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https://en.wikipedia.org/wiki/Frege%3A%20Philosophy%20of%20Mathematics | Frege: Philosophy of Mathematics is a 1991 book about the philosopher Gottlob Frege by the British philosopher Michael Dummett.
Reception
Frege: Philosophy of Mathematics has been highly influential. Together with Frege: Philosophy of Language (1973), it is Dummett's chief contribution to Frege scholarship.
References
Bibliography
Books
1991 non-fiction books
Books about Gottlob Frege
Books by Michael Dummett
English-language books
Logic books
Philosophy of mathematics literature |
https://en.wikipedia.org/wiki/List%20of%20bird%20species%20described%20in%20the%202010s | See also parent article Bird species new to science
This page details the bird species described as new to science in the years 2010 to 2019:
Summary statistics
Number of species described per year
Countries with high numbers of newly described species
Brazil
Peru
Philippines
Indonesia
The birds, year-by-year
2010
Limestone leaf warbler, Phylloscopus calciatilis:
Fenwick's antpitta or Urrao antpitta, Grallaria fenwickorum:
Socotra buzzard, Buteo socotraensis:
Willard's sooty boubou, Laniarius willardi:
Rock tapaculo, Scytalopus petrophilus:
2011
Tsingy wood rail, Canirallus beankaensis:
Bryan's shearwater, Puffinus bryani:
Várzea thrush, Turdus sanchezorum:
2012
Alta Floresta antpitta, Hylopezus whittakeri:
Antioquia wren, Thryophilus sernai:
Sira barbet, Capito fitzpatricki:
†Bermuda towhee, Pipilo naufragus:
Cipó cinclodes, Cinclodes pabsti espinhacensis: . Lumped with long-tailed cinclodes (Cinclodes pabsti) in 2013
Camiguin hawk-owl, Ninox leventisi:. Split from Philippine hawk-owl.
Cebu hawk-owl, Ninox rumseyi: Split from Philippine hawk-owl.
2013
Rinjani scops owl, Otus jolandae:
Pincoya storm petrel, Oceanites pincoyae:
Delta Amacuro softtail, Thripophaga amacurensis:
†Bermuda flicker, Colaptes oceanicus:
†Sao Miguel scops owl, Otus frutuosoi :
Seram masked owl, Tyto almae:
Junin tapaculo, Scytalopus gettyae:
Cambodian tailorbird Orthotomus chaktomuk:
Tropeiro seedeater, Sporophila beltoni:
Sierra Madre ground warbler Robsonius thompsoni:
Guerrero brush-finch Arremon kuehnerii:
Omani owl Strix omanensis:
†New Caledonia snipe, Coenocorypha neocaledonica:
The following 15 Brazilian species are described in the 17th volume of the Handbook of the Birds of the World:
Western striolated-puffbird, Nystalus obamai
Xingu woodcreeper, Dendrocolaptes retentus
Inambari woodcreeper, Lepidocolaptes fatimalimae
Tupana scythebill, Campylorhamphus gyldenstolpei
Tapajós scythebill, Campylorhamphus cardosoi
Roosevelt stipple-throated antwren, Epinecrophylla dentei
Bamboo antwren, Myrmotherula oreni
Predicted antwren, Herpsilochmus praedictus
Aripuana antwren, Herpsilochmus stotzi
Manicoré warbling antbird, Hypocnemis rondoni
Chico's tyrannulet, Zimmerius chicomendesi
Acre tody-tyrant, Hemitriccus cohnhafti
Sucunduri yellow-margined flycatcher, Tolmomyias sucunduri
Inambari gnatcatcher, Polioptila attenboroughi
Campina jay, Cyanocorax hafferi
2014
São Paulo marsh antwren Formicivora paludicola:
[the last issue of RBO 21, from "Dec 2013", was released only in March 2014]
Wakatobi flowerpecker Dicaeum kuehni:
†Cryptic treehunter Cichlocolaptes mazarbarnetti
Bahian mouse-colored tapaculo Scytalopus gonzagai:
Sulawesi streaked flycatcher (Muscicapa sodhii) :
2015
Desert owl Strix hadorami:
Perijá tapaculo Scytalopus perijanus:
Sichuan bush warbler Locustella chengi:
2016
Himalayan forest thrush Zoothera salimalii:
Dahomey forest robin Stiphrornis dahomeyensis: Voelker, G. |
https://en.wikipedia.org/wiki/Guidelines%20for%20Assessment%20and%20Instruction%20in%20Statistics%20Education | The Guidelines for Assessment and Instruction in Statistics Education (GAISE) are a framework for statistics education in grades Pre-K–12 published by the American Statistical Association (ASA) in 2007. The foundations for this framework are the Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics (NCTM) in 2000. A second report focused on statistics education at the collegiate level, the GAISE College Report, was published in 2005. Both reports were endorsed by the ASA. Several grants awarded by the National Science Foundation explicitly reference the GAISE documents as influencing or guiding the projects, and several popular introductory statistics textbooks have cited the GAISE documents as informing their approach.
The GAISE Report (pre-K–12)
The GAISE document provides a two-dimensional framework, specifying four components used in statistical problem solving (formulating questions, collecting data, analyzing data, and interpreting results) and three levels of conceptual understanding through which a student should progress (Levels A, B, and C). A direct parallel between these conceptual levels and grade levels is not made because most students would begin at Level A when they are first exposed to statistics regardless of whether they are in primary, middle, or secondary school. A student's level of statistical maturity is based on experience rather than age.
The GAISE College Report
The GAISE College Report begins by synthesizing the history and current understanding of introductory statistics courses and then lists goals for students based on statistical literacy. Six recommendations for introductory statistics courses are given, namely:
Emphasize statistical thinking and literacy over other outcomes
Use real data where possible
Emphasize conceptual rather than procedural understanding
Take an active learning approach
Analyze data using technology rather than by hand
Focus on supporting student learning with assessments
Examples and suggestions for how these recommendations could be implemented are included in several appendices.
References
External links
The GAISE framework documents
Statistics education
2007 introductions |
https://en.wikipedia.org/wiki/Furrow%20profilometer | A furrow profilometer is used for the measurement of the cross-sectional geometry of furrows and corrugations, and is important in furrow assessments. For each furrow, the cross-sectional geometry should be measured at two to three locations before and after the irrigation. A profilometer for determining the cross-sections of furrows is shown in Figure. Individual scales located on the horizontal rod of the profilometer provide data to plot furrow depth as a function of the lateral distance and the data can be numerically integrated. This gives geometric relationships such as area verses depth, wetted perimeter versus depth and top-width verses depth.
References
Measuring instruments |
https://en.wikipedia.org/wiki/Markus%20L%C3%B6w | Markus Löw (born 4 April 1961) is a former footballer who played as a midfielder or defender. He is the brother of Joachim Löw.
References
External links
Player statistics at statistik-klein.de from the saisons 1982/83, 1983/84, 1984/85, 1985/86, 1986/87, 1987/88, 1988/89, 1989/90 and 1992/93
1961 births
Living people
People from Schönau im Schwarzwald
Footballers from Freiburg (region)
German men's footballers
Men's association football midfielders
Men's association football defenders
SC Freiburg players
FV Biberach players
2. Bundesliga players |
https://en.wikipedia.org/wiki/Ontario%20Mathematics%20Olympiad | The Ontario Mathematics Olympiad (OMO) is an annual mathematics competition for Grade 7s and 8s across Ontario, hosted by the Ontario Association for Mathematics Education (OAME).
Format
Each school can send one team to the OMO, which qualifies by placing in the top two in a regional competition. Each team consists of one Grade 7 boy, one Grade 7 girl, one Grade 8 girl, and one Grade 8 boy. The competition has four stages; individual, pairs, team, and relay. The individual stage is completed by writing a test. The two pairs in the pairs stage are the Grade 7 boy and the Grade 8 girl, and the Grade 8 boy and the Grade 7 girl. Each pair has a set amount of questions to solve. The team stage is completed similarly, with a series of questions. The relay stage is when each member of the team receives a question to answer, but the solution for that question depends on the solution of a previous teammate's question. Thus, if any member of the team does their question wrong, the solutions afterwards are wrong as well.
Awards
There are awards for the top three teams in each of the pairs, team, and relay stages of the competition. For the individual stage, the top three Grade 7 boys, Grade 7 girls, Grade 8 girls, and Grade 8 boys are awarded. There is also a grand prize for the overall winning team.
Past competitions
A table of previous competitions:
References
Educational organizations based in Ontario
Mathematics competitions |
https://en.wikipedia.org/wiki/Log%20structure | In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
Motivation
The idea is to study some algebraic variety (or scheme) U which is smooth but not necessarily proper by embedding it into X, which is proper, and then looking at certain sheaves on X. The problem is that the subsheaf of consisting of functions whose restriction to U is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of , multiplicatively. Remembering this additional structure on X corresponds to remembering the inclusion , which likens X with this extra structure to a variety with boundary (corresponding to ).
Definition
Let X be a scheme. A pre-log structure on X consists of a sheaf of (commutative) monoids on X together with a homomorphism of monoids , where is considered as a monoid under multiplication of functions.
A pre-log structure is a log structure if in addition induces an isomorphism .
A morphism of (pre-)log structures consists in a homomorphism of sheaves of monoids commuting with the associated homomorphisms into .
A log scheme is simply a scheme furnished with a log structure.
Examples
For any scheme X, one can define the trivial log structure on X by taking and to be the inclusion.
The motivating example for the definition of log structure comes from semistable schemes. Let X be a scheme, the inclusion of an open subscheme of X, with complement a divisor with normal crossings. Then there is a log structure associated to this situation, which is , with simply the inclusion morphism into . This is called the canonical (or standard) log structure on X associated to D.
Let R be a discrete valuation ring, with residue field k and fraction field K. Then the canonical log structure on consists of the inclusion of (and not !) inside . This is in fact an instance of the previous construction, but taking .
With R as above, one can also define the hollow log structure on by taking the same sheaf of monoids as previously, but instead sending the maximal ideal of R to 0.
Applications
One application of log structures is the ability to define logarithmic forms (also called differential forms with log poles) on any log scheme. From this, one can for instance define log-smoothness and log-étaleness, generalizing the notions of smooth morphisms and étale morphisms. This then allows the study of deformation theory.
In addition, log structures serve to define the mixed Hodge structure on any smooth complex variety X, by taking a compactification with boundary a normal crossings divisor D, and writing down the corresponding logarithmic de Rham complex.
Log objects also naturally appear as |
https://en.wikipedia.org/wiki/Left%20and%20right%20%28algebra%29 | In algebra, the terms left and right denote the order of a binary operation (usually, but not always, called "multiplication") in non-commutative algebraic structures.
A binary operation ∗ is usually written in the infix form:
The argument is placed on the left side, and the argument is on the right side. Even if the symbol of the operation is omitted, the order of and does matter (unless ∗ is commutative).
A two-sided property is fulfilled on both sides. A one-sided property is related to one (unspecified) of two sides.
Although the terms are similar, left–right distinction in algebraic parlance is not related either to left and right limits in calculus, or to left and right in geometry.
Binary operation as an operator
A binary operation may be considered as a family of unary operators through currying:
,
depending on as a parameter – this is the family of right operations. Similarly,
defines the family of left operations parametrized with .
If for some , the left operation is the identity operation, then is called a left identity. Similarly, if , then is a right identity.
In ring theory, a subring which is invariant under any left multiplication in a ring is called a left ideal. Similarly, a right multiplication-invariant subring is a right ideal.
Left and right modules
Over non-commutative rings, the left–right distinction is applied to modules, namely to specify the side where a scalar (module element) appears in the scalar multiplication.
The distinction is not purely syntactical because one gets two different associativity rules (the lowest row in the table) which link multiplication in a module with multiplication in a ring.
A bimodule is simultaneously a left and right module, with two different scalar multiplication operations, obeying an associativity condition on them.
Other examples
Left eigenvectors
Left and right group actions
In category theory
In category theory the usage of "left" and "right" has some algebraic resemblance, but refers to left and right sides of morphisms. See adjoint functors.
See also
Operator associativity
External links
Abstract algebra
Mathematical terminology |
https://en.wikipedia.org/wiki/Brian%20Westlake | Brian Westlake is a former footballer who played as a centre forward in the Football League for Colchester United, Doncaster Rovers, Halifax Town and Tranmere Rovers.
Career statistics
Source:
References
1943 births
Living people
Footballers from Newcastle-under-Lyme
Men's association football forwards
English men's footballers
Stoke City F.C. players
Doncaster Rovers F.C. players
Halifax Town A.F.C. players
Tranmere Rovers F.C. players
Colchester United F.C. players
Macclesfield Town F.C. players
English Football League players
English expatriate men's footballers
Expatriate men's footballers in Belgium |
https://en.wikipedia.org/wiki/Critical%20exponent%20of%20a%20word | In mathematics and computer science, the critical exponent of a finite or infinite sequence of symbols over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated. For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3.
If w is an infinite word over the alphabet A and x is a finite word over A, then x is said to occur in w with exponent α, for positive real α, if there is a factor y of w with y = xax0 where x0 is a prefix of x, a is the integer part of α, and the length |y| = α |x|: we say that y is an α-power. The word w is α-power-free if it contains no factors which are β-powers for any β ≥ α.
The critical exponent for w is the supremum of the α for which w has α-powers, or equivalently the infimum of the α for which w is α-power-free.
Definition
If is a word (possibly infinite), then the critical exponent of is defined to be
where .
Examples
The critical exponent of the Fibonacci word is (5 + )/2 ≈ 3.618.
The critical exponent of the Thue–Morse sequence is 2. The word contains arbitrarily long squares, but in any factor xxb the letter b is not a prefix of x.
Properties
The critical exponent can take any real value greater than 1.
The critical exponent of a morphic word over a finite alphabet is either infinite or an algebraic number of degree at most the size of the alphabet.
Repetition threshold
The repetition threshold of an alphabet A of n letters is the minimum critical exponent of infinite words over A: clearly this value RT(n) depends only on n. For n=2, any binary word of length four has a factor of exponent 2, and since the critical exponent of the Thue–Morse sequence is 2, the repetition threshold for binary alphabets is RT(2) = 2. It is known that RT(3) = 7/4, RT(4) = 7/5 and that for n≥5 we have RT(n) ≥ n/(n-1). It is conjectured that the latter is the true value, and this has been established for 5 ≤ n ≤ 14 and for n ≥ 33.
See also
Critical exponent of a physical system
Notes
References
Formal languages
Combinatorics on words |
https://en.wikipedia.org/wiki/Mittag-Leffler%20summation | In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by
Definition
Let
be a formal power series in z.
Define the transform of by
Then the Mittag-Leffler sum of y is given by
if each sum converges and the limit exists.
A closely related summation method, also called Mittag-Leffler summation, is given as follows .
Suppose that the Borel transform converges to an analytic function near 0 that can be analytically continued along the positive real axis to a function growing sufficiently slowly that the following integral is well defined (as an improper integral). Then the Mittag-Leffler sum of y is given by
When α = 1 this is the same as Borel summation.
See also
Mittag-Leffler distribution
Mittag-Leffler function
Nachbin's theorem
References
Summability methods |
https://en.wikipedia.org/wiki/Test%20functions%20for%20optimization | In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:
Convergence rate.
Precision.
Robustness.
General performance.
Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given.
The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, Haupt et al. and from Rody Oldenhuis software. Given the number of problems (55 in total), just a few are presented here.
The test functions used to evaluate the algorithms for MOP were taken from Deb, Binh et al. and Binh. The software developed by Deb can be downloaded, which implements the NSGA-II procedure with GAs, or the program posted on Internet, which implements the NSGA-II procedure with ES.
Just a general form of the equation, a plot of the objective function, boundaries of the object variables and the coordinates of global minima are given herein.
Test functions for single-objective optimization
Test functions for constrained optimization
Test functions for multi-objective optimization
See also
Ackley function
Himmelblau's function
Rastrigin function
Rosenbrock function
Shekel function
Binh function
References
Constraint programming
Convex optimization
Types of functions
Test items |
https://en.wikipedia.org/wiki/European%20Cup%20and%20EHF%20Champions%20League%20records%20and%20statistics | This page details statistics of the European Cup and Champions League.
General performances
By club
By nation
Some countries ceased to exist during the early 1990s. SC Magdeburg is the only handball club who has won the European club title while representing two different countries (e.g. East Germany and Germany).
Notes
Results until the Dissolution of the Soviet Union in 1991. Three out of five titles were won by clubs from present day Belarus, while two titles and the additional three times runners-up were achieved by clubs from present day Russia.
Results until the Breakup of Yugoslavia in the early 1990s. Clubs from present day Serbia won the title two times and were runners-up additional two times, clubs from present day Croatia won the title once and were runners-up three times, clubs from present day Bosnia and Herzegovina won the title once and were runners-up once, while clubs from present day Slovenia were runners-up one time.
Results until the Dissolution of Czechoslovakia in 1993. Three titles and two times runners-up were all achieved by HC Dukla Prague.
Number of participating clubs of the Champions League era
The following is a list of clubs that have played in or qualified for the Champions League group stages.
Clubs
Performance review (from 1993–present)
By semi-final appearances (European Cup and EHF Champions League)
Note: In the 1994, 1995 and 1996 seasons there were no semi-finals as the finalists qualified via a group stage. The winners (Braga and TEKA Santander in 1994, Zagreb and Bidasoa Irún in 1995, Bidasoa Irún and Barcelona in 1996) and runners-up (Nîmes and UHK West Wien in 1994, TEKA Santander and THW Kiel in 1995, THW Kiel and Pfadi Winterthur in 1996) of the two groups are still marked as semi-finalists in the table.
By quarter-final and semi-final appearances (EHF Champions League)
EHF Champions League Final Four
The history of the EHF Champions League Final Four system, which was permanently introduced in the 2009–10 season.
By season
Performance by club
Countries
Only on eight occasions has the final of the tournament involved two teams from the same country:
1996 Spain: Barcelona vs Bidasoa Irún 46–38 (23–15, 23–23)
2001 Spain: Portland San Antonio vs Barcelona 52–49 (30–24, 22–25)
2005 Spain: Barcelona vs Ciudad Real 56–55 (27–28, 29–27)
2006 Spain: Ciudad Real vs Portland San Antonio 62–47 (25–19, 37–28)
2007 Germany: THW Kiel vs Flensburg-Handewitt 57–55 (28–28, 29–27)
2011 Spain: Barcelona vs Ciudad Real 27–24
2014 Germany: Flensburg-Handewitt vs THW Kiel 30–28
2018 France: Montpellier vs HBC Nantes 32–26
The country providing the highest number of wins is Germany with 15 victories, shared by eight teams, VfL Gummersbach (5), THW Kiel (3), Frisch Auf Göppingen (2), TV Grosswallstadt (2), Magdeburg (1), HSV Hamburg (1) and Flensburg-Handewitt (1)
See also
EHF Champions League
Women's EHF Champions League
References
External links
Official website
EHF Champions League
Sports recor |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20HNK%20Cibalia%20season | This article shows statistics of individual players for the Cibalia football club. It also lists all matches that Cibalia played in the 2012–13 season.
First-team squad
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Matches
Prva HNL
Croatian Cup
Sources: Prva-HNL.hr
Player seasonal records
Competitive matches only. Updated to games played 8 December 2012.
Top scorers
Source: Competitive matches
References
2012-13
Croatian football clubs 2012–13 season |
https://en.wikipedia.org/wiki/Chaviano | Chaviano is a surname. Notable people with the surname include:
Daína Chaviano (born 1957), Cuban-American novelist
Francisco Chaviano, Cuban human rights activist and mathematics professor
Flores Chaviano (born 1946), Cuban composer, guitarist, professor, and orchestral conductor |
https://en.wikipedia.org/wiki/Horrocks%20construction | In mathematics, the Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by . His original construction gave an example of an indecomposable rank 2 vector bundle over 3-dimensional projective space, and generalizes to give examples of vector bundles of higher ranks over other projective spaces. The Horrocks construction is used in the ADHM construction to construct instantons over the 4-sphere.
References
Vector bundles |
https://en.wikipedia.org/wiki/Bobby%20Griffiths | Robert William Griffithss (born 15 September 1942) is an English footballer, who played as a wing half in the Football League for Chester.
Career statistics
Source:
References
Chester City F.C. players
Bangor City F.C. players
Men's association football wing halves
English Football League players
1942 births
Living people
English men's footballers
People from Aldridge
Sportspeople from the Metropolitan Borough of Walsall
Rhyl F.C. players
Stoke City F.C. players |
https://en.wikipedia.org/wiki/Craig%20Hawtin | Craig Scott Hawtin (born 29 March 1970) is an English former professional footballer who played as a full-back in the Football League for Chester City.
Career statistics
Source:
References
1970 births
Living people
English men's footballers
Footballers from Buxton
Men's association football fullbacks
Port Vale F.C. players
Chester City F.C. players
Burnley F.C. players
Runcorn F.C. Halton players
English Football League players
National League (English football) players |
https://en.wikipedia.org/wiki/George%20Casella | George Casella (January 22, 1951 – June 17, 2012) was a Distinguished Professor in the Department of Statistics at the University of Florida. He died from multiple myeloma.
Academic career
Casella completed his undergraduate education at Fordham University and graduate education at Purdue University. He served on the faculty of Rutgers University, Cornell University, and the University of Florida. His contributions focused on the area of statistics including Monte Carlo methods, model selection, and genomic analysis. He was particularly active in Bayesian and empirical Bayes methods, with works connecting with the Stein phenomenon, on assessing and accelerating the convergence of Markov chain Monte Carlo methods, as in his Rao-Blackwellisation technique, and recasting lasso as Bayesian posterior mode estimation with independent Laplace priors.
Awards
Casella was named as a Fellow of the American Statistical Association and the Institute of Mathematical Statistics in 1988, and he was made an Elected Fellow of the International Statistical Institute in 1989. In 2009, he was made a Foreign Member of the Spanish Royal Academy of Sciences.
Selected bibliography
References
External links
1951 births
2012 deaths
Deaths from multiple myeloma
American statisticians
Fellows of the American Statistical Association
University of Florida faculty
Fordham University alumni
Purdue University alumni
Fellows of the Institute of Mathematical Statistics
Elected Members of the International Statistical Institute
Cornell University faculty
The Bronx High School of Science alumni
Computational statisticians
Mathematical statisticians |
https://en.wikipedia.org/wiki/Lee%20Albert%20Rubel | Lee Albert Rubel ( – ) was a mathematician known for his contributions to analog computing.
Career
Originally from New York, he held a Doctorate of Mathematics degree from University of Wisconsin-Madison, and was professor of Mathematics at University of Illinois at Urbana-Champaign since 1954.
He wrote for several scientific publications like the Complex Variables and Elliptic Equations International Journal, the Constructive Approximation mathematical journal, the American Mathematical Monthly, the Journal of Differential Equations, the Journal of Approximation Theory, the Journal of Symbolic Logic, the Journal of the Australian Mathematical Society. He also collaborated to the Functional Analysis periodical, the Tohoku Mathematical, the Mathematical Proceedings of the Cambridge Philosophical Society, the Franklin Institute-engineering and Applied Mathematics, Combinatorica, Israel Journal of Mathematics, and Journal of Theoretical Neurobiology, among others.
He was a member of the American Mathematical Society for 43 years, which published many of his papers in the Proceedings of the AMS.
He died on March 25, 1995, in Urbana, Illinois.
Academic publications
References
20th-century American mathematicians
University of Illinois Urbana-Champaign faculty
University of Wisconsin–Madison College of Letters and Science alumni
1928 births
1995 deaths |
https://en.wikipedia.org/wiki/Jumping%20line | In mathematics, a jumping line or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in other words the structure of its restriction to the line "jumps". Jumping lines were introduced by . The jumping lines of a vector bundle form a proper closed subset of the Grassmannian of all lines of projective space.
The Birkhoff–Grothendieck theorem classifies the n-dimensional vector bundles over a projective line as corresponding to unordered n-tuples of integers. This phenomenon cannot be generalized to higher dimensional projective spaces, namely, one cannot decompose an arbitrary bundle in terms of a Whitney sum of powers of the Tautological bundle, or in fact of line bundles in general. Still one can gain information of this type by using the following method. Given a bundle on , , we may take a line in , or equivalently, a 2-dimensional subspace of . This forms a variety equivalent to embedded in , so we can the restriction of to , and it will decompose by the Birkhoff–Grothendieck theorem as a sum of powers of the Tautological bundle. It can be shown that the unique tuple of integers specified by this splitting is the same for a 'generic' choice of line. More technically, there is a non-empty, open sub-variety of the Grassmannian of lines in , with decomposition of the same type. Lines such that the decomposition differs from this generic type are called 'Jumping Lines'. If the bundle is generically trivial along lines, then the Jumping lines are precisely the lines such that the restriction is nontrivial.
Example
Suppose that V is a 4-dimensional complex vector space with a non-degenerate skew-symmetric form. There is a rank 2 vector bundle over the 3-dimensional complex projective space associated to V, that assigns to each line L of V the 2-dimensional vector space L⊥/L. Then a plane of V corresponds to a jumping line of this vector bundle if and only if it is isotropic for the skew-symmetric form.
References
Vector bundles |
https://en.wikipedia.org/wiki/Kempf%20vanishing%20theorem | In algebraic geometry, the Kempf vanishing theorem, introduced by , states that the higher cohomology group Hi(G/B,L(λ)) (i > 0) vanishes whenever λ is a dominant weight of B. Here G is a reductive algebraic group over an algebraically closed field, B a Borel subgroup, and L(λ) a line bundle associated to λ. In characteristic 0 this is a special case of the Borel–Weil–Bott theorem, but unlike the Borel–Weil–Bott theorem, the Kempf vanishing theorem still holds in positive characteristic.
and found simpler proofs of the Kempf vanishing theorem using the Frobenius morphism.
References
Algebraic groups
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Ruth%20Lyttle%20Satter%20Prize%20in%20Mathematics | The Ruth Lyttle Satter Prize in Mathematics, also called the Satter Prize, is one of twenty-one prizes given out by the American Mathematical Society (AMS). It is presented biennially in recognition of an outstanding contribution to mathematics research by a woman in the previous six years. The award was funded in 1990 using a donation from Joan Birman, in memory of her sister, Ruth Lyttle Satter, who worked primarily in biological sciences, and was a proponent for equal opportunities for women in science. First awarded in 1991, the award is intended to "honor [Satter's] commitment to research and to encourage women in science". The winner is selected by the council of the AMS, based on the recommendation of a selection committee. The prize is awarded at the Joint Mathematics Meetings during odd numbered years, and has always carried a modest cash reward. Since 2003, the prize has been $5,000, while from 1997 to 2001, the prize came with $1,200, and prior to that it was $4,000. If a joint award is made, the prize money is split between the recipients.
, the award has been given 15 times, to 16 different individuals. Dusa McDuff was the first recipient of the award, for her work on symplectic geometry. A joint award was made for the only time in 2001, when Karen E. Smith and Sijue Wu shared the award. The 2013 prize winner was Maryam Mirzakhani, who, in 2014, was the first woman to be awarded the Fields Medal. This is considered to be the highest honor a mathematician can receive. She won both awards for her work on "the geometry of Riemann surfaces and their moduli spaces". The most recent winner is Kaisa Matomäki, who was awarded the prize in 2021 for her "work (much of it joint with Maksym Radziwiłł) opening up the field of multiplicative functions in short intervals in a completely unexpected and very fruitful way".
The Association for Women in Science have a similarly titled award, the Ruth Satter Memorial Award, which is a cash prize of $1,000 for "an outstanding graduate student who interrupted her education for at least 3 years to raise a family".
Recipients
See also
List of mathematics awards
References
Awards of the American Mathematical Society
Science awards honoring women
American science and technology awards
Awards established in 1991
Lists of women scientists
United States science-related lists
Lists of mathematicians by award
International awards
Women in mathematics
1991 establishments in the United States |
https://en.wikipedia.org/wiki/List%20of%20software%20reliability%20models | Software reliability is the probability of the software causing a system failure over some specified operating time. Software does not fail due to wear out but does fail due to faulty functionality, timing, sequencing, data, and exception handling. The software fails as a function of operating time as opposed to calendar time. Over 225 models have been developed since early 1970s, however, several of them have similar if not identical assumptions. The models have two basic types - prediction modeling and estimation modeling.
1.0 Overview of Software Reliability Prediction Models
These models are derived from actual historical data from real software projects. The user answers a list of questions which calibrate the historical data to yield a software reliability prediction. The accuracy of the prediction depends on how many parameters (questions) and datasets are in the model, how current the data is, and how confident the user is of their inputs. One of the earliest prediction models was the Rome Laboratory TR-92-52. It was developed in 1987 and last updated in 1992 and was geared towards software in avionics systems. Due to the age of the model and data it's no longer recommended but is the basis for several modern models such as the Shortcut model, Full-scale model, and Neufelder assessment model. There are also lookup tables for software defect density based on the capability maturity or the application type. These are very simple models but are generally not as accurate as the assessment based models.
2.0 Overview of Software Reliability Growth (Estimation) Models
Software reliability growth (or estimation) models use failure data from testing to forecast the failure rate or MTBF into the future. The models depend on the assumptions about the fault rate during testing which can either be increasing, peaking, decreasing or some combination of decreasing and increasing. Some models assume that there is a finite and fixed number of inherent defects while others assume that it's infinite. Some models require effort for parameter estimation while others have only a few parameters to estimate. Some models require the exact time in between each failure found in testing, while others only need to have the number of failures found during any given time interval such as a day.
Software reliability tools implementing some of these models include CASRE (Computer-Aided Software Reliability Estimation) and an open source SFRAT (Software Failure and Reliability Assessment Tool).
References
Software testing |
https://en.wikipedia.org/wiki/Johnson%27s%20SU-distribution | The Johnson's SU-distribution is a four-parameter family of probability distributions first investigated by N. L. Johnson in 1949. Johnson proposed it as a transformation of the normal distribution:
where .
Generation of random variables
Let U be a random variable that is uniformly distributed on the unit interval [0, 1]. Johnson's SU random variables can be generated from U as follows:
where Φ is the cumulative distribution function of the normal distribution.
Johnson's SB-distribution
N. L. Johnson firstly proposes the transformation :
where .
Johnson's SB random variables can be generated from U as follows:
The SB-distribution is convenient to Platykurtic distributions (Kurtosis).
To simulate SU, sample of code for its density and cumulative distribution function is available here
Applications
Johnson's -distribution has been used successfully to model asset returns for portfolio management.
This comes as a superior alternative to using the Normal distribution to model asset returns. An R package, JSUparameters, was developed in 2021 to aid in the estimation of the parameters of the best-fitting Johnson's -distribution for a given dataset. Johnson distributions are also sometimes used in option pricing, so as to accommodate an observed volatility smile; see Johnson binomial tree.
An alternative to the Johnson system of distributions is the quantile-parameterized distributions (QPDs). QPDs can provide greater shape flexibility than the Johnson system. Instead of fitting moments, QPDs are typically fit to empirical CDF data with linear least squares.
Johnson's -distribution is also used in the modelling of the invariant mass of some heavy mesons in the field of B-physics.
References
Further reading
( Preprint)
Continuous distributions |
https://en.wikipedia.org/wiki/Sebastian%20Mladen | Sebastian Mladen (born 11 December 1991) is a Romanian professional footballer who plays as a defensive midfielder or a defender for Greek Super League club Panetolikos.
Career statistics
Club
Honours
Viitorul Constanța
Liga I: 2016–17
Cupa României: 2018–19
Supercupa României: 2019
Farul Constanța
Liga I: 2022–23
References
External links
1991 births
Living people
People from Calafat
Men's association football defenders
Romanian men's footballers
Romanian expatriate men's footballers
Romania men's under-21 international footballers
Expatriate men's footballers in Italy
Expatriate men's footballers in Portugal
Expatriate men's footballers in Greece
Romanian expatriate sportspeople in Italy
Romanian expatriate sportspeople in Portugal
Romanian expatriate sportspeople in Greece
AS Roma players
AFC Chindia Târgoviște players
S.C. Olhanense players
FC Südtirol players
FC Viitorul Constanța players
FCV Farul Constanța players
Panetolikos F.C. players
Liga I players
Primeira Liga players
Serie C players
Super League Greece players
Footballers from Dolj County |
https://en.wikipedia.org/wiki/Inverse%20distribution | In probability theory and statistics, an inverse distribution is the distribution of the reciprocal of a random variable. Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables, inverse distributions are special cases of the class of ratio distributions, in which the numerator random variable has a degenerate distribution.
Relation to original distribution
In general, given the probability distribution of a random variable X with strictly positive support, it is possible to find the distribution of the reciprocal, Y = 1 / X. If the distribution of X is continuous with density function f(x) and cumulative distribution function F(x), then the cumulative distribution function, G(y), of the reciprocal is found by noting that
Then the density function of Y is found as the derivative of the cumulative distribution function:
Examples
Reciprocal distribution
The reciprocal distribution has a density function of the form.
where means "is proportional to".
It follows that the inverse distribution in this case is of the form
which is again a reciprocal distribution.
Inverse uniform distribution
If the original random variable X is uniformly distributed on the interval (a,b), where a>0, then the reciprocal variable Y = 1 / X has the reciprocal distribution which takes values in the range (b−1 ,a−1), and the probability density function in this range is
and is zero elsewhere.
The cumulative distribution function of the reciprocal, within the same range, is
For example, if X is uniformly distributed on the interval (0,1), then Y = 1 / X has density and cumulative distribution function when
Inverse t distribution
Let X be a t distributed random variate with k degrees of freedom. Then its density function is
The density of Y = 1 / X is
With k = 1, the distributions of X and 1 / X are identical (X is then Cauchy distributed (0,1)). If k > 1 then the distribution of 1 / X is bimodal.
Reciprocal normal distribution
If variable follows a normal distribution , then the inverse or reciprocal follows a reciprocal normal distribution:
If variable X follows a standard normal distribution , then Y = 1/X follows a reciprocal standard normal distribution,
heavy-tailed and bimodal,
with modes at and density
and the first and higher-order moments do not exist. For such inverse distributions and for ratio distributions, there can still be defined probabilities for intervals, which can be computed either by Monte Carlo simulation or, in some cases, by using the Geary–Hinkley transformation.
However, in the more general case of a shifted reciprocal function , for following a general normal distribution, then mean and variance statistics do exist in a principal value sense, if the difference between the pole and the mean is real-valued. The mean of this transformed random variable (reciprocal shifted normal di |
https://en.wikipedia.org/wiki/Resolvent%20%28Galois%20theory%29 | In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root.
Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are
where is the discriminant, which is a resolvent for the alternating group. In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation.
The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements.
The Cayley resolvent is a resolvent for the maximal resoluble Galois group in degree five. It is a polynomial of degree 6.
These three resolvents have the property of being always separable, which means that, if they have a multiple root, then the polynomial p is not irreducible. It is not known if there is an always separable resolvent for every group of permutations.
For every equation the roots may be expressed in terms of radicals and of a root of a resolvent for a resoluble group, because, the Galois group of the equation over the field generated by this root is resoluble.
Definition
Let be a positive integer, which will be the degree of the equation that we will consider, and an ordered list of indeterminates.
According to Vieta's formulas this defines the generic monic polynomial of degree
where is the th elementary symmetric polynomial.
The symmetric group acts on the by permuting them, and this induces an action on the polynomials in the . The stabilizer of a given polynomial under this action is generally trivial, but some polynomials have a bigger stabilizer. For example, the stabilizer of an elementary symmetric polynomial is the whole group . If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial subgroup ; it is said to be an invariant of . Conversely, given a subgroup of , an invariant of is a resolvent invariant for if it is not an invariant of any bigger subgroup of .
Finding invariants for a given subgroup of is relatively easy; one can sum the orbit of a monomial under the action of . However, it may occur that the resulting polynomial is an invariant for a larger group. For example, consider the case of the subgroup of of order 4, consisting of , , and the identity (for the notation, see Permutation group). The monomial gives the invariant . It is not a resolvent invariant for , because being invariant by , it is in fact a resolvent invariant for the larger dihedral subgroup : , |
https://en.wikipedia.org/wiki/Log%20semiring | In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary algebraic operations on real numbers, and then take the logarithm to reverse the initial exponentiation. Such operations are also known as, e.g., logarithmic addition, etc. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base for the exponent and logarithm ( is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base is equivalent to using a negative sign and using the inverse . If not qualified, the base is conventionally taken to be or , which corresponds to with a negative.
The log semiring has the tropical semiring as limit ("tropicalization", "dequantization") as the base goes to infinity (max-plus semiring) or to zero (min-plus semiring), and thus can be viewed as a deformation ("quantization") of the tropical semiring. Notably, the addition operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log semiring has applications in mathematical optimization, since it replaces the non-smooth maximum and minimum by a smooth operation. The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see ), log probability, or log-likelihoods.
Definition
The operations on the log semiring can be defined extrinsically by mapping them to the non-negative real numbers, doing the operations there, and mapping them back. The non-negative real numbers with the usual operations of addition and multiplication form a semiring (there are no negatives), known as the probability semiring, so the log semiring operations can be viewed as pullbacks of the operations on the probability semiring, and these are isomorphic as rings.
Formally, given the extended real numbers } and a base , one defines:
Regardless of base, log multiplication is the same as usual addition, , since logarithms take multiplication to addition; however, log addition depends on base. The units for usual addition and multiplication are 0 and 1; accordingly, the unit for log addition is for and for , and the unit for log multiplication is , regardless of base.
More concisely, the unit log semiring can be defined for base as:
with additive unit and multiplicative unit 0; this corresponds to the max convention.
The opposite convention is also common, and corresponds to the base , the minimum convention:
with additive unit and multiplicative unit 0.
Properties
A log s |
https://en.wikipedia.org/wiki/2011%20Nepal%20census | Nepal conducted a widespread national census in 2011 by the Nepal Central Bureau of Statistics. Working with the 58 municipalities and the 3915 Village Development Committees at a district level, they recorded data from all the municipalities and villages of each district. The data included statistics on population size, households, sex and age distribution, place of birth, residence characteristics, literacy, marital status, religion, language spoken, caste/ethnic group, economically active population, education, number of children, employment status, and occupation.
Total population in 2011: 26,494,504
Increase since last census 2001: 3,343,081
Annual population growth rate (exponental growth): 1.35
Number of households: 5,427,302
Average Household Size: 4.88
Population in Mountain: 6.73%, Hill: 43.00% and Terai: 50.27%.
Nepalese caste/ethnic groups
The population wise ranking of 126 Nepalese castes/ethnic groups as per 2011 Nepal census.
See also
List of village development committees of Nepal (Former)
1991 Nepal census
2001 Nepal census
2021 Nepal census
Notes
References
External links
Central Bureau of Statistics
National Population and Housing Census 2011
Censuses in Nepal
Nepal |
https://en.wikipedia.org/wiki/2001%20Nepal%20census | The 2001 Nepal census () was conducted by the Nepal Central Bureau of Statistics.
According to the census, the population of Nepal in 2001 was 23,151,423.
Working with Nepal's Village Development Committees at a district level, they recorded data from all the main towns and villages of each district of Nepal. The data included statistics on population size, households, sex and age distribution, place of birth, residence characteristics, literacy, marital status, religion, language spoken, caste/ethnic group, economically active population, education, number of children, employment status, and occupation.
See also
List of village development committees of Nepal
1991 Nepal census
2011 Nepal census
References
Censuses in Nepal
Census
Nepal |
https://en.wikipedia.org/wiki/Monad%20%28homological%20algebra%29 | In homological algebra, a monad is a 3-term complex
A → B → C
of objects in some abelian category whose middle term B is projective, whose first map A → B is injective, and whose second map B → C is surjective. Equivalently, a monad is a projective object together with a 3-step filtration B ⊃ ker(B → C) ⊃ im(A → B). In practice A, B, and C are often vector bundles over some space, and there are several minor extra conditions that some authors add to the definition. Monads were introduced by .
See also
ADHM construction
References
Vector bundles
Homological algebra |
https://en.wikipedia.org/wiki/Splicing%20rule | In mathematics and computer science, a splicing rule is a transformation on formal languages which formalises the action of gene splicing in molecular biology. A splicing language is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the regular languages.
Definition
Let A be an alphabet and L a language, that is, a subset of the free monoid A∗. A splicing rule is a quadruple r = (a,b,c,d) of elements of A∗, and the action of the rule r on L is to produce the language
If R is a set of rules then R(L) is the union of the languages produced by the rules of R. We say that R respects L if R(L) is a subset of L. The R-closure of L is the union of L and all iterates of R on L: clearly it is respected by R. A splicing language is the R-closure of a finite language.
A rule set R is reflexive if (a,b,c,d) in R implies that (a,b,a,b) and (c,d,c,d) are in R. A splicing language is reflexive if it is defined by a reflexive rule set.
Examples
Let A = {a,b,c}. The rule (caba,a,cab,a) applied to the finite set {cabb,cabab,cabaab} generates the regular language caba∗b.
Properties
All splicing languages are regular.
Not all regular languages are splicing. An example is (aa)∗ over {a,b}.
If L is a regular language on the alphabet A, and z is a letter not in A, then the language { zw : w in L } is a splicing language.
There is an algorithm to determine whether a given regular language is a reflexive splicing language.
The set of splicing rules that respect a regular language can be determined from the syntactic monoid of the language.
References
Semigroup theory
Formal languages
Combinatorics on words |
https://en.wikipedia.org/wiki/Edgar%20Lorch | Edgar Raymond Lorch (July 22, 1907 – March 5, 1990) was a Swiss American mathematician. Described by The New York Times as "a leader in the development of modern mathematics theory", he was a professor of mathematics at Columbia University. He contributed to the fields general topology, especially metrizable and Baire spaces, group theory of permutation groups and functional analysis, especially spectral theory, convexity in Hilbert spaces and normed rings.
Biography
Born in Switzerland, Lorch emigrated with his family to the United States in 1917 and became a citizen in 1932. He joined the faculty of Columbia University in 1935 and retired in 1976, although he continued to write and lecture as professor emeritus. For his reminiscences of Szeged, Edgar R. Lorch posthumously received in 1994 the Lester R. Ford Award, with Reuben Hersh as editor.
References
External links
1907 births
1990 deaths
Swiss emigrants to the United States
20th-century American mathematicians
Columbia University faculty
Columbia College (New York) alumni |
https://en.wikipedia.org/wiki/Wonderful%20compactification | In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group is a -equivariant compactification such that the closure of each orbit is smooth. constructed a wonderful compactification of any symmetric variety given by a quotient of an algebraic group by the subgroup fixed by some involution of over the complex numbers, sometimes called the De Concini–Procesi compactification, and generalized this construction to arbitrary characteristic. In particular, by writing a group itself as a symmetric homogeneous space, (modulo the diagonal subgroup), this gives a wonderful compactification of the group itself.
References
Algebraic groups
Compactification (mathematics) |
https://en.wikipedia.org/wiki/Locally%20finite%20operator | In mathematics, a linear operator is called locally finite if the space is the union of a family of finite-dimensional -invariant subspaces.
In other words, there exists a family of linear subspaces of , such that we have the following:
Each is finite-dimensional.
An equivalent condition only requires to be the spanned by finite-dimensional -invariant subspaces. If is also a Hilbert space, sometimes an operator is called locally finite when the sum of the is only dense in .
Examples
Every linear operator on a finite-dimensional space is trivially locally finite.
Every diagonalizable (i.e. there exists a basis of whose elements are all eigenvectors of ) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of .
The operator on , the space of polynomials with complex coefficients, defined by , is not locally finite; any -invariant subspace is of the form for some , and so has infinite dimension.
The operator on defined by is locally finite; for any , the polynomials of degree at most form a -invariant subspace.
References
Abstract algebra
Functions and mappings
Linear algebra
Transformation (function) |
https://en.wikipedia.org/wiki/Dominique%20de%20Caen | Dominique de Caen ( – ) was a mathematician, Doctor of Mathematics, and professor of Mathematics, who specialized in graph theory, probability, and information theory. He is renowned for his research on Turán's extremal problem for hypergraphs.
Career
He studied mathematics at McGill University, where he earned a Bachelor of Science degree in 1977.
In 1979, he obtained a Master of Science degree from Queen's University with a thesis on Prime Boolean matrices.
In 1982, he earned the Doctorate of Mathematics degree from University of Toronto with a thesis entitled On Turán's Hypergraph Problem which was supervised by Eric Mendelsohn.
Most of his academic papers have been published in the journals Discrete Mathematics, Designs, Codes and Cryptography, the Journal of Combinatorial Theory, and the European Journal of Combinatorics, among others.
Academic research
References
20th-century Canadian mathematicians
1956 births
2002 deaths
21st-century Canadian mathematicians
McGill University Faculty of Science alumni
Queen's University at Kingston alumni
University of Toronto alumni
Graph theorists |
https://en.wikipedia.org/wiki/Matsushima%27s%20formula | In mathematics, Matsushima's formula, introduced by , is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of the group G.
The Matsushima–Murakami formula is a generalization giving dimensions of spaces of automorphic forms, introduced by .
References
Differential geometry
Algebraic topology
Topological graph theory
Generating functions |
https://en.wikipedia.org/wiki/Dianalytic%20manifold | In mathematics, dianalytic manifolds are possibly non-orientable generalizations of complex analytic manifolds. A dianalytic structure on a manifold is given by an atlas of
charts such that the transition maps are either complex analytic maps or complex conjugates of complex analytic maps. Every dianalytic manifold is given by the quotient of an analytic manifold (possibly non-connected) by a fixed-point-free involution changing the complex structure to its complex conjugate structure. Dianalytic manifolds were introduced by , and dianalytic manifolds of 1 complex dimension are sometimes called Klein surfaces.
References
Riemann surfaces |
https://en.wikipedia.org/wiki/Ihara%27s%20lemma | In mathematics, Ihara's lemma, introduced by and named by , states that the kernel of the sum of the two p-degeneracy maps from J0(N)×J0(N) to J0(Np) is Eisenstein whenever the prime p does not divide N. Here J0(N) is the Jacobian of the compactification of the modular curve of Γ0(N).
References
Lemmas in number theory |
https://en.wikipedia.org/wiki/Yasutaka%20Ihara | Yasutaka Ihara (伊原 康隆, Ihara Yasutaka; born 1938, Tokyo Prefecture) is a Japanese mathematician and professor emeritus at the Research Institute for Mathematical Sciences. His work in number theory includes Ihara's lemma and the Ihara zeta function.
Career
Ihara received his PhD at the University of Tokyo in 1967 with thesis Hecke polynomials as congruence zeta functions in elliptic modular case.
From 1965 to 1966, Ihara worked at the Institute for Advanced Study. He was a professor at the University of Tokyo and then at the Research Institute for Mathematical Science (RIMS) of the University of Kyōto. In 2002 he retired from RIMS as professor emeritus and then became a professor at Chūō University.
In 1970, he was an invited speaker (with lecture Non abelian class fields over function fields in special cases) at the International Congress of Mathematicians (ICM) in Nice. In 1990, Ihara gave a plenary lecture Braids, Galois groups and some arithmetic functions at the ICM in Kyōto.
His doctoral students include Kazuya Katō.
Research
Ihara has worked on geometric and number theoretic applications of Galois theory. In the 1960s, he introduced the eponymous Ihara zeta function. In graph theory the Ihara zeta function has an interpretation, which was conjectured by Jean-Pierre Serre and proved by Toshikazu Sunada in 1985. Sunada also proved that a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.
Selected works
On Congruence Monodromy Problems, Mathematical Society of Japan Memoirs, World Scientific 2009 (based on lectures in 1968/1969)
with Michael Fried (ed.): Arithmetic fundamental groups and noncommutative Algebra, American Mathematical Society, Proc. Symposium Pure Math. vol.70, 2002
as editor: Galois representations and arithmetic algebraic geometry, North Holland 1987
with Kenneth Ribet, Jean-Pierre Serre (eds.): Galois Groups over Q, Springer 1989 (Proceedings of a Workshop 1987)
References
External links
Yasutaka Ihara's homepage at RIMS
The Ihara Zeta Function and the Riemann Zeta Function by Mollie Stein, Amelia Wallace
Living people
20th-century Japanese mathematicians
21st-century Japanese mathematicians
1938 births
Number theorists
University of Tokyo alumni
Academic staff of the University of Tokyo |
https://en.wikipedia.org/wiki/Glaeser%27s%20composition%20theorem | In mathematics, Glaeser's theorem, introduced by , is a theorem giving conditions for a smooth function to be a composition of F and θ for some given smooth function θ. One consequence is a generalization of Newton's theorem that every symmetric polynomial is a polynomial in the elementary symmetric polynomials, from polynomials to smooth functions.
References
Theorems in real analysis |
https://en.wikipedia.org/wiki/Alexander%20Chuprov | Alexander Chuprov may refer to:
Alexander Ivanovich Chuprov (1841–1908), Russian professor of political economy and statistics at Moscow University
Alexander Alexandrovich Chuprov (1874–1926), his son, Russian statistician and professor at the St. Petersburg Polytechnical Institute |
https://en.wikipedia.org/wiki/Alexander%20Ivanovich%20Chuprov | Alexander Ivanovich Chuprov (Александр Иванович Чупров; 1841–1908) was a professor of political economy and statistics at Moscow University whose lectures provided the standard introduction to economics for late 19th-century Russian students.
Chuprov's father was an Orthodox priest based in Mosalsk. Alexander attended the Law Department of the Moscow University where he became interested in Wilhelm Roscher's research. He founded the Moscow Society to Disseminate Technical Knowledge in 1869 and was elected into the Russian Academy of Sciences in 1887.
Chuprov has been described as the founder of transport economics and the multiple-factor analysis of economic regions. In his landmark work The Railway Economy (1875–78) he analyzed statistics on railway traffic. He distinguished one region from another according to "the value of the market, the cost of transportation, and demographic indicators".
Chuprov became known as "the heart and soul" of zemstvo statistical investigations and sample surveys in the Russian Empire. His researchers are said to have interviewed 4.5 million Russian muzhiks. Their mission was to provide a modern statistical description of the Russian peasant commune, or obschina. Chuprov viewed the Russian obshchina as a valuable social institution which should be preserved.
Chuprov was a lifelong friend of another prominent statistician, Ivan Yanzhul. One of his students at the Moscow University was Wassily Kandinsky. His son Alexander A. Chuprov (1874–1926) is said to have given "much impetus to statistics in pre-revolutionary Russia".
References
External links
Statisticians from the Russian Empire
Economists from the Russian Empire
Transport economists
Moscow State University alumni
Professorships at the Imperial Moscow University
1841 births
1908 deaths |
https://en.wikipedia.org/wiki/Enoch%20Beery%20Seitz | Enoch Beery Seitz (24 August 1846 in Fairfield County, Ohio – 8 October 1883 in Adair, Missouri) was an American mathematician who was Chair of Mathematics at North Missouri State Normal School).
Seitz was elected to the London Mathematical Society on 11 March 1880, only the fifth American to be so honored. Over 500 of his solutions were published in the Analyst, the Mathematical Visitor, the Mathematical Magazine, the School Visitor and the Educational Times of London, England.
References
External links
Enoch Beery Seitz
19th-century American mathematicians
1846 births
1883 deaths
Mathematicians from Missouri |
https://en.wikipedia.org/wiki/Abel%E2%80%93Plana%20formula | In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that
For the case we have
It holds for functions ƒ that are holomorphic in the region Re(z) ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that |ƒ| is bounded by C/|z|1+ε in this region for some constants C, ε > 0, though the formula also holds under much weaker bounds. .
An example is provided by the Hurwitz zeta function,
which holds for all , . Another powerful example is applying the formula to the function : we obtain
where is the gamma function, is the polylogarithm and .
Abel also gave the following variation for alternating sums:
which is related to the Lindelöf summation formula
Proof
Let be holomorphic on , such that , and for , . Taking with the residue theorem
Then
Using the Cauchy integral theorem for the last one. thus obtaining
This identity stays true by analytic continuation everywhere the integral converges, letting we obtain the Abel–Plana formula
The case ƒ(0) ≠ 0 is obtained similarly, replacing by two integrals following the same curves with a small indentation on the left and right of 0.
See also
Euler–Maclaurin summation formula
Euler–Boole summation
Ramanujan summation
References
External links
Summability methods |
https://en.wikipedia.org/wiki/Hyper-finite%20field | In mathematics, a hyper-finite field is an uncountable field similar in many ways to finite fields. More precisely a field F is called hyper-finite if it is uncountable and quasi-finite, and for every subfield E, every absolutely entire E-algebra (regular field extension of E) of smaller cardinality than F can be embedded in F. They were introduced by . Every hyper-finite field is a pseudo-finite field, and is in particular a model for the first-order theory of finite fields.
References
Field (mathematics) |
https://en.wikipedia.org/wiki/Bochner%E2%80%93Kodaira%E2%80%93Nakano%20identity | In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its complex conjugate and the curvature of the bundle and the torsion of the metric of the manifold. It is named after Salomon Bochner, Kunihiko Kodaira, and Shigeo Nakano.
References
Theorems in differential geometry
Vector bundles
Mathematical identities |
https://en.wikipedia.org/wiki/Ordinary%20singularity | In mathematics, an ordinary singularity of an algebraic curve is a singular point of multiplicity r where the r tangents at the point are distinct .
In higher dimensions the literature on algebraic geometry contains many inequivalent definitions of ordinary singular points.
References
Algebraic curves |
https://en.wikipedia.org/wiki/Solid%20Modeling%20Solutions | Solid Modeling Solutions is a software company which specializes in 3D geometry software.
History
NURBS got started with seminal work at Boeing and SDRC (Structural Dynamics Research Corporation), a leading company in mechanical computer-aided engineering in the 1980s and '90's. The history of NURBS at Boeing goes back to 1979 when Boeing began to staff up for the purpose of developing their own comprehensive CAD/CAM system, TIGER, to support the wide variety of applications needed by their various aircraft and aerospace engineering groups. Three basic decisions were critical to establishing an environment conducive to developing NURBS. The first was Boeing's need to develop their own in-house geometry capability. Boeing had special, rather sophisticated, surface geometry needs, especially for wing design, that could not be found in any commercially available CAD/CAM system. As a result, the TIGER Geometry Development Group was established in 1979 and has been strongly supported for many years. The second decision critical to NURBS development was the removal of the constraint of upward geometrical compatibility with the two systems in use at Boeing at that time. One of these systems had evolved as a result of the iterative process inherent to wing design. The other was best suited for adding to the constraints imposed by manufacturing, such as cylindrical and planar regions. The third decision was simple but crucial and added the 'R' to 'NURBS'. Circles were to be represented exactly: no cubic approximations would be allowed.
By late 1979 there were 5 or 6 well-educated mathematicians (PhD's from Stanford, Harvard, Washington and Minnesota) and some had many years of software experience, but none of them had any industrial, much less CAD, geometry experience. Those were the days of the oversupply of math PhDs. The task was to choose the representations for the 11 required curve forms, which included everything from lines and circles to Bézier and B-spline curves.
By early 1980, the staff were busy choosing curve representations and developing the geometry algorithms for TIGER. One of the major tasks was curve/curve intersection. It was noticed very quickly that one could solve the general intersection problem if one could solve it for the Bézier/Bézier case, since everything could be represented in Bézier form at the lowest level. It was soon realized that the geometry development task would be substantially simplified if a way could be found to represent all of the curves using a single form.
With this motivation, the staff started down the road toward what became NURBS. Consider: the design of a wing demands free-form, C2 continuous, cubic splines to satisfy the needs of aerodynamic analysis, yet the circle and cylinders of manufacturing require at least rational Bézier curves. The properties of Bézier curves and uniform B-splines were well known, but the staff had to gain an understanding of non-uniform B-splines and rat |
https://en.wikipedia.org/wiki/Fricke%20involution | In mathematics, a Fricke involution is the involution of the modular curve X0(N) given by τ → –1/Nτ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular curve, such as spaces of modular forms and the Jacobian J0(N) of the modular curve.
See also
Atkin–Lehner involution
References
Modular forms |
https://en.wikipedia.org/wiki/Poles%20in%20Sweden | Poles in Sweden () are citizens and residents of Sweden who emigrated from Poland.
Demographics
According to Statistics Sweden, as of 2016, there are a total 88,704 Poland-born immigrants living in Sweden. They include both native Poles, as well as descendants of Polish Jewish immigrants from Poland.
Education
In 2010, there were 4,186 students with Polish as their mother tongue who participated in the state-run Swedish for Immigrants adult language program. Of these pupils, 251 had 0–6 years of education in their home country (Antal utbildningsår i hemlandet), 241 had 7–9 years of education in their home country, and 3,694 had 10 years education or more in their home country. As of 2012, 5,100 pupils with Polish as their mother tongue and 5,079 Poland-born students were enrolled in the language program.
Organizations
There are several Polish organizations in Sweden, incl. the Polish Institute in Stockholm, the Polish Cultural Association in Gothenburg, and Polonia Center in Gothenburg.
Notable people
Anna Anka
Dorotea Bromberg
Paula Bieler
Anitha Bondestam
Paweł Cibicki
Wonna I DeJong
Jerzy Einhorn
Greekazo
Peter Jablonski
Catherine Jagiellon
Peter Jewszczewski
Katrine Marcal (born Kielos to Polish immigrant parents)
Kissie (Alexandra Nilsson-Petroniak, her mother is a Polish immigrant)
Oscar Lewicki
Henryk Lipp
Stefan Liv
Jerzy Luczak-Szewczyk
Bea Malecki
Dominika Peczynski
Martin Rolinski
Eliza Roszkowska Öberg
Thomas Rusiak
Jerzy Sarnecki
Danny Saucedo
Izabella Scorupco
Sebastian Siemiatkowski
Czeslaw Slania
Amanda Sokolnicki (political editor of DN)
Bea Szenfeld
Robert Wahlström
Cissi Wallin born in Sweden to Polish parents
Michael Winiarski (journalist in DN)
Peter Wolodarski
Michal Zajkowski
Maciej Zaremba
Małgorzata Pieczyńska
Z.E. (born to Polish parents as Józef Wojciechowicz)
Katrin Zytomierska
See also
Poland–Sweden relations
Polish diaspora
Immigration to Sweden
Poles in Finland
References
Poles
Polish minorities
Poles |
https://en.wikipedia.org/wiki/Homeschooling%20in%20New%20Zealand |
Homeschooling in New Zealand is legal. The Ministry of Education reports annually on the population, age, ethnicity, and turnover of students being educated at home.
Statistics
The 2017 statistics showed:"As at 1 July 2017, there were 6,008 home schooled students recorded in the Ministry of Education's Homeschooling database. These students belong to 3,022 families and represent 0.8% of total school enrolments as at 1 July 2017. Out of the 6,008 homeschoolers 67.3% were the aged 12 or under, 68.3% had been home-schooled for less than 5 years, and only 4.2% had been home-schooled for 10 years or more.
European/Pākehā students are more likely to be homeschooled than any other ethnic group with 80.2% of all homeschoolers identifying as European/Pākehā compared to 50.1% of the total school population. Only 8.7% of homeschoolers identify as Māori compared to 24.0% of the total school population, 2.6% of homeschoolers identify as Pasifika compared to 9.8% of the total school population, and 2.2% of homeschoolers identify as Asian compared to 11.8% of the total school population. The ethnicity of 2.0% of homeschoolers is unknown."
Regulations
Under New Zealand law, all children aged six or over must be enrolled in a registered school unless they have been issued an exemption by the Ministry of Education (MoE). Application must be made to the MoE for a Certificate of Exemption for each child and a statutory declaration signed and sent to the Ministry every six months. In the initial application, the parent or caregiver "must satisfy the Ministry that [the] child will be taught at least as regularly and as well as they would be in a registered school." Parents or caregivers who homeschool may choose to receive a "home education supervision allowance" from the MoE for each exempted child.
Motivations
There are a variety of complex reasons why parents choose to educate their children at home, including wanting to customise the education to the individual child and concern or disagreement with the teaching offered by registered schools. Homeschooling is also done for religious reasons and for special needs children (i.e. those who are gifted, problematic or have learning disabilities).
In February 2023, Radio New Zealand (RNZ) reported that a growing number of New Zealand parents were planning to homeschool their children due to a Government COVID-19 mandate requiring children in Year 4 and above to wear facemasks indoors. RNZ reported that the Ministry of Education had received 867 homeschooling applications in November 2021, 800 in December 2021, and 735 in January 2022. The Education Ministry had also approved 2,655 homeschool applications in 2021, declined 78 and was processing 983 applications by February 2023. Of these applications, 900 were from the Auckland Region, 500 from the Canterbury Region, and 400 in Waikato.
Public opinion
As elsewhere in the world, home education is considered something of an alternative lifestyle. Concerns are some |
https://en.wikipedia.org/wiki/Pseudo-finite%20field | In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite.
Pseudo-finite fields were introduced by .
References
Model theory
Field (mathematics) |
https://en.wikipedia.org/wiki/Nagata%27s%20compactification%20theorem | In algebraic geometry, Nagata's compactification theorem, introduced by , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme S can be factored into an open immersion followed by a proper morphism.
Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow.
Deligne showed, in unpublished notes expounded by Conrad, that Nagata's proof can be translated into scheme theory and that the condition that S is Noetherian can be replaced by the much weaker condition that S is quasi-compact and quasi-separated. gave another scheme-theoretic proof of Nagata's theorem.
An important application of Nagata's theorem is in defining the analogue in algebraic geometry of cohomology with compact support, or more generally higher direct image functors with proper support. The idea is that given a compactifiable morphism one defines by choosing a factorization by an open immersion j and proper morphism p, and then setting
,
where is the extension by zero functor. One then shows the independence of the definition from the choice of compactification.
In the context of étale sheaves, this idea was carried out by Deligne in SGA 4, Exposé XVII. In the context of coherent sheaves, the statements are more delicate since for an open immersion j, the inverse image functor does not usually admit a left adjoint. Nonetheless, exists as a pro-left adjoint, and Deligne was able to define the functor as valued in the pro-derived category of coherent sheaves.
References
Stacks Project - Nagata compactification - See Lemma 38.33.8 first, then backtrack
Stacks Project - Derived lower shriek via compactifications
Stacks Project - Compactly supported cohomology for coherent modules
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Michael%20Christopher%20Wendl | Michael Christopher Wendl is a mathematician and biomedical engineer who has worked on DNA sequencing theory, covering and matching problems in probability, theoretical fluid mechanics, and co-wrote Phred. He was a scientist on the Human Genome Project and has done bioinformatics and biostatistics work in cancer. Wendl is of ethnic German heritage and is the son of the aerospace engineer Michael J. Wendl.
Research Work
Theoretical Fluid Mechanics
The problem of low Reynolds number flow in the gap between 2 infinite cylinders, so-called Couette flow, was solved in 1845 by Stokes. Wendl reported the generalization of this solution for finite-length cylinders, which can actually be built for experimental work, in 1999, as a series of modified Bessel functions and . He also examined a variety of other low Reynolds number rotational devices and shear-driven devices, including a general form of the unsteady disk flow problem, for which the velocity profile is:
where , , , and are physical parameters, are eigen-values, and are coordinates. This result united prior-published special cases for steady flow, infinite disks, etc.
Covering and Matching Problems in Probability
Wendl examined a number of matching and covering problems in combinatorial probability, especially as these problems apply to molecular biology. He determined the distribution of match counts of pairs of integer multisets in terms of Bell polynomials, a problem directly relevant to physical mapping of DNA. Prior to this, investigators had used a number of ad-hoc quantifiers, like the Sulston score, which idealized match trials as being independent. His result for the multiple-group birthday proposition solves various related "collision problems", e.g. some types of P2P searching. He has also examined a variety of 1-dimensional covering problems (see review by Cyril Domb), generalizing the basic configuration to forms relevant to molecular biology. His covering investigation of rare DNA variants with Richard K. Wilson played a role in designing the 1000 Genomes Project.
Bioinformatics and Biostatistics
Wendl co-wrote Phred, a widely used DNA trace analyzer that converted raw output stream of early DNA sequence machines to sequence strings. He has also contributed extensively to biostatistical analysis of cancer studies.
Personal life
Wendl's heritage is ethnic German, originating from the Banat region of the old Austro-Hungarian Empire and he is the son of the aerospace engineer Michael J. Wendl. He is married to the former Pamela Bjerkness of Chicago
External links
Michael Wendl at Engineering Tree database
References
American bioinformaticians
Human Genome Project scientists
Living people
20th-century American mathematicians
Year of birth missing (living people)
21st-century American mathematicians
Washington University in St. Louis alumni |
https://en.wikipedia.org/wiki/ZFC%20%28disambiguation%29 | ZFC — Zermelo–Fraenkel set theory — is one of the foundations of modern mathematics.
ZFC may also refer to:
Zeyashwemye F.C., an association football club from Myanmar
ZFC Meuselwitz, a football club in Germany
Zico Football Center, a sports complex in Brazil
Zambia Forestry College
Nikon Z fc, a digital camera model |
https://en.wikipedia.org/wiki/Protorus | In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group.
Some examples of protori are given by solenoid groups.
See also
Duocylinder - Cartesian product of two disks
Proprism
References
Topological groups |
https://en.wikipedia.org/wiki/F-crystal | In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter F stands for Frobenius, indicating that F-crystals have an action of Frobenius on them. F-isocrystals are crystals "up to isogeny".
F-crystals and F-isocrystals over perfect fields
Suppose that k is a perfect field, with ring of Witt vectors W and let K be the quotient field of W, with Frobenius automorphism σ.
Over the field k, an F-crystal is a free module M of finite rank over the ring W of Witt vectors of k, together with a σ-linear injective endomorphism of M. An F-isocrystal is defined in the same way, except that M is a module for the quotient field K of W rather than W.
Dieudonné–Manin classification theorem
The Dieudonné–Manin classification theorem was proved by and . It describes the structure of F-isocrystals over an algebraically closed field k. The category of such F-isocrystals is abelian and semisimple, so every F-isocrystal is a direct sum of simple F-isocrystals. The simple F-isocrystals are the modules Es/r where r and s are coprime integers with r>0. The F-isocrystal Es/r has a basis over K of the form v, Fv, F2v,...,Fr−1v for some element v, and Frv = psv. The rational number s/r is called the slope of the F-isocrystal.
Over a non-algebraically closed field k the simple F-isocrystals are harder to describe explicitly, but an F-isocrystal can still be written as a direct sum of subcrystals that are isoclinic, where an F-crystal is called isoclinic if over the algebraic closure of k it is a sum of F-isocrystals of the same slope.
The Newton polygon of an F-isocrystal
The Newton polygon of an F-isocrystal encodes the dimensions of the pieces of given slope. If the F-isocrystal is a sum of isoclinic pieces with slopes s1 < s2 < ... and dimensions (as Witt ring modules) d1, d2,... then the Newton polygon has vertices (0,0), (x1, y1), (x2, y2),... where the nth line segment joining the vertices has slope sn = (yn−yn−1)/(xn−xn−1) and projection onto the x-axis of length dn = xn − xn−1.
The Hodge polygon of an F-crystal
The Hodge polygon of an F-crystal M encodes the structure of M/FM considered as a module over the Witt ring. More precisely since the Witt ring is a principal ideal domain, the module M/FM can be written as a direct sum of indecomposable modules of lengths n1 ≤ n2 ≤ ... and the Hodge polygon then has vertices (0,0), (1,n1), (2,n1+ n2), ...
While the Newton polygon of an F-crystal depends only on the corresponding isocrystal, it is possible for two F-crystals corresponding to the same F-isocrystal to have different Hodge polygons. The Hodge polygon has edges with integer slopes, while the Newton polygon has edges with rational slopes.
Isocrystals over more general schemes
Suppose that A is a complete discrete valuation ring of characteristic 0 with quotient field k of characteristic p>0 and perfect. An affine enlargement of a scheme X0 over k consists of a torsio |
https://en.wikipedia.org/wiki/Eigencurve | In number theory, an eigencurve is a rigid analytic curve that parametrizes certain p-adic families of modular forms, and an eigenvariety is a higher-dimensional generalization of this. Eigencurves were introduced by , and the term "eigenvariety" seems to have been introduced around 2001 by .
References
Modular forms |
https://en.wikipedia.org/wiki/Shimura%20subgroup | In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X0(N) of level N, given by the kernel of the natural map to the Jacobian of X1(N). It is named after Goro Shimura. There is a similar subgroup Σ(N,D) associated to Shimura curves of quaternion algebras.
References
Abelian varieties
Modular forms |
https://en.wikipedia.org/wiki/228%20%28number%29 | 228 (two hundred [and] twenty-eight) is the natural number following 227 and preceding 229.
In mathematics
228 is a refactorable number
and a practical number.
There are 228 matchings in a ladder graph with five rungs.
228 is the smallest even number n such that the numerator of the nth Bernoulli number is divisible by a nontrivial square number that is relatively prime to n.
The binary form of 228 contains all the two digit binary numbers in sequence from highest to lowest (11 10 01 00).
References
Integers |
https://en.wikipedia.org/wiki/Cadabra%20%28computer%20program%29 | Cadabra is a computer algebra system designed specifically for the solution of problems encountered in classical field theory, quantum field theory and string theory.
The first version of Cadabra was developed around 2001 for computing higher-derivative string theory correction to supergravity.
Released under the GNU General Public License, Cadabra is free software.
Cadabra has extensive functionality for tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.
A Java program inspired by Cadabra called Redberry was developed between 2013 and 2016. It achieved faster speeds for most index contractions with an approach based on the graph isomorphism problem rather than canonicalisation.
See also
List of computer algebra systems
References
Further reading
Kasper Peeters (2007), "Introducing Cadabra: A Symbolic computer algebra system for field theory problems", hep-th/0701238
Kasper Peeters (2006), "A Field-theory motivated approach to symbolic computer algebra", Comput. Phys. Commun. 176 (2007) 550, [cs/0608005 [cs.SC]]
External links
Free computer algebra systems
Free physics software
Free software programmed in C++ |
https://en.wikipedia.org/wiki/Algebrator | Algebrator (also called Softmath) is a computer algebra system (CAS), which was developed in the late 1990s by Neven Jurkovic of Softmath, San Antonio, Texas. This is a CAS specifically geared towards algebra education. Beside the computation results, it shows step by step the solution process and context sensitive explanations.
See also
List of computer algebra systems
References
External links
Computer algebra systems
Educational math software |
https://en.wikipedia.org/wiki/Symmetric%20variety | In algebraic geometry, a symmetric variety is an algebraic analogue of a symmetric space in differential geometry, given by a quotient G/H of a reductive algebraic group G by the subgroup H fixed by some involution of G.
See also
Wonderful compactification
Homogeneous variety
Spherical variety
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Tami%20Reller | Tami L. Reller (born 1963 or 1964) is an American businesswoman. Reller is a native of Grand Forks, North Dakota. She earned a bachelor's degree in mathematics from Minnesota State University Moorhead and a master's degree in business administration from Saint Mary's College of California. In 1984, while still attending college, she began her career at Great Plains Software. She joined Microsoft Corporation in 2001 as part of the acquisition of Great Plains Software, where she served as a chief financial officer (CFO). In 2011, after the departure of Steven Sinofsky, she was promoted to the corporate vice president and the CFO of the company's Windows division. In July 2013, she was promoted to executive vice president, marketing for Microsoft Corporation.
In March 2014, Reller left Microsoft.
References
Microsoft employees
Women corporate executives
American women business executives
Minnesota State University Moorhead alumni
Saint Mary's College of California alumni
1960s births
Living people
20th-century American businesspeople
20th-century American businesswomen
21st-century American businesspeople
21st-century American businesswomen |
https://en.wikipedia.org/wiki/Implicit%20Shape%20Model | An Implicit Shape Model for a given object category consists of a class-specific alphabet (codebook) of local appearances that are prototypical for the object category, and of a spatial probability distribution which specifies where each codebook entry may be found on the object.
References
Image processing |
https://en.wikipedia.org/wiki/SBI%20ring | In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI was introduced by Irving Kaplansky and stands for "suitable for building idempotent elements" .
Examples
Any ring with nil radical is SBI.
Any Banach algebra is SBI: more generally, so is any compact topological ring.
The ring of rational numbers with odd denominator, and more generally, any local ring, is SBI.
References
Ring theory |
https://en.wikipedia.org/wiki/Modular%20symbol | In mathematics, modular symbols, introduced independently by Bryan John Birch and by , span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms.
Definition
The abelian group of (universal weight 2) modular symbols is spanned by symbols {α,β} for α, β in the rational projective line Q ∪ {∞} subject to the relations
{α,β} + {β,γ} = {α,γ}
Informally, {α,β} represents a homotopy class of paths from α to β in the upper half-plane.
The group GL2(Q) acts on the rational projective line, and this induces an action on the modular symbols.
There is a pairing between cusp forms f of weight 2 and modular symbols given by integrating the cusp form, or rather fdτ, along the path corresponding to the symbol.
References
Modular forms |
https://en.wikipedia.org/wiki/Track%20geometry | Track geometry is concerned with the properties and relations of points, lines, curves, and surfaces in the three-dimensional positioning of railroad track. The term is also applied to measurements used in design, construction and maintenance of track. Track geometry involves standards, speed limits and other regulations in the areas of track gauge, alignment, elevation, curvature and track surface. Standards are usually separately expressed for horizontal and vertical layouts although track geometry is three-dimensional.
Layout
Horizontal layout
Horizontal layout is the track layout on the horizontal plane. This can be thought of as the plan view which is a view of a 3-dimensional track from the position above the track. In track geometry, the horizontal layout involves the layout of three main track types: tangent track (straight line), curved track, and track transition curve (also called transition spiral or spiral) which connects between a tangent and a curved track. Curved track can also be categorized into three types. The first type is simple curve which has the same radius throughout that curved track. The second type is compound curve which comprises two or more simple curves of different radii that have the same direction of curvature. The third type is reverse curve which comprises two or more simple curves that has the opposite direction of curvature (sometime known as "S" curve or serpentine curve).
In Australia, there is a special definition for a bend (or a horizontal bend) which is a connection between two tangent tracks at almost 180 degrees (with deviation not more than 1 degree 50 minutes) without an intermediate curve. There is a set of speed limits for the bends separately from normal tangent track.
Vertical layout
Vertical layout is the track layout on the vertical plane. This can be thought of as the elevation view which is the side view of the track to show track elevation. In track geometry, the vertical layout involves concepts such as crosslevel, cant and gradient.
Reference rail
The reference rail is the base rail that is used as a reference point for the measurement. It can vary in different countries. Most countries use one of the rails as the reference rail. For example, North America uses the reference rail as the line rail which is the east rail of tangent track running north and south, the north rail of tangent track running east and west, the outer rail (the rail that is further away from the center) on curves, or the outside rails in multiple track territory. For Swiss railroad, the reference rail for tangent track is the center line between two rails, but it is the outside rail for curved track.
Track gauge
Track gauge or rail gauge (also known as track gage in North America) is the distance between the inner sides (gauge sides) of the heads of the two load bearing rails that make up a single railway line. Each country uses different gauges for different types of trains. However, the gauge is the bas |
https://en.wikipedia.org/wiki/Parshin%20chain | In number theory, a Parshin chain is a higher-dimensional analogue of a place of an algebraic number field. They were introduced by in order to define an analogue of the idele class group for 2-dimensional schemes.
A Parshin chain of dimension s on a scheme is a finite sequence of points p0, p1, ..., ps such that pi has dimension i and each point is contained in the closure of the next one.
References
Algebraic number theory |
https://en.wikipedia.org/wiki/Robert%20Horton%20Cameron | Robert Horton Cameron (May 17, 1908 – July 17, 1989) was an American mathematician, who worked on analysis and probability theory. He is known for the Cameron–Martin theorem.
Education and career
Cameron received his Ph.D. in 1932 from Cornell University under the direction of W. A. Hurwitz. He studied under a National Research Council postdoc at the Institute for Advanced Study in Princeton from 1933 to 1935. Cameron was a faculty member at MIT from 1935 to 1945. He was then a faculty member at the University of Minnesota until his retirement. He spent the academic year 1953–1954 on sabbatical leave at the Institute for Advanced Study. His doctoral students include Monroe D. Donsker and Elizabeth Cuthill. He had a total of 35 Ph.D. students at the University of Minnesota — his first two graduated in 1946 and his last one in 1977. Cameron published a total of 72 papers — his first in 1934 and his last, posthumously, in 1990.
At MIT, he did some work with Norbert Wiener. During the 1940s Cameron and W. T. Martin, who was from 1943 to 1946 the chair of the mathematics department at Syracuse University, engaged in an ambitious program of extending Norbert Wiener's early work on mathematical models of Brownian motion. In 1944, Cameron was awarded the Chauvenet Prize for '"Some Introductory Exercises in the Manipulation of Fourier Transforms", which appeared in National Mathematics Magazine, 1941, vol. 15, pages 331–356.
References
20th-century American mathematicians
Cornell University alumni
Institute for Advanced Study visiting scholars
Massachusetts Institute of Technology faculty
Probability theorists
University of Minnesota faculty
1908 births
1989 deaths
People from Brooklyn |
https://en.wikipedia.org/wiki/Asymmetric%20simple%20exclusion%20process | In probability theory, the asymmetric simple exclusion process (ASEP) is an interacting particle system introduced in 1970 by Frank Spitzer. Many articles have been published on it in the physics and mathematics literature since then, and it has become a "default stochastic model for transport phenomena".
The process with parameters is a continuous-time Markov process on ,
the 1s being thought of as particles and the 0s as empty sites. Each particle waits a random amount of time having the distribution of an exponential random variable with mean one and then attempts a jump, one site to the right
with probability and one site to the left with probability . However, the jump is performed only if there is no particle at the target site. Otherwise, nothing happens and the particle waits another exponential time. All particles are doing this independently of each other.
The model is related to the Kardar–Parisi–Zhang equation in the weakly asymmetric limit, i.e. when tends to zero under some particular scaling. Recently, progress has been made to understand the statistics of the current of particles and it appears that the Tracy–Widom distribution plays a key role.
Sources
References
.
.
Statistical mechanics |
https://en.wikipedia.org/wiki/Indranil%20Biswas | Indranil Biswas (born 19 October 1964) is an Indian mathematician. He is professor of mathematics at the Tata Institute of Fundamental Research, Mumbai. He is known for his work in the areas of algebraic geometry, differential geometry, and deformation quantization.
In 2006, the Government of India awarded him the Shanti Swarup Bhatnagar Prize in mathematical sciences for his contributions to "algebraic geometry, centering around moduli problems of vector bundles."
Biography
Biswas is an Indian citizen. He received a Ph.D. in mathematics from the University of Mumbai.
Selected publications
Awards and honours
Shanti Swarup Bhatnagar Prize for Science and Technology, 2006.
Fellow, Indian Academy of Sciences (2003).
References
External links
1964 births
20th-century Indian mathematicians
21st-century Indian mathematicians
Algebraic geometers
Fellows of the Indian Academy of Sciences
Living people
Scientists from Mumbai
Tata Institute of Fundamental Research alumni
Academic staff of Tata Institute of Fundamental Research
University of Mumbai alumni
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science |
https://en.wikipedia.org/wiki/Diamond%20operator | In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix in Γ0(N) where δ ≈ d mod N. The diamond operators form an abelian group and commute with the Hecke operators.
Unicode
In Unicode, the diamond operator is represented by the character .
Notes
References
Modular forms |
https://en.wikipedia.org/wiki/Lie%E2%80%93Palais%20theorem | In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. For manifolds with boundary the action must preserve the boundary, in other words the vector fields on the boundary must be tangent to the boundary. proved it as a global form of an earlier local theorem due to Sophus Lie.
The example of the vector field d/dx on the open unit interval shows that the result is false for non-compact manifolds.
Without the assumption that the Lie algebra is finite dimensional the result can be false. gives the following example due to Omori: the Lie algebra is all vector fields f(x,y)∂/∂x + g(x,y)∂/∂y acting on the torus R2/Z2 such that g(x, y) = 0 for 0 ≤ x ≤ 1/2. This Lie algebra is not the Lie algebra of any group. gives an infinite dimensional generalization of the Lie–Palais theorem for Banach–Lie algebras with finite-dimensional center.
References
Reprinted in collected works volume 5.
Lie algebras
Theorems in differential geometry |
https://en.wikipedia.org/wiki/Lukas%20K%C3%BCbler | Lukas Kübler (born 30 August 1992) is a German professional footballer who plays as a full-back for Bundesliga club SC Freiburg.
Career statistics
References
1992 births
Living people
Footballers from Bonn
German men's footballers
Men's association football defenders
Bonner SC players
1. FC Köln players
1. FC Köln II players
SC Freiburg players
SC Freiburg II players
Bundesliga players
2. Bundesliga players
Regionalliga players |
https://en.wikipedia.org/wiki/Lee%20Chang-geun | Lee Chang-geun (; born 30 August 1993) is a South Korean footballer who plays as a goalkeeper for Daejeon Hana Citizen.
Club career statistics
Honours
South Korea U-20
AFC U-19 Championship: 2012
South Korea U-23
King's Cup: 2015
References
External links
1993 births
Living people
Men's association football goalkeepers
South Korean men's footballers
South Korea men's youth international footballers
South Korea men's under-20 international footballers
South Korea men's under-23 international footballers
South Korea men's international footballers
Busan IPark players
Suwon FC players
Jeju United FC players
Gimcheon Sangmu FC players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Suicide%20in%20Bangladesh | Suicide in Bangladesh is a common cause of unnatural death and a long term social issue. Of all the people reported dead due to suicide worldwide every year, 2.06% are Bangladeshi.
Statistics
According to a report by the World Health Organization 19,697 people died by suicide in Bangladesh in 2011. According to Police Headquarters 11,095 people died by suicide in Bangladesh in 2017. According to a report by Shaheed Suhrawardy Medical College Hospital, Dhaka, published in 2010, around 6,500,000 people of Bangladesh are prone to suicide. The rate is 128.08 people per 100,000 dying by suicide in Bangladesh every year. The six-member team led by Dr AHM Feroz and Dr SM Nurul Islam of the medical college conducted the survey at Mominpur union of Chuadanga district from January to April 2010. But this has been noted that the above rate was assumed only based on survey conducted in one union which is not a good reflector of the total suicide rate in the whole country as total suicide rate per year is far less than what was mentioned in the report in the last couple of years.
According to a report by The Daily Star, from 2002 to 2009, 73,389 people died by suicide in Bangladesh. Of these 73,389 people, 31,857 people hanged themselves and 41,532 swallowed poison to kill themselves. Bangladesh Manabadhikar Bastabayan Sangstha, a human rights group of Bangladesh shows that from January 2011 to August 2011, 258 people died by suicide, and of them, 158 were women and the remainder were men.
Suicide of women
According to a 2010 report by Shaheed Suhrawardy Medical College Hospital, of the 128.08 per 100,000 people who died by suicide in 2010, 89% were women and most of them were unmarried. Statistics from Jatiya Mahila Ainjibi Samity, a Bangladeshi women's organization, show that from 2006 to 2010, 40 girls who died by suicide were victims of stalking. From 2001 to 2010, 4,747 women and girls died by suicide because of physical and domestic violence.
According to a report published in the Lancet published in BBC News, suicidal tendency among women in Bangladesh is higher, because they have inferior status in society. Another factor is a higher rate of illiteracy and their economic dependence on men.
2007 Adam House cult suicide
In 2007, in Mymensingh, a family of nine died by mass suicide by hurling themselves onto a train. According to the diaries recovered from their home, they wanted a pure life as lived by Adam and Eve, freeing themselves from bondage to any religion.
Prevention
Non-profit Kaan Pete Roi provides mental health support via a free crisis helpline.
Common methods
Hanging is the most common method of suicide in Bangladesh. There is no cost involvement in this method other than ligature material, i.e., a rope, and thus that is why it is the preferred method. Swallowing poison is another common method in Bangladesh to die by suicide. In urban areas, people follow other methods to die by suicide, such as by an overdose of barbiturate tab |
https://en.wikipedia.org/wiki/Kuga%20fiber%20variety | In algebraic geometry, a Kuga fiber variety, introduced by , is a fiber space whose fibers are abelian varieties and whose base space is an arithmetic quotient of a Hermitian symmetric space.
References
Algebraic geometry
Abelian varieties |
https://en.wikipedia.org/wiki/Tate%20topology | In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Period%20domain | In mathematics, a period domain is a parameter space for a polarized Hodge structure. They can often be represented as the quotient of a Lie group by a compact subgroup.
See also
Period mapping
References
Complex manifolds |
https://en.wikipedia.org/wiki/Faltings%27%20product%20theorem | In arithmetic geometry, Faltings' product theorem gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introduced by in his proof of Lang's conjecture that subvarieties of an abelian variety containing no translates of non-trivial abelian subvarieties have only finitely many rational points.
and gave explicit versions of Faltings' product theorem.
References
Diophantine approximation
Theorems in number theory |
https://en.wikipedia.org/wiki/Chevalley%27s%20structure%20theorem | In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety. It was proved by (though he had previously announced the result in 1953), , and .
Chevalley's original proof, and the other early proofs by Barsotti and Rosenlicht, used the idea of mapping the algebraic group to its Albanese variety. The original proofs were based on Weil's book Foundations of algebraic geometry and are hard to follow for anyone unfamiliar with Weil's foundations, but later gave an exposition of Chevalley's proof in scheme-theoretic terminology.
Over non-perfect fields there is still a smallest normal connected linear subgroup such that the quotient is an abelian variety, but the linear subgroup need not be smooth.
A consequence of Chevalley's theorem is that any algebraic group over a field is quasi-projective.
Examples
There are several natural constructions that give connected algebraic groups that are neither affine nor complete.
If C is a curve with an effective divisor m, then it has an associated generalized Jacobian Jm. This is a commutative algebraic group that maps onto the Jacobian variety J0 of C with affine kernel. So J is an extension of an abelian variety by an affine algebraic group. In general this extension does not split.
The reduced connected component of the relative Picard scheme of a proper scheme over a perfect field is an algebraic group, which is in general neither affine nor proper.
The connected component of the closed fiber of a Neron model over a discrete valuation ring is an algebraic group, which is in general neither affine nor proper.
For analytic groups some of the obvious analogs of Chevalley's theorem fail. For example, the product of the additive group C and any elliptic curve has a dense collection of closed (analytic but not algebraic) subgroups isomorphic to C so there is no unique "maximal affine subgroup", while the product of two copies of the multiplicative group C* is isomorphic (analytically but not algebraically) to a non-split extension of any given elliptic curve by C.
Applications
Chevalley's structure theorem is used in the proof of the Néron–Ogg–Shafarevich criterion.
References
Algebraic groups
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Steinberg%20formula | In mathematical representation theory, Steinberg's formula, introduced by , describes the multiplicity of an irreducible representation of a semisimple complex Lie algebra in a tensor product of two irreducible representations.
It is a consequence of the Weyl character formula, and for the Lie algebra sl2 it is essentially the Clebsch–Gordan formula.
Steinberg's formula states that the multiplicity of the irreducible representation of highest weight ν in the tensor product of the irreducible representations with highest weights λ and μ is given by
where W is the Weyl group, ε is the determinant of an element of the Weyl group, ρ is the Weyl vector, and P is the Kostant partition function giving the number of ways of writing a vector as a sum of positive roots.
References
Representation theory
Theorems in harmonic analysis |
https://en.wikipedia.org/wiki/Null%20model | In mathematics, for example in the study of statistical properties of graphs, a null model is a type of random object that matches one specific object in some of its features, or more generally satisfies a collection of constraints, but which is otherwise taken to be an unbiasedly random structure. The null model is used as a term of comparison, to verify whether the object in question displays some non-trivial features (properties that wouldn't be expected on the basis of chance alone or as a consequence of the constraints), such as community structure in graphs. An appropriate null model behaves in accordance with a reasonable null hypothesis for the behavior of the system under investigation.
One null model of utility in the study of complex networks is that proposed by Newman and Girvan, consisting of a randomized version of an original graph , produced through edges being rewired at random, under the constraint that the expected degree of each vertex matches the degree of the vertex in the original graph.
The null model is the basic concept behind the definition of modularity, a function which evaluates the goodness of partitions of a graph into clusters. In particular, given a graph and a specific community partition (an assignment of a community-index (here taken as an integer from to ) to each vertex in the graph), the modularity measures the difference between the number of links from/to each pair of communities, from that expected in a graph that is completely random in all respects other than the set of degrees of each of the vertices (the degree sequence). In other words, the modularity contrasts the exhibited community structure in with that of a null model, which in this case is the configuration model (the maximally random graph subject to a constraint on the degree of each vertex).
See also
Null hypothesis
References
Graph theory |
https://en.wikipedia.org/wiki/Katz%E2%80%93Lang%20finiteness%20theorem | In number theory, the Katz–Lang finiteness theorem, proved by , states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, then Ker(X/K) is finite if K has characteristic 0, and the part of the kernel coprime to p is finite if K has characteristic p > 0.
References
Theorems in number theory |
https://en.wikipedia.org/wiki/W.%20T.%20Martin | William Ted Martin (June 4, 1911 – May 30, 2004) was an American mathematician, who worked on mathematical analysis, several complex variables, and probability theory. He is known for the Cameron–Martin theorem and for his 1948 book Several complex variables, co-authored with Salomon Bochner.
Biography
He was born on June 4, 1911, in Arkansas.
W. T. Martin received his B.A. in mathematics from the University of Arkansas in 1930. He did graduate work at the University of Illinois at Urbana–Champaign, where he received his M.A. in 1931 and his Ph.D. in 1934 under the direction of Robert Carmichael. He studied under a National Research Council postdoctoral fellowship at the Institute for Advanced Study in Princeton from 1934 to 1936. In 1936 Martin became an instructor at MIT and in 1938 a faculty member there. He collaborated with several fellow MIT faculty members, notably Norbert Wiener, R. H. Cameron, Stefan Bergman, and Salomon Bochner. During the 1940s Martin and R. H. Cameron wrote a series of papers extending Norbert Wiener's early work on mathematical models of Brownian motion. During the 1950s W. T. Martin wrote with Salomon Bochner a series of papers that proved basic results in the theory of several complex variables.
Martin was the department head for the MIT mathematics department from 1947 to 1968. During this time he oversaw the hiring of 24 faculty members in the mathematics department. He initiated MIT's C. L. E. Moore Instructorship Program in 1949. He spent his entire career at MIT, except for the years from 1943 to 1946, when he left MIT to become the head of the mathematics department of Syracuse University and, in the academic year 1951–1952, when he was on sabbatical at the Institute for Advanced Study. Martin did important editorial work and co-authored three influential books: Several complex variables (1948), Elementary differential equations (1956), and Differential space, quantum space, and prediction (1966). Beginning in 1961, Martin involved himself in developing math curricula for English-speaking African nations, serving as chair of the Steering Committee of the Education Development Center's African Mathematics Program and visited Africa regularly from 1961 to 1975.
He retired to Block Island and died on May 30, 2004.
Selected publications
with Norbert Wiener:
with Norbert Wiener:
with Stefan Bergman:
with R. H. Cameron: (2nd most cited of all Cameron and Martin's papers)
with R. H. Cameron: (most cited of all Cameron and Martin's papers)
with Salomon Bochner: (216 pages)
with Eric Reissner: ; ;
as co-editor with editors Norbert Wiener, Armand Siegel, and Bayard Rankin: (176 pages, essays)
References
1911 births
2004 deaths
20th-century American mathematicians
21st-century American mathematicians
Institute for Advanced Study visiting scholars
Probability theorists
Massachusetts Institute of Technology School of Science faculty
Syracuse University faculty
University of Arkansas alumni
Univers |
https://en.wikipedia.org/wiki/Section%20conjecture | In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism , where is a complete smooth curve of genus at least 2 over a field that is finitely generated over , in terms of decomposition groups of rational points of . The conjecture was introduced by in a 1983 letter to Gerd Faltings.
References
External links
Algebraic geometry
Unsolved problems in geometry
Arithmetic geometry |
https://en.wikipedia.org/wiki/Brandt%20matrix | In mathematics, Brandt matrices are matrices, introduced by , that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra.
calculated the traces of the Brandt matrices.
Let O be an order in a quaternion algebra with class number H, and Ii,...,IH invertible left O-ideals representing the classes. Fix an integer m. Let ej denote the number of units in the right order of Ij and let Bij denote the number of α in Ij−1Ii with reduced norm N(α) equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent of the choice of representatives Ij; it is dependent only on the level of the order O.
References
Number theory
Matrices |
https://en.wikipedia.org/wiki/Atiyah%E2%80%93Hitchin%E2%80%93Singer%20theorem | In differential geometry and gauge theory, the Atiyah–Hitchin–Singer theorem, introduced by , states that the space of SU(2) anti self dual Yang–Mills fields on a 4-sphere with index k > 0 has dimension 8k – 3.
References
Differential geometry |
https://en.wikipedia.org/wiki/Null%20hypersurface | In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to the local metric tensor). A light cone is an example.
An alternative characterization is that the tangent space at every point of a hypersurface contains a nonzero vector such that the metric applied to such a vector and any vector in the tangent space is zero. Another way of saying this is that the pullback of the metric onto the tangent space is degenerate.
For a Lorentzian metric, all the vectors in such a tangent space are space-like except in one direction, in which they are null. Physically, there is exactly one lightlike worldline contained in a null hypersurface through each point that corresponds to the worldline of a particle moving at the speed of light, and no contained worldlines that are time-like. Examples of null hypersurfaces include a light cone, a Killing horizon, and the event horizon of a black hole.
References
.
James B. Hartle, Gravity: an Introduction To Einstein's General Relativity.
General relativity
Lorentzian manifolds |
https://en.wikipedia.org/wiki/Arithmetico-geometric%20sequence | In mathematics, arithmetico-geometric sequence is the result of term-by-term multiplication of a geometric progression with the corresponding terms of an arithmetic progression. Put plainly, the nth term of an arithmetico-geometric sequence is the product of the nth term of an arithmetic sequence
and the nth term of a geometric one. Arithmetico-geometric sequences arise in various applications, such as the computation of expected values in probability theory. For instance, the sequence
is an arithmetico-geometric sequence. The arithmetic component appears in the numerator (in blue), and the geometric one in the denominator (in green).
The summation of this infinite sequence is known as an arithmetico-geometric series, and its most basic form has been called Gabriel's staircase:
The denomination may also be applied to different objects presenting characteristics of both arithmetic and geometric sequences; for instance the French notion of arithmetico-geometric sequence refers to sequences of the form , which generalise both arithmetic and geometric sequences. Such sequences are a special case of linear difference equations.
Terms of the sequence
The first few terms of an arithmetico-geometric sequence composed of an arithmetic progression (in blue) with difference and initial value and a geometric progression (in green) with initial value and common ratio
are given by:
Example
For instance, the sequence
is defined by , , and .
Sum of the terms
The sum of the first terms of an arithmetico-geometric sequence has the form
where and are the th terms of the arithmetic and the geometric sequence, respectively.
This sum has the closed-form expression
Proof
Multiplying,
by , gives
Subtracting from , and using the technique of telescoping series gives
where the last equality results of the expression for the sum of a geometric series. Finally dividing through by gives the result.
Infinite series
If −1 < r < 1, then the sum S of the arithmetico-geometric series, that is to say, the sum of all the infinitely many terms of the progression, is given by
If r is outside of the above range, the series either
diverges (when r > 1, or when r = 1 where the series is arithmetic and a and d are not both zero; if both a and d are zero in the later case, all terms of the series are zero and the series is constant)
or alternates (when r ≤ −1).
Example: application to expected values
For instance, the sum
,
being the sum of an arithmetico-geometric series defined by , , and , converges to .
This sequence corresponds to the expected number of coin tosses before obtaining "tails". The probability of obtaining tails for the first time at the kth toss is as follows:
.
Therefore, the expected number of tosses is given by
.
References
Further reading
Integer sequences
Mathematical series |
https://en.wikipedia.org/wiki/Laplacian%20of%20the%20indicator | In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function. It is analogous to the second derivative of the Heaviside step function in one dimension. It can be obtained by letting the Laplace operator work on the indicator function of some domain D.
The Laplacian of the indicator can be thought of as having infinitely positive and negative values when evaluated very near the boundary of the domain D. From a mathematical viewpoint, it is not strictly a function but a generalized function or measure. Similarly to the derivative of the Dirac delta function in one dimension, the Laplacian of the indicator only makes sense as a mathematical object when it appears under an integral sign; i.e. it is a distribution function. Just as in the formulation of distribution theory, it is in practice regarded as a limit of a sequence of smooth functions; one may meaningfully take the Laplacian of a bump function, which is smooth by definition, and let the bump function approach the indicator in the limit.
History
Paul Dirac introduced the Dirac -function, as it has become known, as early as 1930. The one-dimensional Dirac -function is non-zero only at a single point. Likewise, the multidimensional generalisation, as it is usually made, is non-zero only at a single point. In Cartesian coordinates, the d-dimensional Dirac -function is a product of d one-dimensional -functions; one for each Cartesian coordinate (see e.g. generalizations of the Dirac delta function).
However, a different generalisation is possible. The point zero, in one dimension, can be considered as the boundary of the positive halfline. The function 1x>0 equals 1 on the positive halfline and zero otherwise, and is also known as the Heaviside step function. Formally, the Dirac -function and its derivative (i.e. the one-dimensional surface delta prime function) can be viewed as the first and second derivative of the Heaviside step function, i.e. ∂x1x>0 and .
The analogue of the step function in higher dimensions is the indicator function, which can be written as 1x∈D, where D is some domain. The indicator function is also known as the characteristic function. In analogy with the one-dimensional case, the following higher-dimensional generalisations of the Dirac -function and its derivative have been proposed:
Here n is the outward normal vector. Here the Dirac -function is generalised to a surface delta function on the boundary of some domain D in d ≥ 1 dimensions. This definition gives the usual one-dimensional case, when the domain is taken to be the positive halfline. It is zero except on the boundary of the domain D (where it is infinite), and it integrates to the total surface area enclosing D, as shown below.
The one-dimensional Dirac -function is generalised to a multidimensional surface delta prime |
https://en.wikipedia.org/wiki/Translational%20bioinformatics | Translational bioinformatics (TBI) is a field that emerged in the 2010s to study health informatics, focused on the convergence of molecular bioinformatics, biostatistics, statistical genetics and clinical informatics. Its focus is on applying informatics methodology to the increasing amount of biomedical and genomic data to formulate knowledge and medical tools, which can be utilized by scientists, clinicians, and patients. Furthermore, it involves applying biomedical research to improve human health through the use of computer-based information system. TBI employs data mining and analyzing biomedical informatics in order to generate clinical knowledge for application. Clinical knowledge includes finding similarities in patient populations, interpreting biological information to suggest therapy treatments and predict health outcomes.
History
Translational bioinformatics is a relatively young field within translational research. Google trends indicate the use of "bioinformatics" has decreased since the mid-1990s when it was suggested as a transformative approach to biomedical research. It was coined, however, close to ten years earlier. TBI was then presented as means to facilitate data organization, accessibility and improved interpretation of the available biomedical research. It was considered a decision support tool that could integrate biomedical information into decision-making processes that otherwise would have been omitted due to the nature of human memory and thinking patterns.
Initially, the focus of TBI was on ontology and vocabulary designs for searching the mass data stores. However, this attempt was largely unsuccessful as preliminary attempts for automation resulted in misinformation. TBI needed to develop a baseline for cross-referencing data with higher order algorithms in order to link data, structures and functions in networks. This went hand in hand with a focus on developing curriculum for graduate level programs and capitalization for funding on the growing public acknowledgement of the potential opportunity in TBI.
When the first draft of the human genome was completed in the early 2000s, TBI continued to grow and demonstrate prominence as a means to bridge biological findings with clinical informatics, impacting the opportunities for both industries of biology and healthcare. Expression profiling, text mining for trends analysis, population-based data mining providing biomedical insights, and ontology development has been explored, defined and established as important contributions to TBI. Achievements of the field that have been used for knowledge discovery include linking clinical records to genomics data, linking drugs with ancestry, whole genome sequencing for a group with a common disease, and semantics in literature mining. There has been discussion of cooperative efforts to create cross-jurisdictional strategies for TBI, particularly in Europe. The past decade has also seen the development of personalized medici |
https://en.wikipedia.org/wiki/Zolt%C3%A1n%20Horv%C3%A1th%20%28footballer%2C%20born%201989%29 | Zoltán Horváth (born 30 July 1989) is a Hungarian football player who plays for Nemzeti Bajnokság II club Tiszakécske.
Career
On 7 July 2022, Horváth joined Tiszakécske.
Club statistics
References
External links
Profile
Zoltán Horváth at ÖFB
1989 births
People from Kisvárda
Footballers from Szabolcs-Szatmár-Bereg County
21st-century Hungarian people
Living people
Hungarian men's footballers
Hungarian expatriate men's footballers
Men's association football forwards
Egri FC players
Debreceni VSC players
Cigánd SE players
Győri ETO FC players
Kisvárda FC players
Diósgyőri VTK players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Austrian Regionalliga players
Hungarian expatriate sportspeople in Austria
Expatriate men's footballers in Austria |
https://en.wikipedia.org/wiki/Censuses%20in%20Ukraine | Censuses in Ukraine () is a sporadic event that since 2001 has been conducted by the State Statistics Committee of Ukraine under the jurisdiction of the Government of Ukraine.
History
The first steps
The first official census in the territory of Ukraine took place in 1818 when Western Ukraine was part of the Austrian Empire. However a modern census did not take place until 1857. Since then the next censuses took place in the dual-power state of the Austria-Hungary in 1869, 1880, 1890, 1900, 1910. Those last five censuses also included the territory of the today Zakarpattia Oblast which was part of the Kingdom of Hungary. The further censuses discontinued as the country fell apart. The rest of Ukraine which was part of Russian Empire conducted its first census as part of the 1897 Russian Census. The next national census in Russia did not take place until after World War I and the formation of the Soviet Union. A city-census of Kyiv took place in March 1919, after the Bolsheviks occupied the city. In 1920, a census was conducted only in those areas of Ukraine that were not involved in the Russian Civil War.
Interwar censuses
The next census conducted in most of the territory of Western Ukraine (Eastern Galicia) was the Polish census of 1921, while the 1921 Czechoslovakia Census took place on the territory of the Zakarpattia Oblast. In 1930 another census took place in both regions as part of their respective national censuses that were conducted in the same year. Also the area of today Chernivtsi Oblast saw its first national census in 1930 for the first time since the last one that was conducted in the Austria-Hungary in 1910, while the area of Budjak of today Odessa Oblast along with the rest of Bessarabia had the Russian demographic statistic data back from 1897. Already during the World War II one more census took place in 1941 in Hungary which previously sacked and occupied the territory of Carpatho-Ukraine (today Zakarpattia Oblast). As it was mentioned before, the first national Russian Census since 1897 took place only in 1926 as part of the First All-Union Census in the USSR. The next census in the Soviet Union took place in 1937, but it was recognized as unofficial and was never disclosed. The census was also recognized as a conspiracy against the Soviet regime. Just before the World War II in 1939, the Soviet Union conducted another census that was accepted as the official one.
Post-War censuses
After World War II, Ukraine was united in its current borders (including Crimea) and within the Soviet Union. The first Soviet Census after the war took place in 1959, followed by three more in 1970, 1979 and 1989. The next planned census never took place as the Soviet Union dissolved in 1991.
Post-Soviet
The first (and so far only) national census of Ukraine took place in 2001. It was originally planned that the next one would follow in 2010, but it was postponed until 2020. In April 2020 Minister of the Cabinet of Ministers Oleh Nemchi |
https://en.wikipedia.org/wiki/Remmert%E2%80%93Stein%20theorem | In complex analysis, a field in mathematics, the Remmert–Stein theorem, introduced by , gives conditions for the closure of an analytic set to be analytic.
The theorem states that if F is an analytic set of dimension less than k in some complex manifold D, and M is an analytic subset of D – F with all components of dimension at least k, then the closure of M is either analytic or contains F.
The condition on the dimensions is necessary: for example, the set of points (1/n,0) in the complex plane is analytic in the complex plane minus the origin, but its closure in the complex plane is not.
Relations to other theorems
A consequence of the Remmert–Stein theorem (also treated in their paper), is Chow's theorem stating that any projective complex analytic space is necessarily a projective algebraic variety.
The Remmert–Stein theorem is implied by a proper mapping theorem due to , see .
References
Complex manifolds
Theorems in complex analysis |
https://en.wikipedia.org/wiki/Mazur%27s%20control%20theorem | In number theory, Mazur's control theorem, introduced by , describes the behavior in Zp extensions of the Selmer group of an abelian variety over a number field.
References
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Al-Muabbada | Al-Muabbada (; ) is a town in al-Hasakah Governorate, Syria. According to the Syria Central Bureau of Statistics (CBS), Al-Muabbada had a population of 15,759 in the 2004 census. According to the Kurdish news agency "Rudaw", the Ba'athist Party under President Hafez al-Assad changed the name of the town to Al-Muabbada. The town is 35 kilometres from the Iraqi border and 15 kilometres from the Turkish border. As of 2004, Al-Muabbada is the eighth largest town in Al-Hasakah governorate. The majority of the inhabitants of the town are Kurds with a large Arab minority.
Syrian Civil War
On 24 July 2012, the PYD announced that Syrian security forces withdrew from Al-Muabbada. The YPG forces afterwards took control of all government institutions and the town came fully under the PYD's control.
On 27 September 2022, 2 SDF fighters were killed in a Turkish drone strike on their car in the town.
References
Populated places in al-Malikiyah District
Towns in Syria
Kurdish communities in Syria |
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