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https://en.wikipedia.org/wiki/Iganmode%20Grammar%20School | Iganmode Grammar School is a secondary school in Ota, Ogun State, Nigeria that was established in 1960. official website www.igs.com.ng
Cowbell National Secondary School Mathematics Competition
Igamode Grammar School has lived up to the dictates of its well crafted anthem by winning the coveted Cowbell National Secondary School Mathematics Competition (NASSMAC) for three consecutive seasons in 2011, 2012 and 2013. Though, a public school, students of Iganmode Grammar School, fondly called IGS by its alumnus have defended the coveted Promasidor NASSMAC mathematics prize back-to-back thereby gathering accolades for the school. For the very first time in the history of Cowbell National Mathematics competition in Nigeria Public Schools, students of Iganmode Grammar School (SNR) Ota won the overall best prize for three consecutive years, 2011, 2012 and 2013
Notable alumni
Olalekan Olude, entrepreneur, the co-founder and COO of Jobberman
References
Secondary schools in Nigeria
Schools in Ogun State
Educational institutions established in 1960
1960 establishments in Nigeria |
https://en.wikipedia.org/wiki/Statistics%20department%20%28Anguilla%29 | The Anguilla Statistics Department, subject to the Statistics Act of 2000, reports to the Minister charged with responsibility for the subject of Statistics; the Head of the Department is referred to as the Statistician also designated as the Chief Statistician. It was created mainly to facilitate the development of a statistical system for Anguilla which is also a component of both the regional statistical systems of CARICOM and OECS, together with other member countries of these two regional institutions.
Mission
As stated by the Statistics Law, the mission of the Anguilla Statistics Department is:
to collect, compile, analyses abstract and publish statistical information relative to the commercial, industrial, social, economic and general activities and conditions of the people who are the inhabitants of Anguilla:
to collaborate with all other departments of Government and with local authorities in the collection, computation and publication of statistical records of administration;
to take any census in Anguilla; and
generally to organize a coordinated scheme of social and economic statistics and intelligence pertaining to Anguilla;
The Law lists matters the Statistics department shall collect statistics on, with prior approval of the Governor in Council.
Organisation
Under the Chief Statistician authority, the Department activities are distributed according to the following areas:
Economic Statistics
Social Statistics
Tourism and International Trade Statistics
Administration
History
{| class="wikitable"
|+ Previous heads of the Statistics department
! width=300 | Name
! width=100 | Period
|-
|Ms Penny Hope-Ross
|
|}
See also
Sub-national autonomous statistical services
United Nations Statistics Division
External links
Ministry of Finance, Economic Development, Investments & Tourism
Anguilla Statistics Department
Anguilla statistics indicators on CARICOM Statical indicators publication
Anguilla statistical system on United Nations Statistics Division website
CARICOM Statistics
OECS Statistics
References
Official statistics
Anguilla |
https://en.wikipedia.org/wiki/Grothendieck%20trace%20formula | In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf.
The Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology.
One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups. This is one of the steps used in the proof of the Weil conjectures.
Behrend's trace formula generalizes the formula to algebraic stacks.
Formal statement for L-functions
Let k be a finite field, l a prime number invertible in k, X a smooth k-scheme of dimension n, and a constructible -sheaf on X. Then the following cohomological expression for the L-function of holds:
where F is everywhere a geometric Frobenius action on l-adic cohomology with compact supports of the sheaf . Taking logarithmic derivatives of both formal power series produces a statement on sums of traces for each finite field extension E of the base field k:
For a constant sheaf (viewed as to qualify as an l-adic sheaf) the left hand side of this formula is the number of E-points of X.
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Sensors%20for%20arc%20welding | Sensors for arc welding are devices which – as a part of a fully mechanised welding equipment – are capable to acquire information about position and, if possible, about the geometry of the intended weld at the workpiece and to provide respective data in a suitable form for the control of the weld torch position and, if possible, for the arc welding process parameters.
Introduction
The quality of a weld depends, besides the weld parameters which are important for the welding process (e.g. voltage, current, wire feed and weld speed) also mainly from the type of input of process energy and of the used filler material. The positioning of the torch exerts a direct influence on the material flow. The heat input for the melting of the component edges and the steady heat flow are, furthermore, directly connected with the torch guidance and exert substantial influence on the weld quality and on the resulting residual stresses. In fully mechanised and automated shielded gas welding, the inaccuracies of torch guidance, workpiece handling, groove preparation and thermal distortion are adding to the variations of the edge position and edge geometry. In fully mechanised welding, the information which is required for the weld quality is detected via sensors. Sensors are applied for checking the position of the component (detection of weld start and end of weld), for joint tracking and for the adaptation of the process parameters to changes of the joints/grooves. It is possible to use the sensors online (together/at the same time with the welding process) or offline (in a separate working step before welding). Sensors are mainly used in online joint tracking.
Principles
All physical principles which are capable to provide information about the position of an object are suitable to serve as the starting basis for a sensor function. The ambient conditions prevailing during arc welding and also the requirements which are made by fully mechanised equipments have, however, many restrictions as a consequence. Figure 1 depicts the system overview. The monitoring strategy of the sensor (process or geometry) has been chosen as the superordinate criterion, the further subdivision is orientated on the measuring principle. A further distinctive feature of sensor systems is their design. Leading sensors are, thus, marked by the fact that measuring point and joining point are not located in the same position. Here, the measuring and joining process are mainly running in sequence. For making position-relevant statements about the welding process, those systems require calibration of the relative position. If process-oriented sensors are used, the measuring point and the joining point are identical.
What the measuring principles all have in common is the fact that through the evaluation of the sensor signal, geometrical information about the joint and its relative position to the measuring head is provided. The individual active principles allow different processing speed |
https://en.wikipedia.org/wiki/Darmon | Darmon may refer to:
Gérard Darmon (born in 1948), French-Moroccan movie actor and singer
Henri Darmon (born in 1965), French Canadian mathematician specializing in number theory
Jean-Charles Darmon (born in 1961), French literary critic
Pierre Darmon (born in 1934), former French tennis player, husband of Rosa Maria Darmon
Ron Darmon (born in 1992), Israeli Olympic triathlete
Rosa Maria Reyes Darmon (born in 1939), former French tennis player, wife of Pierre Darmon |
https://en.wikipedia.org/wiki/Sofic%20group | In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two elements of the group have distance 1. They were introduced by as a common generalization of amenable and residually finite groups. The name "sofic", from the Hebrew word meaning "finite", was later applied by , following Weiss's earlier use of the same word to indicate a generalization of finiteness in sofic subshifts.
The class of sofic groups is closed under the operations of taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.
As Gromov proved, Sofic groups are surjunctive. That is, they obey a form of the Garden of Eden theorem for cellular automata defined over the group (dynamical systems whose states are mappings from the group to a finite set and whose state transitions are translation-invariant and continuous) stating that every injective automaton is surjective and therefore also reversible.
Notes
References
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.
.
.
Properties of groups |
https://en.wikipedia.org/wiki/2012%20Third%20Division%20Football%20Tournament | Statistics of Third Division Football Tournament in the 2012 season. According to the FAM Calendar 2012, Third Division Football Tournament will start on October 15.
Teams
43 teams are competition in the 2012 Third Division Football Tournament, and these teams were divided into 14 groups.
Group 1
Kelaa Naalhi Sports
Zeal Sports Club
Vaikaradhoo Football Club
Group 2
MS Helping Hand Sports
L.Q. Sports Club
Kodey Sports Club
Group 3
Sports Club Henveyru Dhekunu
Society for Alifushi Youth
Club Green Streets
Group 4
Our Recreation Club
Club New Oceans
Rakeedhoo Ijuthimaaee Gulhun
Group 5
Sealand
Kudahuvadhoo Sports Club
Teenage Juniors
Group 6
Kuda Henveiru United
Tent Sports Club
Buru Sports Club
Group 7
Club Amigos
Club PK
S.T.E.L.C.O. Recreation Club
Group 8
Hilaaly New Generation
Ilhaar
The Bows Sports Club
Group 9
West Sports Club
Falcon Sports Club
Youth Revolution Club
Group 10
Fiyoree Sports Club
Offu Football Club
Club O1O
Group 11
Maaenboodhoo Zuvaanunge Jamiyyaa
TC Sports Club
Mahibadhoo Sports Club
Group 12
Iramaa Youth Association
Sports Club Velloxia
Veyliant Sports Club
Group 13
Lorenzo Sports Club
Veyru Sports Club
Sent Sports Club
Group 14
Naivaadhoo Trainers Sports Club
Lagoons Sports Club
Sports Club Rivalsa
Muiveyo Friends Club
Group stage
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Group 11
Group 12
Group 13
Group 14
Ranking of second-placed teams
Second round
Quarter-finals
Semi-finals
Final
Awards
External links
Mahibadhoo and Green Street to play in the final (DHIVEHI) at Haveeru Online
Mahibadhoo claims champion (DHIVEHI) at Haveeru Online
Mahibadhoo & Sent to Semis (DHIVEHI) at Haveeru Online
ORC & Green Street to semis (DHIVEHI) at Haveeru Online
References
Maldivian Third Division Football Tournament seasons
3 |
https://en.wikipedia.org/wiki/Surjunctive%20group | In mathematics, a surjunctive group is a group such that every injective cellular automaton with the group elements as its cells is also surjective. Surjunctive groups were introduced by . It is unknown whether every group is surjunctive.
Definition
A cellular automaton consists of a regular system of cells, each containing a symbol from a finite alphabet, together with a uniform rule called a transition function for updating all cells simultaneously based on the values of neighboring cells. Most commonly the cells are arranged in the form of a line or a higher-dimensional integer grid, but other arrangements of cells are also possible. What is required of the cells is that they form a structure in which every cell "looks the same as" every other cell: there is a symmetry of both the arrangement of cells and the rule set that takes any cell to any other cell. Mathematically, this can be formalized by the notion of a group, a set of elements together with an associative and invertible binary operation. The elements of the group can be used as the cells of an automaton, with symmetries generated by the group operation. For instance, a one-dimensional line of cells can be described in this way as the additive group of the integers, and the higher-dimensional integer grids can be described as the free abelian groups.
The collection of all possible states of a cellular automaton over a group can be described as the functions that map each group element to one of the symbols in the alphabet.
As a finite set, the alphabet has a discrete topology, and the collection of states can be given the product topology (called a prodiscrete topology because it is the product of discrete topologies).
To be the transition function of a cellular automaton, a function from states to states must be a continuous function for this topology, and must also be equivariant with the group action, meaning that shifting the cells prior to applying the transition function produces the same result as applying the function and then shifting the cells. For such functions, the Curtis–Hedlund–Lyndon theorem ensures that the value of the transition function at each group element depends on the previous state of only a finite set of neighboring elements.
A state transition function is a surjective function when every state has a predecessor (there can be no Garden of Eden). It is an injective function when no two states have the same successor. A surjunctive group is a group with the property that, when its elements are used as the cells of cellular automata, every injective transition function of a cellular automaton is also surjective.
Equivalently, summarizing the definitions above, a group is surjunctive if, for every finite set , every continuous equivariant injective function is also surjective. The implication from injectivity to surjectivity is a form of the Garden of Eden theorem, and the cellular automata defined from injective and surjective transition functions are r |
https://en.wikipedia.org/wiki/Central%20Bureau%20of%20Statistics%20%28Aruba%29 | The Central Bureau of Statistics of Aruba, is in charge of the collection, processing and publication of statistics and reports to the Minister charged with responsibility for the subject of Statistics. It was created mainly to facilitate the development of a statistical system for Aruba which is also a component of the CARICOM regional statistical system together with other member countries of this regional institution.
Mission
Become the leading organization to compile and publish undisputed, coherent and up-to-date statistical information that is relevant for practice, policy and research.
Legislation
Statistics Aruba is governed by the Statistics Act, GT 1991.
Organisation
The Central Bureau of Statistics (CBS) is the institution officially assigned with the collection, processing and publication of statistics to be used by policymakers, in practice and for research in different areas. It is a government department that resorts under the jurisdiction of the Ministry of Finance and Economic Affairs.
Objective statistical information is vital to an open and democratic society. It provides a solid foundation for informed decision-making by elected representatives, businesses, unions and non-profit organizations, as well as the general individual Aruban population. It’s also recognized as an inexhaustible source for scientific research.
Leadership
The director of Statistics Aruba is the Chief Statistician of Aruba. Since 1986, the former directors of the Central Bureau of Statistics Aruba were:
Enrique Jaccopucci (1986–1994)
Randolf Lee (1996-2009)
History
{| class="wikitable"
|+ Previous heads of the statistics service in Aruba
! width=300 | Name:
! width=100 | Period
|-
| Randolf Lee
| 1996-2009
|-
| Enrique Jaccopucci
| 1986–1994
|-
|
| -1986
|}
Publications
Statistics Aruba publishes numerous documents covering a range of statistical information about Aruba, including census data, demography, Tourism, economic and health indicators,
See also
Statistics Netherlands
Sub-national autonomous statistical services
United Nations Statistics Division
References
External links
Ministerie van Economische Zaken, Sociale Zaken en Cultuur
Anguilla statistics indicators on CARICOM Statical indicators publication
Aruba statistical system on United Nations Statistics Division website
CARICOM Statistics
Official statistics
Aruba |
https://en.wikipedia.org/wiki/4-Chlorophenyl%20azide | 4-Chlorophenyl azide is an organic aryl azide compound with the chemical formula C6H4ClN3. The geometry between the nitrogen atoms in the azide functional group is approximately linear while the geometry between the nitrogen and the carbon of the benzene is trigonal planar.
Preparation
There are various methods to synthesize aryl azides. One such method would be to set use react aniline with sodium nitrite (NaNO2) and hydrazine hydrate in the presence of acetic acid. This reaction will give moderate to good yield of the desired aryl azide. The best solvent for this reaction is dichloromethane. Dichloromethane is most effective because it is only slightly polar whereas highly polar solvents give significantly lower yields in this reaction. The reactants dissolve in a less polar solvent better and the reaction proceeds more fully towards completion. Two equivalents of sodium nitrite should be used with five equivalents of hydrazine hydrate to get a high yield of aryl azide. To form 4-chlorophenyl azide specifically, an aniline with a chloride group in the para position is used. The sodium nitrite reacts with aniline to form a diazonium salt that performs nucleophilic substitution with the azide ion formed by another reaction between sodium nitrite and hydrazine hydrate in an acidic medium. Such a reaction takes around 30 minutes to complete and gives around an 80% yield. This is an effective method of synthesis because of the short reaction time, easy work-up and inexpensive reagents. A picture of the preparation reaction is shown below:
Another synthesis method that was researched was the Wong Synthesis which makes use of the reagents NaN3 and Tf2O. The study of this synthesis method was very detailed because NaN3 is known to be explosive so careful attention to the synthesis procedures must be used.
Reactions
Azides are used in a variety of useful reactions and syntheses. In many different reactions they act as an intermediate step to convert a substituent group to an amine. The reason why using azides is useful in this process is because one of the products of reaction is nitrogen gas (N2). When a reaction produces gas there is a thermodynamically favorable push towards the products of the reaction. This relates to 4-chlorophenyl azide because this molecule is an intermediate during the formation of 4-chlorophenyl amine. In many instances lithium aluminium hydride (LAH) is used to reduce the azide functional group. An example of this reaction is the following:
Upon further reactions from the above synthesis, 4-chlorophenyl azide can also lead to the useful transformation of the iminium ion. The iminium ion is important in organic syntheses because it reacts similarly like a carbonyl compound. A partial positive charge builds up on the carbon that is doubly bound to the nitrogen which provides an excellent site for nucleophilic attack. A simple way to make the iminium ion is to react an amine with formaldehyde so water leaves t |
https://en.wikipedia.org/wiki/Descending%20wedge | The descending wedge symbol ∨ may represent:
Logical disjunction in propositional logic
Join in lattice theory
The wedge sum in topology
The V sign, a symbol representing peace among other things
The vertically reflected symbol, ∧, is a wedge, and often denotes related or dual operators.
The ∨ symbol was introduced by Russell and Whitehead in Principia Mathematica, where they called it the Logical Sum or Disjunctive Function.
In Unicode the symbol is encoded . In TeX, it is \vee or \lor.
One motivation and the most probable explanation for the choice of the symbol ∨ is the latin word "vel" meaning "or" in the inclusive sense. Several authors use "vel" as name of the "or" function.
References
See also
List of mathematical symbols
List of logic symbols
Logic symbols |
https://en.wikipedia.org/wiki/Kempf%E2%80%93Ness%20theorem | In algebraic geometry, the Kempf–Ness theorem, introduced by , gives a criterion for the stability of a vector in a representation of a complex reductive group. If the complex vector space is given a norm that is invariant under a maximal compact subgroup of the reductive group, then the Kempf–Ness theorem states that a vector is stable if and only if the norm attains a minimum value on the orbit of the vector.
The theorem has the following consequence: If X is a complex smooth projective variety and if G is a reductive complex Lie group, then (the GIT quotient of X by G) is homeomorphic to the symplectic quotient of X by a maximal compact subgroup of G.
References
Invariant theory
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Horseshoe%20%28symbol%29 | Horseshoe (⊃, \supset in TeX) is a symbol used to represent:
Material conditional in propositional logic
Superset in set theory
It was used by Whitehead and Russell in Principia Mathematica. In Unicode the symbol is encoded .
See also
List of mathematical symbols
List of logic symbols
⊂
ʊ
Ω
References
Logic symbols |
https://en.wikipedia.org/wiki/Nick%20Wormald | Nicholas Charles Wormald (born 1953) is an Australian mathematician and professor of mathematics at Monash University. He specializes in probabilistic combinatorics, graph theory, graph algorithms, Steiner trees, web graphs, mine optimization, and other areas in combinatorics.
In 1979, Wormald earned a Ph.D. in mathematics from the University of Newcastle with a dissertation titled Some problems in the enumeration of labelled graphs.
In 2006, he won the Euler Medal from the Institute of Combinatorics and its Applications. He has held the Canada Research Chair in Combinatorics and Optimization at the University of Waterloo. In 2012, he was recognized with an Australian Laureate Fellowship for his achievements. In 2017, he was elected as a Fellow of the Australian Academy of Science.
In 2018, Wormald was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro.
Selected publications
References
1953 births
Living people
Australian mathematicians
University of Newcastle (Australia) alumni
Academic staff of the University of Waterloo
Graph theorists
Canada Research Chairs
Fellows of the Australian Academy of Science |
https://en.wikipedia.org/wiki/Robert%20Coveyou | Robert R. Coveyou (February 9, 1915 – February 19, 1996) was an American research mathematician who worked at the Oak Ridge National Laboratory. He also taught mathematics part-time for several years at Knoxville College and worked at the International Atomic Energy Agency in Vienna, Austria, while on leave from the Oak Ridge National Laboratory from 1968 until 1971.
An expert on pseudo-random number generators, today he is probably best known for the title of an article published around 1970: "Random Number Generation is too Important to be Left to Chance".
Coveyou was an original member of the small group of radiation protection specialists at the University of Chicago assembled under the leadership of Ernest O. Wollan in 1942/43 and moved to Oak Ridge, Tennessee as part of the Manhattan Project.
After the end of World War II he returned to Chicago to finish his undergraduate degree in Mathematics, and in the following year he received his master's degree from the University of Tennessee, both while employed at the Oak Ridge National Laboratory. He then returned to the laboratory for the remainder of his career, retiring in 1976.
In the early 1950s, Coveyou was one of the scientists and engineers involved in the early introduction of computers to the Oak Ridge National Laboratory and has been credited with naming the first computer housed at the laboratory: the ORACLE (Oak Ridge Automatic Computer and Logical Engine). In preparation for working on the computer in Oak Ridge, he spent two stretches of several weeks each at Remington Rand Corporation in New York City working with their staff to learn how they used the new UNIVAC computer.
Coveyou was a tournament chess player, and was Tennessee State Champion eight times. He is a member of the Tennessee Chess Hall of Fame, having been inducted with the inaugural class in 1990. He also mentored many young Oak Ridge and Tennessee chess players, with an unusual and effective approach to tutoring young players, emphasizing the mastery of simple end games before tackling more complex aspects of the game, including openings.
One of Coveyou's memorable chess experiences was hosting 13-year-old Bobby Fischer at his hotel room in Cleveland, Ohio, after Fischer had just won the 1957 U.S. Open. Coveyou, Fischer, and Edmar Mednis, a chess master from New York and friend of Fischer's, played informal games of chess for hours after the conclusion of the tournament, lasting into the early morning hours of the next day.
Bob Coveyou was also active politically and in the civil rights movement. He helped lead an effort to establish Scarboro High School in the African-American neighborhood of Oak Ridge. Prior to the school's opening, African American children there had had to bus to Knoxville, 30 miles away, to attend Austin High School. The school operated from 1950 until Oak Ridge High School was desegregated in the fall of 1955.
Notes
External links
HISTORY OF THE ENGINEERING PHYSICS AND MATHEMATIC |
https://en.wikipedia.org/wiki/Osgood%27s%20lemma | In mathematics, Osgood's lemma, introduced by , is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous can be dropped, but that form of the lemma is much harder to prove and is known as Hartogs' theorem.
There is no analogue of this result for real variables. If we assume that a function is globally continuous and separately differentiable on each variable (all partial derivatives exist everywhere), it is not true that will necessarily be differentiable. A counterexample in two dimensions is given by
If in addition we define , this function is everywhere continuous and has well-defined partial derivatives in and everywhere (also at the origin), but is not differentiable at the origin.
References
Theorems in complex analysis |
https://en.wikipedia.org/wiki/Tangent%20Lie%20group | In mathematics, a tangent Lie group is a Lie group whose underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian subgroup with underlying space the Lie algebra of G, and G itself.
References
Lie groups |
https://en.wikipedia.org/wiki/Engel%20subalgebra | In mathematics, an Engel subalgebra of a Lie algebra with respect to some element x is the subalgebra of elements annihilated by some power of ad x. Engel subalgebras are named after Friedrich Engel. For finite-dimensional Lie algebras over infinite fields the minimal Engel subalgebras are the Cartan subalgebras.
See also
Engel's theorem
References
Lie algebras |
https://en.wikipedia.org/wiki/Convergent%20matrix | In linear algebra, a convergent matrix is a matrix that converges to the zero matrix under matrix exponentiation.
Background
When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi-convergent.
Definition
We call an n × n matrix T a convergent matrix if
for each i = 1, 2, ..., n and j = 1, 2, ..., n.
Example
Let
Computing successive powers of T, we obtain
and, in general,
Since
and
T is a convergent matrix. Note that ρ(T) = , where ρ(T) represents the spectral radius of T, since is the only eigenvalue of T.
Characterizations
Let T be an n × n matrix. The following properties are equivalent to T being a convergent matrix:
for some natural norm;
for all natural norms;
;
for every x.
Iterative methods
A general iterative method involves a process that converts the system of linear equations
into an equivalent system of the form
for some matrix T and vector c. After the initial vector x(0) is selected, the sequence of approximate solution vectors is generated by computing
for each k ≥ 0. For any initial vector x(0) ∈ , the sequence defined by (), for each k ≥ 0 and c ≠ 0, converges to the unique solution of () if and only if ρ(T) < 1, that is, T is a convergent matrix.
Regular splitting
A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations () above, with A non-singular, the matrix A can be split, that is, written as a difference
so that () can be re-written as () above. The expression () is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0, that is, and C have only nonnegative entries. If the splitting () is a regular splitting of the matrix A and A−1 ≥ 0, then ρ(T) < 1 and T is a convergent matrix. Hence the method () converges.
Semi-convergent matrix
We call an n × n matrix T a semi-convergent matrix if the limit
exists. If A is possibly singular but () is consistent, that is, b is in the range of A, then the sequence defined by () converges to a solution to () for every x(0) ∈ if and only if T is semi-convergent. In this case, the splitting () is called a semi-convergent splitting of A.
See also
Gauss–Seidel method
Jacobi method
List of matrices
Nilpotent matrix
Successive over-relaxation
Notes
References
.
.
.
Limits (mathematics)
Matrices
Numerical linear algebra
Relaxation (iterative methods) |
https://en.wikipedia.org/wiki/Killing%E2%80%93Hopf%20theorem | In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space forms. The Killing–Hopf theorem was proved by and .
References
Riemannian geometry
Theorems in Riemannian geometry |
https://en.wikipedia.org/wiki/Lelong%20number | In mathematics, the Lelong number is an invariant of a point of a complex analytic variety that in some sense measures the local density at that point. It was introduced by . More generally a closed positive (p,p) current u on a complex manifold has a Lelong number n(u,x) for each point x of the manifold. Similarly a plurisubharmonic function also has a Lelong number at a point.
Definitions
The Lelong number of a plurisubharmonic function φ at a point x of Cn is
For a point x of an analytic subset A of pure dimension k, the Lelong number ν(A,x) is the limit of the ratio of the areas of A ∩ B(r,x) and a ball of radius r in Ck as the radius tends to zero. (Here B(r,x) is a ball of radius r centered at x.) In other words the Lelong number is a sort of measure of the local density of A near x. If x is not in the subvariety A the Lelong number is 0, and if x is a regular point the Lelong number is 1. It can be proved that the Lelong number ν(A,x) is always an integer.
References
Complex manifolds |
https://en.wikipedia.org/wiki/Norman%20J.%20Pullman | Norman J. Pullman ( – ) was a mathematician, professor of mathematics, and Doctor of Mathematics, who specialized in number theory, matrix theory, linear algebra, and theory of tournaments.
Career
He earned an M.A. degree in mathematics from Harvard University, and in 1962, he was awarded the Doctorate degree of Mathematics from Syracuse University.
From 1962 to 1965, he was professor of Mathematics at McGill University. And in 1965 he was awarded a postdoctoral fellowship at University of Alberta.
In 1965 he started to work at the faculty of Queen's University, and held a professorship position since 1971.
He lectured in professional meetings for the American Mathematical Society and the Australian Mathematical Society.
He was a visiting scholar for Curtin University of Technology in a great many occasions, and had a professional association with the institution.
During his career, he supervised mathematicians like Dominique de Caen, Rolf S. Rees, and Bill Jackson, among others.
His research included contributions in matrix theory, linear algebra, and theory of tournaments.
Academic publications
References
20th-century American mathematicians
1931 births
1999 deaths
Number theorists
Harvard University alumni
Syracuse University alumni
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Local%20language%20%28formal%20language%29 | In mathematics, a local language is a formal language for which membership of a word in the language can be determined by looking at the first and last symbol and each two-symbol substring of the word. Equivalently, it is a language recognised by a local automaton, a particular kind of deterministic finite automaton.
Formally, a language L over an alphabet A is defined to be local if there are subsets R and S of A and a subset F of A×A such that a word w is in L if and only if the first letter of w is in R, the last letter of w is in S and no factor of length 2 in w is in F. This corresponds to the regular expression
More generally, a k-testable language L is one for which membership of a word w in L depends only on the prefix, suffix and the set of factors of w of length k; a language is locally testable if it is k-testable for some k. A local language is 2-testable.
Examples
Over the alphabet {a,b,[,]}
Properties
The family of local languages over A is closed under intersection and Kleene star, but not complement, union or concatenation.
Every regular language not containing the empty string is the image of a local language under a strictly alphabetic morphism.
References
Semigroup theory
Formal languages
Combinatorics on words |
https://en.wikipedia.org/wiki/William%20Payne%20%28mathematician%29 | William Payne (unknown – c. 1779) was an English mathematician and the author of books about mathematics, draughts, and whist. Payne was the brother of prominent London bookseller Thomas Payne, who sold his works and published some of them.
Payne's first book, An Introduction to the Game of Draughts, was published in 1756. The dedication and preface were written by Samuel Johnson.
Payne's second book, An Introduction to Geometry: Containing the Most Useful Propositions in Euclid, & Other Authors, was published in 1767.
The book Maxims for Playing the Game of Whist; With All Necessary Calculations, and Laws of the Game was published anonymously in 1773; published by his brother Thomas, it is believed to have been written by William Payne.
References
Sources
Courtney, William Prideaux (1894). English whist and English whist players. Richard Bentley and Son.
Courtney, William Prideaux; Smith, David Nichol (ed.) (1915). A bibliography of Samuel Johnson. Clarendon Press.
Boswell, James (1888). The Life of Samuel Johnson. Swan Schonnenheim, Lowrey & Co.
Hanley, Brian J (2001). Samuel Johnson as Book Reviewer: A Duty to Examine the Labors of the Learned. University of Delaware Press.
1779 deaths
18th-century English mathematicians
Year of birth unknown |
https://en.wikipedia.org/wiki/Dixmier%20mapping | In mathematics, the Dixmier mapping describes the space Prim(U(g)) of primitive ideals of the universal enveloping algebra U(g) of a finite-dimensional solvable Lie algebra g over an algebraically closed field of characteristic 0 in terms of coadjoint orbits. More precisely, it is a homeomorphism from the space of orbits g*/G of the dual g* of g (with the Zariski topology) under the action of the adjoint group G to Prim(U(g)) (with the Jacobson topology). The Dixmier map is closely related to the orbit method, which relates the irreducible representations of a nilpotent Lie group to its coadjoint orbits. introduced the Dixmier map for nilpotent Lie algebras and then in extended it to solvable ones.
describes the Dixmier mapping in detail.
Construction
Suppose that g is a completely solvable Lie algebra, and f is an element of the dual g*. A polarization of g at f is a subspace h of maximal dimension subject to the condition that f vanishes on [h,h], that is also a subalgebra. The Dixmier map I is defined by letting I(f) be the kernel of the twisted induced representation Ind~(f|h,g) for a polarization h.
References
Lie algebras |
https://en.wikipedia.org/wiki/Cornish%E2%80%93Fisher%20expansion | The Cornish–Fisher expansion is an asymptotic expansion used to approximate the quantiles of a probability distribution based on its cumulants.
It is named after E. A. Cornish and R. A. Fisher, who first described the technique in 1937.
Definition
For a random variable X with mean μ, variance σ², and cumulants κn, its quantile yp at order-of-quantile p can be estimated as where:
where Hen is the nth probabilists' Hermite polynomial. The values γ1 and γ2 are the random variable's skewness and (excess) kurtosis respectively. The value(s) in each set of brackets are the terms for that level of polynomial estimation, and all must be calculated and combined for the Cornish–Fisher expansion at that level to be valid.
Example
Let X be a random variable with mean 10, variance 25, skew 5, and excess kurtosis of 2. We can use the first two bracketed terms above, which depend only on skew and kurtosis, to estimate quantiles of this random variable. For the 95th percentile, the value for which the standard normal cumulative distribution function is 0.95 is 1.644854, which will be x. The w weight can be calculated as:
or about 2.55621. So the estimated 95th percentile of X is 10 + 5×2.55621 or about 22.781. For comparison, the 95th percentile of a normal random variable with mean 10 and variance 25 would be about 18.224; it makes sense that the normal random variable has a lower 95th percentile value, as the normal distribution has no skew or excess kurtosis, and so has a thinner tail than the random variable X.
References
Logical expressions
Statistical deviation and dispersion
Statistical approximations
Asymptotic theory (statistics) |
https://en.wikipedia.org/wiki/Hessian%20automatic%20differentiation | In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD)
that calculate the second derivative of an -dimensional function, known as the Hessian matrix.
When examining a function in a neighborhood of a point, one can discard many complicated global aspects of the function and accurately approximate it with simpler functions. The quadratic approximation is the best-fitting quadratic in the neighborhood of a point, and is frequently used in engineering and science. To calculate the quadratic approximation, one must first calculate its gradient and Hessian matrix.
Let , for each the Hessian matrix is the second order derivative and is a symmetric matrix.
Reverse Hessian-vector products
For a given , this method efficiently calculates the Hessian-vector product . Thus can be used to calculate the entire Hessian by calculating , for .
The method works by first using forward AD to perform , subsequently the method then calculates the gradient of using Reverse AD to yield . Both of these two steps come at a time cost proportional to evaluating the function, thus the entire Hessian can be evaluated at a cost proportional to n evaluations of the function.
Reverse Hessian: Edge_Pushing
An algorithm that calculates the entire Hessian with one forward and one reverse sweep of the computational graph is Edge_Pushing. Edge_Pushing is the result of applying the reverse gradient to the computational graph of the gradient. Naturally, this graph has n output nodes, thus in a sense one has to apply the reverse gradient method to each outgoing node. Edge_Pushing does this by taking into account overlapping calculations.
The algorithm's input is the computational graph of the function. After a preceding forward sweep where all intermediate values in the computational graph are calculated, the algorithm initiates a reverse sweep of the graph. Upon encountering a node that has a corresponding nonlinear elemental function, a new nonlinear edge is created between the node's predecessors indicating there is nonlinear interaction between them. See the example figure on the right. Appended to this nonlinear edge is an edge weight that is the second-order partial derivative of the nonlinear node in relation to its predecessors. This nonlinear edge is subsequently pushed down to further predecessors in such a way that when it reaches the independent nodes, its edge weight is the second-order partial derivative of the two independent nodes it connects.
Graph colouring techniques for Hessians
The graph colouring techniques explore sparsity patterns of the Hessian matrix and cheap Hessian vector products to obtain the entire matrix. Thus these techniques are suited for large, sparse matrices. The general strategy of any such colouring technique is as follows.
Obtain the global sparsity pattern of
Apply a graph colouring algorithm that allows us to compact the sparsity structure.
For each desired point |
https://en.wikipedia.org/wiki/NCAA%20Division%20I%20men%27s%20soccer%20tournament%20all-time%20team%20records | The following is a list of National Collegiate Athletic Association (NCAA) Division I college soccer team statistics through the 2017 NCAA Division I Men's Soccer Championship, including all-time number of wins, losses, and draws; number of tournaments played; and percent of games won.
Team Records
Most Single Team Goals, Single Game: 9
Saint Louis (vs. Stanford–3, 1962)
Michigan State (vs. Howard–1, 1962)
Howard (vs. Duke-0, 1972)
Appalachian State (vs. George Washington–3, 1978)
Most Combined Goals, Single Game: 12
Saint Louis 9, Stanford 3, 1962
Appalachian State 9, George Washington 3, 1978
Most Goals, Tournament: 17
Clemson, 1976
Clemson, 1978
UCLA, 2002
Most Shots, Game: 54
Connecticut 2, Rhode Island 3 (4OT), 1979
Most Corner Kicks, Game: 20
Connecticut 2, Hartford 1 (4OT), 1999
Most Fouls, Game: 50
Connecticut 2, Rhode Island 3 (4OT), 1979
Indiana 1, UC Santa Barbara 1 (2OT/PK), 2004
Goals Per Game, Tournament (minimum 2 games): 5.00
Saint Louis (15 goals, 3 games), 1959
Saint Louis (15 goals, 3 games), 1962
Appalachian State (10 goals, 2 games), 1978
Lowest Goals-Against Average Tournament (minimum 3 games): 0.00
San Francisco (0 goals against, 4 games), 1976
Wisconsin (0 goals against, 5 games), 1995
Akron (0 goals against, 5 games), 2009
Stanford (0 goals against, 5 games) 2016, 2017
Overtime Games, Tournament: 5
Alabama A&M, 1981
Longest Game: 166 minutes, 5 seconds
UCLA 1, American 0 (8OT), 1985
Ranking by number of wins, then winning percentage (Minimum of 4 wins) (As of 1960)
Schools in Italics no longer compete in Division I.
References
External links
NCAA Men's Soccer
records |
https://en.wikipedia.org/wiki/Rank%20of%20a%20partition | In mathematics, particularly in the fields of number theory and combinatorics, the rank of a partition of a positive integer is a certain integer associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition.
Definition
By a partition of a positive integer n we mean a finite multiset λ = { λk, λk − 1, . . . , λ1 } of positive integers satisfying the following two conditions:
λk ≥ . . . ≥ λ2 ≥ λ1 > 0.
λk + . . . + λ2 + λ1 = n.
If λk, . . . , λ2, λ1 are distinct, that is, if
λk > . . . > λ2 > λ1 > 0
then the partition λ is called a strict partition of n.
The integers λk, λk − 1, ..., λ1 are the parts of the partition. The number of parts in the partition λ is k and the largest part in the partition is λk. The rank of the partition λ (whether ordinary or strict) is defined as λk − k.
The ranks of the partitions of n take the following values and no others:
n − 1, n −3, n −4, . . . , 2, 1, 0, −1, −2, . . . , −(n − 4), −(n − 3), −(n − 1).
The following table gives the ranks of the various partitions of the number 5.
Ranks of the partitions of the integer 5
Notations
The following notations are used to specify how many partitions have a given rank. Let n, q be a positive integers and m be any integer.
The total number of partitions of n is denoted by p(n).
The number of partitions of n with rank m is denoted by N(m, n).
The number of partitions of n with rank congruent to m modulo q is denoted by N(m, q, n).
The number of strict partitions of n is denoted by Q(n).
The number of strict partitions of n with rank m is denoted by R(m, n).
The number of strict partitions of n with rank congruent to m modulo q is denoted by T(m, q, n).
For example,
p(5) = 7 , N(2, 5) = 1 , N(3, 5) = 0 , N(2, 2, 5) = 5 .
Q(5) = 3 , R(2, 5) = 1 , R(3, 5) = 0 , T(2, 2, 5) = 2.
Some basic results
Let n, q be a positive integers and m be any integer.
Ramanujan's congruences and Dyson's conjecture
Srinivasa Ramanujan in a paper published in 1919 proved the following congruences involving the partition function p(n):
p(5n + 4) ≡ 0 (mod 5)
p(7n + 5) ≡ 0 (mod 7)
p(11n + 6) ≡ 0 (mod 11)
In commenting on this result, Dyson noted that " . . . although we can prove that the partitions of 5n + 4 can be divided into five equally numerous subclasses, it is unsatisfactory to receive from the proofs no |
https://en.wikipedia.org/wiki/Verification%20bias | In statistics, verification bias is a type of measurement bias in which the results of a diagnostic test affect whether the gold standard procedure is used to verify the test result. This type of bias is also known as "work-up bias" or "referral bias".
In clinical practice, verification bias is more likely to occur when a preliminary diagnostic test is negative. Because many gold standard tests can be invasive, expensive, and carry a higher risk (e.g. angiography, biopsy, surgery), patients and physicians may be more reluctant to undergo further work-up if a preliminary test is negative.
In cohort studies, obtaining a gold standard test on every patient may not always be ethical, practical, or cost effective. These studies can thus be subjected to verification bias. One method to limit verification bias in clinical studies is to perform gold standard testing in a random sample of study participants.
In most situations, verification bias introduces a sensitivity estimate that is too high and a specificity that is too low.
References
Epidemiology
Medical statistics
Bias |
https://en.wikipedia.org/wiki/2012%20CECAFA%20Cup%20statistics | The following are the statistics for the 2012 CECAFA Cup, which took place in Kampala, Uganda from 24 November to 8 December 2012. All statistics are correct as of 20:00 UTC+3 on 8 December 2012. Goals scored from penalty shoot-outs are not counted.
Goalscorers
5 goals
John Bocco
Mrisho Ngassa
Robert Ssentongo
3 goals
Selemani Ndikumana
Christophe Nduwarugira
Chiukepo Msowoya
Geoffrey Kizito
Brian Umony
Khamis Mcha Khamis
2 goals
Clifton Miheso
David Ochieng
Mike Baraza
Dady Birori
1 goal
Yusuf Ndikumana
Yosief Ghide
Hermon Tecleab
Yonathan Kebede
Elias Mamo
Edwin Lavatsa
Rama Salim
Ndaziona Chatsalira
Miciam Mhone
Jean-Baptiste Mugiraneza
Haruna Niyonzima
Tumaine Ntamuhanga
Jabril Hassan Mohammed
Farid Mohamed Najeeb
Mwinyi Kazimoto
Amri Kiemba
Hamis Kizza
Emmanuel Okwi
Aggrey Morris
Abdallah Othman
Scoring
Wins and losses
Disciplinary record
By match
By referee
By team
By individual
Overall statistics
Bold numbers indicate the maximum values in each column.
See also
2012 CECAFA Cup scorers
2012 CECAFA Cup schedule
References
statistics |
https://en.wikipedia.org/wiki/Albert%20Turner%20Bharucha-Reid | Albert Turner Bharucha-Reid (November 13, 1927 February 26, 1985) was an American mathematician and theorist who worked extensively on probability theory, Markov chains, and statistics. The author of more than 70 papers and 6 books, his work touched on such diverse fields as economics, physics, and biology.
Life
Bharucha-Reid was born Albert Turner Reid, the son of William Thaddeus Reid and Mae Marie Beamon Reid of Hampton, Virginia. He studied math and biology at Iowa State University, where completed a BS in 1949. He continued his studies at the University of Chicago from 1950 to 1953, where he began to focus more intensely on statistics and probability. He published eight papers during his time at the University of Chicago, but he did not finish his PhD dissertation because he felt it was a waste of time. In 1954, he married Rodabe Phiroze Bharucha, and he legally changed his name to Albert Turner Bharucha-Reid. He had two children, Kurush Feroze Bharucha-Reid, and Rustam William Bharucha-Reid.
Work
Bharucha-Reid published his first paper, a work on mathematical biology, when he was only 18 years old. He went on to teach and lecture in the United States, Europe, and India. He held professorships or research positions at Columbia University, the University of California, Berkeley, the University of Oregon, Wayne State University, the Polish Academy of Sciences, and Atlanta University. In particular, in 1970 he was appointed Dean of the School of Arts and Sciences at Wayne State University.
Legacy
A lecture series has been named in his honor by the National Association of Mathematicians.
Selected publications
Bharucha-Reid is the author or coauthor of:
Elements of the Theory of Markov Processes and Their Application (McGraw Hill, 1960; Dover, 1997)
Random Integral Equations (Academic Press, 1972)
Random Polynomials (with M. Sambandham, Academic Press, 1986)
He is also the editor of:
Probabilistic Methods in Applied Mathematics (edited, Academic Press, Vol. I, 1968; Vol. II, 1970)
Probabilistic Analysis and Related Topics (edited, Academic Press, Vol. I, 1978; Vol. II, 1979; Vol. III, 1983)
Approximate Solution of Random Equations (North-Holland, 1979)
References
1927 births
1985 deaths
African-American mathematicians
20th-century African-American scientists
20th-century American mathematicians
American statisticians
Clark Atlanta University faculty
Iowa State University alumni
People from Hampton, Virginia
Probability theorists
University of Chicago alumni
University of Oregon faculty
Wayne State University faculty
University of California, Berkeley faculty
Academics from Virginia
Mathematicians from Virginia
African-American statisticians
20th-century African-American academics
20th-century American academics |
https://en.wikipedia.org/wiki/Adolf%20Piltz | Adolf Piltz (8 December 1855 – 1940) was a German mathematician who contributed to number theory. Piltz was arguably the first to formulate a generalized Riemann hypothesis, in 1884.
Notes
References
Davenport, Harold. Multiplicative number theory. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. xiv+177 pp. .
Further reading
External links
1855 births
1940 deaths
19th-century German mathematicians
Humboldt University of Berlin alumni
20th-century German mathematicians
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Crank%20of%20a%20partition | In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathematics Society of Cambridge University. Dyson then gave a list of properties this yet-to-be-defined quantity should have. In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties hypothesized for it by Dyson.
Dyson's crank
Let n be a non-negative integer and let p(n) denote the number of partitions of n (p(0) is defined to be 1). Srinivasa Ramanujan in a paper published in 1918 stated and proved the following congruences for the partition function p(n), since known as Ramanujan congruences.
p(5n + 4) ≡ 0 (mod 5)
p(7n + 5) ≡ 0 (mod 7)
p(11n + 6) ≡ 0 (mod 11)
These congruences imply that partitions of numbers of the form 5n + 4 (respectively, of the forms 7n + 5 and 11n + 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size.
In his Eureka paper Dyson proposed the concept of the rank of a partition. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = −1. Denoting by N(m, q, n), the number of partitions of n whose ranks are congruent to m modulo q, Dyson considered N(m, 5, 5 n + 4) and N(m, 7, 7n + 5) for various values of n and m. Based on empirical evidences Dyson formulated the following conjectures known as rank conjectures.
For all non-negative integers n we have:
N(0, 5, 5n + 4) = N(1, 5, 5n + 4) = N(2, 5, 5n + 4) = N(3, 5, 5n + 4) = N(4, 5, 5n + 4).
N(0, 7, 7n + 5) = N(1, 7, 7n + 5) = N(2, 7, 7n + 5) = N(3, 7, 7n + 5) = N(4, 7, 7n + 5) = N(5, 7, 7n + 5) = N(6, 7, 7n + 5)
Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5n + 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the form 7n + 6 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11n + 6 into 11 classes of the same size, as the following table shows.
Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,
I hold in fact :
that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the "crank" of the partition and denote by M(m, q, n) the number of partitions of n whose crank is congruent to m modulo q;
that M(m, q, n) = |
https://en.wikipedia.org/wiki/Baxter%20permutation | In combinatorial mathematics, a Baxter permutation is a permutation which satisfies the following generalized pattern avoidance property:
There are no indices i < j < k such that σ(j + 1) < σ(i) < σ(k) < σ(j) or σ(j) < σ(k) < σ(i) < σ(j + 1).
Equivalently, using the notation for vincular patterns, a Baxter permutation is one that avoids the two dashed patterns 2-41-3 and 3-14-2.
For example, the permutation σ = 2413 in S4 (written in one-line notation) is not a Baxter permutation because, taking i = 1, j = 2 and k = 4, this permutation violates the first condition.
These permutations were introduced by Glen E. Baxter in the context of mathematical analysis.
Enumeration
For n = 1, 2, 3, ..., the number an of Baxter permutations of length n is
1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586,...
This is sequence in the OEIS. In general, an has the following formula:
In fact, this formula is graded by the number of descents in the permutations, i.e., there are Baxter permutations in Sn with k – 1 descents.
Other properties
The number of alternating Baxter permutations of length 2n is (Cn)2, the square of a Catalan number, and of length 2n + 1 is CnCn+1.
The number of doubly alternating Baxter permutations of length 2n and 2n + 1 (i.e., those for which both σ and its inverse σ−1 are alternating) is the Catalan number Cn.
Baxter permutations are related to Hopf algebras, planar graphs, and tilings.
Motivation: commuting functions
Baxter introduced Baxter permutations while studying the fixed points of commuting continuous functions. In particular, if f and g are continuous functions from the interval [0, 1] to itself such that f(g(x)) = g(f(x)) for all x, and f(g(x)) = x for finitely many x in [0, 1], then:
the number of these fixed points is odd;
if the fixed points are x1 < x2 < ... < x2k + 1 then f and g act as mutually-inverse permutations on {x1, x3, ..., x2k + 1} and {x2, x4, ..., x2k};
the permutation induced by f on {x1, x3, ..., x2k + 1} uniquely determines the permutation induced by f on {x2, x4, ..., x2k};
under the natural relabeling x1 → 1, x3 → 2, etc., the permutation induced on {1, 2, ..., k + 1} is a Baxter permutation.
See also
Enumerations of specific permutation classes
References
External links
Permutation patterns |
https://en.wikipedia.org/wiki/Television%20in%20Egypt | Television in Egypt is mainly received through free satellite, while analog terrestrial represents 41% of total viewers. The Central Agency for Public Mobilisation and Statistics (CAPMAS) said the average time an Egyptian spends watching television a day is 180 minutes (3 hours), while Egyptian channels recorded 170,000 hours of broadcast in 2019.
Since Egypt launched its first broadcasts in 1960, state-run channels have held a monopoly on terrestrial broadcast. The Ministry of Information strictly regulated private satellite channels as well. The Egyptian Radio and Television Union, a government entity, owns all 17 terrestrial channels. Channel 1 and Channel 2 are the network’s main channels and broadcast across Egypt. The state-owned Nile TV is the main foreign language channel, aims at promoting Egypt's state point of view and promote tourism. There are 6 regional terrestrial channels, which used to all be broadcast to Greater Cairo, but as of 2007, only Greater Cairo channel (Channel 3) of the regional channels is broadcast to Greater Cairo. Most terrestrial channels were in fact satellite channels owned by ERTU, but simulcasted to Greater Cairo, since 2007.
The state's 23 channels are reported to have, as of 2012, "a small and dwindling viewership".
There are also many private satellite stations. As of 2002, there used to be only two, Al-Mehwar and Dream, though the government has a financial stake in both channels. Since the 2011 revolution, more channels have launched, including Capital Broadcasting Center, Al Nahar and Al Tahrir (now TeN), which have managed to attract significant viewership. Rotana launched Rotana Masriya, a channel broadcasting programs aimed at the Egyptian market.
Subscription television penetration is low, estimated to be 9% in 2011, which consists of OSN and Arab Radio and Television Network. OSN was formed in 2009 by merging Orbit and Showtime Arabia. All of which are not owned by Egyptian companies, but by Persian Gulf companies.
In the 1990s, there used to be an Egyptian company called CNE (Cable Network of Egypt) which provided a few foreign pay TV stations broadcast terrestrially over the air (CNN International, MTV Europe, Al Jazeera English with one show made for Arab League viewers, and other defunct channels), but needed a special receiver and a card.
The overwhelming number of private satellite stations launched during 2008 till 2012 has changed the Egyptian TV production market drastically, lifting the dominant hand of state-run channels off the market. Over 50 TV series have been broadcast annually during Ramadan – Main TV viewership season.
Introduction of dubbed TV shows – from Turkey and India mainly- on Egyptian TV channels made the market more competitive. Egyptian TV productions companies started to adapt in efforts to match the foreign offerings which started to dominate the market. Companies like Egyptian Arts Group, El Adl Group and many others started doubling their annual production budg |
https://en.wikipedia.org/wiki/Grothendieck%E2%80%93Teichm%C3%BCller%20group | In mathematics, the Grothendieck–Teichmüller group GT is a group closely related to (and possibly equal to) the absolute Galois group of the rational numbers. It was introduced by and named after Alexander Grothendieck and Oswald Teichmüller, based on Grothendieck's suggestion in his 1984 essay Esquisse d'un Programme to study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of Teichmüller groupoids Tg,n, the fundamental groupoids of moduli stacks of genus g curves with n points removed. There are several minor variations of the group: a discrete version, a pro-l version, a k-pro-unipotent version, and a profinite version; the first three versions were defined by Drinfeld, and the version most often used is the profinite version.
References
Translation in Leningrad Math. J. 2 (1991), no. 4, 829–860 .
Number theory |
https://en.wikipedia.org/wiki/Oleg%20Novachuk | Oleg Novachuk (born 9 February 1971) is a Kazakh businessman, and the chief executive (CEO) of KAZ Minerals.
Early life
He has a master's degree in applied mathematics from Kazakh State University.
Career
Novachuk has been CEO of Kazakhmys since 15 March 2007, having been Finance Director from 23 September 2005 to 15 March 2007, and joining the company in 2001.
From 1998 to 2001, he was deputy chairman, then chairman, of JSC Kazprombank, at that time one of the largest private banks in Kazakhstan.
Following the completion of the Restructuring of Kazakhmys PLC Novachuk continued as the CEO of KAZ Minerals PLC.
As at April 2021, he owns 39.4% of KAZ Minerals.
References
1971 births
Living people
Kazakhstani businesspeople
Al-Farabi Kazakh National University alumni
Kazakhmys |
https://en.wikipedia.org/wiki/Confidence%20accounting | Confidence accounting is a method of accounting whereby some of the figures are expressed not as single point estimates, but rather as probability distributions. Under Confidence Accounting, the end results of audits would be presentations of distributions for major entries in the profit & loss, balance sheet and cashflow statements. The proposed benefits of Confidence Accounting include a fairer representation of financial results, reduced footnotes, more measurable audit quality and a mitigation of mark-to-market perturbations.
History
This method is in the discussion stage and has not yet been adopted by any accountancy body, though events and publications have been sponsored by the Association of Chartered Certified Accountants and some other events by the Institute of Chartered Accountants of Scotland.
The term "confidence accounting" was first adopted in the mid-2000s by the Long Finance initiative, having grown out of earlier publications that referred to "stochastic accounting".
Advantages
Confidence accounting has the alleged advantages of:
Being more scientific. Most scientific experimental results are expressed as expected values and some quantisation of the error involved, accounts are not.
Giving a fairer view of the risk associated with the accounts. For example a company may own drilling rights to a potential oilfield. The value of this asset could be zero, or could be immense. If this is not known then a distribution would be a more faithful representation of the asset value, rather than a single estimate of the mean value.
Holding the accountants responsible Accounts from previous years can be retrospectively checked against the forecasts, to see if, for example an accountancy firm actually got 90% of its accounts within the 90% error band forecast.
Criticism
Confidence accounting can be criticized for:
adding further complexity to an already complex subject, although its supporters claim that this method could actually reduce the size of accounts.
the alleged advantages of holding the accountants responsible can be evaded by accountants claiming (with some justification) that the forecast distributions do not take into account unforeseen macroeconomic factors.
no external party having both the power and the interest to make this happen.
External links
Confidence Accounting: A Proposal published in 2012 by ACCA, CISI and Long Finance
Confidence Accounting discussion and reports
References
Accounting systems |
https://en.wikipedia.org/wiki/Takiff%20algebra | In mathematics, a Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the form g[x]/(xn+1) = g⊗kk[x]/(xn+1) for some positive integer n. Sometimes these are called generalized Takiff algebras, and the name Takiff algebra is used for the case when n = 1. These algebras (for n = 1) were studied by , who in some cases described the ring of polynomials on these algebras invariant under the action of the adjoint group.
References
Lie algebras
1971 in science |
https://en.wikipedia.org/wiki/Masao%20Haji | was a Japanese political activist, mathematics lecturer and critic. He also wrote books under the name . He was chair of mathematics at the correspondence-course "Z-kai", and taught at the three top exam preparation schools (juku): Yoyogi Seminar, Sundai Preparatory School and Kawai Juku Groupwork.
Career
For many years, he was responsible for the "basic mathematics seminar" at Yoyogi exam preparation school. He was a lecturer in Oubun-Sha company's "University entrance exam radio". Oubun-Sha was the author of the "fundamental issues lecture" and "standard issue lecture" series and the author of "training" series of the Zoushin-kai Publishing House, such as Aleph's "Calculus Lab".
He gave counseling to young people, including homeless youth.
Personal life
He graduated from the Department of Engineering of the University of Tokyo.
Haji graduated from the kyu-sei-ichi-koukou (currently, University of Tokyo liberal arts).
In his youth he suffered from tuberculosis and recovered but sometimes had hoarseness or cough.
He is buried in Aoyama cemetery in Minato-ku, Tokyo.
Haji was involved in the "Peace to Vietnam" committee.
Publications
Masao Haji. "Han-sugaku-riron" (Futohsha, 1977) ISBN B000J8S3KC
Masao Haji. "Koukou-muyou-no-Daigaku-Shingaku-Hou" (Sanichi-syobou, 1983) ISBN B000J7BLYS
Masao Haji. "Sugaku1 Hyoujun-Mondai-seikou" (Oubun-sha, 1985)
Masao Haji. "Kisokaiseki-hyoujyun-mondai-seikou" (Oubun-sha, 1996)
References
External links
Memorial texts collection of works from "44fragments of the surrounding Mr. Haji masao." By Yuichi Yoshikawa (in Japanese)
"Meeting in memory of Mr. Haji masao." success. Facsimile edition of the book also 05 / July / 99 (in Japanese)
1925 births
1998 deaths
Japanese educators
20th-century Japanese mathematicians
Japanese activists
University of Tokyo alumni |
https://en.wikipedia.org/wiki/P-adic%20modular%20form | In mathematics, a p-adic modular form is a p-adic analog of a modular form, with coefficients that are p-adic numbers rather than complex numbers. introduced p-adic modular forms as limits of ordinary modular forms, and shortly afterwards gave a geometric and more general definition. Katz's p-adic modular forms include as special cases classical p-adic modular forms, which are more or less p-adic linear combinations of the usual "classical" modular forms, and overconvergent p-adic modular forms, which in turn include Hida's ordinary modular forms as special cases.
Serre's definition
Serre defined a p-adic modular form to be a formal power series with p-adic coefficients that is a p-adic limit of classical modular forms with integer coefficients. The weights of these classical modular forms need not be the same; in fact, if they are then the p-adic modular form is nothing more than a linear combination of classical modular forms. In general the weight of a p-adic modular form is a p-adic number, given by the limit of the weights of the classical modular forms (in fact a slight refinement gives a weight in Zp×Z/(p–1)Z).
The p-adic modular forms defined by Serre are special cases of those defined by Katz.
Katz's definition
A classical modular form of weight k can be thought of roughly as a function f from pairs (E,ω) of a complex elliptic curve with a holomorphic 1-form ω to complex numbers, such that f(E,λω) = λ−kf(E,ω), and satisfying some additional conditions such as being holomorphic in some sense.
Katz's definition of a p-adic modular form is similar, except that E is now an elliptic curve over some algebra R (with p nilpotent) over the ring of integers R0 of a finite extension of the p-adic numbers, such that E is not supersingular, in the sense that the Eisenstein series Ep–1 is invertible at (E,ω). The p-adic modular form f now takes values in R rather than in the complex numbers. The p-adic modular form also has to satisfy some other conditions analogous to the condition that a classical modular form should be holomorphic.
Overconvergent forms
Overconvergent p-adic modular forms are similar to the modular forms defined by Katz, except that the form has to be defined on a larger collection of elliptic curves. Roughly speaking, the value of the Eisenstein series Ek–1 on the form is no longer required to be invertible, but can be a smaller element of R. Informally the series for the modular form converges on this larger collection of elliptic curves, hence the name "overconvergent".
References
Modular forms
p-adic numbers |
https://en.wikipedia.org/wiki/Po%C5%A1torn%C3%A1 | Poštorná is a municipal district located in the town of Břeclav, South Moravia, Czech Republic.
Former football club SK Tatran Poštorná was based in the district.
External links
Poštorná statistics at Ministry of the Interior website
Databáze statistických obvodů (Statistical database of districts)
Populated places in Břeclav District
Neighbourhoods in the Czech Republic |
https://en.wikipedia.org/wiki/Credal%20network | Credal networks are probabilistic graphical models based on imprecise probability. Credal networks can be regarded as an extension of Bayesian networks, where credal sets replace probability mass functions in the specification of the local models for the network variables given their parents. As a Bayesian network defines a joint probability mass function over its variables, a credal network defines a joint credal set. The way this credal set is defined depends on the particular notion of independence for imprecise probability adopted. Most of the research on credal networks focused on the case of strong independence. Given strong independence the joint credal set associated to a credal network is called its strong extension. Let denote a collection of categorical variables and . If is, for each , a conditional credal set over , then the strong extension of a credal network is defined as follows:
where denote the convex hull.
Inference
Inference on a credal network is intended as the computation of the bounds of an expectation with respect to its strong extensions. When computing the bounds of a conditional event, inference is called updating. Say that the queried variable and the observed variables are , the lower bound to be evaluated is:
Being a generalization of the same problem for Bayesian networks, updating with credal networks is a NP-hard task. Yet a number of algorithm have been specified.
See also
Imprecise probability
Credal set
Bayesian network
References
Cozman, F.G., 2000. Credal networks. Artificial intelligence, 120(2), pp. 199–233.
Bayesian inference |
https://en.wikipedia.org/wiki/Steven%20Takiff | Steven Joel Takiff is an American mathematician who introduced what became Takiff algebras in 1971.
Publications
References
External links
Doctoral graduates from 1903-present, Department of Mathematics, University of Illinois at Urbana-Champaign
LinkedIn account
Living people
20th-century American mathematicians
21st-century American mathematicians
University of Illinois Urbana-Champaign alumni
Michigan State University faculty
Algebraists
Scientists from Illinois
Scientists from Michigan
People from Dayton, Ohio
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Matrix%20Toolkit%20Java | Matrix Toolkit Java (MTJ) is an open-source Java software library for performing numerical linear algebra. The library contains a full set of standard linear algebra operations for dense matrices based on BLAS and LAPACK code. Partial set of sparse operations is provided through the Templates project. The library can be configured to run as a pure Java library or use BLAS machine-optimized code through the Java Native Interface.
MTJ was originally developed by Bjørn-Ove Heimsund, who has taken a step back due to other commitments. The project webpage states that "(The new maintainers are) primarily concerned with keeping the library maintained, and fixing bugs as they are discovered. There is no road plan for future releases".
Several citations for MTJ can be found in scientific literature, including which uses its LU preconditioner. Performance of MTJ has been compared to other libraries, which can be found at Java Matrix Benchmark's site.
Capabilities
The following is an overview of MTJ's capabilities, as listed on the project's website:
Datastructures for dense and structured sparse matrices in the following formats:
Dense, column major.
Banded matrices, which store only a few diagonals.
Packed matrices, storing only half the matrices (for triangular or symmetric matrices).
Tridiagonal and symmetric tridiagonal matrices.
Transparent support for symmetric and triangular storage.
Datastructures for unstructured sparse matrices in these formats:
Compressed row or column storage (CRS/CCS).
Flexible CRS/CCS, using growable sparse vectors.
Compressed diagonal storage (CDS).
The dense and structured sparse matrices are built on top of BLAS and LAPACK, and include the following intrinsic operations:
Matrix/vector multiplication.
Matrix/matrix multiplication.
Rank updates by matrices or vectors.
Direct matrix solvers.
The unstructured sparse matrices supports the same operations as the structured ones, except they do not have direct solvers. However, their matrix/vector multiplication methods are optimised for use in iterative solvers.
Matrix decompositions of dense and structured sparse matrices:
LU and Cholesky.
Eigenvalue decompositions for unsymmetrical dense matrices.
Singular value decompositions for unsymmetrical dense matrices.
Eigenvalue decompositions for symmetrical matrices (tridiagonal, banded, packed and dense).
Orthogonal matrix decompositions for dense matrices (QR, RQ, LQ, and QL).
Iterative solvers for unstructured sparse matrices from the Templates project:
BiConjugate gradients.
BiConjugate gradients stabilized.
Conjugate gradients.
Conjugate gradients squared.
Chebyshev iteration.
Generalized minimal residual (GMRES).
Iterative refinement (Richardson's method).
Quasi-minimal residual.
A selection of algebraic preconditioners:
Diagonal preconditioning.
Symmetrical successive overrelaxation.
Incomplete Cholesky.
Incomplete LU.
Incomplete LU with fill-in using thresholding.
Algebrai |
https://en.wikipedia.org/wiki/Oxford%20Bulletin%20of%20Economics%20and%20Statistics | Oxford Bulletin of Economics and Statistics is a bimonthly peer-reviewed academic journal published by John Wiley & Sons on behalf of the Department of Economics, University of Oxford. The journal was established in 1939 as the Bulletin of the Oxford University Institute of Economics and Statistics and became the Oxford Bulletin of Economics and Statistics in 1973. The journal publishes articles on applied economics with emphasis placed on the practical importance, theoretical interest and policy-relevance of their results. General topics include macroeconomics, microeconomics, derivatives, investment and interest rates.
According to the Journal Citation Reports, the journal has a 2020 impact factor of 1.791, ranking it 33rd out of 52 journals in the category "Social Sciences, Mathematical Methods", 53rd out of 125 journals in the category "Statistics & Probability" and 204th out of 378 journals in the category "Economics".
References
External links
Wiley-Blackwell academic journals
English-language journals
Bimonthly journals
Economics journals |
https://en.wikipedia.org/wiki/Hayato%20Nakamura | Hayato Nakamura (中村 隼, born November 18, 1991) is a Japanese football player.
Club statistics
Updated to 23 February 2016.
References
External links
1991 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Montedio Yamagata players
V-Varen Nagasaki players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Koki%20Takenaka | is a Japanese football player for Tochigi Uva FC.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Tochigi SC
1992 births
Living people
Association football people from Osaka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Tokyo Verdy players
Briobecca Urayasu players
Tochigi SC players
Vanraure Hachinohe players
Tochigi City FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Soichi%20Tanaka | Soichi Tanaka (田中 奏一, born June 27, 1989) is a Japanese football player.
Club statistics
Updated to end of 2018 season.
References
External links
Profile at Kagoshima United FC
1989 births
Living people
Keio University alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
J3 League players
Fagiano Okayama players
Kagoshima United FC players
Nara Club players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Masashi%20Wakasa | is a Japanese football player for Vegalta Sendai.
Club statistics
Updated to end of 2022 season.
1Includes J1 Promotion Playoffs and J2/J3 Relegation Playoffs.
References
External links
Profile at Vegalta Sendai
Profile at JEF United Chiba
1989 births
Living people
Toyo University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
JEF United Chiba players
Tokyo Verdy players
Vegalta Sendai players
Men's association football defenders |
https://en.wikipedia.org/wiki/Naoya%20Fuji | is a Japanese football player.
Club statistics
Updated to 23 February 2016.
References
External links
Profile at Ehime FC
1993 births
Living people
Association football people from Ehime Prefecture
Japanese men's footballers
J2 League players
J3 League players
Ehime FC players
J.League U-22 Selection players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Ilan%20Sadeh | Ilan Sadeh (born June 1, 1953) is an Israeli IT theoretician, entrepreneur, and human rights activist. He holds the position of Associate Professor of Computer Sciences and Mathematics at the University for Information Science and Technology "St. Paul The Apostle" in Ohrid, North Macedonia.
Biography
Background and activities
Sadeh was the first to claim publicly in the Israeli media that Israel has no right to be called the "heir" to Holocaust victims and no right to represent Holocaust survivors. According to him, Zionist leaders have little cause for pride in their actions during the Second World War – Zionist financiers withheld funds, while the JDC refused to help save Europe's Jewry, instead prioritizing the needs of the Yishuv in Palestine.
The situation in Israel brought Sadeh to the conclusion that the political system must be replaced. He entered politics and led a movement in behalf of Holocaust survivors. He published a few articles in Israeli newspapers and had a public impact. Sadeh was elected a representative of that community and ran in the preliminary election of the Labor Party for the Knesset, or Israeli Parliament (1996), but was not elected. Following his activities, Sadeh was recently threatened and accused of being a traitor. Sadeh has taken libel action over the charges in Israeli Court (2011).
Mathematical background and Sadeh's contribution
The asymptotic equipartition property (AEP) or "Shannon–McMillan–Breiman theorem" is a general property of the output samples of a stochastic source and is the basis of Information Theory. It is fundamental to the concept of typical sequences used in theories of coding theory. AEP was first introduced by Shannon (1948), proved in weak convergence by McMillan (1953) and later refined to strong convergence by Breiman (1957, 1960).
Shannon Theorems are based on AEP. Shannon provided in 1959 the first source-compression coding theorems. But neither he nor his successors could present any algorithm that attains Shannon bound.
Only in 1990, Ornstein and Shields have proposed an algorithm that attains Shannon bound. They proved the convergence to Shannon bound known as "rate-distortion function". But their algorithm is far from being useful and assumes a-priori knowledge of source distribution.
In Sadeh's Ph.D. research (1990–1992) he proposed a universal algorithm that attains Shannon bound. That is, it does not require a priori knowledge of source distribution and asymptotically has some computational advantages. The algorithm is a generalization and merging of Ornstein Shields Algorithm and Wiener Ziv Algorithm (1989).
When he tried to prove convergence to Shannon bound, known also as "Rate Distortion Function", he realized that he could not rely on AEP or Shannon McMillan Breiman Theory.
So in 1992, he presented and proved a new "Limit Theorem" and named it "Lossy AEP" or "Extended Shannon McMillan Breiman Theorem".
That means that the basis of "Information theory" |
https://en.wikipedia.org/wiki/Shigeru%20Mukai | is a Japanese mathematician at Kyoto University specializing in algebraic geometry.
Work
He introduced the Fourier–Mukai transform in 1981 in a paper on abelian varieties, which also made up his doctoral thesis. His research since has included work on vector bundles on K3 surfaces, three-dimensional Fano varieties, moduli theory, and non-commutative Brill-Noether theory. He also found a new counterexample to Hilbert's 14th problem (the first counterexample was found by Nagata in 1959).
Publications
References
External links
1953 births
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Algebraic geometers
Kyoto University alumni
Academic staff of Kyoto University
Living people
Academic staff of Nagoya University |
https://en.wikipedia.org/wiki/Argument%20shift%20method | In mathematics, the argument shift method is a method for constructing functions in involution with respect to Poisson–Lie brackets, introduced by . They used it to prove that the Poisson algebra of a finite-dimensional semisimple Lie algebra contains a complete commuting set of polynomials.
References
English translation: Math. USSR-Izv. 12 (1978), no. 2, 371–389
Lie algebras |
https://en.wikipedia.org/wiki/Interacting%20particle%20system | In probability theory, an interacting particle system (IPS) is a stochastic process on some configuration space given by a site space, a countably-infinite-order graph and a local state space, a compact metric space . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata.
Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model.
IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates where is a finite set of sites and with for all . The rates describe exponential waiting times of the process to jump from configuration into configuration . More generally the transition rates are given in form of a finite measure on .
The generator of an IPS has the following form. First, the domain of is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space . Then for any observable in the domain of , one has
.
For example, for the stochastic Ising model we have , , if for some and
where is the configuration equal to except it is flipped at site . is a new parameter modeling the inverse temperature.
The Voter model
The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform.
Discrete time process
In the discrete time voter model in one dimension, represents the state of particle at time . Informally each individual is arranged on a line and can "see" other individuals that are within a radius, . If more than a certain proportion, of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value such that if most individuals never change, and for in the limit most sites agree. (Both of these results assume the probability of is one half.)
This process has a natural generalization to more dimensions, some results for this are discussed in Durrett and Steif (1993).
Continuous time process
The contin |
https://en.wikipedia.org/wiki/Tanque%20%28footballer%29 | Tanque (born 18 July 1991 in Goiânia) is a Hungarian football player who plays for Zalaegerszegi TE.
Club statistics
External links
Profile
1991 births
Living people
Footballers from Goiânia
Brazilian men's footballers
Men's association football forwards
Egri FC players
Zalaegerszegi TE players
Nemzeti Bajnokság I players
Brazilian expatriate men's footballers
Expatriate men's footballers in Hungary
Brazilian expatriate sportspeople in Hungary |
https://en.wikipedia.org/wiki/Astrostatistics | Astrostatistics is a discipline which spans astrophysics, statistical analysis and data mining. It is used to process the vast amount of data produced by automated scanning of the cosmos, to characterize complex datasets, and to link astronomical data to astrophysical theory. Many branches of statistics are involved in astronomical analysis including nonparametrics, multivariate regression and multivariate classification, time series analysis, and especially Bayesian inference. The field is closely related to astroinformatics.
Professional association
Practitioners are represented by the International Astrostatistics Association affiliated with the International Statistical Institute, the International Astronomical Union Working Group in Astrostatistics and Astroinformatics, the American Astronomical Society Working Group in Astroinformatics and Astrostatistics, the American Statistical Association Interest Group in Astrostatistics, and the Cosmostatistics Initiative. All of these organizations participate in the Astrostatistics and Astroinformatics Portal Web site.
References
Astrophysics
Applied statistics
Data mining
Machine learning |
https://en.wikipedia.org/wiki/Botond%20Kir%C3%A1ly | Botond Király (born 26 October 1994) is a Hungarian professional footballer who plays for Pápa.
Club statistics
Updated to games played as of 6 December 2014.
References
1994 births
People from Pápa
Footballers from Veszprém County
Living people
Hungarian men's footballers
Men's association football midfielders
Pápai FC footballers
Rákosmenti KSK players
Vasas SC players
Aqvital FC Csákvár players
Győri ETO FC players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/India%E2%80%93Malaysia%20field%20hockey%20record | India and Malaysia have played 125 hockey matches out of which 87 have been won by India and 17 have been won by Malaysia. The remaining 21 matches are draws.
Statistics
Major Tournament matches
The following table show India vs Malaysia in major tournaments and their finishing in the tournament:
See also
Indian field hockey team in Malaya and Singapore
References
AZLAN SHAH 2000 HIGHLIGHTS
India vs Malaysia (bharatiyahockey.org)
M
Field hockey in Malaysia |
https://en.wikipedia.org/wiki/Gelfand%E2%80%93Zeitlin%20integrable%20system | In mathematics, the Gelfand–Zeitlin system (also written Gelfand–Zetlin system, Gelfand–Cetlin system, Gelfand–Tsetlin system) is an integrable system on conjugacy classes of Hermitian matrices. It was introduced by , who named it after the Gelfand–Zeitlin basis, an early example of canonical basis, introduced by I. M. Gelfand and M. L. Cetlin in 1950s. introduced a complex version of this integrable system.
References
External links
http://ncatlab.org/nlab/show/Gelfand-Tsetlin+basis
Integrable systems |
https://en.wikipedia.org/wiki/North%20Shore%20%28Nova%20Scotia%29 | The North Shore is a region of Nova Scotia, Canada. Although it has no formal identity and is variously defined by geographic, county and other political boundaries, it is defined by Statistics Canada as an economic region, composed of Antigonish County, Colchester County, Cumberland County, Guysborough County, and Pictou County.
References
Geography of Cumberland County, Nova Scotia
Geography of Antigonish County, Nova Scotia
Geography of Guysborough County, Nova Scotia
Geography of Pictou County
Region of Queens Municipality
Geography of Colchester County
Geographic regions of Nova Scotia |
https://en.wikipedia.org/wiki/Binary%20Goppa%20code | In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups.
Construction and properties
An irreducible binary Goppa code is defined by a polynomial of degree over a finite field with no repeated roots, and a sequence of distinct elements from that are not roots of .
Codewords belong to the kernel of the syndrome function, forming a subspace of :
The code defined by a tuple has dimension at least and
distance at least , thus it can encode messages of length at least using codewords of size while correcting at least errors. It possesses a convenient parity-check matrix in form
Note that this form of the parity-check matrix, being composed of a Vandermonde matrix and diagonal matrix , shares the form with check matrices of alternant codes, thus alternant decoders can be used on this form. Such decoders usually provide only limited error-correcting capability (in most cases ).
For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the -by- matrix over to a -by- binary matrix by writing polynomial coefficients of elements on successive rows.
Decoding
Decoding of binary Goppa codes is traditionally done by Patterson algorithm, which gives good error-correcting capability (it corrects all design errors), and is also fairly simple to implement.
Patterson algorithm converts a syndrome to a vector of errors. The syndrome of a binary word is expected to take a form of
Alternative form of a parity-check matrix based on formula for can be used to produce such syndrome with a simple matrix multiplication.
The algorithm then computes . That fails when , but that is the case when the input word is a codeword, so no error correction is necessary.
is reduced to polynomials and using the extended euclidean algorithm, so that , while and .
Finally, the error locator polynomial is computed as . Note that in binary case, locating the errors is sufficient to correct them, as there's only one other value possible. In non-binary cases a separate error correction polynomial has to be computed as well.
If the original codeword was decodable and the was the binary error vector, then
Factoring or evaluating all roots of therefore gives enough information to recover the error vector and fix the errors.
Properties and usage
Binary Goppa codes viewed as a special case of Goppa codes have the interesting property that they correct full errors, while only errors in ternary and all other cases. Asympt |
https://en.wikipedia.org/wiki/Q-expansion%20principle | In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients in M. It was introduced by .
References
Modular forms |
https://en.wikipedia.org/wiki/WIRIS | WIRIS is a company, legally registered as Maths for More, providing a set of proprietary HTML-based JavaScript tools which can author and edit mathematical formulas, execute mathematical problems and show mathematical graphics on the Cartesian coordinate system.
WIRIS equation editor is a native browser application, with a light server-side, that supports both MathML and LaTeX. Since 2017, after buying Design Science, a US-based a developer of MathType desktop software, WIRIS rebranded their web equation editor as MathType by WIRIS.
WIRIS is based in Barcelona, Spain and was founded by teachers and former students from the Technical University of Catalonia (Barcelona Tech) coordinated by Professor Sebastià Xambó.
References
External links
Interactive geometry software
Mathematical software
Formula editors
Formula manipulation languages
Computer algebra systems
Cross-platform software |
https://en.wikipedia.org/wiki/Koszul%20cohomology | In mathematics, the Koszul cohomology groups are groups associated to a projective variety X with a line bundle L. They were introduced by , and named after Jean-Louis Koszul as they are closely related to the Koszul complex.
surveys early work on Koszul cohomology, gives an introduction to Koszul cohomology, and gives a more advanced survey.
Definitions
If M is a graded module over the symmetric algebra of a vector space V, then the Koszul cohomology of M is the cohomology of the sequence
If L is a line bundle over a projective variety X, then the Koszul cohomology is given by the Koszul cohomology of the graded module , viewed as a module over the symmetric algebra of the vector space .
References
Algebraic geometry
Cohomology theories |
https://en.wikipedia.org/wiki/Ostrowski%20numeration | In mathematics, Ostrowski numeration, named after Alexander Ostrowski, is either of two related numeration systems based on continued fractions: a non-standard positional numeral system for integers and a non-integer representation of real numbers.
Fix a positive irrational number α with continued fraction expansion [a0; a1, a2, ...]. Let (qn) be the sequence of denominators of the convergents pn/qn to α: so qn = anqn−1 + qn−2. Let αn denote Tn(α) where T is the Gauss map T(x) = {1/x}, and write βn = (−1)n+1 α0 α1 ... αn: we have βn = anβn−1 + βn−2.
Real number representations
Every positive real x can be written as
where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.
Integer representations
Every positive integer N can be written uniquely as
where the integer coefficients 0 ≤ bn ≤ an and if bn = an then bn−1 = 0.
If α is the golden ratio, then all the partial quotients an are equal to 1, the denominators qn are the Fibonacci numbers and we recover Zeckendorf's theorem on the Fibonacci representation of positive integers as a sum of distinct non-consecutive Fibonacci numbers.
See also
Complete sequence
References
.
Non-standard positional numeral systems |
https://en.wikipedia.org/wiki/M/D/1%20queue | In queueing theory, a discipline within the mathematical theory of probability, an M/D/1 queue represents the queue length in a system having a single server, where arrivals are determined by a Poisson process and job service times are fixed (deterministic). The model name is written in Kendall's notation. Agner Krarup Erlang first published on this model in 1909, starting the subject of queueing theory. An extension of this model with more than one server is the M/D/c queue.
Model definition
An M/D/1 queue is a stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of entities in the system, including any currently in service.
Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1.
Service times are deterministic time D (serving at rate μ = 1/D).
A single server serves entities one at a time from the front of the queue, according to a first-come, first-served discipline. When the service is complete the entity leaves the queue and the number of entities in the system reduces by one.
The buffer is of infinite size, so there is no limit on the number of entities it can contain.
Transition Matrix
The transition probability matrix for an M/D/1 queue with arrival rate λ and service time 1, such that λ <1 (for stability of the queue) is given by P as below:
， , n = 0,1,....
Classic performance metrics
The following expressions present the classic performance metrics of a single server queuing system such as M/D/1, with:
arrival rate ,
service rate , and
utilization
The average number of entities in the system, L is given by:
The average number of entities in the queue (line), LQ is given by:
The average waiting time in the system, ω is given by:
The average waiting time in the queue (line), ω Q is given by:
Example
Considering a system that has only one server, with an arrival rate of 20 entities per hour and the service rate is at a constant of 30 per hour.
So the utilization of the server is: ρ=20/30=2/3. Using the metrics shown above, the results are as following: 1) Average number in line LQ= 0.6667; 2) Average number in system L =1.333; 3) Average time in line ωQ = 0.033 hour; 4) Average time in system ω = 0.067 hour.
Relations for Mean Waiting Time in M/M/1 and M/D/1 queues
For an equilibrium M/G/1 queue, the expected value of the time W spent by a customer in the queue are given by Pollaczek-Khintchine formula as below:
where τ is the mean service time; σ2 is the variance of service time; and ρ=λτ < 1, λ being the arrival rate of the customers.
For M/M/1 queue, the service times are exponentially distributed, then σ2 = τ2 and the mean waiting time in the queue denoted by WM is given by the following equation:
Using this, the corresponding equation for M/D/1 queue can be derived, assuming constant service times. Then the variance of service time becomes zero, i.e. σ2 = 0. The mean waiting time in the M/D/1 queue d |
https://en.wikipedia.org/wiki/List%20of%20IF%20Elfsborg%20records%20and%20statistics | IF Elfsborg is a Swedish professional football club based in Borås. In 2012 Elfsborg, played their 69th season in Allsvenskan from its inception in 1924 up to and including the 2012 season. This placing them on a 5th place of those teams who have participated in total seasons. They have also played top flight football continually in Sweden since 1997, which currently is the 3rd longest top flight tenure of any club in Sweden—the longest one is IFK Göteborg, since 1977.
The club is currently placed 5th in the all-time Allsvenskan table, "maratontabellen" in Swedish, which is a cumulative record of all match results, points, and goals of every team that has played in Allsvenskan since its inception in 1924–25. Furthermore, Elfsborg is placed 6th in the all-time medal table, with a total of 24 medals of different value – the most recent received in 2012.
The player in IF Elfsborg who holds most club and national records is Sven Jonasson. In 409 games, most appearances ever in Elfsborg, he scored a total of 252 goals in Allsvenskan, which makes it an all-time record. He played in Elfsborg throughout his career, which started in 1927 and ended 1947, a total of 20 years. The most remarkable achievement is his unbroken record of 344 consecutive games. A sequence that was broken when he missed his first game in 14 years because he failed to receive furlough during his military service. He was also the first goalscorer ever for the Sweden national football team in a World Cup. It was in the 1934 FIFA World Cup, where he scored two goals in Sweden's 3–2 victory against Argentina.
This article is about the statistics and records of the football section of IF Elfsborg.
Honours
Domestic
Swedish Champions:
Winners (6): 1935–36, 1938–39, 1939–40, 1961, 2006, 2012
League
Allsvenskan:
Winners (6): 1935–36, 1938–39, 1939–40, 1961, 2006, 2012
Runners-up (6): 1942–43, 1943–44, 1944–45, 1965, 1977, 2008
Division 1 Södra:
Winners (1): 1996
Runners-up (1): 1991
Division 2 Götaland:
Winners (1): 1960
Runners-up (4): 1955–56, 1956–57, 1957–58, 1959
Division 2 Västsvenska Serien:
Winners (1): 1925–26
Runners-up (2): 1914–15, 1923–24
Cups
Svenska Cupen:
Winners (2): 2001, 2003
Runners-up (3): 1942, 1980–81, 1996–97
Svenska Supercupen:
Winners (1): 2007
European
UEFA Intertoto Cup:
Winners (2): 1980, 2008
Runners-up (1): 1961
The Atlantic Cup:
Winners (1): 2011
Player records
Appearances
Most Allsvenskan appearances: Sven Jonasson, 409
Most consecutive appearances: Sven Jonasson, 344 (1927–41)
Most appearances
Competitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored.
Goalscorers
Most Allsvenskan goals: Sven Jonasson, 252
Top goalscorers
Competitive matches only. Numbers in brackets indicate appearances made.
International
Most capped Elfsborg player for Sweden while playing for the club: Anders Svensson, 89 caps whilst an Elfsborg player
Club records
Matches
Biggest victory, Svenska cupen: 1 |
https://en.wikipedia.org/wiki/Hermitian%20Yang%E2%80%93Mills%20connection | In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons.
The Kobayashi–Hitchin correspondence proved by Donaldson, Uhlenbeck and Yau asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is slope polystable.
Hermitian Yang–Mills equations
Hermite-Einstein connections arise as solutions of the Hermitian Yang-Mills equations. These are a system of partial differential equations on a vector bundle over a Kähler manifold, which imply the Yang-Mills equations. Let be a Hermitian connection on a Hermitian vector bundle over a Kähler manifold of dimension . Then the Hermitian Yang-Mills equations are
for some constant . Here we have
Notice that since is assumed to be a Hermitian connection, the curvature is skew-Hermitian, and so implies .
When the underlying Kähler manifold is compact, may be computed using Chern-Weil theory. Namely, we have
Since and the identity endomorphism has trace given by the rank of , we obtain
where is the slope of the vector bundle , given by
and the volume of is taken with respect to the volume form .
Due to the similarity of the second condition in the Hermitian Yang-Mills equations with the equations for an Einstein metric, solutions of the Hermitian Yang-Mills equations are often called Hermite-Einstein connections, as well as Hermitian Yang-Mills connections.
Examples
The Levi-Civita connection of a Kähler–Einstein metric is Hermite-Einstein with respect to the Kähler-Einstein metric. (These examples are however dangerously misleading, because there are compact Einstein manifolds, such as the Page metric on , that are Hermitian, but for which the Levi-Civita connection is not Hermite-Einstein.)
When the Hermitian vector bundle has a holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection. For the Chern connection, the condition that is automatically satisfied. The Hitchin-Kobayashi correspondence asserts that a holomorphic vector bundle admits a Hermitian metric such that the associated Chern connection satisfies the Hermitian Yang-Mills equations if and only if the vector bundle is polystable. From this perspective, the Hermitian Yang-Mills equations can be seen as a system of equations for the metric rather than the associated Chern connection, and such metrics solving the equations are called Hermite-Einstein metrics.
The Hermite-Einstein condition on Chern connections was first |
https://en.wikipedia.org/wiki/Trace%20inequality | In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.
Basic definitions
Let denote the space of Hermitian matrices, denote the set consisting of positive semi-definite Hermitian matrices and denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.
For any real-valued function on an interval one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as
given the spectral decomposition
Operator monotone
A function defined on an interval is said to be operator monotone if for all and all with eigenvalues in the following holds,
where the inequality means that the operator is positive semi-definite. One may check that is, in fact, not operator monotone!
Operator convex
A function is said to be operator convex if for all and all with eigenvalues in and , the following holds
Note that the operator has eigenvalues in since and have eigenvalues in
A function is if is operator convex;=, that is, the inequality above for is reversed.
Joint convexity
A function defined on intervals is said to be if for all and all
with eigenvalues in and all with eigenvalues in and any the following holds
A function is if − is jointly convex, i.e. the inequality above for is reversed.
Trace function
Given a function the associated trace function on is given by
where has eigenvalues and stands for a trace of the operator.
Convexity and monotonicity of the trace function
Let : ℝ → ℝ be continuous, and let be any integer. Then, if is monotone increasing, so
is on Hn.
Likewise, if is convex, so is on Hn, and
it is strictly convex if is strictly convex.
See proof and discussion in, for example.
Löwner–Heinz theorem
For , the function is operator monotone and operator concave.
For , the function is operator monotone and operator concave.
For , the function is operator convex. Furthermore,
is operator concave and operator monotone, while
is operator convex.
The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for to be operator monotone. An elementary proof of the theorem is discussed in and a more general version of it in.
Klein's inequality
For all Hermitian × matrices and and all differentiable convex functions
: ℝ → ℝ with derivative , or for all positive-definite Hermitian × matrices and , and all differentiable
convex functions :(0,∞) → ℝ, the following inequality holds,
In either case, if is strictly convex, equality holds if and only if = .
A popular choice in applications is , see below.
Proof
Let so |
https://en.wikipedia.org/wiki/Kobayashi%E2%80%93Hitchin%20correspondence | In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same.
This was proven by Simon Donaldson for projective algebraic surfaces and later for projective algebraic manifolds, by Karen Uhlenbeck and Shing-Tung Yau for compact Kähler manifolds, and independently by Buchdahl for non-Kahler compact surfaces, and by Jun Li and Yau for arbitrary compact complex manifolds.
The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem concerned with the case of compact Riemann surfaces, and has been influential in the development of differential geometry, algebraic geometry, and gauge theory since the 1980s. In particular the Hitchin–Kobayashi correspondence inspired conjectures leading to the nonabelian Hodge correspondence for Higgs bundles, as well as the Yau–Tian–Donaldson conjecture about the existence of Kähler–Einstein metrics on Fano varieties, and the Thomas–Yau conjecture about existence of special Lagrangians inside isotopy classes of Lagrangian submanifolds of a Calabi–Yau manifold.
History
In 1965, M. S. Narasimhan and C. S. Seshadri proved the Narasimhan–Seshadri theorem, which relates stable holomorphic (or algebraic) vector bundles over compact Riemann surfaces (or non-singular projective algebraic curves), to projective unitary representations of the fundamental group of the Riemann surface. It was realised in the 1970s by Michael Atiyah, Raoul Bott, Hitchin and others that such representation theory of the fundamental group could be understood in terms of Yang–Mills connections, notions arising out of then-contemporary mathematical physics. Inspired by the Narasimhan–Seshadri theorem, around this time a folklore conjecture formed that slope polystable vector bundles admit Hermitian Yang–Mills connections. This is partially due to the argument of Fedor Bogomolov and the success of Yau's work on constructing global geometric structures in Kähler geometry. This conjecture was first shared explicitly by Kobayashi and Hitchin independently in the early 1980s.
The explicit relationship between Yang–Mills connections and stable vector bundles was made concrete in the early 1980s. A direct correspondence when the dimension of the base complex manifold is one was explained in the work of Atiyah and Bott in 1982 on the Yang–Mills equations over compact Riemann surfaces, and in Donaldson's new proof of the Narasimhan–Seshadri theorem from the perspective of gauge theory in 1983. In that setting, a Hermitian Yang–Mills connection could be simply understood as a (projectively) flat connection over the Riem |
https://en.wikipedia.org/wiki/Ramanujan%27s%20ternary%20quadratic%20form | In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression with integral values for x, y and z. Srinivasa Ramanujan considered this expression in a footnote in a paper published in 1916 and briefly discussed the representability of integers in this form. After giving necessary and sufficient conditions that an integer cannot be represented in the form for certain specific values of a, b and c, Ramanujan observed in a footnote: "(These) results may tempt us to suppose that there are similar simple results for the form whatever are the values of a, b and c. It appears, however, that in most cases there are no such simple results." To substantiate this observation, Ramanujan discussed the form which is now referred to as Ramanujan's ternary quadratic form.
Properties discovered by Ramanujan
In his 1916 paper Ramanujan made the following observations about the form .
The even numbers that are not of the form are 4λ(16μ + 6).
The odd numbers that are not of the form , viz. do not seem to obey any simple law.
Odd numbers beyond 391
By putting an ellipsis at the end of the list of odd numbers not representable as x2 + y2 + 10z2, Ramanujan indicated that his list was incomplete. It was not clear whether Ramanujan intended it to be a finite list or infinite list. This prompted others to look for such odd numbers. In 1927, Burton W. Jones and Gordon Pall discovered that the number 679 could not be expressed in the form and they also verified that there were no other such numbers below 2000. This led to an early conjecture that the seventeen numbers – the sixteen numbers in Ramanujan's list and the number discovered by them – were the only odd numbers not representable as . However, in 1941, H Gupta showed that the number 2719 could not be represented as . He also verified that there were no other such numbers below 20000. Further progress in this direction took place only after the development of modern computers. W. Galway wrote a computer program to determine odd integers not expressible as . Galway verified that there are only eighteen numbers less than 2 × 1010 not representable in the form . Based on Galway's computations, Ken Ono and K. Soundararajan formulated the following conjecture:
The odd positive integers which are not of the form x2 + are: .
Some known results
The conjecture of Ken Ono and Soundararajan has not been fully resolved. However, besides the results enunciated by Ramanujan, a few more general results about the form have been established. The proofs of some of them are quite simple while those of the others involve quite complicated concepts and arguments.
Every integer of the form 10n + 5 is represented by Ramanujan's ternary quadratic form.
If n is an odd integer which is not square-free then it can be represented in the form .
There are only a finite number of odd integers which cannot be represented in the form x2 + y2 + 10z2.
If the generalized Riemann hypothesi |
https://en.wikipedia.org/wiki/Constant%20scalar%20curvature%20K%C3%A4hler%20metric | In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is Kähler–Einstein metric, and a more general case is extremal Kähler metric.
, Tian and Yau conjectured that the existence of a cscK metric on a polarised projective manifold is equivalent to the polarised manifold being K-polystable. Recent developments in the field suggest that the correct equivalence may be to the polarised manifold being uniformly K-polystable . When the polarisation is given by the (anti)-canonical line bundle (i.e. in the case of Fano or Calabi–Yau manifolds) the notions of K-stability and K-polystability coincide, cscK metrics are precisely Kähler-Einstein metrics and the Yau-Tian-Donaldson conjecture is known to hold .
Extremal Kähler metrics
Constant scalar curvature Kähler metrics are specific examples of a more general notion of canonical metric on Kähler manifolds, extremal Kähler metrics. Extremal metrics, as the name suggests, extremise a certain functional on the space of Kähler metrics, the Calabi functional, introduced by Calabi.
Calabi functional
The Calabi functional is a functional defined on the space of Kähler potentials in a specific Kähler de Rham cohomology class on a compact Kähler manifold. Namely, let be a Kähler class on a compact Kähler manifold , and let be any Kähler metric in this class, which differs from by the potential . The Calabi functional is defined by
where is the scalar curvature of the associated Riemannian metric to and . This functional is essentially the norm squared of the scalar curvature for Kähler metrics in the Kähler class . Understanding the flow of this functional, the Calabi flow, is a key goal in understanding the existence of canonical Kähler metrics.
Extremal metrics
By definition, an extremal Kähler metric is a critical point of the Calabi functional., either local or global minimizers. In this sense extremal Kähler metrics can be seen as the best or canonical choice of Kähler metric on any compact Kähler manifold.
Constant scalar curvature Kähler metrics are examples of extremal Kähler metrics which are absolute minimizers of the Calabi functional. In this sense the Calabi functional is similar to the Yang–Mills functional and extremal metrics are similar to Yang–Mills connections. The role of constant scalar curvature metrics are played by certain absolute minimizers of the Yang–Mills functional, anti-self dual connections or Hermitian Yang–Mills connections.
In some circumstances constant scalar curvature Kähler metrics may not exist on a compact Kähler manifold, but extremal metrics may still exist. For example some manifolds may admit Kähler–Ricci solitons, which are examples of extremal Kähler metrics, and explicit extremal metrics can be constructed in the case of surfaces.
The absolute minimizers of the Calabi functional, the constant scalar curv |
https://en.wikipedia.org/wiki/Rice%27s%20formula | In probability theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u. Adler and Taylor describe the result as "one of the most important results in the applications of smooth stochastic processes." The formula is often used in engineering.
History
The formula was published by Stephen O. Rice in 1944, having previously been discussed in his 1936 note entitled "Singing Transmission Lines."
Formula
Write Du for the number of times the ergodic stationary stochastic process x(t) takes the value u in a unit of time (i.e. t ∈ [0,1]). Then Rice's formula states that
where p(x,x') is the joint probability density of the x(t) and its mean-square derivative x'''(t).
If the process x(t) is a Gaussian process and u = 0 then the formula simplifies significantly to give
where ρ'' is the second derivative of the normalised autocorrelation of x(t) at 0.
Uses
Rice's formula can be used to approximate an excursion probability
as for large values of u'' the probability that there is a level crossing is approximately the probability of reaching that level.
References
Ergodic theory |
https://en.wikipedia.org/wiki/P%C3%A9ter%20Kiss%20%28mathematician%29 | Péter Kiss ( – ) was a Hungarian mathematician, Doctor of Mathematics, and professor of mathematics at Eszterházy Károly College, who specialized in number theory. In 1992 he won the Albert Szent-Györgyi Prize for his achievements.
Life
He was born in Nagyréde, Hungary, in 1937.
He majored in Mathematics and Physics from Eötvös Loránd University. After graduation, he taught mathematics at Gárdonyi Géza Secondary School in Eger. In 1971 he was appointed to Teacher's College, and in 1972 he began teaching at the Department of Mathematics of Eszterházy Károly University.
He earned the Doctorate of Mathematics degree from the Hungarian Academy of Sciences in 1999.
He was the doctoral advisor for mathematicians like Ferenc Mátyás, Sándor Molnár, Béla Zay, Kálman Liptai, László Szalay. He also assisted other colleagues like Bui Minh Phong, Lászlo Gerőcs, and Pham Van Chung, in the writing of their dissertations.
He was a member of the János Bolyai Mathematical Society, where he held different positions.
Many of his academic papers have been published in the zbMATH database in the Periodica Mathematica Hungarica, in the Proceedings of the Japan Academy, Series A, in Mathematics of Computation, in the Fibonacci Quarterly, and in the American Mathematical Society journals.
Academic papers
References
20th-century Hungarian mathematicians
1937 births
2002 deaths
Number theorists |
https://en.wikipedia.org/wiki/Coxeter%20matroid | In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group W and a parabolic subgroup P. Ordinary matroids correspond to the case when P is a maximal parabolic subgroup of a symmetric group W. They were introduced by , who named them after H. S. M. Coxeter.
give a detailed account of Coxeter matroids.
Definition
Suppose that W is a Coxeter group, generated by a set S of involutions, and P is a parabolic subgroup (the subgroup generated by some subset of S). A Coxeter matroid is a subset M of W/P that for every w in W, M contains a unique minimal element with respect to the w-Bruhat order.
Relation to matroids
Suppose that the Coxeter group W is the symmetric group Sn and P is the parabolic subgroup Sk×Sn–k. Then W/P can be identified with the k-element subsets of the n-element set {1,2,...,n} and the elements w of W correspond to the linear orderings of this set. A Coxeter matroid consists of k elements sets such that for each w there is a unique minimal element in the corresponding Bruhat ordering of k-element subsets. This is exactly the definition of a matroid of rank k on an n-element set in terms of bases: a matroid can be defined as some k-element subsets called bases of an n-element set such that for each linear ordering of the set there is a unique minimal base in the Gale ordering of k-element subsets.
References
– English translation in Russian Mathematical Surveys 42 (1987), no. 2, 133–168
Matroid theory
Coxeter groups |
https://en.wikipedia.org/wiki/Kostant%27s%20convexity%20theorem | In mathematics, Kostant's convexity theorem, introduced by , states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set. It is a special case of a more general result for symmetric spaces. Kostant's theorem is a generalization of a result of , and for hermitian matrices. They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λn) is the convex polytope with vertices all permutations of the coordinates of Λ.
Kostant used this to generalize the Golden–Thompson inequality to all compact groups.
Compact Lie groups
Let K be a connected compact Lie group with maximal torus T and Weyl group W = NK(T)/T. Let their Lie algebras be and . Let P be the orthogonal projection of onto for some Ad-invariant inner product on . Then for X in , P(Ad(K)⋅X) is the convex polytope with vertices w(X) where w runs over the Weyl group.
Symmetric spaces
Let G be a compact Lie group and σ an involution with K a compact subgroup fixed by σ and containing the identity component of the fixed point subgroup of σ. Thus G/K is a symmetric space of compact type. Let and be their Lie algebras and let σ also denote the corresponding involution of . Let be the −1 eigenspace of σ and let be a maximal Abelian subspace. Let Q be the orthogonal projection of onto for some Ad(K)-invariant inner product on . Then for X in , Q(Ad(K)⋅X) is the convex polytope with vertices the w(X) where w runs over the restricted Weyl group (the normalizer of in K modulo its centralizer).
The case of a compact Lie group is the special case where G = K × K, K is embedded diagonally and σ is the automorphism of G interchanging the two factors.
Proof for a compact Lie group
Kostant's proof for symmetric spaces is given in . There is an elementary proof just for compact Lie groups using similar ideas, due to : it is based on a generalization of the Jacobi eigenvalue algorithm to compact Lie groups.
Let K be a connected compact Lie group with maximal torus T. For each positive root α there is a homomorphism of SU(2) into K. A simple calculation with 2 by 2 matrices shows that if Y is in and k varies in this image of SU(2), then P(Ad(k)⋅Y) traces a straight line between P(Y) and its reflection in the root α. In particular the component in the α root space—its "α off-diagonal coordinate"—can be sent to 0. In performing this latter operation, the distance from P(Y) to P(Ad(k)⋅Y) is bounded above by size of the α off-diagonal coordinate of Y. Let m be the number of positive roots, half the dimension of K/T. Starting from an arbitrary Y1 take the largest off-diagonal coordinate and send it to zero to get Y2. Continue in this way, to get a sequence (Yn). Then
Thus P⊥(Yn) tends to 0 and
Hence Xn = P(Yn) is a Cauchy sequence, so tends to X in . Since Yn = P(Yn) ⊕ P⊥(Yn), Yn tends to X. On the other hand, Xn lies on the |
https://en.wikipedia.org/wiki/2012%20Swedish%20Football%20Division%202 | Statistics of Swedish football Division 2 for the 2012 season.
League standings
Norrland 2012
Norra Svealand 2012
Södra Svealand 2012
Östra Götaland 2012
Västra Götaland 2012
Södra Götaland 2012
Player of the year awards
Ever since 2003 the online bookmaker Unibet have given out awards at the end of the season to the best players in Division 2. The recipients are decided by a jury of sportsjournalists, coaches and football experts. The names highlighted in green won the overall national award.
References
Swedish Football Division 2 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/Excursion%20probability | In probability theory, an excursion probability is the probability that a stochastic process surpasses a given value in a fixed time period. It is the probability
Numerous approximation methods for the situation where u is large and f(t) is a Gaussian process have been proposed such as Rice's formula. First-excursion probabilities can be used to describe deflection to a critical point experienced by structures during "random loadings, such as earthquakes, strong gusts, hurricanes, etc."
References
Stochastic processes |
https://en.wikipedia.org/wiki/Ross%27s%20conjecture | In queueing theory, a discipline within the mathematical theory of probability, Ross's conjecture gives a lower bound for the average waiting-time experienced by a customer when arrivals to the queue do not follow the simplest model for random arrivals. It was proposed by Sheldon M. Ross in 1978 and proved in 1981 by Tomasz Rolski. Equality can be obtained in the bound; and the bound does not hold for finite buffer queues.
Bound
Ross's conjecture is a bound for the mean delay in a queue where arrivals are governed by a doubly stochastic Poisson process
or by a non-stationary Poisson process. The conjecture states that the average amount of time that a customer spends waiting in a queue is greater than or equal to
where S is the service time and λ is the average arrival rate (in the limit as the length of the time period increases).
References
Probabilistic inequalities
Queueing theory |
https://en.wikipedia.org/wiki/Quadratic-linear%20algebra | In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by . An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – [X, Y] = 0.
References
Algebra |
https://en.wikipedia.org/wiki/Joseph%20Dennis%20%28mathematician%29 | Joseph James Dennis (11 April 1905 in Gainesville, Florida – April 1977) was an African-American mathematician. He served as the chairman of the Clark College mathematics department from 1930 to 1974.
Dennis gained his B.A. from Clark College in 1929, and his M.A. from Northwestern University in 1935. He earned his Ph.D. at Northwestern University in 1944. This is the same year as two other African-American men earned a Ph.D in mathematics, Wade Ellis Sr. and Warren Hill Brothers (both from the University of North Western University). His thesis was "Some Points in the Theory of Positive Definite J-Fractions" (related to continued fractions), supervised by H. S. Wall. He was one of the first African Americans to earn a PhD.
A building, and a scholarship fund for junior and senior mathematics majors at Clark University, are named in J.J. Dennis' honor.
External links
http://www.math.buffalo.edu/mad/PEEPS/dennis_josephj.html
http://ufdc.ufl.edu/UF00073707/00005
Thesis by Sherese LaTrelle Williams on Dennis and his contributions to Clark University
References
1905 births
1977 deaths
20th-century American mathematicians
20th-century African-American academics
20th-century American academics
Northwestern University alumni
People from Gainesville, Florida |
https://en.wikipedia.org/wiki/Statistics%20Mauritius | Statistics Mauritius formerly known as the Central Statistics Office (CSO) is the national statistical agency of Mauritius. It operates under the aegis of the Ministry of Finance and Economic Development and is responsible for all statistical activities except for fisheries and health statistics which fall under the responsibility of the respective ministry.
According to the Statistics Act No. 38 of 2000, "Statistics Mauritius shall constitute the central statistical authority and depository of all officials statistics produced in Mauritius and as such, shall collect, compile, analyse and disseminate accurate, relevant, timely and high quality statistics and related information on social, demographic, economic and financial activities to serve the needs of public and private users." The headquarters of the statistics office is located in Port Louis.
See also
List of national and international statistical services
References
External links
Government agencies of Mauritius
Government agencies established in 1945
Mauritius |
https://en.wikipedia.org/wiki/Tate%20vector%20space | In mathematics, a Tate vector space is a vector space obtained from finite-dimensional vector spaces in a way that makes it possible to extend concepts such as dimension and determinant to an infinite-dimensional situation. Tate spaces were introduced by , who named them after John Tate.
Introduction
A typical example of a Tate vector space over a field k are the Laurent power series
It has two characteristic features:
as n grows, V is the union of its submodules , where denotes the power series ring. These submodules are referred to as lattices.
Even though each lattice is an infinite-dimensional vector space, the quotients of any individual lattices,
are finite-dimensional k-vector spaces.
Tate modules
Tate modules were introduced by to serve as a notion of infinite-dimensional vector bundles. For any ring R, Drinfeld defined elementary Tate modules to be topological R-modules of the form
where P and Q are projective R-modules (of possibly infinite rank) and * denotes the dual.
For a field, Tate vector spaces in this sense are equivalent to locally linearly compact vector spaces, a concept going back to Lefschetz. These are characterized by the property that they have a base of the topology consisting of commensurable sub-vector spaces.
Tate objects
Tate objects can be defined in the context of any exact category C. Briefly, an exact category is way to axiomatize certain features of short exact sequences. For example, the category of finite-dimensional k-vector spaces, or the category of finitely generated projective R-modules, for some ring R, is an exact category, with its usual notion of short exact sequences.
The extension of the above example to a more general situation is based on the following observation: there is an exact sequence
whose outer terms are an inverse limit and a direct limit, respectively, of finite-dimensional k-vector spaces
In general, for an exact category C, there is the category Pro(C) of pro-objects and the category Ind(C) of ind-objects. This construction can be iterated and yields an exact category Ind(Pro(C)). The category of elementary Tate objects
is defined to be the smallest subcategory of those Ind-Pro objects V such that there is a short exact sequence
where L is a pro-object and L' is an ind-object.
It can be shown that this condition on V is equivalent to that requiring for an ind-presentation
the quotients are in C (as opposed to Pro(C)).
The category Tate(C) of Tate objects is defined to be the closure under retracts (idempotent completion) of elementary Tate objects.
showed that Tate objects (for C the category of finitely generated projective R-modules, and subject to the condition that the indexing families of the Ind-Pro objects are countable) are equivalent to countably generated Tate R-modules in the sense of Drinfeld mentioned above.
Related notions and applications
A Tate Lie algebra is a Tate vector space with an additional Lie algebra structure. An example of a Tate |
https://en.wikipedia.org/wiki/Zero-inflated%20model | In statistics, a zero-inflated model is a statistical model based on a zero-inflated probability distribution, i.e. a distribution that allows for frequent zero-valued observations.
Introduction to Zero-Inflated Models
Zero-inflated models are commonly used in the analysis of count data, such as the number of visits a patient makes to the emergency room in one year, or the number of fish caught in one day in one lake. Count data can take values of 0, 1, 2, … (non-negative integer values). Other examples of count data are the number of hits recorded by a Geiger counter in one minute, patient days in the hospital, goals scored in a soccer game, and the number of episodes of hypoglycemia per year for a patient with diabetes.
For statistical analysis, the distribution of the counts is often represented using a Poisson distribution or a negative binomial distribution. Hilbe notes that "Poisson regression is traditionally conceived of as the basic count model upon which a variety of other count models are based." In a Poisson model, "… the random variable is the count response and parameter (lambda) is the mean. Often, is also called the rate or intensity parameter… In statistical literature, is also expressed as (mu) when referring to Poisson and traditional negative binomial models."
In some data, the number of zeros is greater than would be expected using a Poisson distribution or a negative binomial distribution. Data with such an excess of zero counts are described as Zero-inflated.
Example histograms of zero-inflated Poisson distributions with mean of 5 or 10 and proportion of zero inflation of 0.2 or 0.5 are shown below, based on the R program ZeroInflPoiDistPlots.R from Bilder and Laughlin.
Examples of Zero-inflated count data
Fish counts "… suppose we recorded the number of fish caught on various lakes in 4-hour fishing trips to Minnesota. Some lakes in Minnesota are too shallow for fish to survive the winter, so fishing in those lakes will yield no catch. On the other hand, even on a lake where fish are plentiful, we may or may not catch any fish due to conditions or our own competence. Thus, the number of fish caught will be zero if the lake does not support fish, and will be zero, one or more if it does."
Number of wisdom teeth extracted. The number of wisdom teeth that a person has had extracted can range from 0 to 4. Some individuals, about one-third of the population, do not have any wisdom teeth. For these individuals, the number of wisdom teeth extracted will always be zero. For other individuals, the number extracted will be between 0 and 4, where a 0 indicates that the subject has not yet, and may never, have any of their 4 wisdom teeth extracted.
Publications by PhD candidates. Long examined the number of publications by 915 doctoral candidates in biochemistry in the last three years of their PhD studies. The proportion of candidates with zero publications exceeded the number predicted by a Poisson model. "Long |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Isaac%20Newton | This is a list of things named after Sir Isaac Newton.
Science and mathematics
Newtonianism, the philosophical principle of applying Newton's methods in a variety of fields
Mathematics
Physics
Places
Schools
Artwork
Other
See also
Newtonian (disambiguation)
Newton
Newton
Named after |
https://en.wikipedia.org/wiki/Skorokhod%20problem | In probability theory, the Skorokhod problem is the problem of solving a stochastic differential equation with a reflecting boundary condition.
The problem is named after Anatoliy Skorokhod who first published the solution to a stochastic differential equation for a reflecting Brownian motion.
Problem statement
The classic version of the problem states that given a càdlàg process {X(t), t ≥ 0} and an M-matrix R, then stochastic processes {W(t), t ≥ 0} and {Z(t), t ≥ 0} are said to solve the Skorokhod problem if for all non-negative t values,
W(t) = X(t) + R Z(t) ≥ 0
Z(0) = 0 and dZ(t) ≥ 0
.
The matrix R is often known as the reflection matrix, W(t) as the reflected process and Z(t) as the regulator process.
See also
List of things named after Anatoliy Skorokhod
References
Stochastic calculus |
https://en.wikipedia.org/wiki/Lie-%2A%20algebra | In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld (), and are similar to the conformal algebras discussed by and to vertex Lie algebras.
References
Lie algebras |
https://en.wikipedia.org/wiki/Tutte%20homotopy%20theorem | In mathematics, the Tutte homotopy theorem, introduced by , generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
Statement
A matroid on a set Q is specified by a class of non-empty subsets M of Q, called circuits, such that no element of M contains another, and if X and Y are in M, a is in X and Y, b is in X but not in Y, then there is some Z in M containing b but not a and contained in X∪Y.
The subsets of Q that are unions of circuits are called flats (this is the language used in Tutte's original paper, however in modern usage the flats of a matroid mean something different). The elements of M are called 0-flats, the minimal non-empty flats that are not 0-flats are called 1-flats, the minimal nonempty flats that are not 0-flats or 1-flats are called 2-flats, and so on.
A path is a finite sequence of 0-flats such that any two consecutive elements of the path lie in some 1-flat.
An elementary path is one of the form (X,Y,X), or (X,Y,Z,X) with X,Y,Z all lying in some 2-flat.
Two paths P and Q such that the last 0-flat of P is the same as the first 0-flat of Q can be composed in the obvious way to give a path PQ.
Two paths are called homotopic if one can be obtained from the other by the operations of adding or removing elementary paths inside a path, in other words changing a path PR to PQR or vice versa, where Q is elementary.
A weak form of Tutte's homotopy theorem states that any closed path is homotopic to the trivial path. A stronger form states a similar result for paths not meeting certain "convex" subsets.
References
Matroid theory |
https://en.wikipedia.org/wiki/Altruism%20theory%20of%20voting | The altruism theory of voting is a model of voter behavior which states that if citizens in a democracy have "social" preferences for the welfare of others, the extremely low probability of a single vote determining an election will be outweighed by the large cumulative benefits society will receive from the voter's preferred policy being enacted, such that it is rational for an “altruistic” citizen, who receives utility from helping others, to vote. Altruistic voting has been compared to purchasing a lottery ticket, in which the probability of winning is extremely low but the payoff is large enough that the expected benefit outweighs the cost.
Since the failure of standard rational choice models—which assume voters have "selfish" preferences—to explain voter turnout in large elections, public choice economists and social scientists have increasingly turned to altruism as a way to explain why rational individuals would choose to vote despite its apparent lack of individual benefit, explaining the paradox of voting. The theory suggests that individual voters do, in fact, derive personal utility from influencing the outcome of elections in favor of the candidate that they believe will implement policies for the greater good of the entire population.
The rational calculus of voting
The "selfish" rationale for voting
The standard model of voter calculus was articulated by Riker and Ordeshook in their 1968 article "A Theory of the Calculus of Voting" in The American Political Science Review. The basic utility hypothesis for the calculus of voting they gave was:
R = (BP) − C
Where B is the expected differential utility a voter personally receives from his preferred candidate winning; P is the probability of the voter bringing about B (that is, turning the election for his preferred candidate); C is the individual's cost of voting in the election; and R is the individual's expected reward from voting. If R > 0, then the expected utility of voting outweighs its costs, and it is reasonable to vote. But if R ≤ 0, the costs outweigh the benefits and a strictly rational individual would not be expected to vote.
Because P, the probability of any one vote determining the outcome, is extremely small for any large election, the expected benefits of voting under the traditional rational choice model is always roughly equal to zero. This leads to the so-called paradox of voting, in which rational choice models of voter behavior predict tiny turnouts which simply do not occur. In all democracies, voter turnout exceeds what the basic rational choice models predict.
Expressive versus instrumental voting
Because simple selfishness cannot explain why large numbers of people consistently choose to vote, Riker and Ordeshook introduced another term to the equation, D, to symbolize the personal or social benefits conferred by the act of voting itself, rather than by affecting the outcome of the election.
R = (BP) − C + D
This drew a distinction between expressiv |
https://en.wikipedia.org/wiki/Giuseppe%20Bruno | Giuseppe Bruno can refer to:
Giuseppe Bruno (mathematician) (1828–1893), Italian mathematician and professor of geometry.
Giuseppe Bruno (cardinal) (1875–1954), Italian cardinal of the Catholic Church.
Giuseppe Bruno (photographer) (1836–1904), Italian photographer.
Giuseppe Bruno (photographer) (1926-1999), Italian photographer. |
https://en.wikipedia.org/wiki/Ge%20Jun | Ge Jun (; born in October 1964 in Nantong, Jiangsu), is the associate professor and master instructor of College of Mathematics and Computer Science of Nanjing Normal University. Ge took part in the composing and designing process of the mathematics paper of the National College Entrance Examination for several times. The papers of the year 2003, 2010 and 2012 were regarded as “extremely difficult” by examinees. As a result, he is called "the Emperor of Mathematics" () by netizens in China. Some media also use this nickname.
Composing of the mathematics papers
The 2003 mathematics examination paper composed by Ge Jun averaged only 68 (out of 150). Some teachers, students and parents believed that the paper of 2010 was more difficult than that of 2003 but actually the average score was 83 (out of 160). In 2012, he joined the composing process of the mathematics paper following the new curriculum standard and the paper was once more considered difficult by examinees.
Since 2011, whenever a difficult mathematics examination paper occurs in China, its composers will be human flesh searched by netizens. Ge was supposed to participate in the composing process of 2011 college entrance examination in Guangdong Province. The Guangdong Provincial Education Examination Authority denied the rumors later. According to the report of Modern Express, Ge Jun himself claimed that he did not participate in the work of Jiangsu or Guangdong.
Evaluations of the mathematics papers of college entrance examination composed by Ge Jun were mixed. For instance, experts believe that the paper was innovative, differentiated, and appropriate while many students and their parents were apparently dissatisfied because of the low marks examinees received. Professor Tu Rongbao (), an expert of the go-over group of mathematics entrance paper of Jiangsu believed the 2010 paper to have a wide coverage of knowledge, to meet the requirements of the exam instructions and to contain a lot of innovations.
Academic areas
Ge Jun mainly engaged in mathematics competitions, teaching theory of mathematics curriculum, network curriculum, school education and research. He has published more than 60 papers and edited more than 30 textbooks and works.
Ge Jun is the senior coach of Chinese Mathematical Olympiad, deputy director of the Centre for Teacher Training in Colleges and Universities of Jiangsu Province, executive director of Primary and Secondary School Science and Technology Education Association of Jiangsu Province, vice chairman of the Council of Mathematics Teaching in Middle Schools of Jiangsu Province, general secretary of Nanjing Institute of Mathematics.
Ge is currently the vice president of the College of teacher education of Nanjing Normal University. His main fields of research include methodology of mathematical thought, teaching theory of mathematics, curriculum evaluation, education of mathematics, mathematical problem solving and mathematical communication.
On 16 Novemb |
https://en.wikipedia.org/wiki/Juan%20Giuria | Juan Giuria (1880-1957) was a Uruguayan architect and architectural historian.
Biography
He was a student of the old Faculty of Mathematics of Montevideo, where he obtained his degree in Architecture. He devoted himself to lecturing and investigation. He was one of the founders of the Institute of Architectural History; among his collaborators were Aurelio Lucchini and Elzeario Boix.
Projects
Pavilion of Hospital Pereira Rossell (1915)
Hospital Pedro Visca (now: Facultad de Ciencias Económicas y Administración, 1923)
Selected works
La obra arquitectónica hecha por los maestros jesuitas Andrés Blanqui y Juan Prímoli, Revista de la Sociedad de Amigos de la Arqueología, Tomo X, Montevideo, 1947.
La arquitectura en el Paraguay, Instituto de Arte Americano e Investigaciones Estéticas, Buenos Aires, 1950.
Modalidades de la arquitectura colonial peruana, El Siglo Ilustrado, Montevideo, 1952.
La arquitectura en el Uruguay, Imprenta Universal, Montevideo, 1955-1958 (4 vol.).
References
1880 births
1957 deaths
Uruguayan people of Italian descent
University of the Republic (Uruguay) alumni
Academic staff of the University of the Republic (Uruguay)
Uruguayan architects
Uruguayan architectural historians |
https://en.wikipedia.org/wiki/List%20of%20FK%20Partizan%20records%20and%20statistics | Fudbalski klub Partizan is a Serbian professional association football club based in Belgrade, Serbia, who currently play in the Serbian SuperLiga. They have played at their current home ground, Partizan Stadium, since 1949.
This list include the major honours won by Partizan, records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Partizan players on the international stage.
The club's record appearance maker is Saša Ilić, who made 800 appearances. Stjepan Bobek is the club's record goalscorer, scoring 425 goals during his career in Partizan.
Honours
Domestic
National Championships – 27
Yugoslav First League
Winners (11): 1946–47, 1948–49, 1960–61, 1961–62, 1962–63, 1964–65, 1975–76, 1977–78, 1982–83, 1985–86, 1986–87
Runners-up (9): 1953–54, 1955–56, 1957–58, 1958–59, 1967–68, 1969–70, 1983–84, 1987–88, 1991–92
FR Yugoslavia First League/Serbia and Montenegro First League
Winners (8): 1992–93, 1993–94, 1995–96, 1996–97, 1998–99, 2001–02, 2002–03, 2004–05
Runners-up (5): 1994–95, 1999–2000, 2000–01, 2003–04, 2005–06
Serbian SuperLiga
Winners (8): 2007–08, 2008–09, 2009–10, 2010–11, 2011–12, 2012–13, 2014–15, 2016–17
Runners-up (3): 2006–07, 2013–14, 2015–16
National Cups – 16
Yugoslav Cup
Winners (6): 1946–47, 1952, 1953–54, 1956–57, 1988–89, 1991–92
Runners-up (4): 1947–48, 1958–59, 1959–60, 1978–79
FR Yugoslavia Cup/Serbia and Montenegro Cup
Winners (3): 1993–94, 1997–98, 2000–01
Runners-up (3): 1992–93, 1995–96, 1998–99
Serbian Cup
Winners (7): 2007–08, 2008–09, 2010–11, 2015–16, 2016–17, 2017–18, 2018–19
Runners-up (3): 2014–15, 2019–20 ,2020–21
Yugoslav Supercup – 1
Winners (1): 1989
European
European Cup/UEFA Champions League
Runners-up (1): 1965–66
Quarter–finals (2): 1955–56, 1963–64
Round of 16 (1): 1961–62
Group stage (2): 2003–04, 2010–11
UEFA Cup/UEFA Europa League
Third round/Round of 16 (4): 1974–75, 1984–85, 1990–91, 2004–05
European Cup Winners' Cup
Quarter–finals (1): 1989–90
UEFA Conference League
Round of 16 (1): 2021-22
Mitropa Cup
Winner (1): 1978
Friendly Tournaments
Trofeo Mohamed V (1): 1963
Torneo Pentagonal Internacional de la Ciudad de México (1): 1970
Torneo Pentagonal Internacional de la Ciudad de Bogotá (1): 1971
Trofeo Colombino de fútbol (1): 1976
Lunar New Year Cup (1): 1984
40th Anniversary FK Partizan (1): 1985
Uhrencup (1): 1989
Player records
Most appearances
Top goalscorers
All matches
Individual awards
Domestic
Yugoslavian First League top scorers
FR Yugoslavia First League top scorers/Serbia and Montenegro top scorers
Serbian SuperLiga top scorers
Serbian SuperLiga Team of the Season
2008–09 Mladen Božović, Ivan Stevanović, Nenad Đorđević, Ivan Obradović, Ljubomir Fejsa, Nemanja Tomić, Almami Moreira, Lamine Diarra
2009–10 Mladen Krstajić, Marko Lomić, Lju |
https://en.wikipedia.org/wiki/Denis%20Higgs | Denis A. Higgs ( – ) was a British mathematician, Doctor of Mathematics, and professor of mathematics who specialised in combinatorics, universal algebra, and category theory. He wrote one of the most influential papers in category theory entitled A category approach to boolean valued set theory, which introduced many students to topos theory.
He was a member of the National Committee of Liberation and was an outspoken critic against the apartheid in South Africa.
Life
He earned degrees from Cambridge University, St John's College, in England, University of the Witwatersrand in South Africa, and McMaster University in Canada.
In 1962, he became a member of the National Committee of Liberation, a movement whose main objective was to dismantle the apartheid in South Africa.
On 28 August 1964, he was kidnapped from his home in Lusaka, Zambia. Then South Africa's Justice Minister John Vorster, who later became Prime Minister, denied any involvement by either the South African government or the police, but accused Higgs of being an accessory to the bombing that killed Ethel Rhys and wounded many others.
On 1 September, an unidentified man who claimed to be part of British Protectorates called the Rand Daily Mail newspaper and gave specific details of Denis Higgs's whereabouts. On 2 September, police authorities found him. He was blindfolded and bound in a van over by the Zoo Park area. Although Higgs was wanted for the Johannesburg railway bombing that killed Rhys, authorities declined to prosecute since Higgs had been returned to South Africa via an extrajudicial kidnapping rather than proper extradition channels.
After first leaving South Africa to return to Zambia, on 6 September 1964, Higgs fled to London, accompanied by his family. He later stated that he feared for his safety and that of his family, since a day before his departure, the South African government had begun proceedings of extradition for his alleged participation in the explosion at the Johannesburg Railway Station that killed Rhys.
Career
He emigrated to Canada in 1966, earning a doctorate from McMaster, and held a position as a professor of Pure Mathematics at the University of Waterloo, where he wrote one of the most influential papers in category theory entitled A category approach to boolean valued set theory, which introduced many students to topos theory.
In 1973, he generalised the Rasiowa-Sikorski Boolean models to the case of category theory.
His academic papers were published in Algebra Universalis, the Journal of Pure and Applied Algebra, the Journal of the Australian Mathematical Society, the Journal of the London Mathematical Society, and Mathematics of Computation, among other journals.
He died on 25 February 2011.
Academic publications
References
20th-century British mathematicians
21st-century British mathematicians
South African mathematicians
1932 births
2011 deaths
Place of birth missing
Alumni of St John's College, Cambridge
University of the Witw |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20Bury%20F.C.%20season | During the 2012–13 season Bury competed in the third tier of English football, Football League One.
League table
First-team squad
Out on loan
Reserve squad
Squad statistics
Appearances and goals
|-
|colspan="14"|Players played for Bury this season who are no longer at the club:
|-
|colspan="14"|Players who played for Bury on loan and returned to their parent club:
|}
Top scorers
Disciplinary record
Results and fixtures
Pre-season friendlies
League One
bury 2 Fleetwood 0
FA Cup
League Cup
Football League Trophy
Transfers
Awards
References
Bury F.C. seasons
Bury |
https://en.wikipedia.org/wiki/Salih%20Zeki | Salih Zeki Bey (1864, Istanbul – 1921, Istanbul) was an Ottoman mathematician, astronomer and the founder of the mathematics, physics, and astronomy departments of Istanbul University.
He was sent by the Post and Telegraph Ministry to study electrical engineering at the École Polytechnique in Paris. He returned to Istanbul in 1887 and started working at the Ministry as an electrical engineer and inspector.
He was appointed as the director of the state observatory () (now Kandilli Observatory) after Coumbary in 1895.
In 1912, he became Under Secretary of the Ministry of Education and in 1913 the president of Istanbul University. In 1917, he resigned as the president but continued teaching at the University in the Faculty of Sciences until his death.
Works
Astronomy
New Cosmography
Abridged Cosmography
Physics
Hikmet-i Tabiiyye
Mebhas-ı Elektrik-i Miknatisi
Mebhas-ı Hararet-i Harekiye
History of science
Asar-ı Bakiye
Mathematics
Kamus-i Riyaziyat
Hendese-i Tahliliye
Hesab-i Ihtimali
References
Hüseyin Gazi Topdemir, "Salih Zeki" in The Biographical Encyclopedia of Astronomers, Thomas Hockey, Virginia Trimble, Thomas R. Williams, Katherine Bracher, Richard A. Jarrell, Jordan D. Marché, II, F. Jamil Ragep, JoAnn Palmeri, Marvin Bolt (Eds.), Springer Science+Business Media, LLC, 2007, pp. 1007-1008.
Salih Zeki Özel Sayısı (=Osmanlı Bilimi Araştırmaları, vol. 7, no. 1), 2005. (Special issue on Salih Zeki of the journal 'Studies in Ottoman Science', in Turkish.)
1864 births
1921 deaths
Scientists from the Ottoman Empire
Turkish mathematicians
Academic staff of Istanbul University
École Polytechnique alumni
Engineers from the Ottoman Empire
Astronomers from the Ottoman Empire |
https://en.wikipedia.org/wiki/Jacobson%E2%80%93Bourbaki%20theorem | In algebra, the Jacobson–Bourbaki theorem is a theorem used to extend Galois theory to field extensions that need not be separable. It was introduced by for commutative fields and extended to non-commutative fields by , and who credited the result to unpublished work by Nicolas Bourbaki. The extension of Galois theory to normal extensions is called the Jacobson–Bourbaki correspondence, which replaces the correspondence between some subfields of a field and some subgroups of a Galois group by a correspondence between some sub division rings of a division ring and some subalgebras of an associative algebra.
The Jacobson–Bourbaki theorem implies both the usual Galois correspondence for subfields of a Galois extension, and Jacobson's Galois correspondence for subfields of a purely inseparable extension of exponent at most 1.
Statement
Suppose that L is a division ring.
The Jacobson–Bourbaki theorem states that there is a natural 1:1 correspondence between:
Division rings K in L of finite index n (in other words L is a finite-dimensional left vector space over K).
Unital K-algebras of finite dimension n (as K-vector spaces) contained in the ring of endomorphisms of the additive group of K.
The sub division ring and the corresponding subalgebra are each other's commutants.
gave an extension to sub division rings that might have infinite index, which correspond to closed subalgebras in the finite topology.
References
Field (mathematics)
Theorems in algebra |
https://en.wikipedia.org/wiki/Bakul%20Kayastha | Bakul Kayastha (born c. 1400) was a mathematician from Kamrup. He was especially known for his masterpiece in the field of mathematics named Kitabat Manjari, written in 1434, and Lilavati.
Kitabat Manjari is a poetical treatise on arithmetic, surveying and bookkeeping. The book teaches how accounts are to be kept under different heads and how stores belonging to the royal treasury are to be classified and entered into a stock book.
The works of Bakul Kayastha were regarded as standards in his time to be followed by other Kayasthas in maintaining royal accounts.
See also
Bhattadeva
Hema Saraswati
References
15th-century births
1450s deaths
15th-century Indian mathematicians
Kamrupi people
Scientists from Assam
Scholars from Assam |
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