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https://en.wikipedia.org/wiki/Center%20manifold
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In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modelling.
Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold and in multiscale mathematics because the long time dynamics of the micro-scale often are attracted to a relatively simple center manifold involving the coarse scale variables.
Informal description
Saturn's rings capture much center-manifold geometry. Dust particles in the rings are subject to tidal forces, which act characteristically to "compress and stretch". The forces compress particle orbits into the rings, stretch particles along the rings, and ignore small shifts in ring radius. The compressing direction defines the stable manifold, the stretching direction defining the unstable manifold, and the neutral direction is the center manifold.
While geometrically accurate, one major difference distinguishes Saturn's rings from a physical center manifold. Like most dynamical systems, particles in the rings are governed by second-order laws. Understanding trajectories requires modeling position and a velocity/momentum variable, to give a tangent manifold structure called phase space. Physically speaking, the stable, unstable and neutral manifolds of Saturn's ring system do not divide up the coordinate space for a particle's position; they analogously divide up phase space instead.
The center manifold typically behaves as an extended collection of saddle points. Some position-velocity pairs are driven towards the center manifold, while others are flung away from it. Small perturbations that generally push them about randomly, and often push them out of the center manifold. There are, however, dramatic counterexamples to instability at the center manifold, called Lagrangian coherent structures. The entire unforced rigid body dynamics of a ball is a center manifold.
A much more sophisticated example is the Anosov flow on tangent bundles of Riemann surfaces. In that case, the tangent space splits very explicitly and precisely into three parts: the unstable and stable bundles, with the neutral manifold wedged between.
Definition
The center manifold of a dynamical system is based upon an equilibrium point of that system. A center manifold of the equilibrium then consists of those nearby orbits that neither decay nor grow exponentially quickly.
Mathematically, the first step when studying equilibrium points of dynamical systems is to linearize the system, and then compute its eigenvalues and eigenvectors. The eigenvectors (and generalized eigenvectors if they occur) corresponding to eigenvalues with negative real part form a basis for the stable eigenspace. The (generalized) eigenvectors corresponding to eigenvalues with positive real part form the unstable eig
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https://en.wikipedia.org/wiki/Balanced%20prime
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In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, given a prime number , where is its index in the ordered set of prime numbers,
For example, 53 is the sixteenth prime; the fifteenth and seventeenth primes, 47 and 59, add up to 106, and half of that is 53; thus 53 is a balanced prime.
Examples
The first few balanced primes are
5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903 .
Infinitude
It is conjectured that there are infinitely many balanced primes.
Three consecutive primes in arithmetic progression is sometimes called a CPAP-3. A balanced prime is by definition the second prime in a CPAP-3. the largest known CPAP-3 has 15004 digits and was found by Serge Batalov. It is:
The value of n (its rank in the sequence of all primes) is not known.
Generalization
The balanced primes may be generalized to the balanced primes of order n. A balanced prime of order n is a prime number that is equal to the arithmetic mean of the nearest n primes above and below. Algebraically, given a prime number , where k is its index in the ordered set of prime numbers,
Thus, an ordinary balanced prime is a balanced prime of order 1. The sequences of balanced primes of orders 2, 3, and 4 are given as sequences , , and in the OEIS respectively.
See also
Strong prime, a prime that is greater than the arithmetic mean of its two neighboring primes
Interprime, a composite number balanced between two prime neighbours
References
Classes of prime numbers
Unsolved problems in number theory
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https://en.wikipedia.org/wiki/Boolean%20domain
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In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or
The algebraic structure that naturally builds on a Boolean domain is the Boolean algebra with two elements. The initial object in the category of bounded lattices is a Boolean domain.
In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many programming languages do not have a Boolean datatype in the strict sense. In C or BASIC, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
Generalizations
The Boolean domain {0, 1} can be replaced by the unit interval , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with conjunction (AND) is replaced with multiplication (), and disjunction (OR) is defined via De Morgan's law to be .
Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
See also
Boolean-valued function
GF(2)
References
Further reading
(455 pages) (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2012-09-19/21.)
(480 pages) (NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)
(536 pages) (NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)
(vii+265+7 pages) (NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held in Bremen, Germany on 2018-09-19/21.)
(204 pages) (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held virtually on 2020-09-24/25.)
Boolean algebra
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https://en.wikipedia.org/wiki/Monoid%20%28category%20theory%29
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In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object M together with two morphisms
μ: M ⊗ M → M called multiplication,
η: I → M called unit,
such that the pentagon diagram
and the unitor diagram
commute. In the above notation, is the identity morphism of , is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C.
Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop.
Suppose that the monoidal category C has a symmetry γ. A monoid M in C is commutative when .
Examples
A monoid object in Set, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense.
A monoid object in Top, the category of topological spaces (with the monoidal structure induced by the product topology), is a topological monoid.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument.
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
A monoid object in (Ab, ⊗Z, Z), the category of abelian groups, is a ring.
For a commutative ring R, a monoid object in
(R-Mod, ⊗R, R), the category of modules over R, is a R-algebra.
the category of graded modules is a graded R-algebra.
the category of chain complexes of R-modules is a differential graded algebra.
A monoid object in K-Vect, the category of K-vector spaces (again, with the tensor product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra.
For any category C, the category [C,C] of its endofunctors has a monoidal structure induced by the composition and the identity functor IC. A monoid object in [C,C] is a monad on C.
For any category with a terminal object and finite products, every object becomes a comonoid object via the diagonal morphism . Dually in a category with an initial object and finite coproducts every object becomes a monoid object via .
Categories of monoids
Given two monoids (M, μ, η) and (M', μ', η') in a monoidal category C, a morphism f : M → M ' is a morphism of monoids when
f o μ = μ''' o (f ⊗ f),
f o η = η.
In other words, the following diagrams
,
commute.
The category of monoids in C and their monoid morphisms is written Mon'''C.
See also
Act-S, the category of monoids acting on sets
References
Monoidal categories
Objects (category theory)
Categories in category theory
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https://en.wikipedia.org/wiki/Archimedean%20principle
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Archimedean principle may refer to:
Archimedes' principle, a principle relating buoyancy with displacement
Archimedean property, a mathematical property of numbers and other algebraic structures
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https://en.wikipedia.org/wiki/Join%20and%20meet
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In mathematics, specifically order theory, the join of a subset of a partially ordered set is the supremum (least upper bound) of denoted and similarly, the meet of is the infimum (greatest lower bound), denoted In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are dual to one another with respect to order inversion.
A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.
The join/meet of a subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists.
If a subset of a partially ordered set is also an (upward) directed set, then its join (if it exists) is called a directed join or directed supremum. Dually, if is a downward directed set, then its meet (if it exists) is a directed meet or directed infimum.
Definitions
Partial order approach
Let be a set with a partial order and let An element of is called the (or or ) of and is denoted by if the following two conditions are satisfied:
(that is, is a lower bound of ).
For any if then (that is, is greater than or equal to any other lower bound of ).
The meet need not exist, either since the pair has no lower bound at all, or since none of the lower bounds is greater than all the others. However, if there is a meet of then it is unique, since if both are greatest lower bounds of then and thus If not all pairs of elements from have a meet, then the meet can still be seen as a partial binary operation on
If the meet does exist then it is denoted If all pairs of elements from have a meet, then the meet is a binary operation on and it is easy to see that this operation fulfills the following three conditions: For any elements
(commutativity),
(associativity), and
(idempotency).
Joins are defined dually with the join of if it exists, denoted by
An element of is the (or or ) of in if the following two conditions are satisfied:
(that is, is an upper bound of ).
For any if then (that is, is less than or equal to any other upper bound of ).
Universal algebra approach
By definition, a binary operation on a set is a if it satisfies the three conditions a, b, and c. The pair is then a meet-semilattice. Moreover, we then may define a binary relation on A, by stating that if and only if In fact, this relation is a partial order on Indeed, for any elements
since by c;
if then by a; and
if then since then by b.
Both meets and joins equally satisfy this definition: a couple
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https://en.wikipedia.org/wiki/List%20of%20West%20Ham%20United%20F.C.%20records%20and%20statistics
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This article lists records and statistics associated with West Ham United.
Team records
Scoring records
Biggest victory: 10–0 v Bury, Football League Cup (25 October 1983)
Biggest league win: 8–0 v Rotherham United (8 March 1958), and v Sunderland (19 October 1968)
Biggest defeat: 0–7 v Barnsley (1 September 1919), v Everton (22 October 1927), and v Sheffield Wednesday (28 November 1959)
League sequences
Wins: 9 (19 October to 4 December 1985)
Draws: 5 (7 September to 5 October 1968, and 15 October to 1 November 2003)
Defeats: 9 (28 March to 29 August 1932)
(source:)
Goalscorers
Leading first class goalscorers
Other top goalscorers
Top goalscorers by season
For a list of top scorers by season see List of West Ham United F.C. seasons.
Penalty shoot-outs
Honours
source;
European
UEFA Cup Winners' Cup
Winners: 1964–65
Runners-up: 1975–76
UEFA Europa Conference League
Winners: 2022–23
UEFA Intertoto Cup
Winners: 1999
Anglo-Italian League Cup
Runners-up: 1975
Domestic
Leagues
First Division/Premier League (Tier 1)
Highest placing: 3rd, 1985–86
Second Division/Championship (Tier 2)
Champions (2): 1957–58, 1980–81
Runners-up: 1922–23, 1990–91, 1992–93
Play-off winners: 2005, 2012
Play-off runners-up: 2004
Cups
FA Cup
Winners (3): 1963–64, 1974–75, 1979–80
Runners-up: 1922–23, 2005–06
EFL Cup/Football League Cup
Runners-up: 1965–66, 1980–81
FA Community/Charity Shield
Winners: 1964 (shared)
Runners-up: 1975, 1980
Football League War Cup
Winners: 1939–40
Other
Southern League Division One:
Highest placing: 3rd, 1912–13
Southern League Division Two
Champions: 1898–99
Western League:
Champions: 1906–07
London League:
Champions (2): 1897–98, 1901–02
Runners-up: 1896–97, 1902–03
Southern Floodlit Cup:
Winners: 1956
Runners-up: 1960
London Challenge Cup
Winners (9): 1925, 1926, 1930, 1947, 1949, 1953, 1957, 1968, 1969
Essex Professional Cup:
Winners (3): 1951, 1955 (shared), 1959
Runners-up: 1952, 1958
Southern Charity Cup
Runners-up: 1902
West Ham Charity Cup
Winners: 1896
Runners-up: 1897
Norfolk & Norwich Hospital Cup
Winners: 1924
Runners-up: 1925
Wartime
London Combination:
Champions: 1916–17
Runners-up: 1915–16 (Supplementary Tournament), 1917–18
Regional League South:
Runners-up: 1939–40, 1940–41
London League:
Runners-up: 1943–44, 1944–45
Indoor
London Fives
Winners: 1967, 1970, 1984
Runners-up: 1955, 1957, 1960, 1971, 1974, 1977, 1981
International
International Soccer League
Winners: 1963
American Challenge Cup
Runners-up: 1963
Friendly
Bobby Moore Cup:
Winners: 2008
SBOBET Cup:
Winners: 2010
Ciutat de Barcelona Trophy:
Winners: 2013
Betway Cup:
Winners: 2018, 2021
Other awards
BBC Sports Personality of the Year Team Award: 1965
Honorary Degree (awarded to the club) in 2009 by the University of East London
References
Statistics
West Ham
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https://en.wikipedia.org/wiki/Kleisli%20category
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In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question Does every monad arise from an adjunction? The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli.
Formal definition
Let 〈T, η, μ〉 be a monad over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by
That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in CT (but with codomain Y). Composition of morphisms in CT is given by
where f: X → T Y and g: Y → T Z. The identity morphism is given by the monad unit η:
.
An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane. We use very slightly different notation for this presentation. Given the same monad and category as above, we associate with each object in a new object , and for each morphism in a morphism . Together, these objects and morphisms form our category , where we define
Then the identity morphism in is
Extension operators and Kleisli triples
Composition of Kleisli arrows can be expressed succinctly by means of the extension operator (–)# : Hom(X, TY) → Hom(TX, TY). Given a monad 〈T, η, μ〉 over a category C and a morphism f : X → TY let
Composition in the Kleisli category CT can then be written
The extension operator satisfies the identities:
where f : X → TY and g : Y → TZ. It follows trivially from these properties that Kleisli composition is associative and that ηX is the identity.
In fact, to give a monad is to give a Kleisli triple 〈T, η, (–)#〉, i.e.
A function ;
For each object in , a morphism ;
For each morphism in , a morphism
such that the above three equations for extension operators are satisfied.
Kleisli adjunction
Kleisli categories were originally defined in order to show that every monad arises from an adjunction. That construction is as follows.
Let 〈T, η, μ〉 be a monad over a category C and let CT be the associated Kleisli category. Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F: C → CT by
and a functor G : CT → C by
One can show that F and G are indeed functors and that F is left adjoint to G. The counit of the adjunction is given by
Finally, one can show that T = GF and μ = GεF so that 〈T, η, μ〉 is the monad associated to the adjunction 〈F, G, η, ε〉.
Showing that GF = T
For any object X in category C:
For any in category C:
Since is true for any object X in C and is true for any morphism f in C, then . Q.E.D.
References
External links
Adjoint functors
Categories in category theory
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https://en.wikipedia.org/wiki/Solving%20the%20geodesic%20equations
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Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of (usually ideal) particles with no proper acceleration, their motion satisfying the geodesic equations. Because the particles are subject to no proper acceleration, the geodesics generally represent the straightest path between two points in a curved spacetime.
The differential geodesic equation
On an n-dimensional Riemannian manifold , the geodesic equation written in a coordinate chart with coordinates is:
where the coordinates xa(s) are regarded as the coordinates of a curve γ(s) in and are the Christoffel symbols. The Christoffel symbols are functions of the metric and are given by:
where the comma indicates a partial derivative with respect to the coordinates:
As the manifold has dimension , the geodesic equations are a system of ordinary differential equations for the coordinate variables. Thus, allied with initial conditions, the system can, according to the Picard–Lindelöf theorem, be solved. One can also use a Lagrangian approach to the problem: defining
and applying the Euler–Lagrange equation.
Heuristics
As the laws of physics can be written in any coordinate system, it is convenient to choose one that simplifies the geodesic equations. Mathematically, this means a coordinate chart is chosen in which the geodesic equations have a particularly tractable form.
Effective potentials
When the geodesic equations can be separated into terms containing only an undifferentiated variable and terms containing only its derivative, the former may be consolidated into an effective potential dependent only on position. In this case, many of the heuristic methods of analysing energy diagrams apply, in particular the location of turning points.
Solution techniques
Solving the geodesic equations means obtaining an exact solution, possibly even the general solution, of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions implicitly, but in many examples does yield the general solution in explicit form.
In general relativity, to obtain timelike geodesics it is often simplest to start from the spacetime metric, after dividing by to obtain the form
where the dot represents differentiation with respect to . Because timelike geodesics are maximal, one may apply the Euler–Lagrange equation directly, and thus obtain a set of equations equivalent to the geodesic equations. This method has the advantage of bypassing a tedious calculation of Christoffel symbols.
See also
Geodesics of the Schwarzschild vacuum
Mathematics of general relativity
Transition from special relativity to general relativity
References
General relativity
Mathematical methods in general relativity
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https://en.wikipedia.org/wiki/Joichi%20Suetsuna
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Joichi Suetsuna (Japanese: 末綱 恕一 Suetsuna Joichi; alternative Romanziation: Zyoiti Suetuna; November 28, 1898 – August 6, 1970) was a Japanese mathematician who worked mainly on number theory. In addition to working in Japan, where he held a chair at Tokyo University and was eventually selected to the Japan Academy, Suetsuna also spent time studying in Europe and introduced to Japan research styles he witnessed there. Later in life, especially after World War II, he studied Buddhist philosophy.
He was a teacher of Hirofumi Uzawa.
References
External links
Suetsuna at MacTutor
5th Director-General of the Institute of Statistical Mathematics in Tokyo
1898 births
1970 deaths
20th-century Japanese mathematicians
Number theorists
Japanese Buddhists
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https://en.wikipedia.org/wiki/Hong%20Kong%20Mathematics%20Olympiad
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Hong Kong Mathematics Olympiad (HKMO, ) is a Mathematics Competition held in Hong Kong every year, jointly organized by The Education University of Hong Kong and Education Bureau. At present, more than 250 secondary schools send teams of 4-6 students of or below Form 5 to enter the competition. It is made up of a Heat Event and a Final Event, which both forbid the usage of calculators and calculation assisting equipments (e.g. printed mathematical table). Though it bears the term Mathematics Olympiad, it has no relationship with the International Mathematical Olympiad.
History
The predecessor of HKMO is the Inter-school Mathematics Olympiad initiated by the Mathematics Society of Northcote College of Education in 1974, which had attracted 20 secondary schools to participate. Since 1983, the competition is jointly conducted by the Mathematics Department of Northcote College of Education and the Mathematics Section of the Advisory Inspectorate Division of the Education Department. Also in 1983, the competition is formally renamed as Hong Kong Mathematics Olympiad.
Format and Scoring in the Heat Event
The Heat Event is usually held in four venues, for contestants from schools on Hong Kong Island, and in Kowloon, New Territories East and New Territories West respectively. It comprises an individual event and a group event. Each team sends 4 contestants among 4-6 team members for each event.
For the individual event, 1 mark and 2 marks will be given to each correct answer in Part A and Part B respectively. The maximum score for a team should be 80.
For the group event, 2 marks will be given to each correct answer. The maximum score for a team should be 20.
For the geometric construction event, the maximum score for a team should be 20 (all working, including construction work, must be clearly shown).
In other words, a contesting school may earn 120 marks at most in the Heat Event. The top 50 may enter the Final Event.
Format and Scoring in the Final Event
The Final Event is usually held at the Education University of Hong Kong in Tai Po. It comprises 4 individual events and 4 group events. Before the real events begin, there is a mock event which carries no marks. Each team may send any 4 students for the individual events, and any 4 students for the group events. For every events, only answers are required.
There are 4 questions in each Final Individual Event. The questions have to be solved by alternate contestants independently, and no discussions are allowed. For each event, the questions are interrelated, i.e. to solve the second question, the answer of the first question is needed, and to solve the third, the answer from the second is needed, etc..
There are also 4 questions in each Final Group Event, which may be interrelated or not. The four contestants shall complete each event together, and discussion is allowed.
For each event, 5 minutes is given. There are timekeepers to report the time taken used for each team in each event. The
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https://en.wikipedia.org/wiki/Hkmo
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HKMO may refer to:
Hong Kong Mathematics Olympiad
ICAO-Code for Mombasa Moi International Airport
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https://en.wikipedia.org/wiki/Frobenius%20theorem%20%28real%20division%20algebras%29
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In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following:
(the real numbers)
(the complex numbers)
(the quaternions).
These algebras have real dimension , and , respectively. Of these three algebras, and are commutative, but is not.
Proof
The main ingredients for the following proof are the Cayley–Hamilton theorem and the fundamental theorem of algebra.
Introducing some notation
Let be the division algebra in question.
Let be the dimension of .
We identify the real multiples of with .
When we write for an element of , we imply that is contained in .
We can consider as a finite-dimensional -vector space. Any element of defines an endomorphism of by left-multiplication, we identify with that endomorphism. Therefore, we can speak about the trace of , and its characteristic and minimal polynomials.
For any in define the following real quadratic polynomial:
Note that if then is irreducible over .
The claim
The key to the argument is the following
Claim. The set of all elements of such that is a vector subspace of of dimension . Moreover as -vector spaces, which implies that generates as an algebra.
Proof of Claim: Let be the dimension of as an -vector space, and pick in with characteristic polynomial . By the fundamental theorem of algebra, we can write
We can rewrite in terms of the polynomials :
Since , the polynomials are all irreducible over . By the Cayley–Hamilton theorem, and because is a division algebra, it follows that either for some or that for some . The first case implies that is real. In the second case, it follows that is the minimal polynomial of . Because has the same complex roots as the minimal polynomial and because it is real it follows that
Since is the characteristic polynomial of the coefficient of in is up to a sign. Therefore, we read from the above equation we have: if and only if , in other words if and only if .
So is the subset of all with . In particular, it is a vector subspace. The rank–nullity theorem then implies that has dimension since it is the kernel of . Since and are disjoint (i.e. they satisfy ), and their dimensions sum to , we have that .
The finish
For in define . Because of the identity , it follows that is real. Furthermore, since , we have: for . Thus is a positive definite symmetric bilinear form, in other words, an inner product on .
Let be a subspace of that generates as an algebra and which is minimal with respect to this property. Let be an orthonormal basis of with respect to . Then orthonormality implies that:
If , then is isomorphic to .
If , then is generated by and subject to the relation . Hence it is isomorphic to .
If , it has been shown above that is generated
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https://en.wikipedia.org/wiki/Reflective%20subcategory
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In mathematics, a full subcategory A of a category B is said to be reflective in B when the inclusion functor from A to B has a left adjoint. This adjoint is sometimes called a reflector, or localization. Dually, A is said to be coreflective in B when the inclusion functor has a right adjoint.
Informally, a reflector acts as a kind of completion operation. It adds in any "missing" pieces of the structure in such a way that reflecting it again has no further effect.
Definition
A full subcategory A of a category B is said to be reflective in B if for each B-object B there exists an A-object and a B-morphism such that for each B-morphism to an A-object there exists a unique A-morphism with .
The pair is called the A-reflection of B. The morphism is called the A-reflection arrow. (Although often, for the sake of brevity, we speak about only as being the A-reflection of B).
This is equivalent to saying that the embedding functor is a right adjoint. The left adjoint functor is called the reflector. The map is the unit of this adjunction.
The reflector assigns to the A-object and for a B-morphism is determined by the commuting diagram
If all A-reflection arrows are (extremal) epimorphisms, then the subcategory A is said to be (extremal) epireflective. Similarly, it is bireflective if all reflection arrows are bimorphisms.
All these notions are special case of the common generalization—-reflective subcategory, where is a class of morphisms.
The -reflective hull of a class A of objects is defined as the smallest -reflective subcategory containing A. Thus we can speak about reflective hull, epireflective hull, extremal epireflective hull, etc.
An anti-reflective subcategory is a full subcategory A such that the only objects of B that have an A-reflection arrow are those that are already in A.
Dual notions to the above-mentioned notions are coreflection, coreflection arrow, (mono)coreflective subcategory, coreflective hull, anti-coreflective subcategory.
Examples
Algebra
The category of abelian groups Ab is a reflective subcategory of the category of groups, Grp. The reflector is the functor that sends each group to its abelianization. In its turn, the category of groups is a reflective subcategory of the category of inverse semigroups.
Similarly, the category of commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the commutator ideal. This is used in the construction of the symmetric algebra from the tensor algebra.
Dually, the category of anti-commutative associative algebras is a reflective subcategory of all associative algebras, where the reflector is quotienting out by the anti-commutator ideal. This is used in the construction of the exterior algebra from the tensor algebra.
The category of fields is a reflective subcategory of the category of integral domains (with injective ring homomorphisms as morphisms). The reflector is the functor th
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https://en.wikipedia.org/wiki/Normal%20score
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The term normal score is used with two different meanings in statistics. One of them relates to creating a single value which can be treated as if it had arisen from a standard normal distribution (zero mean, unit variance). The second one relates to assigning alternative values to data points within a dataset, with the broad intention of creating data values than can be interpreted as being approximations for values that might have been observed had the data arisen from a standard normal distribution.
The first meaning is as an alternative name for the standard score or z score, where values are standardised by subtracting the sample or estimated mean and dividing by the sample or other estimate of the standard deviation. Particularly in applications where the name "normal score" is used, there is usually a presumption that the value can be referred to a table of standard normal probabilities as a means of providing a significance test of some hypothesis, such as a difference in means.
The second meaning of normal score is associated with data values derived from the ranks of the observations within the dataset. A given data point is assigned a value which is either exactly, or an approximation, to the expectation of the order statistic of the same rank in a sample of standard normal random variables of the same size as the observed data set. Thus the meaning of a normal score of this type is essentially the same as a rankit, although the term "rankit" is becoming obsolete. In this case the transformation creates a set of values which is matched in a certain way to what would be expected had the original set of data values arisen from a normal distribution.
See also
Normalization (statistics)
Normal probability plot
Q–Q plot
References
Nonparametric statistics
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https://en.wikipedia.org/wiki/Taylor%20expansions%20for%20the%20moments%20of%20functions%20of%20random%20variables
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In probability theory, it is possible to approximate the moments of a function f of a random variable X using Taylor expansions, provided that f is sufficiently differentiable and that the moments of X are finite.
First moment
Given and , the mean and the variance of , respectively, a Taylor expansion of the expected value of can be found via
Since the second term vanishes. Also, is . Therefore,
.
It is possible to generalize this to functions of more than one variable using multivariate Taylor expansions. For example,
Second moment
Similarly,
The above is obtained using a second order approximation, following the method used in estimating the first moment. It will be a poor approximation in cases where is highly non-linear. This is a special case of the delta method.
Indeed, we take .
With , we get . The variance is then computed using the formula
.
An example is,
The second order approximation, when X follows a normal distribution, is:
First product moment
To find a second-order approximation for the covariance of functions of two random variables (with the same function applied to both), one can proceed as follows. First, note that . Since a second-order expansion for has already been derived above, it only remains to find . Treating as a two-variable function, the second-order Taylor expansion is as follows:
Taking expectation of the above and simplifying—making use of the identities and —leads to . Hence,
Random vectors
If X is a random vector, the approximations for the mean and variance of are given by
Here and denote the gradient and the Hessian matrix respectively, and is the covariance matrix of X.
See also
Propagation of uncertainty
WKB approximation
Delta method
Notes
Further reading
Statistical approximations
Algebra of random variables
Moment (mathematics)
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https://en.wikipedia.org/wiki/Ancient%20Egyptian%20multiplication
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In mathematics, ancient Egyptian multiplication (also known as Egyptian multiplication, Ethiopian multiplication, Russian multiplication, or peasant multiplication), one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands (preferably the smaller) into a set of numbers of powers of two and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication.
This method may be called mediation and duplation, where mediation means halving one number and duplation means doubling the other number. It is still used in some areas.
The second Egyptian multiplication and division technique was known from the hieratic Moscow and Rhind Mathematical Papyri written in the seventeenth century B.C. by the scribe Ahmes.
Although in ancient Egypt the concept of base 2 did not exist, the algorithm is essentially the same algorithm as long multiplication after the multiplier and multiplicand are converted to binary. The method as interpreted by conversion to binary is therefore still in wide use today as implemented by binary multiplier circuits in modern computer processors.
Method
The ancient Egyptians had laid out tables of a great number of powers of two, rather than recalculating them each time. The decomposition of a number thus consists of finding the powers of two which make it up. The Egyptians knew empirically that a given power of two would only appear once in a number. For the decomposition, they proceeded methodically; they would initially find the largest power of two less than or equal to the number in question, subtract it out and repeat until nothing remained. (The Egyptians did not make use of the number zero in mathematics.)
After the decomposition of the first multiplicand, the person would construct a table of powers of two times the second multiplicand (generally the smaller) from one up to the largest power of two found during the decomposition.
The result is obtained by adding the numbers from the second column for which the corresponding power of two makes up part of the decomposition of the first multiplicand.
Example
25 × 7 = ?
Decomposition of the number 25:
{|
| The largest power of two less than or equal to 25 || is 16: || style="text-align:center;"| 25 − 16 || = 9.
|-
| The largest power of two less than or equal to 9 || is 8: || style="text-align:center;"| 9 − 8 || = 1.
|-
| The largest power of two less than or equal to 1 || is 1: || style="text-align:center;"| 1 − 1 || = 0.
|-
| colspan="3" | 25 is thus the sum of: 16, 8 and 1.
|}
The largest power of two is 16 and the second multiplicand is 7.
As 25 = 16 + 8 + 1, the corresponding multiples of 7 are added to get 25 × 7 = 112 + 56 + 7 = 175.
Russian peasant multiplication
In the Russian peasant met
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https://en.wikipedia.org/wiki/Baire%20set
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In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets.
There are several inequivalent definitions of Baire sets, but in the most widely used, the Baire sets of a locally compact Hausdorff space form the smallest σ-algebra such that all compactly supported continuous functions are measurable. Thus, measures defined on this σ-algebra, called Baire measures, are a convenient framework for integration on locally compact Hausdorff spaces. In particular, any compactly supported continuous function on such a space is integrable with respect to any finite Baire measure.
Every Baire set is a Borel set. The converse holds in many, but not all, topological spaces. Baire sets avoid some pathological properties of Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use of regular Borel measures on Borel sets.
Baire sets were introduced by , and , who named them after Baire functions, which are in turn named after René-Louis Baire.
Basic definitions
There are at least three inequivalent definitions of Baire sets on locally compact Hausdorff spaces, and even more definitions for general topological spaces, though all these definitions are equivalent for locally compact σ-compact Hausdorff spaces. Moreover, some authors add restrictions on the topological space that Baire sets are defined on, and only define Baire sets on spaces that are compact Hausdorff, or locally compact Hausdorff, or σ-compact.
First definition
Kunihiko Kodaira defined what we call Baire sets (although he confusingly calls them "Borel sets") of certain topological spaces to be the sets whose characteristic function is a Baire function (the smallest class of functions containing all continuous real-valued functions and closed under pointwise limits of sequences).
gives an equivalent definition and defines Baire sets of a topological space to be elements of the smallest σ-algebra such that all continuous real-valued functions are measurable. For locally compact σ-compact Hausdorff spaces this is equivalent to the following definitions, but in general the definitions are not equivalent.
Conversely, the Baire functions are exactly the real-valued functions that are Baire measurable. For metric spaces, the Baire sets coincide with the Borel sets.
Second definition
defined Baire sets of a locally compact Hausdorff space to be the elements of the σ-ring generated by the compact Gδ sets. This definition is no longer used much, as σ-rings are somewhat out of fashion. When the space is σ-compact, this definition is equivalent to the next definition.
One reason for working with compact Gδ sets rather than closed Gδ sets is that Baire measures are then automatically regular .
Third definition
The third and most widely used definition is similar to Halmos's definition, modi
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https://en.wikipedia.org/wiki/Pointwise%20product
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In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If and are both functions with domain and codomain , and elements of can be multiplied (for instance, could be some set of numbers), then the pointwise product of and is another function from to which maps in to in .
Formal definition
Let and be sets such that has a notion of multiplication — that is, there is a binary operation
given by
Then given two functions the pointwise product is defined by
for all in . Just as we often omit the symbol for the binary operation ⋅ (i.e. we write instead of ), we often write for .
Examples
The most common case of the pointwise product of two functions is when the codomain is a ring (or field), in which multiplication is well-defined.
Algebraic application of pointwise products
Let be a set and let be a ring. Since addition and multiplication are defined in , we can construct an algebraic structure known as an algebra out of the functions from to by defining addition, multiplication, and scalar multiplication of functions to be done pointwise.
If denotes the set of functions from to , then we say that if are elements of , then , , and — the last of which is defined by
for all in — are all elements of .
Generalization
If both and have as their domain all possible assignments of a set of discrete variables, then their pointwise product is a function whose domain is constructed by all possible assignments of the union of both sets. The value of each assignment is calculated as the product of the values of both functions given to each one the subset of the assignment that is in its domain.
For example, given the function of the boolean variables and , and of the boolean variables and , both with the range in the pointwise product of and is shown in the next table:
See also
Pointwise
Elementary algebra
Binary operations
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https://en.wikipedia.org/wiki/Petter%20Jakob%20Bjerve
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Petter Jakob Bjerve (27 September 1913 – 12 January 2004) was a Norwegian economist, statistician and politician for the Labour Party. Prominent positions include director of Statistics Norway from 1949 to 1980, Norwegian Minister of Finance from 1960 to 1963, and president of the International Statistical Institute from 1971 to 1975.
Career
He was born in Stjørdal as a son of farmers Petter Jakob Bjerve, Sr. (1869–1928) and Kristine Arnstad (1870–1961). He married Rannveig Bremer, a daughter of Anders H. Bremer.
Bjerve attended secondary school in Orkdal, and was active in Clarté before joining the Labour Party. He studied under Ragnar Frisch at the University of Oslo, and graduated with the cand.oecon. degree in 1941. He studied at The American University, Washington DC from 1938 to 1939 and again in the US with a Rockefeller Foundation grant from 1947 to 1949. At the University of Oslo he worked as a research assistant from 1939 to 1940, teacher from 1941 to 1943 and research fellow from 1945 to 1949. He also held sporadic lectures between 1945 and 1960. He was also a visiting professor at Stanford University from 1954 to 1955. He was also a secretary in Statistics Norway from 1944 to 1945 and assistant secretary in the Norwegian Ministry of Finance. In 1949 he was hired as a director in Statistics Norway. He remained here until 1980. The exception was his period as Minister of Finance in Gerhardsen's Third Cabinet, from 23 April 1960 until his resignation on 4 February 1963. His doctorate thesis Planning in Norway 1947–1953 was finished in 1959, and he defended his thesis for the dr.philos. degree in 1962 while serving as Minister of Finance.
He was a prolific writer throughout his career. He was president of the International Statistical Institute from 1971 to 1975, and honorary member from 1986. He was an honorary member of Statistiska Föreningen in Stockholm from 1951, the Finnish Statistical Society from 1960, the American Statistical Association from 1964 and the Royal Statistical Society from 1967.
In 1964 he was elected as a Fellow of the American Statistical Association.
He was a board member of Norsk Regnesentral from 1958 to 1960, of Institutt for anvendt sosialvitenskapelig forskning from 1966 to 1981, of Statsøkonomisk Forening from 1950 to 1953, 1963 to 1967 and 1971 to 1974. He chaired Sosialistiske økonomers forening from 1965 to 1967 and the Chr. Michelsen Institute department of social sciences from 1982 to 1984.
Bjerve was also an adviser for governments and banks of Zambia, Pakistan, Sri Lanka, Bangladesh, Portugal, Italy and Zimbabwe, partly with a United Nations and International Labour Organization connection. He also held other positions within the United Nations and OECD systems.
He was also a board member of Riksskattestyret from 1959 to 1963, Dag og Tid from 1965 to 1967 and Ja til EF 1972, and chairman of Livstrygdelaget Andvake from 1979 to 1985.
References
1913 births
2004 deaths
People from Stjørdal
Unive
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https://en.wikipedia.org/wiki/Correlation%20sum
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In chaos theory, the correlation sum is the estimator of the correlation integral, which reflects the mean probability that the states at two different times are close:
where is the number of considered states , is a threshold distance, a norm (e.g. Euclidean norm) and the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem):
where is the time series, the embedding dimension and the time delay.
The correlation sum is used to estimate the correlation dimension.
See also
Recurrence quantification analysis
References
Chaos theory
Dynamical systems
Dimension theory
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https://en.wikipedia.org/wiki/Epsilon%20number
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In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation
in which ω is the smallest infinite ordinal.
The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals:
where is the supremum function, which is equivalent to set union in the case of the von Neumann representation of ordinals.
Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in . The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal).
The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic).
Many larger epsilon numbers can be defined using the Veblen function.
A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx.
defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see multiplicatively indecomposable ordinals) to be numbers δ>1 such that αδ=δ whenever 0<α<δ, and epsilon numbers to be numbers ε>2 such that αε=ε whenever 1<α<ε. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ.
Ordinal ε numbers
The standard definition of ordinal exponentiation with base α is:
when has an immediate predecessor .
, whenever is a limit ordinal.
From this definition, it follows that for any fixed ordinal , the mapping is a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions. When , these fixed points are precisely the ordinal epsilon numbers.
when has an immediate predecessor .
, whenever is a limit ordinal.
Because
a different sequence with the same supremum, , is obtained by starting from 0 and exponentiating with bas
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https://en.wikipedia.org/wiki/Lattice%20of%20subgroups
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In mathematics, the lattice of subgroups of a group is the lattice whose elements are the subgroups of , with the partial order relation being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.
Example
The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of order-two elements. The lattice formed by these ten subgroups is shown in the illustration.
This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" N5 as a sublattice.
Properties
For any A, B, and C subgroups of a group with A ≤ C (A subgroup of C) then AB ∩ C = A(B ∩ C); the multiplication here is the product of subgroups. This property has been called the modular property of groups or (Dedekind's) modular law (, ). Since for two normal subgroups the product is actually the smallest subgroup containing the two, the normal subgroups form a modular lattice.
The Lattice theorem establishes a Galois connection between the lattice of subgroups of a group and that of its quotients.
The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups.
In general, there is no restriction on the shape of the lattice of subgroups, in the sense that every lattice is isomorphic to a sublattice of the subgroup lattice of some group. Furthermore, every finite lattice is isomorphic to a sublattice of the subgroup lattice of some finite group .
Characteristic lattices
Subgroups with certain properties form lattices, but other properties do not.
Normal subgroups always form a modular lattice. In fact, the essential property that guarantees that the lattice is modular is that subgroups commute with each other, i.e. that they are quasinormal subgroups.
Nilpotent normal subgroups form a lattice, which is (part of) the content of Fitting's theorem.
In general, for any Fitting class F, both the subnormal F-subgroups and the normal F-subgroups form lattices. This includes the above with F the class of nilpotent groups, as well as other examples such as F the class of solvable groups. A class of groups is called a Fitting class if it is closed under isomorphism, subnormal subgroups, and products of subnormal subgroups.
Central subgroups form a lattice.
However, neither finite subgroups nor torsion subgroups form a lattice: for instance, the free product is generated by two torsion elements, but is infinite and contains elements of infinite order.
The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in
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https://en.wikipedia.org/wiki/Mathematics%20of%20Computation
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Mathematics of Computation is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as Mathematical Tables and Other Aids to Computation, obtaining its current name in 1960. Articles older than five years are available electronically free of charge.
Abstracting and indexing
The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the Journal Citation Reports, the journal has a 2020 impact factor of 2.417.
References
External links
Delayed open access journals
English-language journals
Mathematics journals
Academic journals established in 1943
American Mathematical Society academic journals
Bimonthly journals
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https://en.wikipedia.org/wiki/Linear%20group
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In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithful, finite-dimensional representation over K).
Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behavior (for example, finitely generated infinite torsion groups).
Definition and basic examples
A group G is said to be linear if there exists a field K, an integer d and an injective homomorphism from G to the general linear group GLd(K) (a faithful linear representation of dimension d over K): if needed one can mention the field and dimension by saying that G is linear of degree d over K. Basic instances are groups which are defined as subgroups of a linear group, for example:
The group GLn(K) itself;
The special linear group SLn(K) (the subgroup of matrices with determinant 1);
The group of invertible upper (or lower) triangular matrices
If gi is a collection of elements in GLn(K) indexed by a set I, then the subgroup generated by the gi is a linear group.
In the study of Lie groups, it is sometimes pedagogically convenient to restrict attention to Lie groups that can be faithfully represented over the field of complex numbers. (Some authors require that the group be represented as a closed subgroup of the GLn(C).) Books that follow this approach include Hall (2015) and Rossmann (2002).
Classes of linear groups
Classical groups and related examples
The so-called classical groups generalize the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined by a finite number of equations. Basic examples are orthogonal, unitary and symplectic groups but it is possible to construct more using division algebras (for example the unit group of a quaternion algebra is a classical group). Note that the projective groups associated to these groups are also linear, though less obviously. For example, the group PSL2(R) is not a group of 2 × 2 matrices, but it has a faithful representation as 3 × 3 matrices (the adjoint representation), which can be used in the general case.
Many Lie groups are linear, but not all of them. The universal cover of SL2(R) is not linear, as are many solvable groups, for instance the quotient of the Heisenberg group by a central cyclic subgroup.
Discrete subgroups of classical Lie groups (for example lattices or thin groups) are also examples of interesting linear groups.
Finite groups
A finite group G of order n is linear of degree at most n over any field K. This statement is sometimes called Cayley's theorem, and simply results from the fact th
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https://en.wikipedia.org/wiki/Cofunction
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In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles (pairs that sum to one right angle). This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).
For example, sine (Latin: sinus) and cosine (Latin: cosinus, sinus complementi) are cofunctions of each other (hence the "co" in "cosine"):
The same is true of secant (Latin: secans) and cosecant (Latin: cosecans, secans complementi) as well as of tangent (Latin: tangens) and cotangent (Latin: cotangens, tangens complementi):
These equations are also known as the cofunction identities.
This also holds true for the versine (versed sine, ver) and coversine (coversed sine, cvs), the vercosine (versed cosine, vcs) and covercosine (coversed cosine, cvc), the haversine (half-versed sine, hav) and hacoversine (half-coversed sine, hcv), the havercosine (half-versed cosine, hvc) and hacovercosine (half-coversed cosine, hcc), as well as the exsecant (external secant, exs) and excosecant (external cosecant, exc):
See also
Hyperbolic functions
Lemniscatic cosine
Jacobi elliptic cosine
Cologarithm
Covariance
List of trigonometric identities
References
Trigonometry
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https://en.wikipedia.org/wiki/Syndetic%20set
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In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.
Definition
A set is called syndetic if for some finite subset of
where . Thus syndetic sets have "bounded gaps"; for a syndetic set , there is an integer such that for any .
See also
Ergodic Ramsey theory
Piecewise syndetic set
Thick set
References
Semigroup theory
Ergodic theory
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https://en.wikipedia.org/wiki/Bochner%27s%20formula
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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement
If is a smooth function, then
,
where is the gradient of with respect to , is the Hessian of with respect to and is the Ricci curvature tensor. If is harmonic (i.e., , where is the Laplacian with respect to the metric ), Bochner's formula becomes
.
Bochner used this formula to prove the Bochner vanishing theorem.
As a corollary, if is a Riemannian manifold without boundary and is a smooth, compactly supported function, then
.
This immediately follows from the first identity, observing that the integral of the left-hand side vanishes (by the divergence theorem) and integrating by parts the first term on the right-hand side.
Variations and generalizations
Bochner identity
Weitzenböck identity
References
Differential geometry
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https://en.wikipedia.org/wiki/The%20Music%20of%20the%20Primes
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The Music of the Primes (British subtitle: Why an Unsolved Problem in Mathematics Matters; American subtitle: Searching to Solve the Greatest Mystery in Mathematics) is a 2003 book by Marcus du Sautoy, a professor in mathematics at the University of Oxford, on the history of prime number theory. In particular he examines the Riemann hypothesis, the proof of which would revolutionize our understanding of prime numbers. He traces the prime number theorem back through history, highlighting the work of some of the greatest mathematical minds along the way.
The cover design for the hardback version of the book contains several pictorial depictions of prime numbers, such as the number 73 bus. It also has an image of a clock, referring to clock arithmetic, which is a significant theme in the text.
References
2003 non-fiction books
Mathematics books
Analytic number theory
Prime numbers
Fourth Estate books
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https://en.wikipedia.org/wiki/Siegel%27s%20theorem%20on%20integral%20points
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In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0.
The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For g > 1 it was superseded by Faltings's theorem in 1983.
History
In 1929, Siegel proved the theorem by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from diophantine geometry (required in Weil's version, to apply to the Jacobian variety of C).
In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new method based on the subspace theorem.
Effective versions
Siegel's result was ineffective (see effective results in number theory), since Thue's method in diophantine approximation also is ineffective in describing possible very good rational approximations to algebraic numbers. Effective results in some cases derive from Baker's method.
See also
Diophantine geometry
References
Diophantine equations
Theorems in number theory
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https://en.wikipedia.org/wiki/Chiral%20Potts%20curve
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The chiral Potts curve is an algebraic curve defined over the complex numbers that occurs in the study of the chiral Potts model of statistical mechanics. For an integer N, the parameters in the Boltzmann weights of the model are constrained to lie on the intersection of two algebraic surfaces of degree N in projective 3-space.
The equation is
.
The curve has been known since papers published in 1987 and 1988. It has genus that is quadratic in N.
External links
PDF paper by Brian Davies and Amnon Neeman
Algebraic curves
Exactly solvable models
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https://en.wikipedia.org/wiki/161%20%28number%29
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161 (one hundred [and] sixty-one) is the natural number following 160 and preceding 162.
In mathematics
161 is the sum of five consecutive prime numbers: 23, 29, 31, 37, and 41
161 is a hexagonal pyramidal number.
161 is a semiprime. Since its prime factors 7 and 23 are both Gaussian primes, 161 is a Blum integer.
161 is a palindromic number
is a commonly used rational approximation of the square root of 5 and is the closest fraction with denominator <300 to that number.
In the military
was a U.S. Navy Type T2 tanker during World War II
was a U.S. Navy during World War II
was a U.S. Navy Trefoil-class concrete barge during World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War II
was a U.S. Navy wooden yacht during World War I
was a U.S. Navy during World War II
was a U.S. Navy Achomawi-class fleet ocean tug following World War II
was a U.S. Navy fourth-group S-class submarine between 1920 and 1931
is a fictional U.S. Navy diesel engine submarine featured in the 1996 film Down Periscope
The 161st Intelligence Squadron unit of the Kansas Air National Guard. Its parent unit is the 184th Intelligence Wing
In music
The Bose 161 Speaker System (2001)
The Kay K-161 ThinTwin guitar
In transportation
MTA Maryland commuter bus 161
New Jersey Bus Route 161
London Bus route 161
In other fields
161 is also:
The year AD 161 or 161 BC
161 AH is a year in the Islamic calendar that corresponds to 777 – 778 CE
161 Athor is an M-type Main belt asteroid
E.161 is an ITU-T assigns letters to the 12-key telephone keypad
Fiorina Fury 161 is a foundry facility and penal colony from the film Alien 3
161 is used by Anti Fascist Action as a code for AFA (A=1, F=6, by order of the alphabet), sometimes used in 161>88 (88 is code for Heil Hitler among neo-nazis, as H=8)
See also
Anti-Fascist Action
List of highways numbered 161
United Nations Security Council Resolution 161
United States Supreme Court cases, Volume 161
External links
Number Facts and Trivia: 161
The Number 161
VirtueScience: 161
References
Integers
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https://en.wikipedia.org/wiki/Maine%20School%20of%20Science%20and%20Mathematics
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The Maine School of Science and Mathematics (MSSM) is a public residential magnet high school in Limestone, Maine, United States.
MSSM serves students from all over the state of Maine, as well as youth from other states and international students. It is a public high school for students in grades 9–12, and its summer program is for boys and girls from grades 5–9. MSSM is an all-residential boarding school with a total capacity of 156 students.
The school is a member of the National Consortium of Secondary STEM Schools (NCSSS).
History
After the announcement that Loring Air Force Base would be closed, funding from the Defense Reauthorization Bill provided for the creation of the Maine School of Science and Mathematics at the site of Limestone High School, which was going to lose many of its students upon the closure of the base. The town's elementary school was eventually converted into dormitories for the school, as they are located on the same property. MSSM continues to share the former Limestone High School building with the local Limestone Community School. Each school occupies approximately half of the building. Due to their small size and physical proximity, the two schools also share most of their sports teams. In 2014, the school acquired a new dormitory, dubbed "Limestone Manor", in the center of town. The building housed a nursing home until the business relocated in 2013. As of 2014, the Limestone Manor, a male-only dormitory, houses close to 30 students.
Chartered and funded by the Maine Legislature, MSSM opened with a pioneer class in 1995. At that time, it was only the eleventh statewide residential magnet school specializing in mathematics and science education in the United States and the only school of its kind in New England. Both remain true today. It is the only magnet school currently operating in Maine.
National ranking
In 2013, U.S. News & World Report ranked MSSM 13th on its list of "America's Best High Schools," a ranking of public high schools in the United States. In 2017, it was ranked 19th; and in 2019, it was ranked second. In 2022, it was ranked 2,355th.
References
Further reading
Maine Statute establishing the school
U.S. News & World Report 2013 ranking
External links
Official MSSM Website
MSSM Parents Association Website
Public high schools in Maine
Magnet schools in Maine
Schools in Aroostook County, Maine
Boarding schools in Maine
Limestone, Maine
Public boarding schools in the United States
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https://en.wikipedia.org/wiki/G.%20B.%20Halsted
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George Bruce Halsted (November 25, 1853 – March 16, 1922), usually cited as G. B. Halsted, was an American mathematician who explored foundations of geometry and introduced non-Euclidean geometry into the United States through his translations of works by Bolyai, Lobachevski, Saccheri, and Poincaré. He wrote an elementary geometry text, Rational Geometry, based on Hilbert's axioms, which was translated into French, German, and Japanese. Halsted produced original works in synthetic geometry, first with an elementary text in 1896, and with a text on synthetic projective geometry in 1906.
Life
Halsted was a tutor and instructor at Princeton University. He held a mathematical fellowship while a student at Princeton. Halsted was a fourth generation Princeton graduate, earning his bachelor's degree in 1875 and his Master's in 1878. He went on to Johns Hopkins University where he was J. J. Sylvester's first student, receiving his Ph.D. in 1879. After graduation, Halsted served as an instructor in mathematics at Princeton until beginning his post at the University of Texas at Austin in 1884.
From 1884 to 1903, Halsted was a member of the University of Texas at Austin Department of Pure and Applied Mathematics, eventually becoming its chair. He taught mathematicians R. L. Moore and L. E. Dickson, among other students. He explored the foundations of geometry and many alternatives to Euclid's development, culminating with his Rational Geometry.
In the interest of hyperbolic geometry in 1891 he translated the work of Nicolai Lobachevsky on theory of parallels. In 1893 in Chicago, Halsted read a paper Some salient points in the history of non-Euclidean and hyper-spaces at the International Mathematical Congress held in connection with the World's Columbian Exposition. Halsted frequently contributed to the early American Mathematical Monthly. In one article he championed the role of J. Bolyai in the development of non-Euclidean geometry and criticized C. F. Gauss. See also the letter from Robert Gauss to Felix Klein on 3 September 1912.
In 1903, Halsted was fired from UT Austin after having published several articles that criticized the university for having passed over R. L. Moore, at that time a young and promising mathematician whom Halsted hoped to have as an assistant, for an instructor post in favor of a well-connected but less qualified candidate with roots in the area.
He completed his teaching career at St. John's College, Annapolis; Kenyon College, Gambier, Ohio (1903-1906); and the Colorado State Teachers College, Greeley (1906-1914).
In 1913 Science Press published three translations by Halsted of popular science works by Henri Poincaré. In a preface, Poincaré paid tribute to Halsted's inter-continental reach: He "has already taken the trouble to translate many European treatises and thus powerfully contributed to make the new continent understand the thought of the old."
Halsted was a member of the American Mathematical Society and served
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https://en.wikipedia.org/wiki/Subcompact%20cardinal
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In mathematics, a subcompact cardinal is a certain kind of large cardinal number.
A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality hereditarily less than κ+) with critical point μ and j(μ) = κ.
Analogously, κ is a quasicompact cardinal if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(κ+), A) → (H(μ+), B) with critical point κ and j(κ) = μ.
H(λ) consists of all sets whose transitive closure has cardinality less than λ.
Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is analogous to that of extendible versus supercompact cardinals. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many superstrong cardinals. Existence of a 2κ-supercompact cardinal κ implies existence of many quasicompact cardinals.
Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the square principle. If κ is subcompact, then the square principle fails at κ. Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.)
Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point. This prevents them from witnessing even a κ+ strongly compact cardinal κ.
Subcompact and quasicompact cardinals were defined by Ronald Jensen.
References
"Square in Core Models" in the September 2001 issue of the Bulletin of Symbolic Logic
Large cardinals
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https://en.wikipedia.org/wiki/Piecewise%20syndetic%20set
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In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.
A set is called piecewise syndetic if there exists a finite subset G of such that for every finite subset F of there exists an such that
where . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of where the gaps in S are bounded by b.
Properties
A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set.
If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.
A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of , the Stone–Čech compactification of the natural numbers.
Partition regularity: if is piecewise syndetic and , then for some , contains a piecewise syndetic set. (Brown, 1968)
If A and B are subsets of with positive upper Banach density, then is piecewise syndetic.
Other notions of largeness
There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers:
Cofiniteness
IP set
member of a nonprincipal ultrafilter
positive upper density
syndetic set
thick set
See also
Ergodic Ramsey theory
Notes
References
Semigroup theory
Ergodic theory
Ramsey theory
Combinatorics
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https://en.wikipedia.org/wiki/Thick%20set
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In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set , for every , there is some such that .
Examples
Trivially is a thick set. Other well-known sets that are thick include non-primes and non-squares. Thick sets can also be sparse, for example:
Generalisations
The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup and , is said to be thick if for any finite subset , there exists such that
It can be verified that when the semigroup under consideration is the natural numbers with the addition operation , this definition is equivalent to the one given above.
See also
Cofinal (mathematics)
Cofiniteness
Ergodic Ramsey theory
Piecewise syndetic set
Syndetic set
References
J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (Summer 2000), pp. 317-332.
Vitaly Bergelson, "Minimal Idempotents and Ergodic Ramsey Theory", Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
Vitaly Bergelson, N. Hindman, "Partition regular structures contained in large sets are abundant", Journal of Combinatorial Theory, Series A 93 (2001), pp. 18-36
N. Hindman, D. Strauss. Algebra in the Stone-Čech Compactification. p104, Def. 4.45.
Semigroup theory
Ergodic theory
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https://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s%20principle
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In mathematics, Vopěnka's principle is a large cardinal axiom.
The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings.
Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by H. Jerome Keisler, and was written up by .
According to , Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof he found a flaw in it.
Definition
Vopěnka's principle asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. This cannot be stated as a single sentence of ZFC as it involves a quantification over classes. A cardinal κ is called a Vopěnka cardinal if it is inaccessible and Vopěnka's principle holds in the rank Vκ (allowing arbitrary S ⊂ Vκ as "classes").
Many equivalent formulations are possible.
For example, Vopěnka's principle is equivalent to each of the following statements.
For every proper class of simple directed graphs, there are two members of the class with a homomorphism between them.
For any signature Σ and any proper class of Σ-structures, there are two members of the class with an elementary embedding between them.
For every predicate P and proper class S of ordinals, there is a non-trivial elementary embedding j:(Vκ, ∈, P) → (Vλ, ∈, P) for some κ and λ in S.
The category of ordinals cannot be fully embedded in the category of graphs.
Every subfunctor of an accessible functor is accessible.
(In a definable classes setting) For every natural number n, there exists a C(n)-extendible cardinal.
Strength
Even when restricted to predicates and proper classes definable in first order set theory, the principle implies existence of Σn correct extendible cardinals for every n.
If κ is an almost huge cardinal, then a strong form of Vopěnka's principle holds in Vκ:
There is a κ-complete ultrafilter U such that for every {Ri: i < κ} where each Ri is a binary relation and Ri ∈ Vκ, there is S ∈ U and a non-trivial elementary embedding j: Ra → Rb for every a < b in S.
References
External links
gives a number of equivalent definitions of Vopěnka's principle.
Large cardinals
Mathematical principles
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https://en.wikipedia.org/wiki/Mercator%20series
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In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:
In summation notation,
The series converges to the natural logarithm (shifted by 1) whenever .
History
The series was discovered independently by Johannes Hudde and Isaac Newton. It was first published by Nicholas Mercator, in his 1668 treatise Logarithmotechnia.
Derivation
The series can be obtained from Taylor's theorem, by inductively computing the nth derivative of at , starting with
Alternatively, one can start with the finite geometric series ()
which gives
It follows that
and by termwise integration,
If , the remainder term tends to 0 as .
This expression may be integrated iteratively k more times to yield
where
and
are polynomials in x.
Special cases
Setting in the Mercator series yields the alternating harmonic series
Complex series
The complex power series
is the Taylor series for , where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number . In fact, as seen by the ratio test, it has radius of convergence equal to 1, therefore converges absolutely on every disk B(0, r) with radius r < 1. Moreover, it converges uniformly on every nibbled disk , with δ > 0. This follows at once from the algebraic identity:
observing that the right-hand side is uniformly convergent on the whole closed unit disk.
See also
John Craig
References
Anton von Braunmühl (1903) Vorlesungen über Geschichte der Trigonometrie, Seite 134, via Internet Archive
Eriksson, Larsson & Wahde. Matematisk analys med tillämpningar, part 3. Gothenburg 2002. p. 10.
Some Contemporaries of Descartes, Fermat, Pascal and Huygens from A Short Account of the History of Mathematics (4th edition, 1908) by W. W. Rouse Ball
Mathematical series
Logarithms
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https://en.wikipedia.org/wiki/Fundamental%20unit%20%28number%20theory%29
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In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. ).
Real quadratic fields
For the real quadratic field (with d square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of K, then the fundamental unit is
where (a, b) is the smallest solution to
in positive integers. This equation is basically Pell's equation or the negative Pell equation and its solutions can be obtained similarly using the continued fraction expansion of .
Whether or not x2 − Δy2 = −4 has a solution determines whether or not the class group of K is the same as its narrow class group, or equivalently, whether or not there is a unit of norm −1 in K. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then K does not have a unit of norm −1. However, the converse does not hold as shown by the example d = 34. In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if D(X) is the number of real quadratic fields whose discriminant Δ < X is not divisible by a prime congruent to 3 modulo 4 and D−(X) is those who have a unit of norm −1, then
In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners who show that the converse fails between 33% and 59% of the time. In 2022, Peter Koymans and Carlo Pagano claimed a complete proof of Stevenhagen's conjecture.
Cubic fields
If K is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding. If the discriminant Δ of K satisfies |Δ| ≥ 33, then
For example, the fundamental unit of is and whereas the discriminant of this field is −108 thus
so .
Notes
References
External links
Algebraic number theory
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https://en.wikipedia.org/wiki/William%20Edge%20%28mathematician%29
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William Leonard Edge FRSE (8 November 1904 – 27 September 1997) was a British mathematician most known for his work in finite geometry. Students knew him as WLE.
Life
Born in Stockport to schoolteacher parents (his father William Henry Edge being a headmaster), Edge attended Stockport Grammar School before winning a place at Trinity College, Cambridge in 1923 with an entrance scholarship, later graduating MA DSc. In 1928 Trinity College made him a Research Fellow and he was also an Allen Scholar.
William Edge was a geometry student of H. F. Baker at Cambridge. Edge's dissertation extended Luigi Cremona’s 1868 delineation of the quadric ruled surfaces in projective 3-space RP3. Edge made a "systematic classification of the quintic and sextic ruled surfaces of three-dimensional projective space."
In 1932 E. T. Whittaker invited Edge to lecture at University of Edinburgh. An anachronism, Edge never drove a motor car and disdained the mass-media of radio and television; he was distressed by the decline of school geometry. In 1949 he became Reader, and professor in 1969.
In the 1950s Edge began to explore vector spaces over Galois fields as an entry to finite geometry. Points and lines of finite projective geometry arise as lines and planes in these spaces, and the projectivities of these spaces provide representation of some finite groups. For example, in 1954 he described the space S over GF(3): 40 points, 13 in each plane and 4 on each line. In S he described a 16-point quadric with two reguli of four lines each.
He also extended work of Moore, Jordan and Dickson on the alternating group A8 as represented by the projective special linear group PSL(4,2). The next year he parametrized the lines of the space S over GF(3) in analogy to the Klein quadric description of lines in RP3.
Edge's student James Hirschfeld has advanced the science of finite geometry also.
In 1934 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, Herbert Westren Turnbull, Edward Thomas Copson and David Gibb. He won the Society's Keith Prize for 1943–45.
Edge retired in 1975. A lifelong bachelor and devout Roman Catholic, Edge spent his final years in the care of the Sisters of Nazareth House in Bonnyrigg, just south of Edinburgh, and died there on 27 September 1997.
Since 2013, every year the School of Mathematics of the University of Edinburgh celebrates the EDGE Days, an annual one-week workshop in algebraic geometry named after Edge.
References
1904 births
1997 deaths
20th-century British mathematicians
British geometers
Alumni of Trinity College, Cambridge
Academics of the University of Edinburgh
People from Stockport
Fellows of the Royal Society of Edinburgh
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https://en.wikipedia.org/wiki/IP%20set
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In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set.
The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D.
The set of all finite sums over D is often denoted as FS(D). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni).
A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A. Equivalently, one may require that A contains all finite sums FS((ni)) of a sequence (ni).
Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset.
The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent" (a set is an IP if and only if it is a member of an idempotent ultrafilter).
Hindman's theorem
If is an IP set and , then at least one is an IP set.
This is known as Hindman's theorem or the finite sums theorem. In different terms, Hindman's theorem states that the class of IP sets is partition regular.
Since the set of natural numbers itself is an IP set and partitions can also be seen as colorings, one can reformulate a special case of Hindman's theorem in more familiar terms: Suppose the natural numbers are "colored" with n different colors; each natural number gets one and only one color. Then there exists a color c and an infinite set D of natural numbers, all colored with c, such that every finite sum over D also has color c.
Hindman's theorem is named for mathematician Neil Hindman, who proved it in 1974.
The Milliken–Taylor theorem is a common generalisation of Hindman's theorem and Ramsey's theorem.
Semigroups
The definition of being IP has been extended from subsets of the special semigroup of natural numbers with addition to subsets of semigroups and partial semigroups in general. A variant of Hindman's theorem is true for arbitrary semigroups.
See also
Ergodic Ramsey theory
Piecewise syndetic set
Syndetic set
Thick set
References
Further reading
Vitaly Bergelson, I. J. H. Knutson, R. McCutcheon "Simultaneous diophantine approximation and VIP Systems" Acta Arith. 116, Academia Scientiarum Polona, (2005), 13-23
Vitaly Bergelson, "Minimal Idempotents and Ergodic Ramsey Theory" Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 317–332
Semigroup theory
Ergodic theory
Ramsey theory
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https://en.wikipedia.org/wiki/Partition%20regularity
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In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets.
Given a set , a collection of subsets is called partition regular if every set A in the collection has the property that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is,
for any , and any finite partition , there exists an i ≤ n such that belongs to . Ramsey theory is sometimes characterized as the study of which collections are partition regular.
Examples
The collection of all infinite subsets of an infinite set X is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.)
Sets with positive upper density in : the upper density of is defined as (Szemerédi's theorem)
For any ultrafilter on a set , is partition regular: for any , if , then exactly one .
Sets of recurrence: a set R of integers is called a set of recurrence if for any measure-preserving transformation of the probability space (Ω, β, μ) and of positive measure there is a nonzero so that .
Call a subset of natural numbers a.p.-rich if it contains arbitrarily long arithmetic progressions. Then the collection of a.p.-rich subsets is partition regular (Van der Waerden, 1927).
Let be the set of all n-subsets of . Let . For each n, is partition regular. (Ramsey, 1930).
For each infinite cardinal , the collection of stationary sets of is partition regular. More is true: if is stationary and for some , then some is stationary.
The collection of -sets: is a -set if contains the set of differences for some sequence .
The set of barriers on : call a collection of finite subsets of a barrier if:
and
for all infinite , there is some such that the elements of X are the smallest elements of I; i.e. and .
This generalizes Ramsey's theorem, as each is a barrier. (Nash-Williams, 1965)
Finite products of infinite trees (Halpern–Läuchli, 1966)
Piecewise syndetic sets (Brown, 1968)
Call a subset of natural numbers i.p.-rich if it contains arbitrarily large finite sets together with all their finite sums. Then the collection of i.p.-rich subsets is partition regular (Folkman–Rado–Sanders, 1968).
(m, p, c)-sets (Deuber, 1973)
IP sets (Hindman, 1974, see also Hindman, Strauss, 1998)
MTk sets for each k, i.e. k-tuples of finite sums (Milliken–Taylor, 1975)
Central sets; i.e. the members of any minimal idempotent in , the Stone–Čech compactification of the integers. (Furstenberg, 1981, see also Hindman, Strauss, 1998)
Diophantine equations
A Diophantine equation is called partition regular if the collection of all infinite subsets of containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations are partition regular. Much progress has been made recently on classifying nonline
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https://en.wikipedia.org/wiki/Truncated%2024-cells
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In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell.
There are two degrees of truncations, including a bitruncation.
Truncated 24-cell
The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure.
Construction
The truncated 24-cell can be constructed from polytopes with three symmetry groups:
F4 [3,4,3]: A truncation of the 24-cell.
B4 [3,3,4]: A cantitruncation of the 16-cell, with two families of truncated octahedral cells.
D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of truncated octahedral cells.
Zonotope
It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,−1,0,0).
Cartesian coordinates
The Cartesian coordinates of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of:
(0,1,2,3) [4!×23 = 192 vertices]
The dual configuration has coordinates at all coordinate permutation and signs of
(1,1,1,5) [4×24 = 64 vertices]
(1,3,3,3) [4×24 = 64 vertices]
(2,2,2,4) [4×24 = 64 vertices]
Structure
The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces.
Projections
The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout:
The projection envelope is a truncated cuboctahedron.
Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope.
Six cuboidal volumes join the square faces of this central truncated octahedron to the center of the octagonal faces of the great rhombicuboctahedron. These are the images of 12 of the cubical cells, a pair of cells to each image.
The 12 square faces of the great rhombicuboctahedron are the images of the remaining 12 cubes.
The 6 octagonal faces of the great rhombicuboctahedron are the images of 6 of the truncated octahedra.
The 8 (non-uniform) truncated octahedral volumes lying between the hexagonal faces of the projection envelope and the central truncated octahedron are the images of the remaining 16 truncated octahedra, a pair of cells to each image.
Images
Related polytopes
The convex hull of the truncated 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 480 cells: 48 cubes, 144 square antiprisms, 288 tetrahedra (as tetragonal disphenoids), and 384 vertices. Its vertex figure is a hexakis triangular cupola.
Vertex figure
Bitruncated 24-cell
The bitruncated 24-cell. 48-cell, or tetracontoctachoron is a 4-dimensional unifo
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https://en.wikipedia.org/wiki/Polycyclic%20group
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In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.
Terminology
Equivalently, a group G is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say G0, ..., Gn such that
Gn coincides with G
G0 is the trivial subgroup
Gi is a normal subgroup of Gi+1 (for every i between 0 and n - 1)
and the quotient group Gi+1 / Gi is a cyclic group (for every i between 0 and n - 1)
A metacyclic group is a polycyclic group with n ≤ 2, or in other words an extension of a cyclic group by a cyclic group.
Examples
Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent groups, and finite solvable groups. Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices. The holomorph of a polycyclic group is also such a group of integer matrices.
Strongly polycyclic groups
A polycyclic group G is said to be strongly polycyclic if each quotient Gi+1 / Gi is infinite. Any subgroup of a strongly polycyclic group is strongly polycyclic.
Polycyclic-by-finite groups
A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite.
In the textbook and some papers, an M-group refers to what is now called a polycyclic-by-finite group, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite cyclic group.
These groups are particularly interesting because they are the only known examples of Noetherian group rings , or group rings of finite injective dimension.
Hirsch length
The Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series.
If G is a polycyclic-by-finite group, then the Hirsch length of G is the Hirsch length of a polycyclic normal subgroup H of G, where H has finite index in G. This is independent of choice of subgroup, as all such subgroups will have the same Hirsch length.
See also
Group theory
Supersolvable group
References
Notes
Properties of groups
Solvable groups
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https://en.wikipedia.org/wiki/Logarithmic%20growth
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In mathematics, logarithmic growth describes a phenomenon whose size or cost can be described as a logarithm function of some input. e.g. y = C log (x). Any logarithm base can be used, since one can be converted to another by multiplying by a fixed constant. Logarithmic growth is the inverse of exponential growth and is very slow.
A familiar example of logarithmic growth is a number, N, in positional notation, which grows as logb (N), where b is the base of the number system used, e.g. 10 for decimal arithmetic. In more advanced mathematics, the partial sums of the harmonic series
grow logarithmically. In the design of computer algorithms, logarithmic growth, and related variants, such as log-linear, or linearithmic, growth are very desirable indications of efficiency, and occur in the time complexity analysis of algorithms such as binary search.
Logarithmic growth can lead to apparent paradoxes, as in the martingale roulette system, where the potential winnings before bankruptcy grow as the logarithm of the gambler's bankroll. It also plays a role in the St. Petersburg paradox.
In microbiology, the rapidly growing exponential growth phase of a cell culture is sometimes called logarithmic growth. During this bacterial growth phase, the number of new cells appearing is proportional to the population. This terminological confusion between logarithmic growth and exponential growth may be explained by the fact that exponential growth curves may be straightened by plotting them using a logarithmic scale for the growth axis.
See also
(an even slower growth model)
References
Logarithms
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https://en.wikipedia.org/wiki/Axiom%20%28disambiguation%29
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An axiom is a proposition in mathematics and epistemology that is taken to be self-evident or is chosen as a starting point of a theory.
Axiom may also refer to:
Music
Axiom (band), a 1970s Australian rock band featuring Brian Cadd and Glenn Shorrock
Axiom (record label), best known for Bill Laswell releases
Axiom (Archive album), 2014
Axiom (Christian Scott album), 2020
"Axiom", a song by British blackened death metal band Akercocke
Axiom (rapper), rapper, beatmaker and record producer
Axioms (album), a 1999 album by Asia
Computers and information technology
Axiom (computer algebra system), a free, general-purpose computer algebra system
AXIOM (camera), a professional grade open hardware and free software digital cinema camera
Axiom Engine, 3D computer graphics engine
Apache Axiom, a library providing a lightweight XML object model
Other uses
Axiom, the name of the luxury starship in the film WALL-E and in the home short BURN-E
Axiom Space, a company planning to build a private space station
Axiom Research Labs, an aerospace company also known as TeamIndus
Axioms (journal), an academic journal
Isuzu Axiom, a sport utility vehicle produced 2001–2004
A-Kid, professional wrestler who wrestles with the current ring name Axiom
See also
Axiomatic (disambiguation)
Axion (disambiguation)
Acxiom (disambiguation)
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https://en.wikipedia.org/wiki/Multiscale%20modeling
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Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion).
An example of such problems involve the Navier–Stokes equations for incompressible fluid flow.
In a wide variety of applications, the stress tensor is given as a linear function of the gradient . Such a choice for has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation.
History
Horstemeyer 2009, 2012 presented a historical review of the different disciplines (mathematics, physics, and materials science) for solid materials related to multiscale materials modeling.
The aforementioned DOE multiscale modeling efforts were hierarchical in nature. The first concurrent multiscale model occurred when Michael Ortiz (Caltech) took the molecular dynamics code, Dynamo, (developed by Mike Baskes at Sandia National Labs) and with his students embedded it into a finite element code for the first time. Martin Karplus, Michael Levitt, Arieh Warshel 2013 were awarded a Nobel Prize in Chemistry for the development of a multiscale model method using both classical and quantum mechanical theory which were used to model large complex chemical systems and reactions.
Areas of research
In physics and chemistry, multiscale modeling is aimed at the calculation of material properties or system behavior on one level using information or models from different levels. On each level, particular approaches are used for the description of a system. The following levels are usually distinguished: level of quantum mechanical models (information about electrons is included), level of molecular dynamics models (information about individual atoms is included), coarse-grained models (information about atoms and/or groups of atoms is included), mesoscale or nano-level (information about large groups of atoms and/or molecule positions is included), level of continuum models, level of device models. Each level addresses a phenomenon over a specific window of length and time. Multiscale modeling is particularly important in integrated computational materials engineering since it allows the prediction of material properties or system behavior based on knowledge of the process-structure-property relationships.
In operations research, multiscale modeling addresses challenges for decision-makers that come from multiscale phenomena across organizational, temporal, and spatial scales. This theory fuses decision theory and multiscale
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https://en.wikipedia.org/wiki/Quotient%20category
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In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.
Definition
Let C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalence relation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if
are related in Hom(X, Y) and
are related in Hom(Y, Z), then g1f1 and g2f2 are related in Hom(X, Z).
Given a congruence relation R on C we can define the quotient category C/R as the category whose objects are those of C and whose morphisms are equivalence classes of morphisms in C. That is,
Composition of morphisms in C/R is well-defined since R is a congruence relation.
Properties
There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).
Every functor F : C → D determines a congruence on C by saying f ~ g iff F(f) = F(g). The functor F then factors through the quotient functor C → C/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for categories.
Examples
Monoids and groups may be regarded as categories with one object. In this case the quotient category coincides with the notion of a quotient monoid or a quotient group.
The homotopy category of topological spaces hTop is a quotient category of Top, the category of topological spaces. The equivalence classes of morphisms are homotopy classes of continuous maps.
Let k be a field and consider the abelian category Mod(k) of all vector spaces over k with k-linear maps as morphisms. To "kill" all finite-dimensional spaces, we can call two linear maps f,g : X → Y congruent iff their difference has finite-dimensional image. In the resulting quotient category, all finite-dimensional vector spaces are isomorphic to 0. [This is actually an example of a quotient of additive categories, see below.]
Related concepts
Quotients of additive categories modulo ideals
If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f1, f2, g1 and g2 are morphisms from X to Y with f1 ~ f2 and g1 ~g2, then f1 + g1 ~ f2 + g2), then the quotient category C/~ will also be additive, and the quotient functor C → C/~ will be an additive functor.
The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all f ∈ I(X,Y), g ∈ HomC(Y, Z) and h∈ HomC(W, X), we have gf ∈ I(X,Z) and fh ∈ I(W,Y). Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).
Every unital ring may be viewed as an additive category with a single object, and the quotient of
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https://en.wikipedia.org/wiki/H%C3%B6lder%20condition
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In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that
for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition. For any α > 0, the condition implies the function is uniformly continuous. The condition is named after Otto Hölder.
We have the following chain of strict inclusions for functions defined on a closed and bounded interval [a, b] of the real line with a < b :
Continuously differentiable ⊂ Lipschitz continuous ⊂ α-Hölder continuous ⊂ uniformly continuous ⊂ continuous,
where 0 < α ≤ 1.
Hölder spaces
Hölder spaces consisting of functions satisfying a Hölder condition are basic in areas of functional analysis relevant to solving partial differential equations, and in dynamical systems. The Hölder space Ck,α(Ω), where Ω is an open subset of some Euclidean space and k ≥ 0 an integer, consists of those functions on Ω having continuous derivatives up through order k and such that the kth partial derivatives are Hölder continuous with exponent α, where 0 < α ≤ 1. This is a locally convex topological vector space. If the Hölder coefficient
is finite, then the function f is said to be (uniformly) Hölder continuous with exponent α in Ω. In this case, the Hölder coefficient serves as a seminorm. If the Hölder coefficient is merely bounded on compact subsets of Ω, then the function f is said to be locally Hölder continuous with exponent α in Ω.
If the function f and its derivatives up to order k are bounded on the closure of Ω, then the Hölder space can be assigned the norm
where β ranges over multi-indices and
These seminorms and norms are often denoted simply and or also and in order to stress the dependence on the domain of f. If Ω is open and bounded, then is a Banach space with respect to the norm .
Compact embedding of Hölder spaces
Let Ω be a bounded subset of some Euclidean space (or more generally, any totally bounded metric space) and let 0 < α < β ≤ 1 two Hölder exponents. Then, there is an obvious inclusion map of the corresponding Hölder spaces:
which is continuous since, by definition of the Hölder norms, we have:
Moreover, this inclusion is compact, meaning that bounded sets in the ‖ · ‖0,β norm are relatively compact in the ‖ · ‖0,α norm. This is a direct consequence of the Ascoli-Arzelà theorem. Indeed, let (un) be a bounded sequence in C0,β(Ω). Thanks to the Ascoli-Arzelà theorem we can assume without loss of generality that un → u uniformly, and we can also assume u = 0. Then
because
Examples
If 0 < α ≤ β ≤ 1 then all Hölder continuous functions on a bounded set Ω are also Hölder continuous. This also i
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https://en.wikipedia.org/wiki/Syndetic
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Syndetic may refer one of the following
Syndetic set, in mathematics
Syndetic coordination, in linguistics
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https://en.wikipedia.org/wiki/Media%20in%20Richmond%2C%20Virginia
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According to Nielsen Media statistics for 2015–2016, the Richmond, Virginia market area is the 56th largest Designated Market Area in the United States, with 549,730 TV households. Richmond is served by a variety of communication media:
Print media
Daily
The local daily newspaper in Richmond is the Richmond Times-Dispatch.
Weekly
Style Weekly (alternative weekly)
Chesterfield Observer
Monthly / bi-monthly / quarterly
NORTH of the JAMES Magazine (monthly)
Boomer Magazine (bi-monthly)
Chesterfield Living, West Ends Best, Hanover Lifestyle (bi-monthly)
Greater Richmond Grid Magazine (bi-monthly)
OurHealth Richmond Magazine (bi-monthly)
Richmond Magazine (monthly)
RVA Magazine (quarterly)
Virginia Business (monthly)
Whurk (monthly)
News and newsmagazines
The Richmond Free Press and the Richmond Voice are weekly newspapers that cover the news from a predominantly African American perspective. The only Hispanic magazine in the state, La Voz Hispana de Virginia provides significant cultural and news content in both English and Spanish. There are also two major publications from the Jewish community of Richmond, published in English; The Reflector is the semi-weekly newspaper of the Jewish Federation of Richmond and Virginia Jewish Life (formerly Virginia Jewish News) is an independent monthly magazine published by the Chabad community of Richmond, but highlighting stories of general Jewish interest in Virginia. City Edition was a civic-minded newspaper that listed municipal and council related events, issues, and results, which stopped publication in October 2007. . Richmond.com is an online newsmagazine with a wide readership. Other local topical publications include Richmond Parents Magazine and V Magazine for Women. the voice of women in Richmond. Richmond Guide is a quarterly that is targeted toward visitors. The Virginia Defender is a quarterly statewide community newspaper with a press run of 16,000 distributed through nearly 300 distribution sites in Richmond, plus 16 other Virginia cities and five counties.
Richmond's leading African American newspaper at the turn of the century was the Richmond Planet which ran from 1883 to 1996 and was edited by John Mitchel, Jr. from 1884 until his death in 1929.
Regional and county newspapers include the following:
The Amelia Bulletin Monitor for Amelia County, Virginia
Capital News Service at VCU for regional and national news
The Chesterfield Observer for Chesterfield County, Virginia
The Goochland Courier for Goochland County, Virginia
The Goochland Gazette for Goochland County, Virginia
The Henrico Citizen for Henrico County, Virginia
The Hopewell News (defunct) for Hopewell, Virginia
Petersburg Progress-Index for Petersburg, Virginia
RVA Magazine
Virginia Living is a glossy magazine published bi-monthly that covers Virginia events.
Student Operated
Many colleges and universities in Richmond have student operated new services including The Richard T. Robertson Sch
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https://en.wikipedia.org/wiki/Cheeger%20bound
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In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs.
Let be a finite set and let be the transition probability for a reversible Markov chain on . Assume this chain has stationary distribution .
Define
and for define
Define the constant as
The operator acting on the space of functions from to , defined by
has eigenvalues . It is known that . The Cheeger bound is a bound on the second largest eigenvalue .
Theorem (Cheeger bound):
See also
Stochastic matrix
Cheeger constant
References
J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in Analysis, Papers dedicated to Salomon Bochner, 1969, Princeton University Press, Princeton, 195-199.
P. Diaconis, D. Stroock, Geometric bounds for eigenvalues of Markov chains, Annals of Applied Probability, vol. 1, 36-61, 1991, containing the version of the bound presented here.
Probabilistic inequalities
Stochastic processes
Statistical inequalities
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https://en.wikipedia.org/wiki/DCAS
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DCAS may be:
DCAS keys, control keys on the computer keyboard, see
Deputy Chief of the Air Staff (disambiguation)
Derive computer algebra system
Double compare-and-swap
Downloadable Conditional Access System
New York City Department of Citywide Administrative Services
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https://en.wikipedia.org/wiki/SageMath
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SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics.
The first version of SageMath was released on 24 February 2005 as free and open-source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, was a mathematician at the University of Washington.
SageMath uses a syntax resembling Python's, supporting procedural, functional and object-oriented constructs.
Development
Stein realized when designing Sage that there were many open-source mathematics software packages already written in different languages, namely C, C++, Common Lisp, Fortran and Python.
Rather than reinventing the wheel, Sage (which is written mostly in Python and Cython) integrates many specialized CAS software packages into a common interface, for which a user needs to know only Python. However, Sage contains hundreds of thousands of unique lines of code adding new functions and creating the interfaces among its components.
SageMath uses both students and professionals for development. The development of SageMath is supported by both volunteer work and grants. However, it was not until 2016 that the first full-time Sage developer was hired (funded by an EU grant). The same year, Stein described his disappointment with a lack of academic funding and credentials for software development, citing it as the reason for his decision to leave his tenured academic position to work full-time on the project in a newly founded company, SageMath, Inc.
Achievements
2007: first prize in the scientific software division of Les Trophées du Libre, an international competition for free software.
2012: one of the projects selected for the Google Summer of Code.
2013: ACM/SIGSAM Jenks Prize.
Performance
Both binaries and source code are available for SageMath from the download page. If SageMath is built from source code, many of the included libraries such as OpenBLAS, FLINT, GAP (computer algebra system), and NTL will be tuned and optimized for that computer, taking into account the number of processors, the size of their caches, whether there is hardware support for SSE instructions, etc.
Cython can increase the speed of SageMath programs, as the Python code is converted into C.
Licensing and availability
SageMath is free software, distributed under the terms of the GNU General Public License version 3.
Although Microsoft was sponsoring a native version of SageMath for the Windows operating system, prior to 2016 there were no plans for a native port, and users of Windows had to use virtualization technology such as VirtualBox to run SageMath. SageMath 8.0 (July 2017), with devel
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https://en.wikipedia.org/wiki/Peirce%27s%20criterion
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In robust statistics, Peirce's criterion is a rule for eliminating outliers from data sets, which was devised by Benjamin Peirce.
Outliers removed by Peirce's criterion
The problem of outliers
In data sets containing real-numbered measurements, the suspected outliers are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust.
In the presence of outliers, the statistician has two options. First, the statistician may remove the suspected outliers from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median statistic.
Peirce's criterion is a statistical procedure for eliminating outliers.
Uses of Peirce's criterion
The statistician and historian of statistics Stephen M. Stigler wrote the following about Benjamin Peirce:
"In 1852 he published the first significance test designed to tell an investigator whether an outlier should be rejected (Peirce 1852, 1878). The test, based on a likelihood ratio type of argument, had the distinction of producing an international debate on the wisdom of such actions (Anscombe, 1960, Rider, 1933, Stigler, 1973a)."
Peirce's criterion is derived from a statistical analysis of the Gaussian distribution. Unlike some other criteria for removing outliers, Peirce's method can be applied to identify two or more outliers.
"It is proposed to determine in a series of observations the limit of error, beyond which all observations involving so great an error may be rejected, provided there are as many as such observations. The principle upon which it is proposed to solve this problem is, that the proposed observations should be rejected when the probability of the system of errors obtained by retaining them is less than that of the system of errors obtained by their rejection multiplied by the probability of making so many, and no more, abnormal observations."
Hawkins provides a formula for the criterion.
Peirce's criterion was used for decades at the United States Coast Survey.
"From 1852 to 1867 he served as the director of the longitude determinations of the U. S. Coast Survey and from 1867 to 1874 as superintendent of the Survey. During these years his test was consistently employed by all the clerks of this, the most active and mathematically inclined statistical organization of the era."
Peirce's criterion was discussed in William Chauvenet's book.
Applications
An application for Peirce's criterion is removing poor data points from observation pairs in order to perform a regression between the two observations (e.g., a linear regression). Peirce's criterion does not depend on observation
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https://en.wikipedia.org/wiki/Reciprocal%20gamma%20function
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In mathematics, the reciprocal gamma function is the function
where denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is of order 1 (meaning that grows no faster than ), but of infinite type (meaning that grows faster than any multiple of , since its growth is approximately proportional to in the left-half plane).
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function.
Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
Infinite product expansion
Following from the infinite product definitions for the gamma function, due to Euler and Weierstrass respectively, we get the following infinite product expansion for the reciprocal gamma function:
where is the Euler–Mascheroni constant. These expansions are valid for all complex numbers .
Taylor series
Taylor series expansion around 0 gives:
where is the Euler–Mascheroni constant. For , the coefficient for the term can be computed recursively as
where is the Riemann zeta function. An integral representation for these coefficients was recently found by Fekih-Ahmed (2014):
For small values, these give the following values:
Fekih-Ahmed (2014) also gives an approximation for :
where and is the minus-first branch of the Lambert W function.
The Taylor expansion around has the same (but shifted) coefficients, i.e.:
(the reciprocal of Gauss' pi-function).
Asymptotic expansion
As goes to infinity at a constant we have:
Contour integral representation
An integral representation due to Hermann Hankel is
where is the Hankel contour, that is, the path encircling 0 in the positive direction, beginning at and returning to positive infinity with respect for the branch cut along the positive real axis. According to Schmelzer & Trefethen, numerical evaluation of Hankel's integral is the basis of some of the best methods for computing the gamma function.
Integral representations at the positive integers
For positive integers , there is an integral for the reciprocal factorial function given by
Similarly, for any real and we have the next integral for the reciprocal gamma function along the real axis in the form of:
where the particular case when provides a corresponding relation for the reciprocal double factorial function,
Integral along the real axis
Integration of the reciprocal gamma function along the positive real axis gives the value
which is known as the Fransén–Robinson constant.
We have the following formula ( chapter 9, exercise 100)
See also
Bessel–Clifford function
Inverse-gamma distribution
References
Mette Lund, An integral for the reciprocal Gamma function
Milton Abramowitz & Irene A. Stegun, Handbook of Ma
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https://en.wikipedia.org/wiki/Algebraic%20specification
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Algebraic specification is a software engineering technique for formally specifying system behavior. It was a very active subject of computer science research around 1980.
Overview
Algebraic specification seeks to systematically develop more efficient programs by:
formally defining types of data, and mathematical operations on those data types
abstracting implementation details, such as the size of representations (in memory) and the efficiency of obtaining outcome of computations
formalizing the computations and operations on data types
allowing for automation by formally restricting operations to this limited set of behaviors and data types.
An algebraic specification achieves these goals by defining one or more data types, and specifying a collection of functions that operate on those data types. These functions can be divided into two classes:
Constructor functions: Functions that create or initialize the data elements, or construct complex elements from simpler ones. The set of available constructor functions is implied by the specification's signature. Additionally, a specification can contain equations defining equivalences between the objects constructed by these functions. Whether the underlying representation is identical for different but equivalent constructions is implementation-dependent.
Additional functions: Functions that operate on the data types, and are defined in terms of the constructor functions.
Examples
Consider a formal algebraic specification for the boolean data type.
One possible algebraic specification may provide two constructor functions for the data-element: a true constructor and a false constructor. Thus, a boolean data element could be declared, constructed, and initialized to a value. In this scenario, all other connective elements, such as XOR and AND, would be additional functions. Thus, a data element could be instantiated with either "true" or "false" value, and additional functions could be used to perform any operation on the data element.
Alternatively, the entire system of boolean data types could be specified using a different set of constructor functions: a false constructor and a not constructor. In that case, an additional function true could be defined to yield the value not false, and an equation should be added.
The algebraic specification therefore describes all possible states of the data element, and all possible transitions between states.
For a more complicated example, the integers can be specified (among many other ways, and choosing one of the many formalisms) with two constructors
1 : Z
(_ - _) : Z × Z -> Z
and three equations:
(1 - (1 - p)) = p
((1 - (n - p)) - 1) = (p - n)
((p1 - n1) - (n2 - p2)) = (p1 - (n1 - (p2 - n2)))
It is easy to verify that the equations are valid, given the usual interpretation of the binary "minus" function. (The variable names have been chosen to hint at positive and negative contributions to the value.) Wi
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https://en.wikipedia.org/wiki/Jensen%E2%80%93Shannon%20divergence
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In probability theory and statistics, the Jensen–Shannon divergence is a method of measuring the similarity between two probability distributions. It is also known as information radius (IRad) or total divergence to the average. It is based on the Kullback–Leibler divergence, with some notable (and useful) differences, including that it is symmetric and it always has a finite value. The square root of the Jensen–Shannon divergence is a metric often referred to as Jensen–Shannon distance.
Definition
Consider the set of probability distributions where is a set provided with some σ-algebra of measurable subsets. In particular we can take to be a finite or countable set with all subsets being measurable.
The Jensen–Shannon divergence (JSD) is a symmetrized and smoothed version of the Kullback–Leibler divergence . It is defined by
where is a mixture distribution of and .
The geometric Jensen–Shannon divergence (or G-Jensen–Shannon divergence) yields a closed-form formula for divergence between two Gaussian distributions by taking the geometric mean.
A more general definition, allowing for the comparison of more than two probability distributions, is:
where
and are weights that are selected for the probability distributions , and is the Shannon entropy for distribution . For the two-distribution case described above,
Hence, for those distributions
Bounds
The Jensen–Shannon divergence is bounded by 1 for two probability distributions, given that one uses the base 2 logarithm:
.
With this normalization, it is a lower bound on the total variation distance between P and Q:
.
With base-e logarithm, which is commonly used in statistical thermodynamics, the upper bound is . In general, the bound in base b is :
.
A more general bound, the Jensen–Shannon divergence is bounded by for more than two probability distributions:
.
Relation to mutual information
The Jensen–Shannon divergence is the mutual information between a random variable associated to a mixture distribution between and and the binary indicator variable that is used to switch between and to produce the mixture. Let be some abstract function on the underlying set of events that discriminates well between events, and choose the value of according to if and according to if , where is equiprobable. That is, we are choosing according to the probability measure , and its distribution is the mixture distribution. We compute
It follows from the above result that the Jensen–Shannon divergence is bounded by 0 and 1 because mutual information is non-negative and bounded by in base 2 logarithm.
One can apply the same principle to a joint distribution and the product of its two marginal distribution (in analogy to Kullback–Leibler divergence and mutual information) and to measure how reliably one can decide if a given response comes from the joint distribution or the product distribution—subject to the assumption that these are the only two possibilities.
Quantu
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https://en.wikipedia.org/wiki/Elizabeth%20Scott%20%28mathematician%29
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Elizabeth Leonard Scott (November 23, 1917 – December 20, 1988) was an American mathematician specializing in statistics.
Scott was born in Fort Sill, Oklahoma. Her family moved to Berkeley, California when she was 4 years old. She attended the University of California, Berkeley where she studied astronomy. She earned her Ph.D. in 1949 in astronomy, and received a permanent position in the Department of Mathematics at Berkeley in 1951.
She wrote over 30 papers on astronomy and 30 on weather modification research analysis, incorporating and expanding the use of statistical analyses in these fields. She also used statistics to promote equal opportunities and equal pay for female academics.
In 1957 Scott noted a bias in the observation of galaxy clusters. She noticed that for an observer to find a very distant cluster, it must contain brighter-than-normal galaxies and must also contain a large number of galaxies. She proposed a correction formula to adjust for (what came to be known as) the Scott effect.
Dr. Scott was a Fellow of the Institute of Mathematical Statistics.
The Committee of Presidents of Statistical Societies awards a prize in her honor, the Elizabeth L. Scott Award, for "fostering opportunities in statistics for women".
References
1917 births
1988 deaths
American statisticians
Women statisticians
Fellows of the Institute of Mathematical Statistics
Presidents of the Institute of Mathematical Statistics
20th-century American mathematicians
20th-century women mathematicians
University of California, Berkeley alumni
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https://en.wikipedia.org/wiki/Berlin%20population%20statistics
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Berlin is the most populous city in the European Union, as calculated by city-proper population (not metropolitan area).
Demographics
Population by borough
Historical development of Berlin's population
The spike in population in 1920 is a result of the Greater Berlin Act.
Population by nationality
On 31 December 2010 the largest groups by foreign nationality were citizens from Turkey (104,556), Poland (40,988), Serbia (19,230), Italy (15,842), Russia (15,332), United States (12,733), France (13,262), Vietnam (13,199), Croatia (10,104), Bosnia and Herzegovina (10,198), UK (10,191), Greece (9,301), Austria (9,246), Ukraine (8,324), Lebanon (7,078), Spain (7,670), Bulgaria (9,988), the People's Republic of China (5,632), Thailand (5,037). There is also a large Arabic community, mostly from Lebanon, Palestine and Iraq. Additionally, Berlin has one of the largest Vietnamese communities outside Vietnam, with about 83,000 people of Vietnamese origin.
See also
Demographics of Berlin
Demographics of Cologne
Demographics of Hamburg
Demographics of Munich
References
External links
Berlin State Statistical Office
Berlin State Statistical Office (old homepage)
Schwenk, Herbert, Berliner Stadtentwicklung von A bis Z: Kleines Handbuch zum Werden und Wachsen der deutschen Hauptstadt, 2nd edition. Berlin: Luisenstädtischer Bildungsverein, 1998.
Geography of Berlin
History of Berlin
Demographics of Germany
Demographics by city
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https://en.wikipedia.org/wiki/Cho%20Hyun
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Cho Hyun is a football player from South Korea.
He was a member of the South Korea U-20 team in early 1990s and went on to play as a professional in the K-League.
Club career statistics
External links
1974 births
Living people
Men's association football midfielders
South Korean men's footballers
Suwon Samsung Bluewings players
Ulsan Hyundai FC players
K League 1 players
Dongguk University alumni
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https://en.wikipedia.org/wiki/Truncated%20tesseract
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In geometry, a truncated tesseract is a uniform 4-polytope formed as the truncation of the regular tesseract.
There are three truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell.
Truncated tesseract
The truncated tesseract is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra.
Alternate names
Truncated tesseract (Norman W. Johnson)
Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers)
Construction
The truncated tesseract may be constructed by truncating the vertices of the tesseract at of the edge length. A regular tetrahedron is formed at each truncated vertex.
The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:
Projections
In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:
The projection envelope is a cube.
Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope.
The other 6 truncated cubes project onto the square faces of the envelope.
The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.
Images
Related polytopes
The truncated tesseract, is third in a sequence of truncated hypercubes:
Bitruncated tesseract
The bitruncated tesseract, bitruncated 16-cell, or tesseractihexadecachoron is constructed by a bitruncation operation applied to the tesseract. It can also be called a runcicantic tesseract with half the vertices of a runcicantellated tesseract with a construction.
Alternate names
Bitruncated tesseract/Runcicantic tesseract (Norman W. Johnson)
Tesseractihexadecachoron (Acronym tah) (George Olshevsky, and Jonathan Bowers)
Construction
A tesseract is bitruncated by truncating its cells beyond their midpoints, turning the eight cubes into eight truncated octahedra. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other.
The Cartesian coordinates of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of:
Structure
The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.
Projections
Stereographic projections
The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled
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https://en.wikipedia.org/wiki/Penelope%20Maddy
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Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Emerita UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on mathematical realism (especially set-theoretic realism) and mathematical naturalism.
Education and career
Maddy received her Ph.D. from Princeton University in 1979. Her dissertation, Set Theoretical Realism, was supervised by John P. Burgess. She taught at the University of Notre Dame and University of Illinois, Chicago before joining Irvine in 1987.
She was elected a Fellow of the American Academy of Arts and Sciences in 1998.
The German Mathematical Society awarded her a Gauss Lectureship in 2006.
Philosophical work
Maddy's early work, culminating in Realism in Mathematics, defended Kurt Gödel's position that mathematics is a true description of a mind-independent realm that we can access through our intuition. However, she suggested that some mathematical entities are in fact concrete, unlike, notably, Gödel, who assumed all mathematical objects are abstract. She suggested that sets can be causally efficacious, and in fact share all the causal and spatiotemporal properties of their elements. Thus, when one sees three cups on a table, one also sees the set. She used contemporary work in cognitive science and psychology to support this position, pointing out that just as at a certain age we begin to see objects rather than mere sense perceptions, there is also a certain age at which we begin to see sets rather than just objects.
In the 1990s, she moved away from this position, towards a position described in Naturalism in Mathematics. Her "naturalist" position, like Quine's, suggests that since science is our most successful project so far for knowing about the world, philosophers should adopt the methods of science in their own discipline, and especially when discussing science. As Maddy stated in an interview, "If you're a 'naturalist', you think that science shouldn't be held to extra-scientific standards, that it doesn't require extra-scientific ratification." However, rather than a unified picture of the sciences like Quine's, her picture has mathematics as separate. That is, mathematics is neither supported nor undermined by the needs and goals of science but is allowed to obey its own criteria. This means that traditional metaphysical and epistemological concerns of the philosophy of mathematics are misplaced. Like Wittgenstein, she suggests that many of these puzzles arise merely because of the application of language outside its proper domain of significance.
She has been dedicated to understanding and explaining the methods that set theorists use in agreeing on axioms, especially those that go beyond ZFC.
Selected publications
(a copy with corrections is available at the author's web page)
Realism in Mathematics,
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https://en.wikipedia.org/wiki/Rectified%20tesseract
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In geometry, the rectified tesseract, rectified 8-cell is a uniform 4-polytope (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a runcic tesseract.
It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract, rr{3,31,1}, the second alternating with two types of tetrahedral cells.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC8.
Construction
The rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges.
The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:
Images
Projections
In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:
The projection envelope is a cube.
A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells.
The remaining 6 cuboctahedral cells are projected to the square faces of the cube.
The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image.
Alternative names
Rit (Jonathan Bowers: for rectified tesseract)
Ambotesseract (Neil Sloane & John Horton Conway)
Rectified tesseract/Runcic tesseract (Norman W. Johnson)
Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
Related uniform polytopes
Runcic cubic polytopes
Tesseract polytopes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
4-polytopes
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https://en.wikipedia.org/wiki/Perfect%20power
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In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k).
Examples and sums
A sequence of perfect powers can be generated by iterating through the possible values for m and k. The first few ascending perfect powers in numerical order (showing duplicate powers) are :
The sum of the reciprocals of the perfect powers (including duplicates such as 34 and 92, both of which equal 81) is 1:
which can be proved as follows:
The first perfect powers without duplicates are:
(sometimes 0 and 1), 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, ...
The sum of the reciprocals of the perfect powers p without duplicates is:
where μ(k) is the Möbius function and ζ(k) is the Riemann zeta function.
According to Euler, Goldbach showed (in a now-lost letter) that the sum of over the set of perfect powers p, excluding 1 and excluding duplicates, is 1:
This is sometimes known as the Goldbach–Euler theorem.
Detecting perfect powers
Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to . So if the divisors of are then one of the values must be equal to n if n is indeed a perfect power.
This method can immediately be simplified by instead considering only prime values of k. This is because if for a composite where p is prime, then this can simply be rewritten as . Because of this result, the minimal value of k must necessarily be prime.
If the full factorization of n is known, say where the are distinct primes, then n is a perfect power if and only if where gcd denotes the greatest common divisor. As an example, consider n = 296·360·724. Since gcd(96, 60, 24) = 12, n is a perfect 12th power (and a perfect 6th power, 4th power, cube and square, since 6, 4, 3 and 2 divide 12).
Gaps between perfect powers
In 2002 Romanian mathematician Preda Mihăilescu proved that the only pair of consecutive perfect powers is 23 = 8 and 32 = 9, thus proving Catalan's conjecture.
Pillai's conjecture states that for any given positive integer k there are only a finite number of pairs of perfect powers whose difference is k. This is an unsolved problem.
See also
Prime power
References
External links
Lluís Bibiloni, Pelegrí Viade
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https://en.wikipedia.org/wiki/Universal%20hashing
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In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarantees a low number of collisions in expectation, even if the data is chosen by an adversary. Many universal families are known (for hashing integers, vectors, strings), and their evaluation is often very efficient. Universal hashing has numerous uses in computer science, for example in implementations of hash tables, randomized algorithms, and cryptography.
Introduction
Assume we want to map keys from some universe into bins (labelled ). The algorithm will have to handle some data set of keys, which is not known in advance. Usually, the goal of hashing is to obtain a low number of collisions (keys from that land in the same bin). A deterministic hash function cannot offer any guarantee in an adversarial setting if , since the adversary may choose to be precisely the preimage of a bin. This means that all data keys land in the same bin, making hashing useless. Furthermore, a deterministic hash function does not allow for rehashing: sometimes the input data turns out to be bad for the hash function (e.g. there are too many collisions), so one would like to change the hash function.
The solution to these problems is to pick a function randomly from a family of hash functions. A family of functions is called a universal family if, .
In other words, any two different keys of the universe collide with probability at most when the hash function is drawn uniformly at random from . This is exactly the probability of collision we would expect if the hash function assigned truly random hash codes to every key.
Sometimes, the definition is relaxed by a constant factor, only requiring collision probability rather than . This concept was introduced by Carter and Wegman in 1977, and has found numerous applications in computer science (see, for .
If we have an upper bound of on the collision probability, we say that we have -almost universality. So for example, a universal family has -almost universality.
Many, but not all, universal families have the following stronger uniform difference property:
, when is drawn randomly from the family , the difference is uniformly distributed in .
Note that the definition of universality is only concerned with whether , which counts collisions. The uniform difference property is stronger.
(Similarly, a universal family can be XOR universal if , the value is uniformly distributed in where is the bitwise exclusive or operation. This is only possible if is a power of two.)
An even stronger condition is pairwise independence: we have this property when we have the probability that will hash to any pair of hash values is as if they were perfectly random: . Pairwise independence is sometimes called strong universality.
Another property is uniformity. We say
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https://en.wikipedia.org/wiki/Kaplansky%27s%20conjectures
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The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures.
Group rings
Let be a field, and a torsion-free group. Kaplansky's zero divisor conjecture states:
The group ring does not contain nontrivial zero divisors, that is, it is a domain.
Two related conjectures are known as, respectively, Kaplansky's idempotent conjecture:
does not contain any non-trivial idempotents, i.e., if , then or .
and Kaplansky's unit conjecture (which was originally made by Graham Higman and popularized by Kaplansky):
does not contain any non-trivial units, i.e., if in , then for some in and in .
The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved for fields of positive characteristic by Giles Gardam in February 2021: he published a preprint on the arXiv that constructs a counterexample. The field is of characteristic 2. (see also: Fibonacci group)
There are proofs of both the idempotent and zero-divisor conjectures for large classes of groups. For example, the zero-divisor conjecture is known for all torsion-free elementary amenable groups (a class including all virtually solvable groups), since their group algebras are known to be Ore domains. It follows that the conjecture holds more generally for all residually torsion-free elementary amenable groups. Note that when is a field of characteristic zero, then the zero-divisor conjecture is implied by the Atiyah conjecture, which has also been established for large classes of groups.
The idempotent conjecture has a generalisation, the Kadison idempotent conjecture, also known as the Kadison–Kaplansky conjecture, for elements in the reduced group C*-algebra. In this setting, it is known that if the Farrell–Jones conjecture holds for , then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all hyperbolic groups.
The unit conjecture is also known to hold in many groups, but its partial solutions are much less robust than the other two. For example, there is a torsion-free 3-dimensional crystallographic group for which it is not known whether all units are trivial. This conjecture is not known to follow from any analytic statement like the other two, and so the cases where it is known to hold have all been established via a direct combinatorial approach involving the so-called unique products property. By Gardam's work mentioned above, it is now known to not be true in general.
Banach algebras
This conjecture states that every algebra homomorphism from the Banach algebra C(X) (continuous complex-valued functions on X, where X is a compact Hausdorff space) into any other Banach algebra, is necessari
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https://en.wikipedia.org/wiki/Reflection%20formula
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In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation, and it is very common in the literature to use the term "functional equation" when "reflection formula" is meant.
Reflection formulas are useful for numerical computation of special functions. In effect, an approximation that has greater accuracy or only converges on one side of a reflection point (typically in the positive half of the complex plane) can be employed for all arguments.
Known formulae
The even and odd functions satisfy by definition simple reflection relations around a = 0. For all even functions,
and for all odd functions,
A famous relationship is Euler's reflection formula
for the gamma function , due to Leonhard Euler.
There is also a reflection formula for the general n-th order polygamma function ψ(n)(z),
which springs trivially from the fact that the polygamma functions are defined as the derivatives of and thus inherit the reflection formula.
The Riemann zeta function ζ(z) satisfies
and the Riemann Xi function ξ(z) satisfies
References
Calculus
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https://en.wikipedia.org/wiki/Relation%20construction
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In logic and mathematics, relation construction and relational constructibility have to do with the ways that one relation is determined by an indexed family or a sequence of other relations, called the relation dataset. The relation in the focus of consideration is called the faciendum. The relation dataset typically consists of a specified relation over sets of relations, called the constructor, the factor, or the method of construction, plus a specified set of other relations, called the faciens, the ingredients, or the makings.
Relation composition and relation reduction are special cases of relation constructions.
See also
Projection
Relation
Relation composition
Mathematical relations
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https://en.wikipedia.org/wiki/Generalized%20polygon
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In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized n-gons encompass as special cases projective planes (generalized triangles, n = 3) and generalized quadrangles (n = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the Moufang property have been completely classified by Tits and Weiss. Every generalized n-gon with n even is also a near polygon.
Definition
A generalized 2-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line.
For a generalized n-gon is an incidence structure (), where is the set of points, is the set of lines and is the incidence relation, such that:
It is a partial linear space.
It has no ordinary m-gons as subgeometry for .
It has an ordinary n-gon as a subgeometry.
For any there exists a subgeometry () isomorphic to an ordinary n-gon such that .
An equivalent but sometimes simpler way to express these conditions is: consider the bipartite incidence graph with the vertex set and the edges connecting the incident pairs of points and lines.
The girth of the incidence graph is twice the diameter n of the incidence graph.
From this it should be clear that the incidence graphs of generalized polygons are Moore graphs.
A generalized polygon is of order (s,t) if:
all vertices of the incidence graph corresponding to the elements of have the same degree s + 1 for some natural number s; in other words, every line contains exactly s + 1 points,
all vertices of the incidence graph corresponding to the elements of have the same degree t + 1 for some natural number t; in other words, every point lies on exactly t + 1 lines.
We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order.
The dual of a generalized n-gon (), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the converse relation of . It can easily be shown that this is again a generalized n-gon.
Examples
The incidence graph of a generalized digon is a complete bipartite graph Ks+1,t+1.
For any natural n ≥ 3, consider the boundary of the ordinary polygon with n sides. Declare the vertices of the polygon to be the points and the sides to be the lines, with set inclusion as the incidence relation. This results in a generalized n-gon with s = t = 1.
For each group of Lie type G of rank 2 there is an associated generalized n-gon X with n equal to 3, 4, 6 or 8 such that G acts transitively on the set of flags of X. In the finite case, for n=6, one obtains the Split Cayley hexagon of order (q, q) for G2(q) and the twisted triality hexagon of order (q3, q) for 3D4(q3), and for n=8, one obtains the Ree-Tits octagon of order (q, q2) for 2F4(q)
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https://en.wikipedia.org/wiki/Planar%20ternary%20ring
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In mathematics, an algebraic structure consisting of a non-empty set and a ternary mapping may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation is defined by . Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication.
There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ternary ring" often means a planar ternary ring, but it can also simply mean a ternary system.
Definition
A planar ternary ring is a structure where is a set containing at least two distinct elements, called 0 and 1, and is a mapping which satisfies these five axioms:
;
;
, there is a unique such that : ;
, there is a unique , such that ; and
, the equations have a unique solution .
When is finite, the third and fifth axioms are equivalent in the presence of the fourth.
No other pair (0', 1') in can be found such that still satisfies the first two axioms.
Binary operations
Addition
Define . The structure is a loop with identity element 0.
Multiplication
Define . The set is closed under this multiplication. The structure is also a loop, with identity element 1.
Linear PTR
A planar ternary ring is said to be linear if .
For example, the planar ternary ring associated to a quasifield is (by construction) linear.
Connection with projective planes
Given a planar ternary ring , one can construct a projective plane with point set P and line set L as follows: (Note that is an extra symbol not in .)
Let
, and
.
Then define, , the incidence relation in this way:
Every projective plane can be constructed in this way, starting with an appropriate planar ternary ring. However, two nonisomorphic planar ternary rings can lead to the construction of isomorphic projective planes.
Conversely, given any projective plane π, by choosing four points, labelled o, e, u, and v, no three of which lie on the same line, coordinates can be introduced in π so that these special points are given the coordinates: o = (0,0), e = (1,1), v = () and u = (0). The ternary operation is now defined on the coordinate symbols (except ) by y = T(x,a,b) if and only if the point (x,y) lies on the line which joins (a) with (0,b). The axioms defining a projective plane are used to show that this gives a planar ternary ring.
Linearity of the PTR is equivalent to a geometric condition holding in the associated projective plane.
Related algebraic structures
PTR's which satisfy additional algebraic conditions are given other names. These names
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https://en.wikipedia.org/wiki/Petersson%20inner%20product
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In mathematics the Petersson inner product is an inner product defined on the space
of entire modular forms. It was introduced by the German mathematician Hans Petersson.
Definition
Let be the space of entire modular forms of weight and
the space of cusp forms.
The mapping ,
is called Petersson inner product, where
is a fundamental region of the modular group and for
is the hyperbolic volume form.
Properties
The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.
For the Hecke operators , and for forms of level , we have:
This can be used to show that the space of cusp forms of level has an orthonormal basis consisting of
simultaneous eigenfunctions for the Hecke operators and the Fourier coefficients of these
forms are all real.
References
T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer Verlag Berlin Heidelberg New York 1990,
M. Koecher, A. Krieg, Elliptische Funktionen und Modulformen, Springer Verlag Berlin Heidelberg New York 1998,
S. Lang, Introduction to Modular Forms, Springer Verlag Berlin Heidelberg New York 2001,
Modular forms
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https://en.wikipedia.org/wiki/Homotopy%20extension%20property
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In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations.
Definition
Let be a topological space, and let . We say that the pair has the homotopy extension property if, given a homotopy and a map such that then there exists an extension of to a homotopy such that .
That is, the pair has the homotopy extension property if any map can be extended to a map (i.e. and agree on their common domain).
If the pair has this property only for a certain codomain , we say that has the homotopy extension property with respect to .
Visualisation
The homotopy extension property is depicted in the following diagram
If the above diagram (without the dashed map) commutes (this is equivalent to the conditions above), then pair (X,A) has the homotopy extension property if there exists a map which makes the diagram commute. By currying, note that homotopies expressed as maps are in natural bijection with expressions as maps .
Note that this diagram is dual to (opposite to) that of the homotopy lifting property; this duality is loosely referred to as Eckmann–Hilton duality.
Properties
If is a cell complex and is a subcomplex of , then the pair has the homotopy extension property.
A pair has the homotopy extension property if and only if is a retract of
Other
If has the homotopy extension property, then the simple inclusion map is a cofibration.
In fact, if you consider any cofibration , then we have that is homeomorphic to its image under . This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.
See also
Homotopy lifting property
References
Homotopy theory
Algebraic topology
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https://en.wikipedia.org/wiki/Quasifield
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In mathematics, a quasifield is an algebraic structure where and are binary operations on , much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields.
Definition
A quasifield is a structure, where and are binary operations on , satisfying these axioms:
is a group
is a loop, where
(left distributivity)
has exactly one solution for ,
Strictly speaking, this is the definition of a left quasifield. A right quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry.
Although not assumed, one can prove that the axioms imply that the additive group is abelian. Thus, when referring to an abelian quasifield, one means that is abelian.
Kernel
The kernel of a quasifield is the set of all elements such that:
Restricting the binary operations and to , one can shown that is a division ring.
One can now make a vector space of over , with the following scalar multiplication :
As a finite division ring is a finite field by Wedderburn's theorem, the order of the kernel of a finite quasifield is a prime power. The vector space construction implies that the order of any finite quasifield must also be a prime power.
Examples
All division rings, and thus all fields, are quasifields.
A (right) near-field that is a (right) quasifield is called a "planar near-field".
The smallest quasifields are abelian and unique. They are the finite fields of orders up to and including eight. The smallest quasifields that are not division rings are the four non-abelian quasifields of order nine; they are presented in and .
Projective planes
Given a quasifield , we define a ternary map by
One can then verify that satisfies the axioms of a planar ternary ring. Associated to is its corresponding projective plane. The projective planes constructed this way are characterized as follows;
the details of this relationship are given in .
A projective plane is a translation plane with respect to the line at infinity if and only if any (or all) of its associated planar ternary rings are right quasifields. It is called a shear plane if any (or all) of its ternary rings are left quasifields.
The plane does not uniquely determine the ring; all 4 nonabelian quasifields of order 9 are ternary rings for the unique non-Desarguesian translation plane of order 9. These differ in the fundamental quadrilateral used to construct the plane (see Weibel 2007).
History
Quasifields were called "Veblen–Wedderburn systems" in the literature before 1975, since they were first studied in the
1907 paper (Veblen-Wedderburn 1907) by O. Veblen and J. Wedderburn. Surveys of quasifields and their applications to projective planes may be found in and .
References
.
See also
Near-field
Semifield
Alternative division ring
Hall systems (Hall planes)
M
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https://en.wikipedia.org/wiki/List%20of%20Sunderland%20A.F.C.%20records%20and%20statistics
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Sunderland Association Football Club, are a professional football club based in Sunderland, North East England. They were announced to the world by the local newspaper, The Sunderland Daily Echo and Shipping Gazette on 27 September 1880 as Sunderland & District Teachers Association Football Club following a meeting of the Teachers at Rectory Park school in Sunderland on 25 September 1880. The football club changed their name to the current form on 16 October 1880, just 20 days after the September announcement. They were elected into The Football League in the 1890–91 season, becoming the first team to join the league since its inauguration in the 1889–90 season, replacing Stoke F.C.
Transfers
All figures are based on the maximum potential fee and are correct as at 1 September 2013.
In
Out
Honours and achievements
Sunderland have won a total of six Football League Championships including three in the space of four seasons, along with being runners-up five times. Sunderland have also experienced success in the FA Cup, winning it twice; in 1937 and 1973. They have never won the League Cup but finished as finalists in 1985 and 2014.
League
First Division (level 1):
Winners (6): 1891–92, 1892–93, 1894–95, 1901–02, 1912–13, 1935–36
Runners-up (5): 1893–94, 1897–98, 1900–01, 1922–23, 1934–35
Football League Championship (level 2):
Winners (2): 2004–05, 2006–07
Second Division (level 2):
Winners (1): 1975–76
Runners-up (2): 1963–64, 1979–80
Promotion (1): 1989–90
First Division (level 2):
Winners (1): 1995–96, 1998–99
Third Division (level 3):
Winners (1): 1987–88
Cup
FA Cup:
Winners (2): 1937, 1973
Finalists (2): 1913, 1992
Football League Cup:
Finalists (2): 1985, 2014
FA Charity Shield:
Winners (1): 1936
Finalists (1): 1937
Sheriff of London Charity Shield:
Winners (1): 1903
Football League War Cup:
Finalists (1): 1942
Durham Challenge Cup:
Winners (4): 1884, 1887, 1888, 1890,
Northern Temperance Festival Cup:
Winners (1): 1884,
Durham and Northumberland Championship:
Winners (1): 1888
British Cup:
Runners Up (1): 1902
Dewar Sheriff of London Shield:
Winners (1): 1903
Newcastle and Sunderland Hospitals Cup:
Winners (3): 1912, 1913, 1914
Runners Up (1): 1911
Durham Senior Cup:
Winners (11): 1919, 1923, 1924, 1927, 1929, 1931, 1932, 1935, 1936, 1937, 1939
Runners Up (3): 1925, 1926, 1928
Northern Victory League:
Runners Up (1): 1919
North East Counties Cup:
Winners (2): 1920, 1921
Northumberland and Durham Challenge Cup:
Runners Up (1): 1883
Player records
Appearances
Youngest first-team player: Derek Forster, 15 years 185 days (vs Leicester City, 22 August 1964).
Oldest first-team player: Jermain Defoe, 39 years 163 days (vs Lincoln City, 19 March 2022).Charles Thompson was 41 when he played his last game for Sunderland in 1919
Most appearances
Competitive matches only. Each column contains appearances in the starting eleven, followed by appearances as substitute in brackets.
Goalscorers
Top goalscorers
Competitive matches only, a
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https://en.wikipedia.org/wiki/Reciprocal%20difference
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In mathematics, the reciprocal difference of a finite sequence of numbers on a function is defined inductively by the following formulas:
See also
Divided differences
References
Finite differences
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https://en.wikipedia.org/wiki/Q%E2%80%93Q%20plot
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In statistics, a Q–Q plot (quantile–quantile plot) is a probability plot, a graphical method for comparing two probability distributions by plotting their quantiles against each other. A point on the plot corresponds to one of the quantiles of the second distribution (-coordinate) plotted against the same quantile of the first distribution (-coordinate). This defines a parametric curve where the parameter is the index of the quantile interval.
If the two distributions being compared are similar, the points in the Q–Q plot will approximately lie on the identity line . If the distributions are linearly related, the points in the Q–Q plot will approximately lie on a line, but not necessarily on the line . Q–Q plots can also be used as a graphical means of estimating parameters in a location-scale family of distributions.
A Q–Q plot is used to compare the shapes of distributions, providing a graphical view of how properties such as location, scale, and skewness are similar or different in the two distributions. Q–Q plots can be used to compare collections of data, or theoretical distributions. The use of Q–Q plots to compare two samples of data can be viewed as a non-parametric approach to comparing their underlying distributions. A Q–Q plot is generally more diagnostic than comparing the samples' histograms, but is less widely known. Q–Q plots are commonly used to compare a data set to a theoretical model. This can provide an assessment of goodness of fit that is graphical, rather than reducing to a numerical summary statistic. Q–Q plots are also used to compare two theoretical distributions to each other. Since Q–Q plots compare distributions, there is no need for the values to be observed as pairs, as in a scatter plot, or even for the numbers of values in the two groups being compared to be equal.
The term "probability plot" sometimes refers specifically to a Q–Q plot, sometimes to a more general class of plots, and sometimes to the less commonly used P–P plot. The probability plot correlation coefficient plot (PPCC plot) is a quantity derived from the idea of Q–Q plots, which measures the agreement of a fitted distribution with observed data and which is sometimes used as a means of fitting a distribution to data.
Definition and construction
A Q–Q plot is a plot of the quantiles of two distributions against each other, or a plot based on estimates of the quantiles. The pattern of points in the plot is used to compare the two distributions.
The main step in constructing a Q–Q plot is calculating or estimating the quantiles to be plotted. If one or both of the axes in a Q–Q plot is based on a theoretical distribution with a continuous cumulative distribution function (CDF), all quantiles are uniquely defined and can be obtained by inverting the CDF. If a theoretical probability distribution with a discontinuous CDF is one of the two distributions being compared, some of the quantiles may not be defined, so an interpolated quantil
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https://en.wikipedia.org/wiki/Thiele%27s%20interpolation%20formula
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In mathematics, Thiele's interpolation formula is a formula that defines a rational function from a finite set of inputs and their function values . The problem of generating a function whose graph passes through a given set of function values is called interpolation. This interpolation formula is named after the Danish mathematician Thorvald N. Thiele. It is expressed as a continued fraction, where ρ represents the reciprocal difference:
Be careful that the -th level in Thiele's interpolation formula is
while the -th reciprocal difference is defined to be
.
The two terms are different and can not be cancelled!
References
Finite differences
Articles with example ALGOL 68 code
Interpolation
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https://en.wikipedia.org/wiki/Churchill%20Eisenhart
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Churchill Eisenhart (1913–1994) was a United States mathematician. He was Chief of the Statistical Engineering Laboratory (SEL), Applied Mathematics Division of the National Bureau of Standards (NBS).
Biography
Eisenhart was the son of Luther Eisenhart, a prominent mathematician in his own right.
Churchill Eisenhart was brought to the NBS from the University of Wisconsin–Madison in 1946 by Edward Condon, Director of the NBS, to establish a statistical consulting group to "substitute sound mathematical analysis for costly experimentation." He was allowed to recruit his own staff and, over the years, he brought many notable and accomplished statisticians to SEL. He served as its Chief from 1947 until his appointment as Senior Research Fellow in 1963. He retired in 1983 after which he formed the Standards Alumni Association, which he headed until his death in 1994.
Over his career, Eisenhart was awarded the U.S. Department of Commerce Exceptional Service Award in 1957; the Rockefeller Public Service Award in 1958; and the Wildhack Award of the National Conference of Standards Laboratories in 1982. He was elected President of the American Statistical Association (ASA) in 1971 and received the Association's Wilks Memorial Medal in 1977. Eisenhart was honored with an Outstanding Achievements Award of the Princeton University Class of 1934 and with Fellowships in the ASA, the American Association for the Advancement of Science, and the Institute of Mathematical Sciences. He was a long-time member of the Cosmos Club.
References
Ingram Olkin (1992) A Conversation with Churchill Eisenhart. Statistical Science, 7, 512–530.
Joseph M. Cameron; Joan R. Rosenblatt (1995) Churchill Eisenhart, 1913–1994, The American Statistician, 49, 243–244.
Samuel S. Wilks Award Citation for 1977
An interview with Joseph Daly and Churchill Eisenhart about their experiences at Princeton 10 July 1984
An interview with Churchill Eisenhart about his experience at Princeton 10 July 1984
External sources
Obituary in the Washington Post
Churchill Eisenhart on the page Portraits of Statisticians
1913 births
1994 deaths
American statisticians
20th-century American mathematicians
Princeton University alumni
Presidents of the American Statistical Association
Fellows of the American Statistical Association
University of Wisconsin people
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https://en.wikipedia.org/wiki/Near-field%20%28mathematics%29
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In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse.
Definition
A near-field is a set together with two binary operations, (addition) and (multiplication), satisfying the following axioms:
A1: is an abelian group.
A2: = for all elements , , of (The associative law for multiplication).
A3: for all elements , , of (The right distributive law).
A4: contains an element 1 such that for every element of (Multiplicative identity).
A5: For every non-zero element of there exists an element such that (Multiplicative inverse).
Notes on the definition
The above is, strictly speaking, a definition of a right near-field. By replacing A3 by the left distributive law we get a left near-field instead. Most commonly, "near-field" is taken as meaning "right near-field", but this is not a universal convention.
A (right) near-field is called "planar" if it is also a right quasifield. Every finite near-field is planar, but infinite near-fields need not be.
It is not necessary to specify that the additive group is abelian, as this follows from the other axioms, as proved by B.H. Neumann and J.L. Zemmer. However, the proof is quite difficult, and it is more convenient to include this in the axioms so that progress with establishing the properties of near-fields can start more rapidly.
Sometimes a list of axioms is given in which A4 and A5 are replaced by the following single statement:
A4*: The non-zero elements form a group under multiplication.
However, this alternative definition includes one exceptional structure of order 2 which fails to satisfy various basic theorems (such as for all ). Thus it is much more convenient, and more usual, to use the axioms in the form given above. The difference is that A4 requires 1 to be an identity for all elements, A4* only for non-zero elements.
The exceptional structure can be defined by taking an additive group of order 2, and defining multiplication by for all and .
Examples
Any division ring (including any field) is a near-field.
The following defines a (right) near-field of order 9. It is the smallest near-field which is not a field.
Let be the Galois field of order 9. Denote multiplication in by ' '. Define a new binary operation ' · ' by:
If is any element of which is a square and is any element of then .
If is any element of which is not a square and is any element of then .
Then is a near-field with this new multiplication and the same addition as before.
History and applications
The concept of a near-field was first introduced by Leonard Dickson in 1905. He took division rings and modified their multiplication, while leaving addition as it was, and thus produced the first known examples of near-fields that were not division rings. The ne
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https://en.wikipedia.org/wiki/Pui%20Ching%20Invitational%20Mathematics%20Competition
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Pui Ching Invitational Mathematics Competition (Traditional Chinese: 培正數學邀請賽), is held yearly by Pui Ching Middle School since 2002. It was formerly named as Pui Ching Middle School Invitational Mathematics Competition for the first three years. At present, more than 130 secondary schools send teams to participate in the competition.
See also
List of mathematics competitions
Education in Hong Kong
External links
Official website (in Traditional Chinese)
Site with past papers (in Traditional Chinese and English)
Competitions in Hong Kong
Mathematics competitions
Recurring events established in 2002
2002 establishments in Hong Kong
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https://en.wikipedia.org/wiki/Class%20number
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In mathematics, class number may refer to
Class number (group theory), in group theory, is the number of conjugacy classes of a group
Class number (number theory), the size of the ideal class group of a number ring
Class number (binary quadratic forms), the number of equivalence classes of binary quadratic forms of a given discriminant
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