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https://en.wikipedia.org/wiki/Multiplicative%20partition | In number theory, a multiplicative partition or unordered factorization of an integer is a way of writing as a product of integers greater than 1, treating two products as equivalent if they differ only in the ordering of the factors. The number is itself considered one of these products. Multiplicative partitions closely parallel the study of multipartite partitions, which are additive partitions of finite sequences of positive integers, with the addition made pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been introduced by . The Latin name "factorisatio numerorum" had been used previously. MathWorld uses the term unordered factorization.
Examples
The number 20 has four multiplicative partitions: 2 × 2 × 5, 2 × 10, 4 × 5, and 20.
3 × 3 × 3 × 3, 3 × 3 × 9, 3 × 27, 9 × 9, and 81 are the five multiplicative partitions of 81 = 34. Because it is the fourth power of a prime, 81 has the same number (five) of multiplicative partitions as 4 does of additive partitions.
The number 30 has five multiplicative partitions: 2 × 3 × 5 = 2 × 15 = 6 × 5 = 3 × 10 = 30.
In general, the number of multiplicative partitions of a squarefree number with prime factors is the th Bell number, .
Application
describe an application of multiplicative partitions in classifying integers with a given number of divisors. For example, the integers with exactly 12 divisors take the forms , , , and , where , , and are distinct prime numbers; these forms correspond to the multiplicative partitions , , , and respectively. More generally, for each multiplicative partition
of the integer , there corresponds a class of integers having exactly divisors, of the form
where each is a distinct prime. This correspondence follows from the multiplicative property of the divisor function.
Bounds on the number of partitions
credits with the problem of counting the number of multiplicative partitions of ; this problem has since been studied by others under the Latin name of factorisatio numerorum. If the number of multiplicative partitions of is , McMahon and Oppenheim observed that its Dirichlet series generating function has the product representation
The sequence of numbers begins
Oppenheim also claimed an upper bound on , of the form
but as showed, this bound is erroneous and the true bound is
Both of these bounds are not far from linear in : they are of the form .
However, the typical value of is much smaller: the average value of , averaged over an interval , is
a bound that is of the form .
Additional results
observe, and prove, that most numbers cannot arise as the number of multiplicative partitions of some : the number of values less than which arise in this way is . Additionally, Luca et al. show that most values of are not multiples of : the number of values such that divides is .
See also
Partition (number theory)
Divisor
References
Further rea |
https://en.wikipedia.org/wiki/Radix%20%28disambiguation%29 | A radix, or base, is the number of unique digits, including zero, used to represent numbers in a positional numeral system.
Radix may also refer to:
Mathematics and science
Radix (gastropod), a genus of freshwater snails
Radical symbol (√), used to indicate a root
Root (Latin: ), in biology
Computing
Radix point, a symbol used in mathematics to separate the integral part of the number from its fractional part
Radix sort, a computer sorting algorithm
Radix tree, a type of set data structure
DEC Radix-50, a character encoding
Radix-64, a character encoding
Entertainment
Radix Ace Entertainment, a Japanese Animation studio
Radix Tetrad, a science fiction novel series by A. A. Attanasio
Radix (novel), the first novel, published in 1981
Radix: Beyond the Void, a 1995 first-person shooter video game
Other
Radix (Company), founded by Bhavin Turakhia
Radix Journal, an online periodical published by the National Policy Institute
See also
Base (disambiguation)
Root (disambiguation) |
https://en.wikipedia.org/wiki/Multinomial%20probit | In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial logit model as one method of multiclass classification. It is not to be confused with the multivariate probit model, which is used to model correlated binary outcomes for more than one independent variable.
General specification
It is assumed that we have a series of observations Yi, for i = 1...n, of the outcomes of multi-way choices from a categorical distribution of size m (there are m possible choices). Along with each observation Yi is a set of k observed values x1,i, ..., xk,i of explanatory variables (also known as independent variables, predictor variables, features, etc.). Some examples:
The observed outcomes might be "has disease A, has disease B, has disease C, has none of the diseases" for a set of rare diseases with similar symptoms, and the explanatory variables might be characteristics of the patients thought to be pertinent (sex, race, age, blood pressure, body-mass index, presence or absence of various symptoms, etc.).
The observed outcomes are the votes of people for a given party or candidate in a multi-way election, and the explanatory variables are the demographic characteristics of each person (e.g. sex, race, age, income, etc.).
The multinomial probit model is a statistical model that can be used to predict the likely outcome of an unobserved multi-way trial given the associated explanatory variables. In the process, the model attempts to explain the relative effect of differing explanatory variables on the different outcomes.
Formally, the outcomes Yi are described as being categorically-distributed data, where each outcome value h for observation i occurs with an unobserved probability pi,h that is specific to the observation i at hand because it is determined by the values of the explanatory variables associated with that observation. That is:
or equivalently
for each of m possible values of h.
Latent variable model
Multinomial probit is often written in terms of a latent variable model:
where
Then
That is,
Note that this model allows for arbitrary correlation between the error variables, so that it doesn't necessarily respect independence of irrelevant alternatives.
When is the identity matrix (such that there is no correlation or heteroscedasticity), the model is called independent probit.
Estimation
For details on how the equations are estimated, see the article Probit model.
References
Regression analysis
Statistical classification |
https://en.wikipedia.org/wiki/Cholponbek%20Esenkul%20Uulu | Cholponbek Esenkul Uulu (born 15 January 1986) is a former Kyrgyzstani footballer who played as a striker.
Career statistics
International
Statistics accurate as of match played 5 September 2014
International Goals
References
External links
1986 births
Living people
Kyrgyzstani men's footballers
Kyrgyzstan men's international footballers
Kyrgyzstani expatriate men's footballers
Footballers at the 2014 Asian Games
Men's association football forwards
Asian Games competitors for Kyrgyzstan |
https://en.wikipedia.org/wiki/Jean-Fran%C3%A7ois%20Le%20Gall | Jean-François Le Gall (born 15 November 1959) is a French mathematician working in areas of probability theory such as Brownian motion, Lévy processes, superprocesses and their connections with partial differential equations, the Brownian snake, random trees, branching processes, stochastic coalescence and random planar maps. He received his Ph.D. in 1982 from Pierre and Marie Curie University (Paris VI) under the supervision of Marc Yor. He is currently professor at the University of Paris-Sud in Orsay and is a senior member of the Institut universitaire de France. He was elected to French academy of sciences, December 2013.
He was awarded the Rollo Davidson Prize in 1986, the Loève Prize in 1997, and the Fermat Prize in 2005. He was the thesis advisor of at least 11 students including Wendelin Werner. For 2019 he received the Wolf Prize in Mathematics. and for 2021 he was awarded the BBVA Foundation Frontiers of Knowledge Award in Basic Sciences.
Books
Le Gall, Jean-François, Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1999). 163 pp.
Le Gall, Jean-François, Brownian Motion, Martingales, and Stochastic Calculus. Graduate Texts in Mathematics. Springer International Publishing Switzerland (2016).
References
External links
professional page
20th-century French mathematicians
21st-century French mathematicians
Academic staff of Paris-Sud University
Probability theorists
École Normale Supérieure alumni
Members of the French Academy of Sciences
Living people
1959 births
People from Morlaix
Pierre and Marie Curie University alumni
Probability Theory and Related Fields editors |
https://en.wikipedia.org/wiki/Hector%20Munro%20Macdonald | Prof Hector Munro Macdonald FRAS FRSE LLD (19 January 1865 – 16 May 1935) was a Scottish mathematician, born in Edinburgh in 1865. He researched pure mathematics at Cambridge University after graduating from Aberdeen University with an honours degree.
Life
Both of Hector Macdonald's parents, his mother Annie Munro and his father Donald Macdonald, were from Kiltearn. Hector was the older of his parents' two sons and, as a young child, he lived in Edinburgh. However, not long after he began his schooling in the Scottish capital, the family moved to a farm near Hill of Fearn, in Easter Ross. After arriving, Hector attended the local school before attending the Royal Academy in Tain. He completed his school education at the Old Aberdeen Grammar School before entering Aberdeen University in 1882.
After studying mathematics at Aberdeen University, he graduated with First Class Honours in 1886 and won a Fullerton Scholarship. Macdonald proceeded to Cambridge to take the Mathematical Tripos after completing his first degree in Scotland. Entering Clare College, Cambridge, as a foundation scholar, he graduated as fourth Wrangler in the Mathematical Tripos of 1889, was awarded a fellowship at Clare in the following year and, in 1891, was awarded the second Smith's Prize.
In 1901 he received the Adams Prize and was elected a Fellow of the Royal Society of London (FRS). He was awarded the Royal Society's Royal Medal in 1916.
Macdonald held his fellowship at Clare College until 1908 and in 1914 he was awarded an honorary fellowship of his former College. From 1916 to 1918 he served as President of the London Mathematical Society. During World War I, Macdonald did war service in London attached to the Ministry of Munitions where he dealt with wages. He was transferred to the Ministry of Labour in 1916, where he remained until 1919. For his services he was appointed an Officer of the Order of the British Empire in the 1918 Birthday Honours.
Macdonald worked on electric waves and solved difficult problems regarding diffraction of these waves by summing series of Bessel functions. He corrected his 1903 solution to the problem of a perfectly conducting sphere embedded in an infinite homogeneous dielectric in 1904 after a subtle error was pointed out by Poincaré. The major problem which he tackled was that of wireless waves. About the time that Macdonald published his prize winning essay on electric waves, Guglielmo Marconi was successful in the transmission of the first wireless signals across the Atlantic. However this posed a major problem at first because wireless signals, like light, should not be capable of being bent round the surface of the earth as apparently Marconi wireless signals were. Macdonald suggested that the wireless waves were being refracted by the atmosphere. It is now known that in fact the waves are reflected by the ionosphere.
Macdonald became Professor of Mathematics at the University of Aberdeen in 1905 and remained at the universi |
https://en.wikipedia.org/wiki/Susan%20Brown | Susan or Sue Brown may refer to:
Susan Brown (mathematician) (1937–2017), British professor of mathematics
L. Susan Brown (born 1959), Canadian anarcha-feminist writer
Susan Brown (minister) (born 1958), Scottish minister
Susan Brown (English actress) (born 1946)
Susan Brown (American actress) (1932–2018)
Susan E. Brown, American medical anthropologist and nutritionist
Sue Brown (cricketer) (born 1958), New Zealand cricketer
Sue Brown (rowing) (born 1958), first woman to take part in The Boat Race (Oxford cox in 1981 and 1982)
Sue Ellen Brown (born 1954), American artist
Sue K. Brown (born 1948), American ambassador to Montenegro
Sue M. Wilson Brown (1877–1941), African-American activist for women's suffrage
Susan Brown (judge), judge of the Supreme Court of Queensland, Australia |
https://en.wikipedia.org/wiki/Communications%20on%20Pure%20and%20Applied%20Mathematics | Communications on Pure and Applied Mathematics is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or mathematical physics. The journal was established in 1948 as the Communications on Applied Mathematics, obtaining its current title the next year. According to the Journal Citation Reports, the journal has a 2020 impact factor of 3.219.
References
External links
Mathematics journals
Monthly journals
Wiley (publisher) academic journals
Academic journals established in 1948
English-language journals |
https://en.wikipedia.org/wiki/Coherent%20ring | In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.
Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings.
Every left Noetherian ring is left coherent. The ring of polynomials in an infinite number of variables over a left Noetherian ring is an example of a left coherent ring that is not left Noetherian.
A ring is left coherent if and only if every direct product of flat right modules is flat , . Compare this to: A ring is left Noetherian if and only if every direct sum of injective left modules is injective.
References
Ring theory |
https://en.wikipedia.org/wiki/CPAM | CPAM may refer to:
Caisse primaire d'assurances maladie, a primary health insurance fund in France.
Center for Performing Arts Medicine
Center for Pure and Applied Mathematics
Certified Patient Account Manager
Certified Public Accountant in Malawi
Christian Petersen Art Museum at Iowa State University
College Park Aviation Museum
Communications on Pure and Applied Mathematics
Community Pesticide Action Monitoring
Concrete Paving Association of Minnesota
Concrete Pipe Association of Michigan
Congenital pulmonary airway malformation
Continental Polar Air Mass
Continuous particulate air monitor
Council of Presidential Awardees of Mathematics, an organization of recipients of the Presidential Award for Excellence in Mathematics and Science Teaching
Crime Prevention Association of Michigan
Cross-Platform Application Management
See also
CJWI on 1410 AM also known as CPAM Radio Union or CPAM 1410 - a French-language Canadian radio station located in Montreal, Quebec with mainly Haitian programming, but also the Latin American and French-speaking African communities |
https://en.wikipedia.org/wiki/Coxeter%E2%80%93Todd%20lattice | In mathematics, the Coxeter–Todd lattice K12, discovered by , is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice).
Properties
The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over the Eisenstein integers. The automorphism group of this complex lattice has index 2 in the full automorphism group of the Coxeter–Todd lattice and is a complex reflection group (number 34 on the list) with structure 6.PSU4(F3).2, called the Mitchell group.
The genus of the Coxeter–Todd lattice was described by and has 10 isometry classes: all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
Construction
Based on Nebe web page we can define K12 using following 6 vectors in 6-dimensional complex coordinates. ω is complex number of order 3 i.e. ω3=1.
(1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0),
½(1,ω,ω,1,0,0), ½(ω,1,ω,0,1,0), ½(ω,ω,1,0,0,1),
By adding vectors having scalar product -½ and multiplying by ω we can obtain all lattice vectors. We have 15 combinations of two zeros times 16 possible signs gives 240 vectors; plus 6 unit vectors times 2 for signs gives 240+12=252 vectors. Multiply it by 3 using multiplication by ω we obtain 756 unit vectors in K12 lattice.
Further reading
The Coxeter–Todd lattice is described in detail in and .
References
External links
Coxeter–Todd lattice in Sloane's lattice catalogue
Quadratic forms |
https://en.wikipedia.org/wiki/Barnes%E2%80%93Wall%20lattice | In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter–Todd lattice.
The automorphism group of the Barnes–Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2). There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice).
The genus of the Barnes–Wall lattice was described by and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16.
The Barnes–Wall lattice is described in detail in .
References
External links
Barnes–Wall lattice at Sloane's lattice catalogue.
Quadratic forms |
https://en.wikipedia.org/wiki/David%20Aldous | David John Aldous FRS (born 13 July 1952) is a mathematician known for his research on probability theory and its applications, in particular in topics such as exchangeability, weak convergence, Markov chain mixing times, the continuum random tree and stochastic coalescence. He entered St. John's College, Cambridge, in 1970 and received his Ph.D. at the University of Cambridge in 1977 under his advisor, D. J. H. Garling. Aldous was on the faculty at University of California, Berkeley from 1979 until his retirement in 2018.
He was awarded the Rollo Davidson Prize in 1980, the Loève Prize in 1993, and was elected a Fellow of the Royal Society in 1994. In 2004, Aldous was elected a Fellow of the American Academy of Arts and Sciences. From 2004 to 2010, Aldous was an Andrew Dickson White Professor-at-Large at Cornell University. He was an invited speaker at the International Congress of Mathematicians (ICM) in 1998 in Berlin and a plenary speaker at the ICM in 2010 in Hyderabad. In 2012 he became a fellow of the American Mathematical Society. He discovered (independently from Andrei Broder) an algorithm for generating a uniform spanning tree of a given graph.
Selected publications
Books
As editor
(pbk reprint of 1995 original)
(pbk reprint of 1996 original)
Papers
Aldous, David, "Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists". Bernoulli 5 (1999) pp. 3–48.
Aldous, David, "Exchangeability and related topics". Lecture Notes in Math., 1117 (1985) pp 1–198. Springer, Berlin.
References
External links
Dynkin Collection Interview
1952 births
Living people
Probability theorists
Alumni of St John's College, Cambridge
University of California, Berkeley College of Letters and Science faculty
Fellows of the Royal Society
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Foreign associates of the National Academy of Sciences
International Mathematical Olympiad participants |
https://en.wikipedia.org/wiki/Kazuhiko%20Chiba | is a Japanese footballer. He currently plays for the J2 League club Albirex Niigata.
Club statistics
Updated to 5 May 2021.
1Includes Japanese Super Cup, FIFA Club World Cup and J. League Championship.
National team statistics
Honours
Club
Sanfrecce Hiroshima
J1 League: 2012, 2013, 2015
Japanese Super Cup: 2013, 2014, 2016
International
Japan
EAFF East Asian Cup: 2013
References
External links
Japan National Football Team Database
Profile at Sanfrecce Hiroshima
Profile at Albirex Niigata
1985 births
Living people
Association football people from Hokkaido
Japanese men's footballers
Japan men's international footballers
Eerste Divisie players
J1 League players
J2 League players
AGOVV players
FC Dordrecht players
Albirex Niigata players
Sanfrecce Hiroshima players
Nagoya Grampus players
Japanese expatriate men's footballers
Expatriate men's footballers in the Netherlands
Japanese expatriate sportspeople in the Netherlands
Men's association football defenders
People from Kushiro, Hokkaido |
https://en.wikipedia.org/wiki/Linn%20Nyr%C3%B8nning | Linn Nyrønning (born 4 June 1981) is a Norwegian football midfielder who currently plays for Trondheims-Ørn.
References
Profile at club site
National team statistics
1981 births
Living people
Norwegian women's footballers
Norway women's international footballers
SK Trondheims-Ørn players
Women's association football midfielders |
https://en.wikipedia.org/wiki/Superstatistics | Superstatistics is a branch of statistical mechanics or statistical physics devoted to the study of non-linear and non-equilibrium systems. It is characterized by using the superposition of multiple differing statistical models to achieve the desired non-linearity. In terms of ordinary statistical ideas, this is equivalent to compounding the distributions of random variables and it may be considered a simple case of a doubly stochastic model.
Consider an extended thermodynamical system which is locally in equilibrium and has a Boltzmann distribution, that is the probability of finding the system in a state with energy is proportional to . Here is the local inverse temperature. A non-equilibrium thermodynamical system is modeled by considering macroscopic fluctuations of the local inverse temperature. These fluctuations happen on time scales which are much larger than the microscopic relaxation times to the Boltzmann distribution. If the fluctuations of are characterized by a distribution , the superstatistical Boltzmann factor of the system is given by
This defines the superstatistical partition function
for system that can assume discrete energy states . The probability of finding the system in state is then given by
Modeling the fluctuations of leads to a description in terms of statistics of Boltzmann statistics, or "superstatistics". For example, if follows a Gamma distribution, the resulting superstatistics corresponds to Tsallis statistics. Superstatistics can also lead to other statistics such as power-law distributions or stretched exponentials. One needs to note here that the word super here is short for superposition of the statistics.
This branch is highly related to the exponential family and Mixing. These concepts are used in many approximation approaches, like particle filtering (where the distribution is approximated by delta functions) for example.
See also
Maxwell–Boltzmann statistics
E.G.D. Cohen
References
Statistical mechanics
Nonlinear systems |
https://en.wikipedia.org/wiki/Tate%27s%20algorithm | In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent fp of p in the conductor of E, the type of reduction at p, the local index
where is the group of -points
whose reduction mod p is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal.
Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces: in turn this determines the exponent fp of the conductor E.
Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and c and f can be read off from the valuations of j and Δ (defined below).
Tate's algorithm was introduced by as an improvement of the description of the Néron model of an elliptic curve by .
Notation
Assume that all the coefficients of the equation of the curve lie in a complete discrete valuation ring R with perfect residue field K and maximal ideal generated by a prime π.
The elliptic curve is given by the equation
Define:
the p-adic valuation of in , that is, exponent of in prime factorization of , or infinity if
The algorithm
Step 1: If π does not divide Δ then the type is I0, c=1 and f=0.
Step 2: If π divides Δ but not c4 then the type is Iv with v = v(Δ), c=v, and f=1.
Step 3. Otherwise, change coordinates so that π divides a3,a4,a6. If π2 does not divide a6 then the type is II, c=1, and f=v(Δ);
Step 4. Otherwise, if π3 does not divide b8 then the type is III, c=2, and f=v(Δ)−1;
Step 5. Otherwise, let Q1 be the polynomial
.
If π3 does not divide b6 then the type is IV, c=3 if has two roots in K and 1 if it has two roots outside of K, and f=v(Δ)−2.
Step 6. Otherwise, change coordinates so that π divides a1 and a2, π2 divides a3 and a4, and π3 divides a6. Let P be the polynomial
If has 3 distinct roots modulo π then the type is I0*, f=v(Δ)−4, and c is 1+(number of roots of P in K).
Step 7. If P has one single and one double root, then the type is Iν* for some ν>0, f=v(Δ)−4−ν, c=2 or 4: there is a "sub-algorithm" for dealing with this case.
Step 8. If P has a triple root, change variables so the triple root is 0, so that π2 divides a2 and π3 divides a4, and π4 divides a6. Let Q2 be the polynomial
.
If has two distinct roots modulo π then the type is IV*, f=v(Δ)−6, and c is 3 if the roots are in K, 1 otherwise.
Step 9. If has a double root, change variables so the double root is 0. Then π3 divides a3 and π5 divides a6. If π4 does not divide a4 then the type is III* and f=v(Δ)−7 and c = 2.
Step 10. Otherwise if π6 does not divide a6 then the type is II* and f=v(Δ)−8 and c = 1.
Step 11. Otherwise the equation is not mini |
https://en.wikipedia.org/wiki/Compound%20of%20five%20cuboctahedra | In geometry, this uniform polyhedron compound is a composition of 5 cuboctahedra. It has icosahedral symmetry Ih.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
(±2, 0, ±2)
(±τ, ±τ−1, ±(2τ−1))
(±1, ±τ−2, ±τ2)
where τ = (1+)/2 is the golden ratio (sometimes written φ).
References
.
Polyhedral compounds |
https://en.wikipedia.org/wiki/Compound%20of%20five%20octahemioctahedra | In geometry, this uniform polyhedron compound is a composition of 5 octahemioctahedra, in the same vertex arrangement as in the compound of 5 cuboctahedra.
Filling
There is some controversy on how to colour the faces of this polyhedron compound. Although the common way to fill in a polygon is to just colour its whole interior, this can result in some filled regions hanging as membranes over empty space. Hence, the "neo filling" is sometimes used instead as a more accurate filling. In the neo filling, orientable polyhedra are filled traditionally, but non-orientable polyhedra have their faces filled with the modulo-2 method (only odd-density regions are filled in). In addition, overlapping regions of coplanar faces can cancel each other out. Usage of the "neo filling" makes the compound of five octahemioctahedra a hollow polyhedron compound.
References
.
Polyhedral compounds |
https://en.wikipedia.org/wiki/Theaker%20Wilder | Theaker Wilder (1717–1778) was an Anglo-Irish academic with expertise in mathematics and Greek. He was the first Regius Professor of Greek, Senior Register and Senior Fellow at Trinity College Dublin. He is remembered for being Oliver Goldsmith's 'learned savage' of a tutor.
Family
Born in 1717 at Castle Wilder, Abbeyshrule, County Longford, Wilder was the youngest son of Mathew Wilder (d.1719) of Castle Wilder, High Sheriff of Longford, and Eleanor Steuart (d.1729), co-heiress of her uncle General Sir William Steuart. His mother was a daughter of Captain James Steuart (d.1689), but after his death at the Siege of Londonderry, she and her brothers and sisters were brought up by their father's younger brother, General Steuart and his first wife, Katherine FitzGerald, Viscountess Grandison. Theaker Wilder's uncles included Admiral James Steuart M.P., Admiral of the Fleet; Charles Steaurt (d.1740) of Bailieborough Castle, County Cavan; and Brigadier-General The Hon. William Steuart (d.1737) M.P., of Ballylane, County Waterford, who through his marriage to his uncle's stepdaughter, the Hon. Mary FitzGerald-Villiers, came to be the uncle of Prime Minister William Pitt, 1st Earl of Chatham.
Education
Wilder was tutored at home by Dr. Elwood before entering Trinity College Dublin as a pensioner (ordinary student) on 8 July 1734. He became a Scholar in 1736, earned a BA degree in 1738, got an M.A. in 1741, and became a Fellow in 1744. He also received a BD in 1748 and a D.D. in 1753. He was appointed Donegall Lecturer in Mathematics at TCD in 1759 and the first Regius Professor of Greek in 1761. He was succeeded in this latter post by John Stokes in 1760 and 1764, respectively.
Academics at Trinity College led an affluent life. According to Thomas D'Arcy McGee in his book A Popular History of Ireland: from the Earliest Period to the Emancipation of the Catholics:
The Established Church continued, of course, to monopolise University honours, and to enjoy its princely revenues and all political advantages. Trinity College continued annually to farm its 200,000 acres at a rental averaging 100,000 pounds sterling. Its wealth, and the uses to which it is put, are thus described by a recent writer:
"Some of Trinity's senior fellows enjoy higher incomes than Cabinet ministers; many of her tutors have revenues above those of cardinals; and junior fellows, of a few days' standing, frequently decline some of her thirty-one church livings with benefices which would shame the poverty of scores of continental, not to say Irish, Catholic archbishops. Even eminent judges hold her professorships; some of her chairs are vacated for the
Episcopal bench only; and majors and field officers would acquire increased pay by being promoted to the rank of head porter, first menial, in Trinity College. Apart from her princely fellowships and professorships, her seventy Foundation, and sixteen non-Foundation Scholarships, her thirty Sizarships, and her fourteen valuable Student |
https://en.wikipedia.org/wiki/Greg%20Lawler | Gregory Francis Lawler (born July 14, 1955) is an American mathematician working in probability theory and best known for his work since 2000 on the Schramm–Loewner evolution.
He received his PhD from Princeton University in 1979 under the supervision of Edward Nelson. He was on the faculty of Duke University from 1979 to 2001, of Cornell University from 2001 to 2006, and since 2006 is at the University of Chicago.
Awards and honors
He received the 2006 SIAM George Pólya Prize with Oded Schramm and Wendelin Werner.
In 2019 he received the Wolf Prize in Mathematics.
Lawler is a member of the National Academy of Sciences (since 2013)
and the American Academy of Arts and Sciences (since 2005).
Since 2012, he has been a fellow of the American Mathematical Society.
He gave an invited lecture at the International Congress of Mathematicians in Beijing (2002)
and a plenary lecture at the ICM in Rio de Janeiro (2018).
References
External links
Personal home page
(Plenary Lecture 5)
1955 births
20th-century American mathematicians
21st-century American mathematicians
Cornell University faculty
Probability theorists
University of Chicago faculty
Fellows of the American Mathematical Society
Living people
Annals of Probability editors |
https://en.wikipedia.org/wiki/Compound%20of%20three%20cubes | In geometry, the compound of three cubes is a uniform polyhedron compound formed from three cubes arranged with octahedral symmetry. It has been depicted in works by Max Brückner and M.C. Escher.
History
This compound appears in Max Brückner's book Vielecke und Vielflache (1900), and in the lithograph print Waterfall (1961) by M.C. Escher, who learned of it from Brückner's book. Its dual, the compound of three octahedra, forms the central image in an earlier Escher woodcut, Stars.
In the 15th-century manuscript De quinque corporibus regularibus, Piero della Francesca includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three cubes inscribed in this way within a single octahedron would form the compound of three cubes, but della Francesca does not depict the compound.
Construction and coordinates
This compound can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis (that passes through the centres of two opposite faces).
Cartesian coordinates for the vertices of this compound can be chosen as all the permutations of .
References
Polyhedral compounds |
https://en.wikipedia.org/wiki/Mitchell%27s%20group | In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by . It has the structure 6.PSU4(F3).2. As a complex reflection group it has 126 reflections of order 2, and its ring of invariants is a polynomial algebra with generators of degrees 6, 12, 18, 24, 30, 42. Coxeter gives it group symbol [1 2 3]3 and Coxeter-Dynkin diagram .
Mitchell's group is an index 2 subgroup of the automorphism group of the Coxeter–Todd lattice.
References
Finite groups |
https://en.wikipedia.org/wiki/Minlos%27s%20theorem | In the mathematics of topological vector spaces, Minlos's theorem states that a cylindrical measure on the dual of a nuclear space is a Radon measure if its Fourier transform is continuous. It is named after Robert Adol'fovich Minlos and can be proved using Sazonov's theorem.
References
Theorems in functional analysis
Theorems regarding stochastic processes |
https://en.wikipedia.org/wiki/Artin%E2%80%93Zorn%20theorem | In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.
The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane is the classical projective plane over a finite field.
References
Theorems in ring theory |
https://en.wikipedia.org/wiki/Coherence%20condition | In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.
An illustrative example: a monoidal category
Part of the data of a monoidal category is a chosen morphism
, called the associator:
for each triple of objects in the category. Using compositions of these , one can construct a morphism
Actually, there are many ways to construct such a morphism as a composition of various . One coherence condition that is typically imposed is that these compositions are all equal.
Typically one proves a coherence condition using a coherence theorem, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects , the following diagram commutes.
Any pair of morphisms from to constructed as compositions of various are equal.
Further examples
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.
Identity
Let be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms and . By composing these with f, we construct two morphisms:
, and
.
Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:
.
Associativity of composition
Let , and be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways:
, and
.
We have now the following coherence statement:
.
In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.
References
Category theory |
https://en.wikipedia.org/wiki/Trace%20identity | In mathematics, a trace identity is any equation involving the trace of a matrix.
Properties
Trace identities are invariant under simultaneous conjugation.
Uses
They are frequently used in the invariant theory of matrices to find the generators and relations of the ring of invariants, and therefore are useful in answering questions similar to that posed by Hilbert's fourteenth problem.
Examples
The Cayley–Hamilton theorem says that every square matrix satisfies its own characteristic polynomial. This also implies that all square matrices satisfy where the coefficients are given by the elementary symmetric polynomials of the eigenvalues of .
All square matrices satisfy
See also
References
.
Invariant theory
Linear algebra |
https://en.wikipedia.org/wiki/Johann%20Friedrich%20Hennert | Johann Friedrich Hennert (19 October 1733 – 30 March 1813) was German-born and lectured in mathematics and physics at the University of Utrecht. He was a significant student of Leonhard Euler. He was known for his inclination towards the British school of philosophy.
Work
Hennert held the chair of mathematics at the University of Utrecht until 1805.
Hennert was an important figure in the history of Dutch mathematics. He wrote a number of textbooks on differential calculus.
Jan van Swinden was one of his most important students.
References
Helmers Dini M., Timmerman, Petronella Johanna de, in: Digitaal Vrouwenlexicon van Nederland.
External links
18th-century German mathematicians
19th-century German mathematicians
Number theorists
1733 births
1813 deaths
Mathematical analysts |
https://en.wikipedia.org/wiki/McShane | McShane may refer to:
McShane (name)
McShane Bell Foundry, church bell manufacturer, located in Glen Burnie, Maryland, USA
See also
McShane's identity, geometric topology
Shane (disambiguation)
MacShane
O'Shane |
https://en.wikipedia.org/wiki/Dominance%20order | In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of a positive integer n that plays an important role in algebraic combinatorics and representation theory, especially in the context of symmetric functions and representation theory of the symmetric group.
Definition
If p = (p1,p2,…) and q = (q1,q2,…) are partitions of n, with the parts arranged in the weakly decreasing order, then p precedes q in the dominance order if for any k ≥ 1, the sum of the k largest parts of p is less than or equal to the sum of the k largest parts of q:
In this definition, partitions are extended by appending zero parts at the end as necessary.
Properties of the dominance ordering
Among the partitions of n, (1,…,1) is the smallest and (n) is the largest.
The dominance ordering implies lexicographical ordering, i.e. if p dominates q and p ≠ q, then for the smallest i such that pi ≠ qi one has pi > qi.
The poset of partitions of n is linearly ordered (and is equivalent to lexicographical ordering) if and only if n ≤ 5. It is graded if and only if n ≤ 6. See image at right for an example.
A partition p covers a partition q if and only if pi = qi + 1, pk = qk − 1, pj = qj for all j ≠ i,k and either (1) k = i + 1 or (2) qi = qk (Brylawski, Prop. 2.3). Starting from the Young diagram of q, the Young diagram of p is obtained from it by first removing the last box of row k and then appending it either to the end of the immediately preceding row k − 1, or to the end of row i < k if the rows i through k of the Young diagram of q all have the same length.
Every partition p has a conjugate (or dual) partition p′, whose Young diagram is the transpose of the Young diagram of p. This operation reverses the dominance ordering:
if and only if
The dominance ordering determines the inclusions between the Zariski closures of the conjugacy classes of nilpotent matrices.
Lattice structure
Partitions of n form a lattice under the dominance ordering, denoted Ln, and the operation of conjugation is an antiautomorphism of this lattice. To explicitly describe the lattice operations, for each partition p consider the associated (n + 1)-tuple:
The partition p can be recovered from its associated (n+1)-tuple by applying the step 1 difference, Moreover, the (n+1)-tuples associated to partitions of n are characterized among all integer sequences of length n + 1 by the following three properties:
Nondecreasing,
Concave,
The initial term is 0 and the final term is n,
By the definition of the dominance ordering, partition p precedes partition q if and only if the associated (n + 1)-tuple of p is term-by-term less than or equal to the associated (n + 1)-tuple of q. If p, q, r are partitions then if and only if The componentwise minimum of two nondecreasing concave integer sequences is also nondecreasing and concave. Therefore, for any two partitions of n, p and q, |
https://en.wikipedia.org/wiki/Covering%20relation | In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours. The covering relation is commonly used to graphically express the partial order by means of the Hasse diagram.
Definition
Let be a set with a partial order .
As usual, let be the relation on such that if and only if and .
Let and be elements of .
Then covers , written ,
if and there is no element such that . Equivalently, covers if the interval is the two-element set .
When , it is said that is a cover of . Some authors also use the term cover to denote any such pair in the covering relation.
Examples
In a finite linearly ordered set {1, 2, ..., n}, i + 1 covers i for all i between 1 and n − 1 (and there are no other covering relations).
In the Boolean algebra of the power set of a set S, a subset B of S covers a subset A of S if and only if B is obtained from A by adding one element not in A.
In Young's lattice, formed by the partitions of all nonnegative integers, a partition λ covers a partition μ if and only if the Young diagram of λ is obtained from the Young diagram of μ by adding an extra cell.
The Hasse diagram depicting the covering relation of a Tamari lattice is the skeleton of an associahedron.
The covering relation of any finite distributive lattice forms a median graph.
On the real numbers with the usual total order ≤, the cover set is empty: no number covers another.
Properties
If a partially ordered set is finite, its covering relation is the transitive reduction of the partial order relation. Such partially ordered sets are therefore completely described by their Hasse diagrams. On the other hand, in a dense order, such as the rational numbers with the standard order, no element covers another.
References
.
.
.
Binary relations
Order theory |
https://en.wikipedia.org/wiki/Algebraic%20combinatorics | Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
History
The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of symmetries (association schemes, strongly regular graphs, posets with a group action) or possessed a rich algebraic structure, frequently of representation theoretic origin (symmetric functions, Young tableaux). This period is reflected in the area 05E, Algebraic combinatorics, of the AMS Mathematics Subject Classification, introduced in 1991.
Scope
Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be enumerative in nature or involve matroids, polytopes, partially ordered sets, or finite geometries. On the algebraic side, besides group theory and representation theory, lattice theory and commutative algebra are commonly used.
Important topics
Symmetric functions
The ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number n of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the representation theory of the symmetric groups.
Association schemes
An association scheme is a collection of binary relations satisfying certain compatibility conditions. Association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
Strongly regular graphs
A strongly regular graph is defined as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:
Every two adjacent vertices have λ common neighbours.
Every two non-adjacent vertices have μ common neighbours.
A graph of this kind is sometimes said to be a srg(v, k, λ, μ).
Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs, and their complements, the Turán graphs.
Young tableaux
A Young tableau (pl.: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups |
https://en.wikipedia.org/wiki/Journal%20of%20Algebraic%20Combinatorics | Journal of Algebraic Combinatorics is a peer-reviewed scientific journal covering algebraic combinatorics. It was established in 1992 and is published by Springer Science+Business Media. The editor-in-chief is Ilias S. Kotsireas (Wilfrid Laurier University).
In 2017, the journal's four editors-in-chief and editorial board resigned to protest the publisher's high prices and limited accessibility. They criticized Springer for "double-dipping", that is, charging large subscription fees to libraries in addition to high fees for authors who wished to make their publications open access. The board subsequently started their own open access journal, Algebraic Combinatorics.
Abstracting and indexing
The journal is abstracted and indexed in:
According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.875.
References
External links
Combinatorics journals
Academic journals established in 1992
Springer Science+Business Media academic journals
English-language journals |
https://en.wikipedia.org/wiki/Young%27s%20lattice | In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed the representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of . It is also closely connected with the crystal bases for affine Lie algebras.
Definition
Young's lattice is a lattice (and hence also a partially ordered set) Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).
Significance
The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from Sn +1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn +1 with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of Sn with partition p, which is indexed by the standard Young tableaux of shape p.
Properties
The poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank.
The poset Y is a lattice. The meet and join of two partitions are given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice in which the meet and join operations are represented by intersections and unions, it is a distributive lattice.
If a partition p covers k elements of Young's lattice for some k then it is covered by k + 1 elements. All partitions covered by p can be found by removing one of the "corners" of its Young diagram (boxes at the end both of their row and of their column). All partitions covering p can be found by adding one of the "dual corners" to its Young diagram (boxes outside the diagram that are the first such box both in their row and in their column). There is always a dual corner in the first row, and for each other dual corner there is a corner in the previous row, whence the stated property.
If distinct partitions p and q both cover k elements of Y then k is 0 or 1, and p and q are covered by k elements. In plain language: two partitions can have at most one (third) partition covered by both (their respective diagrams then each have one box not belonging to the othe |
https://en.wikipedia.org/wiki/Journal%20of%20Algebra | Journal of Algebra (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. Journal of Algebra was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief.
In 2004, Journal of Algebra announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005).
The Editor-in-Chief of the Journal of Algebra is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen (RWTH) is Editor of the computational algebra section.
See also
Susan Montgomery, an editor of the journal
External links
Journal of Algebra at ScienceDirect
Mathematics journals
Academic journals established in 1964 |
https://en.wikipedia.org/wiki/Lusin%20space | In mathematics, a Lusin space or Luzin space, named for N. N. Luzin, may mean:
In general topology, Polish space #Lusin spaces, image of a Polish space under a bijective continuous map
In descriptive set theory and general topology, Luzin space or Luzin set, a hypothetical uncountable topological T1 space without isolated points in which every nowhere-dense subset is at most countable |
https://en.wikipedia.org/wiki/Arnaud%20Vincent | Arnaud Vincent (born 30 November 1974) is a French former Grand Prix motorcycle road racer. He was the 2002 F.I.M. 125cc world champion.
Career statistics
Grand Prix motorcycle racing
Races by year
(key) (Races in bold indicate pole position, races in italics indicate fastest lap)
Supersport World Championship
Races by year
(key)
References
1974 births
Living people
Sportspeople from Meurthe-et-Moselle
French motorcycle racers
250cc World Championship riders
125cc World Championship riders
Supersport World Championship riders
125cc World Riders' Champions |
https://en.wikipedia.org/wiki/Prismatic%20compound%20of%20antiprisms | In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.
Infinite family
This infinite family can be enumerated as follows:
For each positive integer n≥1 and for each rational number p/q>3/2 (expressed with p and q coprime), there occurs the compound of n p/q-gonal antiprisms, with symmetry group:
Dnpd if nq is odd
Dnph if nq is even
Where p/q=2, the component is the tetrahedron (or dyadic antiprism). In this case, if n=2 then the compound is the stella octangula, with higher symmetry (Oh).
Compounds of two antiprisms
Compounds of two n-antiprisms share their vertices with a 2n-prism, and exist as two alternated set of vertices.
Cartesian coordinates for the vertices of an antiprism with n-gonal bases and isosceles triangles are
with k ranging from 0 to 2n−1; if the triangles are equilateral,
Compound of two trapezohedra (duals)
The duals of the prismatic compound of antiprisms are compounds of trapezohedra:
Compound of three antiprisms
For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.
References
.
Polyhedral compounds |
https://en.wikipedia.org/wiki/Composition%20ring | In mathematics, a composition ring, introduced in , is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation
such that, for any three elements one has
It is not generally the case that , nor is it generally the case that (or ) has any algebraic relationship to and .
Examples
There are a few ways to make a commutative ring R into a composition ring without introducing anything new.
Composition may be defined by for all f,g. The resulting composition ring is a rather uninteresting.
Composition may be defined by for all f,g. This is the composition rule for constant functions.
If R is a boolean ring, then multiplication may double as composition: for all f,g.
More interesting examples can be formed by defining a composition on another ring constructed from R.
The polynomial ring R[X] is a composition ring where for all .
The formal power series ring R also has a substitution operation, but it is only defined if the series g being substituted has zero constant term (if not, the constant term of the result would be given by an infinite series with arbitrary coefficients). Therefore, the subset of R formed by power series with zero constant coefficient can be made into a composition ring with composition given by the same substitution rule as for polynomials. Since nonzero constant series are absent, this composition ring does not have a multiplicative unit.
If R is an integral domain, the field R(X) of rational functions also has a substitution operation derived from that of polynomials: substituting a fraction g1/g2 for X into a polynomial of degree n gives a rational function with denominator , and substituting into a fraction is given by
However, as for formal power series, the composition cannot always be defined when the right operand g is a constant: in the formula given the denominator should not be identically zero. One must therefore restrict to a subring of R(X) to have a well-defined composition operation; a suitable subring is given by the rational functions of which the numerator has zero constant term, but the denominator has nonzero constant term. Again this composition ring has no multiplicative unit; if R is a field, it is in fact a subring of the formal power series example.
The set of all functions from R to R under pointwise addition and multiplication, and with given by composition of functions, is a composition ring. There are numerous variations of this idea, such as the ring of continuous, smooth, holomorphic, or polynomial functions from a ring to itself, when these concepts makes sense.
For a concrete example take the ring , considered as the ring of polynomial maps from the integers to itself. A ring endomorphism
of is determined by the image under of the variable , which we denote by
and this image can be any element of . Therefore, one may consider the elements as endomorphisms and assign , accordingly. One easily verifies |
https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss%20lemma | In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection.
The lemma has applications in compressed sensing, manifold learning, dimensionality reduction, and graph embedding. Much of the data stored and manipulated on computers, including text and images, can be represented as points in a high-dimensional space (see vector space model for the case of text). However, the essential algorithms for working with such data tend to become bogged down very quickly as dimension increases. It is therefore desirable to reduce the dimensionality of the data in a way that preserves its relevant structure. The Johnson–Lindenstrauss lemma is a classic result in this vein.
Also, the lemma is tight up to a constant factor, i.e. there exists a set of points of size m that needs dimension
in order to preserve the distances between all pairs of points within a factor of .
Lemma
Given , a set of points in (), and an integer , there is a linear map such that
for all .
The formula can be rearranged:
Alternatively, for any and any integer there exists a linear function such that the restriction is -bi-Lipschitz.
One proof of the lemma takes ƒ to be a suitable multiple of the orthogonal projection onto a random subspace of dimension in , and exploits the phenomenon of concentration of measure.
An orthogonal projection will, in general, reduce the average distance between points, but the lemma can be viewed as dealing with relative distances, which do not change under scaling. In a nutshell, you roll the dice and obtain a random projection, which will reduce the average distance, and then you scale up the distances so that the average distance returns to its previous value. If you keep rolling the dice, you will, in polynomial random time, find a projection for which the (scaled) distances satisfy the lemma.
Alternate statement
A related lemma is the distributional JL lemma. This lemma states that for any and positive integer , there exists a distribution over from which the matrix is drawn such that for and for any unit-length vector , the claim below holds.
One can obtain the JL lemma from the distributional version by setting and for some pair u,v both in X. Then the JL lemma follows by a union bound over all such pairs.
Speeding up the JL transform
Given A, computing the matrix vector product takes time. There has been some work in deriving distributions for which the matrix vector product can be computed in less than time.
There are two major lines of work. The first, Fast Joh |
https://en.wikipedia.org/wiki/Moufang%20plane | In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a translation line, that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line. A translation plane is Moufang if every line of the plane is a translation line.
Characterizations
A Moufang plane can also be described as a projective plane in which the little Desargues theorem holds. This theorem states that a restricted form of Desargues' theorem holds for every line in the plane.
For example, every Desarguesian plane is a Moufang plane.
In algebraic terms, a projective plane over any alternative division ring is a Moufang plane, and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and of Moufang planes.
As a consequence of the algebraic Artin–Zorn theorem, that every finite alternative division ring is a field, every finite Moufang plane is Desarguesian, but some infinite Moufang planes are non-Desarguesian planes. In particular, the Cayley plane, an infinite Moufang projective plane over the octonions, is one of these because the octonions do not form a division ring.
Properties
The following conditions on a projective plane P are equivalent:
P is a Moufang plane.
The group of automorphisms fixing all points of any given line acts transitively on the points not on the line.
Some ternary ring of the plane is an alternative division ring.
P is isomorphic to the projective plane over an alternative division ring.
Also, in a Moufang plane:
The group of automorphisms acts transitively on quadrangles.
Any two ternary rings of the plane are isomorphic.
See also
Moufang loop
Moufang polygon
Notes
References
Further reading
Projective geometry |
https://en.wikipedia.org/wiki/Cecil%20County%20Public%20Schools | Cecil County Public Schools is a public school system serving the residents of Cecil County, Maryland. Demographics, assessments, and statistics are available on the Maryland Report Card website.
It is the school district for the entire county.
In 2023, it was concluded that Cecil County Public Schools was the least funded school in the state of Maryland per capita, much in part to the county executive's allocation of funding to reduce taxes.
High schools
Bohemia Manor High School, Chesapeake City, MD
Elkton High School, Elkton, MD
North East High School, North East, MD
Perryville High School, Perryville, MD
Rising Sun High School, North East, MD
Middle schools
Bohemia Manor, Chesapeake City, MD
Cherry Hill, Elkton, MD
Elkton, Elkton, MD
North East, North East, MD
Perryville, Perryville, MD
Rising Sun, Rising Sun, MD
Elementary schools
Bainbridge, Port Deposit
Bay View, North East
Calvert, Rising Sun
Cecil Manor, Elkton
Cecilton, Cecilton
Charlestown, Charlestown
Chesapeake City, Chesapeake City
The previous facility, about in size, is in the southern part of Chesapeake City, along the Chesapeake & Delaware Canal. In 2019 a groundbreaking for the new school facility, along Augustine Herman Highway at the midpoint between the Bohemia Manor secondary schools and the Cheseapeake City fire department facility, was imminent. The facility, with about in area, is designed to look like the area bridge. The building's model is Gilpin Manor Elementary School.
Conowingo, Conowingo
Elk Neck, Elkton
Gilpin Manor, Elkton
Holly Hall, Elkton
Kenmore, Elkton
Leeds, Elkton
North East, North East
Perryville, Perryville
Port Deposit, Port Deposit
Rising Sun, Rising Sun
Thomson Estates, Elkton
Other
Cecil Alternative Program at Providence, Elkton, MD
Cecil County School of Technology, Elkton, MD
News
In the fall of 2015, the new Cecil County School of Technology opened in Elkton, MD.
In 2018, the New Gilpin Manor Elementary School opened next to the original building which was over 60 years old. The original building has since been razed.
In 2021, the New Chesapeake City Elementary School opened as part of the same campus where Bohemia Manor Middle School and High School exist. The original building was located closer to downtown Chesapeake City.
In 2021, Cecil County Public Schools announced they would construct a new shared campus for North East Middle School and High School which will be located near the current high school.
Controversies
On October 27, 2023, Cecil County Public Schools alerted affected fine arts teachers their programs may be cut the following school year because the county executive was developing a budget that would slash CCPS funding, in turn cutting funding for the Fine Arts to the minimum required by Maryland law. This caused outrage amongst students, parents, and the greater community, including actions taken across social media and planning mass public uproar at upcoming Board of Education meetings and town hall events.
|
https://en.wikipedia.org/wiki/Wintner | Wintner is a surname. Notable people with the surname include:
Aurel Wintner (1903–1958), mathematician; one of the founders of probabilistic number theory
Robert Wintner, author and entrepreneur
See also
Tintner
Winter (surname) |
https://en.wikipedia.org/wiki/Senador%20Vasconcelos | Senador Vasconcelos is a neighborhood in the West Zone of Rio de Janeiro, Brazil.
Neighborhood statistics
Total area (2003): 644.18 hectares.
Total population (2010): 30,600
Total of domiciles (2010): 9,826
Administrative region: XVIII - Campo Grande.
References
Neighbourhoods in Rio de Janeiro (city) |
https://en.wikipedia.org/wiki/Brocard%20triangle | In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points constructed using different combinations of vertices and Brocard points. This triangle is also called the first Brocard triangle, as further triangles can be formed by forming the Brocard triangle of the Brocard triangle and continuing this pattern. The Brocard triangle is inscribed in the Brocard circle. It is named for Henri Brocard.
See also
Henri Brocard
Brocard points
Notes
Triangles |
https://en.wikipedia.org/wiki/Weyl%20connection | In differential geometry, a Weyl connection (also called a Weyl structure) is a generalization of the Levi-Civita connection that makes sense on a conformal manifold. They were introduced by Hermann Weyl in an attempt to unify general relativity and electromagnetism. His approach, although it did not lead to a successful theory, lead to further developments of the theory in conformal geometry, including a detailed study by Élie Cartan . They were also discussed in .
Specifically, let be a smooth manifold, and a conformal class of (non-degenerate) metric tensors on , where iff for some smooth function (see Weyl transformation). A Weyl connection is a torsion free affine connection on such that, for any ,
where is a one-form depending on .
If is a Weyl connection and , then
so the one-form transforms by
Thus the notion of a Weyl connection is conformally invariant, and the change in one-form is mediated by a de Rham cocycle.
An example of a Weyl connection is the Levi-Civita connection for any metric in the conformal class , with . This is not the most general case, however, as any such Weyl connection has the property that the one-form is closed for all belonging to the conformal class. In general, the Ricci curvature of a Weyl connection is not symmetric. Its skew part is the dimension times the two-form , which is independent of in the conformal class, because the difference between two is a de Rham cocycle. Thus, by the Poincaré lemma, the Ricci curvature is symmetric if and only if the Weyl connection is locally the Levi-Civita connection of some element of the conformal class.
Weyl's original hope was that the form could represent the vector potential of electromagnetism (a gauge dependent quantity), and the field strength (a gauge invariant quantity). This synthesis is unsuccessful in part because the gauge group is wrong: electromagnetism is associated with a gauge field, not an gauge field.
showed that an affine connection is a Weyl connection if and only if its holonomy group is a subgroup of the conformal group. The possible holonomy algebras in Lorentzian signature were analyzed in .
A Weyl manifold is a manifold admitting a global Weyl connection. The global analysis of Weyl manifolds is actively being studied. For example, considered complete Weyl manifolds such that the Einstein vacuum equations hold, an Einstein–Weyl geometry, obtaining a complete characterization in three dimensions.
Weyl connections also have current applications in string theory and holography.
Weyl connections have been generalized to the setting of parabolic geometries, of which conformal geometry is a special case, in .
Citations
References
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Further reading
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See also
Einstein–Weyl geometry
External links
Conformal geometry
Connection (mathematics) |
https://en.wikipedia.org/wiki/Conformal%20dimension | In mathematics, the conformal dimension of a metric space X is the infimum of the Hausdorff dimension over the conformal gauge of X, that is, the class of all metric spaces quasisymmetric to X.
Formal definition
Let X be a metric space and be the collection of all metric spaces that are quasisymmetric to X. The conformal dimension of X is defined as such
Properties
We have the following inequalities, for a metric space X:
The second inequality is true by definition. The first one is deduced from the fact that the topological dimension T is invariant by homeomorphism, and thus can be defined as the infimum of the Hausdorff dimension over all spaces homeomorphic to X.
Examples
The conformal dimension of is N, since the topological and Hausdorff dimensions of Euclidean spaces agree.
The Cantor set K is of null conformal dimension. However, there is no metric space quasisymmetric to K with a 0 Hausdorff dimension.
See also
Anomalous scaling dimension
References
Fractals
Metric geometry
Dimension theory |
https://en.wikipedia.org/wiki/George%20Anderson%20%28footballer%2C%20born%201904%29 | George Russell Anderson (24 October 1904 – December 1974) was a Scottish professional footballer who played in the Football League for a number of clubs as a forward.
Career statistics
Sources
Canary Citizens by Mike Davage, John Eastwood, Kevin Platt, published by Jarrold Publishing, (2001),
99 Years & Counting – Stats & Stories – Huddersfield Town History
References
1904 births
1974 deaths
People from Saltcoats
Scottish men's footballers
Men's association football forwards
Brentford F.C. players
Chelsea F.C. players
Norwich City F.C. players
Gillingham F.C. players
Bury F.C. players
Huddersfield Town A.F.C. players
Mansfield Town F.C. players
English Football League players
Footballers from North Ayrshire
Scottish Football League players
Dalry Thistle F.C. players
Airdrieonians F.C. (1878) players
Carlisle United F.C. players
Cowdenbeath F.C. players
Yeovil Town F.C. players
Southern Football League players
Midland Football League players
Ayr United F.C. players
Saltcoats Victoria F.C. players
Newark F.C. players |
https://en.wikipedia.org/wiki/Ren%C3%A9%20de%20Saussure | René de Saussure (17 March 1868 – 2 December 1943) was a Swiss Esperantist and professional mathematician (he defended a doctoral thesis on a subject in geometry at the Johns Hopkins University in 1895 and until 1899 he was professor at the Catholic University of America in Washington, D.C., and later in Geneva and Berne), who composed important works about Esperanto and interlinguistics from a linguistic viewpoint. He was born in Geneva, Switzerland. His chef d'oeuvre is an analysis on the logic of word construction in Esperanto, Fundamentaj reguloj de la vortteorio en Esperanto ("Fundamental rules of word theory in Esperanto"), defending the language against several Idist critiques. He developed the concept of neceso kaj sufiĉo ("necessity and sufficience") by which he opposed the criticism of Louis Couturat that Esperanto lacks recursion.
In 1907, de Saussure proposed the international currency spesmilo (₷). It was used by the Ĉekbanko esperantista and other British and Swiss banks until the First World War.
Beginning in 1919, de Saussure proposed a series of Esperanto reforms, and in 1925, he renounced Esperanto in favor of his language Esperanto II. He later became a consultant for the International Auxiliary Language Association, the linguistic research body that standardized and presented Interlingua. He died on 2 December 1943 in Berne, Switzerland.
René was the brother of the linguist Ferdinand de Saussure and the scholar of ancient Chinese astronomy, Léopold de Saussure. His father was the scientist, Henri Louis Frédéric de Saussure.
A new silver Esperanto coin for 100 Steloj was struck in 2018 for the 150th birthday of René de Saussure.
References
1868 births
1943 deaths
Swiss Esperantists
Swiss mathematicians
Morphologists
Writers from Geneva |
https://en.wikipedia.org/wiki/Ordered%20weighted%20averaging | In applied mathematics, specifically in fuzzy logic, the ordered weighted averaging (OWA) operators provide a parameterized class of mean type aggregation operators. They were introduced by Ronald R. Yager.
Many notable mean operators such as the max, arithmetic average, median and min, are members of this class. They have been widely used in computational intelligence because of their ability to model linguistically expressed aggregation instructions.
Definition
An OWA operator of dimension is a mapping that has an associated collection of weights lying in the unit interval and summing to one and with
where is the jth largest of the .
By choosing different W one can implement different aggregation operators. The OWA operator is a non-linear operator as a result of the process of determining the bj.
Notable OWA operators
if and for
if and for
if for all
Properties
The OWA operator is a mean operator. It is bounded, monotonic, symmetric, and idempotent, as defined below.
Characterizing features
Two features have been used to characterize the OWA operators. The first is the attitudinal character, also called orness. This is defined as
It is known that .
In addition A − C(max) = 1, A − C(ave) = A − C(med) = 0.5 and A − C(min) = 0. Thus the A − C goes from 1 to 0 as we go from Max to Min aggregation. The attitudinal character characterizes the similarity of aggregation to OR operation(OR is defined as the Max).
The second feature is the dispersion. This defined as
An alternative definition is The dispersion characterizes how uniformly the arguments are being used.
Type-1 OWA aggregation operators
The above Yager's OWA operators are used to aggregate the crisp values. Can we aggregate fuzzy sets in the OWA mechanism? The
Type-1 OWA operators have been proposed for this purpose.
So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.
The type-1 OWA operator is defined according to the alpha-cuts of fuzzy sets as follows:
Given the n linguistic weights in the form of fuzzy sets defined on the domain of discourse , then for each , an -level type-1 OWA operator with -level sets to aggregate the -cuts of fuzzy sets is given as
where , and is a permutation function such that , i.e., is the th largest
element in the set .
The computation of the type-1 OWA output is implemented by computing the left end-points and right end-points of the intervals :
and
where . Then membership function of resulting aggregation fuzzy set is:
For the left end-points, we need to solve the following programming problem:
while for the right end-points, we need to solve the following programming problem:
This paper has presented a fast method to solve two programming problem so that the type-1 OWA aggregation operation can be performed |
https://en.wikipedia.org/wiki/Awarta | Awarta () is a Palestinian town located southeast of Nablus, in the northern West Bank. According to the Palestinian Central Bureau of Statistics, the town had a population of 7,054 inhabitants in 2017. Awarta's built-up area consists of and it is governed by a village council.
Awarta was an important Samaritan center between the 4th and the 12th century and housed one of their major synagogues. It is home to several holy sites revered by Samaritans, Jews and Muslims, the most well-known of which is the traditional tomb of Phinehas, grandson of Aaron.
Etymology
According to Palestinian historian Mustafa Dabbagh, the name "'Awarta" derives from the Syriac word 'awra, meaning "windowless" or "hidden". According to E. H. Palmer, "Awarta" is a personal name or unknown word. In Samaritan text, the town was called "Caphar Abearthah". An earlier Arabic name for the village was "'Awert".
History
Awarta has been inhabited in the First Temple period, Byzantine and Early Islamic period, and again during Ottoman rule. Between the 4th and 12th centuries, the town was an important Samaritan center, being the location of one of their main synagogues. According to Ben-Zvi, the last Samaritan family, who are of priestly Cohen lineage from the tribe of Levi, left Awarta during the 17th century. He also mentions that the Samaritans of his time (1920s) believed that the majority of the village's residents or all of them are of Samaritan ancestry but were forced to convert to Islam.
The Hill of Phinehas related in the Bible is associated with the location of the town of Awarta. Three large monuments in the town are attributed to the priestly family of Aaron. According to tradition, they are the burial sites of his sons Ithamar and Eleazar. His grandson Phinehas is believed to be buried at the site alongside his son Abishua — the latter is especially revered by the Samaritans, who believe that he wrote the Torah. The seventy Elders are believed to be buried in a cave near Phinehas' tomb. On the western side of Awarta lies the tomb Muslims attribute to Nabi Uzeir, Ezra the scribe.
Arab geographer Yaqut al-Hamawi records in 1226, while Awarta was under Ayyubid rule, that it was a "village or small town, on the road from Nablus to Jerusalem. According to the Samaritan Chronicle, in Kefr Ghuweirah (now called Awarta) is found the tomb of Yusha (Joshua) ibn Nun. Mufaddal, the son of Aaron's uncle, is also said to be buried here. These lie in a cave, where the seventy prophets are also buried." Conder and Kitchener, citing another Samaritan tradition, say rather that it was Eleazer the priest who was buried a "little way west of Awarta (at al 'Azeir)," while Joshua bin Nun was buried at Kefr Haris.
Ottoman era
Awarta was incorporated into the Ottoman Empire in 1517 with all of Palestine, and in 1596 it appeared in the tax registers as being in the Nahiya of Jabal Qubal, part of Nablus Sanjak. It had a population of 50 households, all Muslim. The villagers paid a fi |
https://en.wikipedia.org/wiki/Fuhrmann%20circle | __notoc__
In geometry, the Fuhrmann circle of a triangle, named after the German Wilhelm Fuhrmann (1833–1904), is the circle with a diameter of the line segment between the orthocenter and the Nagel point . This circle is identical with the circumcircle of the Fuhrmann triangle.
The radius of the Fuhrmann circle of a triangle with sides a, b, and c and circumradius R is
which is also the distance between the circumcenter and incenter.
Aside from the orthocenter the Fuhrmann circle intersects each altitude of the triangle in one additional point. Those points all have the distance from their associated vertices of the triangle. Here denotes the radius of the triangles incircle.
Notes
Further reading
Nguyen Thanh Dung: "The Feuerbach Point and the Fuhrmann Triangle". Forum Geometricorum, Volume 16 (2016), pp. 299–311.
J. A. Scott: An Eight-Point Circle. In: The Mathematical Gazette, Volume 86, No. 506 (Jul., 2002), pp. 326–328 (JSTOR)
External links
Fuhrmann circle
Circles defined for a triangle |
https://en.wikipedia.org/wiki/Center%20for%20Research%20in%20Economics%20and%20Statistics | The Center for Research in Economics and Statistics (CREST) is the center of research of the INSEE, the French National Institute for Statistics and Economic Studies.
The research center is affiliated with the ENSAE graduate school.
It has been directed by Arnak Dalalyan since 2020. Before Dalalyan, it had been directed by Francis Kramarz since 2007.
It includes several laboratories in econometrics, macroeconomics, microeconomics, statistics, finance, mathematics, sociology.
It is ranked 9th research center in Economics in the world, 2nd in Europe and 1st in France by www.econphd.net
See ranking here
Objectives
The general objective of CREST is to play an active role in the international development of research in two main fields:
economic and social modeling
conception and implementation of statistical methods
This general goal can be broken down into four more specific objectives:
The training of graduate students
The production and diffusion of research
The exchange and management of visitors programs
The links with firms and public administrations
The training of graduate students
CREST organizes advanced PhD courses and participates in the supervision of dissertations. Over forty research students are currently working on their dissertations at the center. French and foreign students and apply for a CREST fellowship.
The CREST maintains very close relationships with the ENSAE, the Graduate School of INSEE.
The production and diffusion of research
The CREST production is important both in the theoretical and the applied fields. The center organizes eight regular seminars and many international conferences (more than ten conferences over the last five years). The Centre also publishes a series of working papers, an information letter and an annual report.
Exchange and visitors programs
Invitations of junior and senior researchers are part of the Center's policy. The length of the visits varies between one week and one year and financial support is available. In addition, CREST is involved in several international networks, both in the doctoral field (the European Doctoral Program in quantitative economics, for example) and in research (Human Capital and Mobility networks, for instance).
Links with firms and public administrations
In particular with INSEE. Several members of CREST are advisors for French firms or administrations. CREST also has strong links with INSEE: on the one hand, CREST benefits from the statistical data available at INSEE, and, on the other hand, the Center provides expertise for various studies and training programs.
External links
Official CREST (Center for Research in Economics and Statistics) website
Econphd.net: Ranking of CREST
Economic research institutes
Institut national de la statistique et des études économiques |
https://en.wikipedia.org/wiki/Sava%20Grozdev | Sava Grozdev () (born July 13, 1950, in Sofia, Bulgaria) is a Bulgarian mathematician and educator. He currently holds positions as Professor in Mathematics (Mathematical Analysis) and Professor in Mathematical Education.
Biography
Grozdev has PhD degree in mathematics (1980) and DSc degree in Pedagogical Sciences (2003).
His teaching and research activities are in the field of Mathematics and Pedagogical Sciences. He has given courses in Mathematical Analysis, Analytical Mechanics, Generalized Functions, Operational Calculus, Content of Geometry, History of Mathematics, Methodology, Control and Stability of Mechanical Systems, Nonlinear Oscillations, and Chosen Chapters of Mathematics. He has authored more than 150 scientific publications, several books and handbooks, and more than 200 Olympic problems.
Grozdev is most notable for his pedagogical work in mathematics education, both for students and for teachers. He has developed a mathematical model for high achievements in Mathematics.
In 2003 Sava Grozdev was the leader of a group of teachers which took a prize in the International Competition of the American Organization “Best Practices in Education”. In the period 1994–2003 he was the scientific leader of the Bulgarian National Team in Mathematics. Under his leadership during Balkan and International Mathematical Olympiads the Bulgarian students won 59 medals out of 60 possible: 33 gold, 24 silver and 2 bronze. The highest achievement was in 2003 when Bulgaria became World Champion at the International Mathematical Olympiad in Tokyo.
Sava Grozdev has been awarded with prestigious foreign and international prizes. In 2003 he received the Sign of Honour of the President of the Republic of Bulgaria. He has been awarded with the Sign of Honour of the Bulgarian Academy of Sciences (2003), the Sign of Honour of Sofia Community (2003), the Jubilee Medal of the Latvian Mathematical Society (2004), the Honourable Professorship of the South-West University (2006), the medal “St. Cyril and Methodius” – 1st Degree (2007).
References
EUROPEAN MATHEMATICAL SOCIETY ARTICLE COMPETITION - THIRD PRIZE
Professors Sava Grozdev, Ivan Derzhanski and Evgenia Sendova, Union of
Bulgarian Mathematicians, Sofia, Bulgaria.
For the article For those who think mathematics dreary, published in the
Bulgarian daily newspaper Dnevnik, 27 December 2001.
The prize winning article by Grozdev, Derzhanski and Sendova can be found
in Bulgarian and English at:
20th-century Bulgarian mathematicians
21st-century Bulgarian mathematicians
Bulgarian educators
Living people
1950 births
Scientists from Sofia |
https://en.wikipedia.org/wiki/Perspective%20%28geometry%29 | Two figures in a plane are perspective from a point O, called the center of perspectivity, if the lines joining corresponding points of the figures all meet at O. Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions.
Terminology
The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaically perspector. The figures are said to be perspective from this center.
Perspectivity
If each of the perspective figures consists of all the points on a line (a range) then transformation of the points of one range to the other is called a central perspectivity. A dual transformation, taking all the lines through a point (a pencil) to another pencil by means of an axis of perspectivity is called an axial perspectivity.
Triangles
An important special case occurs when the figures are triangles. Two triangles that are perspective from a point are called a central couple and two triangles that are perspective from a line are called an axial couple.
Notation
Karl von Staudt introduced the notation to indicate that triangles ABC and abc are perspective.
Related theorems and configurations
Desargues' theorem states that a central couple of triangles is axial. The converse statement, that an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the real projective plane, and with suitable modifications for special cases, in the Euclidean plane. Projective planes in which this equivalence is true are called Desarguesian planes.
There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity. Dually, there are also ten lines associated with two perspective triangles: three sides of the triangles, three lines through the center of perspectivity, and the axis of perspectivity. These ten points and ten lines form an instance of the Desargues configuration.
If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of Pappus's (hexagon) theorem. When this happens, the nine associated points (six triangle vertices an |
https://en.wikipedia.org/wiki/Periphery%20%28France%29 | A periphery, (Fr: couronne) is an INSEE (French demographic statistics institution) statistical area designating a commuter belt around an urban unit (Fr: unité urbaine). Together these complete the INSEE urban area statistical area.
Based on France's commune system (interlocking administrative subdivisions often comparable to civil parishes, towns or cities), a commune is considered part of a couronne when
it is not densely constructed enough or is too isolated to be part of any unité urbaine (or "pôle urbain" if it is the core of the agglomeration), and
at least 40% of its population commutes to workplaces in a unité urbaine or pôle urbain, or to another commune connected to a unité urbaine through the same criteria.
References
Demographics of France
Subdivisions of France
INSEE concepts |
https://en.wikipedia.org/wiki/List%20of%20Cardiff%20City%20F.C.%20records%20and%20statistics | Cardiff City Football Club is a Welsh professional association football club based in Cardiff, Wales. The club was founded in 1899 and initially played in local amateur leagues before joining the English football league system. After spending a decade in the Southern Football League, Cardiff joined the Football League in 1920. Since then, the club has played in all four professional divisions of the Football League, spending 17 seasons in the top tier since its formation. Cardiff has also reached the final of the FA Cup on three occasions, winning the trophy in the 1927 final, and the League Cup once. The team currently plays in the second tier of the English league system, the EFL Championship.
Billy Hardy is the club's record appearance holder having played in 590 first team matches between 1911 and 1931. Phil Dwyer made the most appearances for the club in the Football League with 471. The club's goalscoring record is held by Len Davies who scored 179 times between 1919 and 1931. Davies is one of only eight players to have scored 100 or more goals in the club's history.
The list encompasses the major honours won by Cardiff City, records set by the club, its managers and players, and details of its performance in European competition. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records achievements by Cardiff City players on the international stage, and the club's highest transfer fees. Attendance records at Ninian Park and the Cardiff City Stadium, the club's home grounds since 1910 and 2009 respectively, are also included.
Honours
Cardiff City was originally founded in 1899 as Riverside A.F.C., initially playing in local amateur competitions. The club won its first trophy under the guise by winning the Bevan Shield, an amateur cup competition, in 1905. The club changed its name to Cardiff City in 1908 and entered the Southern Football League in 1910. The club was the first side based in South Wales to win the Welsh Cup after defeating Pontypridd in the 1912 final. The side won its first league honour by winning the Southern Football League Second Division title the following year, in the 1912–13 season. Cardiff entered the Football League in 1920 and enjoyed the most successful period in its history. Cardiff finished as First Division runners-up in the 1923–24 season and reached two FA Cup finals, losing the first in 1925 before becoming the only non-English side to win the cup two years later in 1927, defeating Arsenal 1–0. The club reached a third FA Cup final 82 years later in 2008 but suffered a 1–0 defeat to Portsmouth. The club is the second most successful side in the history of the Welsh Cup having won the competition on 22 occasions, one fewer than Wrexham. The most recent honour won by the club was the Championship title during the 2012–13 season.
Cardiff City's list of competition victories includes:
League titles
Southern F |
https://en.wikipedia.org/wiki/Scheff%C3%A9%27s%20method | In statistics, Scheffé's method, named after American statistician Henry Scheffé, is a method for adjusting significance levels in a linear regression analysis to account for multiple comparisons. It is particularly useful in analysis of variance (a special case of regression analysis), and in constructing simultaneous confidence bands for regressions involving basis functions.
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method. It works on similar principles as the Working–Hotelling procedure for estimating mean responses in regression, which applies to the set of all possible factor levels.
The method
Let be the means of some variable in disjoint populations.
An arbitrary contrast is defined by
where
If are all equal to each other, then all contrasts among them are . Otherwise, some contrasts differ from .
Technically there are infinitely many contrasts. The simultaneous confidence coefficient is exactly , whether the factor level sample sizes are equal or unequal. (Usually only a finite number of comparisons are of interest. In this case, Scheffé's method is typically quite conservative, and the family-wise error rate (experimental error rate) will generally be much smaller than .)
We estimate by
for which the estimated variance is
where
is the size of the sample taken from the th population (the one whose mean is ), and
is the estimated variance of the errors.
It can be shown that the probability is that all confidence limits of the type
are simultaneously correct, where as usual is the size of the whole population. Norman R. Draper and Harry Smith, in their 'Applied Regression Analysis' (see references), indicate that should be in the equation in place of . The slip with is a result of failing to allow for the additional effect of the constant term in many regressions. That the result based on is wrong is readily seen by considering , as in a standard simple linear regression. That formula would then reduce to one with the usual -distribution, which is appropriate for predicting/estimating for a single value of the independent variable, not for constructing a confidence band for a range of values of the independent value. Also note that the formula is for dealing with the mean values for a range of independent values, not for comparing with individual values such as individual observed data values.
Denoting Scheffé significance in a table
Frequently, superscript letters are used to indicate which values are significantly different using the Scheffé method. For example, when mean values of variables that have been analyzed using an ANOVA are presented in a table, they are assigned a different letter superscript based on a Scheffé contrast. Values that are not significantly different based on the post-hoc Scheffé contrast will have the sa |
https://en.wikipedia.org/wiki/Kim%20Tae-yeon%20%28footballer%29 | Kim Tae-Yeon (born 27 June 1988 in Seoul) is a South Korean footballer who plays as a defensive midfielder, he has also been used as a centre-back.
Club career statistics
References
External links
Living people
1988 births
Men's association football midfielders
South Korean men's footballers
South Korean expatriate men's footballers
Vissel Kobe players
Ehime FC players
Mito HollyHock players
Fagiano Okayama players
Tokyo Verdy players
Roasso Kumamoto players
Daejeon Hana Citizen players
Gwangju FC players
Busan IPark players
J1 League players
J2 League players
K League 1 players
K League 2 players
China League One players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Expatriate men's footballers in China
South Korean expatriate sportspeople in China
Footballers from Seoul |
https://en.wikipedia.org/wiki/Explained%20variation | In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used.
The complementary part of the total variation is called unexplained or residual variation.
Definition in terms of information gain
Information gain by better modelling
Following Kent (1983), we use the Fraser information (Fraser 1965)
where is the probability density of a random variable , and with () are two families of parametric models. Model family 0 is the simpler one, with a restricted parameter space .
Parameters are determined by maximum likelihood estimation,
The information gain of model 1 over model 0 is written as
where a factor of 2 is included for convenience. Γ is always nonnegative; it measures the extent to which the best model of family 1 is better than the best model of family 0 in explaining g(r).
Information gain by a conditional model
Assume a two-dimensional random variable where X shall be considered as an explanatory variable, and Y as a dependent variable. Models of family 1 "explain" Y in terms of X,
,
whereas in family 0, X and Y are assumed to be independent. We define the randomness of Y by , and the randomness of Y, given X, by . Then,
can be interpreted as proportion of the data dispersion which is "explained" by X.
Special cases and generalized usage
Linear regression
The fraction of variance unexplained is an established concept in the context of linear regression. The usual definition of the coefficient of determination is based on the fundamental concept of explained variance.
Correlation coefficient as measure of explained variance
Let X be a random vector, and Y a random variable that is modeled by a normal distribution with centre . In this case, the above-derived proportion of explained variation equals the squared correlation coefficient .
Note the strong model assumptions: the centre of the Y distribution must be a linear function of X, and for any given x, the Y distribution must be normal. In other situations, it is generally not justified to interpret as proportion of explained variance.
In principal component analysis
Explained variance is routinely used in principal component analysis. The relation to the Fraser–Kent information gain remains to be clarified.
Criticism
As the fraction of "explained variance" equals the squared correlation coefficient , it shares all the disadvantages of the latter: it reflects not only the quality of the regression, but also the distribution of the independent (conditioning) variables.
In the words of one critic: "Thus gives the 'percentage of variance explained' by the regression, an expression that, for most social scientists, is of doubtful meaning but great rhetorical value. If this number is large, the regression gives a good fit, and there is little point in searching for addit |
https://en.wikipedia.org/wiki/Expectation%20propagation | Expectation propagation (EP) is a technique in Bayesian machine learning.
EP finds approximations to a probability distribution. It uses an iterative approach that uses the factorization structure of the target distribution. It differs from other Bayesian approximation approaches such as variational Bayesian methods.
More specifically, suppose we wish to approximate an intractable probability distribution with a tractable distribution . Expectation propagation achieves this approximation by minimizing the Kullback-Leibler divergence . Variational Bayesian methods minimize instead.
If is a Gaussian , then is minimized with and being equal to the mean of and the covariance of , respectively; this is called moment matching.
Applications
Expectation propagation via moment matching plays a vital role in approximation for indicator functions that appear when deriving the message passing equations for TrueSkill.
References
External links
Minka's EP papers
List of papers using EP.
Machine learning
Bayesian statistics |
https://en.wikipedia.org/wiki/Decagrammic%20prism | In geometry, the decagrammic prism is one of an infinite set of nonconvex prisms formed by squares sides and two regular star polygon caps, in this case two decagrams.
It has 12 faces (10 squares and 2 decagrams), 30 edges, and 20 vertices.
Prismatoid polyhedra |
https://en.wikipedia.org/wiki/Armenians%20in%20Azerbaijan | Armenians in Azerbaijan (; ) are the Armenians who lived in great numbers in the modern state of Azerbaijan and its precursor, Soviet Azerbaijan. According to the statistics, about 500,000 Armenians lived in Soviet Azerbaijan prior to the outbreak of the First Nagorno-Karabakh War in 1988. Most of the Armenian-Azerbaijanis however had to flee the republic, like Azerbaijanis in Armenia, in the events leading up to the First Nagorno-Karabakh War, a result of the ongoing Armenian-Azerbaijani conflict. Atrocities directed against the Armenian population took place in Sumgait (February 1988), Ganja (Kirovabad, November 1988) and Baku (January 1990). Today the vast majority of Armenians in Azerbaijan live in territory controlled by the break-away region Nagorno-Karabakh which declared its unilateral act of independence in 1991 under the name Nagorno-Karabakh Republic but has not been recognised by any country, including Armenia.
Non-official sources estimate that the number Armenians living on Azerbaijani territory outside Nagorno-Karabakh is around 2,000 to 3,000, and almost exclusively comprises persons married to Azerbaijanis or of mixed Armenian-Azerbaijani descent. The number of Armenians who are likely not married to Azerbaijanis and are not of mixed Armenian-Azerbaijani descent are estimated at 645 (36 men and 609 women) and more than half (378 or 59 per cent of Armenians in Azerbaijan outside Nagorno-Karabakh) live in Baku and the rest in rural areas. They are likely to be the elderly and sick, and probably have no other family members. Armenians in Azerbaijan are at a great risk as long as the Nagorno-Karabakh conflict remains unsettled. In Azerbaijan, the status of Armenians is precarious. Armenian churches remain closed, because of the large emigration of Armenians and fear of Azerbaijani attacks.
History
Armenians in Baku
Armenians in Nagorno-Karabakh
Armenians have lived in the Karabakh region since the period of antiquity. In the beginning of the 2nd century BC. Karabakh became a part of Armenian Kingdom as province of Artsakh. In the 14th century, a local Armenian leadership emerged, consisting of five noble dynasties led by princes, who held the titles of meliks and were referred to as Khamsa (five in Arabic). The Armenian meliks maintained control over the region until the 18th century. In the early 16th century, control of the region passed to the Safavid dynasty, which created the Ganja-Karabakh province (beylerbeydom, bəylərbəylik). Despite these conquests, the population of Upper Karabakh remained largely Armenian.
Karabakh passed to Imperial Russia by the Kurekchay Treaty, signed between the Khan of Karabakh and Tsar Alexander I of Russia in 1805, and later further formalized by the Russo-Persian Treaty of Gulistan in 1813, before the rest of Transcaucasia was incorporated into the Empire in 1828 by the Treaty of Turkmenchay. In 1822, the Karabakh khanate was dissolved, and the area became part of the Elizavetpol Governorat |
https://en.wikipedia.org/wiki/Cleaver%20%28geometry%29 | In geometry, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. They are not to be confused with splitters, which also bisect the perimeter, but with an endpoint on one of the triangle's vertices instead of its sides.
Construction
Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle.
The broken chord theorem of Archimedes provides another construction of the cleaver. Suppose the triangle to be bisected is , and that one endpoint of the cleaver is the midpoint of side . Form the circumcircle of and let be the midpoint of the arc of the circumcircle from to through . Then the other endpoint of the cleaver is the closest point of the triangle to , and can be found by dropping a perpendicular from to the longer of the two sides and .
Related figures
The three cleavers concur at a point, the center of the Spieker circle.
See also
Splitter (geometry)
References
External links
Straight lines defined for a triangle |
https://en.wikipedia.org/wiki/Ibrahim%20Umar | Ibrahim Khalil Umar is a Nigerian scientist and university administrator. He was Vice-Chancellor of Bayero University, Kano, Nigeria from 1979 to 1986. He holds a B.Sc. in physics and mathematics from Ahmadu Bello University in Zaria, Nigeria, a M.Sc. in physics from Northern Illinois University, USA and a Ph.D. (1974) in physics at the University of East Anglia, United Kingdom. In 1976 he became the first Nigerian academic in physics to teach at Bayero University. In 1978 he served on the national constitutional assembly that drafted the Constitution of the 2nd Republic.
Between 1994 and 1997, Umar served as Sole Administrator of the Federal University of Technology, Minna.
He represented Nigeria at the Executive Assembly of the World Energy Council from 1990. He was a member of the Nigerian delegation to the International Atomic Energy Agency (IAEA) General Conference from 1989 and was appointed Director-General of the Energy Commission of Nigeria in 1989. He served as Chairman of the Board of Governors of the IAEA for 2000–2001. In 2004 he was the Director of the Centre for Energy Research and Training, where the first Nigerian research nuclear reactor is located.
In 2007, he was on the international advisory committee for the international workshop on Renewable Energy for Sustainable Development in Africa, held at the University of Nigeria, Nsukka in Nigeria.
Honour
As one of the former Vice chancellor of Bayero University Kano state a lecture Twin theatre was name after him at the university old campus along Biochemistry department Road.
Death
On the 30th of January 2023 his son Faruk Ibrahim Umar announced his death.
References
Year of birth missing (living people)
Living people
Ahmadu Bello University alumni
Northern Illinois University alumni
Alumni of the University of East Anglia
Vice-Chancellors of Nigerian universities
Nigerian scientists
People from Kano State
International Atomic Energy Agency officials
Academic staff of Bayero University Kano
Academic staff of the Federal University of Technology, Minna
Vice-Chancellors of Federal University of Technology, Minna |
https://en.wikipedia.org/wiki/Florence%20Marie%20Mears | Florence Marie Mears (May 18, 1896 – December 3, 1995) was a professor of Mathematics at The George Washington University.
Background and education
Mears was born in Baltimore, Maryland and attended Baltimore public schools. She received her undergraduate degree in Mathematics at Goucher College, earning a Phi Beta Kappa Key. She received a master's degree from Cornell University in 1924 after completing her thesis on "A Special Function of One Variable." She then went on to achieve her doctorate from Cornell in 1927, completing her thesis on the "Riesz Summability for Double Series" (published in Transactions of the American Mathematical Society in 1928) with thesis advisor Wallie Abraham Hurwitz.
Career
Mears's first job as a college professor was at the Women’s College of Alabama; she soon left to become an associate professor of mathematics at Pennsylvania State College, and then joined The George Washington University in 1929.
At George Washington University, Mears was known for her popularity among both her students and fellow faculty members. She was considered by the university’s President as "one of the greatest teachers of mathematics in the entire country." During her first year at The George Washington University, Mears taught as an assistant professor of Mathematics in the Columbian College of Arts and Sciences. Her office could be found directly on campus at 2033 G Street. At the beginning of her career at GW, Mears was the only woman in the Mathematics department, which included seven other professors. Additionally, it was reported that she was paid substantially less than her male co-workers.
Mears earned the title of being a master teacher. According to the criteria set by the Columbian College of Arts and Sciences, Mears earned this title due to her excellent teaching skills, and her contributions in both research and the Mathematics Department. Among the many classes that she taught were Advanced Calculus, Introduction to Analysis, Introduction to Infinite Series, and Fourier Series and Spherical Harmonics. In 1955, George Washington University awarded her an Alumni Citation for twenty-five years of distinguished service. In 1958, the University of California selected her as one of ten female mathematicians to work on a project studying creativity. In 1962, she became a member of an examining committee, which focused on selecting doctoral dissertations in mathematics for the University of Allahabad in India.
Mears specialized in the findings of definitions and values assigned to various infinite series of numbers. An "infinite series" is an endless series of numbers, each succeeding the other that is a certain amount lesser or greater than the proceeding one. An example set of an infinite series includes is 1 + ½ + ¼ etc. in which the definition of the series can be defined as the number two. As a result, Mears created several theorems about these definitions, many of which provided truth for many practicing mathematician |
https://en.wikipedia.org/wiki/Topology%20of%20the%20World%20Wide%20Web | World Wide Web topology is the network topology of the World Wide Web, as seen as a network of web pages connected by hyperlinks.
The Jellyfish and Bow Tie models are two attempts at modeling the topology of hyperlinks between web pages.
Models of web page topology
Jellyfish Model
The simplistic Jellyfish model of the World Wide Web centers around a large strongly connected core of high-degree web pages that form a clique; pages such that there is a path from any page within the core to any other page. In other words, starting from any node within the core, it is possible to visit any other node in the core just by clicking hyperlinks. From there, a distinction is made between pages of single degree and those of higher order degree. Pages with many links form rings around the center, with all such pages that are a single link away from the core making up the first ring, all such pages that are two links away from the core making up the second ring, and so on. Then from each ring, pages of single degree are depicted as hanging downward, with a page linked by the core hanging from the center, for example. In this manner, the rings form a sort of dome away from the center that is reminiscent of a jellyfish, with the hanging nodes making up the creature's tentacles.
Bow Tie Model
The Bow Tie model comprises four main groups of web pages, plus some smaller ones. Like the Jellyfish model there is a strongly connected core. There are then two other large groups, roughly of equal size. One consists of all pages that link to the strongly connected core, but which have no links from the core back out to them. This is the "Origination" or "In" group, as it contains links that lead into the core and originate outside it. The counterpart to this is the group of all pages that the strongly connected core links to, but which have no links back into the core. This is the "Termination" or "Out" group, as it contains links that lead out of the core and terminate outside it. A fourth group is all the disconnected pages, which neither link to the core nor are linked from it.
The Bow Tie model has additional, smaller groups of web pages. Both the "In" and "Out" groups have smaller "Tendrils" leading to and from them. These consist of pages that link to and from the "In" and "Out" group but are not part of either to begin with, in essence the "Origination" and "Termination" groups of the larger "In" and "Out". This can be carried on ad nauseam, adding tendrils to the tendrils, and so on. Additionally, there is another important group known as "Tubes". This group consists of pages accessible from "In" and which link to "Out", but which are not part of the large core. Visually, they form alternative routes from "In" to "Out", like tubes bending around the central strongly connected component.
See also
Webgraph
References
External links
Internet Topology Mapping Tools (Mehmet Engin Tozal)
The Workshop on Internet Topology (WIT) Report
Perl program that gen |
https://en.wikipedia.org/wiki/Woo%20Sung-yong | Woo Sung-yong (born August 18, 1974) is a South Korean former professional footballer who played as a forward.
Career statistics
Club
International
Results list South Korea's goal tally first.
References
External links
1974 births
Living people
South Korean men's footballers
Men's association football forwards
South Korea men's under-23 international footballers
South Korea men's international footballers
2007 AFC Asian Cup players
Busan IPark players
Pohang Steelers players
Seongnam FC players
Ulsan Hyundai FC players
Incheon United FC players
K League 1 players
Ajou University alumni
South Korean Buddhists
South Korean football managers
Seoul E-Land FC managers
Footballers from Gangwon Province, South Korea
Danyang Woo clan |
https://en.wikipedia.org/wiki/Mittenpunkt | In geometry, the (from German: middle point) of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It was identified in 1836 by Christian Heinrich von Nagel as the symmedian point of the excentral triangle of the given triangle.
Coordinates
The mittenpunkt has trilinear coordinates
where , , and are the side lengths of the given triangle.
Expressed instead in terms of the angles , , and , the trilinears are
The barycentric coordinates are
Collinearities
The mittenpunkt is at the intersection of the line connecting the centroid and the Gergonne point, the line connecting the incenter and the symmedian point and the line connecting the orthocenter with the Spieker center, thus establishing three collinearities involving the mittenpunkt.
Related figures
The three lines connecting the excenters of the given triangle to the corresponding edge midpoints all meet at the mittenpunkt; thus, it is the center of perspective of the excentral triangle and the median triangle, with the corresponding axis of perspective being the trilinear polar of the Gergonne point. The mittenpunkt is also the centroid of the Mandart inellipse of the given triangle, the ellipse tangent to the triangle at its extouch points.
Notes
The Mittenpunkt also serves as the Gergonne point of the Medial triangle.
References
External links
Triangle centers |
https://en.wikipedia.org/wiki/Fitting%20length | In mathematics, specifically in the area of algebra known as group theory, the Fitting length (or nilpotent length) measures how far a solvable group is from being nilpotent. The concept is named after Hans Fitting, due to his investigations of nilpotent normal subgroups.
Definition
A Fitting chain (or Fitting series or ) for a group is a subnormal series with nilpotent quotients. In other words, a finite sequence of subgroups including both the whole group and the trivial group, such that each is a normal subgroup of the previous one, and such that the quotients of successive terms are nilpotent groups.
The Fitting length or nilpotent length of a group is defined to be the smallest possible length of a Fitting chain, if one exists.
Upper and lower Fitting series
Just as the upper central series and lower central series are extremal among central series, there are analogous series extremal among nilpotent series.
For a finite group H, the Fitting subgroup Fit(H) is the maximal normal nilpotent subgroup, while the minimal normal subgroup such that the quotient by it is nilpotent is γ∞(H), the intersection of the (finite) lower central series, which is called the nilpotent residual.
These correspond to the center and the commutator subgroup (for upper and lower central series, respectively). These do not hold for infinite groups, so for the sequel, assume all groups to be finite.
The upper Fitting series of a finite group is the sequence of characteristic subgroups Fitn(G) defined by Fit0(G) = 1, and Fitn+1(G)/Fitn(G) = Fit(G/Fitn(G)). It is an ascending nilpotent series, at each step taking the maximal possible subgroup.
The lower Fitting series of a finite group G is the sequence of characteristic subgroups Fn(G) defined by F0(G) = G, and Fn+1(G) = γ∞(Fn(G)). It is a descending nilpotent series, at each step taking the minimal possible subgroup.
Examples
A nontrivial group has Fitting length 1 if and only if it is nilpotent.
The symmetric group on three points has Fitting length 2.
The symmetric group on four points has Fitting length 3.
The symmetric group on five or more points has no Fitting chain at all, not being solvable.
The iterated wreath product of n copies of the symmetric group on three points has Fitting length 2n.
Properties
A group has a Fitting chain if and only if it is solvable.
The lower Fitting series is a Fitting chain if and only if it eventually reaches the trivial subgroup, if and only if G is solvable.
The upper Fitting series is a Fitting chain if and only if it eventually reaches the whole group, G, if and only if G is solvable.
The lower Fitting series descends most quickly amongst all Fitting chains, and the upper Fitting series ascends most quickly amongst all Fitting chains. Explicitly: For every Fitting chain, 1 = H0 ⊲ H1 ⊲ … ⊲ Hn = G, one has that Hi ≤ Fiti(G), and Fi(G) ≤ Hn−i.
For a solvable group, the length of the lower Fitting series is equal to length of the upper Fitting series, an |
https://en.wikipedia.org/wiki/Hiroshima%20Mathematical%20Journal | The Hiroshima Mathematical Journal is an open-access mathematics journal that continues the Journal of Science of the Hiroshima University, Series A (1930–1960) and Journal of Science of the Hiroshima University, Series A - I (1961–1970). The journal contains original research papers in pure and applied mathematics. Each annual volume has had three issues since 1974.
External links
Official website
Mathematics journals
Hiroshima University
Triannual journals
Open access journals
Academic journals established in 1930 |
https://en.wikipedia.org/wiki/Free%20ideal%20ring | In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not.
Properties and examples
It turns out that a left and right fir is a domain. Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain. These last facts are not generally true for noncommutative rings, however .
Every principal right ideal domain R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R. In the same way, a right Bézout domain is a semifir.
Since all right ideals of a right fir are free, they are projective. So, any right fir is a right hereditary ring, and likewise a right semifir is a right semihereditary ring. Because projective modules over local rings are free, and because local rings have invariant basis number, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir.
Unlike a principal right ideal domain, a right fir is not necessarily right Noetherian, however in the commutative case, R is a Dedekind domain since it is a hereditary domain, and so is necessarily Noetherian.
Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings .
Semifirs have invariant basis number and every semifir is a Sylvester domain.
References
Further reading
Ring theory |
https://en.wikipedia.org/wiki/Elman%20Sultanov | Elman Sultanov (born 6 May 1974) is an Azerbaijani-Israeli-Ukrainian retired professional footballer and current Reserve team coach for Sabail FK.
National team statistics
External links
1974 births
Living people
Israeli men's footballers
Azerbaijani men's footballers
FC Krystal Kherson players
FC Vorskla Poltava players
FC Torpedo Zaporizhzhia players
Neftçi PFK players
MOIK Baku players
FC Baku players
Hapoel Tzafririm Holon F.C. players
FK Žalgiris players
Qarabağ FK players
Simurq PIK players
Ukrainian Premier League players
Ukrainian First League players
Ukrainian Second League players
Israeli Premier League players
A Lyga players
Azerbaijan men's international footballers
Men's association football midfielders
Azerbaijani expatriate men's footballers
Expatriate men's footballers in Ukraine
Azerbaijani expatriate sportspeople in Ukraine
Expatriate men's footballers in Lithuania
Azerbaijani expatriate sportspeople in Lithuania |
https://en.wikipedia.org/wiki/Sz%C3%A1sz%E2%80%93Mirakyan%20operator | In functional analysis, a discipline within mathematics, the Szász–Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mirakjan in 1941. They are defined by
=
where and .
Basic results
In 1964, Cheney and Sharma showed that if is convex and non-linear, the sequence decreases with (). They also showed that if is a polynomial of degree , then so is for all .
A converse of the first property was shown by Horová in 1968 (Altomare & Campiti 1994:350).
Theorem on convergence
In Szász's original paper, he proved the following:
If is continuous on , having a finite limit at infinity, then converges uniformly to as .
This is analogous to a theorem stating that Bernstein polynomials approximate continuous functions on [0,1].
Generalizations
A Kantorovich-type generalization is sometimes discussed in the literature. These generalizations are also called the Szász–Mirakjan–Kantorovich operators.
In 1976, C. P. May showed that the Baskakov operators can reduce to the Szász–Mirakyan operators.
References
(See also: Favard operators)
Footnotes
Approximation theory |
https://en.wikipedia.org/wiki/Sz%C3%A1sz%E2%80%93Mirakjan%E2%80%93Kantorovich%20operator | In functional analysis, a discipline within mathematics, the Szász–Mirakjan–Kantorovich operators are defined by
where and .
See also
Szász–Mirakyan operator
Notes
References
Approximation theory |
https://en.wikipedia.org/wiki/List%20of%20vector%20spaces%20in%20mathematics | This is a list of vector spaces in abstract mathematics, by Wikipedia page.
Banach space
Besov space
Bochner space
Dual space
Euclidean space
Fock space
Fréchet space
Hardy space
Hilbert space
Hölder space
LF-space
Lp space
Minkowski space
Montel space
Morrey–Campanato space
Orlicz space
Riesz space
Schwartz space
Sobolev space
Tsirelson space
Linear algebra
Mathematics-related lists |
https://en.wikipedia.org/wiki/Hanner%27s%20inequalities | In mathematics, Hanner's inequalities are results in the theory of Lp spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of Lp spaces for p ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936.
Statement of the inequalities
Let f, g ∈ Lp(E), where E is any measure space. If p ∈ [1, 2], then
The substitutions F = f + g and G = f − g yield the second of Hanner's inequalities:
For p ∈ [2, +∞) the inequalities are reversed (they remain non-strict).
Note that for the inequalities become equalities which are both the parallelogram rule.
References
Banach spaces
Inequalities
Measure theory |
https://en.wikipedia.org/wiki/Trisected%20perimeter%20point | In geometry, given a triangle ABC, there exist unique points A´, B´, and C´ on the sides BC, CA, AB respectively, such that:
A´, B´, and C´ partition the perimeter of the triangle into three equal-length pieces. That is,
.
The three lines AA´, BB´, and CC´ meet in a point, the trisected perimeter point.
This is point X369 in Clark Kimberling's Encyclopedia of Triangle Centers. Uniqueness and a formula for the trilinear coordinates of X369 were shown by Peter Yff late in the twentieth century. The formula involves the unique real root of a cubic equation.
See also
Bisected perimeter point
References
Triangle centers |
https://en.wikipedia.org/wiki/Magic%20circle%20%28mathematics%29 | Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameters are identical. One of his magic circles was constructed from the natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center.
Yang Hui magic circles
Yang Hui's magic circle series was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》(Sequel to Excerpts of Mathematical Wonders) of 1275. His magic circle series includes: magic 5 circles in square, 6 circles in ring, magic eight circle in square
magic concentric circles, magic 9 circles in square.
Yang Hui magic concentric circle
Yang Hui's magic concentric circle has the following properties
The sum of the numbers on four diameters = 147,
28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147
The sum of 8 numbers plus 9 at the center = 147;
28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 + 9 = 147
The sum of eight radius without 9 = magic number 69: such as 27 + 15 + 3 + 24 = 69
The sum of all numbers on each circle (not including 9) = 2 × 69
There exist 8 semicircles, where the sum of numbers = magic number 69; there are 16 line segments (semicircles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers.
Yang Hui magic eight circles in a square
64 numbers arrange in circles of eight numbers, total sum 2080, horizontal / vertical sum = 260.
From NW corner clockwise direction, the sum of 8-number circles are:
40 + 24 + 9 + 56 + 41 + 25 + 8 + 57 = 260
14 + 51 + 46 + 30 + 3 + 62 + 35 + 19 = 260
45 + 29 + 4 + 61 + 36 + 20 + 13 + 52 = 260
37 + 21 + 12 + 53 + 44 + 28 + 5 + 60 = 260
47 + 31 + 2 + 63 + 34 + 18 + 15 + 50 = 260
7 + 58 + 39 + 23 + 10 + 55 + 42 + 26 = 260
38 + 22 + 11 + 54 + 43 + 27 + 6 + 59 = 260
48 + 32 + 1 + 64 + 33 + 17 + 16 + 49 = 260
Also the sum of the eight numbers along the WE/NS axis
14 + 51 + 62 + 3 + 7 + 58 + 55 + 10 = 260
49 + 16 + 1 + 64 + 60 + 5 + 12 + 53 = 260
Furthermore, the sum of the 16 numbers along the two diagonals equals to 2 times 260:
40 + 57 + 41 + 56 + 50 + 47 + 34 + 63 + 29 + 4 + 13 + 20 + 22 + 11 + 6 + 27 = 2 × 260 = 520
Yang Hui magic nine circles in a square
72 numbers from 1 to 72, arranged in nine circles of eight numbers in a square; with neighbouring numbers forming four additional eight number circles: thus making a total of 13 eight number circles:
Extra circle x1 contains numbers from circles NW, N, C, and W; x2 contains numbers from N, NE, E, and C; x3 contains numbers from W, C, S, and SW; x4 contains numbers from C, E, SE, and S.
Total sum of 72 numbers = 2628;
sum of numbers in any eight number circle = 292;
sums of three circles along horizontal lines = 876;
sum of three circles along vertical lines = 876;
sum of three circles along the diagonals = 876.
Ding Yidong magic circles
Ding Yidong was a mathematician contemporary with Ya |
https://en.wikipedia.org/wiki/Splitter%20%28geometry%29 | In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle. They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides.
Properties
The opposite endpoint of a splitter to the chosen triangle vertex lies at the point on the triangle's side where one of the excircles of the triangle is tangent to that side. This point is also called a splitting point of the triangle. It is additionally a vertex of the extouch triangle and one of the points where the Mandart inellipse is tangent to the triangle side.
The three splitters concur at the Nagel point of the triangle, which is also called its splitting center.
Generalization
Some authors have used the term "splitter" in a more general sense, for any line segment that bisects the perimeter of the triangle. Other line segments of this type include the cleavers, which are perimeter-bisecting segments that pass through the midpoint of a triangle side, and the equalizers, segments that bisect both the area and perimeter of a triangle.
References
External links
Straight lines defined for a triangle |
https://en.wikipedia.org/wiki/Iwasawa%20manifold | In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An
Iwasawa manifold is a nilmanifold, of real dimension 6.
Iwasawa manifolds give examples where the first two terms E1 and E2 of the Frölicher spectral sequence are not isomorphic.
As a complex manifold, such an Iwasawa manifold is an important example of
a compact complex manifold which does not admit any Kähler metric.
References
.
Differential geometry
Lie groups
Homogeneous spaces
Complex manifolds |
https://en.wikipedia.org/wiki/List%20of%20Manchester%20City%20F.C.%20records%20and%20statistics | This article lists various statistics related to Manchester City Football Club.
Club honours
Premier League/First Division (highest tier)
Winners (9): 1936–37, 1967–68, 2011–12, 2013–14, 2017–18, 2018–19, 2020–21, 2021–22, 2022–23
Runners-up (6): 1903–04, 1920–21, 1976–77, 2012–13, 2014–15, 2019–20
First Division/Second Division (second tier)
Winners (7, joint record): 1898–99, 1902–03, 1909–10, 1927–28, 1946–47, 1965–66, 2001–02
Runners-up (4): 1895–96, 1950–51, 1988–89, 1999–2000
Promoted third place (1): 1984–85
Second Division (third tier)
(Best) Promoted third place (1): 1998–99
FA Cup
Winners (7): 1903–04, 1933–34, 1955–56, 1968–69, 2010–11, 2018–19, 2022–23
Runners-up (5): 1925–26, 1932–33, 1954–55, 1980–81, 2012–13
EFL Cup/Football League Cup
Winners (8): 1969–70, 1975–76, 2013–14, 2015–16, 2017–18, 2018–19, 2019–20, 2020–21
Runners-up (1): 1973–74
Full Members' Cup
(Best) Runners-up (1): 1985–86
FA Community Shield/FA Charity Shield
Winners (6): 1937, 1968, 1972, 2012, 2018, 2019
Runners-up (9): 1934, 1956, 1969, 1973, 2011, 2014, 2021, 2022, 2023
UEFA Champions League/European Cup
Winners (1): 2022–23
Runners-up (1): 2020–21
UEFA Europa League/UEFA Cup
(Best) Quarter-finals (2): 1978–79, 2008–09
European Cup Winners' Cup
Winners (1): 1969–70
UEFA Super Cup
Winners (1): 2023
Competitive record
The table that follows is accurate as of the end of the 2022–23 season for all competitions. It excludes war competitions and league performances prior to City's joining the English football league in 1892.
Current first-team squad statistics
Ordered by squad number.Appearances include league and cup appearances, including as substitute.Includes EDS players who train regularly with the first team, having made at least one previous league appearance.
Club records
Matches
Record league victory – 11–3 v. Lincoln City (23 March 1895, most goals scored), 10–0 v. Darwen (18 February 1899, widest margin of victory)
Record FA Cup victory – 12–0 v. Liverpool Stanley (4 October 1890)
Record European victory – 7–0 v. Schalke 04, UEFA Champions League round of 16 second leg (12 March 2019), 7–0 v. RB Leipzig, UEFA Champions League round of 16 second leg (14 March 2023)
Record league defeat – 0–8 v. Burton Wanderers (26 December 1894), 0–8 v. Wolverhampton Wanderers (23 December 1933), 1–9 v. Everton (3 September 1906), 2–10 v. Small Heath (17 March 1893)
Record FA Cup defeat – 0–6 v. Preston North End (30 January 1897), 2–8 v. Bradford Park Avenue (30 January 1946)
Record European defeat – 0–4 v Barcelona, UEFA Champions League group stage, 19 October 2016
Streaks
Winning runs
Longest winning run in all competitions: 21, 19 December 2020 – 2 March 2021 (all games won over 90 minutes) (national record)
Longest league winning run: 18, 26 August 2017 – 27 December 2017 (joint national record)
Longest league winning run from the start of a calendar year: 13, 3 January 2021 – 2 March 2021 (national record)
Longest run of games without going behind in the |
https://en.wikipedia.org/wiki/Toroid | In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1-g).
The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids.
Toroidal structures occur in both natural and synthetic materials.
Equations
A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):
Square toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
Circular toroid
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
See also
Toroidal inductors and transformers
Toroidal propellers
Annulus
Solenoid
Helix
Notes
External links
Topology
Geometric shapes |
https://en.wikipedia.org/wiki/Mandart%20inellipse | In geometry, the Mandart inellipse of a triangle is an ellipse that is inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century.
Parameters
As an inconic, the Mandart inellipse is described by the parameters
where a, b, and c are sides of the given triangle.
Related points
The center of the Mandart inellipse is the mittenpunkt of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the Nagel point of the triangle.
See also
Steiner inellipse, a different ellipse tangent to a triangle at the midpoints of its sides
Notes
External links
Curves defined for a triangle |
https://en.wikipedia.org/wiki/Rabinowitsch%20trick | In mathematics, the Rabinowitsch trick, introduced by ,
is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called weak Nullstellensatz), by introducing an extra variable.
The Rabinowitsch trick goes as follows. Let K be an algebraically closed field. Suppose the polynomial f in K[x1,...xn] vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the weak Nullstellensatz for K[x0, ..., xn] they generate the unit ideal of K[x0 ,..., xn]. Spelt out, this means there are polynomials such that
as an equality of elements of the polynomial ring . Since are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting that
as elements of the field of rational functions , the field of fractions of the polynomial ring . Moreover, the only expressions that occur in the denominators of the right hand side are f and powers of f, so rewriting that right hand side to have a common denominator results in an equality on the form
for some natural number r and polynomials . Hence
which literally states that lies in the ideal generated by f1,....,fm. This is the full version of the Nullstellensatz for K[x1,...,xn].
References
Commutative algebra |
https://en.wikipedia.org/wiki/Tur%C3%A1n%20number | In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by , and the problem for general r was introduced in . The paper gives a survey of Turán numbers.
Definitions
Fix a set X of n vertices. For given r, an r-edge or block is a set of r vertices. A set of blocks is called a Turán (n,k,r) system (n ≥ k ≥ r) if every k-element subset of X contains a block.
The Turán number T(n,k,r) is the minimum size of such a system.
Example
The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7.
Relations to other combinatorial designs
It can be shown that
Equality holds if and only if there exists a Steiner system S(n - k, n - r, n).
An (n,r,k,r)-lotto design is an (n, k, r)-Turán system. Thus, T(n,k, r) = L(n,r,k,r).
See also
Forbidden subgraph problem
Combinatorial design
References
Bibliography
Extremal graph theory
Combinatorial design |
https://en.wikipedia.org/wiki/MathFest | MathFest is a mathematics conference hosted annually in late summer by the Mathematical Association of America. It is known for its dual focus on teaching and research in mathematics, as well as for student participation.
MathFest Locations
The 2015 meeting in Washington, D.C. was an extra day long in order to include events to mark the centennial anniversary of the MAA.
The 2020 meeting in Philadelphia, PA was cancelled due to the COVID-19 pandemic.
The 2021 meeting was held virtually due to the COVID-19 pandemic.
Events
MathFest features many annual lectures, such as the Earle Raymond Hedrick Lecture Series, which consists of up to three lectures by the same presenter, on three consecutive days,
and the AWM-MAA Falconer Lecture, which is given by a distinguished female mathematician or mathematics educator.
Notes
External links
List of national Mathfest meetings of the MAA
Mathematics conferences
Mathematical Association of America
Mathematics education in the United States
Festival organizations in North America |
https://en.wikipedia.org/wiki/Joint%20Mathematics%20Meetings | The Joint Mathematics Meetings (JMM) is a mathematics conference hosted annually in early January by the American Mathematical Society (AMS). Frequently, several other national mathematics organizations also participate. The meeting is the largest gathering of mathematicians in the United States, and the largest annual meeting of mathematicians in the world. For example, more than 6000 people attended the 2017 JMM. Several thousand talks, panels, minicourses, and poster sessions are held each year.
The JMM also hosts an Employment Center, which is a focal point for the hiring process of academic mathematicians, especially for liberal arts colleges. Many employers conduct their preliminary interview process at the meeting. Often these interviews take place outside the confines of the conference, so the employers may not appear on the official Employment Center listing.
Future Meetings
San Francisco, CA, January 3–6, 2024
Seattle, WA, January 8-11, 2025
Washington, D.C., January 4-7, 2026
Past Meetings
Boston, MA, January 4-7, 2023 link
Virtual, April 6-9, 2022 (the originally scheduled meeting in Seattle, WA, January 5-8, had been postponed due to concerns of the Omicron variant of COVID-19)
Virtual, January 6–9, 2021 (the originally scheduled in-person meeting in Washington, D.C. was canceled due to COVID-19)
Denver, CO, January 15–18, 2020 link
Baltimore, MD, January 16–19, 2019 link
San Diego, CA, January 10-13, 2018 link
Atlanta, GA, January 4–7, 2017 link
Seattle, WA, January 6–9, 2016, link
San Antonio, TX, January 10–13, 2015, link
Baltimore, MD, January 15–18, 2014 link
San Diego, CA, January 9–12, 2013 link
Boston, MA, January 4–7, 2012 link
New Orleans, LA, January 6–9, 2011 link
San Francisco, CA, January 13–16, 2010 link
Washington, D.C., January 5–8, 2009 link
San Diego, CA, January 6–9, 2008 link
New Orleans, LA, January 5–8, 2007 link
San Antonio, TX, January 12–15, 2006 link
Atlanta, GA, January 5–8, 2005 link
Phoenix, AZ, January 7–10, 2004 link
Baltimore, MD, January 15–18, 2003 link
San Diego, CA, January 6–9, 2002 link
New Orleans, LA, January 10–13, 2001 link
Washington, D.C., January 19–22, 2000 link
See also
American Mathematical Society
External links
Listing of national AMS conferences
Information about the Employment Center
References
American Mathematical Society
Mathematics conferences |
https://en.wikipedia.org/wiki/History%20of%20statistics | Statistics, in the modern sense of the word, began evolving in the 18th century in response to the novel needs of industrializing sovereign states.
In early times, the meaning was restricted to information about states, particularly demographics such as population. This was later extended to include all collections of information of all types, and later still it was extended to include the analysis and interpretation of such data. In modern terms, "statistics" means both sets of collected information, as in national accounts and temperature record, and analytical work which requires statistical inference. Statistical activities are often associated with models expressed using probabilities, hence the connection with probability theory. The large requirements of data processing have made statistics a key application of computing. A number of statistical concepts have an important impact on a wide range of sciences. These include the design of experiments and approaches to statistical inference such as Bayesian inference, each of which can be considered to have their own sequence in the development of the ideas underlying modern statistics.
Introduction
By the 18th century, the term "statistics" designated the systematic collection of demographic and economic data by states. For at least two millennia, these data were mainly tabulations of human and material resources that might be taxed or put to military use. In the early 19th century, collection intensified, and the meaning of "statistics" broadened to include the discipline concerned with the collection, summary, and analysis of data. Today, data is collected and statistics are computed and widely distributed in government, business, most of the sciences and sports, and even for many pastimes. Electronic computers have expedited more elaborate statistical computation even as they have facilitated the collection and aggregation of data. A single data analyst may have available a set of data-files with millions of records, each with dozens or hundreds of separate measurements. These were collected over time from computer activity (for example, a stock exchange) or from computerized sensors, point-of-sale registers, and so on. Computers then produce simple, accurate summaries, and allow more tedious analyses, such as those that require inverting a large matrix or perform hundreds of steps of iteration, that would never be attempted by hand. Faster computing has allowed statisticians to develop "computer-intensive" methods which may look at all permutations, or use randomization to look at 10,000 permutations of a problem, to estimate answers that are not easy to quantify by theory alone.
The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relation between statistics and probability theory developed rather late, however. In the 19th century, statistics increasingly used probability theory, w |
https://en.wikipedia.org/wiki/Knaster%E2%80%93Kuratowski%20fan | In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point makes it totally disconnected. It is also known as Cantor's leaky tent or Cantor's teepee (after Georg Cantor), depending on the presence or absence of the apex.
Let be the Cantor set, let be the point , and let , for , denote the line segment connecting to . If is an endpoint of an interval deleted in the Cantor set, let ; for all other points in let ; the Knaster–Kuratowski fan is defined as equipped with the subspace topology inherited from the standard topology on .
The fan itself is connected, but becomes totally disconnected upon the removal of .
See also
Antoine's necklace
References
Topological spaces |
https://en.wikipedia.org/wiki/M.%20Ram%20Murty | Maruti Ram Pedaprolu Murty, FRSC (born 16 October 1953)
is an Indo-Canadian mathematician at Queen's University, where he holds a Queen's Research Chair in mathematics.
Biography
M. Ram Murty is the brother of mathematician V. Kumar Murty.
Murty graduated with a B.Sc. from Carleton University in 1976. He received his Ph.D. in 1980 from the Massachusetts Institute of Technology, supervised by Harold Stark and Dorian Goldfeld. He was on the faculty of McGill University from 1982 until 1996, when he joined Queen's University. Murty is also cross-appointed as a professor of philosophy at Queen's, specialising in Indian philosophy.
Research
Specializing in number theory, Murty is a researcher in the areas of modular forms, elliptic curves, and sieve theory.
Murty has Erdős number 1 and frequently collaborates with his brother, V. Kumar Murty.
Awards
Murty received the Coxeter–James Prize in 1988. He was elected a Fellow of the Royal Society of Canada in 1990, was elected to the Indian National Science Academy (INSA) in 2008, and became a fellow of the American Mathematical Society in 2012.
Selected publications
.
.
References
External links
Murty's home page at Queen's
1953 births
20th-century Indian mathematicians
21st-century Indian mathematicians
Canada Research Chairs
Canadian Hindus
Canadian mathematicians
Carleton University alumni
Fellows of the American Mathematical Society
Fellows of the Royal Society of Canada
Foreign Fellows of the Indian National Science Academy
Indian emigrants to Canada
People from Guntur
Living people
Massachusetts Institute of Technology alumni
Indian number theorists
Academic staff of Queen's University at Kingston
Academic staff of McGill University
Scientists from Andhra Pradesh
Number theorists |
https://en.wikipedia.org/wiki/Tur%C3%A1n%27s%20inequalities | In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors.
If is the th Legendre polynomial, Turán's inequalities state that
For , the th Hermite polynomial, Turán's inequalities are
whilst for Chebyshev polynomials they are
See also
Askey–Gasper inequality
Sturm Chain
References
Orthogonal polynomials
Inequalities |
https://en.wikipedia.org/wiki/Ruin%20theory | In actuarial science and applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
Classical model
The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or Poisson risk process) was introduced in 1903 by the Swedish actuary Filip Lundberg. Lundberg's work was republished in the 1930s by Harald Cramér.
The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate c > 0 from customers and claims arrive according to a Poisson process with intensity λ and are independent and identically distributed non-negative random variables with distribution F and mean μ (they form a compound Poisson process). So for an insurer who starts with initial surplus x, the aggregate assets are given by:
The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as
where the time of ruin is with the convention that . This can be computed exactly using the Pollaczek–Khinchine formula as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an M/G/1 queue)
where is the transform of the tail distribution of ,
and denotes the -fold convolution.
In the case where the claim sizes are exponentially distributed, this simplifies to
Sparre Andersen model
E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.
where the claim number process is a renewal process and are independent and identically distributed random variables.
The model furthermore assumes that almost surely and that and are independent. The model is also known as the renewal risk model.
Expected discounted penalty function
Michael R. Powers and Gerber and Shiu analyzed the behavior of the insurer's surplus through the expected discounted penalty function, which is commonly referred to as Gerber-Shiu function in the ruin literature and named after actuarial scientists Elias S.W. Shiu and Hans-Ulrich Gerber. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.
In Powers' notation, this is defined as
,
where is the discounting force of interest, is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation corresponds to the probability measure . The function is called expected discounted cost of insolvency by Powers.
In Gerber and Shiu |
https://en.wikipedia.org/wiki/Sl2-triple | In the theory of Lie algebras, an sl2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl2. This notion plays an important role in the theory of semisimple Lie algebras, especially in regard to their nilpotent orbits.
Definition
Elements {e,h,f} of a Lie algebra g form an sl2-triple if
These commutation relations are satisfied by the generators
of the Lie algebra sl2 of 2 by 2 matrices with zero trace. It follows that sl2-triples in g are in a bijective correspondence with the Lie algebra homomorphisms from sl2 into g.
The alternative notation for the elements of an sl2-triple is {H, X, Y}, with H corresponding to h, X corresponding to e, and Y corresponding to f. H is called a neutral, X is called a nilpositive, and Y is called a nilnegative.
Properties
Assume that g is a finite dimensional Lie algebra over a field of characteristic zero.
From the representation theory of the Lie algebra sl2, one concludes that the Lie algebra g decomposes into a direct sum of finite-dimensional subspaces, each of which is isomorphic to Vj, the (j + 1)-dimensional simple sl2-module with highest weight j. The element h of the sl2-triple is semisimple, with the simple eigenvalues j, j − 2, …, −j on a submodule of g isomorphic to Vj . The elements e and f move between different eigenspaces of h, increasing the eigenvalue by 2 in case of e and decreasing it by 2 in case of f. In particular, e and f are nilpotent elements of the Lie algebra g.
Conversely, the Jacobson–Morozov theorem states that any nilpotent element e of a semisimple Lie algebra g can be included into an sl2-triple {e,h,f}, and all such triples are conjugate under the action of the group ZG(e), the centralizer of e in the adjoint Lie group G corresponding to the Lie algebra g.
The semisimple element h of any sl2-triple containing a given nilpotent element e of g is called a characteristic of e.
An sl2-triple defines a grading on g according to the eigenvalues of h:
The sl2-triple is called even if only even j occur in this decomposition, and odd otherwise.
If g is a semisimple Lie algebra, then g0 is a reductive Lie subalgebra of g (it is not semisimple in general). Moreover, the direct sum of the eigenspaces of h with non-negative eigenvalues is a parabolic subalgebra of g with the Levi component g0.
If the elements of an sl2-triple are regular, then their span is called a principal subalgebra.
See also
Affine Weyl group
Finite Coxeter group
Hasse diagram
Linear algebraic group
Nilpotent orbit
Root system
Special linear Lie algebra
Weyl group
References
A. L. Onishchik, E. B. Vinberg, V. V. Gorbatsevich, Structure of Lie groups and Lie algebras. Lie groups and Lie algebras, III. Encyclopaedia of Mathematical Sciences, 41. Springer-Verlag, Berlin, 1994. iv+248 pp. (A translation of Current problems in mathematics. Fundamental directions. Vol. 41, Akad. Nauk SSSR, Vse |
https://en.wikipedia.org/wiki/Steinberg%20group%20%28K-theory%29 | In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of .
It is named after Robert Steinberg, and it is connected with lower -groups, notably and .
Definition
Abstractly, given a ring , the Steinberg group is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).
Presentation using generators and relations
A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form , where is the identity matrix, is the matrix with in the -entry and zeros elsewhere, and — satisfy the following relations, called the Steinberg relations:
The unstable Steinberg group of order over , denoted by , is defined by the generators , where and , these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by , is the direct limit of the system . It can also be thought of as the Steinberg group of infinite order.
Mapping yields a group homomorphism . As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.
Interpretation as a fundamental group
The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of .
Relation to K-theory
K1
is the cokernel of the map , as is the abelianization of and the mapping is surjective onto the commutator subgroup.
K2
is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher -groups.
It is also the kernel of the mapping . Indeed, there is an exact sequence
Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: .
K3
showed that .
References
K-theory |
https://en.wikipedia.org/wiki/Victor%20Ginzburg | Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalence of categories). He is currently a Professor of Mathematics at the University of Chicago.
Career
Ginzburg received his Ph.D. at Moscow State University in 1985, under the direction of Alexandre Kirillov and Israel Gelfand.
Ginzburg wrote a textbook Representation theory and complex geometry with Neil Chriss on geometric representation theory.
A paper by Alexander Beilinson, Ginzburg, and Wolfgang Soergel introduced the concept of Koszul duality (cf. Koszul algebra) and the technique of "mixed categories" to representation theory. Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for operads.
In noncommutative geometry, Ginzburg defined, following earlier ideas of Maxim Kontsevich, the notion of Calabi–Yau algebra. An important role in the theory of motivic Donaldson–Thomas invariants is played by the so-called "Ginzburg dg algebra", a Calabi-Yau (dg)-algebra of dimension 3 associated with any cyclic potential on the path algebra of a quiver.
Selected publications
References
External links
1957 births
Living people
Russian emigrants to the United States
American people of Russian-Jewish descent
Soviet Jews
Soviet mathematicians
20th-century American mathematicians
21st-century American mathematicians
University of Chicago faculty
Moscow State University alumni |
https://en.wikipedia.org/wiki/Vaughan%27s%20identity | In mathematics and analytic number theory, Vaughan's identity is an identity found by that can be used to simplify Vinogradov's work on trigonometric sums. It can be used to estimate summatory functions of the form
where f is some arithmetic function of the natural integers n, whose values in applications are often roots of unity, and Λ is the von Mangoldt function.
Procedure for applying the method
The motivation for Vaughan's construction of his identity is briefly discussed at the beginning of Chapter 24 in Davenport. For now, we will skip over most of the technical details motivating the identity and its usage in applications, and instead focus on the setup of its construction by parts. Following from the reference, we construct four distinct sums based on the expansion of the logarithmic derivative of the Riemann zeta function in terms of functions which are partial Dirichlet series respectively truncated at the upper bounds of and , respectively. More precisely, we define and , which leads us to the exact identity that
This last expansion implies that we can write
where the component functions are defined to be
We then define the corresponding summatory functions for to be
so that we can write
Finally, at the conclusion of a multi-page argument of technical and at times delicate estimations of these sums, we obtain the following form of Vaughan's identity when we assume that , , and :
It is remarked that in some instances sharper estimates can be obtained from Vaughan's identity by treating the component sum more carefully by expanding it in the form of
The optimality of the upper bound obtained by applying Vaughan's identity appears to be application-dependent with respect to the best functions and we can choose to input into equation (V1). See the applications cited in the next section for specific examples that arise in the different contexts respectively considered by multiple authors.
Applications
Vaughan's identity has been used to simplify the proof of the Bombieri–Vinogradov theorem and to study Kummer sums (see the references and external links below).
In Chapter 25 of Davenport, one application of Vaughan's identity is to estimate an important prime-related exponential sum of Vinogradov defined by
In particular, we obtain an asymptotic upper bound for these sums (typically evaluated at irrational ) whose rational approximations satisfy
of the form
The argument for this estimate follows from Vaughan's identity by proving by a somewhat intricate argument that
and then deducing the first formula above in the non-trivial cases when and with .
Another application of Vaughan's identity is found in Chapter 26 of Davenport where the method is employed to derive estimates for sums (exponential sums) of three primes.
Examples of Vaughan's identity in practice are given as the following references / citations in this informative post:.
Generalizations
Vaughan's identity was generalized by .
Notes
References |
https://en.wikipedia.org/wiki/Stefan%20Glarner | Stefan Glarner (born 21 November 1987) is a Swiss footballer who plays for FC Köniz in the Swiss 1. Liga.
Career statistics
References
External links
Weltfussball profile
1987 births
Living people
People from Interlaken-Oberhasli District
Swiss men's footballers
Switzerland men's under-21 international footballers
Men's association football midfielders
FC Thun players
FC Sion players
FC Zürich players
FC Köniz players
Swiss Super League players
Swiss Challenge League players
Swiss Promotion League players
Swiss 1. Liga (football) players
Footballers from the canton of Bern |
https://en.wikipedia.org/wiki/Alejandro%20Gavatorta | Alejandro Roberto Gavatorta (born 21 March 1980 in Gálvez, Santa Fe) is an Argentine retired football midfielder.
References
External links
Alejandro Gavatorta – Argentine Primera statistics at Fútbol XXI
Alejandro Gavatorta at BDFA.com.ar
1980 births
Living people
Argentine men's footballers
Club Atlético Colón footballers
Expatriate men's footballers in Romania
Liga I players
FC Politehnica Iași (1945) players
AEK Larnaca FC players
Men's association football midfielders
Footballers from Santa Fe Province |
https://en.wikipedia.org/wiki/Encyclopedia%20of%20Triangle%20Centers | The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the list identifies 54,031 triangle centers.
Each point in the list is identified by an index number of the form X(n)—for example, X(1) is the incenter. The information recorded about each point includes its trilinear and barycentric coordinates and its relation to lines joining other identified points. Links to The Geometer's Sketchpad diagrams are provided for key points. The Encyclopedia also includes a glossary of terms and definitions.
Each point in the list is assigned a unique name. In cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, the 770th point in the list is named point Acamar.
Notable points
The first 10 points listed in the Encyclopedia are:
{| class="wikitable"
|-
! ETC reference !! Name !! Definition
|-
! X(1)
| Incenter
|| center of the incircle
|-
! X(2)
| Centroid
|| intersection of the three medians
|-
! X(3)
| Circumcenter
|| center of the circumscribed circle
|-
! X(4)
| orthocenter
|| intersection of the three altitudes
|-
! X(5)
| nine-point center
|| center of the nine-point circle
|-
! X(6)
| symmedian point
|| intersection of the three symmedians
|-
! X(7)
| Gergonne point
|| symmedian point of contact triangle
|-
! X(8)
| Nagel point
|| intersection of lines from each vertex to the corresponding semiperimeter point
|-
! X(9)
| Mittenpunkt
|| symmedian point of the triangle formed by the centers of the three excircles
|-
! X(10)
| Spieker center
|| center of the Spieker circle
|}
Other points with entries in the Encyclopedia include:
{| class="wikitable"
|-
! ETC reference !! Name
|-
! X(11)
| Feuerbach point
|-
! X(13)
| Fermat point
|-
! X(15), X(16)
| first and second isodynamic points
|-
! X(17), X(18)
| first and second Napoleon points
|-
! X(19)
| Clawson point
|-
! X(20)
| de Longchamps point
|-
! X(21)
| Schiffler point
|-
! X(22)
| Exeter point
|-
! X(39)
| Brocard midpoint
|-
! X(40)
| Bevan point
|-
! X(175)
| Isoperimetric point
|-
! X(176)
| Equal detour point
|}
Similar, albeit shorter, lists exist for quadri-figures (quadrilaterals and systems of four lines) and polygon geometry.
See also
Catalogue of Triangle Cubics
List of triangle topics
Triangle center
The Secrets of Triangles
Modern triangle geometry
References
External links
Implementation of ETC points as Perl subroutines by Jason Cantarella
Encyclopedia of Quadri-figures
Encyclopedia of Polygon Geometry
Triangle centers
Mathematical databases
20th-century encyclopedias
21st-century encyclopedias |
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