population (within some tolerance). A number of statistical tests, such as the Student's t-test and the one-way and two-way ANOVA, require a normally distributed sample population.
Graphical methods
An informal approach to testing normality is to compare a histogram of the sample data to a normal probability curve. The empirical distribution of the data (the histogram) should be bell-shaped and resemble the normal distribution. This might be difficult to see if the sample is small. In this case one might proceed by regressing the data against the quantiles of a normal distribution with the same mean and variance as the sample. Lack of fit to the regression line suggests a departure from normality (see Anderson Darling coefficient and minitab).
A graphical tool for assessing normality is the normal probability plot, a quantile-quantile plot (QQ plot) of the standardized data against the standard normal distribution. Here the correlation between the sample data and normal quantiles (a measure of the goodness of fit) measures how well the data are modeled by a normal distribution. For normal data the points plotted in the QQ plot should fall approximately on a straight line, indicating high positive correlation. These plots are easy to interpret and also have the benefit that outliers are easily identified.
Back-of-the-envelope test
Simple back-of-the-envelope test takes the sample maximum and minimum and computes their z-score, or more properly t-statistic
(number of sample standard deviations that a sample is above or below the sample mean), and compares it to the 68–95–99.7 rule
|
https://en.wikipedia.org/wiki/Mesaad%20Al-Hamad
|
Mesaad Ali Al-Hamad (born February 11, 1986) is a Qatari footballer who plays as a right defender . He is a member of the Qatar national football team. He was born in Qatar.
Club career statistics
Statistics accurate as of 21 August 2011
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
References
External links
Player Tactical Profile at football-lineups.com
Player profile - doha-2006.com
1986 births
Living people
Qatari men's footballers
Qatar men's international footballers
Al Sadd SC players
Al Ahli SC (Doha) players
Umm Salal SC players
Al-Wakrah SC players
Al Shahaniya SC players
Muaither SC players
Yemeni emigrants to Qatar
2007 AFC Asian Cup players
2011 AFC Asian Cup players
Qatar Stars League players
Qatari Second Division players
Naturalised citizens of Qatar
Asian Games medalists in football
Footballers at the 2006 Asian Games
Asian Games gold medalists for Qatar
Men's association football defenders
Medalists at the 2006 Asian Games
|
https://en.wikipedia.org/wiki/Mohammed%20Rabia%20Al-Noobi
|
Mohammed Rabia Jamaan Al-Noobi (, born 10 May 1981), commonly known as Mohammed Rabia, is an Omani footballer who plays for Dhofar S.C.S.C.
Club career statistics
International career
Mohammed was part of the first team squad of the Oman national football team from 2001 to 2010. He was selected for the national team for the first time in 2001. He has made appearances in the 2003 Gulf Cup of Nations, the 2004 Gulf Cup of Nations, the 2004 AFC Asian Cup qualification, the 2004 AFC Asian Cup, the 2007 Gulf Cup of Nations, the 2007 AFC Asian Cup qualification, the 2007 AFC Asian Cup, the 2010 Gulf Cup of Nations and the 2011 AFC Asian Cup qualification.
FIFA World Cup Qualification
Mohammed has made seven appearances in the 2002 FIFA World Cup qualification, five in the 2006 FIFA World Cup qualification and five in the 2010 FIFA World Cup qualification.
References
External links
1981 births
Living people
Omani men's footballers
Oman men's international footballers
Omani expatriate men's footballers
Men's association football defenders
2004 AFC Asian Cup players
2007 AFC Asian Cup players
Dhofar Club players
Kazma SC players
Al Wehda FC players
Al Sadd SC players
Al Ahli SC (Doha) players
Saudi Pro League players
Qatar Stars League players
Expatriate men's footballers in Kuwait
Omani expatriate sportspeople in Kuwait
Expatriate men's footballers in Saudi Arabia
Omani expatriate sportspeople in Saudi Arabia
Expatriate men's footballers in Qatar
Omani expatriate sportspeople in Qatar
Footballers at the 2006 Asian Games
Asian Games competitors for Oman
Kuwait Premier League players
|
https://en.wikipedia.org/wiki/Lynn%20Steen
|
Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a professor of mathematics at St. Olaf College, Northfield, Minnesota, in the U.S. He wrote numerous books and articles on the teaching of mathematics. He was a past president of the Mathematics Association of America (MAA) and served as chairman of the Conference Board of the Mathematical Sciences.
Biography
Steen was born in Chicago, Illinois, but was raised in Staten Island, New York. His mother was a singer at the N.Y. City Center Opera and his father conducted the Wagner College Choir. In 1961, Steen graduated from Luther College with a degree in mathematics and a minor in physics. In 1965 Steen graduated from MIT with a Ph.D. in mathematics. He then joined the faculty of St. Olaf College.
At the beginning of Steen's career, he mainly focused on teaching and helping develop research experiences for undergraduates. His teaching led Steen to begin to investigate the links between mathematics and other fields. He wrote many articles aimed for a non-mathematical audience about new developments in mathematics. The majority of his work in the 1970s was regarding mathematical exposition, communicating mathematical research to students, teachers, and the public.
In the 1980s, Steen helped lead national efforts to modernize the teaching of calculus and other areas in undergraduate mathematics. He helped broaden the mathematics major at St. Olaf by focusing the students work on inquiry and investigation. With the help of his mathematical colleagues, Steen made mathematics one of the five top majors for St. Olaf. St Olaf also became one of the nation's largest undergraduate producers in mathematical sciences.
In 1992, Steen went on leave from St. Olaf, he served as executive director of the Mathematical Sciences Education Board at the National Academy of Sciences in Washington, DC. In 1995, he returned to St. Olaf and began working on special projects for the Provost office. In the late 1990s, Steen worked as a writer and editor in pioneering grade-by-grade standards that helped meet the mathematical requirements of college as well as careers. The campaign for similar standards that is seen nowadays is an evolution of his former efforts.
In 2009 Steen retired from St. Olaf. He died June 21, 2015, of heart failure. He was survived by his wife of 52 years, Mary Steen.
Publications
'"The 'Gift' of Mathematics in the Era of Biology."' Math and Bio 2010: Linking Undergraduate Disciplines, Mathematical Association of America, 2005, pp. 13–25.
"Mathematics and Biology: New Challenges for Both Disciplines." The Chronicle Review, 4 March 2005, p. B12.
"Analysis 2000: Challenges and Opportunities." One Hundred Years of L'Enseignement Mathématique: Moments of Mathematics Education in the Twentieth Century, Daniel Coray et al., editors. Genéve: L'Enseignement Mathématique 2003, pp. 191–211.
"A Mind for Math." Review of The Math Gene: How Mathematical Thinking Evolv
|
https://en.wikipedia.org/wiki/Computational%20logic
|
Computational logic is the use of logic to perform or reason about computation. It bears a similar relationship to computer science and engineering as mathematical logic bears to mathematics and as philosophical logic bears to philosophy. It is synonymous with "logic in computer science".
The term “computational logic” came to prominence with the founding of the ACM Transactions on Computational Logic in 2000. However, the term was introduced much earlier, by J.A. Robinson in 1970. The expression is used in the second paragraph with a footnote claiming that "computational logic" is "surely a better phrase than 'theorem proving', for the branch of artificial intelligence which deals with how to make machines do deduction efficiently".
In 1972 the Metamathematics Unit at the University of Edinburgh was renamed “The Department of Computational Logic” in the School of Artificial Intelligence. The term was then used by Robert S. Boyer and J Strother Moore, who worked in the Department in the early 1970s, to describe their work on program verification and automated reasoning. They also founded Computational Logic Inc.
Computational logic has also come to be associated with logic programming, because much of the early work in logic programming in the early 1970s also took place in the Department of Computational Logic in Edinburgh. It was reused in the early 1990s to describe work on extensions of logic programming in the EU Basic Research Project "Compulog" and in the associated Network of Excellence. Krzysztof Apt, who was the co-ordinator of the Basic Research Project Compulog-II, reused and generalized the term when he founded the ACM Transactions on Computational Logic in 2000 and became its first Editor-in-Chief.
See also
Logic programming
Automated theorem proving
Type theory
Formal verification
References
Further reading
Logic in computer science
Computational fields of study
|
https://en.wikipedia.org/wiki/Mohammed%20Al-Hinai
|
Mohammed Mubarak Suwaid Al-Hinai (; born 19 July 1984), is an Omani footballer who plays for Fanja SC.
Club career statistics
International career
Mohammed was selected for the national team for the first time in 1999. He has made appearances in the 2003 Gulf Cup of Nations, the 2004 Gulf Cup of Nations, the 2004 AFC Asian Cup qualification, the 2004 AFC Asian Cup, the 2007 Gulf Cup of Nations and the 2007 AFC Asian Cup qualification and the 2007 AFC Asian Cup.
He also played at the 2001 FIFA U-17 World Championship in Trinidad and Tobago and scored two goals, one in a 1-2 loss against Spain and another in a 1-1 draw against Burkina Faso
FIFA World Cup Qualification
Mohammed has made three appearances in the 2006 FIFA World Cup qualification and six in the 2010 FIFA World Cup qualification.
In the 2006 FIFA World Cup qualification, he scored a brace in the 2006 FIFA World Cup qualification – AFC second round|second round]] in a 5-1 win over India.
In the 2010 FIFA World Cup qualification, he scored one goal in the 2010 FIFA World Cup qualification – AFC first round|first round]] in a 2-0 win over Nepal.
National team career statistics
Goals for Senior National Team
Honours
Club
With Fanja
Oman Professional League (1): 2011–12; Runner-Up 2012–13, 2013-14
Sultan Qaboos Cup (1): 2013-14
Oman Professional League Cup (1): 2014-15
Oman Super Cup (1): 2012; Runner-Up 2013, 2014
References
External links
Mohamed Al Hinai at Goal.com
1984 births
Living people
Sportspeople from Muscat, Oman
Omani men's footballers
Men's association football midfielders
2004 AFC Asian Cup players
Oman Club players
Al-Nasr SC (Salalah) players
Fanja SC players
Oman Professional League players
Expatriate men's footballers in Kuwait
Omani expatriate sportspeople in Kuwait
Footballers at the 2006 Asian Games
Asian Games competitors for Oman
Oman men's international footballers
Al-Sahel SC (Kuwait) players
Qadsia SC players
Kuwait Premier League players
Al-Tadamon SC (Kuwait) players
|
https://en.wikipedia.org/wiki/Leray%E2%80%93Hirsch%20theorem
|
In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the Leray spectral sequence.
Statement
Setup
Let
be a fibre bundle with fibre . Assume that for each degree , the singular cohomology rational vector space
is finite-dimensional, and that the inclusion
induces a surjection in rational cohomology
.
Consider a section of this surjection
,
by definition, this map satisfies
.
The Leray–Hirsch isomorphism
The Leray–Hirsch theorem states that the linear map
is an isomorphism of -modules.
Statement in coordinates
In other words, if for every , there exist classes
that restrict, on each fiber , to a basis of the cohomology in degree , the map given below is then an isomorphism of modules.
where is a basis for and thus, induces a basis for
Notes
Fiber bundles
Theorems in algebraic topology
|
https://en.wikipedia.org/wiki/Dot%20plot%20%28statistics%29
|
A dot chart or dot plot is a statistical chart consisting of data points plotted on a fairly simple scale, typically using filled in circles. There are two common, yet very different, versions of the dot chart. The first has been used in hand-drawn (pre-computer era) graphs to depict distributions going back to 1884. The other version is described by William S. Cleveland as an alternative to the bar chart, in which dots are used to depict the quantitative values (e.g. counts) associated with categorical variables.
Of a distribution
The dot plot as a representation of a distribution consists of group of data points plotted on a simple scale. Dot plots are used for continuous, quantitative, univariate data. Data points may be labelled if there are few of them.
Dot plots are one of the simplest statistical plots, and are suitable for small to moderate sized data sets. They are useful for highlighting clusters and gaps, as well as outliers. Their other advantage is the conservation of numerical information. When dealing with larger data sets (around 20–30 or more data points) the related stemplot, box plot or histogram may be more efficient, as dot plots may become too cluttered after this point. Dot plots may be distinguished from histograms in that dots are not spaced uniformly along the horizontal axis.
Although the plot appears to be simple, its computation and the statistical theory underlying it are not simple. The algorithm for computing a dot plot is closely related to kernel density estimation. The size chosen for the dots affects the appearance of the plot. Choice of dot size is equivalent to choosing the bandwidth for a kernel density estimate.
In the R programming language this type of plot is also referred to as a stripchart
or stripplot.
Cleveland dot plots
Dot plot may also refer to plots of points that each belong to one of several categories. They are an alternative to bar charts or pie charts, and look somewhat like a horizontal bar chart where the bars are replaced by a dots at the values associated with each category. Compared to (vertical) bar charts and pie charts, Cleveland argues that dot plots allow more accurate interpretation of the graph by readers by making the labels easier to read, reducing non-data ink (or graph clutter) and supporting table look-up.
Dot Chart in process mapping
The term Dot chart is also used in the area of process mapping. This is a simplified flowchart process flow chart in which columns are tasks, rows are roles, and dots that are inserted at the intersection of tasks and roles represent a sequence of steps. In other words, it is an extensive RACI table with additional information about the sequence of steps in the process.Example of tool for dot chart process mapping
See also
Data and information visualization
Scatter plot
References
Other references
Wild, C. and Seber, G. (2000) Chance Encounters: A First Course in Data Analysis and Inference John Wiley and Sons.
External links
|
https://en.wikipedia.org/wiki/Hassan%20Zaher%20Al-Maghni
|
Hassan Zaher Al-Maghni (; born 7 January 1985), commonly known as Hassan Zaher, is an Omani footballer who plays for Al-Nasr S.C.S.C.
Club career statistics
International career
Hassan was selected for the national team for the first time in 2006. He has made appearances in the 2007 AFC Asian Cup qualification.
References
External links
Hassan Zaher Al Maghni at Goal.com
1985 births
Living people
Omani men's footballers
Oman men's international footballers
Omani expatriate men's footballers
Men's association football forwards
Al-Nasr SC (Salalah) players
Salalah SC players
Expatriate men's footballers in Bahrain
Omani expatriate sportspeople in Bahrain
Footballers at the 2006 Asian Games
Asian Games competitors for Oman
Sportspeople from Muscat, Oman
|
https://en.wikipedia.org/wiki/Indiana%20University%20Mathematics%20Journal
|
The Indiana University Mathematics Journal is a journal of mathematics published by Indiana University. Its first volume was published in 1952, under the name Journal of Rational Mechanics and Analysis and edited by Zachery D. Paden and Clifford Truesdell. In 1957, Eberhard Hopf became editor, the journal name changed to the Journal of Mathematics and Mechanics, and Truesdell founded a separate successor journal, the Archive for Rational Mechanics and Analysis, now published by Springer-Verlag. The Journal of Mathematics and Mechanics later changed its name again to the present name.
The full text of all articles published under the various incarnations of this journal is available online from the journal's web site. The web site lists all such papers under the Indiana University Mathematics Journal name, but other bibliographies generally use the name of the journal as of the date each paper was published.
External links
Indiana University Mathematics Journal, official site and archive
Information about the history of the IUMJ from the USC library
Mathematics journals
Academic journals established in 1952
Indiana University
|
https://en.wikipedia.org/wiki/Hajime%20Tanabe
|
was a Japanese philosopher of science, particularly of mathematics and physics. His work brought together elements of Buddhism, scientific thought, Western philosophy, Christianity, and Marxism. In the postwar years, Tanabe coined the concept of metanoetics, proposing that the limits of speculative philosophy and reason must be surpassed by metanoia.
Tanabe was a key member of what has become known in the West as the Kyoto School, alongside philosophers Kitaro Nishida (also Tanabe's teacher) and Keiji Nishitani. He taught at Tōhoku Imperial University beginning in 1913 and later at Kyōto Imperial University, and studied at the universities of Berlin, Leipzig, and Freiburg in the 1920s under figures such as Edmund Husserl and Martin Heidegger. In 1947 he became a member of the Japan Academy, and in 1950 he received the Order of Cultural Merit.
Biography
Tanabe was born on February 3, 1885, in Tokyo to a household devoted to education. His father, the principal of Kaisei Academy, was a scholar of Confucius, whose teachings may have influenced Tanabe's philosophical and religious thought. Tanabe enrolled at Tokyo Imperial University, first as a mathematics student before moving to literature and philosophy. After graduation, he worked as a lecturer at Tohoku University and taught English at Kaisei Academy.
In 1916, Tanabe translated Henri Poincaré’s La Valeur de la science. In 1918, he received his doctorate from Kyoto Imperial University with a dissertation entitled ‘Investigations into the Philosophy of Mathematics’ (predecessor to the 1925 book with the same title).
In 1919, at Nishida’s invitation, Tanabe accepted the position of associate professor at Kyoto Imperial University. From 1922 to 23, he studied in Germany — first, under Alois Riehl at the University of Berlin and then under Edmund Husserl at the University of Freiburg. At Freiburg, he befriended the young Martin Heidegger and Oskar Becker. One can recognise the influence of these philosophers in Tanabe.
In September 1923, soon after the Great Kantō Earthquake, the Home Ministry ordered his return, so Tanabe used the little time he had left — about a couple of months — to visit London and Paris, before boarding his return ship at Marseille. He arrived back in Japan in 1924.
In 1928, Tanabe translated Max Planck’s 1908 lecture, ‘Die Einheit des physikalischen Weltbildes’ for the Philosophical Essays [哲学論叢] translation series, which he co-edited, for his publisher Iwanami Shoten. The same series published translations of essays by Bruno Bauch, Adolf Reinach, Wilhelm Windelband, Siegfried Marck, Max Planck, Franz Brentano, Paul Natorp, Nicolai Hartmann, Kazimierz Twardowski, Ernst Cassirer, Hermann Cohen, Emil Lask, Victor Brochard, Ernst Troeltsch, Theodor Lipps, Konrad Fiedler, Wincenty Lutosławski, Sergei Rubinstein, Hermann Bonitz, Max Weber, Émile Durkheim, Martin Grabmann, Heinrich Rickert, Alexius Meinong, Karl von Prantl and Wilhelm Dilthey (the series ended before the pla
|
https://en.wikipedia.org/wiki/Joan%20Birman
|
Joan Sylvia Lyttle Birman (born May 30, 1927, in New York City) is an American mathematician, specializing in low-dimensional topology. She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems. Birman is research professor emerita at Barnard College, Columbia University, where she has been since 1973.
Family
Her parents were George and Lillian Lyttle, both Jewish immigrants. Her father was from Russia but grew up in Liverpool, England. Her mother was born in New York and her parents were Russian-Polish immigrants. At age 17, George emigrated to the US and became a successful dress manufacturer. He appreciated the opportunities from having a business but he wanted his daughters to focus on education. She has three children, Kenneth P. Birman, Deborah Birman Shlider, and Carl David Birman. Her late husband, Joseph Birman, was a physicist and a leading advocate for human rights for scientists.
Education
After high school, Birman entered Swarthmore College, a coeducational institution in Swarthmore, Pennsylvania, and majored in mathematics. However, she disliked living in the dorms so she transferred to Barnard College, a women's only college affiliated to Columbia University, to live at home.
Birman received her B.A. (1948) in mathematics from Barnard College and an M.A. (1950) in physics from Columbia University.
After working in the industry from 1950 to 1960, she did a PhD in mathematics at the Courant Institute (NYU) under the supervision of Wilhelm Magnus, graduating in 1968. Her dissertation was titled Braid groups and their relationship to mapping class groups.
Career
After she earned her bachelor's degree from Barnard, Birman accepted a position at the Polytechnic Research and Development Co., which was affiliated with Brooklyn Polytechnic University. She later worked from the Technical Research Group and the W. L. Maxson Corporation.
Birman's first academic position was at the Stevens Institute of Technology (1968–1973). When she joined, she was the only female professor out of 160. In 1969 she published "On Braid Groups", which introduced a mapping class group of a surface called the Birman Exact Sequence, which became one of the most important tools in the study of braids and surfaces. During the later part of this period she published a monograph, 'Braids, links, and mapping class groups' based on a graduate course she taught as a visiting professor at Princeton University in 1971–72. This book is considered the first comprehensive treatment of braid theory, introducing the modern theory to the field, and contains the first complete proof of the Markov theorem on braids.
In 1973, she joined the faculty at Barnard College, where she served as Chairman of the Mathematics Department from 1973 to 1987, 1989 to 1991, and 1995 to 1998. She was a visiting scholar at the Institute for Advanced Study in the summer of 1988.
She supervi
|
https://en.wikipedia.org/wiki/V%20Gymnasium
|
The Fifth Gymnasium () is a high school in Zagreb, Croatia specialising in science and mathematics. It was opened on 7 November 1938. Today it has about 900 students in 28 classes. It is considered to be the most prestigious gymnasium in Zagreb alongside the XV Gymnasium.
Students are known for often excelling in mathematics, physics, chemistry, biology, Latin, computer science, history, geography and logic.
Also, students have taken part in the International Mathematical Olympiad and the International Chemistry Olympiad, where they have received various medals, and also the International Physics Olympiad.
History
The secondary school was initially founded in 1938 as a single-sex general gymnasium. Locally, it known as the Fifth Male Gymnasium of Zagreb. In its first year of existence, there were 762 students enrolled, with 45 students per class.
In 1946, a year after Croatia's entrance into Yugoslavia, its name was changed to the Fifth Gymnasium of Bogdan Ogrizović. was a prominent physicist in Zagreb who was best known as a high school teacher of physics and mathematics. He died during WWII.
In the period of Croatia's scrapping of the single-sex model and the creation co-educational institutions, the school merged in 1960 with the Seventh Woman's Gymnasium.
After this, it transformed into a high school primarily specialising in a program of mathematics and scientific subjects.
In 1977, after a reform on education and gymnasiums, the school changed its name to the Pedagogical Education Centre. However, it still retained its emphasis and reputation for natural sciences and mathematics.
After the collapse of Yugoslavia, it has again been known as the Fifth Gymnasium.
Notable alumni
Sven Medvešek, a famous theatre actor in Croatia and stage director
Rene Medvešek, a well-known theatre actor in Croatia, and additionally a stage director
Zoran Čutura, a professional basketball player and also sports columnist
Žarko Puhovski, a political analyst and Marxist theorist
Davor Pavun, a physicist, university professor, inventor, and educational speaker
References
External links
Gymnasiums in Croatia
Gymnasium, 05
Education in Zagreb
Donji grad, Zagreb
Educational institutions established in 1938
1938 establishments in Croatia
School buildings completed in 1938
Neoclassical architecture in Croatia
|
https://en.wikipedia.org/wiki/Regular%20Polytopes%20%28book%29
|
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973.
The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.
Overview
The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. It has 14 chapters, along with multiple appendices, providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject.
The first chapter discusses regular polygons, regular polyhedra, basic concepts of graph theory, and the Euler characteristic. Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their duals to generate related polyhedra, including the semiregular polyhedra, and discusses zonohedra and Petrie polygons. Here and throughout the book, the shapes it discusses are identified and classified by their Schläfli symbols.
Chapters 3 through 5 describe the symmetries of polyhedra, first as permutation groups and later, in the most innovative part of the book, as the Coxeter groups, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra.
The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds.
Later editions
The second edition was published in paperback; it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups, appends a new definition of polytopes at the end of the book, and makes minor corrections throughout. The photographic plates were also enlarged for this printing, and some figures were redrawn. The nomenclature of these editions was occasionally cumbersome, and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the electron micr
|
https://en.wikipedia.org/wiki/List%20of%20Hereford%20United%20F.C.%20records%20and%20statistics
|
Below are statistics and records related to Hereford United Football Club.
Competition history
Football League Second Division (second tier) – finished 22nd in only season (1976–77)
Football League Third Division (third tier) – Champions (1975–76)
Football League Fourth Division (fourth tier) – Runners-up (1972–73); Third Place (2007–08)
Conference National – Runners-up (2003–04, 2004–05, 2005–06); Playoff Winners (2005–06)
Southern Football League – Runners-up (1945–46, 1950–51, 1971–72); Regional Champions (1958–59)
FA Cup – reached Fourth Round (1971–72, 1973–74, 1976–77, 1981–82, 1989–90, 1991–92, 2007–08, 2010–11)
Football League Cup – reached Third Round (1973–74)
Football League Trophy – reached Area Final (1985–86)
Welsh Cup – Winners (1989–90); Runners-up (1967–68, 1975–76, 1980–81)
FA Trophy – reached semi-final (1970–71, 2000–01)
Southern League Cup – Winners (1951–52, 1956–57, 1958–59)
Overall Football League/Conference Record to 2013:
Division 2: 1976-7 (1 season)
Division 3/League 1: 1973–6, 1977–8, 2008-9 (5 seasons)
Division 4/League 2: 1972–3, 1978–97, 2006–8, 2009-12 (25 seasons)
Conference/Conference National: 1997–2006, 2012-3 (10 completed seasons)
Prior to 1972 the club played in various minor leagues. See List of Hereford United F.C. seasons.
FA Cup
Record
Since 1924 Hereford United have reached the Fourth Round on seven occasions, losing each time.
Longest FA Cup Run – 10 matches (Fourth Qualifying Round – fourth round proper, 1971–72)
Giantkilling
Hereford United beat the following Football League teams whilst a non-league club.
Exeter City (1952–53)
Aldershot (1956–57)
Queens Park Rangers (1957–58)
Millwall (1965–66)
Northampton Town (1970–71)
Northampton Town (1971–72)
Newcastle United (1971–72)
Brighton & Hove Albion (1997–98)
Colchester United (1997–98)
York City (1999-00)
Hartlepool United (1999-00)
Wrexham (2001–02)
Shrewsbury Town (2012–2013)
Records
National
Competed in Highest Attended Match Between Non-League Clubs – 24,526 v Wigan Athletic (1953–54 FA Cup second round at The Hawthorns)
Biggest Win by a Non-League Club over a League Club – 6–1 v Queens Park Rangers (1957–58 FA Cup second round)This record is shared with Wigan Athletic and Boston United
Club
Record Competitive win – 11–0 v Thynnes Athletic (1947–48 FA Cup preliminary round)
Record Competitive defeat* – 7–0 v Middlesbrough (1996–97 League Cup) - 7-0 v Luton "(2013-14 Conference)"
Record Football League win – 6–0 v Burnley (1986–87 Fourth Division)
Record Football League defeat – 7–1 v Mansfield Town (1994–95 Division Three)
Record Conference win – 9–0 v Dagenham & Redbridge (2003–04)
Record Conference defeat – 4–0 v Doncaster Rovers (2001–02)
Record League Cup win – 4–0 v Bristol Rovers (1978–79), 5–1 v Bristol City (1985–86)
Record Welsh Cup win – 10–1 v BSC Shotton (1985–86)
Record Football League Trophy win – 4–0 v Yeovil Town (2000–01)
Record Football League Trophy defeat – 4–0 v Millwall (1996–
|
https://en.wikipedia.org/wiki/Weakly%20contractible
|
In mathematics, a topological space is said to be weakly contractible if all of its homotopy groups are trivial.
Property
It follows from Whitehead's Theorem that if a CW-complex is weakly contractible then it is contractible.
Example
Define to be the inductive limit of the spheres . Then this space is weakly contractible. Since is moreover a CW-complex, it is also contractible. See Contractibility of unit sphere in Hilbert space for more.
The Long Line is an example of a space which is weakly contractible, but not contractible. This does not contradict Whitehead theorem since the Long Line does not have the homotopy type of a CW-complex.
Another prominent example for this phenomenon is the Warsaw circle.
References
Topology
Homotopy theory
|
https://en.wikipedia.org/wiki/Pompeiu%27s%20theorem
|
Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following:
Given an equilateral triangle ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a (maybe, degenerate) triangle.
The proof is quick. Consider a rotation of 60° about the point B. Assume A maps to C, and P maps to P '. Then , and . Hence triangle PBP ' is equilateral and . Then . Thus, triangle PCP ' has sides equal to PA, PB, and PC and the proof by construction is complete (see drawing).
Further investigations reveal that if P is not in the interior of the triangle, but rather on the circumcircle, then PA, PB, PC form a degenerate triangle, with the largest being equal to the sum of the others, this observation is also known as Van Schooten's theorem.
Generally, by the point P and the lengths to the vertices of the equilateral triangle - PA, PB, and PC two equilateral triangles ( the larger and the smaller) with sides and are defined:
.
The symbol △ denotes the area of the triangle whose sides have lengths PA, PB, PC.
Pompeiu published the theorem in 1936, however August Ferdinand Möbius had published a more general theorem about four points in the Euclidean plane already in 1852. In this paper Möbius also derived the statement of Pompeiu's theorem explicitly as a special case of his more general theorem. For this reason the theorem is also known as the Möbius-Pompeiu theorem.
External links
MathWorld's page on Pompeiu's Theorem
Pompeiu's theorem at cut-the-knot.org
Notes
Elementary geometry
Theorems about equilateral triangles
Theorems about triangles and circles
Articles containing proofs
|
https://en.wikipedia.org/wiki/Ackermann%20set%20theory
|
In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.
AST differs from Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets.
It replaces several of the standard ZF axioms for constructing new sets with a principle known as Ackermann's schema. Intuitively, the schema allows a new set to be constructed if it can be defined by a formula which does not refer to the class of all sets.
In its use of classes, AST differs from other alternative set theories such as Morse–Kelley set theory and Von Neumann–Bernays–Gödel set theory in that a class may be an element of another class.
William N. Reinhardt established in 1970 that AST is effectively equivalent in strength to ZF, putting it on equal foundations. In particular, AST is consistent if and only if ZF is consistent.
Preliminaries
AST is formulated in first-order logic. The language of AST contains one binary relation denoting set membership and one constant denoting the class of all sets. Ackermann used a predicate instead of ; this is equivalent as each of and can be defined in terms of the other.
We will refer to elements of as sets, and general objects as classes. A class that is not a set is called a proper class.
Axioms
The following formulation is due to Reinhardt.
The five axioms include two axiom schemas.
Ackermann's original formulation included only the first four of these, omitting the axiom of regularity.
1. Axiom of extensionality
If two classes have the same elements, then they are equal.
This axiom is identical to the axiom of extensionality found in many other set theories, including ZF.
2. Heredity
Any element or a subset of a set is a set.
3. Comprehension schema
For any property, we can form the class of sets satisfying that property. Formally, for any formula where is not free:
That is, the only restriction is that comprehension is restricted to objects in . But the resulting object is not necessarily a set.
4. Ackermann's schema
For any formula with free variables and no occurrences of :
Ackermann's schema is a form of set comprehension that is unique to AST. It allows constructing a new set (not just a class) as long as we can define it by a property that does not refer to the symbol . This is the principle that replaces ZF axioms such as pairing, union, and power set.
5. Regularity
Any set contains an element disjoint from itself:
Here, is shorthand for . This axiom is identical to the axiom of regularity in ZF.
This axiom is conservative in the sense that without it, we can simply use comprehension (axiom schema 3) to restrict our attention to the subclass of sets that are regular.
Alternative formulations
Ackermann's original axioms did not include regularity, and used a predicate symbol instead of the constant symbol . We follow Lévy and Reinhardt in replacing instances of with . This is eq
|
https://en.wikipedia.org/wiki/Weinstein%E2%80%93Aronszajn%20identity
|
In mathematics, the Weinstein–Aronszajn identity states that if and are matrices of size and respectively (either or both of which may be infinite) then,
provided (and hence, also ) is of trace class,
where is the identity matrix.
It is closely related to the matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses.
Proof
The identity may be proved as follows.
Let be a matrix consisting of the four blocks , , and :
Because is invertible, the formula for the determinant of a block matrix gives
Because is invertible, the formula for the determinant of a block matrix gives
Thus
Substituting for then gives the Weinstein–Aronszajn identity.
Applications
Let . The identity can be used to show the somewhat more general statement that
It follows that the non-zero eigenvalues of and are the same.
This identity is useful in developing a Bayes estimator for multivariate Gaussian distributions.
The identity also finds applications in random matrix theory by relating determinants of large matrices to determinants of smaller ones.
References
Determinants
Matrix theory
Theorems in linear algebra
|
https://en.wikipedia.org/wiki/Herman%20March
|
Herman William March (1878 – 1953) was a mathematician and physicist.
March studied physics and mathematics at the University of Munich under Wilhelm Röntgen and Arnold Sommerfeld. He received his doctorate in 1911. He had a position at the University of Wisconsin–Madison no later than circa 1920. He died in 1953.
Partial literature
1917: Calculus. Herman W. March and Henry C. Wolff. McGraw-Hill, New York.
1925: The Deflection of a Rectangular Plate Fixed at the Edges, Transactions of the American Mathematical Society, 27(3): 307–317
1927: The Heaviside Operational Calculus, Bulletin of the American Mathematical Society 33: 311–8.
1928: (with Warren Weaver) The Diffusion Problem for a Solid in Contact with a Stirred Liquid, Physical Review 31: 1072 - 1082.
1936: Bending of a Centrally Loaded Rectangular Strip of Plywood, Journal of Applied Physics 7(1): 32–41.
1953: The Field of a Magnetic Dipole in the Presence of a Conducting Sphere, Geophysics 18(3): 671–684.
Notes
1878 births
1953 deaths
20th-century German physicists
20th-century American physicists
20th-century American mathematicians
|
https://en.wikipedia.org/wiki/Priestville%2C%20Nova%20Scotia
|
Priestville is a designated place within Pictou County in Nova Scotia, Canada near New Glasgow and the Trans-Canada Highway.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Priestville had a population of 157 living in 73 of its 79 total private dwellings, a change of from its 2016 population of 163. With a land area of , it had a population density of in 2021.
References
Communities in Pictou County
Designated places in Nova Scotia
|
https://en.wikipedia.org/wiki/Dot%20plot
|
Dot plot may refer to:
Dot plot (bioinformatics), for comparing two sequences
Dot plot (statistics), data points on a simple scale
Dot plot graphic for Federal Reserve Open Market Committee polling result
|
https://en.wikipedia.org/wiki/Association%20of%20Mathematics%20Teachers%20of%20India
|
The Association of Mathematics Teachers of India or AMTI is an academically oriented body of professionals and students interested in the fields of mathematics and mathematics education.
The AMTI's main base is Tamil Nadu, but it has recently been spreading its network in other parts of India, particularly in South India.
Examinations and Olympiads
National Mathematics Talent Contest
AMTI conducts a National Mathematics Talent Contest or NMTC at Primary(Gauss Contest) (Standards 4 to 6), Sub-junior (Kaprekar Contest) (Standards 7 and 8), Junior (Bhaskara Contest) (Standards 9 and 10), Inter(Ramanujan Contest) (Standards 11 and 12) and Senior (Aryabhata Contest) (B.Sc.) levels. For students at the Junior and Inter levels from Tamil Nadu, the NMTC also plays the role of Regional Mathematical Olympiad. Although the question papers are different for Junior and Inter levels, students from both levels may be chosen to appear at INMO based on their performance.
The NMTC is usually held around the last week of October. A preliminary examination is conducted earlier (in September) for all levels except B.Sc. students. Students (Junior and Inter) qualifying the preliminary examination are invited for an Orientation Camp one week before the NMTC, where Olympiad problems and theories are taught. This is also useful for those students qualifying further for INMO.
Grand Achievement Test
This test is for students studying in 12th standard under the Tamil Nadu State Board. It is intended to give a perfectly simulated atmosphere of the board's examination.
Training Activities
Ten-week training session
In 2005, AMTI started a ten-week training programme for students for Olympiad-related problems. The training batches were split into:
Primary level: Standards 4 to 6
Sub-junior level: Standards 7 and 8
Junior level: Standards 9 and 10
Inter level: Standards 11 and 12
Around 85 students attended the ten-week training session.
AMTI conducted the programme again in 2006, and received a much better response.
Workshops and conferences
The AMTI has been organizing conferences in different parts of the country to meet and deliberate issues of mathematics education, particularly at the school level.
Notable office bearers
P. K. Srinivasan, a famous teacher of mathematics, was the first Editor of the magazine Junior Mathematician (1990 to 1994) and the Academic Secretary of AMTI from 1981 to 1994.
External links
AMTI official page
Mathematical societies
Indian mathematics
Mathematics education in India
Scientific societies based in India
|
https://en.wikipedia.org/wiki/National%20Mathematics%20Talent%20Contest
|
The National Mathematics Talent Contest or NMTC is a national-level mathematics contest conducted by the Association of Mathematics Teachers of India (AMTI). It is strongest in Tamil Nadu, which is the operating base of the AMTI. The AMTI is a pioneer organisation in promoting, and conducting, Maths Talent Tests in India. In the National level tests 66,066 students, from 332 institutions spread all over India, participated at the screening level. Of these, 10% were selected for the final test. For the benefit of final level contestants, and the chosen few for INMO, special orientation camps were conducted. Merit certificates and prizes were awarded to the deserving students.
Thirty-five among them from Tamil Nadu and Puduchery at the Junior and Inter Levels have been sponsored to write the Indian National Mathematics Olympiad (INMO 2013). From among them 2 have been selected at the national level.
Levels
Primary level: Standard 5 and 6, is called the Gauss Contest
Sub-junior level: Standards 7 and 8, is called the Kaprekar Contest
Junior level: Standards 9 and 10, is called the Bhaskara Contest
Inter level: Standards 11 and 12, is called the Ramanujan Contest
Senior level: B.Sc. students, is called the Aryabhata Contest
Stages
For all levels except the Senior level, there is a preliminary examination comprising multiple choice questions. The preliminary examination is held in the end of August. Students qualifying in the preliminary examination are eligible to sit for the main examination, which is held around the last week
of October. A week before the main examination, students are invited for a two-day orientation camp.
Fee
The fee for the preliminary examination is Rs. 50 in India. No further fee is required for the main examination.
Rs. 75/- per candidate (out of which Rs.15/- will be retained by the institution only for all expenses and Rs.60/- to be sent to AMTI).
Syllabus
No special knowledge of curriculum material is required. Good knowledge of curriculum at the next lower level would be helpful.
The syllabus for Mathematics Olympiad (Regional, National and International) is pre-degree college mathematics. The areas covered are, mainly –
a)Algebra, b) Geometry, c) Number theory and d) Graph theory & combinatorics.
Algebra: Polynomials, Solving equations, inequalities, and complex numbers.
Geometry: Geometry of triangles and circles.
(Trigonometric methods, vector methods, complex number methods, transformation geometry methods can also be used to solve problems)
Number Theory: Divisibility, Diophantine equations, congruence relations, prime numbers and elementary results on prime numbers.
Combinatorics & Graph Theory: Counting techniques, pigeon hole principle, the principle of inclusion and exclusion, basic graph theory.
External links
MATH -n-CODING TECH COMPETITION (MCTC) - 2022
AMTI page on the NMTC
Mathematical Olympiads in India
Science and technology in Tamil Nadu
|
https://en.wikipedia.org/wiki/Convex%20body
|
In mathematics, a convex body in -dimensional Euclidean space is a compact convex set with non-empty interior.
A convex body is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point lies in if and only if its antipode, also lies in Symmetric convex bodies are in a one-to-one correspondence with the unit balls of norms on
Important examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope.
Kinds of convex bodies
A convex body may be defined as:
A Convex set of points.
The Convex Hull of a set of points.
The intersection of Hyperplanes.
The interior of any Convex polygon or Convex polytope.
Polar body
If is a bounded convex body containing the origin in its interior, the polar body is . The polar body has several nice properties including , is bounded, and if then . The polar body is a type of duality relation.
See also
References
Convex geometry
Multi-dimensional geometry
|
https://en.wikipedia.org/wiki/Reciprocity%20%28evolution%29
|
Reciprocity in evolutionary biology refers to mechanisms whereby the evolution of cooperative or altruistic behaviour may be favoured by the probability of future mutual interactions. A corollary is how a desire for revenge can harm the collective and therefore be naturally deselected.
Main types
Three types of reciprocity have been studied extensively:
Direct reciprocity
Indirect
Network reciprocity
Direct reciprocity
Direct reciprocity was proposed by Robert Trivers as a mechanism for the evolution of cooperation. If there are repeated encounters between the same two players in an evolutionary game in which each of them can choose either to "cooperate" or "defect", then a strategy of mutual cooperation may be favoured even if it pays each player, in the short term, to defect when the other cooperates. Direct reciprocity can lead to the
evolution of cooperation only if the probability, w, of another encounter between the same two individuals exceeds the cost-to-benefit ratio of the altruistic act: w > c / b
Indirect reciprocity
"In the standard framework of indirect reciprocity, there are randomly chosen pairwise encounters between members of a population; the same two individuals need not meet again. One individual acts as donor, the other as recipient. The donor can decide whether or not to cooperate. The interaction is observed by a subset of the population who might inform others. Reputation allows evolution of cooperation by indirect reciprocity. Natural selection favors strategies that base the decision to help on the reputation of the recipient: studies show that people who are more helpful are more likely to receive help." In many situations cooperation is favoured and it even benefits an individual to forgive an occasional defection but cooperative societies are always unstable because mutants inclined to defect can upset any balance.
The calculations of indirect reciprocity are complicated, but again a simple rule has emerged. Indirect reciprocity can only promote cooperation if the probability, q, of knowing someone’s reputation exceeds the cost-to-benefit ratio of the altruistic act:
q > c / b
One important problem with this explanation is that individuals may be able to evolve the capacity to obscure their reputation, reducing the probability, q, that it will be known.
Individual acts of indirect reciprocity may be classified as "upstream" or "downstream":
Upstream reciprocity occurs when an act of altruism causes the recipient to perform a later act of altruism in the benefit of a third party. In other words: A helps B, which then motivates B to help C.
Downstream reciprocity occurs when the performer of an act of altruism is more likely to be the recipient of a later act of altruism. In other words: A helps B, making it more likely that C will later help A.
Network reciprocity
Real populations are not well mixed, but have spatial structures or social networks which imply that some individuals interact more often t
|
https://en.wikipedia.org/wiki/William%20Thomas%20Fletcher
|
William Thomas Fletcher is an American mathematician.
Education
He received the B.S.(magna cum laude) and M.S. degrees (major in mathematics) from North Carolina Central University (NCCU), Durham, NC in 1956 and 1958 respectively. He received the Ph.D. degree in mathematics from the University of Idaho in 1966.
Early career
In 1957 Dr. Fletcher accepted his first teaching position in the department of mathematics at LeMoyne-Owen College, Memphis, Tennessee, where he served as chairman until 1972. For the ten-year period 1962–72 Fletcher pursued summer employment as a mathematical applications computer programmer in industry, business, and government at IBM Mohansic Laboratory (Yorktown Heights, NY), Western Electric (Hopewell, NJ), the Lawrence Livermore Laboratory (Livermore, CA), and the US Departments of Commerce (Washington, DC), Agriculture (St. Paul, Minn), and Energy (Livermore, CA).
NCCU
In 1972 Fletcher returned to NCCU as professor and chairman of the mathematics department where he joined his former teacher and mentor, Marjorie Lee Browne, the second African-American woman to earn a Ph.D. in Mathematics. During his 25-year tenure at NCCU, Fletcher instituted a BS degree in computer science; wrote petition to obtain a chapter of Pi Mu Epsilon, honorary national mathematics society promoting scholarly activity in mathematics, organized the Marjorie Lee Browne Distinguished Alumni Lectures Series; developed, with two other alumni the Marjorie Lee Browne Memorial Scholarship; led several summer institutes for science and math teachers; organized a Department Speaker Bureau; developed, with other department members, the Mathematics Department Resource Learning Center;and was the principal writer of a proposal to establish a chapter of Sigma Xi at NCCU.
Professional memberships and awards
He held membership in several professional organizations; served on several boards including the North Carolina State Board of Science and Technology; received several awards including outstanding teacher of the year (1990); the Year 2000 NC State Award in Science, the highest award that the Governor of NC can bestow upon a citizen. He was the recipient of the 2005 Durham County and North Carolina State Jaycees awards for outstanding community service.
Publications
"On the Decomposition of Associative Algebras of Prime Characteristic", Journal of Algebra, Vol. 16, No. 2, 1970, pp. 227–236.
"On Whitehead's Second Lemma for Lie Algebras", International Journal of Mathematics and Mathematical Sciences, Vol. 29, No. 3, 1982, pp. 521–523.
"On the Structure of Associative Algebras Relative to Their Radicals", Rev. Roumaine Math. Pures Appl., Vol. 29, Vol. 4, 1984, pp. 301–307.
"Implementing the Recommendation of CUPM in A Small Mathematics Department", Newsletter, CUPM; The Mathematical Association of America, 1970.
"The Nature of Mathematics and Its Importance in a Liberal Education", The Mathematics Newsletter, Vol. VII, No. 2, University of Idaho,
|
https://en.wikipedia.org/wiki/Technical%20University%20of%20Denmark%20Department%20of%20Mathematics
|
The Department of Mathematics at DTU (, MAT) is an institute at the Technical University of Denmark. It was founded to consolidate all mathematical research and teaching in one institute. All bsc.-students at DTU receive at least 20 ECTS points worth of classes from MAT during their first year.
The institute is located in building 303 S at Matematiktorvet, quadrant 3 on Lundtoftesletten in Lyngby, Denmark.
Mathematical Research
The research at Department of Mathematics covers both theoretical and applications issues and is currently centered on four main areas:
Discrete mathematics
Dynamical systems
Applied functional analysis
Geometry
Significant research is also done in graph theory by Carsten Thomassen.
Research Networks
MAT participate in several research networks.
The applied functional analysis group participate in a research network with Department of Mechanical Engineering, Topology Optimization (TOPOPT).
The goal of TOPOPT is to use topology optimization and other structural optimization methods to develop systematic tools for design of Multiphysics structures. It is sponsored by a NEDO grant and a European Young Investigator award (EURYI).
The group is also heading a mathematical network in modelling, estimation and control of biotechnological systems sponsored by The Danish Research Council for Technology and Production.
Collaborations
MAT co-organises - along with University of Southern Denmark - an annual European Study Group with Industry (ESGI), where a weeklong workshop is centered on finding solutions to mathematical modelling problems brought forth by the industry.
Master thesis projects and PhD projects are also completed in collaboration with private companies.
External links
Homepage of Department of Mathematics, DTU
History of MAT
Topology Optimization
Mathematical Network in Modelling, Estimation and Control of Biotechnological Systems
Technical University of Denmark
Mathematics departments
|
https://en.wikipedia.org/wiki/Hypergeometric%20function%20of%20a%20matrix%20argument
|
In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals.
Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument.
Definition
Let and be integers, and let
be an complex symmetric matrix.
Then the hypergeometric function of a matrix argument
and parameter is defined as
where means is a partition of , is the generalized Pochhammer symbol, and
is the "C" normalization of the Jack function.
Two matrix arguments
If and are two complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:
where is the identity matrix of size .
Not a typical function of a matrix argument
Unlike other functions of matrix argument, such as the matrix exponential, which are matrix-valued, the hypergeometric function of (one or two) matrix arguments is scalar-valued.
The parameter α
In many publications the parameter is omitted. Also, in different publications different values of are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984), whereas in other settings (e.g., in the complex case—see Gross and Richards, 1989), . To make matters worse, in random matrix theory researchers tend to prefer a parameter called instead of which is used in combinatorics.
The thing to remember is that
Care should be exercised as to whether a particular text is using a parameter or and which the particular value of that parameter is.
Typically, in settings involving real random matrices, and thus . In settings involving complex random matrices, one has and .
References
K. I. Gross and D. St. P. Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Theory, 59, no. 2, 224–246, 1989.
J. Kaneko, "Selberg Integrals and hypergeometric functions associated with Jack polynomials", SIAM Journal on Mathematical Analysis, 24, no. 4, 1086-1110, 1993.
Plamen Koev and Alan Edelman, "The efficient evaluation of the hypergeometric function of a matrix argument", Mathematics of Computation, 75, no. 254, 833-846, 2006.
Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
External links
Software for computing the hypergeometric function of a matrix argument by Plamen Koev.
Hypergeometric functions
|
https://en.wikipedia.org/wiki/Generalized%20Pochhammer%20symbol
|
In mathematics, the generalized Pochhammer symbol of parameter and partition generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as
It is used in multivariate analysis.
References
Gamma and related functions
Factorial and binomial topics
|
https://en.wikipedia.org/wiki/John%20R.%20Klauder
|
John Rider Klauder (born January 24, 1932) is an American professor of physics and mathematics, and author of over 250 published articles on physics.
He graduated from University of California, Berkeley in 1953 with a Bachelor of Science. He received his PhD in 1959 from Princeton University where he was a student of John Archibald Wheeler.
A former head of the Theoretical Physics and Solid State Spectroscopy Departments of Bell Telephone Laboratories, he has been a visiting professor at Rutgers University, Syracuse University, and the University of Bern. In 1988 John Klauder was appointed professor of physics and mathematics at the University of Florida. He was awarded the title of distinguished professor in 2006, and became emeritus in 2010.
He was inducted as a Foreign Member of the Royal Norwegian Society of Sciences and Letters, and received the Onsager Medal in 2006 at NTNU (Norway).
He has also served on the Physics Advisory Panel of the National Science Foundation and been Editor of the Journal of Mathematical Physics, president of the International Association of Mathematical Physics, associate secretary-general of the International Union of Pure and Applied Physics.
Bibliography
Beyond Conventional Quantization
"When treated conventionally, certain systems yield trivial and unacceptable results. This book describes enhanced procedures, generally involving extended correspondence rules for the association of a classical and a quantum theory, which, when applied to such systems, yield nontrivial and acceptable results. Requiring only a modest prior knowledge of quantum mechanics and quantum field theory, this book will be of interest to graduate students and researchers in theoretical physics, mathematical physics, and mathematics."
Coherent States - Applications in Physics and Mathematical Physics - A brief introduction to Coherent States followed by a collection of useful articles. Authored with B. S. Skagerstam.
Fundamentals of Quantum Optics - Volume authored jointly with E. C. G. Sudarshan.
On Klauder's Path: A Field Trip - "A volume in celebration of his 60th birthday."
References
21st-century American physicists
1932 births
Living people
University of Florida faculty
Royal Norwegian Society of Sciences and Letters
Presidents of the International Association of Mathematical Physics
|
https://en.wikipedia.org/wiki/Jack%20function
|
In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials, and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.
Definition
The Jack function
of an integer partition , parameter , and arguments can be recursively defined as
follows:
For m=1
For m>1
where the summation is over all partitions such that the skew partition is a horizontal strip, namely
( must be zero or otherwise ) and
where equals if and otherwise. The expressions and refer to the conjugate partitions of and , respectively. The notation means that the product is taken over all coordinates of boxes in the Young diagram of the partition .
Combinatorial formula
In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials in n variables:
The sum is taken over all admissible tableaux of shape and
with
An admissible tableau of shape is a filling of the Young diagram with numbers 1,2,…,n such that for any box (i,j) in the tableau,
whenever
whenever and
A box is critical for the tableau T if and
This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials.
C normalization
The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product:
This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as
where
For is often denoted by and called the Zonal polynomial.
P normalization
The P normalization is given by the identity , where
where and denotes the arm and leg length respectively. Therefore, for is the usual Schur function.
Similar to Schur polynomials, can be expressed as a sum over Young tableaux. However, one need to add an extra weight to each tableau that depends on the parameter .
Thus, a formula for the Jack function is given by
where the sum is taken over all tableaux of shape , and denotes the entry in box s of T.
The weight can be defined in the following fashion: Each tableau T of shape can be interpreted as a sequence of partitions
where defines the skew shape with content i in T. Then
where
and the product is taken only over all boxes s in such that s has a box from in the same row, but not in the same column.
Connection with the Schur polynomial
When the Jack function is a scalar multiple of the Schur polynomial
where
is the product of all hook lengths of .
Properties
If the partition has more parts than the number of variables, then the Jack function is 0:
Matrix argument
In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If is a matrix with eigenvalues
, then
Re
|
https://en.wikipedia.org/wiki/Heptagonal%20antiprism
|
In geometry, the heptagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 7-sided base, one usually considers the case where its copy is twisted by an angle 180°/7. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two heptagonal bases and, connecting those bases, 14 isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
See also
External links
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A7
Prismatoid polyhedra
|
https://en.wikipedia.org/wiki/Henri%20Cohen%20%28number%20theorist%29
|
Henri Cohen (born 8 June 1947) is a number theorist, and a professor at the University of Bordeaux. He is best known for leading the team that created the PARI/GP computer algebra system. He introduced the Rankin–Cohen bracket and has written several textbooks in computational and algebraic number theory.
Selected publications
; 2nd correct. print 1995; 1st printing 1993
References
External links
Personal web page
Number theorists
École Normale Supérieure alumni
20th-century French mathematicians
21st-century French mathematicians
1947 births
Living people
Academic staff of the University of Bordeaux
|
https://en.wikipedia.org/wiki/No%20Surprise%20%28Theory%20of%20a%20Deadman%20song%29
|
"No Surprise" is a song by Canadian hard rock group Theory of a Deadman. It was released in February 2005 as the lead single off their second album, Gasoline.
Content
The song is about a girl who always runs off with guys besides her boyfriend. He says, "It ain't no surprise that bitch is leaving me."
Music video
The video is about a girl waking up and finding a note that says "I'm leaving" from her boyfriend. She goes to trash his apartment, and in the end, she finds another note which says "be back with coffee love you." At this point he walks in and sees what she has done. The video then ends.
The video has been compared to Kelly Clarkson's "Since U Been Gone". This came as a surprise to the band, as they found it quite humorous and said the similarities were not intentional.
Chart positions
References
Theory of a Deadman songs
2005 singles
604 Records singles
Song recordings produced by Howard Benson
Songs written by Tyler Connolly
|
https://en.wikipedia.org/wiki/Joseph%20Sauveur
|
Joseph Sauveur (; 24 March 1653 – 9 July 1716) was a French mathematician and physicist. He was a professor of mathematics and in 1696 became a member of the French Academy of Sciences.
Life
Joseph Sauveur was born in La Flèche, the son of a provincial notary. Despite a hearing and speech impairment that kept him totally mute until he was seven, Joseph benefited from a fine education at the Jesuit College of La Flèche. At seventeen, his uncle agreed to finance his studies in philosophy and theology at Paris. Joseph, however, discovered Euclid and turned to anatomy and botany. He soon met Cordemoy, reader to the son of Louis XIV; and Cordemoy soon sang his praises to Bossuet, preceptor to the Dauphin. Despite his handicap, Joseph promptly began teaching mathematics to the Dauphine's pages and also to a number of princes, among them Eugene of Savoy. By 1680, he was something of a pet at court, where he gave anatomy courses to courtiers and calculated for them the odds in the game called "basset."
In 1681, Sauveur did the mathematical calculations for a waterworks project for the "Grand Condé's" estate at Chantilly, working with Edmé Mariotte, the "father of French hydraulics. Condé became very fond of Sauveur and severely reprimanded anyone who laughed at the mathematician's speech impairment. Condé would invite Saveur to stay at Chantilly. It was there that Sauveur did his work on hydrostatics.
During the summer of 1689, Sauveur was chosen to be the science and mathematics teacher for the Duke of Chartres, Louis XIV's nephew. For the prince, he drew up a manuscript outlining the "elements" of geometry and, in collaboration with Marshal Vauban, a manuscript on the "elements of military fortification." (In 1691 Sauveur and Chartres were present at the siege of Mons by the French.) Another of the prince's teachers was Étienne Loulié, a musician engaged to teach him the "elements" of musical theory and notation. Loulié and Sauveur joined forces to show the prince how mathematics and musical theory were inter-related. Remnants of this joint course have survived in Sauveur's manuscript treatise on the theory of music, and in Loulié's Éléments. In the years that followed, Sauveur taught mathematics to various princes of the royal family. In 1686 he obtained the mathematics chair at the Collège de France, which granted him a rare exemption: since he was incapable of reciting a speech from memory, he was permitted to read his inaugural lecture.
Circa 1694, Sauveur began working with Loulié on "the science of sound", that is, acoustics. As Fontenelle put it, Sauveur laid out a vast plan that amounted to the "discovery of an unknown country", and that created for him a "personal empire", the study of "acoustical sound" (le son acoustique). But, as Fontenelle pointed out, "He had neither a voice nor hearing, yet he could think only of music. He was reduced to borrowing the voice and the ear of someone else. and in return he gave hitherto unknown demonstr
|
https://en.wikipedia.org/wiki/B%C3%BCrgi%E2%80%93Dunitz%20angle
|
The Bürgi–Dunitz angle (BD angle) is one of two angles that fully define the geometry of "attack" (approach via collision) of a nucleophile on a trigonal unsaturated center in a molecule, originally the carbonyl center in an organic ketone, but now extending to aldehyde, ester, and amide carbonyls, and to alkenes (olefins) as well. The angle was named after crystallographers Hans-Beat Bürgi and Jack D. Dunitz, its first senior investigators.
Practically speaking, the Bürgi–Dunitz and Flippin–Lodge angles were central to the development of understanding of chiral chemical synthesis, and specifically of the phenomenon of asymmetric induction during nucleophilic attack at hindered carbonyl centers (see the Cram–Felkin–Anh and Nguyen models).
Additionally, the stereoelectronic principles that underlie nucleophiles adopting a proscribed range of Bürgi–Dunitz angles may contribute to the conformational stability of proteins and are invoked to explain the stability of particular conformations of molecules in one hypothesis of a chemical origin of life.
Definition
In the addition of a nucleophile (Nu) attack to a carbonyl, the BD angle is defined as the Nu-C-O bond angle. The BD angle adopted during an approach by a nucleophile to a trigonal unsaturated electrophile depends primarily on the molecular orbital (MO) shapes and occupancies of the unsaturated carbon center (e.g., carbonyl center), and only secondarily on the molecular orbitals of the nucleophile.
Of the two angles which define the geometry of nucleophilic "attack", the second describes the "offset" of the nucleophile's approach toward one of the two substituents attached to the carbonyl carbon or other electrophilic center, and was named the Flippin–Lodge angle (FL angle) by Clayton Heathcock after his contributing collaborators Lee A. Flippin and Eric P. Lodge.
These angles are generally construed to mean the angle measured or calculated for a given system, and not the historically observed value range for the original Bürgi–Dunitz aminoketones, or an idealized value computed for a particular system (such as hydride addition to formaldehyde, image at left). That is, the BD and FL angles of the hydride-formadehyde system produce a given pair of values, while the angles observed for other systems may vary relative to this simplest of chemical systems.
Measurement
The original Bürgi-Dunitz measurements were of a series of intramolecular amine-ketone carbonyl interactions, in crystals of compounds bearing both functionalities—e.g., methadone and protopine. These gave a narrow range of BD angle values (105 ± 5°); corresponding computations—molecular orbital calculations of the SCF-LCAO-type—describing the approach of the s-orbital of a hydride anion (H−) to the pi-system of the simplest aldehyde, formaldehyde (H2C=O), gave a BD angle value of 107°.
Hence, Bürgi, Dunitz, and thereafter many others noted that the crystallographic measurements of the aminoketones and the computational esti
|
https://en.wikipedia.org/wiki/Liao%20%28surname%29
|
Liao () is a Chinese surname, most commonly found in Taiwan and Southern China. Statistics show it is among the 100 most common surnames in mainland China; figures from the Ministry of Public Security showed it to be the 61st most common surname, shared by around 4.2 million Chinese citizens.
The pinyin romanisation of the Mandarin pronunciation is . Its Cantonese pronunciation is generally transcribed as Liu. Other romanisations of the name include Leo, Leow, Liau, Liaw, Liauw, Leeau, Lio, Liow, Leaw, Leou, Lau, Loh, Liu, Liêu, Liew, Liw and Lew.
Notable people surnamed 廖
People with the surname Liao include:
Ashley Liao (born 2001), American actress
Bernice Liu (, born 1979), Canadian actress and former TVB model
Liao Cheng-hao, Minister of Justice of the Republic of China (1996–1998)
Liao Chengzhi (1908–1983), Chinese politician
Liao Chi-chun (1902–1976), Taiwanese oil painter and sculptor
Liao Feng-teh (1951–2008), Taiwanese politician
Gladys Liu, Hong Kong-born Australian politician
Liao Hua (died 264), military general
Liao Hui (born 1942), Hong Kong politician
Liao Hui (born 1987), Chinese weightlifter
Liao Liou-yi, Magistrate of Taichung County (1989–1997)
Martin Liao (born 1957), Hong Kong politician
Liao Qiuyun (born 1995), Chinese weightlifter
Liao Tianding
Liao Yaoxiang (1906–1968), a high-ranking Kuomintang commander
Liao Yiwu (born 1958), Chinese author and poet
Liao Zhongkai (1877–1925), Kuomintang leader
Liau Huei-fang, Deputy Minister of Labor of the Republic of China (2016–2017)
Liow Han Heng, Singaporean convicted killer
See also
Five Great Clans of the New Territories
References
Chinese-language surnames
Individual Chinese surnames
|
https://en.wikipedia.org/wiki/Clifford%20bundle
|
In mathematics, a Clifford bundle is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold M which is called the Clifford bundle of M.
General construction
Let V be a (real or complex) vector space together with a symmetric bilinear form <·,·>. The Clifford algebra Cℓ(V) is a natural (unital associative) algebra generated by V subject only to the relation
for all v in V. One can construct Cℓ(V) as a quotient of the tensor algebra of V by the ideal generated by the above relation.
Like other tensor operations, this construction can be carried out fiberwise on a smooth vector bundle. Let E be a smooth vector bundle over a smooth manifold M, and let g be a smooth symmetric bilinear form on E. The Clifford bundle of E is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of E:
The topology of Cℓ(E) is determined by that of E via an associated bundle construction.
One is most often interested in the case where g is positive-definite or at least nondegenerate; that is, when (E, g) is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that (E, g) is a Riemannian vector bundle. The Clifford bundle of E can be constructed as follows. Let CℓnR be the Clifford algebra generated by Rn with the Euclidean metric. The standard action of the orthogonal group O(n) on Rn induces a graded automorphism of CℓnR. The homomorphism
is determined by
where vi are all vectors in Rn. The Clifford bundle of E is then given by
where F(E) is the orthonormal frame bundle of E. It is clear from this construction that the structure group of Cℓ(E) is O(n). Since O(n) acts by graded automorphisms on CℓnR it follows that Cℓ(E) is a bundle of Z2-graded algebras over M. The Clifford bundle Cℓ(E) can then be decomposed into even and odd subbundles:
If the vector bundle E is orientable then one can reduce the structure group of Cℓ(E) from O(n) to SO(n) in the natural manner.
Clifford bundle of a Riemannian manifold
If M is a Riemannian manifold with metric g, then the Clifford bundle of M is the Clifford bundle generated by the tangent bundle TM. One can also build a Clifford bundle out of the cotangent bundle T*M. The metric induces a natural isomorphism TM = T*M and therefore an isomorphism Cℓ(TM) = Cℓ(T*M).
There is a natural vector bundle isomorphism between the Clifford bundle of M and the exterior bundle of M:
This is an isomorphism of vector bundles not algebra bundles. The isomorphism is induced from the corresponding isomorphism on each fiber. In this way one can think of sections of the Clifford bundle as differential forms on M equipped with Clifford multiplication rather than the wedge product (which is independent of the metric).
The above isomorphism respects the grading in the sense that
Local description
For a vector at , and a form the C
|
https://en.wikipedia.org/wiki/Yuri%20Burago
|
Yuri Dmitrievich Burago (; born 1936) is a Russian mathematician. He works in differential and convex geometry.
Education and career
Burago studied at Leningrad University, where he obtained his Ph.D. and Habilitation degrees. His advisors were Victor Zalgaller and Aleksandr Aleksandrov.
Yuri is a creator (with his students Perelman and Petrunin, and M. Gromov) of what is known now as Alexandrov Geometry. Also brought geometric inequalities to the state of art.
Burago is the head of the Laboratory of Geometry and Topology that is part of the St. Petersburg Department of Steklov Institute of Mathematics. He took part in a report for the United States Civilian Research and Development Foundation for the Independent States of the former Soviet Union.
Works
His other books and papers include:
Geometry III: Theory of Surfaces (1992)
Potential Theory and Function Theory for Irregular Regions (1969)
Isoperimetric inequalities in the theory of surfaces of bounded external curvature (1970)
Students
He has advised Grigori Perelman, who solved the Poincaré conjecture, one of the seven Millennium Prize Problems. Burago was an advisor to Perelman during the latter's post-graduate research at St. Petersburg Department of Steklov Institute of Mathematics.
Footnotes
External links
Burago's page on the site of Steklov Mathematical Institute
Yuri Dmitrievich Burago in the Oberwolfach Photo Collection
Soviet mathematicians
Geometers
Differential geometers
1936 births
Living people
Saint Petersburg State University alumni
20th-century Russian mathematicians
21st-century Russian mathematicians
|
https://en.wikipedia.org/wiki/Baldassarre%20Boncompagni
|
Prince Baldassarre Boncompagni-Ludovisi (10 May 1821 – 13 April 1894), was an Italian historian of mathematics and aristocrat.
Biography
Boncompagni was born in Rome, into an ancient noble and wealthy Roman family, the Ludovisi-Boncompagni, as the third son of Prince Luigi Boncompagni Ludovisi and Princess Maria Maddalena Odescalchi. He studied under the mathematician Barnaba Tortolini and astronomer Ignazio Calandrelli, developing an interest in the history of science. In 1847 Pope Pius IX appointed him a member of the Accademia dei Lincei. Between 1850-1862 he produced studies on mathematicians of the Middle Ages and in 1868 founded the Bullettino di bibliografia e di storia delle scienze matematiche e fisiche. After the annexation of the Papal States into the Kingdom of Italy (1870), he refused further participation in the new Academy of the Lincei, and did not accept the appointment as Senator of the Kingdom offered by Quintino Sella. He did, however, serve as a member of several other Italian and foreign academies.
Boncompagni edited Bullettino di bibliografia e di storia delle scienze matematiche e fisiche ("The bulletin of bibliography and history of mathematical and physical sciences") (1868–1887), the first Italian periodical entirely dedicated to the history of mathematics. He edited every article that appeared in the journal. He also prepared and published the first modern edition of Fibonacci's Liber Abaci.
Selected works
Recherches sur les integrales définies. Journal für die reine und angewandte Mathematik, 1843, XXV, pagg. 74-96
Intorno ad alcuni avanzamenti della fisica in Italia nei secoli XVI e XVII. Giornale arcadico di scienze, lettere ed arti, 1846, CIX, pagg. 3-48
Della vita e delle opere di Guido Bonatti, astrologo e astronomo del secolo decimoterzo. Roma, 1851
Delle versioni fatte da Platone Tiburtino, traduttore del secolo duodecimo. Atti dell'Accademia Pontificia dei Nuovi Lincei, 1850–51, IV, pagg. 247-286
Della vita e delle opere di Gherardo Cremonese, traduttore del secolo decimosecondo, e di Gherardo da Sabbioneta, astronomo del secolo decimoterzo. Atti dell'Accademia Pontificia dei Nuovi Lincei, 1850–51, IV, pagg. 387-493
Della vita e delle opere di Leonardo Pisano, matematico del secolo decimoterzo. Atti dell'Accademia Pontificia dei Nuovi lincei, 1851–52, V, pagg. 208-245
Intorno ad alcune opere di Leonardo Pisano (Roma : tipografia delle belle arti, 1854)
Opuscoli di Leonardo Pisano, pubblicati da Baldassarre Boncompagni secondo la lezione di un codice della Biblioteca Ambrosiana di Milano, Firenze, 1856
Trattati d'aritmetica pubblicati da Baldassarre Boncompagni, I, Algoritmi de numero Indorum; II, Ioannis Hispalensis liber Algoritmi de practica arismetice. Roma, 1857
Scritti di Leonardo Pisano, matematico, pubblicati da Baldassarre Boncompagni. 2 voll., Roma, 1857–62
Bullettino di bibliografia e di storia delle scienze matematiche e fisiche. Tomi I-XX, Roma, 1868-1887
References
External link
|
https://en.wikipedia.org/wiki/Wolfgang%20M.%20Schmidt
|
Wolfgang M. Schmidt (born 3 October 1933) is an Austrian mathematician working in the area of number theory. He studied mathematics at the University of Vienna, where he received his PhD, which was supervised by Edmund Hlawka, in 1955. Wolfgang Schmidt is a Professor Emeritus from the University of Colorado at Boulder and a member of the Austrian Academy of Sciences and the Polish Academy of Sciences.
Career
He was awarded the eighth Frank Nelson Cole Prize in Number Theory for work on Diophantine approximation. He is known for his subspace theorem.
In 1960, he proved that every normal number in base r is normal in base s if and only if log r / log s is a rational number. He also proved the existence of T numbers. His series of papers on irregularities of distribution can be seen in J.Beck and W.Chen, Irregularities of Distribution, Cambridge University Press.
Schmidt is in a small group of number theorists who have been invited to address the International Congress of Mathematicians three times. The others are Iwaniec, Shimura, and Tate.
In 1986, Schmidt received the Humboldt Research Award and in 2003, he received the Austrian Decoration for Science and Art. Schmidt holds honorary doctorates from the University of Ulm, the Sorbonne, the University of Waterloo, the University of Marburg and the University of York. In 2012 he became a fellow of the American Mathematical Society.
Books
Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000
Equations Over Finite Fields: An Elementary Approach, 2nd edition, Kendrick Press 2004
References
Further reading
Diophantine approximation: festschrift for Wolfgang Schmidt, Wolfgang M. Schmidt, H. P. Schlickewei, Robert F. Tichy, Klaus Schmidt, Springer, 2008,
1933 births
Living people
Number theorists
Recipients of the Austrian Decoration for Science and Art
Institute for Advanced Study visiting scholars
University of Colorado Boulder faculty
Members of the Austrian Academy of Sciences
Members of the Polish Academy of Sciences
Fellows of the American Mathematical Society
|
https://en.wikipedia.org/wiki/Exeter%20Science%20Park
|
Exeter Science Park is an English centre of activity for businesses in science, technology, engineering, maths and medicine (STEMM).
Exeter Science Park is based on a 26 hectare (64 acre site) at Junction of 29 of the M5 motorway on the edge of the city of Exeter. It was established in 2013 and was officially opened in 2015.
Exeter Science Park Ltd (ESPL), the park developer, has four shareholders: Devon County Council, the University of Exeter, East Devon District Council and Exeter City Council. Its two strategic partners are the Heart of the South West Local Enterprise Partnership (LEP) and the Exeter and East Devon Enterprise Zone.
The building of the Science Park Centre was made possible with shareholder equity from Devon County Council, East Devon District Council, Exeter City Council, and the University of Exeter; the Heart of the South West Local Enterprise Partnership (HotSW LEP) which committed a £4.5m loan from the Growing Places Fund; and a £1million grant from the Regional Growth Fund.
Exeter Science Park’s Grow-on Buildings were partly funded by £4.5m from the HotSW LEP Growth Deal Funding. The HotSW LEP also provided £2.5m local Government funding towards the Environmental Futures Campus.
The Ada Lovelace Building is partly funded by £5.5 million from the Heart of the South West Local Enterprise Partnership’s Growth Deal Funding. In addition to this, East Devon District Council’s Cabinet invested £1.1m in the development of the building in conjunction with Devon County Council as part of the Exeter and East Devon Enterprise Zone programme.
Funding for the George Parker Bidder building was secured in August 2020 from the Government’s ‘Getting Building Fund’ and allocated to Exeter Science Park by the Heart of the South West Local Enterprise Partnership (HotSW LEP) from its £35.4 million share of the national pot. The building was one of the first Getting Building Fund projects to begin construction in the area.
Buildings at the Science Park include provision of laboratories, offices, meeting rooms and hotdesking facilities. It is also home to a café which is open to the public.
In 2021, the University of Exeter transferred the business activity of its Innovation Centre to Exeter Science Park and released £2.25m funding to support the provision of innovation services by SETsquared Exeter over the next 18 years.
The park is being developed around four clusters and once complete, it is anticipated it will comprise a million square feet of accommodation and employ around 3,000 people.
Occupants
As of December 2022, the park has 42 tenants. These include CEFAS, Securious, Attomarker, Heart of the South West LEP, Diagnexia UK Limited, Klarian Ltd., the University of Exeter and Remit Zero Limited.
In 2016, the Met Office took delivery of an IT hall and neighbouring office building at the Science Park. The IT hall houses a new supercomputer, the Met Office's third.
Redhayes Bridge
The Redhayes pedestrian and cycle bridge over
|
https://en.wikipedia.org/wiki/Sweep%20line%20algorithm
|
In computational geometry, a sweep line algorithm or plane sweep algorithm is an algorithmic paradigm that uses a conceptual sweep line or sweep surface to solve various problems in Euclidean space. It is one of the critical techniques in computational geometry.
The idea behind algorithms of this type is to imagine that a line (often a vertical line) is swept or moved across the plane, stopping at some points. Geometric operations are restricted to geometric objects that either intersect or are in the immediate vicinity of the sweep line whenever it stops, and the complete solution is available once the line has passed over all objects.
History
This approach may be traced to scanline algorithms of rendering in computer graphics, followed by exploiting this approach in early algorithms of integrated circuit layout design, in which a geometric description of an IC was processed in parallel strips because the entire description could not fit into memory.
Applications
Application of this approach led to a breakthrough in the computational complexity of geometric algorithms when Shamos and Hoey presented algorithms for line segment intersection in the plane, and in particular, they described how a combination of the scanline approach with efficient data structures (self-balancing binary search trees) makes it possible to detect whether there are intersections among segments in the plane in time complexity of . The closely related Bentley–Ottmann algorithm uses a sweep line technique to report all intersections among any segments in the plane in time complexity of and space complexity of .
Since then, this approach has been used to design efficient algorithms for a number of problems, such as the construction of the Voronoi diagram (Fortune's algorithm) and the Delaunay triangulation or boolean operations on polygons.
Generalizations and extensions
Topological sweeping is a form of plane sweep with a simple ordering of processing points, which avoids the necessity of completely sorting the points; it allows some sweep line algorithms to be performed more efficiently.
The rotating calipers technique for designing geometric algorithms may also be interpreted as a form of the plane sweep, in the projective dual of the input plane: a form of projective duality transforms the slope of a line in one plane into the x-coordinate of a point in the dual plane, so the progression through lines in sorted order by their slope as performed by a rotating calipers algorithm is dual to the progression through points sorted by their x-coordinates in a plane sweep algorithm.
The sweeping approach may be generalised to higher dimensions.
References
Geometric algorithms
|
https://en.wikipedia.org/wiki/List%20of%20books%20in%20computational%20geometry
|
This is a list of books in computational geometry.
There are two major, largely nonoverlapping categories:
Combinatorial computational geometry, which deals with collections of discrete objects or defined in discrete terms: points, lines, polygons, polytopes, etc., and algorithms of discrete/combinatorial character are used
Numerical computational geometry, also known as geometric modeling and computer-aided geometric design (CAGD), which deals with modelling of shapes of real-life objects in terms of curves and surfaces with algebraic representation.
Combinatorial computational geometry
General-purpose textbooks
The book is the first comprehensive monograph on the level of a graduate textbook to systematically cover the fundamental aspects of the emerging discipline of computational geometry. It is written by founders of the field and the first edition covered all major developments in the preceding 10 years. In the aspect of comprehensiveness it was preceded only by the 1984 survey paper, Lee, D, T., Preparata, F. P. : "Computational geometry - a survey". IEEE Trans. on Computers. Vol. 33, No. 12, pp. 1072–1101 (1984). It is focused on two-dimensional problems, but also has digressions into higher dimensions.
The initial core of the book was M.I.Shamos' doctoral dissertation, which was suggested to turn into a book by a yet another pioneer in the field, Ronald Graham.
The introduction covers the history of the field, basic data structures, and necessary notions from the theory of computation and geometry.
The subsequent sections cover geometric searching (point location, range searching), convex hull computation, proximity-related problems (closest points, computation and applications of the Voronoi diagram, Euclidean minimum spanning tree, triangulations, etc.), geometric intersection problems, algorithms for sets of isothetic rectangles
The monograph is a rather advanced exposition of problems and approaches in computational geometry focused on the role of hyperplane arrangements, which are shown to constitute a basic underlying combinatorial-geometric structure in certain areas of the field. The primary target audience are active theoretical researchers in the field, rather than application developers. Unlike most of books in computational geometry focused on 2- and 3-dimensional problems (where most applications of computational geometry are), the book aims to treat its subject in the general multi-dimensional setting.
The textbook provides an introduction to computation geometry from the point of view of practical applications. Starting with an introduction chapter, each of the 15 remaining ones formulates a real application problem, formulates an underlying geometrical problem, and discusses techniques of computational geometry useful for its solution, with algorithms provided in pseudocode. The book treats mostly 2- and 3-dimensional geometry. The goal of the book is to provide a comprehensive introduction into methods and approa
|
https://en.wikipedia.org/wiki/Education%20in%20the%20member%20states%20of%20the%20Organisation%20of%20Islamic%20Cooperation
|
The public spending on education in the 57 Organisation of Islamic Cooperation (OIC) countries is one of the lowest in the world.
Statistics
Public expenditure on education (% of GDP)
Scientifically productive countries
Most productive universities in OIC
Notes
See also
List of Organisation of Islamic Cooperation member states by population
Economy of the Organisation of Islamic Cooperation
Islamic Educational, Scientific and Cultural Organization
External links
Islamic Educational, Scientific and Cultural Organization
OIC Standing Committee on Scientific and Technological Cooperation
Organisation of Islamic Cooperation
|
https://en.wikipedia.org/wiki/Indra%27s%20Pearls%20%28book%29
|
Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002 and 2015.
The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity. These patterns were glimpsed by German mathematician Felix Klein, but modern computer graphics allows them to be fully visualised and explored in detail.
Title
The book's title refers to Indra's net, a metaphorical object described in the Buddhist text of the Flower Garland Sutra. Indra's net consists of an infinite array of gossamer strands and pearls. The frontispiece to Indra's Pearls quotes the following description:
In the glistening surface of each pearl are reflected all the other pearls ... In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.
The allusion to Felix Klein's "vision" is a reference to Klein's early investigations of Schottky groups and hand-drawn plots of their limit sets. It also refers to Klein's wider vision of the connections between group theory, symmetry and geometry - see Erlangen program.
Contents
The contents of Indra's Pearls are as follows:
Chapter 1. The language of symmetry – an introduction to the mathematical concept of symmetry and its relation to geometric groups.
Chapter 2. A delightful fiction – an introduction to complex numbers and mappings of the complex plane and the Riemann sphere.
Chapter 3. Double spirals and Möbius maps – Möbius transformations and their classification.
Chapter 4. The Schottky dance – pairs of Möbius maps which generate Schottky groups; plotting their limit sets using breadth-first searches.
Chapter 5. Fractal dust and infinite words – Schottky limit sets regarded as fractals; computer generation of these fractals using depth-first searches and iterated function systems.
Chapter 6. Indra's necklace – the continuous limit sets generated when pairs of generating circles touch.
Chapter 7. The glowing gasket – the Schottky group whose limit set is the Apollonian gasket; links to the modular group.
Chapter 8. Playing with parameters – parameterising Schottky groups with parabolic commutator using two complex parameters; using these parameters to explore the Teichmüller space of Schottky groups.
Chapter 9. Accidents will happen – introducing Maskit's slice, parameterised by a single complex parameter; exploring the boundary between discrete and non-discrete groups.
Chapter 10. Between the cracks – further exploration of the Maskit boundary between discrete and non-discrete groups in another slice of parameter space; identification and exploration of degenerate groups.
Chapter 11. Crossing boundaries – ideas for further exploration, such as adding a third generator.
Chapter 12. Epilogue – concluding overview of non-Euclidean geometry
|
https://en.wikipedia.org/wiki/Ulam%20number
|
In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with U1 = 1 and U2 = 2. Then for n > 2, Un is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way and larger than all earlier terms.
Examples
As a consequence of the definition, 3 is an Ulam number (1 + 2); and 4 is an Ulam number (1 + 3). (Here 2 + 2 is not a second representation of 4, because the previous terms must be distinct.) The integer 5 is not an Ulam number, because 5 = 1 + 4 = 2 + 3. The first few terms are
1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126, 131, 138, 145, 148, 155, 175, 177, 180, 182, 189, 197, 206, 209, 219, 221, 236, 238, 241, 243, 253, 258, 260, 273, 282, ... .
There are infinitely many Ulam numbers. For, after the first n numbers in the sequence have already been determined, it is always possible to extend the sequence by one more element: is uniquely represented as a sum of two of the first n numbers, and there may be other smaller numbers that are also uniquely represented in this way, so the next element can be chosen as the smallest of these uniquely representable numbers.
Ulam is said to have conjectured that the numbers have zero density, but they seem to have a density of approximately 0.07398.
Properties
Apart from 1 + 2 = 3 any subsequent Ulam number cannot be the sum of its two prior consecutive Ulam numbers.
Proof: Assume that for n > 2, Un−1 + Un = Un+1 is the required sum in only one way; then so does Un−2 + Un produce a sum in only one way, and it falls between Un and Un+1. This contradicts the condition that Un+1 is the next smallest Ulam number.
For n > 2, any three consecutive Ulam numbers (Un−1, Un, Un+1) as integer sides will form a triangle.
Proof: The previous property states that for n > 2, Un−2 + Un ≥ Un + 1. Consequently Un−1 + Un > Un+1 and because Un−1 < Un < Un+1 the triangle inequality is satisfied.
The sequence of Ulam numbers forms a complete sequence.
Proof: By definition Un = Uj + Uk where j < k < n and is the smallest integer that is the sum of two distinct smaller Ulam numbers in exactly one way. This means that for all Un with n > 3, the greatest value that Uj can have is Un−3 and the greatest value that Uk can have is Un−1.
Hence Un ≤ Un−1 + Un−3 < 2Un−1 and U1 = 1, U2 = 2, U3 = 3. This is a sufficient condition for Ulam numbers to be a complete sequence.
For every integer n > 1 there is always at least one Ulam number Uj such that n ≤ Uj < 2n.
Proof: It has been proved that there are infinitely many Ulam numbers and they start at 1. Therefore for every integer n > 1 it is possible to find j such that Uj−1 ≤ n ≤ Uj. From the proof above for n > 3, Uj ≤ Uj−1 + Uj−3 < 2Uj−1. Therefore n ≤ Uj < 2Uj−1 ≤ 2n. Also for n = 2 and 3 the property is true by calculation.
|
https://en.wikipedia.org/wiki/Toronto%20Argonauts%20all-time%20records%20and%20statistics
|
The following is a list of Toronto Argonauts all time records and statistics current to the 2023 CFL season. Each category lists the top five players, where known, except for when the fifth place player is tied in which case all players with the same number are listed.
Grey Cup Championships
as a Player
6 - Jack Wedley
5 - Joe Krol, Bill Zock, Les Ascott
4 - Royal Copeland
3 - Teddy Morris, Frankie Morris, Bill Stukus, Wes Cutler, Pinball Clemons, Mike O'Shea, Paul Masotti, Adrion Smith, Noah Cantor
as a Head Coach
3 - Lew Hayman, Teddy Morris
2 - Frank Clair, Don Matthews
Games Played
222 – Don Moen (1982–94)
208 – Noel Prefontaine (1998–2007, 2010–13)
205 – Mike O'Shea (1996–99, 2001–08)
204 – Danny Nykoluk (1955–71)
199 – Jeff Johnson (2000–13)
195 – Chad Folk (1997–2008)
189 – Paul Masotti (1988–99)
186 – Michael Clemons (1989–2000)
175 – Dan Ferrone (1981–92)
166 – Adrion Smith (1996–2005)
Scoring
Most Points – Career
1498 – Lance Chomyc (1985–93)
1228 – Noel Prefontaine (1998–2007, 2010–13)
899 – Zenon Andrusyshyn (1971–77, 1980–82)
550 – Dick Shatto (1954–65)
549 – Mike Vanderjagt (1996–97, 2008)
Most Points – Season
236 – Lance Chomyc – 1991
207 – Lance Chomyc – 1988
200 – Lance Chomyc – 1990
198 – Mike Vanderjagt – 1996
193 – Lance Chomyc – 1987
Most Points – Game
27 – Cookie Gilchrist – versus Montreal Alouettes, October 30, 1960
26 – Lance Chomyc – versus Ottawa Rough Riders, October 14, 1988
24 – six players, seven times, most recently Derrell Mitchell – versus BC Lions, September 12, 1998
Most Touchdowns – Career
91 – Dick Shatto (1954–65)
85 – Michael Clemons (1989–2000)
74 – Derrell Mitchell (1997–2003, 2007)
53 – Darrell K. Smith (1986–92)
50 – Terry Greer (1980–85)
Most Touchdowns – Season
20 – Darrell K. Smith – 1990
18 – Lester Brown – 1984
18 – Robert Drummond – 1997
17 – Robert Drummond – 1996
17 – Derrell Mitchell – 1997
Most Touchdowns – Game
4 – Dick Shatto – versus Hamilton Tiger-Cats, October 3, 1958
4 – Bill Symons – versus Ottawa Rough Riders, September 7, 1970
4 – Darrell K. Smith – versus Hamilton Tiger-Cats, September 29, 1990
4 – Robert Drummond – versus Hamilton Tiger-Cats, November 2, 1996
4 – Robert Drummond – versus Montreal Alouettes, July 31, 1997
4 – Derrell Mitchell – versus BC Lions, September 12, 1998
Most Rushing Touchdowns – Career
39 – Dick Shatto (1954–65)
33 – Bill Symons (1967–1973)
32 – Gill Fenerty (1987–1989)
31 – Michael Clemons (1989–2000)
27 – Robert Drummond (1996–2002)
Most Rushing Touchdowns – Season
14 – James Franklin – 2018
12 – Gill Fenerty – 1987
12 – Robert Drummond – 1997
11 – Robert Drummond – 1996
10 – Lester Brown – 1984
10 – Gill Fenerty – 1987
10 – Gill Fenerty – 1987
Most Rushing Touchdowns – Game
4 – Robert Drummond – at Hamilton Tiger-Cats, November 2, 1996
4 – Robert Drummond – versus Montreal Alouettes, July 31, 1997
3 – Bill Symons – at Ottawa Rough Riders, September 7, 1970
3 – Gill Fenerty – versus Hamilton Tiger-Cats, September 20, 1987
3 – Tracy Ham –
|
https://en.wikipedia.org/wiki/Rafail%20Ostrovsky
|
Rafail Ostrovsky is a distinguished professor of computer science and mathematics at UCLA and a well-known researcher in algorithms and cryptography.
Biography
Rafail Ostrovsky received his Ph.D. from MIT in 1992.
He is a member of the editorial board of Algorithmica
, Editorial Board of Journal of Cryptology and Editorial and Advisory Board of the International Journal of Information and Computer Security .
Awards
2022 W. Wallace McDowell Award "for visionary contributions to computer security theory and practice, including foreseeing new cloud vulnerabilities and then pioneering corresponding novel solutions"
2021 AAAS Fellow
2021 Fellow of the Association for Computing Machinery "for contributions to the foundations of cryptography"
2019 Academia Europaea Foreign Member
2018 RSA Award for Excellence in Mathematics "for contributions to the theory and to new variants of secure multi-party computations"
2017 IEEE Edward J. McCluskey Technical Achievement Award "for outstanding contributions to cryptographic protocols and systems, enhancing the scope of cryptographic applications and of assured cryptographic security."
2017 IEEE Fellow, "for contributions to cryptography”
2013 IACR Fellow "for numerous contributions to the scientific foundations of cryptography and for sustained educational leadership in cryptography"
1993 Henry Taub Prize
Publications
Some of Ostrovsky's contributions to computer science include:
1990 Introduced (with R. Venkatesan and M. Yung) the notion ofhttps://services27.ieee.org/fellowsdirectory/home.html interactive hashing proved essential for constructing statistical zero-knowledge proofs for NP based on any one-way function (see NOVY and ).
1991 Introduced (with M. Yung) the notion of mobile adversary (later renamed proactive security) (see survey of Goldwasser
1990 Introduced the first poly-logarithmic Oblivious RAM (ORAM) scheme.
1993 Proved (with A. Wigderson) equivalence ofone-way functions and zero-knowledge .
1996 Introduced (with R. Canetti, C. Dwork and M. Naor) the notion of deniable encryption .
1997 Introduced (with E. Kushilevitz) the first single server private information retrieval scheme .
1997 Showed (with E. Kushilevitz and Y. Rabani) (1+ε) poly-time and poly-size approximate-nearest neighbor search for high-dimensional data for L1-norm and Euclidean space.
References
External links
Ostrovsky's home page
Some of Ostrovsky's publications
1963 births
Living people
American computer scientists
Jewish American scientists
Modern cryptographers
Theoretical computer scientists
Computer security academics
American cryptographers
University of California, Los Angeles faculty
Massachusetts Institute of Technology alumni
21st-century American Jews
|
https://en.wikipedia.org/wiki/Montreal%20Alouettes%20all-time%20records%20and%20statistics
|
The following is a list of Montreal Alouettes all time records and statistics current to the 2023 CFL season.
This list includes the records for the Montreal Concordes (1982 to 1985) but does not include Baltimore CFLers or Stallions records (1994 to 1995).
Grey Cups
Most Grey Cups Won, Player
3 - Peter Dalla Riva
3 - Sonny Wade
3 - Gordon Judges
3 - Barry Randall
3 - Anthony Calvillo
3 - Ben Cahoon
3 - Anwar Stewart
3 - Scott Flory
Most Grey Cup Appearances, Player
8 - Anthony Calvillo
8 - Ben Cahoon
8 - Scott Flory
7 - Bryan Chiu
Most Grey Cups Won, Head Coach
2 - Marv Levy
2 - Marc Trestman
1 - Lew Hayman
1 - Sam Etcheverry
1 - Don Matthews
Most Grey Cup Appearances, Head Coach
3 - Peahead Walker
3 - Marv Levy
3 - Don Matthews
3 - Marc Trestman
Coaching
Most Seasons Coached
8 - Lew Hayman
6 - Peahead Walker
6 - Jim Popp
5 - Marv Levy
5 - Don Matthews
5 - Marc Trestman
Most Games Coached
108 - Peahead Walker
90 - Marc Trestman
86 - Don Matthews
78 - Marv Levy
72 - Lew Hayman
Most Wins
59 - Peahead Walker
59 - Marc Trestman
58 - Don Matthews
43 - Marv Levy
37 - Lew Hayman
Most Losses
48 - Peahead Walker
41 - Joe Galat
36 - Jim Popp
33 - Lew Hayman
31 - Marv Levy
31 - Marc Trestman
31 - Kay Dalton
Games
Most Games Played
269 – Anthony Calvillo (1998–2013)
242 – Scott Flory (1999–2013)
230 – John Bowman (2006–19)
229 – Chip Cox (2006–18)
224 – Ben Cahoon (1998–2010)
218 – Bryan Chiu (1997–2009)
203 – Glen Weir (1972–84)
Most Seasons Played
16 – Anthony Calvillo (1998–2013)
15 – Scott Flory (1999–2013)
14 – Peter Dalla Riva (1968–81)
14 – John Bowman (2006–19)
13 – Gordon Judges (1968, 1970–82)
13 – Don Sweet (1972–84)
13 – Glen Weir (1972–84)
13 – Bryan Chiu (1997–2009)
13 – Ben Cahoon (1998–2010)
13 – Chip Cox (2006–18)
13 – Kristian Matte (2010–19, 2021–23)
Scoring
Most Points – Career
1342 – Don Sweet (1972–84)
1324 – Terry Baker (1996–2002)
1142 – Damon Duval (2005–10)
Most Points – Season
242 – Damon Duval (2009)
220 – Terry Baker (2000)
206 – Damon Duval (2008)
203 – Terry Baker (1998)
201 – Damon Duval (2006)
197 – Sean Whyte (2011)
Most Points – Game
25 – Terry Baker – versus Calgary Stampeders, October 29, 2000
24 – Pat Abbruzzi – versus Hamilton Tiger-Cats, September 22, 1956
24 – Fob James – versus Hamilton Tiger-Cats, October 20, 1956
24 – George Dixon – versus Ottawa Rough Riders, September 5, 1960
24 – Mike Pringle – versus Saskatchewan Roughriders, August 3, 2000
24 – Mike Pringle – versus Edmonton Eskimos, October 9, 2000
24 – Mike Pringle – versus Calgary Stampeders, July 12, 2001
24 – Autry Denson – versus Ottawa Renegades, July 9, 2004
24 – Damon Duval – versus Toronto Argonauts, November 7, 2009
Most Touchdowns – Career
79 – Mike Pringle (1996–2002)
79 – Virgil Wagner (1946–54)
65 – Ben Cahoon (1998–2010)
59 – George Dixon (1959–65)
54 – Peter Dalla Riva (1968–81)
Most Touchdowns – Season
20 – Pat Abbruzzi (1956)
19 – Pat Abbruzzi (1955)
19 – Mike Pringle (2000)
18 – George Dixon (1960)
17 – Robert Edwar
|
https://en.wikipedia.org/wiki/Pyramid%20puzzle
|
Pyramid puzzle may refer to:
Mathematics
Cannonball problem, a mathematical problem
Tower of Hanoi, a mathematical game
Other
Pyramid puzzle, a type of mechanical puzzle
|
https://en.wikipedia.org/wiki/Discharging%20method%20%28discrete%20mathematics%29
|
The discharging method is a technique used to prove lemmas in structural graph theory. Discharging is most well known for its central role in the proof of the four color theorem. The discharging method is used to prove that every graph in a certain class contains some subgraph from a specified list. The presence of the desired subgraph is then often used to prove a coloring result.
Most commonly, discharging is applied to planar graphs.
Initially, a charge is assigned to each face and each vertex of the graph.
The charges are assigned so that they sum to a small positive number. During the Discharging Phase the charge at each face or vertex may be redistributed to nearby faces and vertices, as required by a set of discharging rules. However, each discharging rule maintains the sum of the charges. The rules are designed so that after the discharging phase each face or vertex with positive charge lies in one of the desired subgraphs. Since the sum of the charges is positive, some face or vertex must have a positive charge. Many discharging arguments use one of a few standard initial charge functions (these are listed below). Successful application of the discharging method requires creative design of discharging rules.
An example
In 1904, Wernicke introduced the discharging method to prove the following theorem, which was part of an attempt to prove the four color theorem.
Theorem: If a planar graph has minimum degree 5, then it either has an edge
with endpoints both of degree 5 or one with endpoints of degrees 5 and 6.
Proof:
We use , , and to denote the sets of vertices, faces, and edges, respectively.
We call an edge light if its endpoints are both of degree 5 or are of degrees 5 and 6.
Embed the graph in the plane. To prove the theorem, it is sufficient to only consider planar triangulations (because, if it holds on a triangulation, when removing nodes to return to the original graph, neither node on either side of the desired edge can be removed without reducing the minimum degree of the graph below 5). We arbitrarily add edges to the graph until it is a triangulation.
Since the original graph had minimum degree 5, each endpoint of a new edge has degree at least 6.
So, none of the new edges are light.
Thus, if the triangulation contains a light edge, then that edge must have been in the original graph.
We give the charge to each vertex and the charge to each face , where denotes the degree of a vertex and the length of a face. (Since the graph is a triangulation, the charge on each face is 0.) Recall that the sum of all the degrees in the graph is equal to twice the number of edges; similarly, the sum of all the face lengths equals twice the number of edges. Using Euler's Formula, it's easy to see that the sum of all the charges is 12:
We use only a single discharging rule:
Each degree 5 vertex gives a charge of 1/5 to each neighbor.
We consider which vertices could have positive final charge.
The only vertice
|
https://en.wikipedia.org/wiki/Fernando%20Kanapkis
|
Fernando Alfredo Kanapkis García (born 6 June 1966) is a Uruguayan retired footballer who played as a defender. He was part of Uruguay national team at 1993 Copa América.
Career statistics
International
References
External links
Profile
1966 births
Living people
Footballers from Montevideo
Men's association football defenders
Uruguayan people of Greek descent
Uruguayan people of Spanish descent
Uruguayan men's footballers
Uruguay men's international footballers
Centro Atlético Fénix players
Danubio F.C. players
Textil Mandiyú footballers
Clube Atlético Mineiro players
Club Nacional de Football players
Rampla Juniors players
Racing Club de Montevideo players
1993 Copa América players
Uruguayan expatriate men's footballers
Expatriate men's footballers in Brazil
Expatriate men's footballers in Argentina
|
https://en.wikipedia.org/wiki/BC%20Lions%20all-time%20records%20and%20statistics
|
The following is a list of BC Lions all time records and statistics current to the 2023 CFL season. Each category lists the top five players, where known, except for when the fifth place player is tied in which case all players with the same number are listed. Games and seasons played lists the top ten players.
Grey Cup championships
As a player:
3 - Lui Passaglia
As a Head Coach:
2 - Wally Buono
Games
Most Games Played
408 – Lui Passaglia (1976–2000)
265 – Jamie Taras (1987–2002)
233 – Al Wilson (1972–86)
223 – Norm Fieldgate (1954–67)
212 – Ryan Phillips (2005–16)
202 – Cory Mantyka (1993–2005)
197 – Jim Young (1967–79)
197 – Angus Reid (2001–12)
194 – Geroy Simon (2001–12)
192 – Glen Jackson (1976–87)
Most Seasons Played
25 – Lui Passaglia (1976–2000)
16 – Jamie Taras (1987–2002)
15 – Al Wilson (1972–86)
14 – Norm Fieldgate (1954–67)
13 – Jim Young (1967–79)
13 – Cory Mantyka (1993–2005)
13 – Bret Anderson (1997–2009)
12 – Greg Findlay (1962–73)
12 – Glen Jackson (1976–87)
12 – Angus Reid (2001–12)
12 – Geroy Simon (2001–12)
12 – Ryan Phillips (2005–16)
12 – Paul McCallum (1993–94, 2006–14, 2016)
Scoring
Most points – career
3991 – Lui Passaglia (1976–2000)
1506 – Paul McCallum (1993–94, 2006–14, 2016)
570 – Ted Gerela (1967–73)
568 – Geroy Simon (2001–12)
523 – Willie Fleming (1959–66)
Most points – season
214 – Lui Passaglia – 1987
210 – Lui Passaglia – 1991
203 – Paul McCallum – 2011
197 – Lui Passaglia – 1998
194 – Lui Passaglia – 1995
194 – Sean Whyte – 2023
Most points – game
25 – Willie Fleming – versus Saskatchewan Roughriders, October 29, 1960
24 – Larry Key – at Calgary Stampeders, Jul 31, 1981
24 – Mervyn Fernandez – 3 times
24 – Lui Passaglia – versus Toronto Argonauts, Sep 6, 1985
24 – Cory Philpot – versus Hamilton Tiger-Cats, Sep 16, 1995
24 – Alfred Jackson – versus Montreal Alouettes, Aug 4, 2001
24 – Sean Millington – versus Hamilton Tiger-Cats, Jul 18, 2002
24 – Geroy Simon – at Hamilton Tiger-Cats, Aug 13, 2004
24 – James Butler – versus Edmonton Elks, Jun 11, 2022
Most touchdowns – career
94 – Geroy Simon (2001–12)
87 – Willie Fleming (1959–66)
78 – Sean Millington (1991–97, 2000–02)
68 – Jim Young (1967–79)
58 – Mervyn Fernandez (1982–86, 94)
Most touchdowns – season
22 – Cory Philpot (1995)
20 – Jon Volpe (1991)
19 – Larry Key (1981)
19 – Joe Smith (2007)
18 – Willie Fleming (1960)
18 – David Williams (1988)
Most touchdowns – game
4 – Willie Fleming – versus Saskatchewan Roughriders, October 29, 1960
4 – Larry Key – at Calgary Stampeders, Jul 31, 1981
4 – Mervyn Fernandez – versus Winnipeg Blue Bombers, Jul 6, 1984
4 – Mervyn Fernandez – at Ottawa Rough Riders, Oct 13, 1984
4 – Mervyn Fernandez – versus Calgary Stampeders, Aug 17, 1985
4 – David Williams – versus Edmonton Eskimos, Oct 29, 1988
4 – Cory Philpot – versus Hamilton Tiger-Cats, Sep 16, 1995
4 – Alfred Jackson – versus Montreal Alouettes, Aug 4, 2001
4 – Sean Millington – versus Hamilton Tiger-Cats, Jul 18, 2002
4 – Geroy Simon – at Hamilton Tig
|
https://en.wikipedia.org/wiki/T-square%20%28disambiguation%29
|
A T-square is a drafting and technical drawing tool.
T-square may also refer to:
T-shaped square (tool), in carpentry
T-square (fractal), in mathematics, a two-dimensional fractal
T-Square (software), an early drafting software program
T-Square (band), a Japanese jazz fusion band
A variation of grand cross in astrology
T-square position, a sexual position
Tsquared (born 1987), professional Halo player
See also
"Square-T", the code for the WW2 USAF 490th Bombardment Group
TT (disambiguation)
2T (disambiguation)
T2 (disambiguation)
Square (disambiguation)
The Square (disambiguation)
|
https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20%28Romania%29
|
The National Institute of Statistics (, INS) is a Romanian government agency which is responsible for collecting national statistics, in fields such as geography, the economy, demographics and society. The institute is also responsible for conducting Romania's census every ten years, with the latest census being organised in 2022.
Leadership
The head of the NIS is currently Tudorel Andrei, while the three vice-presidents are:
Elena Mihaela Iagăr, in charge of economic and social statistics
Marian Chivu, in charge of national accounts and the dissemination of statistical information
Beatrix Gered, in charge of IT activities and statistical infrastructure
History
Romania's first official statistics body was the Central Office for Administrative Statistics (Oficiului Central de Statistică Administrativă), established on July 12, 1859, under the reign of Alexandru Ioan Cuza. The organisation, one of the first national statistics organisations in Europe, conducted its first public census between 1859 and 1860.
The Romanian national statistics organisation was known under various names throughout the country's history, as can be seen in the table below:
See also
Demographic history of Romania
Romanian Statistical Yearbook
External links
National Institute of Statistics - official site
Government of Romania
Romania
|
https://en.wikipedia.org/wiki/Beppo%20Levi
|
Beppo Levi (14 May 1875 – 28 August 1961) was an Italian mathematician. He published high-level academic articles and books, not only on mathematics, but also on physics, history, philosophy, and pedagogy. Levi was a member of the Bologna Academy of Sciences and of the Accademia dei Lincei.
Early years
Beppo Levi was born on May 14, 1875, in Turin, Italy, and he was an older brother of Eugenio Elia Levi. He obtained his laurea in mathematics in 1896 at age 21 from the University of Turin under Corrado Segre. He was appointed an assistant professor at the University of Turin three months later and shortly thereafter became a full-time Scholar. Levi was appointed Professor at the University of Piacenza in 1901, at the University of Cagliari in 1906, at the University of Parma in 1910, and finally at the University of Bologna in 1928. The years that followed his last appointment saw the rise of Benito Mussolini's power and of antisemitism in Italy, and Levi, being Jewish, was soon expelled from his position at the University of Bologna. He emigrated to Argentina, as did many other European Jews at that time.
Life in Argentina
Levi chose Argentina because of an invitation by the engineer Cortés Plá, dean of the Facultad de Ciencias Matemáticas, Físico-Químicas y Naturales Aplicadas a la Industria at the Universidad Nacional del Litoral (currently Facultad de Ciencias Exactas, Ingeniería y Agrimensura at the Universidad Nacional de Rosario) in the city of Rosario. Cortés Plá invited Levi to come to Rosario to head the recently created Instituto de Matemática. It was there that Levi did most of his work from 1939 until his death in 1961.
While living in Rosario, Levi joined a group of mathematicians that included Luis Santaló, Simón Rubinstein, Juan Olguín, Enrique Ferrari, Fernando and Enrique Gaspar, Mario Castagnino and Edmundo Rofman. In 1940 Levi founded Mathematicae Notae, the first mathematical journal in Argentina. In 1956 he was awarded the Feltrinelli Prize.
He died on August 28, 1961, in Rosario, Argentina, and was buried in the Jewish cemetery there.
Mathematical contributions
His early work studied singularities on algebraic curves and surfaces. In particular, he supplied a proof (questioned by some) that a procedure for resolution of singularities on algebraic surfaces terminates in finitely many steps. Later he proved some foundational results concerning Lebesgue integration, including what is commonly known as Beppo Levi's lemma.
He also studied the arithmetic of elliptic curves. He classified them up to isomorphism, not only over C, but also over Q. Next he studied what in modern terminology would be the subgroup of rational torsion points on an elliptic curve over Q: he proved that certain groups were realizable and that others were not. He essentially formulated the torsion conjecture for elliptic curves over the rational numbers, providing a complete list of possibilities should be, which was formulated indepe
|
https://en.wikipedia.org/wiki/Multiplication%20theorem
|
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relation for one special function can be derived from that for the others, and is simply a manifestation of the same identity in different guises.
Finite characteristic
The multiplication theorem takes two common forms. In the first case, a finite number of terms are added or multiplied to give the relation. In the second case, an infinite number of terms are added or multiplied. The finite form typically occurs only for the gamma and related functions, for which the identity follows from a p-adic relation over a finite field. For example, the multiplication theorem for the gamma function follows from the Chowla–Selberg formula, which follows from the theory of complex multiplication. The infinite sums are much more common, and follow from characteristic zero relations on the hypergeometric series.
The following tabulates the various appearances of the multiplication theorem for finite characteristic; the characteristic zero relations are given further down. In all cases, n and k are non-negative integers. For the special case of n = 2, the theorem is commonly referred to as the duplication formula.
Gamma function–Legendre formula
The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is
It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is
for integer k ≥ 1, and is sometimes called Gauss's multiplication formula, in honour of Carl Friedrich Gauss. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg formula.
Polygamma function, harmonic numbers
The polygamma function is the logarithmic derivative of the gamma function, and thus, the multiplication theorem becomes additive, instead of multiplicative:
for , and, for , one has the digamma function:
The polygamma identities can be used to obtain a multiplication theorem for harmonic numbers.
Hurwitz zeta function
For the Hurwitz zeta function generalizes the polygamma function to non-integer orders, and thus obeys a very similar multiplication theorem:
where is the Riemann zeta function. This is a special case of
and
Multiplication formulas for the non-principal characters may be given in the form of Dirichlet L-functions.
Periodic zeta function
The periodic zeta function is sometimes defined as
where Lis(z) is the polylogarithm. It obeys the duplication formula
As such, it is an eigenvector of the Bernoulli operator with eigenvalue 21−s. The multiplication theorem is
The periodic zeta function o
|
https://en.wikipedia.org/wiki/Resolvent%20set
|
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Definitions
Let X be a Banach space and let be a linear operator with domain . Let id denote the identity operator on X. For any , let
A complex number is said to be a regular value if the following three statements are true:
is injective, that is, the corestriction of to its image has an inverse ;
is a bounded linear operator;
is defined on a dense subspace of X, that is, has dense range.
The resolvent set of L is the set of all regular values of L:
The spectrum is the complement of the resolvent set:
The spectrum can be further decomposed into the point/discrete spectrum (where condition 1 fails), the continuous spectrum (where conditions 1 and 3 hold but condition 2 fails) and the residual/compression spectrum (where condition 1 holds but condition 3 fails).
If is a closed operator, then so is each , and condition 3 may be replaced by requiring that is surjective.
Properties
The resolvent set of a bounded linear operator L is an open set.
More generally, the resolvent set of a densely defined closed unbounded operator is an open set.
References
(See section 8.3)
External links
See also
Resolvent formalism
Spectrum (functional analysis)
Decomposition of spectrum (functional analysis)
Linear algebra
Operator theory
|
https://en.wikipedia.org/wiki/Octadecagon
|
In geometry, an octadecagon (or octakaidecagon) or 18-gon is an eighteen-sided polygon.
Regular octadecagon
A regular octadecagon has a Schläfli symbol {18} and can be constructed as a quasiregular truncated enneagon, t{9}, which alternates two types of edges.
Construction
As 18 = 2 × 32, a regular octadecagon cannot be constructed using a compass and straightedge. However, it is constructible using neusis, or an angle trisection with a tomahawk.
The following approximate construction is very similar to that of the enneagon, as an octadecagon can be constructed as a truncated enneagon. It is also feasible with exclusive use of compass and straightedge.
Symmetry
The regular octadecagon has Dih18 symmetry, order 36. There are 5 subgroup dihedral symmetries: Dih9, (Dih6, Dih3), and (Dih2 Dih1), and 6 cyclic group symmetries: (Z18, Z9), (Z6, Z3), and (Z2, Z1).
These 15 symmetries can be seen in 12 distinct symmetries on the octadecagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r36 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g18 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octadecagon, m=9, and it can be divided into 36: 4 sets of 9 rhombs. This decomposition is based on a Petrie polygon projection of a 9-cube, with 36 of 4608 faces. The list enumerates the number of solutions as 112018190, including up to 18-fold rotations and chiral forms in reflection.
Uses
A regular triangle, nonagon, and octadecagon can completely surround a point in the plane, one of 17 different combinations of regular polygons with this property. However, this pattern cannot be extended to an Archimedean tiling of the plane: because the triangle and the nonagon both have an odd number of sides, neither of them can be completely surrounded by a ring alternating the other two kinds of polygon.
The regular octadecagon can tessellate the plane with concave hexagonal gaps. And another tiling mixes in nonagons and octagonal gaps. The first tiling is related to a truncated hexagonal tiling, and the second the truncated trihexagonal tiling.
Related figures
An octadecagram is an 18-sided star polygon, represented by symbol {18/n}. There are two regular star polygons: {18/5} and {18/7}, using the same points, but connecting every fifth or seventh poin
|
https://en.wikipedia.org/wiki/Stirling%20numbers%20and%20exponential%20generating%20functions%20in%20symbolic%20combinatorics
|
The use of exponential generating functions (EGFs) to study the properties of Stirling numbers is a classical exercise in combinatorial mathematics and possibly the canonical example of how symbolic combinatorics is used. It also illustrates the parallels in the construction of these two types of numbers, lending support to the binomial-style notation that is used for them.
This article uses the coefficient extraction operator for formal power series, as well as the (labelled) operators (for cycles) and (for sets) on combinatorial classes, which are explained on the page for symbolic combinatorics. Given a combinatorial class, the cycle operator creates the class obtained by placing objects from the source class along a cycle of some length, where cyclical symmetries are taken into account, and the set operator creates the class obtained by placing objects from the source class in a set (symmetries from the symmetric group, i.e. an "unstructured bag".) The two combinatorial classes (shown without additional markers) are
permutations (for unsigned Stirling numbers of the first kind):
and
set partitions into non-empty subsets (for Stirling numbers of the second kind):
where is the singleton class.
Warning: The notation used here for the Stirling numbers is not that of the Wikipedia articles on Stirling numbers; square brackets denote the signed Stirling numbers here.
Stirling numbers of the first kind
The unsigned Stirling numbers of the first kind count the number of permutations of [n] with k cycles. A permutation is a set of cycles, and hence the set of permutations is given by
where the singleton marks cycles. This decomposition is examined in some detail on the page on the statistics of random permutations.
Translating to generating functions we obtain the mixed generating function of the unsigned Stirling numbers of the first kind:
Now the signed Stirling numbers of the first kind are obtained from the unsigned ones through the relation
Hence the generating function of these numbers is
A variety of identities may be derived by manipulating this generating function:
In particular, the order of summation may be exchanged, and derivatives taken, and then z or u may be fixed.
Finite sums
A simple sum is
This formula holds because the exponential generating function of the sum is
Infinite sums
Some infinite sums include
where (the singularity nearest to
of is at )
This relation holds because
Stirling numbers of the second kind
These numbers count the number of partitions of [n] into k nonempty subsets. First consider the total number of partitions, i.e. Bn where
i.e. the Bell numbers. The Flajolet–Sedgewick fundamental theorem applies (labelled case).
The set of partitions into non-empty subsets is given by ("set of non-empty sets of singletons")
This decomposition is entirely analogous to the construction of the set of permutations from cycles, which is given by
and yields the Stirling numbers of the first ki
|
https://en.wikipedia.org/wiki/Dual%20quaternion
|
In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus, they may be constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form , where A and B are ordinary quaternions and ε is the dual unit, which satisfies and commutes with every element of the algebra.
Unlike quaternions, the dual quaternions do not form a division algebra.
In mechanics, the dual quaternions are applied as a number system to represent rigid transformations in three dimensions. Since the space of dual quaternions is 8-dimensional and a rigid transformation has six real degrees of freedom, three for translations and three for rotations, dual quaternions obeying two algebraic constraints are used in this application. Since unit quaternions are subject to two algebraic constraints, unit quaternions are standard to represent rigid transformations.
Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy), and in applications to 3D computer graphics, robotics and computer vision. Polynomials with coefficients given by (non-zero real norm) dual quaternions have also been used in the context of mechanical linkages design.
History
W. R. Hamilton introduced quaternions in 1843, and by 1873 W. K. Clifford obtained a broad generalization of these numbers that he called biquaternions, which is an example of what is now called a Clifford algebra.
In 1898 Alexander McAulay used Ω with Ω2 = 0 to generate the dual quaternion algebra. However, his terminology of "octonions" did not stick as today's octonions are another algebra.
In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics.
In 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in Geometrie der Dynamen in 1901.
B. L. van der Waerden called the structure "Study biquaternions", one of three eight-dimensional algebras referred to as biquaternions.
Formulas
In order to describe operations with dual quaternions, it is helpful to first consider quaternions.
A quaternion is a linear combination of the basis elements 1, i, j, and k. Hamilton's product rule for i, j, and k is often written as
Compute , to obtain , and or . Now because , we see that this product yields , which links quaternions to the properties of determinants.
A convenient way to work with the quaternion product is to write a quaternion as the sum of a scalar and a vector (strictly speaking a bivector), that is , where a0 is a real number and is a three dimensional vector. The vector dot and cross operations can now be us
|
https://en.wikipedia.org/wiki/Pre-math%20skills
|
Pre-math skills (referred to in British English as pre-maths skills) are math skills learned by preschoolers and kindergarten students, including learning to count numbers (usually from 1 to 10 but occasionally including 0), learning the proper sequencing of numbers, learning to determine which shapes are bigger or smaller, and learning to count objects on a screen or book. Pre-math skills are also tied into literacy skills to learn the correct pronunciations of numbers.
References
External links
Pre-math Skills on Bella Online
Mathematics education
Parenting
|
https://en.wikipedia.org/wiki/Norman%20Riley%20%28professor%29
|
Norman Riley is an Emeritus Professor of Applied Mathematics at the University of East Anglia in Norwich (UK).
Biography
Following High School education at Calder High School, Mytholmroyd he read Mathematics at Manchester University graduating with first class honours in 1956, followed by a PhD in 1959. Norman Riley served for one year as an Assistant Lecturer at Manchester University and then spent four years as a lecturer at Durham University before he joined the then new University of East Anglia in 1964, the year that saw the first significant intake of students to the university. Promotion to Reader in 1966 was followed by promotion to a Personal Chair in 1971. He retired in 1999. Married in 1959 he has one son and one daughter.
Research contributions
His research contributions in the field of fluid mechanics, over five decades, have included: unsteady flows with application to acoustic levitation and the loading on the submerged horizontal pontoons of tethered leg platforms; the aerodynamics of wings including leading-edge separation from slender wings and supercritical flow over multi-element wings; heat transfer and combustion including diffusion flames and detonation-wave generation; vortex ring dynamics; crystal growth, in particular the Czochralski crystal growth process. Throughout these investigations complementary numerical and asymptotic methods have featured.
Books
Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
See also
John Frederick Clarke
References
"Broadview" Aug 1999, page6
The Navier-Stokes Equations
Year of birth missing (living people)
Living people
Fluid dynamicists
Academics of the University of Manchester
Academics of Durham University
Academics of the University of East Anglia
English mathematicians
|
https://en.wikipedia.org/wiki/Aleksey%20Negmatov
|
Aleksey Negmatov (; born on 4 January 1986) is a Tajikistani footballer who plays as a defender for FC Vakhsh and the Tajikistan national football team.
Career statistics
International
Statistics accurate as of match played 11 October 2011
International goals
Honours
Vakhsh Qurghonteppa
Tajik League (1): 2009
Regar-TadAZ
Tajik Cup (1): 2012
Tajikistan
AFC Challenge Cup (1): 2006
References
External links
1986 births
Living people
Tajikistani men's footballers
Tajikistan men's international footballers
Men's association football fullbacks
FC Khatlon players
Footballers at the 2006 Asian Games
Tajikistani people of Russian descent
Asian Games competitors for Tajikistan
|
https://en.wikipedia.org/wiki/Ibrahim%20Rabimov
|
Ibrahim Rabimov (born 3 August 1987) is a retired Tajik professional footballer who played as a midfielder.
Career statistics
International
Statistics accurate as of match played 11 June 2015
International goals
Honors
Club
Aviator Bobojon Ghafurov/Parvoz Bobojon Ghafurov
Tajik Cup (1): 2004
Regar-TadAZ Tursunzoda
Tajik League (3): 2006, 2007, 2008
Tajik Cup (1): 2006
Istiklol
Tajik League (1): 2011
Tajik Cup (1): 2013
AFC President's Cup (1): 2012
International
Tajikistan
AFC Challenge Cup (1): 2006
Individual
CIS Cup top goalscorer: 2009 (shared)
References
External links
Player profile – doha-2006.com
1987 births
Living people
Sportspeople from Dushanbe
Tajikistani men's footballers
Tajikistan men's international footballers
Tajikistan Higher League players
FC Istiklol players
Footballers at the 2006 Asian Games
Men's association football midfielders
Asian Games competitors for Tajikistan
|
https://en.wikipedia.org/wiki/Series%20acceleration
|
In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration. Series acceleration techniques may also be used, for example, to obtain a variety of identities on special functions. Thus, the Euler transform applied to the hypergeometric series gives some of the classic, well-known hypergeometric series identities.
Definition
Given a sequence
having a limit
an accelerated series is a second sequence
which converges faster to than the original sequence, in the sense that
If the original sequence is divergent, the sequence transformation acts as an extrapolation method to the antilimit .
The mappings from the original to the transformed series may be linear (as defined in the article sequence transformations), or non-linear. In general, the non-linear sequence transformations tend to be more powerful.
Overview
Two classical techniques for series acceleration are Euler's transformation of series and Kummer's transformation of series. A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the Aitken delta-squared process, introduced by Alexander Aitken in 1926 but also known and used by Takakazu Seki in the 18th century; the epsilon method given by Peter Wynn in 1956; the Levin u-transform; and the Wilf-Zeilberger-Ekhad method or WZ method.
For alternating series, several powerful techniques, offering convergence rates from all the way to for a summation of terms, are described by Cohen et al.
Euler's transform
A basic example of a linear sequence transformation, offering improved convergence, is Euler's transform. It is intended to be applied to an alternating series; it is given by
where is the forward difference operator, for which one has the formula
If the original series, on the left hand side, is only slowly converging, the forward differences will tend to become small quite rapidly; the additional power of two further improves the rate at which the right hand side converges.
A particularly efficient numerical implementation of the Euler transform is the van Wijngaarden transformation.
Conformal mappings
A series
can be written as f(1), where the function f is defined as
The function f(z) can have singularities in the complex plane (branch point singularities, poles or essential singularities), which limit the radius of convergence of the series. If the point z = 1 is close to or on the boundary of the disk of convergence, the series for S will converge very slowly. One can then improve the convergence of the series by means of a conformal mapping that moves the singularities such that the point that is mapped t
|
https://en.wikipedia.org/wiki/Galileo%27s%20Leaning%20Tower%20of%20Pisa%20experiment
|
Between 1589 and 1592, the Italian scientist Galileo Galilei (then professor of mathematics at the University of Pisa) is said to have dropped two spheres of the same volume but different masses from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass, according to a biography by Galileo's pupil Vincenzo Viviani, composed in 1654 and published in 1717. The basic premise had already been demonstrated by Italian experimenters a few decades earlier.
According to the story, Galileo discovered through this experiment that the objects fell with the same acceleration, proving his prediction true, while at the same time disproving Aristotle's theory of gravity (which states that objects fall at speed proportional to their mass). Most historians consider it to have been a thought experiment rather than a physical test.
Background
The 6th-century Byzantine Greek philosopher and Aristotelian commentator John Philoponus argued that the Aristotelian assertion that objects fall proportionately to their weight was incorrect. By 1544, according to Benedetto Varchi, the Aristotelian premise was disproven experimentally by at least two Italians. In 1551, Domingo de Soto suggested that objects in free fall accelerate uniformly. Two years later, mathematician Giambattista Benedetti questioned why two balls, one made of iron and one of wood, would fall at the same speed. All of this preceded the 1564 birth of Galileo Galilei.
Delft tower experiment
A similar experiment was conducted in Delft in the Netherlands, by the mathematician and physicist Simon Stevin and Jan Cornets de Groot (the father of Hugo de Groot). The experiment is described in Stevin's 1586 book De Beghinselen der Weeghconst (The Principles of Statics), a landmark book on statics:
Let us take (as the highly educated Jan Cornets de Groot, the diligent researcher of the mysteries of Nature, and I have done) two balls of lead, the one ten times bigger and heavier than the other, and let them drop together from 30 feet high, and it will show, that the lightest ball is not ten times longer under way than the heaviest, but they fall together at the same time on the ground. ... This proves that Aristotle is wrong.Asimov, Isaac (1964). Asimov's Biographical Encyclopedia of Science and Technology.
Galileo's experiment
At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of falling bodies. He had, however, formulated an earlier version which predicted that bodies of the same material falling through the same medium would fall at the same speed. This was contrary to what Aristotle had taught: that heavy objects fall faster than the lighter ones, and in direct proportion to their weight. While this story has been retold in popular accounts, there is no account by Galileo himself of such an experiment, and many historians believe that it was a thought experiment. An exception is Stillman Dr
|
https://en.wikipedia.org/wiki/Etemadi%27s%20inequality
|
In probability theory, Etemadi's inequality is a so-called "maximal inequality", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. The result is due to Nasrollah Etemadi.
Statement of the inequality
Let X1, ..., Xn be independent real-valued random variables defined on some common probability space, and let α ≥ 0. Let Sk denote the partial sum
Then
Remark
Suppose that the random variables Xk have common expected value zero. Apply Chebyshev's inequality to the right-hand side of Etemadi's inequality and replace α by α / 3. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side:
References
(Theorem 22.5)
Probabilistic inequalities
Statistical inequalities
|
https://en.wikipedia.org/wiki/Ernest%20S.%20Croot%20III
|
Ernest S. Croot III is a mathematician and professor at the School of Mathematics, Georgia Institute of Technology. He is known for his solution of the Erdős–Graham conjecture, and for contributing to the solution of the cap set problem.
Education
Ernest Croot attended Centre College at Danville, Kentucky, where he received a B.S. in Mathematics and a B.S. in
Computer Science in 1994. In 2000, he completed a Ph.D. in Mathematics at the University of Georgia under the supervision of Andrew Granville.
References
External links
Croot's personal web page at Georgia Tech
Mathematics Genealogy Project profile
20th-century American mathematicians
21st-century American mathematicians
Mathematicians from Georgia (U.S. state)
Georgia Tech faculty
1972 births
Living people
University of Georgia alumni
Centre College alumni
|
https://en.wikipedia.org/wiki/Thierry%20Aubin
|
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry
and non-linear partial differential equations. His fundamental contributions to the theory of the Yamabe equation led, in conjunction with
results of Trudinger and Schoen, to a proof of the Yamabe Conjecture: every compact Riemannian manifold
can be conformally rescaled to produce a manifold of constant scalar curvature. Along with Yau, he also showed
that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the Calabi conjecture. The latter result, established by Yau, provides the largest class of known examples of compact Einstein manifolds. Aubin was the first mathematician to propose the Cartan–Hadamard conjecture.
Aubin was a visiting scholar at the Institute for Advanced Study in 1979. He was elected to the Académie des sciences in 2003.
Research
In 1970, Aubin established that any closed smooth manifold of dimension larger than two has a Riemannian metric of negative scalar curvature. Furthermore, he proved that a Riemannian metric of nonnegative Ricci curvature can be deformed to positive Ricci curvature, provided that its Ricci curvature is strictly positive at one point.
In the same year, Aubin introduced an approach to the Calabi conjecture, in the field of Kähler geometry, via the calculus of variations. Later, in 1976, Aubin established the existence of Kähler–Einstein metrics on Kähler manifolds whose first Chern class is negative. Independently, Shing-Tung Yau proved the more powerful Calabi conjecture, which concerns the general problem of prescribing the Ricci curvature of a Kähler metric, via non-variational methods. As such, the existence of Kähler–Einstein metrics with negative first Chern class is often called the Aubin–Yau theorem. After learning Yau's techniques from Jerry Kazdan, Aubin found some simplifications and modifications of his work, along with Kazdan and Jean-Pierre Bourguignon.
Aubin made a number of fundamental contributions to the study of Sobolev spaces on Riemannian manifolds. He established Riemannian formulations of many classical results for Sobolev spaces, such as the equivalence of various definitions, the density of various subclasses of functions, and the standard embedding theorems. In one of Aubin's best-known works, the analysis of the optimal constant in the Sobolev embedding theorem was carried out. Along with similar results for the Moser–Trudinger inequality, Aubin later proved improvements of the optimal constants when the functions are assumed to satisfy certain orthogonality constraints.
Such results are naturally applicable to many problems in the field of geometric analysis. Aubin considered the Yamabe problem on conformal deformation to constant scalar curvature, which Yamabe had reduced to a problem in the calculus of variations. Following prior
|
https://en.wikipedia.org/wiki/J.League%20records%20and%20statistics
|
This page details J.League records.
J1 League
Ranks
.
In bold the ones who are actually playing in J1. In italic the ones who are still active in other Japanese league.
Individual
Most career goals : 191 goals
Yoshito Okubo
Most career hat-tricks : 8 times
Ueslei
Most career appearances : 672 appearances
Yasuhito Endo
Most goals in a season : 36 goals
Masashi Nakayama (1998)
Most hat-tricks in a season : 5 times
Masashi Nakayama (1998)
Most goals in a game : 5 goals
Koji Noguchi for Bellmare Hiratsuka vs Kashima Antlers (3 May 1995)
Edílson for Kashiwa Reysol vs Gamba Osaka (4 May 1996)
Masashi Nakayama for Jubilo Iwata vs Cerezo Osaka (15 April 1998)
/ Wagner Lopes for Nagoya Grampus Eight vs Urawa Red Diamonds (29 May 1999)
Youngest player : 15 years 10 months and 6 days
Takayuki Morimoto for Tokyo Verdy 1969 vs Jubilo Iwata (13 March 2004)
Youngest goalscorer : 15 years 11 months and 28 days
Takayuki Morimoto for Tokyo Verdy 1969 vs JEF United Ichihara (5 May 2004)
Oldest player : 54 years 12 days
Kazuyoshi Miura for Yokohama FC vs Urawa Red Diamonds (12 March 2021)
Oldest goalscorer : 41 years 3 months and 12 days
Zico for Kashima Antlers vs Jubilo Iwata (15 June 1994)
Fastest goal : 8 seconds
Hisato Sato for Sanfrecce Hiroshima vs Cerezo Osaka (22 April 2006)
Fastest hat-trick : 3 minutes
Yasuo Manaka for Cerezo Osaka vs Kashiwa Reysol (14 July 2001)
First scorer
Henny Meijer for Verdy Kawasaki vs Yokohama Marinos (15 May 1993)
First hat-trick
Zico for Kashima Antlers vs Nagoya Grampus Eight (16 May 1993)
Club
Most League championships : 8 times
Kashima Antlers (1996, 1998, 2000, 2001, 2007, 2008, 2009, 2016)
Longest uninterrupted spell in J1: 30 seasons (1993-Present)
Kashima Antlers
Yokohama F. Marinos
Most goals scored in a season : 107 goals
Jubilo Iwata (1998)
Fewest goals scored in a season : 16 goals
Tokushima Vortis (2014)
Most goals conceded in a season : 111 goals
Yokohama Flügels (1995)
Fewest goals conceded in a season : 24 goals
Oita Trinita (2008)
Biggest goal difference in a season : 68 goals
Jubilo Iwata (1998)
Most points in a season : 108 points
Verdy Kawasaki (1995)
Fewest points in a season : 13 points
Bellmare Hiratsuka (1999)
Most wins in a season : 35 wins
Verdy Kawasaki (1995)
Fewest wins in a season : 2 wins
Oita Trinita (2013)
Most draws in a season : 16 draws
Shonan Bellmare (2021)
Most losses in a season : 34 losses
Gamba Osaka (1995)
Fewest losses in a season : 2 losses
Kawasaki Frontale (2021)
Most goals in a game : 12 goals
Cerezo Osaka 5-7 Kashiwa Reysol (8 August 1998)
Record win : 9–1, 8-0
Jubilo Iwata 9-1 Cerezo Osaka (15 April 1998)
Vissel Kobe 0-8 Oita Trinita (26 July 2003)
Shimizu S-Pulse 0-8 Hokkaido Consadole Sapporo (17 August 2019)
Yokohama F. Marinos 8-0 FC Tokyo (6 November 2021)
Highest scoring draw: 5-5
Vissel Kobe 5-5 JEF United Ichihara (14 October 1998)
Highest average home attendance in a season : 47,609
Urawa Red Diamonds (2008)
Highest home attendance : 63,854 (regular
|
https://en.wikipedia.org/wiki/Ernst%20Ludwig%20Taschenberg
|
Ernst Ludwig Taschenberg (10 January 1818 Naumburg – 19 January 1898 Halle) was a German entomologist.
Life
After 1836 Taschenberg studied mathematics and natural sciences in Leipzig and Berlin. He went, then, as an auxiliary teacher to the Franckesche Stiftungen and dedicated himself to arranging the important beetle collection of professor Germar and with the curation and study of the insect collection of the zoological museum particularly the entomology collections.
He worked as a teacher in Seesen for two years and then in five Zahna for five years but in 1856 he became “Inspektor“ at the zoological museum in Halle and in 1871 he was appointed extraordinary professor. His insect studies were mainly applied to agriculture, horticulture and silviculture and he is an important figure in the history of Economic entomology. He also described new insect species in several orders. His son Ernst Otto Wilhelm Taschenberg was also an entomologist specialising in Hymenoptera.
Works
Was da kriecht und fliegt, Bilder aus dem Insektenleben. (Berlin 1861);
Naturgeschichte der wirbellosen Tiere, die in Deutschland den Feld-, Wiesen- und Weidekulturpflanzen schädlich werden. (Leipzig 1865);
Die Hymenopteren Deutschlands (Leipzig 1866);
Entomologie für Gärtner und Gartenfreunde. (Leipzig 1871);
Schutz der Obstbäume und deren Früchte gegen feindliche Tiere. (2. Aufl., Stuttgart 1879);
Forstwirtschaftliche Insektenkunde. (Leipzig 1873);
Das Ungeziefer der landwirtschaftlichen Kulturgewächse. ( Leipzig 1873);
Praktische Insektenkunde. 5 Bde. (Bremen 1879-80);
Die Insekten nach ihrem Schaden und Nutzen. (Leipzig 1882); Digital edition by the University and State Library Düsseldorf
Die Insekten, Tausendfüker und Spinnen'. (Leipzig und Wien 1892).
He also worked on the insects for Alfred Brehm's Tierleben'' (2. Aufl. 1877) and on zoological and entomological posters for school use.
Collection
Taschenberg's faunistic collection of Hymenoptera and Lepidoptera from Sachsen-Anhalt is in the Martin Luther University of Halle-Wittenberg.
External links
Literature at the German National Library
History of entomology at Halle in German
1818 births
1898 deaths
German entomologists
Hymenopterists
Scientists from the Province of Saxony
People from Naumburg (Saale)
Leipzig University alumni
Humboldt University of Berlin alumni
Academic staff of the Martin Luther University of Halle-Wittenberg
|
https://en.wikipedia.org/wiki/Joseph%20Hilbe
|
Joseph Michael Hilbe (December 30, 1944 – March 12, 2017) was an American statistician and philosopher, founding President of the International Astrostatistics Association(IAA) and one of the most prolific authors of books on statistical modeling in the early twenty-first century. Hilbe was an elected Fellow of the American Statistical Association as well as an elected member of the International Statistical Institute (ISI), for which he founded the ISI astrostatistics committee in 2009. Hilbe was also a Fellow of the Royal Statistical Society and Full Member of the American Astronomical Society.
Hilbe made a number of contributions to the fields of count response models and logistic regression. Among his most influential books are two editions of Negative Binomial Regression (Cambridge University Press, 2007, 2011), Modeling Count Data (Cambridge University Press, 2014), and Logistic Regression Models (Chapman & Hall/CRC, 2009). Modeling Count Data won the 2015 PROSE honorable mention award for books in mathematics as the second best mathematics book published in 2014. Hilbe was also editor-in-chief of the Springer Series in Astrostatistics, which began in 2011, was one of two co-editors for the Astrostatistics and AstroInformatics Portal, a co-ordinated website for the major astrostatistical organizations worldwide, hosted by the Pennsylvania State University Department of Astronomy and Astrophysics, and was coordinating editor of the Cambridge University Press Series on Predictive Analytics in Action, which commenced in 2012. A listing of his books, book chapters and encyclopedia articles are listed below (Publications).
Hilbe was also a two-time national champion track & field athlete,
a US team and NCAA Division 1 head coach, and Olympic Games official. He was also chair of the ISI sports statistics committee from 2007 to 2011 and chair of the 2014 Section on Statistics and Sports of the American Statistical Association.
Life and works
Born in Los Angeles, California, son of Rader John Hilbe (1910–1980) and Nadyne Anderson Hilbe (1910–1988), Hilbe was raised in Arcadia, California, attended three years of high school at Woodside Priory School in Portola Valley, California, before graduating from Paradise, California high school in 1962. Hilbe
attended California State University, Chico, from which he graduated in 1968 with a degree in philosophy. Hilbe studied for his doctorate in philosophy at the University of California, Los Angeles (UCLA) where he was a graduate reader for visiting professor and Nobel Laureate Friedrich Hayek and personal assistant to Rudolf Carnap, one of the founders of the Vienna Circle of Logical Positivism.
Hilbe secured a position at the University of Hawaiʻi, where he retired as an emeritus professor of philosophy in 1990. During this time he authored several texts in philosophy and logic. In 1988 he earned a doctorate in statistics (applied mathematics, UCLA) and in 1990 was hired by the Health Care Finan
|
https://en.wikipedia.org/wiki/School%20of%20Science%20and%20Technology
|
The School of Science and Technology (SST) was an accredited, public high school located in Beaverton, Oregon, United States. It was a magnet program for students who have an interest in mathematics, life and physical sciences, and technology. It is part of the Beaverton School District (BSD). It was established in 1993, as the School of Natural Resources Science and Technology, and later renamed. SST moved at the end of 2015 to expanded and remodeled facilities at a site it shares with BSD's Health and Science School. For the 2020–2021 school year and onward, this school and the neighboring school, the Beaverton Health & Science School have merged to become the Beaverton Academy of Science and Engineering.
In 2017, U.S. News & World Report ranked SST as the best high school in Oregon. For 2015, the magazine had ranked SST second among public high schools in the Beaverton School District (first among schools offering AP programs), fourth in the state of Oregon, and 598th nationally. The Oregonian ranked SST first in its 2015 school performance ratings within the Beaverton School District.
History
SST began as a Certificate of Initial Mastery program called the School of Natural Resources Science and Technology (NRST) in 1993, and was one of the magnet programs at the then-new Merlo Station High School, located on S.W. Merlo Drive just west of the then-planned Merlo Road/SW 158th MAX Light Rail station (which opened in 1998). The school occupied an old warehouse, which was renovated a few months before the program's opening. The name was eventually shortened to School of Science and Technology (SST) in 2001.
At the end of 2015, SST moved from the Merlo Station campus to newly remodeled space about one to two miles to the west, in the Capital Center development at N.W. 185th Avenue and Walker Road. BSD's Health & Science School, another option school, had already been located at Capital Center since its 2007 establishment, so SST's move made the two programs neighbors, allowing them to share some facilities, such as a cafeteria and library.
Students
As of February 2015, 68% of students were Caucasian, 15% Asian or Pacific Islander, 8% Hispanic, 2% Black, and 7% fit into multiple categories. 15% were on free or reduced lunch, 19% were eligible for special education, and 1% were enrolled in ESL. 30% were enrolled in TAG in middle or elementary school, significantly higher than the 11% district average.
Academics
The average SAT score in 2015 was 593 in critical reading, 642 in math, and 582 in writing. The average ACT composite score was 27. In the 2015 Regional Science Fair, SST had eight Special Award Winners and ten Category Winners; and had two Category Winners in the 2015 State Science Fair.
Application process
Until 2007, students were required to follow an application process which involved writing three essays and an interview from an SST teacher. However, in effect from 2007 to 2008 onward, the Beaverton School District simp
|
https://en.wikipedia.org/wiki/Skorokhod%27s%20embedding%20theorem
|
In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A. V. Skorokhod.
Skorokhod's first embedding theorem
Let X be a real-valued random variable with expected value 0 and finite variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of W), τ, such that Wτ has the same distribution as X,
and
Skorokhod's second embedding theorem
Let X1, X2, ... be a sequence of independent and identically distributed random variables, each with expected value 0 and finite variance, and let
Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the have the same joint distributions as the partial sums Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying
and
References
(Theorems 37.6, 37.7)
Probability theorems
Wiener process
Ukrainian inventions
|
https://en.wikipedia.org/wiki/Triaxial
|
Triaxial may refer to:
Triaxial cable (electrical cable)
Triaxial ellipsoid (mathematics, geometric shapes)
Triaxial test (Geotechnical engineering)
|
https://en.wikipedia.org/wiki/Jo%C3%A3o%20Carlos%20%28football%20manager%29
|
João Carlos da Silva Costa, best known as João Carlos (born 15 January 1956), is a Brazilian football manager.
Managerial statistics
Honors
J. League Manager of the Year – 1997
References
External links
Site official
1956 births
Living people
Brazilian football managers
Expatriate football managers in Japan
Expatriate football managers in Saudi Arabia
Expatriate soccer managers in South Africa
J1 League managers
J2 League managers
Campeonato Brasileiro Série A managers
Al Hilal SFC managers
América Futebol Clube (SP) managers
Kashima Antlers managers
Club Athletico Paranaense managers
União São João Esporte Clube managers
Brazil national under-20 football team managers
Nagoya Grampus managers
Cerezo Osaka managers
CR Flamengo managers
Hokkaido Consadole Sapporo managers
Tupi Football Club managers
Clube de Regatas Brasil managers
Esporte Clube Tigres do Brasil managers
Orlando Pirates F.C. managers
Al-Tai FC managers
Saudi Pro League managers
Brazilian expatriate sportspeople in Saudi Arabia
|
https://en.wikipedia.org/wiki/Trofeu%20Individual%20Bancaixa
|
The Trofeu Individual Bancaixa () "Bancaixa one-on-one trophy") is the Escala i corda singles league played by Valencian pilota professionals and promoted by the Bancaixa bank.
Statistics
Relevant facts about the Trofeu Individual Bancaixa
Grau has been the only mitger who won a competition almost always played by dauers.
Specific rules
Matches begin with a draw: 10–10 jocs.
Seasons
Trofeu Individual Bancaixa 2007
See also
Valencian pilota
Escala i corda
Circuit Bancaixa
External links
Youtube: 7 videos featuring the 1993 final match, Genovés I vs. Sarasol I
Google Video: Last games of the 1995 final match, Genovés I vs. Álvaro
Google Video: 2004 Final match, Álvaro vs. Genovés II, complete
1986 establishments in Spain
Valencian pilota competitions
Valencian pilota professional leagues
|
https://en.wikipedia.org/wiki/Michael%20Boardman
|
John Michael Boardman (13 February 193818 March 2021) was a mathematician whose speciality was algebraic and differential topology. He was affiliated with the University of Cambridge, England and the Johns Hopkins University in Baltimore, Maryland. Boardman was most widely known for his construction of the first rigorously correct model of the homotopy category of spectra.
He received his PhD from the University of Cambridge in 1964. His thesis advisor was C. T. C. Wall. In 2012 he became a fellow of the American Mathematical Society. He died on 18 March 2021.
Selected publications
References
Further reading
External links
Alumni of the University of Cambridge
20th-century American mathematicians
21st-century American mathematicians
20th-century British mathematicians
21st-century British mathematicians
Johns Hopkins University faculty
Fellows of the American Mathematical Society
2021 deaths
Topologists
1938 births
|
https://en.wikipedia.org/wiki/Raspall%20team%20championship
|
The Campionat per Equips de Raspall (Valencian for Team Raspall Championship) is the Valencian pilota Raspall modality league played by professional pilotaris.
Statistics
References
See also
Valencian pilota
Raspall
1984 establishments in Spain
Valencian pilota competitions
Valencian pilota professional leagues
|
https://en.wikipedia.org/wiki/Raspall%20singles%20championship
|
The Campionat Individual de Raspall (Valencian for Raspall Singles Championship) is the Valencian pilota Raspall modality singles league played by professional pilotaris.
Statistics
Campionat Individual de Raspall relevant facts
2003: The competition was not played.
2006: Coeter II is the first mitger who wins a competition that looked to be only for dauers (despite the runners-up Agustí, and Moro the year 2000).
See also
Valencian pilota
Raspall
Raspall team championship
External links
XXI Campionat Individual de Raspall
1986 establishments in Spain
Valencian pilota competitions
Valencian pilota professional leagues
|
https://en.wikipedia.org/wiki/Kernel%20%28statistics%29
|
The term kernel is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics.
Bayesian statistics
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated. Otherwise, it may be unnecessary (for example, if the distribution only needs to be sampled from).
For many distributions, the kernel can be written in closed form, but not the normalization constant.
An example is the normal distribution. Its probability density function is
and the associated kernel is
Note that the factor in front of the exponential has been omitted, even though it contains the parameter , because it is not a function of the domain variable .
Pattern analysis
The kernel of a reproducing kernel Hilbert space is used in the suite of techniques known as kernel methods to perform tasks such as statistical classification, regression analysis, and cluster analysis on data in an implicit space. This usage is particularly common in machine learning.
Nonparametric statistics
In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density where they are known as window functions. An additional use is in the estimation of a time-varying intensity for a point process where window functions (kernels) are convolved with time-series data.
Commonly, kernel widths must also be specified when running a non-parametric estimation.
Definition
A kernel is a non-negative real-valued integrable function K. For most applications, it is desirable to define the function to satisfy two additional requirements:
Normalization:
Symmetry:
The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used.
If
|
https://en.wikipedia.org/wiki/Partial%20correlation
|
In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. When determining the numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another confounding variable that is numerically related to both variables of interest. This misleading information can be avoided by controlling for the confounding variable, which is done by computing the partial correlation coefficient. This is precisely the motivation for including other right-side variables in a multiple regression; but while multiple regression gives unbiased results for the effect size, it does not give a numerical value of a measure of the strength of the relationship between the two variables of interest.
For example, given economic data on the consumption, income, and wealth of various individuals, consider the relationship between consumption and income. Failing to control for wealth when computing a correlation coefficient between consumption and income would give a misleading result, since income might be numerically related to wealth which in turn might be numerically related to consumption; a measured correlation between consumption and income might actually be contaminated by these other correlations. The use of a partial correlation avoids this problem.
Like the correlation coefficient, the partial correlation coefficient takes on a value in the range from –1 to 1. The value –1 conveys a perfect negative correlation controlling for some variables (that is, an exact linear relationship in which higher values of one variable are associated with lower values of the other); the value 1 conveys a perfect positive linear relationship, and the value 0 conveys that there is no linear relationship.
The partial correlation coincides with the conditional correlation if the random variables are jointly distributed as the multivariate normal, other elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial, or Dirichlet distribution, but not in general otherwise.
Formal definition
Formally, the partial correlation between X and Y given a set of n controlling variables Z = {Z1, Z2, ..., Zn}, written ρXY·Z, is the correlation between the residuals eX and eY resulting from the linear regression of X with Z and of Y with Z, respectively. The first-order partial correlation (i.e., when n = 1) is the difference between a correlation and the product of the removable correlations divided by the product of the coefficients of alienation of the removable correlations. The coefficient of alienation, and its relation with joint variance through correlation are available in Guilford (1973, pp. 344–345).
Computation
Using linear regression
A simple way to compute the sample partial correlation for some data is to solve the two associated linear regression problems
|
https://en.wikipedia.org/wiki/Kernel%20regression
|
In statistics, kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. The objective is to find a non-linear relation between a pair of random variables X and Y.
In any nonparametric regression, the conditional expectation of a variable relative to a variable may be written:
where is an unknown function.
Nadaraya–Watson kernel regression
Nadaraya and Watson, both in 1964, proposed to estimate as a locally weighted average, using a kernel as a weighting function. The Nadaraya–Watson estimator is:
where is a kernel with a bandwidth such that is of order at least 1, that is .
Derivation
Using the kernel density estimation for the joint distribution f(x,y) and f(x) with a kernel K,
we get
which is the Nadaraya–Watson estimator.
Priestley–Chao kernel estimator
where is the bandwidth (or smoothing parameter).
Gasser–Müller kernel estimator
where
Example
This example is based upon Canadian cross-section wage data consisting of a random sample taken from the 1971 Canadian Census Public Use Tapes for male individuals having common education (grade 13). There are 205 observations in total.
The figure to the right shows the estimated regression function using a second order Gaussian kernel along with asymptotic variability bounds.
Script for example
The following commands of the R programming language use the npreg() function to deliver optimal smoothing and to create the figure given above. These commands can be entered at the command prompt via cut and paste.
install.packages("np")
library(np) # non parametric library
data(cps71)
attach(cps71)
m <- npreg(logwage~age)
plot(m, plot.errors.method="asymptotic",
plot.errors.style="band",
ylim=c(11, 15.2))
points(age, logwage, cex=.25)
detach(cps71)
Related
According to David Salsburg, the algorithms used in kernel regression were independently developed and used in fuzzy systems: "Coming up with almost exactly the same computer algorithm, fuzzy systems and kernel density-based regressions appear to have been developed completely independently of one another."
Statistical implementation
GNU Octave mathematical program package
Julia: KernelEstimator.jl
MATLAB: A free MATLAB toolbox with implementation of kernel regression, kernel density estimation, kernel estimation of hazard function and many others is available on these pages (this toolbox is a part of the book ).
Python: the KernelReg class for mixed data types in the statsmodels.nonparametric sub-package (includes other kernel density related classes), the package kernel_regression as an extension of scikit-learn (inefficient memory-wise, useful only for small datasets)
R: the function npreg of the np package can perform kernel regression.
Stata: npregress, kernreg2
See also
Kernel smoother
Local regression
References
Further reading
External links
Scale-adaptive kernel regression (with Matlab software).
Tutorial of Ker
|
https://en.wikipedia.org/wiki/Isothetic
|
Isothetic (from Greek roots: iso- for "equal, same, similar" and for position, placement) may refer to one of the following.
In computational geometry, see isothetic polygon
Isothetic polyhedra
In data analysis, isothetic lines, isothetic curves, or simply isothetics are contour lines for a data set, where data represent Displacements
|
https://en.wikipedia.org/wiki/Alan%20Davies%20%28mathematician%29
|
Alan Davies (born 22 December 1945) is a British professor emeritus of mathematics at the University of Hertfordshire.
He obtained a first class honours degree in mathematics (1968) from Southampton University. He followed that with a master's degree, with distinction, in structural engineering (1974) and a doctorate in numerical computation (1989) from Imperial College. He has spent most of his working life as an academic at the University of Hertfordshire (UH), formerly the Hatfield Polytechnic. He had short spells in industry working as a research engineer in the aircraft industry and as a process engineer in the food industry.
During his time in Hatfield his major activity has been teaching mathematics to undergraduates and postgraduates in mathematics, science and engineering. He has also been engaged in research in numerical computation. In 1992 he became Head of the Department of Mathematics and was appointed Professor of Mathematics and, in 2004, the Department merged with Physical Sciences and Davies was appointed head of the School of Physics, Astronomy and Mathematics. During his time as head of the department he became increasingly involved with outreach activities with both schools and the general public. He retired from his full-time post in 2006 and is currently Professor Emeritus in mathematics and a London Mathematical Society Holgate Lecturer. His particular interest in teaching is in applied mathematics and numerical computation, particularly to students for whom mathematics is not their main subject, in particular engineering.
In 1991, in collaboration with Ros Crouch, he was awarded the British Nuclear Fuels Partnership Award for Innovative Teaching in Mathematics, in recognition of their undergraduate module in which mathematical modelling was used as a vehicle for the teaching and learning of communication skills. He was a member of the Mew Group which produced materials suitable for teachers to use with sixth formers to consider problems different from the rather idealised versions found in their usual text books. He has also worked with the Open University (OU) as a part-time tutor; he retained that position until 2012 and still teaches in summer schools and mathematics revision weekends.
His research interest is in the area of numerical computation. He collaborated with his wife, Dr Diane Crann, for 15 years developing boundary element solutions to diffusion and heat-conduction problems. They are particularly interested in the use of the Laplace transform and domain decomposition approaches. The two of them co-wrote "A Handbook of Essential Mathematical Formulae" intended for students of mathematics and related fields.
Since his retirement he has concentrated on outreach activities, working with his wife to present masterclasses in mathematics and physical science in collaboration with the Royal Institution (Ri). Over the past twenty years they have both became heavily involved with the Ri and its mastercla
|
https://en.wikipedia.org/wiki/Intrinsic%20hyperpolarizability
|
Intrinsic hyperpolarizability in physics, mathematics and statistics, is a scale invariant quantity that can be used to compare molecules of different sizes. The intrinsic hyperpolarizability is defined as the hyperpolarizability divided by the Kuzyk Limit. This quantity is scale invariant and thus is independent of the energy scale and number of electrons in a molecule that is being evaluated for its nonlinear optical response. Therefore, it can be used to compare molecules of different shapes and sizes.
The Intrinsic Hyperpolarizability can be used as a figure of merit for comparing molecules for their usefulness in electro-optics applications.
See also
Molecular mechanics
Molecular modelling
Quantum chemistry
References
Nonlinear optics
|
https://en.wikipedia.org/wiki/Kharitonov%20region
|
A Kharitonov region is a concept in mathematics. It arises in the study of the stability of polynomials.
Let be a simply-connected set in the complex plane and let be the polynomial family.
is said to be a Kharitonov region if
is a subset of Here, denotes the set of all vertex polynomials of complex interval polynomials and denotes the set of all vertex polynomials of real interval polynomials
See also
Kharitonov's theorem
References
Y C Soh and Y K Foo (1991), “Kharitonov Regions: It Suffices to Check a Subset of Vertex Polynomials”, IEEE Trans. on Aut. Cont., 36, 1102 – 1105.
Polynomials
Stability theory
|
https://en.wikipedia.org/wiki/Copositive%20matrix
|
In mathematics, specifically linear algebra, a real matrix A is copositive if
for every nonnegative vector . The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices.
Copositive matrices find applications in economics, operations research, and statistics.
References
Copositive matrix at PlanetMath
Matrices
Convex analysis
|
https://en.wikipedia.org/wiki/Complementary%20sequences
|
For complementary sequences in biology, see complementarity (molecular biology). For integer sequences with complementary sets of members see Lambek–Moser theorem.
In applied mathematics, complementary sequences (CS) are pairs of sequences with the useful property that their out-of-phase aperiodic autocorrelation coefficients sum to zero. Binary complementary sequences were first introduced by Marcel J. E. Golay in 1949. In 1961–1962 Golay gave several methods for constructing sequences of length 2N and gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method for constructing sequences of length mn from sequences of lengths m and n which allows the construction of sequences of any length of the form 2N10K26M.
Later the theory of complementary sequences was generalized by other authors to polyphase complementary sequences, multilevel complementary sequences, and arbitrary complex complementary sequences. Complementary sets have also been considered; these can contain more than two sequences.
Definition
Let (a0, a1, ..., aN − 1) and (b0, b1, ..., bN − 1) be a pair of bipolar sequences, meaning that a(k) and b(k) have values +1 or −1. Let the aperiodic autocorrelation function of the sequence x be defined by
Then the pair of sequences a and b is complementary if:
for k = 0, and
for k = 1, ..., N − 1.
Or using Kronecker delta we can write:
So we can say that the sum of autocorrelation functions of complementary sequences is a delta function, which is an ideal autocorrelation for many applications like radar pulse compression and spread spectrum telecommunications.
Examples
As the simplest example we have sequences of length 2: (+1, +1) and (+1, −1). Their autocorrelation functions are (2, 1) and (2, −1), which add up to (4, 0).
As the next example (sequences of length 4), we have (+1, +1, +1, −1) and (+1, +1, −1, +1). Their autocorrelation functions are (4, 1, 0, −1) and (4, −1, 0, 1), which add up to (8, 0, 0, 0).
One example of length 8 is (+1, +1, +1, −1, +1, +1, −1, +1) and (+1, +1, +1, −1, −1, −1, +1, −1). Their autocorrelation functions are (8, −1, 0, 3, 0, 1, 0, 1) and (8, 1, 0, −3, 0, −1, 0, −1).
An example of length 10 given by Golay is (+1, +1, −1, +1, −1, +1, −1, −1, +1, +1) and (+1, +1, −1, +1, +1, +1, +1, +1, −1, −1). Their autocorrelation functions are (10, −3, 0, −1, 0, 1,−2, −1, 2, 1) and (10, 3, 0, 1, 0, −1, 2, 1, −2, −1).
Properties of complementary pairs of sequences
Complementary sequences have complementary spectra. As the autocorrelation function and the power spectra form a Fourier pair, complementary sequences also have complementary spectra. But as the Fourier transform of a delta function is a constant, we can write
where CS is a constant.
Sa and Sb are defined as a squared magnitude of the Fourier transform of the sequences. The Fourier transform can be a direct DFT of the sequences, it can be a DFT of zero padded sequences or it can be a continuous Fourie
|
https://en.wikipedia.org/wiki/Absorption%20cross%20section
|
In physics, absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain particle-particle interaction, e.g., scattering, electromagnetic absorption, etc. (Note that light in this context is described as consisting of particles, i.e., photons.) Typical absorption cross section has units of cm2⋅molecule−1. In honor of the fundamental contribution of Maria Goeppert Mayer to this area, the unit for the two-photon absorption cross section is named the "GM". One GM is 10−50 cm4⋅s⋅photon−1.
In the context of ozone shielding of ultraviolet light, absorption cross section is the ability of a molecule to absorb a photon of a particular wavelength and polarization. Analogously, in the context of nuclear engineering it refers to the probability of a particle (usually a neutron) being absorbed by a nucleus. Although the units are given as an area, it does not refer to an actual size area, at least partially because the density or state of the target molecule will affect the probability of absorption. Quantitatively, the number of photons absorbed, between the points and along the path of a beam is the product of the number of photons penetrating to depth times the number of absorbing molecules per unit volume times the absorption cross section :
.
The absorption cross-section is closely related to molar absorptivity and mass absorption coefficient.
For a given particle and its energy, the absorption cross-section of the target material can be calculated from mass absorption coefficient using:
where:
is the mass absorption coefficient
is the molar mass in g/mol
is Avogadro constant
This is also commonly expressed as:
where:
is the absorption coefficient
is the atomic number density
See also
Cross section (physics)
Photoionisation cross section
Nuclear cross section
Neutron cross section
Mean free path
Compton scattering
Transmittance
Attenuation
Beer–Lambert law
High energy X-rays
Attenuation coefficient
Absorption spectroscopy
References
Electromagnetism
Nuclear physics
Scattering, absorption and radiative transfer (optics)
|
https://en.wikipedia.org/wiki/Roger%20Evans%20Howe
|
Roger Evans Howe (born May 23, 1945) is the William R. Kenan, Jr. Professor Emeritus of Mathematics at Yale University, and Curtis D. Robert Endowed Chair in Mathematics Education at Texas A&M University. He is known for his contributions to representation theory, in particular for the notion of a reductive dual pair and the Howe correspondence, and his contributions to mathematics education.
Biography
He attended Ithaca High School, then Harvard University as an undergraduate, becoming a Putnam Fellow in 1964. He obtained his Ph.D. from University of California, Berkeley in 1969. His thesis, titled On representations of nilpotent groups, was written under the supervision of Calvin Moore. Between 1969 and 1974, Howe taught at the State University of New York in Stony Brook before joining the Yale faculty in 1974. His doctoral students include Ju-Lee Kim, Jian-Shu Li, Zeev Rudnick, Eng-Chye Tan, and Chen-Bo Zhu. He moved to Texas A&M University in 2015.
He has been a fellow of the American Academy of Arts and Sciences since 1993, and a member of the National Academy of Sciences since 1994.
Howe received a Lester R. Ford Award in 1984. In 2006 he was awarded the American Mathematical Society Distinguished Public Service Award in recognition of his "multifaceted contributions to mathematics and to mathematics education." In 2012 he became a fellow of the American Mathematical Society. In 2015 he received the inaugural Award for Excellence in Mathematics Education.
A conference in his honor was held at the National University of Singapore in 2006, and at Yale University in 2015.
Selected works
Roger Howe, "Tamely ramified supercuspidal representations of ", Pacific Journal of Mathematics 73 (1977), no. 2, 437–460.
Roger Howe and Calvin C. Moore, "Asymptotic properties of unitary representations", Journal of Functional Analysis 32 (1979), no. 1, 72–96.
Roger Howe, "θ-series and invariant theory", in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., XXXIII, American Mathematical Society), pp. 275–285, (1979).
Roger Howe, "Wave front sets of representations of Lie groups". Automorphic forms, representation theory and arithmetic (Bombay, 1979), pp. 117–140, Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay, 1981.
Roger Howe, "On a notion of rank for unitary representations of the classical groups". Harmonic analysis and group representations, 223–331, Liguori, Naples, 1982.
Roger Howe, "Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond". The Schur lectures (1992) (Tel Aviv), 1–182, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995.
Roger Howe & Eng-Chye Tan, "Nonabelian harmonic analysis. Applications of SL(2,R)". Universitext. Springer-Verlag, New York, 1992. xvi+257 pp. .
Roger Howe & William Barker (2007) Continuous Symmetry: From Euclid to Klein, American Mathematical Society, .
Robin Hartshorne (2011) Review of Continuou
|
https://en.wikipedia.org/wiki/Quasiregular%20polyhedron
|
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.
There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case.
These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2).
More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) sequences of the faces around the vertex.
Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally: (p.q)2, with 1/p + 1/q < 1/2.
Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)q/2, if q is even.
Examples:
The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)4/2 = (3a.3b)2, alternating two colors of triangular faces.
The square tiling, with vertex configuration 44 and 4 being even, can be considered quasiregular, with vertex configuration (4.4)4/2 = (4a.4b)2, colored as a checkerboard.
The triangular tiling, with vertex configuration 36 and 6 being even, can be considered quasiregular, with vertex configuration (3.3)6/2 = (3a.3b)3, alternating two colors of triangular faces.
Wythoff construction
Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r.
The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:
The convex quasiregular polyhedra
There are two uniform convex quasiregular polyhedra:
The cuboctahedro
|
https://en.wikipedia.org/wiki/Defence%20Analytical%20Services%20and%20Advice
|
Defence Analytical Services and Advice (DASA) was a statistical and economic unit within the MoD, initially created in 1992 from various statistics branches within the Ministry of Defence (MoD), as the Defence Analytical Services Agency. DASA was initially an executive agency of the MoD but lost its agency status on 1 April 2008, becoming an administrative unit within the MOD, changing its name to Defence Analytical Services and Advice, retaining the acronym DASA. On 1 April 2013, DASA was split into two separate units within the MoD; Defence Economics and Defence Statistics.
The role of DASA was to compile staffing, financial and logistical statistics to provide professional analytical, economic and statistical services and advice to the MoD, Parliament, Ministers, Senior MoD Officials and other government departments, mainly through its publication of Defence related National Statistics Publications and responses to Parliamentary Questions and ad hoc queries. DASA also provided planning and forecasting models to the British Armed Services to help to inform their decisions. In addition to this, DASA was also responsible for providing members of the public with defence related statistics through the Freedom of Information Act 2000.
DASA employed a mixture of statisticians, economists, IT specialists, other analysts and specialists, and administrative staff.
DASA was part of both the Government Statistical Service and the Government Economic Service. As such, DASA Directors were usually the heads of profession for statistics and economics for the MOD.
DASA had offices in six areas: the two largest were in London and Bath, with smaller offices at MoD Abbey Wood in Bristol. DASA also had personnel co-located with each of the three Armed Services at RAF High Wycombe with HQ Air Command for the RAF, HMS Excellent, Portsmouth with the Navy Command Headquarters for the Royal Navy and Andover with Army Headquarters for the British Army.
Its former Director (previous to that Chief Executive) whilst DASA was still an Agency was Dr Mike McDowall, BSc, PhD. DASA was headed by Dr Prabhat Vaze until late 2012.
References
External links
Official website
Defence agencies of the United Kingdom
Defunct executive agencies of the United Kingdom government
|
https://en.wikipedia.org/wiki/Cuboctahedral%20prism
|
In geometry, a cuboctahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 polyhedral cells: 2 cuboctahedra connected by 8 triangular prisms and 6 cubes.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Alternative names
Cuboctahedral dyadic prism
Rhombioctahedral prism
Rhombioctahedral hyperprism
External links
4-polytopes
|
https://en.wikipedia.org/wiki/Uniform%20antiprismatic%20prism
|
In 4-dimensional geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. The symmetry of a p-gonal antiprismatic prism is [2p,2+,2], order 8p.
A p-gonal antiprismatic prism or p-gonal antiduoprism has 2 p-gonal antiprism, 2 p-gonal prism, and 2p triangular prism cells. It has 4p equilateral triangle, 4p square and 4 regular p-gon faces. It has 10p edges, and 4p vertices.
Convex uniform antiprismatic prisms
There is an infinite series of convex uniform antiprismatic prisms, starting with the digonal antiprismatic prism is a tetrahedral prism, with two of the tetrahedral cells degenerated into squares. The triangular antiprismatic prism is the first nondegenerate form, which is also an octahedral prism. The remainder are unique uniform 4-polytopes.
Star antiprismatic prisms
There are also star forms following the set of star antiprisms, starting with the pentagram {5/2}:
Square antiprismatic prism
A square antiprismatic prism or square antiduoprism is a convex uniform 4-polytope. It is formed as two parallel square antiprisms connected by cubes and triangular prisms. The symmetry of a square antiprismatic prism is [8,2+,2], order 32. It has 16 triangle, 16 square and 4 square faces. It has 40 edges, and 16 vertices.
Pentagonal antiprismatic prism
A pentagonal antiprismatic prism or pentagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel pentagonal antiprisms connected by cubes and triangular prisms. The symmetry of a pentagonal antiprismatic prism is [10,2+,2], order 40. It has 20 triangle, 20 square and 4 pentagonal faces. It has 50 edges, and 20 vertices.
Hexagonal antiprismatic prism
A hexagonal antiprismatic prism or hexagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel hexagonal antiprisms connected by cubes and triangular prisms. The symmetry of a hexagonal antiprismatic prism is [12,2+,2], order 48. It has 24 triangle, 24 square and 4 hexagon faces. It has 60 edges, and 24 vertices.
Heptagonal antiprismatic prism
A heptagonal antiprismatic prism or heptagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel heptagonal antiprisms connected by cubes and triangular prisms. The symmetry of a heptagonal antiprismatic prism is [14,2+,2], order 56. It has 28 triangle, 28 square and 4 heptagonal faces. It has 70 edges, and 28 vertices.
Octagonal antiprismatic prism
A octagonal antiprismatic prism or octagonal antiduoprism is a convex uniform 4-polytope (four-dimensional polytope). It is formed as two parallel octagonal antiprisms connected by cubes and triangular prisms. The symmetry of an octagonal antiprismatic prism is [16,2+,2], order 64. It has 32 triangle, 32 square and 4 octagonal faces. It has 80 edges, and 32 vertices.
See also
Duoprism
References
John H. Conway, Heidi Burgiel, Chaim Go
|
https://en.wikipedia.org/wiki/Octahedral%20prism
|
In geometry, an octahedral prism is a convex uniform 4-polytope. This 4-polytope has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.
Alternative names
Octahedral dyadic prism (Norman W. Johnson)
Ope (Jonathan Bowers, for octahedral prism)
Triangular antiprismatic prism
Triangular antiprismatic hyperprism
Coordinates
It is a Hanner polytope with vertex coordinates, permuting first 3 coordinates:
([±1,0,0]; ±1)
Structure
The octahedral prism consists of two octahedra connected to each other via 8 triangular prisms. The triangular prisms are joined to each other via their square faces.
Projections
The octahedron-first orthographic projection of the octahedral prism into 3D space has an octahedral envelope. The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces.
The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces. One triangular prismic cell projects onto a triangular prism at the center of the envelope, surrounded by the images of 3 other triangular prismic cells to cover the entire volume of the envelope. The remaining four triangular prismic cells are projected onto the entire volume of the envelope as well, in the same arrangement, except with opposite orientation.
Related polytopes
It is the second in an infinite series of uniform antiprismatic prisms.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
It is one of four four-dimensional Hanner polytopes; the other three are the tesseract, the 16-cell, and the dual of the octahedral prism (a cubical bipyramid).
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26)
Norman Johnson Uniform Polytopes, Manuscript (1991)
External links
4-polytopes
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.