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https://en.wikipedia.org/wiki/Regev%27s%20theorem | In abstract algebra, Regev's theorem, proved by , states that the tensor product of two PI algebras is a PI algebra.
References
Theorems in ring theory |
https://en.wikipedia.org/wiki/Alex%20Eskin | Alex Eskin (, born May 19, 1965, Moscow, USSR) is an American mathematician. He is the Arthur Holly Compton Distinguished Service Professor in the Department of Mathematics at the University of Chicago. His research focuses on rational billiards and geometric group theory.
Biography
Eskin was born in Moscow on May 19, 1965. He is the son of a Russian-Jewish mathematician Gregory I. Eskin (b. 1936, Kiev), a professor at the University of California, Los Angeles. The family emigrated to Israel in 1974 and in 1982 to the United States.
Eskin earned his doctorate from Princeton University in 1993, under the supervision of Peter Sarnak.
Eskin has been a professor at the University of Chicago since 1999.
Awards
Eskin gave invited talks at the International Congress of Mathematicians in Berlin in 1998, and in Hyderabad in 2010.
For his contribution to joint work with David Fisher and Kevin Whyte establishing the quasi-isometric rigidity of solvable groups, Eskin was awarded the 2007 Clay Research Award. In 2012, he became a fellow of the American Mathematical Society. In April 2015, Eskin was elected a member of the United States National Academy of Sciences. Eskin won the 2020 Breakthrough Prize in mathematics for his classification of -invariant and stationary measures for the moduli of translation surfaces, in joint work with Maryam Mirzakhani.
Selected publications
References
External links
Website at University of Chicago
Paper about measure classification, joint with Maryam Mirzakhani
1965 births
20th-century American mathematicians
21st-century American mathematicians
Living people
American people of Russian-Jewish descent
Clay Research Award recipients
Dynamical systems theorists
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
Princeton University alumni
Simons Investigator
Stanford University alumni
University of Chicago faculty |
https://en.wikipedia.org/wiki/Bost%E2%80%93Connes%20system | In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. introduced Bost–Connes systems by constructing one for the rational numbers. extended the construction to imaginary quadratic fields.
Such systems have been studied for their connection with Hilbert's Twelfth Problem. In the case of a Bost–Connes system over Q, the absolute Galois group acts on the ground states of the system.
References
Number theory
Dynamical systems |
https://en.wikipedia.org/wiki/Alexander%20Merkurjev | Aleksandr Sergeyevich Merkurjev (, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev is a professor at the University of California, Los Angeles.
Work
Merkurjev's work focuses on algebraic groups, quadratic forms, Galois cohomology, algebraic K-theory and central simple algebras. In the early 1980s Merkurjev proved a fundamental result about the structure of central simple algebras of period dividing 2, which relates the 2-torsion of the Brauer group with Milnor K-theory. In subsequent work with Suslin this was extended to higher torsion as the Merkurjev–Suslin theorem. The full statement of the norm residue isomorphism theorem (also known as the Bloch-Kato conjecture) was proven by Voevodsky.
In the late 1990s Merkurjev gave the most general approach to the notion of essential dimension, introduced by Buhler and Reichstein, and made fundamental contributions to that field. In particular Merkurjev determined the essential p-dimension of central simple algebras of degree (for a prime p) and, in joint work with Karpenko, the essential dimension of finite p-groups.
Awards
Merkurjev won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on algebraic K-theory. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley, California, and his talk was entitled "Milnor K-theory and Galois cohomology". In 1995 he won the Humboldt Prize, an international prize awarded to renowned scholars. Merkurjev gave a plenary talk at the second European Congress of Mathematics in Budapest, Hungary in 1996.
In 2012 he won the Cole Prize in Algebra for his work on the essential dimension of groups.
In 2015 a special volume of Documenta Mathematica was published in honor of Merkurjev's sixtieth birthday.
Bibliography
Books
Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol: The book of involutions, American Mathematical Society 1998.
Skip Garibaldi, Jean-Pierre Serre, Alexander Merkurjev: Cohomological Invariants in Galois Cohomology, American Mathematical Society 2003.
Richard Elman, Nikita Karpenko, Alexander Merkurjev: Algebraic and geometric theory of quadratic forms, American Mathematical Society 2008.
References
External links
Alexander Merkurjev - personal webpage at UCLA
1955 births
Living people
University of California, Los Angeles faculty
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
Algebraists
International Mathematical Olympiad participants |
https://en.wikipedia.org/wiki/Heterogeneous%20random%20walk%20in%20one%20dimension | In dynamics, probability, physics, chemistry and related fields, a heterogeneous random walk in one dimension is a random walk in a one dimensional interval with jumping rules that depend on the location of the random walker in the interval.
For example: say that the time is discrete and also the interval. Namely, the random walker jumps every time step either left or right. A possible heterogeneous random walk draws in each time step a random number that determines the local jumping probabilities and then a random number that determines the actual jump direction. Specifically, say that the interval has 9 sites (labeled 1 through 9), and the sites (also termed states) are connected with each other linearly (where the edges sites are connected their adjacent sites and together). In each time step, the jump probabilities (from the actual site) are determined when flipping a coin; for head we set: probability jumping left =1/3, where for tail we set: probability jumping left = 0.55. Then, a random number is drawn from a uniform distribution: when the random number is smaller than probability jumping left, the jump is for the left, otherwise, the jump is for the right. Usually, in such a system, we are interested in the probability of staying in each of the various sites after t jumps, and in the limit of this probability when t is very large, .
Generally, the time in such processes can also vary in a continuous way, and the interval is also either discrete or continuous. Moreover, the interval is either finite or without bounds. In a discrete system, the connections are among adjacent states. The basic dynamics are either Markovian, semi-Markovian, or even not Markovian depending on the model. In discrete systems, heterogeneous random walks in 1d have jump probabilities that depend on the location in the system, and/or different jumping time (JT) probability density functions (PDFs) that depend on the location in the system.
,,,,,,,,,,,,,,,,,
General solutions for heterogeneous random walks in 1d obey equations ()-(), presented in what follows.
Introduction
Random walks in applications
Random walks can be used to describe processes in biology, chemistry, and physics, including chemical kinetics and polymer dynamics. In ndividual molecules, random walks appear when studying individual molecules, individual channels, individual biomolecules, individual enzymes, and quantum dots. Importantly, PDFs and special correlation functions can be easily calculated from single molecule measurements but not from ensemble measurements. This unique information can be used for discriminating between distinct random walk models that share some properties, and this demands a detailed theoretical analysis of random walk models. In this context, utilizing the information content in single molecule data is a matter of ongoing research.
Formulations of random walks
The actual random walk obeys a stochastic equation of motion, but its probability density function (PDF |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20PFC%20Levski%20Sofia%20season | The 2011–12 season is Levski Sofia's 90th season in the First League. This article shows player statistics and all matches (official and friendly) that the club has played during the 2011–12 season.
Transfers
Summer transfers
In:
Out:
See List of Bulgarian football transfers summer 2011
Winter transfers
In:
Out:
See List of Bulgarian football transfers winter 2011–12
Squad
As of July 6, 2011
Statistics
Goalscorers
Assists
Cards
Pre-season and friendlies
Summer
Winter
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
UEFA Europa League
Third qualifying round
References
PFC Levski Sofia seasons
Levski Sofia
Levski Sofia |
https://en.wikipedia.org/wiki/Tony%20Gardiner | Tony Gardiner (born 1947) is a British mathematician who until 2012 held the position of Reader in Mathematics and Mathematics Education at the University of Birmingham. He was responsible for the foundation of the United Kingdom Mathematics Trust in 1996, one of the UK's largest mathematics enrichment programs, initiating the Intermediate and Junior Mathematical Challenges, creating the Problem Solving Journal for secondary school students and organising numerous masterclasses, summer schools and educational conferences. Gardiner has contributed to many educational articles and internationally circulated educational pamphlets. As well as his involvement with mathematics education, Gardiner has also made contributions to the areas of infinite groups, finite groups, graph theory, and algebraic combinatorics.
In the year 1994–1995, he received the Paul Erdős Award for his contributions to UK and international mathematical challenges and olympiads. In 2011, Gardiner was elected Education Secretary of the London Mathematical Society. In 2016 he received the Excellence in Mathematics Education Award from Texas A&M University.
UK national mathematics competitions
The first national mathematics competition was the National Mathematics Contest, established in 1961 by F. R. Watson. This was run by the Mathematical Association from 1975 until its adoption by the United Kingdom Mathematics Trust (UKMT) in 1996 and has in recent years been known as the Senior Mathematical Challenge. In early years, problems for this were taken from the American Annual High School Mathematics Examination, with papers subsequently formed from those used in other countries until 1988, when the first entirely local paper was produced. In 1987, Gardiner founded the Junior and Intermediate Mathematical Challenges under the name of the United Kingdom Mathematics Foundation, to expand the national mathematics competitions to a wider age range of students. Gardiner worked hard to publicise all of the national mathematics competitions from 1987 to 1995 and UKMT yearbooks state that their enormous increase in popularity was "without doubt due to the drive, energy and leadership of Tony Gardiner". The Junior and Intermediate Challenges continued to be run by Gardiner personally until the foundation of the UKMT, with numbers of entrants reaching 105,000 and 115,000 respectively in the year 1994–1995. Between 1988 and 1997, participation in the Senior Mathematical Challenge increased from around 8,000 entries from 340 schools to 40,000 from nearly 900 schools.
Gardiner also played an important role in establishing the first Primary Mathematics Challenge (PMC) in 1998. Run by the Mathematical Association, in 2010 it received more than 84,000 participants in 2,361 schools.
United Kingdom Mathematics Trust
The United Kingdom Mathematics Trust was founded in 1996 to support the large pyramid of national mathematics competitions that had become well established in the UK. In 1995, Gardiner |
https://en.wikipedia.org/wiki/G%C3%B6k%C3%A7eada%20Airport | Gökçeada Airport () is a public airport in Gökçeada, a town in Çanakkale Province, Turkey. Opened to public/civil air traffic in 2010, the airport is away from Gökçeada town centre.
Statistics
External links
https://web.archive.org/web/20111006194142/http://www.airporthaber.com/readnews.php?newid=32314
https://web.archive.org/web/20111006194214/http://www.airporthaber.com/readnews.php?newid=32111
https://web.archive.org/web/20110717005841/http://www.airporthaber.com/gokceada-yolcu-ucagiyla-tanisti-foto-haber--33707h.html
http://www.flyseabird.com/en-US/our-destinations/destinations/
http://web.shgm.gov.tr/kurumsal.php?page=haberler&id=1&haber_id=2667
References
Airports in Turkey
Buildings and structures in Çanakkale Province
Imbros |
https://en.wikipedia.org/wiki/Bogdan%20Musta%C8%9B%C4%83 | Bogdan Mustață (born 28 July 1990) is a Romanian footballer who plays as a right back. He played in Liga I for Unirea Urziceni.
Club statistics
Statistics accurate as of match played 20 October 2011
References
External links
1990 births
Footballers from Bucharest
Romanian men's footballers
Liga I players
Men's association football defenders
FCSB II players
FC Unirea Urziceni players
Living people
FC Steaua București players
CS Turnu Severin players |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Denmark | This article lists various Danish football records for the various Danish football leagues and competitions.
Team records
Most Championships: 15
Kjøbenhavns Boldklub.
Most Cup wins: 10
Aarhus Gymnastik Forening
Most successful clubs overall
NOTE: *** The Danish Super Cup is now defunct, and has not been played since 2004.
Football in Denmark
Denmark national football team
Danish Cup
Records
Denmark |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Stevenage%20F.C.%20season | The 2011–12 season was Stevenage F.C.'s second season in the Football League, where the club competed in League One. This article shows statistics of the club's players in the season, and also lists all matches that the club played during the season. Their sixth-place finish and subsequent successful play-off campaign meant it was Stevenage's first ever season of playing in League One, having only spent one season in League Two. The season also marked the second season that the club played under its new name – Stevenage Football Club, dropping 'Borough' from its title as of 1 June 2010. The season started out as the third year in charge for manager Graham Westley during his second spell at the club; having previously managed the Hertfordshire side from 2003 to 2006. However, Westley left Stevenage in January 2012, and joined fellow League One side Preston North End. The vacant managerial position was filled by former Colorado Rapids manager Gary Smith, signing a contract until 2014.
Ahead of the club's first season in League One, Westley adopted the same "five in, five out" transfer policy as he had done for the two previous seasons. Strikers Yemi Odubade and Charlie Griffin were the first to leave having been loaned out for much of the previous campaign, joining Conference National sides Gateshead and Forest Green Rovers respectively. Second choice goalkeeper Ashley Bayes opted to leave the club in order to play first-team football at Conference South club Basingstoke Town. Luke Foster and David Bridges also opted to leave Stevenage ahead of the season, both on free transfers, with Foster signing for Rotherham United, and Bridges for his former club, Kettering Town. Stevenage's first signing of the season was striker Guy Madjo, who joined on a free transfer from Albanian Superliga side KS Bylis Ballsh. Former Stevenage goalkeeper Alan Julian re-joined the club following his release by Gillingham, while Phil Edwards rejected a contract extension at Accrington Stanley in order to join the Hertfordshire club on a free transfer. Midfielders Jennison Myrie-Williams and Robin Shroot also signed on free transfers following successful trial periods with the club. In terms of transfers during the 2011–12 campaign, striker Don Cowan joined the club from Longford Town for an undisclosed fee in August 2011, and winger Luke Freeman signed from Arsenal in January 2012, after a successful three-month loan spell with the club. Strikers Byron Harrison and Guy Madjo both departed in January 2012, signing for League Two sides AFC Wimbledon and Aldershot Town for respective undisclosed fees.
Stevenage started their first ever League One campaign brightly, losing just one of their first eight League One fixtures, as well as securing a notable 5–1 home victory over Sheffield Wednesday. However, a run of four successive defeats moved Stevenage into the lower half of the league. A 1–0 victory over then-unbeaten league leaders, Charlton Athletic, would ultimately serv |
https://en.wikipedia.org/wiki/Schubert%20polynomial | In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by and are named after Hermann Schubert.
Background
described the history of Schubert polynomials.
The Schubert polynomials are polynomials in the variables depending on an element of the infinite symmetric group of all permutations of fixing all but a finite number of elements. They form a basis for the polynomial ring in infinitely many variables.
The cohomology of the flag manifold is where is the ideal generated by homogeneous symmetric functions of positive degree. The Schubert polynomial is the unique homogeneous polynomial of degree representing the Schubert cycle of in the cohomology of the flag manifold for all sufficiently large
Properties
If is the permutation of longest length in then
if , where is the transposition and where is the divided difference operator taking to .
Schubert polynomials can be calculated recursively from these two properties. In particular, this implies that .
Other properties are
If is the transposition , then .
If for all , then is the Schur polynomial where is the partition . In particular all Schur polynomials (of a finite number of variables) are Schubert polynomials.
Schubert polynomials have positive coefficients. A conjectural rule for their coefficients was put forth by Richard P. Stanley, and proven in two papers, one by Sergey Fomin and Stanley and one by Sara Billey, William Jockusch, and Stanley.
The Schubert polynomials can be seen as a generating function over certain combinatorial objects called pipe dreams or rc-graphs. These are in bijection with reduced Kogan faces, (introduced in the PhD thesis of Mikhail Kogan) which are special faces of the Gelfand-Tsetlin polytope.
Schubert polynomials also can be written as a weighted sum of objects called bumpless pipe dreams.
As an example
Multiplicative structure constants
Since the Schubert polynomials form a -basis, there are unique coefficients
such that
These can be seen as a generalization of the Littlewood−Richardson coefficients described by the Littlewood–Richardson rule.
For algebro-geometric reasons (Kleiman's transversality theorem of 1974), these coefficients are non-negative integers and it is an
outstanding problem in representation theory and combinatorics to give a combinatorial rule for these numbers.
Double Schubert polynomials
Double Schubert polynomials are polynomials in two infinite sets of variables, parameterized by an element w of the infinite symmetric group, that becomes the usual Schubert polynomials when all the variables are .
The double Schubert polynomial are characterized by the properties
when is the permutation on of longest length.
if .
The double Schubert polynomials can also be defined as
.
Quantum Schubert polynomials
introduced quantum Schubert polynomials, that have the same |
https://en.wikipedia.org/wiki/Xiao-Li%20Meng | Xiao-Li Meng (; born 1963) is a Chinese American statistician and the Whipple V. N. Jones Professor of Statistics at Harvard University. He received the COPSS Presidents' Award in 2001. He has written numerous research papers about Markov chain Monte Carlo algorithms and other statistical methodology.
From 2004 to 2012, Meng was the Chair of Harvard's Department of Statistics, where he helped create innovative new statistics courses designed to give students a more positive impression of the subject. He edited the journals Bayesian Analysis from 2003 to 2005 and Statistica Sinica from 2005 to 2008. On August 14, 2012, Xiao-Li Meng was appointed dean of Harvard Graduate School of Arts and Sciences (GSAS).
Meng received his B.Sc. from Fudan University in 1982 and his Ph.D. in statistics from Harvard University in 1990. He was elected a fellow of the Institute of Mathematical Statistics in 1997 and of the American Statistical Association in 2004. He was elected fellow of the American Academy of Arts and Sciences (AAAS) in 2020.
References
External links
Xiao-Li Meng's home page
1963 births
Living people
Harvard University faculty
Fellows of the American Statistical Association
Fellows of the Institute of Mathematical Statistics
Fudan University alumni
Harvard University alumni
Scientists from Shanghai
Chinese emigrants to the United States
Fellows of the American Academy of Arts and Sciences |
https://en.wikipedia.org/wiki/Jun%20S.%20Liu | Jun S. Liu (; born 1965) is a Chinese-American statistician focusing on Bayesian statistical inference and computational biology. He was Assistant Professor of Statistics at Harvard University from 1991 to 1994. From 1994 to 2004, he was Assistant, Associate, and full Professor of Statistics (promoted while being on leave) at Stanford University. Since 2000, Liu has been Professor of Statistics in the Department of Statistics at Harvard University and held a courtesy appointment at Harvard T.H. Chan School of Public Health.
Liu has written many research papers and a book about Markov chain Monte Carlo algorithms, including their applications in biology. He is also co-author of several early software on biological sequence motif discovery.: MACAW, Gibbs Motif Sampler, BioProspector, Motif regressor, MDScan, Tmod; on genetic data analysis: BLADE, HAPLOTYPER, PL-EM, BEAM; and more recently on, genome structure, gene expression and cell type analysis: HiCNorm, BACH, CLIME, RABIT, CLIC, TIMER, and PhyloAcc.
Education
Liu received his B.Sc. from Peking University in 1985. He was a PhD candidate of mathematics at Rutgers University from 1986 to 1988, and obtained his Ph.D. in statistics under the supervision of Wing Hung Wong and Augustine Kong from the University of Chicago in 1991.
Career and research
Liu was the recipient of the 2002 COPSS Presidents' Award,
which is arguably the most prestigious award in the field of statistics.
He also won the 2010 Morningside Gold Medal in Applied Mathematics; and awarded the 2016 Pao-Lu Hsu award by the International Chinese Statistical Association (given every three years to an individual under age 50).
Liu was an Institute of Mathematical Statistics (IMS) Medallion Lecturer in 2002 and a Bernoulli Lecturer in 2004. He was elected a fellow of the Institute of Mathematical Statistics in 2004, fellow of the American Statistical Association in 2005, and fellow of the International Society for Computational Biology in 2022.
References
1965 births
Living people
American statisticians
Bayesian statisticians
Fellows of the Institute of Mathematical Statistics
Fellows of the American Statistical Association
Harvard University faculty
Peking University alumni
Rutgers University alumni
University of Chicago alumni
People's Republic of China emigrants to the United States |
https://en.wikipedia.org/wiki/Jeff%20Rosenthal | Jeffrey Seth Rosenthal (born October 13, 1967) is a Canadian statistician and nonfiction author. He is a professor in the University of Toronto's department of statistics, cross-appointed with its department of mathematics.
Education and career
Rosenthal graduated from Woburn Collegiate Institute in 1984, received his B.Sc. (in mathematics, physics, and computer science) from the University of Toronto in 1988, and received his Ph.D. in mathematics ("Rates of Convergence for Gibbs Sampler and Other Markov Chains") from Harvard University in 1992, supervised by Persi Diaconis. He was an assistant professor in the Department of Mathematics at the University of Minnesota from 1992 to 1993. Rosenthal began his career in the Department of Statistics at the University of Toronto as an assistant professor in 1993, became an associate professor in 1997, and took on his current () position as full professor in 2000. Rosenthal has written numerous research papers about the theory of Markov chain Monte Carlo and other statistical computation algorithms, many joint with Gareth O. Roberts.
Public engagements
In 2005 Rosenthal wrote a book for the general public, Struck by Lightning: The Curious World of Probabilities, which was a bestseller in Canada and has been published in ten languages. He has also written a graduate textbook on probability theory and co-authored an undergraduate textbook on probability and statistics. He has been interviewed by the media about such diverse topics as crime statistics, pedestrian deaths, gambling probabilities, and television game shows, and has appeared on William Shatner's Weird or What?.
In 2006, Rosenthal did the statistical analysis used by the Canadian Broadcasting Corporation television news magazine The Fifth Estate to expose the Ontario lottery retailer fraud scandal, which was debated in the Ontario provincial legislature. In 2010 his research with Albert H. Yoon about the U.S. Supreme Court was quoted in The New York Times. He has also written about the Monty Hall problem.
Honors and awards
Rosenthal received the CRM-SSC Prize in 2006, the COPSS Presidents' Award in 2007, the Statistical Society of Canada Gold Medal in 2013, and a Faculty of Arts & Science Outstanding Teaching Award in 1998. He was elected a Fellow of the Institute of Mathematical Statistics in 2005, and of the Royal Society of Canada in 2012.
Personal life
Rosenthal's father Peter Rosenthal and mother Helen Stephanie Rosenthal (1942 – 2017) are both math professors at the University of Toronto. Besides his research, Rosenthal performs music and improvisational comedy, including at The Bad Dog Theatre Company.
Bibliography
References
1967 births
Canadian statisticians
Fellows of the Institute of Mathematical Statistics
Fellows of the Royal Society of Canada
Harvard University alumni
Living people
People from Scarborough, Toronto
Academic staff of the University of Toronto
University of Toronto alumni
University of Minnesota faculty
2 |
https://en.wikipedia.org/wiki/Step%20detection | In statistics and signal processing, step detection (also known as step smoothing, step filtering, shift detection, jump detection or edge detection) is the process of finding abrupt changes (steps, jumps, shifts) in the mean level of a time series or signal. It is usually considered as a special case of the statistical method known as change detection or change point detection. Often, the step is small and the time series is corrupted by some kind of noise, and this makes the problem challenging because the step may be hidden by the noise. Therefore, statistical and/or signal processing algorithms are often required.
The step detection problem occurs in multiple scientific and engineering contexts, for example in statistical process control (the control chart being the most directly related method), in exploration geophysics (where the problem is to segment a well-log recording into stratigraphic zones), in genetics (the problem of separating microarray data into similar copy-number regimes), and in biophysics (detecting state transitions in a molecular machine as recorded in time-position traces). For 2D signals, the related problem of edge detection has been studied intensively for image processing.
Algorithms
When the step detection must be performed as and when the data arrives, then online algorithms are usually used,
and it becomes a special case of sequential analysis.
Such algorithms include the classical CUSUM method applied to changes in mean.
By contrast, offline algorithms are applied to the data potentially long after it has been received. Most offline algorithms for step detection in digital data can be categorised as top-down, bottom-up, sliding window, or global methods.
Top-down
These algorithms start with the assumption that there are no steps and introduce possible candidate steps one at a time, testing each candidate to find the one that minimizes some criteria (such as the least-squares fit of the estimated, underlying piecewise constant signal). An example is the stepwise jump placement algorithm, first studied in geophysical problems, that has found recent uses in modern biophysics.
Bottom-up
Bottom-up algorithms take the "opposite" approach to top-down methods, first assuming that there is a step in between every sample in the digital signal, and then successively merging steps based on some criteria tested for every candidate merge.
Sliding window
By considering a small "window" of the signal, these algorithms look for evidence of a step occurring within the window. The window "slides" across the time series, one time step at a time. The evidence for a step is tested by statistical procedures, for example, by use of the two-sample Student's t-test. Alternatively, a nonlinear filter such as the median filter is applied to the signal. Filters such as these attempt to remove the noise whilst preserving the abrupt steps.
Global
Global algorithms consider the entire signal in one go, and attempt to find the step |
https://en.wikipedia.org/wiki/Charles%20George%20Broyden | Charles George Broyden (3 February 1933 – 20 May 2011) was a mathematician who specialized in optimization problems and numerical linear algebra. While a physicist working at English Electric Company from 1961–1965, he adapted the Davidon–Fletcher–Powell formula to solving some nonlinear systems of equations that he was working with, leading to his widely cited 1965 paper, "A class of methods for solving nonlinear simultaneous equations". He was a lecturer at UCW Aberystwyth from 1965–1967. He later became a senior lecturer at University of Essex from 1967–1970, where he independently discovered the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. The BFGS method has then become a key technique in solving nonlinear optimization problems. Moreover, he was among those who derived the symmetric rank-one updating formula, and his name was also attributed to Broyden's methods and Broyden family of quasi-Newton methods. After leaving the University of Essex, he continued his research career in the Netherlands and Italy, being awarded the chair at University of Bologna. In later years, he began focusing on numerical linear algebra, in particular conjugate gradient methods and their taxonomy.
Broyden died from complications of a severe stroke at the age of 78. He was survived by his wife, Joan, and their three children Chris, Jane and Nick.
A Charles Broyden Prize was established in 2009 to "honor this remarkable researcher" by Optimization Methods and Software in the international optimization community.
See also
Broyden's method
BFGS method
ABS methods
References
20th-century British mathematicians
21st-century British mathematicians
People educated at Newport Free Grammar School
1933 births
2011 deaths |
https://en.wikipedia.org/wiki/Nil-Coxeter%20algebra | In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.
Definition
The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u1, u2, u3, ... with the relations
These are just the relations for the infinite braid group, together with the relations u = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u = 0 to the relations of the corresponding generalized braid group.
References
Representation theory |
https://en.wikipedia.org/wiki/Extranatural%20transformation | In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
Definition
Let and be two functors of categories.
A family is said to be natural in a and extranatural in b and c if the following holds:
is a natural transformation (in the usual sense).
(extranaturality in b) , , the following diagram commutes
(extranaturality in c) , , the following diagram commutes
Properties
Extranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting (dually ) constant.
Extranatural transformations can be defined in terms of dinatural transformations, of which they are a special case.
See also
Dinatural transformation
References
External links
Higher category theory |
https://en.wikipedia.org/wiki/Vexillary%20permutation | In mathematics, a vexillary permutation is a permutation μ of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers i < j < k < l with μ(j) < μ(i) < μ(l) < μ(k). They were introduced by . The word "vexillary" means flag-like, and comes from the fact that vexillary permutations are related to flags of modules.
showed that vexillary involutions are enumerated by Motzkin numbers.
See also
Riffle shuffle permutation, a subclass of the vexillary permutations
References
Permutation patterns |
https://en.wikipedia.org/wiki/N-ellipse | In geometry, the -ellipse is a generalization of the ellipse allowing more than two foci. -ellipses go by numerous other names, including multifocal ellipse, polyellipse, egglipse, -ellipse, and Tschirnhaus'sche Eikurve (after Ehrenfried Walther von Tschirnhaus). They were first investigated by James Clerk Maxwell in 1846.
Given focal points in a plane, an -ellipse is the locus of points of the plane whose sum of distances to the foci is a constant . In formulas, this is the set
The 1-ellipse is the circle, and the 2-ellipse is the classic ellipse. Both are algebraic curves of degree 2.
For any number of foci, the -ellipse is a closed, convex curve. The curve is smooth unless it goes through a focus.
The n-ellipse is in general a subset of the points satisfying a particular algebraic equation. If n is odd, the algebraic degree of the curve is , while if n is even the degree is
n-ellipses are special cases of spectrahedra.
See also
Generalized conic
Geometric median
References
Further reading
P.L. Rosin: "On the Construction of Ovals"
B. Sturmfels: "The Geometry of Semidefinite Programming", pp. 9–16.
Conic sections
Algebraic curves |
https://en.wikipedia.org/wiki/Circle%20packing%20in%20an%20equilateral%20triangle | Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack unit circles into the smallest possible equilateral triangle. Optimal solutions are known for and for any triangular number of circles, and conjectures are available for .
A conjecture of Paul Erdős and Norman Oler states that, if is a triangular number, then the optimal packings of and of circles have the same side length: that is, according to the conjecture, an optimal packing for circles can be found by removing any single circle from the optimal hexagonal packing of circles. This conjecture is now known to be true for .
Minimum solutions for the side length of the triangle:
A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.
See also
Circle packing in an isosceles right triangle
Malfatti circles, a construction giving the optimal solution for three circles in an equilateral triangle
References
Circle packing |
https://en.wikipedia.org/wiki/South%20Korea%20national%20under-23%20football%20team%20results | This article is the match statistics of the South Korea national under-23 football team.
Results by year
1990s
2000s
2010s
2020s
Non-international matches
The following matches are non-international matches against clubs, regional teams, and other KFA teams, but these are being included in player records of the KFA website.
See also
South Korea national under-23 football team
References
External links
South Korea U-23 (Olympic) Matches - Details 1991-1999 at Yansfield
South Korea U-23 (Olympic) Matches - Details 2000-2004 at Yansfield
Men's U-23 Squads & Results at KFA |
https://en.wikipedia.org/wiki/Javi%20L%C3%B3pez%20%28footballer%2C%20born%201964%29 | Francisco Javier 'Javi' López Castro (born 3 March 1964 in Barcelona, Catalonia) is a Spanish retired footballer who played as a central defender, currently a manager.
Managerial statistics
References
External links
1964 births
Living people
Spanish men's footballers
Footballers from Barcelona
Men's association football defenders
Segunda División players
Segunda División B players
Tercera División players
CF Damm players
RCD Espanyol B footballers
CD Masnou players
CF Gandía players
CP Mérida footballers
Yeclano CF players
Villarreal CF players
Racing de Ferrol footballers
Spanish football managers
Segunda División managers
Segunda División B managers
Ciudad de Murcia managers
CD Castellón managers
UD Salamanca managers
Gimnàstic de Tarragona managers
Deportivo Alavés managers
Recreativo de Huelva managers
Xerez CD managers
FC Cartagena managers
Girona FC managers
RC Celta Fortuna managers
CD Lugo managers |
https://en.wikipedia.org/wiki/Cantellated%207-cubes | In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.
There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.
Cantellated 7-cube
Alternate names
Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)
Images
Bicantellated 7-cube
Alternate names
Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)
Images
Tricantellated 7-cube
Alternate names
Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)
Images
Cantitruncated 7-cube
Alternate names
Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)
Images
It is fifth in a series of cantitruncated hypercubes:
Bicantitruncated 7-cube
Alternate names
Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)
Images
Tricantitruncated 7-cube
Alternate names
Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)
Images
Related polytopes
These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.
See also
List of B7 polytopes
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Chastity%20Reed | Chastity Reed (born March 28, 1989) is an American professional women's basketball player who last played for Panathinaikos in the Greek League.
Arkansas–Little Rock statistics
Source
WNBA
Reed was selected in the third round of the 2011 WNBA draft (25th overall) by the Tulsa Shock. Reed was with the Shock for thirteen games, playing in eleven, including one start, before being waived by interim coach and GM Teresa Edwards one week after she replaced Nolan Richardson. For the 2012 WNBA season, Reed is in the training camp of the Phoenix Mercury.
References
1989 births
Living people
American women's basketball players
Basketball players from New Orleans
Little Rock Trojans women's basketball players
Tulsa Shock draft picks
Tulsa Shock players
Forwards (basketball)
Panathinaikos WBC players |
https://en.wikipedia.org/wiki/Michael%20Harris%20%28mathematician%29 | Michael Howard Harris (born 1954) is an American mathematician known for his work in number theory. He is a professor of mathematics at Columbia University and professor emeritus of mathematics at Université Paris Cité.
Early life and education
Harris was born in Kingsessing, Philadelphia, Pennsylvannia and is of Jewish descent. He received his B.A. in mathematics from Princeton University in 1973. He received his M.A. and Ph.D. in mathematics from Harvard University under the supervision of Barry Mazur in 1976 and 1977 respectively.
Career
Harris was a faculty member at Brandeis University from 1977 to 1994. In 1994, he became a professor of mathematics at Paris Diderot University and the Institut de mathématiques de Jussieu – Paris Rive Gauche, where he has been emeritus since 2021. He became a professor of mathematics at Columbia University in 2013.
He was a member of the Institute for Advanced Study from 1983 to 1984 and in the fall of 2011. He has held visiting positions at various institutions, including Bethlehem University, the Steklov Institute of Mathematics, the Institut des Hautes Études Scientifiques, Oxford University, and the Mathematical Sciences Research Institute.
His former doctoral students include Laurent Fargues and Gaëtan Chenevier.
Work
Research
Harris's research focuses on arithmetic geometry, automorphic forms, L-functions, and motives. He has developed the theory of coherent cohomology of Shimura varieties and applied it to number theoretic problems on special values of L-functions, Galois representations, and the theta correspondence. His later work focuses on geometric aspects of the Langlands program.
In 2001, Harris and Richard Taylor proved the local Langlands conjecture for GL(n) over a p-adic local field
The Sato–Tate conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011.
Mathematics without Apologies
Harris wrote the book Mathematics without Apologies: Portrait of a Problematic Vocation, published in 2015.
Silicon Reckoner
Since 2021, Harris has written the newsletter Silicon Reckoner exploring questions and issues related to the mechanization of mathematics and artificial intelligence.
Recognition
Harris received the Sophie Germain Prize (2006), the Clay Research Award (joint with Richard Taylor, 2007), the Grand Prix Scientifique de la Fondation Simone et Cino del Duca (2009), He is a three-time invited speaker at the International Congress of Mathematicians (2000, 2002, 2014).
He was a Sloan Research Fellow (1983–1985) and a member of the Institut Universitaire de France (2001–2011) He has been elected a Member of the Academia Europaea (2016), Fellow of the American Mathematical Society (2019),, Member of the American Academy of Arts and Sciences (2019), and Member of the National Academy of Science |
https://en.wikipedia.org/wiki/Cantellated%207-orthoplexes | In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.
There are ten degrees of cantellation for the 7-orthoplex, including truncations. Six are most simply constructible from the dual 7-cube.
Cantellated 7-orthoplex
Alternate names
Small rhombated hecatonicosoctaexon (acronym: sarz) (Jonathan Bowers)
Images
Bicantellated 7-orthoplex
Alternate names
Small birhombated hecatonicosoctaexon (acronym: sebraz) (Jonathan Bowers)
Images
Cantitruncated 7-orthoplex
Alternate names
Great rhombated hecatonicosoctaexon (acronym: garz) (Jonathan Bowers)
Images
Bicantitruncated 7-orthoplex
Alternate names
Great birhombated hecatonicosoctaexon (acronym: gebraz) (Jonathan Bowers)
Images
Related polytopes
These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.
See also
List of B7 polytopes
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. 1966
- o3o3o3o3x3o4x - sarz, o3o3o3x3o3x4o - sebraz, o3o3o3o3x3x4x - garz, o3o3o3x3x3x4o - gebraz
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Runcic%207-cubes | In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.
Runcic 7-cube
A runcic 7-cube, h3{4,35}, has half the vertices of a runcinated 7-cube, t0,3{4,35}.
Alternate names
Small rhombated hemihepteract (Acronym sirhesa) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:
(±1,±1,±1,±3,±3,±3,±3)
with an odd number of plus signs.
Images
Runcicantic 7-cube
A runcicantic 7-cube, h2,3{4,35}, has half the vertices of a runcicantellated 7-cube, t0,1,3{4,35}.
Alternate names
Great rhombated hemihepteract (Acronym girhesa) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:
(±1,±1,±1,±1,±3,±5,±5)
with an odd number of plus signs.
Images
Related polytopes
This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 95 uniform polytopes with D7 symmetry, 63 are shared by the BC6 symmetry, and 32 are unique:
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3o *b3x3o3o3o - sirhesa, x3x3o *b3x3o3o3o - girhesa
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/An%20Introduction%20to%20the%20Theory%20of%20Numbers | An Introduction to the Theory of Numbers is a classic textbook in the field of number theory, by G. H. Hardy and E. M. Wright.
The book grew out of a series of lectures by Hardy and Wright and was first published in 1938.
The third edition added an elementary proof of the prime number theorem, and the sixth edition added a chapter on elliptic curves.
See also
List of important publications in mathematics
References
Mathematics textbooks
Number theory
1938 non-fiction books |
https://en.wikipedia.org/wiki/Ify%20Ibekwe | Ifunanya Debbie "Ify" Ibekwe (born 5 October 1989) is a Nigerian American professional basketball player for the Virtus Eirene Ragusa and the Nigeria women's national team.
Arizona statistics
WNBA
Ibekwe was selected in the second round of the 2011 WNBA draft (24th overall) by the Seattle Storm.
National Team Career
Ify represents the Nigerian Women's National Team.
Personal
Ibekwe is the daughter of Nigerian parents, Agatha and Augustine Ibekwe. She has two brothers who played college basketball, Onye Ibekwe played for Long Beach State and Ekene Ibekwe played for the University of Maryland. She also has one sister, Chinyere.
References
External links
Arizona Wildcats player profile
1989 births
Living people
American expatriate basketball people in New Zealand
American sportspeople of Nigerian descent
American women's basketball players
Arizona Wildcats women's basketball players
Basketball players at the 2020 Summer Olympics
Los Angeles Sparks players
Nigerian women's basketball players
Olympic basketball players for Nigeria
Sportspeople from Carson, California
Basketball players from Los Angeles County, California
Power forwards (basketball)
Seattle Storm draft picks
Seattle Storm players
Small forwards
South East Queensland Stars players
Citizens of Nigeria through descent
African-American basketball players
Nigerian people of African-American descent
American emigrants to Nigeria
21st-century African-American sportspeople
Narbonne High School alumni |
https://en.wikipedia.org/wiki/SIAM%20Journal%20on%20Numerical%20Analysis | The SIAM Journal on Numerical Analysis (SINUM; until 1965: Journal of the Society for Industrial & Applied Mathematics, Series B: Numerical Analysis) is a peer-reviewed mathematical journal published by the Society for Industrial and Applied Mathematics that covers research on the analysis of numerical methods. The journal was established in 1964 and appears bimonthly. The editor-in-chief is Angela Kunoth.
References
External links
Numerical Analysis
Mathematics journals
Bimonthly journals
Academic journals established in 1964
English-language journals |
https://en.wikipedia.org/wiki/Runcinated%207-cubes | In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube.
There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex.
These polytopes are among 127 uniform 7-polytopes with B7 symmetry.
Runcinated 7-cube
Alternate names
Small prismated hepteract (acronym: spesa) (Jonathan Bowers)
Images
Biruncinated 7-cube
Alternate names
Small biprismated hepteract (Acronym sibposa) (Jonathan Bowers)
Images
Runcitruncated 7-cube
Alternate names
Prismatotruncated hepteract (acronym: petsa) (Jonathan Bowers)
Images
Biruncitruncated 7-cube
Alternate names
Biprismatotruncated hepteract (acronym: biptesa) (Jonathan Bowers)
Images
Runcicantellated 7-cube
Alternate names
Prismatorhombated hepteract (acronym: parsa) (Jonathan Bowers)
Images
Biruncicantellated 7-cube
Alternate names
Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)
Images
Runcicantitruncated 7-cube
Alternate names
Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)
Images
Biruncicantitruncated 7-cube
Alternate names
Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
x3o3o3x3o3o4o - spo, o3x3o3o3x3o4o - sibpo, x3x3o3x3o3o4o - patto, o3x3x3o3x3o4o - bipto, x3o3x3x3o3o4o - paro, x3x3x3x3o3o4o - gapo, o3x3x3x3x3o3o- gibpo
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Stericated%207-cubes | In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-cube.
There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations. 10 are more simply constructed from the 7-orthoplex.
This polytope is one of 127 uniform 7-polytopes with B7 symmetry.
Stericated 7-cube
Alternate names
Small cellated hepteract (acronym: ) (Jonathan Bowers)
Images
Bistericated 7-cube
Alternate names
Small bicellated hepteractihecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steritruncated 7-cube
Alternate names
Cellitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Bisteritruncated 7-cube
Alternate names
Bicellitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Stericantellated 7-cube
Alternate names
Cellirhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Bistericantellated 7-cube
Alternate names
Bicellirhombihepteract (acronym: ) (Jonathan Bowers)
Images
Stericantitruncated 7-cube
Alternate names
Celligreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Bistericantitruncated 7-cube
Alternate names
Bicelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Steriruncinated 7-cube
Alternate names
Celliprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Steriruncitruncated 7-cube
Alternate names
Celliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Steriruncicantellated 7-cube
Alternate names
Celliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Bisteriruncitruncated 7-cube
Alternate names
Bicelliprismatotruncated hepteractihecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steriruncicantitruncated 7-cube
Alternate names
Great cellated hepteract (acronym: ) (Jonathan Bowers)
Images
Bisteriruncicantitruncated 7-cube
Alternate names
Great bicellated hepteractihecatonicosoctaexon (Acronym ) (Jonathan Bowers)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3o3o3o3x3o4o - , x3o3x3o3x3o4o - , x3x3o3o3x3o4o - , o3x3x3o3o3x4o - , x3o3x3o3x3o4o - , o3x3o3x3o3x4o - , x3x3x3o3x3o4o - , o3x3x3x3o3x4o - , x3o3o3x3x3o4o - , x3x3x3o3x3o4o - , x3o3x3x3x3o4o - , o3x3x3o3x3x4o - , x3x3x3x3x3o4o - , o3x3x3x3x3x4o -
External links
Polytopes of Various Di |
https://en.wikipedia.org/wiki/Pentellated%207-cubes | In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique of the 7-cube with permutations of truncations, cantellations, runcinations, and . 16 are more simply constructed relative to the 7-orthoplex.
Pentellated 7-cube
Alternate names
Small hepteract (acronym:) (Jonathan Bowers)
Images
Pentitruncated 7-cube
Alternate names
Teritruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Penticantellated 7-cube
Alternate names
Terirhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Penticantitruncated 7-cube
Alternate names
Terigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Pentiruncinated 7-cube
Alternate names
Teriprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentiruncitruncated 7-cube
Alternate names
Teriprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentiruncicantellated 7-cube
Alternate names
Teriprismatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentiruncicantitruncated 7-cube
Alternate names
Terigreatoprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentistericated 7-cube
Alternate names
Tericellated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentisteritruncated 7-cube
Alternate names
Tericellitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentistericantellated 7-cube
Alternate names
Tericellirhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentistericantitruncated 7-cube
Alternate names
Tericelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentisteriruncinated 7-cube
Alternate names
Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
Tericelliprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentisteriruncitruncated 7-cube
Alternate names
Tericelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentisteriruncicantellated 7-cube
Alternate names
Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
Tericelliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Pentisteriruncicantitruncated 7-cube
Alternate names
Great hepteract (acronym:) (Jonathan Bowers)
Images
Related polytopes
These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript |
https://en.wikipedia.org/wiki/Hexicated%207-cubes | In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-cube.
There are 32 hexications for the 7-cube, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 20 are represented here, while 12 are more easily constructed from the 7-orthoplex.
The simple hexicated 7-cube is also called an expanded 7-cube, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-cube. The highest form, the hexipentisteriruncicantitruncated 7-cube is more simply called a omnitruncated 7-cube with all of the nodes ringed.
These polytope are among a family of 127 uniform 7-polytopes with B7 symmetry.
Hexicated 7-cube
In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-cube, or alternately can be seen as an expansion operation.
Alternate names
Small petated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexitruncated 7-cube
Alternate names
Petitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexicantellated 7-cube
Alternate names
Petirhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexiruncinated 7-cube
Alternate names
Petiprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexicantitruncated 7-cube
Alternate names
Petigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexiruncitruncated 7-cube
Alternate names
Petiprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexiruncicantellated 7-cube
In seven-dimensional geometry, a hexiruncicantellated 7-cube is a uniform 7-polytope.
Alternate names
Petiprismatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexisteritruncated 7-cube
Alternate names
Peticellitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexistericantellated 7-cube
Alternate names
Peticellirhombihepteract (acronym: ) (Jonathan Bowers)
Images
Hexipentitruncated 7-cube
Alternate names
Petiteritruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexiruncicantitruncated 7-cube
Alternate names
Petigreatoprismated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexistericantitruncated 7-cube
Alternate names
Peticelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexisteriruncitruncated 7-cube
Alternate names
Peticelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexisteriruncicantellated 7-cube
Alternate names
Peticelliprismatorhombihepteract (acronym: ) (Jonathan Bowers)
Images
Hexipenticantitruncated 7-cube
Alternate names
Petiterigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexipentiruncitruncated 7-cube
Alternate names
Great petacellated hepteract (acronym: ) (Jonathan Bowers)
Images
Hexisteriruncicantitruncated 7-cube
Alternate names
Great petacellated hepteract (acronym: ) (Jonathan Bowers)
Images |
https://en.wikipedia.org/wiki/Backpressure%20routing | In queueing theory, a discipline within the mathematical theory of probability, the backpressure routing algorithm is a method for directing traffic around a queueing network that achieves maximum network throughput, which is established using concepts of Lyapunov drift. Backpressure routing considers the situation where each job can visit multiple service nodes in the network. It is an extension of max-weight scheduling where each job visits only a single service node.
Introduction
Backpressure routing is an algorithm for dynamically routing traffic over a multi-hop network by using congestion gradients. The algorithm can be applied to wireless communication networks, including sensor networks, mobile ad hoc networks (MANETS), and heterogeneous networks with wireless and wireline components.
Backpressure principles can also be applied to other areas, such as to the study of
product assembly systems and processing networks.
This article focuses on communication networks,
where packets from multiple data streams arrive and
must be delivered to appropriate destinations. The backpressure
algorithm operates in slotted time. Every time slot it seeks to route data in directions that
maximize the differential backlog between neighboring nodes. This is similar to how water
flows through a network of pipes via pressure gradients. However, the backpressure algorithm
can be applied to multi-commodity networks (where different packets may have different destinations),
and to networks where transmission rates can be selected
from a set of (possibly time-varying) options. Attractive features
of the backpressure algorithm are: (i) it leads to maximum network throughput, (ii)
it is provably robust to time-varying network conditions, (iii) it
can be implemented without knowing traffic arrival rates or channel state
probabilities. However, the algorithm may introduce large delays, and may
be difficult to implement exactly in networks with interference. Modifications of
backpressure that reduce delay and simplify implementation are described below
under Improving delay and Distributed backpressure.
Backpressure routing has mainly been studied in a theoretical
context. In practice, ad hoc wireless networks have typically
implemented alternative routing methods based on shortest
path computations or network flooding, such as
Ad Hoc on-Demand Distance Vector Routing (AODV),
geographic routing, and extremely opportunistic routing (ExOR).
However, the mathematical optimality properties of backpressure
have motivated recent experimental demonstrations of its use
on wireless testbeds at the University of Southern California
and at North Carolina State University.
Origins
The original backpressure algorithm was developed by Tassiulas and Ephremides. They considered a multi-hop packet radio network with random packet arrivals and a fixed set of link selection options. Their algorithm consisted of a max-weight link selection stage and a differential backlog ro |
https://en.wikipedia.org/wiki/Synopsis%20of%20Pure%20Mathematics | Synopsis of Pure Mathematics is a book by G. S. Carr, written in 1886. The book attempted to summarize the state of most of the basic mathematics known at the time.
The book is noteworthy because it was a major source of information for the legendary and self-taught mathematician Srinivasa Ramanujan who managed to obtain a library loaned copy from a friend in 1903. Ramanujan reportedly studied the contents of the book in detail. The book is generally acknowledged as a key element in awakening the genius of Ramanujan.
Carr acknowledged the main sources of his book in its preface:
Bibliography
References
External links
- archive.org
- archive.org
- rarebooksocietyofindia.org
Mathematics books |
https://en.wikipedia.org/wiki/George%20Hall%20%28academic%29 | George Hall (1753 – 23 November 1811) was an academic at Trinity College Dublin, who served as the fourth Erasmus Smith's Professor of Mathematics from 1799 to 1800, as Provost of the college from 1806 to 1811, and the Church of Ireland Bishop of Dromore for a few days before his death in 1811.
Life
Son of the Rev. Mark Hall, of Northumberland, he was born there, but soon thereafter his family moved to in Ireland. His first employment was as an assistant-master in Dr. Darby's school near Dublin. Having entered Trinity College Dublin, 1 November 1770, under the tutorship of the Rev. Gerald Fitzgerald, he was elected a scholar in 1773; he graduated B.A. 1775, M.A. 1778, B.D. 1786, and D.D. 1790. He was a successful candidate for a fellowship in 1777, and on 14 May 1790 he was co-opted a senior fellow. Along with his fellowship he filled various academical offices, being elected Archbishop King's lecturer in divinity 1790–1, regius professor of Greek 1790 and 1795, Erasmus Smith's Professor of Modern History 1791, and Erasmus Smith's Professor of Mathematics 1799.
Hall resigned his fellowship and professorship in 1800, and on 25 February of that year was presented by his college to the rectory of Ardstraw in the diocese of Derry. In 1806 he returned to Trinity College, having been appointed to the provostship by patent dated 22 January, and held that office until his promotion, on 13 November 1811, to the bishopric of Dromore. He was consecrated in the college chapel on the 17th of the same month, but died on the 23rd in the provost's house, from which he had not had time to move. He was buried in the college chapel, where a monument with a Latin inscription to his memory was erected by his niece, Margaret Stack. There was another memorial to him, in the parish church of Ardstraw.
References
Attribution
1753 births
1811 deaths
Anglican bishops of Dromore
Fellows of Trinity College Dublin
Provosts of Trinity College Dublin
People from Northumberland
18th-century Irish Anglican priests
19th-century Irish Anglican priests |
https://en.wikipedia.org/wiki/Rosen%20Vankov | Rosen Vankov (; born 21 March 1985) is a Bulgarian football defender who currently plays for OFC Etar. He had previously played for Etar 1924 and Botev Vratsa.
Career statistics
As of 1 August 2014
References
External links
1985 births
Living people
Bulgarian men's footballers
FC Etar 1924 Veliko Tarnovo players
POFC Botev Vratsa players
First Professional Football League (Bulgaria) players
Men's association football defenders |
https://en.wikipedia.org/wiki/Amir%20Khalifeh-Asl | Amir Khalifeh-Asl (; born 5 May 1979) is an Iranian football player.
Club career
He served his golden days in Esteghlal Ahvaz.
Club career statistics
References
External links
Persian League Profile
1979 births
Living people
Iranian men's footballers
Men's association football forwards
Foolad F.C. players
Fajr Sepasi Shiraz F.C. players
Esteghlal Ahvaz F.C. players
Persian Gulf Pro League players
Azadegan League players
People from Khorramshahr |
https://en.wikipedia.org/wiki/Runcinated%207-orthoplexes | In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-orthoplex.
There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-cube.
These polytopes are among 127 uniform 7-polytopes with B7 symmetry.
Runcinated 7-orthoplex
Alternate names
Small prismated hecatonicosoctaexon (acronym: spaz) (Jonathan Bowers)
Images
Biruncinated 7-orthoplex
Alternate names
Small biprismated hecatonicosoctaexon (Acronym sibpaz) (Jonathan Bowers)
Images
Runcitruncated 7-orthoplex
Alternate names
Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)
Images
Biruncitruncated 7-orthoplex
Alternate names
Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (Jonathan Bowers)
Images
Runcicantellated 7-orthoplex
Alternate names
Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)
Images
Biruncicantellated 7-orthoplex
Alternate names
Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (Jonathan Bowers)
Images
Runcicantitruncated 7-orthoplex
Alternate names
Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)
Images
Biruncicantitruncated 7-orthoplex
Alternate names
Great biprismated hecatonicosoctaexon (acronym: gibpaz) (Jonathan Bowers)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/John%20Wesley%20Young | John Wesley Young (17 November 1879, Columbus, Ohio – 17 February 1932, Hanover, New Hampshire) was an American mathematician who, with Oswald Veblen, introduced the axioms of projective geometry, coauthored a 2-volume work on them, and proved the Veblen–Young theorem. He was a proponent of Euclidean geometry and held it to be substantially "more convenient to employ" than non-Euclidean geometry. His lectures on algebra and geometry were compiled in 1911 and released as Lectures on Fundamental Concepts of Algebra and Geometry.
John Wesley Young was born in 1879 to William Henry Young and Marie Louise Widenhorn Young. William Henry Young was from West Virginia and was of Native American parentage. After serving in the Civil War he was appointed to the consulate in Germany. He taught Mathematics at Ohio University. Marie Louise Widehorn Young was born in Paris, France and spoke French and German fluently.
John Wesley grew up in both Europe and America due to his father's profession and attended schools in Baden Baden and Karlsruhe, Germany and Columbus, Ohio. Young was awarded a Master's degree in Mathematics from Cornell University in 1903.
John Wesley Young was married to Mary Louise Aston on July 20, 1907. They had one daughter, Mary Elizabeth, later Mrs. Allyn.
Between 1903 and 1911, Young held positions at Northwestern University, Princeton University, the University of Illinois, the University of Kansas, and the University of Chicago. He was head of the department of Mathematics at Dartmouth College from 1911 to 1919 and chair of the department from 1923 to 1925. He continued teaching until two days before he died.
Publications
Projective geometry with Oswald Veblen, Ginn and co., 1910–1918
Lectures on Fundamental Concepts of Algebra and Geometry 1911
References
John Wesley Young at MAA website
19th-century American mathematicians
Dartmouth College faculty
Presidents of the Mathematical Association of America
1879 births
1932 deaths
American people of German descent
American people of French descent
Philosophers of mathematics
History of geometry |
https://en.wikipedia.org/wiki/Stericated%207-orthoplexes | In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-orthoplex.
There are 24 unique sterication for the 7-orthoplex with permutations of truncations, cantellations, and runcinations. 14 are more simply constructed from the 7-cube.
This polytope is one of 127 uniform 7-polytopes with B7 symmetry.
Stericated 7-orthoplex
Alternate names
Small cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steritruncated 7-orthoplex
Alternate names
Cellitruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Bisteritruncated 7-orthoplex
Alternate names
Bicellitruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Stericantellated 7-orthoplex
Alternate names
Cellirhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Stericantitruncated 7-orthoplex
Alternate names
Celligreatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Bistericantitruncated 7-orthoplex
Alternate names
Bicelligreatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steriruncinated 7-orthoplex
Alternate names
Celliprismated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steriruncitruncated 7-orthoplex
Alternate names
Celliprismatotruncated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steriruncicantellated 7-orthoplex
Alternate names
Celliprismatorhombated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Steriruncicantitruncated 7-orthoplex
Alternate names
Great cellated hecatonicosoctaexon (acronym: ) (Jonathan Bowers)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Pentellated%207-orthoplexes | In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex.
There are 32 unique of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and . 16 are more simply constructed relative to the 7-cube.
These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.
Pentellated 7-orthoplex
Alternate names
Small hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,1,1,1,2)
Images
Pentitruncated 7-orthoplex
Alternate names
Teritruncated hecatonicosoctaexon (acronym: Tetaz) (Jonathan Bowers)
Images
Coordinates
Coordinates are permutations of (0,1,1,1,1,2,3).
Penticantellated 7-orthoplex
Alternate names
Terirhombated hecatonicosoctaexon (acronym: Teroz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,1,2,2,3).
Images
Penticantitruncated 7-orthoplex
Alternate names
Terigreatorhombated hecatonicosoctaexon (acronym: Tograz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,1,2,3,4).
Pentiruncinated 7-orthoplex
Alternate names
Teriprismated hecatonicosoctaexon (acronym: Topaz) (Jonathan Bowers)
Coordinates
The coordinates are permutations of (0,1,1,2,2,2,3).
Images
Pentiruncitruncated 7-orthoplex
Alternate names
Teriprismatotruncated hecatonicosoctaexon (acronym: Toptaz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,2,2,3,4).
Images
Pentiruncicantellated 7-orthoplex
Alternate names
Teriprismatorhombated hecatonicosoctaexon (acronym: Toparz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,2,3,3,4).
Images
Pentiruncicantitruncated 7-orthoplex
Alternate names
Terigreatoprismated hecatonicosoctaexon (acronym: Tegopaz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,1,2,3,4,5).
Images
Pentistericated 7-orthoplex
Alternate names
Tericellated hecatonicosoctaexon (acronym: Tocaz) (Jonathan Bowers)
Images
Coordinates
Coordinates are permutations of (0,1,2,2,2,2,3).
Pentisteritruncated 7-orthoplex
Alternate names
Tericellitruncated hecatonicosoctaexon (acronym: Tacotaz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,2,2,2,3,4).
Images
Pentistericantellated 7-orthoplex
Alternate names
Tericellirhombated hecatonicosoctaexon (acronym: Tocarz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,2,2,3,3,4).
Images
Pentistericantitruncated 7-orthoplex
Alternate names
Tericelligreatorhombated hecatonicosoctaexon (acronym: Tecagraz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,2,2,3,4,5).
Images
Pentisteriruncinated 7-orthoplex
Alternate names
Bipenticantitruncated 7-orthoplex as t1,2,3,6{35,4}
Tericelliprismated hecatonicosoctaexon (acronym: Tecpaz) (Jonathan Bowers)
Coordinates
Coordinates are permutations of (0,1,2,3,3,3,4).
Images
Pentisteriruncitruncated |
https://en.wikipedia.org/wiki/Life-time%20of%20correlation | In probability theory and related fields, the life-time of correlation measures the timespan over which there is appreciable autocorrelation or cross-correlation in stochastic processes.
Definition
The correlation coefficient ρ, expressed as an autocorrelation function or cross-correlation function, depends on the lag-time between the times being considered. Typically such functions, ρ(t), decay to zero with increasing lag-time, but they can assume values across all levels of correlations: strong and weak, and positive and negative as in the table.
The life-time of a correlation is defined as the length of time when the correlation coefficient is at the strong level. The durability of correlation is determined by signal (the strong level of correlation is separated from weak and negative levels). The mean life-time of correlation could measure how the durability of correlation depends on the window width size (the window is the length of time series used to calculate correlation).
References
Stochastic processes
Machine learning |
https://en.wikipedia.org/wiki/Wiener%20amalgam%20space | In mathematics, amalgam spaces categorize functions with regard to their local and global behavior. While the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgams, as the term is used today, were introduced by Hans Georg Feichtinger in 1980. The concept is named after Norbert Wiener.
Let be a normed space with norm . Then the Wiener amalgam space with local component and
global component , a weighted space with non-negative weight , is defined by
where is a continuously differentiable, compactly supported function, such that , for all . Again, the space defined is independent of . As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.
References
Function spaces |
https://en.wikipedia.org/wiki/Veblen%E2%80%93Young%20theorem | In mathematics, the Veblen–Young theorem, proved by , states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring.
Non-Desarguesian planes give examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary.
Jacques Tits generalized the Veblen–Young theorem to Tits buildings, showing that those of rank at least 3 arise from algebraic groups.
generalized the Veblen–Young theorem to continuous geometry, showing that a complemented modular lattice of order at least 4 is isomorphic to the principal right ideals of a von Neumann regular ring.
Statement
A projective space S can be defined abstractly as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms :
Each two distinct points p and q are in exactly one line.
Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.
Any line has at least 3 points on it.
The Veblen–Young theorem states that if the dimension of a projective space is at least 3 (meaning that there are two non-intersecting lines) then the projective space is isomorphic with the projective space of lines in a vector space over some division ring K.
References
Theorems in projective geometry
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Stanley%20symmetric%20function | In mathematics and especially in algebraic combinatorics, the Stanley symmetric functions are a family of symmetric functions introduced by in his study of the symmetric group of permutations.
Formally, the Stanley symmetric function Fw(x1, x2, ...) indexed by a permutation w is defined as a sum of certain fundamental quasisymmetric functions. Each summand corresponds to a reduced decomposition of w, that is, to a way of writing w as a product of a minimal possible number of adjacent transpositions. They were introduced in the course of Stanley's enumeration of the reduced decompositions of permutations, and in particular his proof that the permutation w0 = n(n − 1)...21 (written here in one-line notation) has exactly
reduced decompositions. (Here denotes the binomial coefficient n(n − 1)/2 and ! denotes the factorial.)
Properties
The Stanley symmetric function Fw is homogeneous with degree equal to the number of inversions of w. Unlike other nice families of symmetric functions, the Stanley symmetric functions have many linear dependencies and so do not form a basis of the ring of symmetric functions. When a Stanley symmetric function is expanded in the basis of Schur functions, the coefficients are all non-negative integers.
The Stanley symmetric functions have the property that they are the stable limit of Schubert polynomials
where we treat both sides as formal power series, and take the limit coefficientwise.
References
Polynomials
Symmetric functions |
https://en.wikipedia.org/wiki/Integral%20of%20the%20secant%20function | In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
This formula is useful for evaluating various trigonometric integrals. In particular, it can be used to evaluate the integral of the secant cubed, which, though seemingly special, comes up rather frequently in applications.
The definite integral of the secant function starting from is the inverse Gudermannian function, For numerical applications, all of the above expressions result in loss of significance for some arguments. An alternative expression in terms of the inverse hyperbolic sine is numerically well behaved for real arguments
The integral of the secant function was historically one of the first integrals of its type ever evaluated, before most of the development of integral calculus. It is important because it is the vertical coordinate of the Mercator projection, used for marine navigation with constant compass bearing.
Proof that the different antiderivatives are equivalent
Trigonometric forms
Three common expressions for the integral of the secant,
are equivalent because
Proof: we can separately apply the tangent half-angle substitution to each of the three forms, and show them equivalent to the same expression in terms of Under this substitution and
First,
Second,
Third, using the tangent addition identity
So all three expressions describe the same quantity.
The conventional solution for the Mercator projection ordinate may be written without the absolute value signs since the latitude lies between and ,
Hyperbolic forms
Let
Therefore,
History
The integral of the secant function was one of the "outstanding open problems of the mid-seventeenth century", solved in 1668 by James Gregory. He applied his result to a problem concerning nautical tables. In 1599, Edward Wright evaluated the integral by numerical methods – what today we would call Riemann sums. He wanted the solution for the purposes of cartography – specifically for constructing an accurate Mercator projection. In the 1640s, Henry Bond, a teacher of navigation, surveying, and other mathematical topics, compared Wright's numerically computed table of values of the integral of the secant with a table of logarithms of the tangent function, and consequently conjectured that
This conjecture became widely known, and in 1665, Isaac Newton was aware of it.
Evaluations
By a standard substitution (Gregory's approach)
A standard method of evaluating the secant integral presented in various references involves multiplying the numerator and denominator by and then using the substitution . This substitution can be obtained from the derivatives of secant and tangent added together, which have secant as a common factor.
Starting with
adding them gives
The derivative of the sum is thus equal to the sum multiplied by . This enab |
https://en.wikipedia.org/wiki/Square%20trisection | In geometry, a square trisection is a type of dissection problem which consists of cutting a square into pieces that can be rearranged to form three identical squares.
History
The dissection of a square in three congruent partitions is a geometrical problem that dates back to the Islamic Golden Age. Craftsman who mastered the art of zellige needed innovative techniques to achieve their fabulous mosaics with complex geometric figures. The first solution to this problem was proposed in the 10th century AD by the Persian mathematician Abu'l-Wafa' (940-998) in his treatise "On the geometric constructions necessary for the artisan". Abu'l-Wafa' also used his dissection to demonstrate the Pythagorean theorem. This geometrical proof of Pythagoras' theorem would be rediscovered in the years 1835 - 1840 by Henry Perigal and published in 1875.
Search of optimality
The beauty of a dissection depends on several parameters. However, it is usual to search for solutions with the minimum number of parts. Far from being minimal, the square trisection proposed by Abu'l-Wafa' uses 9 pieces. In the 14th century Abu Bakr al-Khalil gave two solutions, one of which uses 8 pieces. In the late 17th century Jacques Ozanam came back to this issue and in the 19th century, solutions using 8 and 7 pieces were found, including one given by the mathematician Édouard Lucas. In 1891 Henry Perigal published the first known solution with only 6 pieces (see illustration below). Nowadays, new dissections are still found (see illustration above) and the conjecture that 6 is the minimal number of necessary pieces remains unproved.
See also
Proofs by dissection and rearrangement of Pythagorean theorem
Dissection puzzle
Tangram
Bibliography
References
External links
Greg N. Frederickson web site
Euclidean plane geometry
Mathematical problems
History of geometry
Area
Geometric dissection |
https://en.wikipedia.org/wiki/Wrapped%20exponential%20distribution | In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle.
Definition
The probability density function of the wrapped exponential distribution is
for where is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range .
Characteristic function
The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:
which yields an alternate expression for the wrapped exponential PDF in terms of the circular variable z=e i (θ-m) valid for all real θ and m:
where is the Lerch transcendent function.
Circular moments
In terms of the circular variable the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:
where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:
The mean angle is
and the length of the mean resultant is
and the variance is then 1-R.
Characterisation
The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range for a fixed value of the expectation .
See also
Wrapped distribution
Directional statistics
References
Continuous distributions
Directional statistics |
https://en.wikipedia.org/wiki/Nicholas%20Bingham | Nicholas Hugh Bingham (born 19 March 1945 in York) is a British mathematician working in the field of probability theory, stochastic analysis and analysis more generally.
Education and career
Bingham is currently a Senior Research Investigator at Imperial College London, and is a Visiting Professor at both the London School of Economics and the University of Liverpool.
After undergraduate studies in mathematics at Trinity College, Oxford, where he achieved a first class honours degree, he was a research student at Churchill College, Cambridge, where he obtained his PhD in 1969 under the supervision of David George Kendall. In 1996 he also obtained a ScD from the University of Cambridge.
He serves as Associate Editor of Expositiones Mathematicae and Obituaries Editor of the London Mathematical Society.
With C.M. Goldie and Jozef L. Teugels, Bingham wrote the book Regular Variation; with Rüdiger Kiesel Risk-neutral Valuation: Pricing and Hedging of Financial Derivatives; with J. M. Fry Regression.
Personal life
Bingham is married to Cecilie (m. 1980). They have 3 children: James (1982), Ruth (1985), and Tom (1993).
He is a competitive runner, with a best marathon time of 2:46:52 in the 1991 Abingdon Marathon, aged 46. He is a member of Barnet and District AC.
References
1945 births
Alumni of Trinity College, Oxford
Alumni of Churchill College, Cambridge
English mathematicians
Living people
Probability theorists
Mathematical statisticians
People from York
Academics of the London School of Economics
Academics of Imperial College London |
https://en.wikipedia.org/wiki/Truncated%2024-cell%20honeycomb | In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.
It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the construction. This truncated 24-cell has Schläfli symbol t{31,1,1,1}, and its snub is represented as s{31,1,1,1}.
Alternate names
Truncated icositetrachoric tetracomb
Truncated icositetrachoric honeycomb
Cantitruncated 16-cell honeycomb
Bicantitruncated tesseractic honeycomb
Symmetry constructions
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.
See also
Regular and uniform honeycombs in 4-space:
Tesseractic honeycomb
16-cell honeycomb
24-cell honeycomb
Rectified 24-cell honeycomb
Snub 24-cell honeycomb
5-cell honeycomb
Truncated 5-cell honeycomb
Omnitruncated 5-cell honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 99
o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99
5-polytopes
Honeycombs (geometry)
Truncated tilings |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Sporting%20CP%20season | The 2011–12 season is Sporting CP's 79th season in the top flight, the Primeira Liga, known as the Liga ZON Sagres for sponsorship purposes. This article shows player statistics and all matches (official and friendly) that the club plays during the 2011–12 season.
Sporting CP's under-19 squad played in the inaugural tournament of the NextGen series.
Season overview
Pre-season
Players
Squad information
Transfers
In
Total spending: €15.275 million
Out
Total income: €0 million
Club
Coaching staff
Kit
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Competitions
Pre-season
Last updated: 28 July 2011
Source: Sporting.pt, footballzz.co.uk
Primeira Liga
League table
Matches
Source: Sporting.pt, Journal Record
Taça da Liga
Third round
Source: Sporting.pt,
UEFA Europa League
Play-off round
Group stage
Knockout phase
Round of 32
Round of 16
Quarter-finals
Semi-finals
Taça de Portugal
Round of 64
Round of 32
Round of 16
Quarter-finals
Semi-finals
|}
Final
Overview
References
External links
Official club website
2011-12
Portuguese football clubs 2011–12 season
Sporting |
https://en.wikipedia.org/wiki/Kosmann%20lift | In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natural lift defined on the bundle of linear frames.
Generalisations exist for any given reductive G-structure.
Introduction
In general, given a subbundle of a fiber bundle over and a vector field on , its restriction to is a vector field "along" not on (i.e., tangent to) . If one denotes by the canonical embedding, then is a section of the pullback bundle , where
and is the tangent bundle of the fiber bundle .
Let us assume that we are given a Kosmann decomposition of the pullback bundle , such that
i.e., at each one has where is a vector subspace of and we assume to be a vector bundle over , called the transversal bundle of the Kosmann decomposition. It follows that the restriction to splits into a tangent vector field on and a transverse vector field being a section of the vector bundle
Definition
Let be the oriented orthonormal frame bundle of an oriented -dimensional
Riemannian manifold with given metric . This is a principal -subbundle of , the tangent frame bundle of linear frames over with structure group .
By definition, one may say that we are given with a classical reductive -structure. The special orthogonal group is a reductive Lie subgroup of . In fact, there exists a direct sum decomposition , where is the Lie algebra of , is the Lie algebra of , and is the -invariant vector subspace of symmetric matrices, i.e. for all
Let be the canonical embedding.
One then can prove that there exists a canonical Kosmann decomposition of the pullback bundle such that
i.e., at each one has being the fiber over of the subbundle of . Here, is the vertical subbundle of and at each the fiber is isomorphic to the vector space of symmetric matrices .
From the above canonical and equivariant decomposition, it follows that the restriction of an -invariant vector field on to splits into a -invariant vector field on , called the Kosmann vector field associated with , and a transverse vector field .
In particular, for a generic vector field on the base manifold , it follows that the restriction to of its natural lift onto splits into a -invariant vector field on , called the Kosmann lift of , and a transverse vector field .
See also
Frame bundle
Orthonormal frame bundle
Principal bundle
Spin bundle
Connection (mathematics)
G-structure
Spin manifold
Spin structure
Notes
References
Fiber bundles
Vector bundles
Riemannian geometry
Structures on manifolds |
https://en.wikipedia.org/wiki/Neil%20O%27Connell | Neil Michael O'Connell is an Irish mathematician from Shannon, County Clare. He attended Trinity College Dublin, and was elected to scholarship in 1987. He earned his bachelor's degree in mathematics and a gold medal in 1989 and completed an M.Sc. in 1990. He obtained his PhD in 1993 at UC Berkeley under the supervision of Steven Neil Evans. He subsequently worked at the Dublin Institute for Advanced Studies, and the University of Warwick.
He works in probability theory, in particular random matrices. He was awarded the inaugural Itô prize in 2002 (together with Ben Hambly and James Martin), and the Rollo Davidson Prize in 2005. In 2013 he was Doob Lecturer at the 36th Conference on Stochastic Processes and Their Applications, in Boulder, Colorado. He is currently Professor at University College Dublin.
References
External links
Academics of University College Dublin
Academics of the Dublin Institute for Advanced Studies
Alumni of Trinity College Dublin
Living people
Place of birth missing (living people)
Probability theorists
Scholars of Trinity College Dublin
University of California, Berkeley alumni
Year of birth missing (living people)
People from Shannon, County Clare
Scientists from County Clare
Scholars and academics from County Clare
20th-century Irish mathematicians
21st-century Irish mathematicians |
https://en.wikipedia.org/wiki/Perfect%20lattice | In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by . A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by .
proved that a lattice is extreme if and only if it is both perfect and eutactic.
The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by
1, 1, 1, 2, 3, 7, 33, 10916 . summarize the properties of perfect lattices of dimension up to 7.
verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.
References
External links
List of perfect lattices in dimension 8
Quadratic forms |
https://en.wikipedia.org/wiki/Eutactic | In mathematics, the word eutactic may refer to:
Eutactic lattice
Eutactic star |
https://en.wikipedia.org/wiki/Eutactic%20lattice | In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients ci such that (v, v) = Σci(v, mi)2 where the sum is over the minimal vectors mi. "Eutactic" is derived from the Greek language, and means "well-situated" or "well-arranged".
proved that a lattice is extreme if and only if it is both perfect and eutactic.
summarize the properties of eutactic lattices of dimension up to 7.
References
Quadratic forms |
https://en.wikipedia.org/wiki/Crime%20in%20Germany | Crime in Germany is handled by the German police forces and other agencies.
Recent trends
Statistics
The official statistics PKS 2018 of 2018 by the Bundeskriminalamt for the year 2017 shows an increase of 39.9% for resistance and attacks against state authority, 13.6% in the spreading of pornographic material, 8.3% in crimes against the German drug law, 6.1% for narcotic-related crimes generally and 5.5% in violations of the German arms law.
On the other hand, there is a decrease by 18.2% in sexual assault, rape, sexual harassment including cases with lethal consequences, 16.3% in burglaries, 9.3% in violations of the immigration laws, 7.6% in fraud, 7.5% in theft and 6% in street crime.
European Union Statistics on Income and Living Conditions (EU-SILC)
In the EU-SILC survey, respondents were questioned about whether they experienced problems with violence, crime, or vandalism in the area where they live. Between 2010 and 2017, the EU crime average dropped by 3%. All countries in the EU except Germany, Sweden, and Lithuania showed a falling trend of criminal incidents.
By type
According to Germany's 2010 crime statistics, 5.93 million criminal acts were committed, which was 2% lower than in 2009. According to the Interior Ministry, this was the first time the figure had fallen below six million offenses since 1991 (the year after reunification), and is the lowest crime level since records began. The rate of crimes solved in 2010 was 56%, a record high from 2009's 55.6%.
In 2010, internet-related crime climbed 8.1%, with around 224,000 reported cases. The number of house burglaries in 2010 also increased by 6.6%.
Domestic violence
According to 2015 statistics, there were 127,000 victims of domestic violence. (German: Häusliche Gewalt) 82% of the victims were female. This represented an increase of 5.5% over 2012 statistics. The most commonly reported crime was bodily harm, defined as a slap or a strike of sufficient force to warrant prosecution. Other common crimes were threats (14.4%), grievous bodily harm (German: schwere Körperverletzung), and injury with a deadly outcome (German: Verletzung mit Todesfolge) at 12%. A fourth of the suspects were reported to be intoxicated from the consumption of alcohol.
Ex-partner victims were mostly targeted by stalking.
Murder
The homicide rate in Germany is similarly low to the EU and other developed countries. Homicides increased rapidly in the early 1990s, increasing from 931 in 1990 to 2,032 - 2.5 per 100,000 in 1995 and gradually decreasing in the next 15 years before stabilizing at lower rates from around 2010 (0.98 per 100,000 or 783) to the present with 782 in 2020 at a rate of 0.93 per 100,000.
Organized crime
In the 1990s, the power balance changed in the red light districts of Germany when Russian, Yugoslav, and Albanian organizations started to operate. In parts of Germany, police asked themselves whether they had suppressed German gangs too much as the gangs that took over wer |
https://en.wikipedia.org/wiki/Reflection%20principle%20%28Wiener%20process%29 | In the theory of probability for stochastic processes, the reflection principle for a Wiener process states that if the path of a Wiener process f(t) reaches a value f(s) = a at time t = s, then the subsequent path after time s has the same distribution as the reflection of the subsequent path about the value a. More formally, the reflection principle refers to a lemma concerning the distribution of the supremum of the Wiener process, or Brownian motion. The result relates the distribution of the supremum of Brownian motion up to time t to the distribution of the process at time t. It is a corollary of the strong Markov property of Brownian motion.
Statement
If is a Wiener process, and is a threshold (also called a crossing point), then the lemma states:
Assuming , due to the continuity of Wiener processes, each path (one sampled realization) of Wiener process on which finishes at or above value/level/threshold/crossing point the time ( ) must have crossed (reached) a threshold ( ) at some earlier time for the first time . (It can cross level multiple times on the interval , we take the earliest.)
For every such path, you can define another path on that is reflected or vertically flipped on the sub-interval symmetrically around level from the original path. These reflected paths are also samples of the Wiener process reaching value on the interval , but finish below . Thus, of all the paths that reach on the interval , half will finish below , and half will finish above. Hence, the probability of finishing above is half that of reaching .
In a stronger form, the reflection principle says that if is a stopping time then the reflection of the Wiener process starting at , denoted , is also a Wiener process, where:
and the indicator function and is defined similarly. The stronger form implies the original lemma by choosing .
Proof
The earliest stopping time for reaching crossing point a, , is an almost surely bounded stopping time. Then we can apply the strong Markov property to deduce that a relative path subsequent to , given by , is also simple Brownian motion independent of . Then the probability distribution for the last time is at or above the threshold in the time interval can be decomposed as
.
By the tower property for conditional expectations, the second term reduces to:
since is a standard Brownian motion independent of and has probability of being less than . The proof of the lemma is completed by substituting this into the second line of the first equation.
.
Consequences
The reflection principle is often used to simplify distributional properties of Brownian motion. Considering Brownian motion on the restricted interval then the reflection principle allows us to prove that the location of the maxima , satisfying , has the arcsine distribution. This is one of the Lévy arcsine laws.
References
Stochastic calculus
Probability theorems |
https://en.wikipedia.org/wiki/Method%20of%20normals | In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that the radius of a circle is always normal to the circle itself. With this in mind Descartes would construct a circle that was tangent to a given curve. He could then use the radius at the point of intersection to find the slope of a normal line, and from this one can easily find the slope of a tangent line.
This was discovered about the same time as Fermat's method of adequality. While Fermat's method had more in common with the infinitesimal techniques that were to be used later, Descartes' method was more influential in the early history of calculus.
One reason Descartes' method fell from favor was the algebraic complexity it involved. On the other hand, this method can be used to rigorously define the derivative for a wide class of functions using neither infinitesimal nor limit techniques. It is also related to a completely general definition of differentiability given by Carathéodory .
References
History of calculus |
https://en.wikipedia.org/wiki/Reciprocity%20theorem | Reciprocity theorem may refer to:
Quadratic reciprocity, a theorem about modular arithmetic
Cubic reciprocity
Quartic reciprocity
Artin reciprocity
Weil reciprocity for algebraic curves
Frobenius reciprocity theorem for group representations
Stanley's reciprocity theorem for generating functions
Reciprocity (engineering), theorems relating signals and the resulting responses
including Reciprocity (electrical networks), a theorem relating voltages and currents in a network
Reciprocity (electromagnetism), theorems relating sources and the resulting fields in classical electromagnetism
Tellegen's theorem, a theorem about the transfer function of passive networks
Reciprocity law for Dedekind sums
Betti's theorem in linear elasticity
See also
Reciprocity (disambiguation) |
https://en.wikipedia.org/wiki/Wahba%27s%20problem | In applied mathematics, Wahba's problem, first posed by Grace Wahba in 1965, seeks to find a rotation matrix (special orthogonal matrix) between two coordinate systems from a set of (weighted) vector observations. Solutions to Wahba's problem are often used in satellite attitude determination utilising sensors such as magnetometers and multi-antenna GPS receivers. The cost function that Wahba's problem seeks to minimise is as follows:
for
where is the k-th 3-vector measurement in the reference frame, is the corresponding k-th 3-vector measurement in the body frame and is a 3 by 3 rotation matrix between the coordinate frames.
is an optional set of weights for each observation.
A number of solutions to the problem have appeared in literature, notably Davenport's q-method, QUEST and methods based on the singular value decomposition (SVD). Several methods for solving Wahba's problem are discussed by Markley and Mortari.
This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the alternative formulation). An elegant derivation of the solution on one and a half page can be found in.
Solution via SVD
One solution can be found using a singular value decomposition (SVD).
1. Obtain a matrix as follows:
2. Find the singular value decomposition of
3. The rotation matrix is simply:
where
Notes
References
Wahba, G. Problem 65–1: A Least Squares Estimate of Satellite Attitude, SIAM Review, 1965, 7(3), 409
Shuster, M. D. and Oh, S. D. Three-Axis Attitude Determination from Vector Observations, Journal of Guidance and Control, 1981, 4(1):70–77
Markley, F. L. Attitude Determination using Vector Observations and the Singular Value Decomposition, Journal of the Astronautical Sciences, 1988, 38:245–258
Markley, F. L. and Mortari, D. Quaternion Attitude Estimation Using Vector Observations, Journal of the Astronautical Sciences, 2000, 48(2):359–380
Markley, F. L. and Crassidis, J. L. Fundamentals of Spacecraft Attitude Determination and Control, Springer 2014
Libbus, B. and Simons, G. and Yao, Y. Rotating Multiple Sets of Labeled Points to Bring Them Into Close Coincidence: A Generalized Wahba Problem, The American Mathematical Monthly, 2017, 124(2):149–160
Lourakis, M. and Terzakis, G. Efficient Absolute Orientation Revisited, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2018, pp. 5813-5818.
See also
Triad method
Kabsch algorithm
Orthogonal Procrustes problem
Applied mathematics |
https://en.wikipedia.org/wiki/Coorbit%20theory | In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990. It provides theory for atomic decomposition of a range of Banach spaces of distributions. Among others the well established wavelet transform and the short-time Fourier transform are covered by the theory.
The starting point is a square integrable representation of a locally compact group on a Hilbert space , with which one can define a transform of a function with respect to by . Many important transforms are special cases of the transform, e.g. the short-time Fourier transform and the wavelet transform for the Heisenberg group and the affine group respectively. Representation theory yields the reproducing formula . By discretization of this continuous convolution integral it can be shown that by sufficiently dense sampling in phase space the corresponding functions will span a frame for the Hilbert space.
An important aspect of the theory is the derivation of atomic decompositions for Banach spaces. One of the key steps is to define the voice transform for distributions in a natural way. For a given Banach space , the corresponding coorbit space is defined as the set of all distributions such that . The reproducing formula is true also in this case and therefore it is possible to obtain atomic decompositions for coorbit spaces.
References
Hilbert spaces |
https://en.wikipedia.org/wiki/Peregrine%20soliton | The Peregrine soliton (or Peregrine breather) is an analytic solution of the nonlinear Schrödinger equation. This solution was proposed in 1983 by Howell Peregrine, researcher at the mathematics department of the University of Bristol.
Main properties
Contrary to the usual fundamental soliton that can maintain its profile unchanged during propagation, the Peregrine soliton presents a double spatio-temporal localization. Therefore, starting from a weak oscillation on a continuous background, the Peregrine soliton develops undergoing a progressive increase of its amplitude and a narrowing of its temporal duration. At the point of maximum compression, the amplitude is three times the level of the continuous background (and if one considers the intensity as it is relevant in optics, there is a factor 9 between the peak intensity and the surrounding background). After this point of maximal compression, the wave's amplitude decreases and its width increases.
These features of the Peregrine soliton are fully consistent with the quantitative criteria usually used in order to qualify a wave as a rogue wave. Therefore, the Peregrine soliton is an attractive hypothesis to explain the formation of those waves which have a high amplitude and may appear from nowhere and disappear without a trace.
Mathematical expression
In the spatio-temporal domain
The Peregrine soliton is a solution of the one-dimensional nonlinear Schrödinger equation that can be written in normalized units as follows :
with the spatial coordinate and the temporal coordinate. being the envelope of a surface wave in deep water. The dispersion is anomalous and the nonlinearity is self-focusing (note that similar results could be obtained for a normally dispersive medium combined with a defocusing nonlinearity).
The Peregrine analytical expression is:
so that the temporal and spatial maxima are obtained for and .
In the spectral domain
It is also possible to mathematically express the Peregrine soliton according to the spatial frequency :
with being the Dirac delta function.
This corresponds to a modulus (with the constant continuous background here omitted) :
One can notice that for any given time , the modulus of the spectrum exhibits a typical triangular shape when plotted on a logarithmic scale. The broadest spectrum is obtained for , which corresponds to the maximum of compression of the spatio-temporal nonlinear structure.
Different interpretations of the Peregrine soliton
As a rational soliton
The Peregrine soliton is a first-order rational soliton.
As an Akhmediev breather
The Peregrine soliton can also be seen as the limiting case of the space-periodic Akhmediev breather when the period tends to infinity.
As a Kuznetsov-Ma soliton
The Peregrine soliton can also be seen as the limiting case of the time-periodic Kuznetsov-Ma breather when the period tends to infinity.
Experimental demonstration
Mathematical predictions by H. Peregrine had initial |
https://en.wikipedia.org/wiki/Ockham%20algebra | In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x ∧ ~y, ~0 = 1, ~1 = 0. They were introduced by , and were named after William of Ockham by . Ockham algebras form a variety.
Examples of Ockham algebras include Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras.
References
(pdf available from GDZ)
Algebraic logic |
https://en.wikipedia.org/wiki/Jacobi%20group | In mathematics, the Jacobi group, introduced by
, is the semidirect product of the symplectic group Sp2n(R) and the Heisenberg group R1+2n. The concept is named after Carl Gustav Jacob Jacobi. Automorphic forms on the Jacobi group are called Jacobi forms.
References
Modular forms
Lie groups |
https://en.wikipedia.org/wiki/1992%E2%80%9393%20Second%20League%20of%20FR%20Yugoslavia | Statistics of Second League of FR Yugoslavia () for the 1992–93 season.
Overview
The league was composed of clubs from Serbia and Montenegro after the other former Yugoslav republics became independent and left the league at the end of the 1991–92 Yugoslav Second League.
The champion and the 2 following teams were promoted into the 1993–94 First League of FR Yugoslavia.
At the end of the season FK Jastrebac Niš became champions.
Club names
Some club names were written in a different way in other sources, and that is because some clubs had in their names the sponsorship company included. These cases were:
Jastrebac Niš / Jastrebac Narvik
Rudar Pljevlja / Rudar Volvoks
Borac Čačak / Borac Cane
Obilić / Obilić Kopeneks
Jedinstvo Bijelo Polje / Jedinstvo Tošpred
Radnički Pirot / Radnički Trikom
Final table
References
External sources
Season tables at FSGZ
Yugoslav Second League seasons
Yugo
2 |
https://en.wikipedia.org/wiki/1993%E2%80%9394%20Second%20League%20of%20FR%20Yugoslavia | Statistics of Second League of FR Yugoslavia () for the 1993–94 season.
Overview
The league was divided into 2 groups, A and B, consisting each of 10 clubs. Both groups were played in league system. By winter break all clubs in each group meet each other twice, home and away, with the bottom four classified from A group moving to the group B, and being replaced by the top four from the B group. At the end of the season the same situation happened with four teams being replaced from A and B groups, adding the fact that the bottom three clubs from the B group were relegated into the third national tier. The champion and the second following team were promoted into the 1994–95 First League of FR Yugoslavia.
At the end of the season FK Borac Čačak became champions, and together with FK Obilić got promoted.
Club names
Some club names were written in a different way in other sources, and that is because some clubs had in their names the sponsorship company included. These cases were:
Obilić / Obilić Kopeneks
Čukarički / Čukarički Stankom
Novi Sad / Novi Sad Gumins
Topličanin / Topličanin RIS
Inđija / Agrounija Inđija
Final table
References
External sources
Season tables at FSGZ
Yugoslav Second League seasons
Yugo
2 |
https://en.wikipedia.org/wiki/Purpose%20%28Algebra%20album%29 | Purpose is the debut album by contemporary R&B singer, Algebra. It stayed on the Billboard Top R&B/Hip-Hop Albums chart for 14 weeks, peaking at No. 56.
Track listing
At This Time
Halfway
Run and Hide
U Do It for Me
ABC's 1, 2, 3's (Interlude)/Happy After
My Pride
Holla Back (Interlude)/Simple Complication
What Happened?
No Idea
Tug of War
Can I Keep U?
I Think I Love U
Come Back
Now & Then
Where R We Now
References
Contemporary R&B albums by American artists
2008 debut albums
Neo soul albums |
https://en.wikipedia.org/wiki/Marcello%20Boldrini | Marcello Boldrini (9 February 1890, in Matelica – 5 March 1969, in Milan) was an Italian statistician.
Biography
Beginning in 1922, he taught courses in statistics, biometry, and demography at Bocconi University of Milan, and then at the University of Rome as Emeritus Professor. He was also a member of several academies and institutes in Italy and abroad, serving for several years as president of the International Statistical Institute. His scientific research was on both methodological and applied statistics, particularly on demography, anthropometry, and economics. As a statistician, he has been particularly interested in the foundations of the method, and he proposed a view of statistics as an empirical history of all the positive sciences. To deepen this research he founded a statistical Laboratory at the Catholic University of Milan. The results are widely discussed in the volumes, Statistica: Teoria e metodi (first edition, 1942), and Teoria della Statistica (1963), which have been studied by thousands of students. For Boldrini, Statistics is a formal science like mathematics and logic, but it is different from them as a scientific inquiry method in the inductive and deductive phases of research.
Beyond academic commitment, he also had several management positions in the state oil industry: he was first named president of Agip (1948–1953), was then vice-president of ENI (1953–1962), before succeeding Enrico Mattei as president in 1962, remaining in the post until 1967.
Academic Positions
Professor of Statistics and Demography at the Universities of Messina, Padua, Milan and Rome. Dean of Political Sciences, Milan Catholic University (1043-1945).
Honours, memberships
Member of the Pontifical Academy, Accademia dei Lincei, Honoris Causa at the University of Rio de Janeiro. Professor emeritus at Rome University.
Selected works
English
Scientific Truth and Statistical Method, translated by Ruth Kendall, Griffin (1972)
Italian
Biometrica, CEDAM (1927);
Demografia, A. Giuffrè (1943);
Statistica in Compendio, A. Giuffrè (1957)
Teoria della Statistica, A. Giuffrè (1963).
Statistica, Teoria e Metodi, A. Giuffrè (5th ed. 1968)
References
Italian statisticians
Presidents of the International Statistical Institute
1890 births
1969 deaths
Academic staff of the University of Messina
Academic staff of the University of Padua
Academic staff of the University of Milan
People from Matelica |
https://en.wikipedia.org/wiki/Giuseppe%20Pompilj | Giuseppe Pompilj (17 July 1913, Rome–9 July 1968, Rome) was an Italian statistician.
Biography
He graduated in mathematics in 1935 and immediately undertook a university career. In 1942 he was lecturer in geometry (he had studied algebraic geometry under the guidance of Federigo Enriques and other Roman mathematicians). After an interruption for military service and a long period of imprisonment, in 1948 Pompilj won the competition for the chair in geometry. He taught geometry and probability theory at the Faculty of Statistical Sciences, University of Rome. Probability theory became the main topic of his interests and he taught this discipline until his premature death.
His research interests are attested by seventy publications and ten printed books, ranging across a broad spectrum of topics: algebraic geometry, the random sampling theory, the probabilistic analysis of experiments and especially the theory of random variables. This topic is certainly one of the most attractive for Pompilj because he found in it a fruitful field for the application of geometric concepts and because it could represent the unification of fields(not merely formal) of probability and statistics. Among the many initiatives taken to facilitate statistical studies in Italy, a special mention must go to "Courses of Statistical Methods for Researchers", which he organized in Rome in 1958. The lessons taught in these courses by a large number of distinguished professors were collected in several volumes, which still provides one of the most comprehensive presentations in this field.
Publications
Sulla regressione, Rend. Mat. E sue applicazioni, (1946);
Sulla media di una distribuzione normale, Statistica, (1947);
Complementi di calcolo delle probabilità, Veschi, Roma (1948);
Metodologia statistica. Integrazione e comparazione dei dati (con C. Gini, 1949);
Sulle medie combinatorie potenziate dei campioni, Rend. Sem. matematico, Padova, (1949);
La teoria affine delle variabili casuali, L’industria, (1956);
Piano degli esperimenti (con G. Dall'Aglio, (1959);
Elaborazione probabilistica dei risultati sperimentali, Atti V Convegno UMI, (1956);
Teoria dei campioni, Veschi, Roma, (1956);
Probability Theory, University of Pittsburgh, (1963–64).
References
Dizionario Biografico Degli Statistici
Enciclopedia Treccani: Brief biography (in Italian)
1913 births
1968 deaths
20th-century Italian mathematicians
Italian statisticians
Scientists from Rome |
https://en.wikipedia.org/wiki/Hermite%27s%20problem | Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.
Motivation
A standard way of writing real numbers is by their decimal representation, such as:
where a0 is an integer, the integer part of x, and a1, a2, a3, … are integers between 0 and 9. Given this representation the number x is equal to
The real number x is a rational number only if its decimal expansion is eventually periodic, that is if there are natural numbers N and p such that for every n ≥ N it is the case that an+p = an.
Another way of expressing numbers is to write them as continued fractions, as in:
where a0 is an integer and a1, a2, a3… are natural numbers. From this representation we can recover x since
If x is a rational number then the sequence (an) terminates after finitely many terms. On the other hand, Euler proved that irrational numbers require an infinite sequence to express them as continued fractions. Moreover, this sequence is eventually periodic (again, so that there are natural numbers N and p such that for every n ≥ N we have an+p = an), if and only if x is a quadratic irrational.
Hermite's question
Rational numbers are algebraic numbers that satisfy a polynomial of degree 1, while quadratic irrationals are algebraic numbers that satisfy a polynomial of degree 2. For both these sets of numbers we have a way to construct a sequence of natural numbers (an) with the property that each sequence gives a unique real number and such that this real number belongs to the corresponding set if and only if the sequence is eventually periodic.
In 1848, Charles Hermite wrote a letter to Carl Gustav Jacob Jacobi asking if this situation could be generalised, that is can one assign a sequence of natural numbers to each real number x such that the sequence is eventually periodic precisely when x is a cubic irrational, that is an algebraic number of degree 3? Or, more generally, for each natural number d is there a way of assigning a sequence of natural numbers to each real number x that can pick out when x is algebraic of degree d?
Approaches
Sequences that attempt to solve Hermite's problem are often called multidimensional continued fractions. Jacobi himself came up with an early example, finding a sequence corresponding to each pair of real numbers (x, y) that acted as a higher-dimensional analogue of continued fractions. He hoped to show that the sequence attached to (x, y) was eventually periodic if and only if both x and y belonged to a cubic number field, but was unable to do so and whether this is the case remains unsolved.
In 2015, for the first time, a periodic representation for any cubic irrational has been provided by means of ternary continued fractions, i.e., the problem of writing cubic irrationals as a periodic sequence of rational |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20GNK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players and lists all matches that Dinamo Zagreb will play in the 2011–12 season.
Current squad
Sources: Prva-HNL.hr, Sportske novosti, Sportnet.hr,
1Played for Dinamo Zagreb in 1997–2000
2Played for Dinamo Zagreb in 1999–2003
3Played for Dinamo Zagreb in 2007–2008
Kramarić, Bručić, Ademi and Šitum loaned to NK Lokomotiva till end of the season. Morales loaned to Universidad de Chile till end of the season.
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2011–12 Prva HNL article
2011–12 Champions league
Group D
Matches
Key
Tournament
1. HNL = 2011–12 Prva HNL
Cup = 2011–12 Croatian Cup
UCL = 2011–12 UEFA Champions League
UEL = 2011–12 UEFA Europa League
Ground
H = Home
A = Away
HR = Home replacement
AR = Away replacement
Round
R1 = Round 1 (round of 32)
R2 = Round 2 (round of 16)
QF = Quarter-finals
SF = Semi-finals
F = Final
QR2 = Second Qualifying Round
QR3 = Third Qualifying Round
Play-off = Play-off Round
Group = Group Stage
Pre-season
Season friendly
Mid-season friendly
Competitive
Last updated 8 April 2012Sources: Prva-HNL.hr, Sportske novosti, Sportnet.hr, gnkdinamo.hr
Statistics
Statistics
Competitive matches only. Updated to games played 12 May 2012.
Source: Competitive matches
Transfers
In
Out
References
External links
GNK Dinamo Zagreb official website
2011-12
Croatian football clubs 2011–12 season
2011–12 UEFA Champions League participants seasons
2011-12 |
https://en.wikipedia.org/wiki/Spin%20geometry | In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics.
An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research.
See also
Contact geometry
Symplectic topology
Spinor
Spinor bundle
Spin manifold
Books
Differential topology
Differential geometry |
https://en.wikipedia.org/wiki/Metaplectic%20structure | In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry.
Symplectic spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in establishing the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for symplectic spin geometry.
Formal definition
A metaplectic structure on a symplectic manifold is an equivariant lift of the symplectic frame bundle with respect to the double covering In other words, a pair is a metaplectic structure on the principal bundle when
a) is a principal -bundle over ,
b) is an equivariant -fold covering map such that
and for all and
The principal bundle is also called the bundle of metaplectic frames over .
Two metaplectic structures and on the same symplectic manifold are called equivalent if there exists a -equivariant map such that
and for all and
Of course, in this case and are two equivalent double coverings of the symplectic frame -bundle of the given symplectic manifold .
Obstruction
Since every symplectic manifold is necessarily of even dimension and orientable, one can prove that the topological obstruction to the existence of metaplectic structures is precisely the same as in Riemannian spin geometry. In other words, a symplectic manifold admits a metaplectic structures if and only if the second Stiefel-Whitney class of vanishes. In fact, the modulo reduction of the first Chern class is the second Stiefel-Whitney class . Hence, admits metaplectic structures if and only if is even, i.e., if and only if is zero.
If this is the case, the isomorphy classes of metaplectic structures on are classified by the first cohomology group of with -coefficients.
As the manifold is assumed to be oriented, the first Stiefel-Whitney class of vanishes too.
Examples
Manifolds admitting a metaplectic structure
Phase spaces any orientable manifold.
Complex projective spaces Since is simply connected, such a structure has to be unique.
Grassmannian etc.
See also
Metaplectic group
Symplectic frame bundle
Symplectic group
Symplectic spinor bundle
Notes
References
Symplectic geometry
Structures on manifolds
Algebraic topology |
https://en.wikipedia.org/wiki/Stone%20algebra | In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by and named after Marshall Harvey Stone.
Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras.
Examples:
The open-set lattice of an extremally disconnected space is a Stone algebra.
The lattice of positive divisors of a given positive integer is a Stone lattice.
See also
De Morgan algebra
Heyting algebra
References
Universal algebra
Lattice theory
Ockham algebras |
https://en.wikipedia.org/wiki/Birkhoff%20factorization | In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is the factorization of an invertible matrix M with coefficients that are Laurent polynomials in z into a product M = M+M0M−, where M+ has entries that are polynomials in z, M0 is diagonal, and M− has entries that are polynomials in z−1. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to .
Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles.
Birkhoff factorization follows from the Bruhat decomposition for affine Kac–Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine general linear group follows from Birkhoff factorization together with the Bruhat decomposition for the ordinary general linear group.
See also
Birkhoff decomposition (disambiguation)
Riemann–Hilbert problem
References
Matrices |
https://en.wikipedia.org/wiki/Rectified%2024-cell%20honeycomb | In four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is constructed by a rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.
Alternate names
Rectified icositetrachoric tetracomb
Rectified icositetrachoric honeycomb
Cantellated 16-cell honeycomb
Bicantellated tesseractic honeycomb
Symmetry constructions
There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored rectified 24-cell and tesseract facets. The tetrahedral prism vertex figure contains 4 rectified 24-cells capped by two opposite tesseracts.
See also
Regular and uniform honeycombs in 4-space:
Tesseractic honeycomb
16-cell honeycomb
24-cell honeycomb
Truncated 24-cell honeycomb
Snub 24-cell honeycomb
5-cell honeycomb
Truncated 5-cell honeycomb
Omnitruncated 5-cell honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 93
, o3o3o4x3o, o4x3o3x4o - ricot - O93
5-polytopes
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Concentration%20inequality | In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The law of large numbers of classical probability theory states that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results show that such behavior is shared by other functions of independent random variables.
Concentration inequalities can be sorted according to how much information about the random variable is needed in order to use them.
Markov's inequality
Let be a random variable that is non-negative (almost surely). Then, for every constant ,
Note the following extension to Markov's inequality: if is a strictly increasing and non-negative function, then
Chebyshev's inequality
Chebyshev's inequality requires the following information on a random variable :
The expected value is finite.
The variance is finite.
Then, for every constant ,
or equivalently,
where is the standard deviation of .
Chebyshev's inequality can be seen as a special case of the generalized Markov's inequality applied to the random variable with .
Vysochanskij–Petunin inequality
Let X be a random variable with unimodal distribution, mean μ and finite, non-zero variance σ2. Then, for any
(For a relatively elementary proof see e.g. ).
One-sided Vysochanskij–Petunin inequality
For a unimodal random variable and , the one-sided Vysochanskij-Petunin inequality holds as follows:
Paley–Zygmund inequality
Cantelli's inequality
Gauss's inequality
Chernoff bounds
The generic Chernoff bound requires the moment generating function of , defined as It always exists, but may be infinite. From Markov's inequality, for every :
and for every :
There are various Chernoff bounds for different distributions and different values of the parameter . See for a compilation of more concentration inequalities.
Bounds on sums of independent bounded variables
Let be independent random variables such that, for all i:
almost surely.
Let be their sum, its expected value and its variance:
It is often interesting to bound the difference between the sum and its expected value. Several inequalities can be used.
1. Hoeffding's inequality says that:
2. The random variable is a special case of a martingale, and . Hence, the general form of Azuma's inequality can also be used and it yields a similar bound:
This is a generalization of Hoeffding's since it can handle other types of martingales, as well as supermartingales and submartingales. See Fan et al. (2015). Note that if the simpler form of Azuma's inequality is used, the exponent in the bound is worse by a factor of 4.
3. The sum function, , is a special case of a function of n variables. This function changes in a bounded way: if variable i is changed, the value of f changes by at |
https://en.wikipedia.org/wiki/Kamran%20Michael | Kamran Michael (Urdu: کامران مائیکل) is a Pakistani politician who served as Minister for Statistics, in Abbasi cabinet from August 2017 to May 2018. He previously served as the Minister for Human Rights in the third Sharif ministry from 2013 to 2017. A member of the Pakistan Muslim League (Nawaz), Michael held the cabinet portfolio of Minister for Ports and Shipping from 2013 to 2016.
Michael has been an elected member of the Senate of Pakistan on minorities seat since 2012 and has served as the Provincial Minister for Minorities Affairs, Human Rights, Women development, Social Welfare and Finance in the Provincial Assembly of Punjab.
In 2016, he moved the Hindu marriage bill in 2016, which later became law.
Political career
Michael started his political career in 2001 after getting elected as Councillor. Later he was elected as Member of Lahore District Council.
Michael was elected as member of the Provincial Assembly of Punjab for the first time in 2002 Pakistani general election on one of the eight seats reserved for minorities. He was re-elected as the member of the Provincial Assembly of the Punjab for a second term in 2008 Pakistani general election. He was appointed as provincial minister of Punjab for human rights, provincial minister for Minority Affairs, provincial minister for Social Welfare and provincial minister for Women development. In 2010, he was appointed as the provincial minister for Finance
Michael was elected member of the Senate of Pakistan in 2012 for the first time on a seat reserved for minorities after assassination of Shahbaz Bhatti. Upon PML-N victory in the 2013 Pakistani general election, Michael was made the Minister for Ports and Shipping in June 2013 where he served until May 2016. In 2016, he was appointed as Minister for Human Rights.
He had ceased to hold ministerial office in July 2017 when the federal cabinet was disbanded following the resignation of Prime Minister Nawaz Sharif after Panama Papers case decision. Following the election of Shahid Khaqan Abbasi as Prime Minister of Pakistan in August 2017, he was inducted into the federal cabinet of Abbasi. He was appointed as the Federal Minister for Statistics.
He was nominated by PML-N as its candidate in 2018 Pakistani Senate election. However the Election Commission of Pakistan declared all PML-N candidates for the Senate election as independent after a ruling of the Supreme Court of Pakistan.
He was re-elected to the Senate as an independent candidate on a reserved seat for non-Muslim from Punjab in Senate election. On 12 March 2018, he ceased to hold the office of Federal Minister for Statistics due to expiration of his term in the Senate. On 15 March 2018, he was re-inducted into the federal cabinet of Prime Minister Shahid Khaqan Abbasi and was re-appointed as Federal Minister for Statistics.
He joined the treasury benches, led by PML-N after assuming the office of Senator. Upon the dissolution of the National Assembly on the expiration of |
https://en.wikipedia.org/wiki/Data%20generating%20process | In statistics and in empirical sciences, a data generating process is a process in the real world that "generates" the data one is interested in. Usually, scholars do not know the real data generating model. However, it is assumed that those real models have observable consequences. Those consequences are the distributions of the data in the population. Those distributors or models can be represented via mathematical functions. There are many functions of data distribution. For example, normal distribution, Bernoulli distribution, Poisson distribution, etc.
References
Probability distributions |
https://en.wikipedia.org/wiki/Double%20affine%20Hecke%20algebra | In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and in chemistry.
References
A. A. Kirillov Lectures on affine Hecke algebras and Macdonald's conjectures Bull. Amer. Math. Soc. 34 (1997), 251–292.
Macdonald, I. G. Affine Hecke algebras and orthogonal polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, 2003. x+175 pp.
Algebras
Representation theory |
https://en.wikipedia.org/wiki/Double%20affine%20braid%20group | In mathematics, a double affine braid group is a group containing the braid group of an affine Weyl group. Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras.
For affine An groups, the double affine braid group is the fundamental group of the space of n distinct points on a 2-dimensional torus.
References
Braid groups
Representation theory |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Romania |
League
Records in this section refer to Liga I from its founding in 1909 through to the present.
Clubs
Titles
Most League titles: 26, Steaua București (1951, 1952, 1953, 1956, 1959–60, 1960–61, 1967–68, 1975–76, 1977–78, 1984–85, 1985–86, 1986–87, 1987–88, 1988–89, 1992–93, 1993–94, 1994–95, 1995–96, 1996–97, 1997–98, 2000–01, 2004–05, 2005–06, 2012–13, 2013–14, 2014–15)
Most consecutive League titles: 6, Chinezul Timișoara (1921–22, 1922–23, 1923–24, 1924–25, 1925–26, 1926–27) and Steaua București (1992–93, 1993–94, 1994–95, 1995–96, 1996–97, 1997–98)
Top flight appearances
Most Appearances: 72, FCSB
Wins
Most Wins Overall: 1224, FCSB
Most Consecutive Wins: 21, Steaua București (1988)
Draws
Most Draws Overall: 508, FCSB
Most Consecutive Draws: 8, FCM Bacău (1979)
Losses
Most Losses Overall: 717, Universitatea Cluj
Most Consecutive Losses: 24, ASA Târgu Mureş (1988)
Points
Most Points Overall: 4180, FCSB
Players
Appearances
Goals
Managers
Referees
Most successful clubs overall (official titles, 1909–present)
Teams in Italics no longer exist.
Teams in Bold compete in the 2023–24 Liga I season.
This table is updated as of 8th of July 2023, following Sepsi Sfântu Gheorghe winning the 2023 Supercupa României.
All-time table
The ranking is computed awarding three points for a win, one for a draw. It includes matches played between the 1932–33 and 2022–23 season including. The teams in bold play in the 2023–24 season of Liga I. The teams in italics no longer exist.
This table lists only league finishes since 1932–33, when a national league was first introduced. Since an official national championship has been awarded since 1909, some teams are not listed with all their championships. For example, Venus București have been champions for seven times, but are listed only with the four titles they have won since 1932.
League or status at 2022–23:
See also
List of Romanian football champions
List of football clubs in Romania by major honors won
References
Football in Romania
All-time football league tables |
https://en.wikipedia.org/wiki/Conic%20Sections%20Rebellion | The Conic Sections Rebellion, also known as the Conic Section Rebellion, refers primarily to an incident which occurred at Yale University in 1830, as a result of changes in the methods of mathematics education. When a policy change dictated that students were required to draw reference diagrams for exams rather than be allowed to refer to diagrams in their textbooks, a number of students staged a rebellion in which they refused to take the exams at all. A precursor incident occurred in 1825; historian Clarence Deming described the 1830 incident as being "much more serious", and stated that the two incidents should be "sharply demarcated".
1825 incident
In 1825, students of the Yale sophomore class claimed that "by explicit contract with (their) mathematical tutor, (they were) exempt from the corollaries of the text-book (on conic sections)", and refused to recite these corollaries. Thirty-eight students out of a class of eighty-seven, including Horace Bushnell, William H. Welch, Henry Hogeboom, and William Adams, were suspended; faculty contacted the students' parents, and the students were pressured into signing a statement of concession:
1830 incident
Prior to the introduction of blackboards, Yale students had been allowed to consult diagrams in their textbooks when solving geometry problems pertaining to conic sections – even on exams. When the students were no longer allowed to consult the text, but were instead required to draw their own diagrams on the blackboard, they refused to take the final exam. As a result, forty-three of the ninety-six students – among them, Alfred Stillé, and Andrew Calhoun, the son of John C. Calhoun – were summarily expelled, and Yale authorities warned neighboring universities against admitting them.
References
Yale University
History of education in the United States
1825 in Connecticut
1830 in Connecticut |
https://en.wikipedia.org/wiki/Trapping%20region | In applied mathematics, a trapping region of a dynamical system is a region such that every trajectory that starts within the trapping region will move to the region's interior and remain there as the system evolves.
More precisely, given a dynamical system with flow defined on the phase space , a subset of the phase space is a trapping region if it is compact and for all .
References
Dynamical systems
Applied mathematics
Systems theory |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20F.C.%20Copenhagen%20season | This article shows statistics of individual players for the football club F.C. Copenhagen. It also lists all matches that F.C. Copenhagen played in the 2011–12 season.
Players
Squad information
This section show the squad as currently, considering all players who are confirmedly moved in and out (see section Players in / out).
Squad stats
Players in / out
In
Out
Club
Coaching staff
Kit
Other information
Competitions
Overall
Danish Superliga
Classification
Results summary
Results by round
UEFA Champions League
Third qualifying round
Play-off round
Results summary
UEFA Europa League
Group B
Classification
Results by round
Results summary
Matches
Competitive
References
External links
F.C. Copenhagen official website
2011-12
Danish football clubs 2011–12 season |
https://en.wikipedia.org/wiki/Affine%20braid%20group | In mathematics, an affine braid group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras. They are subgroups of double affine braid groups.
Definition
References
Macdonald, I. G. Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, Eng., 2003. x+175 pp.
Braid groups
Representation theory |
https://en.wikipedia.org/wiki/Brandt%20semigroup | In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups:
Let G be a group and be non-empty sets. Define a matrix of dimension with entries in
Then, it can be shown that every 0-simple semigroup is of the form with the operation .
As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995).
Thus, a Brandt semigroup has the form with the operation .
Moreover, the matrix is diagonal with only the identity element e of the group G in its diagonal.
Remarks
1) The idempotents have the form (i, e, i) where e is the identity of G.
2) There are equivalent ways to define the Brandt semigroup. Here is another one:
ac = bc ≠ 0 or ca = cb ≠ 0 ⇒ a = b
ab ≠ 0 and bc ≠ 0 ⇒ abc ≠ 0
If a ≠ 0 then there are unique x, y, z for which xa = a, ay = a, za = y.
For all idempotents e and f nonzero, eSf ≠ 0
See also
Special classes of semigroups
References
.
Semigroup theory |
https://en.wikipedia.org/wiki/Free%20function | Free Function may refer to
an Uninterpreted function in mathematics,
a non-member function in the C++ programming language. |
https://en.wikipedia.org/wiki/Munn%20semigroup | In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Walter Douglas Munn (1929–2008).
Construction's steps
Let be a semilattice.
1) For all e in E, we define Ee: = {i ∈ E : i ≤ e} which is a principal ideal of E.
2) For all e, f in E, we define Te,f as the set of isomorphisms of Ee onto Ef.
3) The Munn semigroup of the semilattice E is defined as: TE := { Te,f : (e, f) ∈ U }.
The semigroup's operation is composition of partial mappings. In fact, we can observe that TE ⊆ IE where IE is the symmetric inverse semigroup because all isomorphisms are partial one-one maps from subsets of E onto subsets of E.
The idempotents of the Munn semigroup are the identity maps 1Ee.
Theorem
For every semilattice , the semilattice of idempotents of is isomorphic to E.
Example
Let . Then is a semilattice under the usual ordering of the natural numbers ().
The principal ideals of are then for all .
So, the principal ideals and are isomorphic if and only if .
Thus = {} where is the identity map from En to itself, and if . The semigroup product of and is .
In this example,
References
.
.
Semigroup theory |
https://en.wikipedia.org/wiki/Matsumoto%27s%20theorem%20%28group%20theory%29 | In group theory, Matsumoto's theorem, proved by , gives conditions for two reduced words of a Coxeter group to represent the same element.
Statement
If two reduced words represent the same element of a Coxeter group, then Matsumoto's theorem states that the first word can be transformed into the second by repeatedly transforming
xyxy... to yxyx... (or vice versa)
where
xyxy... = yxyx...
is one of the defining relations of the Coxeter group.
Applications
Matsumoto's theorem implies that there is a natural map (not a group homomorphism) from a Coxeter group to the corresponding braid group, taking any element of the Coxeter group represented by some reduced word in the generators to the same word in the generators of the braid group.
References
Theorems in group theory
Braid groups |
https://en.wikipedia.org/wiki/Hideya%20Matsumoto | Hideya Matsumoto (英也, 松本) is a Japanese mathematician who works on algebraic groups, who proved Matsumoto's theorem about Coxeter groups and Matsumoto's theorem calculating the second K-group of a field.
Publications
References
20th-century Japanese mathematicians
Living people
Year of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Matsumoto%27s%20theorem | In mathematics, Matsumotos's theorem, named for Hideya Matsumoto, may refer to:
Matsumoto's theorem (group theory)
Matsumoto's theorem (K-theory) |
https://en.wikipedia.org/wiki/Josef%20Jindra | Josef Jindra (born June 12, 1980) is a Czech professional ice hockey defenceman. He played with BK Mladá Boleslav in the Czech Extraliga during the 2010–11 Czech Extraliga season.
Career statistics
References
External links
1980 births
Living people
BK Mladá Boleslav players
Czech ice hockey defencemen
HC Tábor players
HKM Zvolen players
IHC Písek players
KLH Vajgar Jindřichův Hradec players
LHK Jestřábi Prostějov players
Motor České Budějovice players
MsHK Žilina players
VHK Vsetín players
Ice hockey people from České Budějovice
Czech expatriate ice hockey players in Slovakia |
https://en.wikipedia.org/wiki/List%20of%20Arizona%20wildfires | This is a list of known wildfires in Arizona.
Statistics
Notable fires
Lesser known fires
References
External links
National Interagency Fire Center
InciWeb - Arizona Incidents
Southwest Coordination Center
Arizona Interagency Wildfire Prevention
US Forest Service
Fire Restrictions - Arizona
Public Lands Information Center - Arizona Fire News
Coconino NF fire history web map
Arizona
Wildfires |
https://en.wikipedia.org/wiki/Heckman%E2%80%93Opdam%20polynomials | In mathematics, Heckman–Opdam polynomials (sometimes called Jacobi polynomials) Pλ(k) are orthogonal polynomials in several variables associated to root systems. They were introduced by .
They generalize Jack polynomials when the roots system is of type A, and are limits of Macdonald polynomials Pλ(q, t) as q tends to 1 and (1 − t)/(1 − q) tends to k.
Main properties of the Heckman–Opdam polynomials have been detailed by Siddhartha Sahi
References
Orthogonal polynomials |
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