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https://en.wikipedia.org/wiki/Canonical%20analysis
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In statistics, canonical analysis (from bar, measuring rod, ruler) belongs to the family of regression methods for data analysis. Regression analysis quantifies a relationship between a predictor variable and a criterion variable by the coefficient of correlation r, coefficient of determination r2, and the standard regression coefficient β. Multiple regression analysis expresses a relationship between a set of predictor variables and a single criterion variable by the multiple correlation R, multiple coefficient of determination R2, and a set of standard partial regression weights β1, β2, etc. Canonical variate analysis captures a relationship between a set of predictor variables and a set of criterion variables by the canonical correlations ρ1, ρ2, ..., and by the sets of canonical weights C and D.
Canonical analysis
Canonical analysis belongs to a group of methods which involve solving the characteristic equation for its latent roots and vectors. It describes formal structures in hyperspace invariant with respect to the rotation of their coordinates. In this type of solution, rotation leaves many optimizing properties preserved, provided it takes place in certain ways and in a subspace of its corresponding hyperspace. This rotation from the maximum intervariate correlation structure into a different, simpler and more meaningful structure increases the interpretability of the canonical weights C and D. In this the canonical analysis differs from Harold Hotelling's (1936) canonical variate analysis (also called the canonical correlation analysis), designed to obtain maximum (canonical) correlations between the predictor and criterion canonical variates. The difference between the canonical variate analysis and canonical analysis is analogous to the difference between the principal components analysis and factor analysis, each with its characteristic set of commonalities, eigenvalues and eigenvectors.
Canonical analysis (simple)
Canonical analysis is a multivariate technique which is concerned with determining the relationships between groups of variables in a data set. The data set is split into two groups X and Y, based on some common characteristics. The purpose of canonical analysis is then to find the relationship between X and Y, i.e. can some form of X represent Y. It works by finding the linear combination of X variables, i.e. X1, X2 etc., and linear combination of Y variables, i.e. Y1, Y2 etc., which are most highly correlated. This combination is known as the "first canonical variates" which are usually denoted U1 and V1, with the pair of U1 and V1 being called a "canonical function". The next canonical functions, U2 and V2 are then restricted so that they are uncorrelated with U1 and V1. Everything is scaled so that the variance equals 1.
One can also construct relationships which are made to agree with constraint restrictions arising from theory or to agree with common sense/intuition. These are called maximum correlation models.
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https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors
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In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor.
Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear. The eigenvalue is the factor by which an eigenvector is stretched. If the eigenvalue is negative, the direction is reversed.
Definition
If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as
where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with .
There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.
If is finite-dimensional, the above equation is equivalent to
where is the matrix representation of and is the coordinate vector of .
Overview
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen (cognate with the English word own) for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.
In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or
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https://en.wikipedia.org/wiki/Padovan%20sequence
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In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values
and the recurrence relation
The first few values of P(n) are
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ...
A Padovan prime is a Padovan number that is prime. The first Padovan primes are:
2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719, ... .
The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan : Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics".
The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.
Recurrence relations
In the spiral, each triangle shares a side with two others giving a visual proof that
the Padovan sequence also satisfies the recurrence relation
Starting from this, the defining recurrence and other recurrences as they are discovered,
one can create an infinite number of further recurrences by repeatedly replacing by
The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.
The Perrin sequence can be obtained from the Padovan sequence by the
following formula:
Extension to negative parameters
As with any sequence defined by a recurrence relation, Padovan numbers P(m) for m<0 can be defined by rewriting the recurrence relation as
Starting with m = −1 and working backwards, we extend P(m) to negative indices:
{| class="wikitable" style="text-align:right"
|-
| P−20
| P−19
| P−18
| P−17
| P−16
| P−15
| P−14
| P−13
| P−12
| P−11
| P−10
| P−9
| P−8
| P−7
| P−6
| P−5
| P−4
| P−3
| P−2
| P−1
| P0
| P1
| P2
|-
| 7
| −7
| 4
| 0
| −3
| 4
| −3
| 1
| 1
| −2
| 2
| −1
| 0
| 1
| −1
| 1
| 0
| 0
| 1
| 0
| 1
| 1
| 1
|-
|}
Sums of terms
The sum of the first n terms in the Padovan sequence is 2 less than P(n + 5), i.e.
Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:
Sums involving products of terms in the Padovan sequence satisfy the following identities:
Other identities
The Padovan sequence also satisfies the identity
The Padovan sequence is related to sums of binomial coefficients by the following identity:
For example, for k = 12, the values for the pair (m, n) with 2m + n = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:
Binet-like formula
The Padovan sequence numbers can be written in terms o
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https://en.wikipedia.org/wiki/Dini%20derivative
|
In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions.
The upper Dini derivative, which is also called an upper right-hand derivative, of a continuous function
is denoted by and defined by
where is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, , is defined by
where is the infimum limit.
If is defined on a vector space, then the upper Dini derivative at in the direction is defined by
If is locally Lipschitz, then is finite. If is differentiable at , then the Dini derivative at is the usual derivative at .
Remarks
The functions are defined in terms of the infimum and supremum in order to make the Dini derivatives as "bullet proof" as possible, so that the Dini derivatives are well-defined for almost all functions, even for functions that are not conventionally differentiable. The upshot of Dini's analysis is that a function is differentiable at the point on the real line (), only if all the Dini derivatives exist, and have the same value.
Sometimes the notation is used instead of and is used instead of .
Also,
and
.
So when using the notation of the Dini derivatives, the plus or minus sign indicates the left- or right-hand limit, and the placement of the sign indicates the infimum or supremum limit.
There are two further Dini derivatives, defined to be
and
.
which are the same as the first pair, but with the supremum and the infimum reversed. For only moderately ill-behaved functions, the two extra Dini derivatives aren't needed. For particularly badly behaved functions, if all four Dini derivatives have the same value () then the function is differentiable in the usual sense at the point .
On the extended reals, each of the Dini derivatives always exist; however, they may take on the values or at times (i.e., the Dini derivatives always exist in the extended sense).
See also
References
.
Generalizations of the derivative
Real analysis
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https://en.wikipedia.org/wiki/Sieve%20%28category%20theory%29
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In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection of open subsets of a fixed open set in topology. In a Grothendieck topology, certain sieves become categorical analogues of open covers in topology. Sieves were introduced by in order to reformulate the notion of a Grothendieck topology.
Definition
Let C be a category, and let c be an object of C. A sieve on c is a subfunctor of Hom(−, c), i.e., for all objects c′ of C, S(c′) ⊆ Hom(c′, c), and for all arrows f:c″→c′, S(f) is the restriction of Hom(f, c), the pullback by f (in the sense of precomposition, not of fiber products), to S(c′); see the next section, below.
Put another way, a sieve is a collection S of arrows with a common codomain that satisfies the condition, "If g:c′→c is an arrow in S, and if f:c″→c′ is any other arrow in C, then gf is in S." Consequently, sieves are similar to right ideals in ring theory or filters in order theory.
Pullback of sieves
The most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This new sieve consists of all the arrows in S that factor through c′.
There are several equivalent ways of defining f*S. The simplest is:
For any object d of C, f*S(d) = { g:d→c′ | fg ∈ S(d)}
A more abstract formulation is:
f*S is the image of the fibered product S×Hom(−, c)Hom(−, c′) under the natural projection S×Hom(−, c)Hom(−, c′)→Hom(−, c′).
Here the map Hom(−, c′)→Hom(−, c) is Hom(f, c′), the pullback by f.
The latter formulation suggests that we can also take the image of S×Hom(−, c)Hom(−, c′) under the natural map to Hom(−, c). This will be the image of f*S under composition with f. For each object d of C, this sieve will consist of all arrows fg, where g:d→c′ is an arrow of f*S(d). In other words, it consists of all arrows in S that can be factored through f.
If we denote by ∅c the empty sieve on c, that is, the sieve for which ∅(d) is always the empty set, then for any f:c′→c, f*∅c is ∅c′. Furthermore, f*Hom(−, c) = Hom(−, c′).
Properties of sieves
Let S and S′ be two sieves on c. We say that S ⊆ S′ if for all objects c′ of C, S(c′) ⊆ S′(c′). For all objects d of C, we define (S ∪ S′)(d) to be S(d) ∪ S′(d) and (S ∩ S′)(d) to be S(d) ∩ S′(d). We can clearly extend this definition to infinite unions and intersections as well.
If we define SieveC(c) (or Sieve(c) for short) to be the set of all sieves on c, then Sieve(c) becomes partially ordered under ⊆. It is easy to see from the definition that the union or intersection of any family of sieves on c is a sieve on c, so Sieve(c) is a complete lattice.
A Grothendieck topology is a collection of sieves subject to certain properties. These sieves are called covering sieves. The set of all covering sieves on an object c is a subset J(c) of Sieve(c). J(c) satisfies several properties in addition to those required by the definitio
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https://en.wikipedia.org/wiki/List%20of%20busiest%20airports%20by%20international%20passenger%20traffic
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The following is a list of the world's largest airports by international passenger traffic.
2022 statistics
Airports Council International's preliminary figures are as follows.
2021 statistics
Airports Council International's preliminary figures are as follows.
2020 statistics
Airports Council International's preliminary figures are as follows.
2019 statistics
Airports Council International's (January–December) preliminary figures are as follows.
2018 statistics
Airports Council International's (January–December) preliminary figures are as follows.
2017 statistics
Airports Council International's (January–December) preliminary figures are as follows.
2016 statistics
Airports Council International's (January–December) preliminary figures are as follows.
2015 statistics
Airports Council International's (January–December) figures are as follows.
2014 statistics
Airports Council International's (January–December) figures are as follows.
2013 statistics
Airports Council International's (January–December) figures are as follows.
2011 statistics
Airports Council International's (January–December) figures are as follows.
See also
List of busiest airports by passenger traffic
List of busiest airports by cargo traffic
List of busiest airports by aircraft movements
List of international airports by country
Airport of entry
Notes
References
International Passenger Traffic, Airports Council International
Busiest airports by international passenger traffic
International
Busiest airports
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https://en.wikipedia.org/wiki/Rosser%27s%20theorem
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In number theory, Rosser's theorem states that the th prime number is greater than , where is the natural logarithm function. It was published by J. Barkley Rosser in 1939.
Its full statement is:
Let be the th prime number. Then for
In 1999, Pierre Dusart proved a tighter lower bound:
See also
Prime number theorem
References
External links
Rosser's theorem article on Wolfram Mathworld.
Theorems about prime numbers
de:John Barkley Rosser#Satz von Rosser
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https://en.wikipedia.org/wiki/Vital%20rates
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Vital rates refer to how fast vital statistics change in a population (usually measured per 1000 individuals). There are 2 categories within vital rates: crude rates and refined rates.
Crude rates measure vital statistics in a general population (overall change in births and deaths per 1000).
Refined rates measure the change in vital statistics in a specific demographic (such as age, sex, race, etc.).
Marriage rates
The national marriage rates since 1972,in the US have fallen by almost 50% at six people per 1000. According to Iran Index and National Organization for Civil Registration of Iran Iranian divorce rate is in the red at its record highest level since 1979, divorce quotas were introduced to curb enthuitasim.
References
Ecology
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https://en.wikipedia.org/wiki/Choice%20%28disambiguation%29
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Choice consists of the mental process of thinking involved with the process of judging the merits of multiple options and selecting one of them for action.
Choice may also refer to:
Mathematics
Binomial coefficient, a mathematical function describing number of possible selections of subsets ('seven choose two')
Axiom of choice
Media
Film and television
Choices (1986 film), a television film directed by David Lowell Rich
Choices (2021 film), an OTT Indian film
"Choices" (Buffy the Vampire Slayer), a 1999 season 3 episode of the TV series Buffy the Vampire Slayer
RTÉ Choice, an Irish digital radio station
BBC Choice, a defunct British digital television channel, replaced in 2003 by BBC Three
Choice TV, a New Zealand television station owned by Discovery New Zealand
Music
Choice (group), a 1990s R&B girl group
Choice (rapper), American female rap artist
Choice, an alias for Laurent Garnier, a French techno music producer
Choice, a 1983 album by British pop group Central Line
Choices – The Singles Collection, the 1989 greatest hits collection from British pop rock band The Blow Monkeys
Choices (Dewey Redman album), 1992
"Choices" (Billy Yates song), a 1997 song, later covered by George Jones
"Choices", a song by Mudvayne from Lost and Found (2005)
Choices (Terence Blanchard album), 2009
"Choices" (The Hoosiers song), a 2010 song by The Hoosiers
Choices (EP), a 2013 EP by Clint Lowery (under the name Hello Demons...Meet Skeletons)
"Choices (Yup)", a 2014 song by E-40
Publications
Choice (Australian magazine), a publication of the Australian consumer organisation of the same name
Choice (American magazine), a publication of the American Library Association
Choices (journal), a food industry policy magazine
Technology
Choice (command), a shell command to prompt a user to select one item from a set of one-character choices
Choices (operating system), object-oriented operating system, developed at University of Illinois
Design choice, in engineering
Other
Choice (Australian consumer organisation), formerly named Australian Consumers Association
Choice (credit card), a former credit card issued by Citibank
Choice, Texas, an unincorporated community in the United States
Choice Hotels, a hospitality holding corporation which owns several hotel brands
Choice modelling, in research
Choice of law, a concept within the field of the conflict of laws
Pro-choice, a term used by abortion rights proponents
Tashard Choice (born 1984), American football running back
See also
The Choice (disambiguation)
Choc ice
TV Choice, a British weekly TV listings magazine
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https://en.wikipedia.org/wiki/Gertrude%20Mary%20Cox
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Gertrude Mary Cox (January 13, 1900 – October 17, 1978) was an American statistician and founder of the department of Experimental Statistics at North Carolina State University. She was later appointed director of both the Institute of Statistics of the Consolidated University of North Carolina and the Statistics Research Division of North Carolina State University. Her most important and influential research dealt with experimental design; In 1950 she published the book Experimental Designs, on the subject with W. G. Cochran, which became the major reference work on the design of experiments for statisticians for years afterwards. In 1949 Cox became the first woman elected into the International Statistical Institute and in 1956 was President of the American Statistical Association.
Early life and education
Gertrude Cox was born in Dayton, Iowa on January 13, 1900. She studied at Perry High School in Perry, Iowa, graduating in 1918. At this time she decided to become a deaconess in the Methodist Church and worked towards that end. However, in 1925, she decided to continue her education at Iowa State College (which was renamed Iowa State University in 1959) in Ames where she studied mathematics and statistics and was awarded a B.S. in 1929 and a Master's degree in statistics in 1931.
From 1931 to 1933 Cox undertook graduate studies in psychological statistics at the University of California at Berkeley, then returned to Iowa State College to assist in establishing the new Statistical Laboratory. Here she worked on the design of experiments.
Academic career
In 1939 Cox was appointed assistant professor of statistics at Iowa State College.
In 1940 Cox was appointed professor of statistics at North Carolina State College (now North Carolina State University) at Raleigh. There she headed the new department of Experimental Statistics, the first female head of any department at this institution. In 1945 she became director of the Institute of Statistics of the Consolidated University of North Carolina, and the Statistics Research Division of the North Carolina State College which was run by William Gemmell Cochran. In the same year of 1945 Cox became the editor of Biometrics Bulletin and of Biometrics and she held this editorship for 10 years. When prolific statistician and eugenicist Ronald Fisher founded the International Biometric Society in 1947, Cox was one of the founding members.
In 1960 she took up her final post as Director of Statistics at the Research Triangle Institute in Durham, North Carolina. She held this post until she retired in 1965. After retirement, then worked as a consultant to promote the development of statistical programs in Egypt and Thailand.
Book
In 1950 she published a joint work with William Cochran, Experimental Designs, which became the major reference work on the design of experiments for statisticians for years afterwards.
Recognition
Cox has received many honors. In 1949, she became the first woman elected
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https://en.wikipedia.org/wiki/F-algebra
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In mathematics, specifically in category theory, F-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor F, the signature.
F-algebras can also be used to represent data structures used in programming, such as lists and trees.
The main related concepts are initial F-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras.
Definition
If is a category, and is an endofunctor of , then an -algebra is a tuple , where is an object of and is a -morphism . The object is called the carrier of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple.
A homomorphism from an -algebra to an -algebra is a -morphism such that , according to the following commutative diagram:
Equipped with these morphisms, -algebras constitute a category.
The dual construction are -coalgebras, which are objects together with a morphism .
Examples
Groups
Classically, a group is a set with a group law , with , satisfying three axioms: the existence of an identity element, the existence of an inverse for each element of the group, and associativity.
To put this in a categorical framework, first define the identity and inverse as functions (morphisms of the set ) by with , and with . Here denotes the set with one element , which allows one to identify elements with morphisms .
It is then possible to write the axioms of a group in terms of functions (note how the existential quantifier is absent):
,
,
.
Then this can be expressed with commutative diagrams:
Now use the coproduct (the disjoint union of sets) to glue the three morphisms in one: according to
Thus a group is a -algebra where is the functor . However the reverse is not necessarily true. Some -algebra where is the functor are not groups.
The above construction is used to define group objects over an arbitrary category with finite products and a terminal object . When the category admits finite coproducts, the group objects are -algebras. For example,
finite groups are -algebras in the category of finite sets and Lie groups are -algebras in the category of smooth manifolds with smooth maps.
Algebraic structures
Going one step ahead of universal algebra, most algebraic structures are F-algebras. For example, abelian groups are F-algebras for the same functor F(G) = 1 + G + G×G as for groups, with an additional axiom for commutativity: m∘t = m, where t(x,y) = (y,x) is the transpose on GxG.
Monoids are F-algebras of signature F(M) = 1 + M×M. In the same vein, semigroups are F-algebras of signature F(S) = S×S
Rings, domains and fields are also F-algebras with a signature involving two laws +,•: R×R → R, an additive identity 0: 1 → R, a multiplicative identity 1: 1 → R, and an
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https://en.wikipedia.org/wiki/Algebra%20%28disambiguation%29
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The word 'algebra' is used for various branches and structures of mathematics. For their overview, see Algebra.
The bare word "algebra"
The bare word "algebra" may refer to:
Elementary algebra
Abstract algebra
Algebra over a field
In universal algebra, algebra has an axiomatic definition, roughly as an instance of any of a number of algebraic structures, such as groups, rings, etc.
Branches of mathematics
Elementary algebra, i.e. "high-school algebra"
Abstract algebra
Linear algebra
Relational algebra
Universal algebra
The term is also traditionally used for the field of:
Computer algebra, dealing with software systems for symbolic mathematical computation, which often offer capabilities beyond what is normally understood to be "algebra"
Mathematical structures
Vector space with multiplication
An "algebra", or to be verbose, an algebra over a field, is a vector space equipped with a bilinear vector product. Some notable algebras in this sense are:
In ring theory and linear algebra:
Algebra over a commutative ring, a module equipped with a bilinear product. Generalization of algebras over a field
Associative algebra, a module equipped with an associative bilinear vector product
Superalgebra, a -graded algebra
Lie algebras, Poisson algebras, and Jordan algebras, important examples of (potentially) nonassociative algebras
In functional analysis:
Banach algebra, an associative algebra A over the real or complex numbers which at the same time is also a Banach space
Operator algebra, continuous linear operators on a topological vector space with multiplication given by the composition
*-algebra, An algebra with a notion of adjoints
C*-algebra, a Banach algebra equipped with a unary involution operation
Von Neumann algebra (or W*-algebra)
See also coalgebra, the dual notion.
Other structures
A different class of "algebras" consists of objects which generalize logical connectives, sets, and lattices.
In logic:
Relational algebra, in which a set of finitary relations that is closed under certain operators
Boolean algebra and Boolean algebra (structure)
Heyting algebra
In measure theory:
Algebra over a set, a collection of sets closed under finite unions and complementation
Sigma algebra, a collection of sets closed under countable unions and complementation
"Algebra" can also describe more general structures:
In category theory and computer science:
F-algebra and F-coalgebra
T-algebra
Other uses
Algebra Blessett, singer from the U.S, goes by the stage name Algebra
See also
Algebraic (disambiguation)
List of all articles whose title begins with "algebra"
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https://en.wikipedia.org/wiki/Evert%20Willem%20Beth
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Evert Willem Beth (7 July 1908 – 12 April 1964) was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics. He was a member of the Significs Group.
Biography
Beth was born in Almelo, a small town in the eastern Netherlands. His father had studied mathematics and physics at the University of Amsterdam, where he had been awarded a PhD. Evert Beth studied the same subjects at Utrecht University, but then also studied philosophy and psychology. His 1935 PhD was in philosophy.
In 1946, he became professor of logic and the foundations of mathematics in Amsterdam. Apart from two brief interruptions – a stint in 1951 as a research assistant to Alfred Tarski, and in 1957 as a visiting professor at Johns Hopkins University – he held the post in Amsterdam continuously until his death in 1964. His was the first academic post in his country in logic and the foundations of mathematics, and during this time he contributed actively to international cooperation in establishing logic as an academic discipline.
In 1953 he became member of the Royal Netherlands Academy of Arts and Sciences.
He died in Amsterdam.
Contributions to logic
Beth definability theorem
The Beth definability theorem states that for first-order logic a property (or function or constant) is implicitly definable if and only if it is explicitly definable. Further explanation is provided under Beth definability.
Semantic tableaux
Beth's most famous contribution to formal logic is semantic tableaux, which are decision procedures for propositional logic and first-order logic. It is a semantic method—like Wittgenstein's truth tables or J. Alan Robinson's resolution—as opposed to the proof of theorems in a formal system, such as the axiomatic systems employed by Frege, Russell and Whitehead, and Hilbert, or even Gentzen's natural deduction. Semantic tableaux are an effective decision procedure for propositional logic, whereas they are only semi-effective for first-order logic, since first-order logic is undecidable, as showed by Church's theorem. This method is considered by many to be intuitively simple, particularly for students who are not acquainted with the study of logic, and it is faster than the truth-table method (which requires a table with 2n rows for a sentence with n propositional letters). For these reasons, Wilfrid Hodges for example presents semantic tableaux in his introductory textbook, Logic, and Melvin Fitting does the same in his presentation of first-order logic for computer scientists, First-order logic and automated theorem proving.
One starts out with the intention of proving that a certain set of formulae entail another formula , given a set of rules determined by the semantics of the formulae's connectives (and quantifiers, in first-order logic). The method is to assume the concurrent truth of every member of and of (the negation of ), and then to apply the rules to branch this list into a tree-like structure of
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https://en.wikipedia.org/wiki/David%20Masser
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David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Diophantine geometry. With Joseph Oesterlé in 1985, Masser formulated the abc conjecture, which has been called "the most important unsolved problem in Diophantine analysis".
Early life and education
Masser was born on 8 November 1948 in London, England. He graduated from Trinity College, Cambridge with a B.A. (Hons) in 1970. In 1974, he obtained his M.A. and Ph.D. at the University of Cambridge, with a doctoral thesis under the supervision of Alan Baker titled Elliptic Functions and Transcendence.
Career
Masser was a Lecturer at the University of Nottingham from 1973 to 1975, before spending the 1975–1976 year as a Research Fellow of Trinity College at the University of Cambridge. He returned to the University of Nottingham to serve as a Lecturer from 1976 to 1979 and then as a Reader from 1979 to 1983. He was a professor at the University of Michigan from 1983 to 1992. He then moved to the Mathematics Institute at the University of Basel and became emeritus there in 2014.
Research
Masser's research focuses on transcendental number theory, Diophantine approximation, and Diophantine geometry. The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves.
Awards
Masser was an invited speaker at the International Congress of Mathematicians in Warsaw in 1983. In 1991, he received the Humboldt Prize. He was elected as a Fellow of the Royal Society in 2005. In 2014, he was elected as a Member of the Academia Europaea.
See also
Analytic subgroup theorem
Bézout's theorem
Zilber–Pink conjecture
References
1948 births
Living people
20th-century British mathematicians
21st-century British mathematicians
Number theorists
Alumni of Trinity College, Cambridge
Fellows of the Royal Society
University of Michigan faculty
Academic staff of the University of Basel
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https://en.wikipedia.org/wiki/Piecewise%20linear%20manifold
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In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.
An isomorphism of PL manifolds is called a PL homeomorphism.
Relation to other categories of manifolds
PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory.
Smooth manifolds
Smooth manifolds have canonical PL structures — they are uniquely triangulizable, by Whitehead's theorem on triangulation — but PL manifolds do not always have smooth structures — they are not always smoothable. This relation can be elaborated by introducing the category PDIFF, which contains both DIFF and PL, and is equivalent to PL.
One way in which PL is better behaved than DIFF is that one can take cones in PL, but not in DIFF — the cone point is acceptable in PL.
A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four — the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere. This last step works in PL but not in DIFF, giving rise to exotic spheres.
Topological manifolds
Not every topological manifold admits a PL structure, and of those that do, the PL structure need not be unique—it can have infinitely many. This is elaborated at Hauptvermutung.
The obstruction to placing a PL structure on a topological manifold is the Kirby–Siebenmann class. To be precise, the Kirby-Siebenmann class is the obstruction to placing a PL-structure on M x R and in dimensions n > 4, the KS class vanishes if and only if M has at least one PL-structure.
Real algebraic sets
An A-structure on a PL manifold is a structure which gives an inductive way of resolving the PL manifold to a smooth manifold. Compact PL manifolds admit A-structures. Compact PL manifolds are homeomorphic to real-algebraic sets. Put another way, A-category sits over the PL-category as a richer category with no obstruction to lifting, that is BA → BPL is a product fibration with BA = BPL × PL/A, and PL manifolds are real algebraic sets because A-manifolds are real algebraic sets.
Combinatorial manifolds and digital manifolds
A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
A digital manifold is a special kind of combinatorial manifold which is defined i
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https://en.wikipedia.org/wiki/Reduced%20product
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In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct.
Let {Si | i ∈ I} be a nonempty family of structures of the same signature σ indexed by a set I, and let U be a proper filter on I. The domain of the reduced product is the quotient of the Cartesian product
by a certain equivalence relation ~: two elements (ai) and (bi) of the Cartesian product are equivalent if
If U only contains I as an element, the equivalence relation is trivial, and the reduced product is just the direct product. If U is an ultrafilter, the reduced product is an ultraproduct.
Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by
For example, if each structure is a vector space, then the reduced product is a vector space with addition defined as (a + b)i = ai + bi and multiplication by a scalar c as (ca)i = c ai.
References
, Chapter 6.
Model theory
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https://en.wikipedia.org/wiki/Loop%20space
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In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative.
The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).
The iterated loop spaces of X are formed by applying Ω a number of times.
There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space X is the space of maps from the circle S1 to X with the compact-open topology. The free loop space of X is often denoted by .
As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. (A related phenomenon in computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality.
Eckmann–Hilton duality
The loop space is dual to the suspension of the same space; this duality is sometimes called Eckmann–Hilton duality. The basic observation is that
where is the set of homotopy classes of maps ,
and is the suspension of A, and denotes the natural homeomorphism. This homeomorphism is essentially that of currying, modulo the quotients needed to convert the products to reduced products.
In general, does not have a group structure for arbitrary spaces and . However, it can be shown that and do have natural group structures when and are pointed, and the aforementioned isomorphism is of those groups. Thus, setting (the sphere) gives the relationship
.
This follows since the homotopy group is defined as and the spheres can be obtained via suspensions of each-other, i.e. .
See also
Eilenberg–MacLane space
Free loop
Fundamental group
Gray's conjecture
List of topologies
Loop group
Path (topology)
Quasigroup
Spectrum (topology)
Path space (algebraic topology)
References
Topology
Homotopy theory
Topological spaces
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https://en.wikipedia.org/wiki/Sylvester%20matrix
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In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain).
Sylvester matrices are named after James Joseph Sylvester.
Definition
Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:
The Sylvester matrix associated to p and q is then the matrix constructed as follows:
if n > 0, the first row is:
the second row is the first row, shifted one column to the right; the first element of the row is zero.
the following n − 2 rows are obtained the same way, shifting the coefficients one column to the right each time and setting the other entries in the row to be 0.
if m > 0 the (n + 1)th row is:
the following rows are obtained the same way as before.
Thus, if m = 4 and n = 3, the matrix is:
If one of the degrees is zero (that is, the corresponding polynomial is a nonzero constant polynomial), then there are zero rows consisting of coefficients of the other polynomial, and the Sylvester matrix is a diagonal matrix of dimension the degree of the non-constant polynomial, with the all diagonal coefficients equal to the constant polynomial. If m = n = 0, then the Sylvester matrix is the empty matrix with zero rows and zero columns.
A variant
The above defined Sylvester matrix appears in a Sylvester paper of 1840. In a paper of 1853, Sylvester introduced the following matrix, which is, up to a permutation of the rows, the Sylvester matrix of p and q, which are both considered as having degree max(m, n).
This is thus a -matrix containing pairs of rows. Assuming it is obtained as follows:
the first pair is:
the second pair is the first pair, shifted one column to the right; the first elements in the two rows are zero.
the remaining pairs of rows are obtained the same way as above.
Thus, if m = 4 and n = 3, the matrix is:
The determinant of the 1853 matrix is, up to sign, the product of the determinant of the Sylvester matrix (which is called the resultant of p and q) by (still supposing ).
Applications
These matrices are used in commutative algebra, e.g. to test if two polynomials have a (non-constant) common factor. In such a case, the determinant of the associated Sylvester matrix (which is called the resultant of the two polynomials) equals zero. The converse is also true.
The solutions of the simultaneous linear equations
where is a vector of size and has size , comprise the coefficient vectors of those and only those pairs of polynomials (of degrees and , respectively) which fulfill
where polynomial multiplication and addition is used.
This means the kernel of the tra
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https://en.wikipedia.org/wiki/Method%20of%20moments%20%28statistics%29
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In statistics, the method of moments is a method of estimation of population parameters. The same principle is used to derive higher moments like skewness and kurtosis.
It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest. Those expressions are then set equal to the sample moments. The number of such equations is the same as the number of parameters to be estimated. Those equations are then solved for the parameters of interest. The solutions are estimates of those parameters.
The method of moments was introduced by Pafnuty Chebyshev in 1887 in the proof of the central limit theorem. The idea of matching empirical moments of a distribution to the population moments dates back at least to Pearson.
Method
Suppose that the problem is to estimate unknown parameters characterizing the distribution of the random variable . Suppose the first moments of the true distribution (the "population moments") can be expressed as functions of the s:
Suppose a sample of size is drawn, resulting in the values . For , let
be the j-th sample moment, an estimate of . The method of moments estimator for denoted by is defined to be the solution (if one exists) to the equations:
The method described here for single random variables generalizes in an obvious manner to multiple random variables leading to multiple choices for moments to be used. Different choices generally lead to different solutions [5], [6].
Advantages and disadvantages
The method of moments is fairly simple and yields consistent estimators (under very weak assumptions), though these estimators are often biased.
It is an alternative to the method of maximum likelihood.
However, in some cases the likelihood equations may be intractable without computers, whereas the method-of-moments estimators can be computed much more quickly and easily. Due to easy computability, method-of-moments estimates may be used as the first approximation to the solutions of the likelihood equations, and successive improved approximations may then be found by the Newton–Raphson method. In this way the method of moments can assist in finding maximum likelihood estimates.
In some cases, infrequent with large samples but less infrequent with small samples, the estimates given by the method of moments are outside of the parameter space (as shown in the example below); it does not make sense to rely on them then. That problem never arises in the method of maximum likelihood Also, estimates by the method of moments are not necessarily sufficient statistics, i.e., they sometimes fail to take into account all relevant information in the sample.
When estimating other structural parameters (e.g., parameters of a utility function, instead of parameters of a known probability distribution), appropriate probability distributions may not be known, and moment-based estimates may be preferred to m
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https://en.wikipedia.org/wiki/Interval%20arithmetic
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[[File:Set of curves Outer approximation.png|345px|thumb|right|Tolerance function (turquoise) and interval-valued approximation (red)]]
Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically correct results. Instead of representing a value as a single number, interval arithmetic or interval mathematics represents each value as a range of possibilities.
Mathematically, instead of working with an uncertain real-valued variable , interval arithmetic works with an interval that defines the range of values that can have. In other words, any value of the variable lies in the closed interval between and . A function , when applied to , produces an interval which includes all the possible values for for all .
Interval arithmetic is suitable for a variety of purposes; the most common use is in scientific works, particularly when the calculations are handled by software, where it is used to keep track of rounding errors in calculations and of uncertainties in the knowledge of the exact values of physical and technical parameters. The latter often arise from measurement errors and tolerances for components or due to limits on computational accuracy. Interval arithmetic also helps find guaranteed solutions to equations (such as differential equations) and optimization problems.
Introduction
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
This treatment is typically limited to real intervals, so quantities in the form
where and are allowed. With one of , infinite, the interval would be an unbounded interval; with both infinite, the interval would be the extended real number line. Since a real number can be interpreted as the interval intervals and real numbers can be freely combined.
Example
Consider the calculation of a person's body mass index (BMI). BMI is calculated as a person's body weight in kilograms divided by the square of their height in meters. Suppose a person uses a scale that has a precision of one kilogram, where intermediate values cannot be discerned, and the true weight is rounded to the nearest whole number. For example, 79.6 kg and 80.3kg are indistinguishable, as the scale can only display values to the nearest kilogram. It is unlikely that when the scale reads 80kg, the person has a weight of exactly 80.0kg. Thus, the scale displaying 80kg indicates a weight between 79.5kg and 80.5kg, or the interval .
The BMI of a man wh
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https://en.wikipedia.org/wiki/Michael%20Resnik
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Michael David Resnik (; born March 20, 1938) is a leading contemporary American philosopher of mathematics.
Biography
Resnik obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his PhD in Philosophy at Harvard University in 1964. He wrote his thesis on Frege. He was appointed Associate Professor at the University of North Carolina at Chapel Hill in 1967, Professor in 1975, and University Distinguished Professor in 1988. He is Professor Emeritus of University of North Carolina at Chapel Hill and currently resides in rural Chatham County, North Carolina.
Publications
Books
Journal articles
References
External links
Philpapers.org
Home page of Michael_Resnik
1938 births
20th-century American philosophers
American logicians
American science writers
Analytic philosophers
Harvard University alumni
Living people
Writers from New Haven, Connecticut
Philosophers of mathematics
Structuralism (philosophy of mathematics)
University of North Carolina at Chapel Hill faculty
Yale University alumni
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https://en.wikipedia.org/wiki/Conical%20intersection
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In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are non-vanishing. In the vicinity of conical intersections, the Born–Oppenheimer approximation breaks down and the coupling between electronic and nuclear motion becomes important, allowing non-adiabatic processes to take place. The location and characterization of conical intersections are therefore essential to the understanding of a wide range of important phenomena governed by non-adiabatic events, such as photoisomerization, photosynthesis, vision and the photostability of DNA. The conical intersection involving the ground electronic state potential energy surface of the C6H3F3+ molecular ion is discussed in connection with the Jahn–Teller effect in Section 13.4.2 on pages 380-388 of the textbook by Bunker and Jensen.
Conical intersections are also called molecular funnels or diabolic points as they have become an established paradigm for understanding reaction mechanisms in photochemistry as important as transitions states in thermal chemistry. This comes from the very important role they play in non-radiative de-excitation transitions from excited electronic states to the ground electronic state of molecules. For example, the stability of DNA with respect to the UV irradiation is due to such conical intersection. The molecular wave packet excited to some electronic excited state by the UV photon follows the slope of the potential energy surface and reaches the conical intersection from above. At this point the very large vibronic coupling induces a non-radiative transition (surface-hopping) which leads the molecule back to its electronic ground state. The singularity of vibronic coupling at conical intersections is responsible for the existence of Geometric phase, which was discovered by Longuet-Higgins in this context.
Degenerate points between potential energy surfaces lie in what is called the intersection or seam space with a dimensionality of 3N-8 (where N is the number of atoms). Any critical points in this space of degeneracy are characterised as minima, transition states or higher-order saddle points and can be connected to each other through the analogue of an intrinsic reaction coordinate in the seam. In benzene, for example, there is a recurrent connectivity pattern where permutationally isomeric seam segments are connected by intersections of a higher symmetry point group. The remaining two dimensions that lift the energetic degeneracy of the system are known as the branching space.
Experimental observation
In order to be able to observe it, the process would need to be slowed down from femtoseconds to milliseconds. A novel 2023 quantum experiment, involving trapped-ion quantum computer, slowed down interference pattern of a single atom (caused by a conical intersection) by a factor
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https://en.wikipedia.org/wiki/Airport%20problem
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In mathematics and especially game theory, the airport problem is a type of fair division problem in which it is decided how to distribute the cost of an airport runway among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973. Their proposed solution is:
Divide the cost of providing the minimum level of required facility for the smallest type of aircraft equally among the number of landings of all aircraft
Divide the incremental cost of providing the minimum level of required facility for the second smallest type of aircraft (above the cost of the smallest type) equally among the number of landings of all but the smallest type of aircraft. Continue thus until finally the incremental cost of the largest type of aircraft is divided equally among the number of landings made by the largest aircraft type.
The authors note that the resulting set of landing charges is the Shapley value for an appropriately defined game.
Introduction
In an airport problem there is a finite population N and a nonnegative function C: N-R. For technical reasons it is assumed that the population is taken from the set of the natural numbers: players are identified with their 'ranking number'. The cost function satisfies the inequality C(i) <C（j）whenever i <j. It is typical for airport problems that the cost C（i）is assumed to be a part of the cost C(j) if i<j, i.e. a coalition S is confronted with costs c(S): =MAX C(i). In this way an airport problem generates an airport game (N,c). As the value of each one-person coalition (i) equals C(i), we can rediscover the airport problem from the airport game theory.
Nash Equilibrium
Nash equilibrium, also known as non-cooperative game equilibrium, is an essential term in game theory described by John Nash in 1951. In a game process, regardless of the opponent's strategy choice, one of the parties will choose a certain strategy, which is called dominant strategy. If any participant chooses the optimal strategy when the strategies of all other participants are determined, then this combination is defined as a Nash equilibrium. A game may include multiple Nash equilibrium or none. In addition, a combination of strategies is called the Nash balance. when each player's balance strategy is to achieve the maximum value of its expected return, at the same time, all other players also follow this strategy.
Shapley value
The Shapley value is a solution concept used in game theory. The Shapley value is mainly applicable to the following situation: the contribution of each actor is not equal, but each participant cooperates with each other to obtain profit or return. The efficiency of the resource allocation and combination of the two distribution methods are more reasonable and fair, and it also reflects the process of mutual game among the league members. However, the benefit distribution plan of the Shapley value method has not considered the risk sharing fact
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https://en.wikipedia.org/wiki/Universal%20bundle
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In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space , such that every bundle with the given structure group over is a pullback by means of a continuous map .
Existence of a universal bundle
In the CW complex category
When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem.
For compact Lie groups
We will first prove:
Proposition. Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibre bundle.
Proof. There exists an injection of into a unitary group for big enough. If we find then we can take to be . The construction of is given in classifying space for .
The following Theorem is a corollary of the above Proposition.
Theorem. If is a paracompact manifold and is a principal -bundle, then there exists a map , unique up to homotopy, such that is isomorphic to , the pull-back of the -bundle by .
Proof. On one hand, the pull-back of the bundle by the natural projection is the bundle . On the other hand, the pull-back of the principal -bundle by the projection is also
Since is a fibration with contractible fibre , sections of exist. To such a section we associate the composition with the projection . The map we get is the we were looking for.
For the uniqueness up to homotopy, notice that there exists a one-to-one correspondence between maps such that is isomorphic to and sections of . We have just seen how to associate a to a section. Inversely, assume that is given. Let be an isomorphism:
Now, simply define a section by
Because all sections of are homotopic, the homotopy class of is unique.
Use in the study of group actions
The total space of a universal bundle is usually written . These spaces are of interest in their own right, despite typically being contractible. For example, in defining the homotopy quotient or homotopy orbit space of a group action of , in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if acts on the space , is to consider instead the action on , and corresponding quotient. See equivariant cohomology for more detailed discussion.
If is contractible then and are homotopy equivalent spaces. But the diagonal action on , i.e. where acts on both and coordinates, may be well-behaved when the action on is not.
Examples
Classifying space for U(n)
See also
Chern class
tautological bundle, a universal bundle for the general linear group.
External links
PlanetMath page of universal bundle examples
Notes
Fiber bundles
Homotopy theory
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https://en.wikipedia.org/wiki/Obstruction%20theory
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In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.
In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle.
In homotopy theory
The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called Eilenberg obstruction theory, after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X (the vertices of X), an extension to the 1-skeleton will be possible whenever the image of the 0-skeleton will belong to the same path-connected component of Y. Extending from the 1-skeleton to the 2-skeleton means defining the mapping on each solid triangle from X, given the mapping already defined on its boundary edges. Likewise, then extending the mapping to the 3-skeleton involves extending the mapping to each solid 3-simplex of X, given the mapping already defined on its boundary.
At some point, say extending the mapping from the (n-1)-skeleton of X to the n-skeleton of X, this procedure might be impossible. In that case, one can assign to each n-simplex the homotopy class of the mapping already defined on its boundary, (at least one of which will be non-zero). These assignments define an n-cochain with coefficients in . Amazingly, this cochain turns out to be a cocycle and so defines a cohomology class in the nth cohomology group of X with coefficients in . When this cohomology class is equal to 0, it turns out that the mapping may be modified within its homotopy class on the (n-1)-skeleton of X so that the mapping may be extended to the n-skeleton of X. If the class is not equal to zero, it is called the obstruction to extending the mapping over the n-skeleton, given its homotopy class on the (n-1)-skeleton.
Obstruction to extending a section of a principal bundle
Construction
Suppose that is a simply connected simplicial complex and that is a fibration with fiber . Furthermore, assume that we have a partially defined section on the -skeleton of .
For every -simplex in , can be restricted to the boundary (which is a topological -sphere). Because sends each back to , defines a map from the -sphere to . Because fibrations satisfy the homotopy lifting property, and is contractible; is homotopy equivalent to . So this partially defined section assigns an element of to every -simplex. This is precisely the data of a -valued simplicial cochain of degree on , i.e. an element of . This c
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https://en.wikipedia.org/wiki/Hauptvermutung
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The Hauptvermutung of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, but it is now known to be false.
History
The non-manifold version was disproved by John Milnor in 1961 using Reidemeister torsion.
The manifold version is true in dimensions . The cases and were proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively.
An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–69 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group .
In dimension , a homeomorphism of m-dimensional piecewise linear manifolds has an invariant such that is isotopic to a piecewise linear (PL) homeomorphism if and only if . In the simply-connected case and with , is homotopic to a PL homeomorphism if and only if .
This quantity is now seen as a relative version of the triangulation obstruction of Robion Kirby and Laurent C. Siebenmann, obtained in 1970. The Kirby–Siebenmann obstruction is defined for any compact m-dimensional topological manifold M
again using the Rochlin invariant. For , the manifold M has a PL structure (i.e., it can be triangulated by a PL manifold) if and only if , and if this obstruction is 0, the PL structures are parametrized by . In particular there are only a finite number of essentially distinct PL structures on M.
For compact simply-connected manifolds of dimension 4, Simon Donaldson found examples with an infinite number of inequivalent PL structures, and Michael Freedman found the E8 manifold which not only has no PL structure, but (by work of Casson) is not even homeomorphic to a simplicial complex.
In 2013, Ciprian Manolescu proved that there exist compact topological manifolds of dimension 5 (and hence of any dimension greater than 5) that are not homeomorphic to a simplicial complex. Thus Casson's example illustrates a more general phenomenon that is not merely limited to dimension 4.
Notes
References
External links
Additional material, including original sources
Disproved conjectures
Geometric topology
Structures on manifolds
Surgery theory
German words and phrases
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https://en.wikipedia.org/wiki/Kirby%E2%80%93Siebenmann%20class
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In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.
The KS-class
For a topological manifold M, the Kirby–Siebenmann class is an element of the fourth cohomology group of M that vanishes if M admits a piecewise linear structure.
It is the only such obstruction, which can be phrased as the weak equivalence of TOP/PL with an Eilenberg–MacLane space.
The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure. Concrete examples of such manifolds are , where stands for Freedman's E8 manifold.
The class is named after Robion Kirby and Larry Siebenmann, who developed the theory of topological and PL-manifolds.
See also
Hauptvermutung
References
Homology theory
Geometric topology
Structures on manifolds
Surgery theory
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https://en.wikipedia.org/wiki/Exotic%20R4
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{{DISPLAYTITLE:Exotic R4}}
In mathematics, an exotic is a differentiable manifold that is homeomorphic (i.e. shape preserving) but not diffeomorphic (i.e. non smooth) to the Euclidean space The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds. There is a continuum of non-diffeomorphic differentiable structures of as was shown first by Clifford Taubes.
Prior to this construction, non-diffeomorphic smooth structures on spheresexotic sphereswere already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2023). For any positive integer n other than 4, there are no exotic smooth structures on in other words, if n ≠ 4 then any smooth manifold homeomorphic to is diffeomorphic to
Small exotic R4s
An exotic is called small if it can be smoothly embedded as an open subset of the standard
Small exotic can be constructed by starting with a non-trivial smooth 5-dimensional h-cobordism (which exists by Donaldson's proof that the h-cobordism theorem fails in this dimension) and using Freedman's theorem that the topological h-cobordism theorem holds in this dimension.
Large exotic R4s
An exotic is called large if it cannot be smoothly embedded as an open subset of the standard
Examples of large exotic can be constructed using the fact that compact 4-manifolds can often be split as a topological sum (by Freedman's work), but cannot be split as a smooth sum (by Donaldson's work).
showed that there is a maximal exotic into which all other can be smoothly embedded as open subsets.
Related exotic structures
Casson handles are homeomorphic to by Freedman's theorem (where is the closed unit disc) but it follows from Donaldson's theorem that they are not all diffeomorphic to In other words, some Casson handles are exotic
It is not known (as of 2022) whether or not there are any exotic 4-spheres; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4. Some plausible candidates are given by Gluck twists.
See also
Akbulut cork - tool used to construct exotic 's from classes in
Atlas (topology)
Notes
References
4-manifolds
Differential structures
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https://en.wikipedia.org/wiki/Donaldson%27s%20theorem
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In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalisable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the . The original version of the theorem required the manifold to be simply connected, but it was later improved to apply to 4-manifolds with any fundamental group.
History
The theorem was proved by Simon Donaldson. This was a contribution cited for his Fields medal in 1986.
Idea of proof
Donaldson's proof utilizes the moduli space of solutions to the anti-self-duality equations on a principal -bundle over the four-manifold . By the Atiyah–Singer index theorem, the dimension of the moduli space is given by
where , is the first Betti number of and is the dimension of the positive-definite subspace of with respect to the intersection form. When is simply-connected with definite intersection form, possibly after changing orientation, one always has and . Thus taking any principal -bundle with , one obtains a moduli space of dimension five.
This moduli space is non-compact and generically smooth, with singularities occurring only at the points corresponding to reducible connections, of which there are exactly many. Results of Clifford Taubes and Karen Uhlenbeck show that whilst is non-compact, its structure at infinity can be readily described. Namely, there is an open subset of , say , such that for sufficiently small choices of parameter , there is a diffeomorphism
.
The work of Taubes and Uhlenbeck essentially concerns constructing sequences of ASD connections on the four-manifold with curvature becoming infinitely concentrated at any given single point . For each such point, in the limit one obtains a unique singular ASD connection, which becomes a well-defined smooth ASD connection at that point using Uhlenbeck's removable singularity theorem.
Donaldson observed that the singular points in the interior of corresponding to reducible connections could also be described: they looked like cones over the complex projective plane , with its orientation reversed.
It is thus possible to compactify the moduli space as follows: First, cut off each cone at a reducible singularity and glue in a copy of . Secondly, glue in a copy of itself at infinity. The resulting space is a cobordism between and a disjoint union of copies of with its orientation reversed. The intersection form of a four-manifold is a cobordism invariant up to isomorphism of quadratic forms, from which one concludes the intersection form of is diagonalisable.
Extensions
Michael Freedman had previously shown that any unimodular symmetric bilinear form is realized as the intersection form of some closed, oriented four-manifold. Combining this result with the Serre classification theorem and Donaldson's theorem, several interest
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https://en.wikipedia.org/wiki/Tarski%27s%20axioms
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Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., that part of Euclidean geometry that is formulable as an elementary theory). Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms.
Overview
Early in his career Tarski taught geometry and researched set theory. His coworker Steven Givant (1999) explained Tarski's take-off point:
From Enriques, Tarski learned of the work of Mario Pieri, an Italian geometer who was strongly influenced by Peano. Tarski preferred Pieri's system [of his Point and Sphere memoir], where the logical structure and the complexity of the axioms were more transparent.
Givant then says that "with typical thoroughness" Tarski devised his system:
What was different about Tarski's approach to geometry? First of all, the axiom system was much simpler than any of the axiom systems that existed up to that time. In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the primitive notions only, without the help of defined notions. Of even greater importance, for the first time a clear distinction was made between full geometry and its elementary — that is, its first order — part.
Like other modern axiomatizations of Euclidean geometry, Tarski's employs a formal system consisting of symbol strings, called sentences, whose construction respects formal syntactical rules, and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects other than points, so a variable or constant cannot refer to a line or an angle. Because points are the only primitive objects, and because Tarski's system is a first-order theory, it is not even possible to define lines as sets of points. The only primitive relations (predicates) are "betweenness" and "congruence" among points.
Tarski's axiomatization is shorter than its rivals, in a sense Tarski and Givant (1999) make explicit. It is more concise than Pieri's because Pieri had only two primitive notions while Tarski introduced three: point, betweenness, and congruence. Such economy of primitive and defined notions means that Tarski's system is not very convenient for doing Euclidean geometry. Rather, Tarski designed his system to facilitate its analysis via the tools of mathematical logic, i.e., to facilitate deriving its metamathematical properties. Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the prenex normal form. This form has all universal quantifiers preceding any existential quantifiers, so that all sentences can be recast in the form Th
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https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck%20process
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In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck.
The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over time, the process tends to drift towards its mean function: such a process is called mean-reverting.
The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the center. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.
Definition
The Ornstein–Uhlenbeck process is defined by the following stochastic differential equation:
where and are parameters and denotes the Wiener process.
An additional drift term is sometimes added:
where is a constant.
The Ornstein–Uhlenbeck process is sometimes also written as a Langevin equation of the form
where , also known as white noise, stands in for the supposed derivative of the Wiener process. However, does not exist because the Wiener process is nowhere differentiable, and so the Langevin equation is, strictly speaking, only heuristic. In physics and engineering disciplines, it is a common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations by tacitly assuming that the noise term is a derivative of a differentiable (e.g. Fourier) interpolation of the Wiener process.
Fokker–Planck equation representation
The Ornstein–Uhlenbeck process can also be described in terms of a probability density function, , which specifies the probability of finding the process in the state at time . This function satisfies the Fokker–Planck equation
where . This is a linear parabolic partial differential equation which can be solved by a variety of techniques. The transition probability, also known as the Green's function, is a Gaussian with mean and variance :
This gives the probability of the state occurring at time given initial state at time . Equivalently, is the solution of the Fokker–Planck equation with initial condition .
Mathematical properties
Conditioned on a particular value of , the mean is
and the covariance is
For the stationary (unconditioned) process, the mean of is , and the covariance of and is .
The Ornstein–Uhlenbeck process is an example of a Gaussian process th
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https://en.wikipedia.org/wiki/Serre%27s%20property%20FA
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In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre.
A group G is said to have property FA if every action of G on a tree has a global fixed point.
Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.
Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup.
Examples of groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2. The group SL2(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C4 and C6 along C2.
Any quotient group of a group with property FA has property FA. If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general. If N is a normal subgroup of G and both N and G/N have property FA, then so does G.
It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.
Examples
The following groups have property FA:
A finitely generated torsion group;
SL3(Z);
The Schwarz group for integers A,B,C ≥ 2;
SL2(R) where R is the ring of integers of an algebraic number field which is not Q or an imaginary quadratic field.
The following groups do not have property FA:
SL2(Z);
SL2(RD) where RD is the ring of integers of an imaginary quadratic field of discriminant not −3 or −4.
References
English translation:
Properties of groups
Trees (graph theory)
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https://en.wikipedia.org/wiki/Uniformization
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Uniformization may refer to:
Uniformization (set theory), a mathematical concept in set theory
Uniformization theorem, a mathematical result in complex analysis and differential geometry
Uniformization (probability theory), a method to find a discrete-time Markov chain analogous to a continuous-time Markov chain
Uniformizable space, a topological space whose topology is induced by some uniform structure
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https://en.wikipedia.org/wiki/List%20of%20University%20of%20G%C3%B6ttingen%20people
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This is a list of people who have taught or studied at the University of Göttingen:
Natural sciences and mathematics
A
Wilhelm Ackermann — Mathematics
Immo Appenzeller — Astrophysics
Cahit Arf — (Doctorate in Mathematics)
B
Heinrich Behmann — Mathematical Logic
Paul Bernays — Mathematics, mathematical logic — (Student, later Professor extraordinarius)
Patrick Blackett — Physics — Nobel Prize in Physics 1948
Johann Friedrich Blumenbach— comparative anatomy
Max Born — Mathematical Physics — (Professor ordinarius) — (1882–1970, in Göttingen 1921–1933) — Nobel Prize in Physics 1954
Walther Bothe — Physics — Nobel Prize in Physics 1954 together with Max Born
Michael Buback — Chemistry
Adolf Butenandt — Chemistry — Nobel Prize in Chemistry 1939
C
Constantin Carathéodory — Mathematics
Alonzo Church — Mathematical Logic (Postdoc)
Richard Courant — Mathematics
Haskell Curry — Mathematical Logic (Postdoc)
D
Peter Debye — Mathematical Physics — (Professor ordinarius) — (1884–1966, in Göttingen 1914–1920) — Nobel Prize in Chemistry 1936
Richard Dedekind — Mathematics
Hans Georg Dehmelt — Nobel Prize in Physics 1989
Max Delbrück — Astronomy, Physics — Nobel Prize in Medicine 1969
Paul Dirac — Physics — Nobel Prize in Physics 1933 (with Erwin Schrödinger)
Peter Gustav Lejeune Dirichlet — Mathematics
E
Manfred Eigen — Biophysical Chemistry — Nobel Prize in Chemistry 1967 (with Ronald G. W. Norrish and George Porter)
Albert Einstein — Physics — Nobel Prize in Physics 1921 — (Guest lecturer, 1915)
Heinz Ellenberg — Biology, Botany — (Professor ordinarius) (1913–1997, in Göttingen 1966–1981 emeritus)
F
William Feller — Mathematics
Enrico Fermi — Physics — Nobel Prize in Physics 1938
James Franck — Physics — Nobel Prize in Physics 1925 (with Gustav Hertz)
Gottlob Frege — Mathematical Logic
Uta Fritze-von Alvensleben — Astrophysicist
Lazarus Immanuel Fuchs — Mathematics
G
Carl Friedrich Gauß — Astronomy, geodesy, mathematics, physics — (Professor ordinarius for astronomy)
Gerhard Gentzen — Mathematics
Kurt Gödel — Mathematical logic — (Non-resident lecturer, 1939)
Maria Goeppert-Mayer — Physics — Nobel Prize in Physics 1963
Hans Grauert — Mathematics
August Grisebach — Botany
H
Alfréd Haar — Mathematics
Otto Hahn — Chemistry — Nobel Prize in Chemistry 1944
Georg Hamel — Mathematics
Jacob Pieter Den Hartog — Fluid Mechanics
Helmut Hasse — Mathematics
Klaus Hasselmann — Physics — Nobel Prize in Physics 2021
Herbert Hawkes — Mathematics
Walter Norman Haworth — Chemistry — Nobel Prize in Chemistry 1937
Stefan W. Hell — Nobel Prize in Chemistry 2014 (affiliated with Heidelberg University but works at the Max-Planck Instiutute for Interdisciplinary Sciences at Göttingen formerly known as Max-Planck Institute for Biophysical Chemistry)
Heinrich Heesch — Mathematics
Andreas J. Heinrich — Physics
Werner Heisenberg — Physics — (Professor ordinarius) — Nobel Prize in Physics 1932
Ernst Hellinger — Mathematics
Gerhard Herzberg — Chemistry — Nobel Prize in Che
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https://en.wikipedia.org/wiki/Karl%20Georg%20Christian%20von%20Staudt
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Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.
Life and influence
Karl was born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who was director of the observatory. Staudt provided an ephemeris for the orbits of Mars and the asteroid Pallas. When in 1821 Comet Nicollet-Pons was observed, he provided the elements of its orbit. These accomplishments in astronomy earned him his doctorate from University of Erlangen in 1822.
Staudt's professional career began as a secondary school instructor in Würzburg until 1827 and then Nuremberg until 1835. He married Jeanette Dreschler in 1832. They had a son Eduard and daughter Mathilda, but Jeanette died in 1848.
The book Geometrie der Lage (1847) was a landmark in projective geometry. As Burau (1976) wrote:
Staudt was the first to adopt a fully rigorous approach. Without exception his predecessors still spoke of distances, perpendiculars, angles and other entities that play no role in projective geometry.
Furthermore, this book (page 43) uses the complete quadrangle to "construct the fourth harmonic associated with three points on a straight line", the projective harmonic conjugate.
Indeed, in 1889 Mario Pieri translated von Staudt, before writing his I Principii della Geometrie di Posizione Composti in un Systema Logico-deduttivo (1898). In 1900 Charlotte Scott of Bryn Mawr College paraphrased much of von Staudt's work in English for The Mathematical Gazette. When Wilhelm Blaschke published his textbook Projective Geometry in 1948, a portrait of the young Karl was placed opposite the Vorwort.
Staudt went beyond real projective geometry and into complex projective space in his three volumes of Beiträge zur Geometrie der Lage published from 1856 to 1860.
In 1922 H. F. Baker wrote of von Staudt's work:
It was von Staudt to whom the elimination of the ideas of distance and congruence was a conscious aim, if, also, the recognition of the importance of this might have been much delayed save for the work of Cayley and Klein upon the projective theory of distance. Generalised, and combined with the subsequent Dissertation of Riemann, v. Staudt's volumes must be held to be the foundation of what, on its geometrical side, the Theory of Relativity, in Physics, may yet become.
Von Staudt is also remembered for his view of conic sections and the relation of pole and polar:
Von Staudt made the important discovery that the relation which a conic establishes between poles and polars is really more fundamental than the conic itself, and can be set up independently. This "polarity" can then be used to define the conic, in a manner that is perfectly symmetrical and immediately self-dual: a conic is simply the locus of points which
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https://en.wikipedia.org/wiki/S.%20K.%20Gurunathan
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S. K. Gurunathan (1 August 1908 – 5 May 1966) was a sports journalist and one of the pioneers of cricket statistics in India.
Gurunathan studied in the Hindu High School in Triplicane, Madras. He started his journalistic career in the advertisement section of The Hindu in 1928. He became a reporter in 1938 and from 1958 till his death, was the sports editor. He founded the Madras Sports Annual which covered local cricket and other sports in the 1940s. While at The Hindu, he started the magazine Sport and Pastime which ran for about twenty years and ceased publication due to labour troubles soon after his death. Gurunathan was the first Honorary Cricket Statistician for the Board of Control for Cricket in India, serving in that post from 1949 to 1950 till his death.
Gurunathan founded the annual Indian Cricket in 1946 on the same lines as the Wisden Cricketers' Almanack and remained its editor till his death. He also regularly contributed to the Indian section of Wisden. He covered more than 50 Test matches including the Indian tours of Australia in 1947–48, England in 1952 and Pakistan in 1954–55, and reported the 1961-62 MCC tour of India for The Times. He authored the books 12 years of Ranji Trophy and three volumes of Story of the Tests. Gurunathan became the Founder-President of the Madras Sports Writers Club in 1963–64.
He was a stylish wicket keeper in his youth and represented the Indians in the Madras Presidency matches. In the Madras League matches, he represented Sundar C.C. He died a few months before he was due to retire from The Hindu.
Notes
The references used for this article differ in several details. According to P.N. Sundaresan (who worked with Gurunathan for twenty years), Gurunathan was born on 1 July 1908 and became the sports reporter in Hindu in 1937.
References
Gurunathan, S.K.
Indian sports journalists
1966 deaths
1908 births
Indian male journalists
Place of birth missing
Place of death missing
Cricket statisticians
Indian cricketers
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https://en.wikipedia.org/wiki/Complex%20Lie%20group
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In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way is holomorphic. Basic examples are , the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group.
The Lie algebra of a complex Lie group is a complex Lie algebra.
Examples
A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way.
A connected compact complex Lie group A of dimension g is of the form , a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra can be shown to be abelian and then is a surjective morphism of complex Lie groups, showing A is of the form described.
is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since , this is also an example of a representation of a complex Lie group that is not algebraic.
Let X be a compact complex manifold. Then, analogous to the real case, is a complex Lie group whose Lie algebra is the space of holomorphic vector fields on X:.
Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) , and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, is the complexification of the unitary group. If K is acting on a compact Kähler manifold X, then the action of K extends to that of G.
Linear algebraic group associated to a complex semisimple Lie group
Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows: let be the ring of holomorphic functions f on G such that spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: ). Then is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation of G. Then is Zariski-closed in .
References
Lie groups
Manifolds
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https://en.wikipedia.org/wiki/Love%2C%20Hell%20or%20Right
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Love Hell or Right (Da Come Up) is an album by the hip hop producer DJ Mathematics, who is a DJ with Wu-Tang Clan.
Completely mixed, arranged and produced by Mathematics himself, Love, Hell or Right was released August 26, 2003, on his own Quewisha Records label in conjunction with High Times Records, and it went on to sell 30,000 units. It was released in CD, vinyl and cassette tape formats.
The title is a reference to the Nation of Gods and Earths's Supreme Alphabet, in which the letter "L" is seen to stand for "Love Hell or Right".
Track listing
Note: Artists marked with an asterisk (*) are not affiliated with the Wu-Tang Clan.
"Love Hell or Right (Da Intro)"
"Pimpology 101" (Buddah Bless*)
"Thank U (Da DJ's Version)" (Method Man, Ghostface Killah and Angela Neal*)
"Message to a Blackman (Skit)" (Queen-Shatiyah*)
"Juscantluv" (Eyes Low*)
"Return of Da Cobra (Skit)" (Buddah Bless*)
"Hav Mercy" (Killa Sin and La the Darkman)
"Respect Mine" (Method Man, Raekwon and Cappadonna)
"Da Heist (Skit)" (Starking*, LEO*, Karim*, and Mouth*)
"Gangsta" (Logic*, Nemy*, Mad Man* and Eyes Low*)
"Da Great Siege" (RZA)
"Message from a Blackman (Skit)"
"Real Talk (Pop's Song)" (Pop Poppa Don*)
"Hip Hop 101" (Prodigal Sunn, H-Speed*, Born Justice, Shacronz and Allah Real)
"Queens Day '88" (Pop Poppa Don* and Eyes Low*)
"Alwayz N.Y." (Masta Killa, U-God, Inspectah Deck, Buddah Bless* and Icarus Da Don*)
"Gun Talk" (Street Life and Buddah Bless*)
"...On Da Radio (Skit)" (Ghostface Killah)
Also produced by The RZA
"Pimp Party" (Almighty Infinite*, Eyes Low*, Boy Big* and Buddah Bless*)
"Outro"
"Da Way We Were"
Mathematics (producer) albums
2003 albums
Albums produced by Mathematics
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https://en.wikipedia.org/wiki/Corner%20solution
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In mathematics and economics, a corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is when the chooser is either unwilling or unable to make a trade-off between goods.
In economics
In the context of economics the corner solution is best characterised by when the highest indifference curve attainable is not tangential to the budget line, in this scenario the consumer puts their entire budget into purchasing as much of one of the goods as possible and none of any other. When the slope of the indifference curve is greater than the slope of the budget line, the consumer is willing to give up more of good 1 for a unit of good 2 than is required by the market. Thus, it follows that if the slope of the indifference curve is strictly greater than the slope of the budget line:
Then the result will be a corner solution intersecting the x-axis. The converse is also true for a corner solution resulting from an intercept through the y-axis.
Examples
Real world examples of a corner solution occur when someone says "I wouldn't buy that at any price", "Why would I buy X when Y is cheaper" or "I will do X no matter the cost" , this could be for any number of reasons e.g. a bad brand experience, loyalty to a specific brand or when a cheaper version of the same good exists.
Another example is "zero-tolerance" policies, such as a parent who is unwilling to expose their children to any risk, no matter how small and no matter what the benefits of the activity might be. "Nothing is more important than my child's safety" is a corner solution in its refusal to admit there might be trade-offs. The term "corner solution" is sometimes used by economists in a more colloquial fashion to refer to these sorts of situations.
Another situation a corner solution may arise is when the two goods in question are perfect substitutes. The word "corner" refers to the fact that if one graphs the maximization problem, the optimal point will occur at the "corner" created by the budget constraint and one axis.
In mathematics
A corner solution is an instance where the "best" solution (i.e. maximizing profit, or utility, or whatever value is sought) is achieved based not on the market-efficient maximization of related quantities, but rather based on brute-force boundary conditions. Such a solution lacks mathematical elegance, and most examples are characterized by externally forced conditions (such as "variables x and y cannot be negative") that put the actual local extrema outside the permitted values.
Another technical way to state it is that a corner solution is a solution to a minimization or maximization problem where the non-corner solution is infeasible, that is, not in the domain. Instead, the solution is a corner solution on an axis where either x or y is equal to zero. For instance, from the example above in economics, if the max
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https://en.wikipedia.org/wiki/Tree%20%28descriptive%20set%20theory%29
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In descriptive set theory, a tree on a set is a collection of finite sequences of elements of such that every prefix of a sequence in the collection also belongs to the collection.
Definitions
Trees
The collection of all finite sequences of elements of a set is denoted .
With this notation, a tree is a nonempty subset of , such that if
is a sequence of length in , and if ,
then the shortened sequence also belongs to . In particular, choosing shows that the empty sequence belongs to every tree.
Branches and bodies
A branch through a tree is an infinite sequence of elements of , each of whose finite prefixes belongs to . The set of all branches through is denoted and called the body of the tree .
A tree that has no branches is called wellfounded; a tree with at least one branch is illfounded. By Kőnig's lemma, a tree on a finite set with an infinite number of sequences must necessarily be illfounded.
Terminal nodes
A finite sequence that belongs to a tree is called a terminal node if it is not a prefix of a longer sequence in . Equivalently, is terminal if there is no element of such that that . A tree that does not have any terminal nodes is called pruned.
Relation to other types of trees
In graph theory, a rooted tree is a directed graph in which every vertex except for a special root vertex has exactly one outgoing edge, and in which the path formed by following these edges from any vertex eventually leads to the root vertex.
If is a tree in the descriptive set theory sense, then it corresponds to a graph with one vertex for each sequence in , and an outgoing edge from each nonempty sequence that connects it to the shorter sequence formed by removing its last element. This graph is a tree in the graph-theoretic sense. The root of the tree is the empty sequence.
In order theory, a different notion of a tree is used: an order-theoretic tree is a partially ordered set with one minimal element in which each element has a well-ordered set of predecessors.
Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences and are ordered by if and only if is a proper prefix of . The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes).
An order-theoretic tree may be represented by an isomorphic tree of sequences if and only if each of its elements has finite height (that is, a finite set of predecessors).
Topology
The set of infinite sequences over (denoted as ) may be given the product topology, treating X as a discrete space.
In this topology, every closed subset of is of the form for some pruned tree .
Namely, let consist of the set of finite prefixes of the infinite sequences in . Conversely, the body of every tree forms a closed set in this topology.
Frequently trees on Cartesian products are considered. In this case, by convention, we consider only the subset of t
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https://en.wikipedia.org/wiki/Decagram
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Decagram may refer to:
10 gram, or 0.01 kilogram, a unit of mass, in SI referred to as a dag
Decagram (geometry), geometric figure
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https://en.wikipedia.org/wiki/Rectification%20%28geometry%29
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In Euclidean geometry, rectification, also known as critical truncation or complete-truncation, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.
A rectification operator is sometimes denoted by the letter with a Schläfli symbol. For example, is the rectified cube, also called a cuboctahedron, and also represented as . And a rectified cuboctahedron is a rhombicuboctahedron, and also represented as .
Conway polyhedron notation uses for ambo as this operator. In graph theory this operation creates a medial graph.
The rectification of any regular self-dual polyhedron or tiling will result in another regular polyhedron or tiling with a tiling order of 4, for example the tetrahedron becoming an octahedron As a special case, a square tiling will turn into another square tiling under a rectification operation.
Example of rectification as a final truncation to an edge
Rectification is the final point of a truncation process. For example, on a cube this sequence shows four steps of a continuum of truncations between the regular and rectified form:
Higher degree rectifications
Higher degree rectification can be performed on higher-dimensional regular polytopes. The highest degree of rectification creates the dual polytope. A rectification truncates edges to points. A birectification truncates faces to points. A trirectification truncates cells to points, and so on.
Example of birectification as a final truncation to a face
This sequence shows a birectified cube as the final sequence from a cube to the dual where the original faces are truncated down to a single point:
In polygons
The dual of a polygon is the same as its rectified form. New vertices are placed at the center of the edges of the original polygon.
In polyhedra and plane tilings
Each platonic solid and its dual have the same rectified polyhedron. (This is not true of polytopes in higher dimensions.)
The rectified polyhedron turns out to be expressible as the intersection of the original platonic solid with an appropriately scaled concentric version of its dual. For this reason, its name is a combination of the names of the original and the dual:
The tetrahedron is its own dual, and its rectification is the tetratetrahedron, better known as the octahedron.
The octahedron and the cube are each other's dual, and their rectification is the cuboctahedron.
The icosahedron and the dodecahedron are duals, and their rectification is the icosidodecahedron.
Examples
In nonregular polyhedra
If a polyhedron is not regular, the edge midpoints surrounding a vertex may not be coplanar. However, a form of rectification is still possible in this case: every polyhedron has a polyhedral graph as its 1-skeleton, and from that graph one may form the medial graph by placing a vertex at each edg
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https://en.wikipedia.org/wiki/Resultant
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In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant.
The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation.
The resultant of n homogeneous polynomials in n variables (also called multivariate resultant, or Macaulay's resultant for distinguishing it from the usual resultant) is a generalization, introduced by Macaulay, of the usual resultant. It is, with Gröbner bases, one of the main tools of elimination theory.
Notation
The resultant of two univariate polynomials and is commonly denoted or
In many applications of the resultant, the polynomials depend on several indeterminates and may be considered as univariate polynomials in one of their indeterminates, with polynomials in the other indeterminates as coefficients. In this case, the indeterminate that is selected for defining and computing the resultant is indicated as a subscript: or
The degrees of the polynomials are used in the definition of the resultant. However, a polynomial of degree may also be considered as a polynomial of higher degree where the leading coefficients are zero. If such a higher degree is used for the resultant, it is usually indicated as a subscript or a superscript, such as or
Definition
The resultant of two univariate polynomials over a field or over a commutative ring is commonly defined as the determinant of their Sylvester matrix. More precisely, let
and
be nonzero polynomials of degrees and respectively. Let us denote by the vector space (or free module if the coefficients belong to a commutative ring) of dimension whose elements are the polynomials of degree strictly less than . The map
such that
is a linear map between two spaces of the same dimension. Over the basis of the powers of (listed in descending order), this map is represented by a square matrix of dimension , which is called the Sylvester matrix of and (for many authors and in the article Sylvester matrix, the Sylvester matrix is defined as the transpose of this matrix; this convention is not used here, as it breaks the usual convention for writing the matrix of a linear map).
The resultant of and is thus the determinant
which has columns of and columns of (the fact that the first column of 's and the
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https://en.wikipedia.org/wiki/Rational%20zeta%20series
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In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function. Specifically, given a real number x, the rational zeta series for x is given by
where qn is a rational number, the value m is held fixed, and ζ(s, m) is the Hurwitz zeta function. It is not hard to show that any real number x can be expanded in this way.
Elementary series
For integer m>1, one has
For m=2, a number of interesting numbers have a simple expression as rational zeta series:
and
where γ is the Euler–Mascheroni constant. The series
follows by summing the Gauss–Kuzmin distribution. There are also series for π:
and
being notable because of its fast convergence. This last series follows from the general identity
which in turn follows from the generating function for the Bernoulli numbers
Adamchik and Srivastava give a similar series
Polygamma-related series
A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is
.
The above converges for |z| < 1. A special case is
which holds for |t| < 2. Here, ψ is the digamma function and ψ(m) is the polygamma function. Many series involving the binomial coefficient may be derived:
where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta
taken at y = −1. Similar series may be obtained by simple algebra:
and
and
and
For integer n ≥ 0, the series
can be written as the finite sum
The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series
may be written as
for integer n ≥ 1. The above follows from the identity Tn + Tn + 1 = Sn. This process may be applied recursively to obtain finite series for general expressions of the form
for positive integers m.
Half-integer power series
Similar series may be obtained by exploring the Hurwitz zeta function at half-integer values. Thus, for example, one has
Expressions in the form of p-series
Adamchik and Srivastava give
and
where are the Bernoulli numbers and are the Stirling numbers of the second kind.
Other series
Other constants that have notable rational zeta series are:
Khinchin's constant
Apéry's constant
References
Zeta and L-functions
Real numbers
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https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin%20distribution
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In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. It is given by the probability mass function
Gauss–Kuzmin theorem
Let
be the continued fraction expansion of a random number x uniformly distributed in (0, 1). Then
Equivalently, let
then
tends to zero as n tends to infinity.
Rate of convergence
In 1928, Kuzmin gave the bound
In 1929, Paul Lévy improved it to
Later, Eduard Wirsing showed that, for λ = 0.30366... (the Gauss–Kuzmin–Wirsing constant), the limit
exists for every s in [0, 1], and the function Ψ(s) is analytic and satisfies Ψ(0) = Ψ(1) = 0. Further bounds were proved by K. I. Babenko.
See also
Khinchin's constant
Lévy's constant
References
Continued fractions
Discrete distributions
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https://en.wikipedia.org/wiki/SYSTAT%20%28statistics%20package%29
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SYSTAT is a statistics and statistical graphics software package, developed by Leland Wilkinson in the late 1970s, who was at the time an assistant professor of psychology at the University of Illinois at Chicago. Systat Software Inc. was incorporated in 1983 and grew to over 50 employees.
In 1995, SYSTAT was sold to SPSS Inc., who marketed the product to a scientific audience under the SPSS Science division. By 2002, SPSS had changed its focus to business analytics and decided to sell SYSTAT to Cranes Software in Bangalore, India. Cranes formed Systat Software, Inc. to market and distribute SYSTAT in the US, and a number of other divisions for global distribution. The headquarters are in Chicago, Illinois.
By 2005, SYSTAT was in its eleventh version having a revamped codebase completely changed from Fortran into C++. Version 13 came out in 2009, with improvements in the user interface and several new features.
See also
Comparison of statistical packages
PeakFit
TableCurve 2D
TableCurve 3D
References
External links
SYSTAT
The story of SYSTAT as told by Wilkinson
C++ software
Statistical software
Windows-only proprietary software
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https://en.wikipedia.org/wiki/Zeta%20function%20universality
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In mathematics, the universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well.
The universality of the Riemann zeta function was first proven by in 1975 and is sometimes known as Voronin's universality theorem.
Formal statement
A mathematically precise statement of universality for the Riemann zeta function ζ(s) follows.
Let U be a compact subset of the strip
such that the complement of U is connected. Let be a continuous function on U which is holomorphic on the interior of U and does not have any zeros in U. Then for any there exists a such that
for all .
Even more: the lower density of the set of values t satisfying the above inequality is positive. Precisely
where denotes the Lebesgue measure on the real numbers and denotes the limit inferior.
Discussion
The condition that the complement of U be connected essentially means that U does not contain any holes.
The intuitive meaning of the first statement is as follows: it is possible to move U by some vertical displacement it so that the function f on U is approximated by the zeta function on the displaced copy of U, to an accuracy of ε.
The function f is not allowed to have any zeros on U. This is an important restriction; if we start with a holomorphic function with an isolated zero, then any "nearby" holomorphic function will also have a zero. According to the Riemann hypothesis, the Riemann zeta function does not have any zeros in the considered strip, and so it couldn't possibly approximate such a function. The function which is identically zero on U can be approximated by ζ: we can first pick the "nearby" function (which is holomorphic and does not have zeros) and find a vertical displacement such that ζ approximates g to accuracy ε/2, and therefore f to accuracy ε.
The accompanying figure shows the zeta function on a representative part of the relevant strip. The color of the point s encodes the value ζ(s) as follows: the hue represents the argument of ζ(s), with red denoting positive real values, and then counterclockwise through yellow, green cyan, blue and purple. Strong colors denote values close to 0 (black = 0), weak colors denote values far away from 0 (white = ∞). The picture shows three zeros of the zeta function, at about , and . Voronin's theorem essentially states that this strip contains all possible "analytic" color patterns that do not use black or white.
The rough meaning of the statement on the lower density is as follows: if a function f and an are given, then there is a positive probability that a randomly picked vertical displacement it will yield an approximation of f to accuracy ε.
The interior of U may be empty, in which case there is no requirement of f being holomorphic. For example, if we take U to be a line segment, then a continuous function is a curve in the complex plan
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https://en.wikipedia.org/wiki/Cross-covariance
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In probability and statistics, given two stochastic processes and , the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation for the expectation operator, if the processes have the mean functions and , then the cross-covariance is given by
Cross-covariance is related to the more commonly used cross-correlation of the processes in question.
In the case of two random vectors and , the cross-covariance would be a matrix (often denoted ) with entries Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector , which is understood to be the matrix of covariances between the scalar components of itself.
In signal processing, the cross-covariance is often called cross-correlation and is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis.
Cross-covariance of random vectors
Cross-covariance of stochastic processes
The definition of cross-covariance of random vectors may be generalized to stochastic processes as follows:
Definition
Let and denote stochastic processes. Then the cross-covariance function of the processes is defined by:
where and .
If the processes are complex-valued stochastic processes, the second factor needs to be complex conjugated:
Definition for jointly WSS processes
If and are a jointly wide-sense stationary, then the following are true:
for all ,
for all
and
for all
By setting (the time lag, or the amount of time by which the signal has been shifted), we may define
.
The cross-covariance function of two jointly WSS processes is therefore given by:
which is equivalent to
.
Uncorrelatedness
Two stochastic processes and are called uncorrelated if their covariance is zero for all times. Formally:
.
Cross-covariance of deterministic signals
The cross-covariance is also relevant in signal processing where the cross-covariance between two wide-sense stationary random processes can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a sub-sampling of one of the signals). For a large number of samples, the average converges to the true covariance.
Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices. For example, for discrete-time signals and the cross-covariance is defined as
where the line indicates that the complex conjugate is taken when the signals are complex-valued.
For continuous functions and the (determinist
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https://en.wikipedia.org/wiki/Front%20velocity
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In physics, front velocity is the speed at which the first rise of a pulse above zero moves forward.
In mathematics, it is used to describe the velocity of a propagating front in the solution of hyperbolic partial differential equation.
Various velocities
Associated with propagation of a disturbance are several different velocities. For definiteness, consider an amplitude modulated electromagnetic carrier wave. The phase velocity is the speed of the underlying carrier wave. The group velocity is the speed of the modulation or envelope. Initially it was thought that the group velocity coincided with the speed at which information traveled. However, it turns out that this speed can exceed the speed of light in some circumstances, causing confusion by an apparent conflict with the theory of relativity. That observation led to consideration of what constitutes a signal.
By definition, a signal involves new information or an element of 'surprise' that cannot be predicted from the wave motion at an earlier time. One possible form for a signal (at the point of emission) is:
where u(t) is the Heaviside step function. Using such a form for a signal, it can be shown, subject to the (expected) condition that the refractive index of any medium tends to one as the frequency tends to infinity, that the wave discontinuity, called the front, propagates at a speed less than or equal to the speed of light c in any medium. In fact, the earliest appearance of the front of an electromagnetic disturbance (the precursor) travels at the front velocity, which is c, no matter what the medium. However, the process always starts from zero amplitude and builds up.
References
Wave mechanics
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https://en.wikipedia.org/wiki/Selberg%20class
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In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis. The class was defined by Atle Selberg in , who preferred not to use the word "axiom" that later authors have employed.
Definition
The formal definition of the class S is the set of all Dirichlet series
absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them):
Comments on definition
The condition that the real part of μi be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μi is negative. Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Peterssen conjecture holds, and have a functional equation, but do not satisfy the Riemann hypothesis.
The condition that θ < 1/2 is important, as the θ = 1 case includes whose zeros are not on the critical line.
Without the condition there would be which violates the Riemann hypothesis.
It is a consequence of 4. that the an are multiplicative and that
Examples
The prototypical example of an element in S is the Riemann zeta function. Another example, is the L-function of the modular discriminant Δ
where and τ(n) is the Ramanujan tau function.
All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.
The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2.
Basic properties
As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s). The other zeroes are referred to as the non-trivial zeroes of F. These will all be located in some strip . Denoting the number of non-trivial zeroes of F with by NF(T), Selberg showed that
Here, dF is called the degree (or dimension) of F. It is given by
It can be shown that F = 1 is the only function in S whose degree is less than 1.
If F and G are in the Selberg class, then so is their product and
A function in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2. If dF = 1, then F is primitive. Every function of S can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique.
Examples of primitive functions include the Riemann zeta function and Dirichl
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https://en.wikipedia.org/wiki/Stickelberger%27s%20theorem
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In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890).
The Stickelberger element and the Stickelberger ideal
Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to (where is an integer). It is a Galois extension of with Galois group isomorphic to the multiplicative group of integers modulo . The Stickelberger element (of level or of ) is an element in the group ring and the Stickelberger ideal (of level or of ) is an ideal in the group ring . They are defined as follows. Let denote a primitive th root of unity. The isomorphism from to is given by sending to defined by the relation
.
The Stickelberger element of level is defined as
The Stickelberger ideal of level , denoted , is the set of integral multiples of which have integral coefficients, i.e.
More generally, if be any Abelian number field whose Galois group over is denoted , then the Stickelberger element of and the Stickelberger ideal of can be defined. By the Kronecker–Weber theorem there is an integer such that is contained in . Fix the least such (this is the (finite part of the) conductor of over ). There is a natural group homomorphism given by restriction, i.e. if , its image in is its restriction to denoted . The Stickelberger element of is then defined as
The Stickelberger ideal of , denoted , is defined as in the case of , i.e.
In the special case where , the Stickelberger ideal is generated by as varies over . This not true for general F.
Examples
If is a totally real field of conductor , then
where is the Euler totient function and is the degree of over .
Statement of the theorem
Stickelberger's Theorem
Let be an abelian number field. Then, the Stickelberger ideal of annihilates the class group of .
Note that itself need not be an annihilator, but any multiple of it in is.
Explicitly, the theorem is saying that if is such that
and if is any fractional ideal of , then
is a principal ideal.
See also
Gross–Koblitz formula
Herbrand–Ribet theorem
Thaine's theorem
Jacobi sum
Gauss sum
Notes
References
Boas Erez, Darstellungen von Gruppen in der Algebraischen Zahlentheorie: eine Einführung
External links
PlanetMath page
Cyclotomic fields
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/David%20Blackwell
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David Harold Blackwell (April 24, 1919 – July 8, 2010) was an American statistician and mathematician who made significant contributions to game theory, probability theory, information theory, and statistics. He is one of the eponyms of the Rao–Blackwell theorem. He was the first African American inducted into the National Academy of Sciences, the first African American full professor (with tenure) at the University of California, Berkeley, and the seventh African American to receive a Ph.D. in mathematics. In 2012, President Obama posthumously awarded Blackwell the National Medal of Science.
Blackwell was also a pioneer in textbook writing. He wrote one of the first Bayesian statistics textbooks, his 1969 Basic Statistics. By the time he retired, he had published over 90 papers and books on dynamic programming, game theory, and mathematical statistics.
Early life and education
David Harold Blackwell was born on April 24, 1919, in Centralia, Illinois, to Mabel Johnson Blackwell, a full-time homemaker, and Grover Blackwell, an Illinois Central Railroad worker. He was the eldest of four children with two brothers, J. W. and Joseph, and one sister, Elizabeth. Growing up in an integrated community, Blackwell attended "mixed" schools, where he distinguished himself in mathematics. During elementary school, his teachers promoted him beyond his grade level on two occasions. It was in a high school geometry course, however, that his passion for math began. An exceptional student, Blackwell graduated high school in 1935 at the age of sixteen.
Blackwell entered the University of Illinois at Urbana-Champaign with the intent to study elementary school mathematics and become a teacher. He was a member of Alpha Phi Alpha, a black fraternity that housed him for his full six years as a student. He earned his bachelor's degree in mathematics in three years in 1938 and, a year later, a master's degree in 1939. He was awarded a Doctor of Philosophy in mathematics in 1941 at the age of 22. His doctoral advisor was Joseph L. Doob. At the time, Blackwell was the seventh African American to earn a Ph.D. in mathematics in the United States and the first at the University of Illinois at Urbana-Champaign. His doctoral thesis was on Markov chains.
Career and research
Postdoctoral study and early career
Blackwell completed one year of postdoctoral research as a fellow at the Institute for Advanced Study (IAS) at Princeton in 1941 after receiving a Rosenwald Fellowship, which was a fund to aid black scholars. There he met John von Neumann, who asked Blackwell to discuss his Ph.D. thesis with him. Blackwell, who believed that von Neumann was just being polite and not genuinely interested in his work, did not approach him until von Neumann himself asked him again a few months later. According to Blackwell, "He (von Neumann) listened to me talk about this rather obscure subject and in ten minutes he knew more about it than I did."
While a postdoc at IAS, Blackwell was
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https://en.wikipedia.org/wiki/Explicit%20formulae%20for%20L-functions
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In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.
Riemann's explicit formula
In his 1859 paper "On the Number of Primes Less Than a Given Magnitude" Riemann sketched an explicit formula (it was not fully proven until 1895 by von Mangoldt, see below) for the normalized prime-counting function which is related to the prime-counting function by
which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. His formula was given in terms of the related function
in which a prime power counts as of a prime. The normalized prime-counting function can be recovered from this function by
where is the Möbius function. Riemann's formula is then
involving a sum over the non-trivial zeros of the Riemann zeta function. The sum is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral
The terms involving the zeros of the zeta function need some care in their definition as has branch points at 0 and 1, and are defined by analytic continuation in the complex variable in the region and . The other terms also correspond to zeros: The dominant term comes from the pole at , considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see .)
The first rigorous proof of the aforementioned formula was given by von Mangoldt in 1895: it started with a proof of the following formula for the Chebyshev's function
where the LHS is an inverse Mellin transform with
and the RHS is obtained from the residue theorem, and then converting it into the formula that Riemann himself actually sketched.
This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part:
where
The error involved in truncating the sum to is always smaller than in absolute value, and when divided by the natural logarithm of , has absolute value smaller than divided by the distance from to the nearest prime power.
Weil's explicit formula
There are several slightly different ways to state the explicit formula. André Weil's form of the explicit formula states
where
ρ runs over the non-trivial zeros of the zeta function
p runs over positive primes
m runs over positive integers
F is a smooth function all of whose derivativ
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https://en.wikipedia.org/wiki/Concurrent%20lines
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In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point. They are in contrast to parallel lines.
Examples
Triangles
In a triangle, four basic types of sets of concurrent lines are altitudes, angle bisectors, medians, and perpendicular bisectors:
A triangle's altitudes run from each vertex and meet the opposite side at a right angle. The point where the three altitudes meet is the orthocenter.
Angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter.
Medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid.
Perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter.
Other sets of lines associated with a triangle are concurrent as well. For example:
Any median (which is necessarily a bisector of the triangle's area) is concurrent with two other area bisectors each of which is parallel to a side.
A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle.
A splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle.
Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter, and each triangle has one, two, or three of these lines. Thus if there are three of them, they concur at the incenter.
The Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle.
The Schiffler point of a triangle is the point of concurrence of the Euler lines of four triangles: the triangle in question, and the three triangles that each share two vertices with it and have its incenter as the other vertex.
The Napoleon points and generalizations of them are points of concurrency. For example, the first Napoleon point is the point of concurrency of the three lines each from a vertex to the centroid of the equilateral triangle drawn on the exterior of the opposite side from the vertex. A generalization of this notion is the Jacobi point.
The de Longchamps point is the point of concurrence of several lines with the Euler line.
Three lines, each formed by drawing an external equilateral triangle on one of the sides of a given triangle and connecting the new vertex to the original triangle's opposite vertex, are concurrent at a point called the first isogonal center. In the case in which the original triangle has no angle greater than 120°, this
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https://en.wikipedia.org/wiki/Einstein%20coefficients
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Einstein coefficients are quantities describing the probability of absorption or emission of a photon by an atom or molecule. The Einstein A coefficients are related to the rate of spontaneous emission of light, and the Einstein B coefficients are related to the absorption and stimulated emission of light. Throughout this article, "light" refers to any electromagnetic radiation, not necessarily in the visible spectrum.
Spectral lines
In physics, one thinks of a spectral line from two viewpoints.
An emission line is formed when an atom or molecule makes a transition from a particular discrete energy level of an atom, to a lower energy level , emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons.
An absorption line is formed when an atom or molecule makes a transition from a lower, , to a higher discrete energy state, , with a photon being absorbed in the process. These absorbed photons generally come from background continuum radiation (the full spectrum of electromagnetic radiation) and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons.
The two states must be bound states in which the electron is bound to the atom or molecule, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom completely ("bound–free" transition) into a continuum state, leaving an ionized atom, and generating continuum radiation.
A photon with an energy equal to the difference between the energy levels is released or absorbed in the process. The frequency at which the spectral line occurs is related to the photon energy by Bohr's frequency condition where denotes the Planck constant.
Emission and absorption coefficients
An atomic spectral line refers to emission and absorption events in a gas in which is the density of atoms in the upper-energy state for the line, and is the density of atoms in the lower-energy state for the line.
The emission of atomic line radiation at frequency may be described by an emission coefficient with units of energy/(time × volume × solid angle). ε dt dV dΩ is then the energy emitted by a volume element in time into solid angle . For atomic line radiation,
where is the Einstein coefficient for spontaneous emission, which is fixed by the intrinsic properties of the relevant atom for the two relevant energy levels.
The absorption of atomic line radiation may be described by an absorption coefficient with units of 1/length. The expression κ' dx gives the fraction of intensity absorbed for a light beam at frequency while traveling distance dx. The absorption coefficient is given by
where and are the Einstein coefficients for photon absorption and induced emission respectively. Like the coefficient , these are also fixed by the intrinsic properties of the releva
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https://en.wikipedia.org/wiki/Calculus%20%28medicine%29
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A calculus (: calculi), often called a stone, is a concretion of material, usually mineral salts, that forms in an organ or duct of the body. Formation of calculi is known as lithiasis (). Stones can cause a number of medical conditions.
Some common principles (below) apply to stones at any location, but for specifics see the particular stone type in question.
Calculi are not to be confused with gastroliths.
Types
Calculi in the inner ear are called otoliths
Calculi in the urinary system are called urinary calculi and include kidney stones (also called renal calculi or nephroliths) and bladder stones (also called vesical calculi or cystoliths). They can have any of several compositions, including mixed. Principal compositions include oxalate and urate.
Calculi of the gallbladder and bile ducts are called gallstones and are primarily developed from bile salts and cholesterol derivatives.
Calculi in the nasal passages (rhinoliths) are rare.
Calculi in the gastrointestinal tract (enteroliths) can be enormous. Individual enteroliths weighing many pounds have been reported in horses.
Calculi in the stomach are called gastric calculi (Not to be confused with gastroliths which are exogenous in nature).
Calculi in the salivary glands are called salivary calculi (sialoliths).
Calculi in the tonsils are called tonsillar calculi (tonsilloliths).
Calculi in the veins are called venous calculi (phleboliths).
Calculi in the skin, such as in sweat glands, are not common but occasionally occur.
Calculi in the navel are called omphaloliths.
Calculi are usually asymptomatic, and large calculi may have required many years to grow to their large size.
Cause
From an underlying abnormal excess of the mineral, e.g., with elevated levels of calcium (hypercalcaemia) that may cause kidney stones, dietary factors for gallstones.
Local conditions at the site in question that promote their formation, e.g., local bacteria action (in kidney stones) or slower fluid flow rates, a possible explanation of the majority of salivary duct calculus occurring in the submandibular salivary gland.
Enteroliths are a type of calculus found in the intestines of animals (mostly ruminants) and humans, and may be composed of inorganic or organic constituents.
Bezoars are lumps of indigestible material in the stomach and/or intestines; most commonly, they consist of hair (in which case they are also known as hairballs). A bezoar may form the nidus of an enterolith.
In kidney stones, calcium oxalate is the most common mineral type (see Nephrolithiasis). Uric acid is the second most common mineral type, but an in vitro study showed uric acid stones and crystals can promote the formation of calcium oxalate stones.
Pathophysiology
Stones can cause disease by several mechanisms:
Irritation of nearby tissues, causing pain, swelling, and inflammation
Obstruction of an opening or duct, interfering with normal flow and disrupting the function of the organ in question
Predisposit
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https://en.wikipedia.org/wiki/Lov%C3%A1sz%20local%20lemma
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In probability theory, if a large number of events are all independent of one another and each has probability less than 1, then there is a positive (possibly small) probability that none of the events will occur. The Lovász local lemma allows one to relax the independence condition slightly: As long as the events are "mostly" independent from one another and aren't individually too likely, then there will still be a positive probability that none of them occurs. It is most commonly used in the probabilistic method, in particular to give existence proofs.
There are several different versions of the lemma. The simplest and most frequently used is the symmetric version given below. A weaker version was proved in 1975 by László Lovász and Paul Erdős in the article Problems and results on 3-chromatic hypergraphs and some related questions. For other versions, see . In 2020, Robin Moser and Gábor Tardos received the Gödel Prize for their algorithmic version of the Lovász Local Lemma, which uses entropy compression to provide an efficient randomized algorithm for finding an outcome in which none of the events occurs.
Statements of the lemma (symmetric version)
Let A1, A2,..., Ak be a sequence of events such that each event occurs with probability at most p and such that each event is independent of all the other events except for at most d of them.
Lemma I (Lovász and Erdős 1973; published 1975) If
then there is a nonzero probability that none of the events occurs.
Lemma II (Lovász 1977; published by Joel Spencer) If
where e = 2.718... is the base of natural logarithms, then there is a nonzero probability that none of the events occurs.
Lemma II today is usually referred to as "Lovász local lemma".
Lemma III (Shearer 1985) If
then there is a nonzero probability that none of the events occurs.
The threshold in Lemma III is optimal and it implies that the bound
is also sufficient.
Asymmetric Lovász local lemma
A statement of the asymmetric version (which allows for events with different probability bounds) is as follows:
Lemma (asymmetric version). Let be a finite set of events in the probability space Ω. For let denote the neighbours of in the dependency graph (In the dependency graph, event is not adjacent to events which are mutually independent). If there exists an assignment of reals to the events such that
then the probability of avoiding all events in is positive, in particular
The symmetric version follows immediately from the asymmetric version by setting
to get the sufficient condition
since
Constructive versus non-constructive
Note that, as is often the case with probabilistic arguments, this theorem is nonconstructive and gives no method of determining an explicit element of the probability space in which no event occurs. However, algorithmic versions of the local lemma with stronger preconditions are also known (Beck 1991; Czumaj and Scheideler 2000). More recently, a constructive version of the local lemma w
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https://en.wikipedia.org/wiki/Johann%20Gottlieb%20N%C3%B6rremberg
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Johann Gottlieb Christian Nörremberg (11 August 1787, in Pustenbach – 20 July 1862) was a German physicist who worked on the polarization of light.
From 1823 he taught classes in mathematics and physics at the military school in Darmstadt. In 1833 he became a professor of mathematics, physics and astronomy at the University of Tübingen, where he worked on surveying and the development of optical instruments. Among his better known creations was a polarization apparatus, a device used in the making of a "Nörremberg polariscope". Most of his scientific articles were published in Poggendorfs Annalen.
Notes
External links
1787 births
1862 deaths
19th-century German physicists
Academic staff of the University of Tübingen
People from Bergneustadt
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https://en.wikipedia.org/wiki/Generic%20filter
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In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC. For example, Paul Cohen used forcing to establish that ZFC, if consistent, cannot prove the continuum hypothesis, which states that there are exactly aleph-one real numbers. In the contemporary re-interpretation of Cohen's proof, it proceeds by constructing a generic filter that codes more than reals, without changing the value of .
Formally, let P be a partially ordered set, and let F be a filter on P; that is, F is a subset of P such that:
F is nonempty
If p, q ∈ P and p ≤ q and p is an element of F, then q is an element of F (F is closed upward)
If p and q are elements of F, then there is an element r of F such that r ≤ p and r ≤ q (F is downward directed)
Now if D is a collection of dense open subsets of P, in the topology whose basic open sets are all sets of the form {q | q ≤ p} for particular p in P, then F is said to be D-generic if F meets all sets in D; that is,
for all E ∈ D.
Similarly, if M is a transitive model of ZFC (or some sufficient fragment thereof), with P an element of M, then F is said to be M-generic, or sometimes generic over M, if F meets all dense open subsets of P that are elements of M.
See also
in computability
References
Forcing (mathematics)
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https://en.wikipedia.org/wiki/Poisson%20kernel
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In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.
Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics.
In practice, the definition of Poisson kernels are often extended to n-dimensional problems.
Two-dimensional Poisson kernels
On the unit disc
In the complex plane, the Poisson kernel for the unit disc is given by
This can be thought of in two ways: either as a function of r and θ, or as a family of functions of θ indexed by r.
If is the open unit disc in C, T is the boundary of the disc, and f a function on T that lies in L1(T), then the function u given by
is harmonic in D and has a radial limit that agrees with f almost everywhere on the boundary T of the disc.
That the boundary value of u is f can be argued using the fact that as , the functions form an approximate unit in the convolution algebra L1(T). As linear operators, they tend to the Dirac delta function pointwise on Lp(T). By the maximum principle, u is the only such harmonic function on D.
Convolutions with this approximate unit gives an example of a summability kernel for the Fourier series of a function in L1(T) . Let f ∈ L1(T) have Fourier series {fk}. After the Fourier transform, convolution with Pr(θ) becomes multiplication by the sequence {r|k|} ∈ ℓ1(Z). Taking the inverse Fourier transform of the resulting product {r|k|fk} gives the Abel means Arf of f:
Rearranging this absolutely convergent series shows that f is the boundary value of g + h, where g (resp. h) is a holomorphic (resp. antiholomorphic) function on D.
When one also asks for the harmonic extension to be holomorphic, then the solutions are elements of a Hardy space. This is true when the negative Fourier coefficients of f all vanish. In particular, the Poisson kernel is commonly used to demonstrate the equivalence of the Hardy spaces on the unit disk, and the unit circle.
The space of functions that are the limits on T of functions in Hp(z) may be called Hp(T). It is a closed subspace of Lp(T) (at least for p ≥ 1). Since Lp(T) is a Banach space (for 1 ≤ p ≤ ∞), so is Hp(T).
On the upper half-plane
The unit disk may be conformally mapped to the upper half-plane by means of certain Möbius transformations. Since the conformal map of a harmonic function is also harmonic, the Poisson kernel carries over to the upper half-plane. In this case, the Poisson integral equation takes the form
The kernel itself is given by
Given a function , the Lp space of integrable functions on the real line, u can be understood as a harmonic extension of f into the upper half-plane. In analogy to the situation for the disk, when u is holomorphic in the upper half-plane, then u is an ele
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https://en.wikipedia.org/wiki/Geometry%20processing
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Geometry processing, or mesh processing, is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation and transmission of complex 3D models. As the name implies, many of the concepts, data structures, and algorithms are directly analogous to signal processing and image processing. For example, where image smoothing might convolve an intensity signal with a blur kernel formed using the Laplace operator, geometric smoothing might be achieved by convolving a surface geometry with a blur kernel formed using the Laplace-Beltrami operator.
Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing.
Geometry processing is a common research topic at SIGGRAPH, the premier computer graphics academic conference, and the main topic of the annual Symposium on Geometry Processing.
Geometry processing as a life cycle
Geometry processing involves working with a shape, usually in 2D or 3D, although the shape can live in a space of arbitrary dimensions. The processing of a shape involves three stages, which is known as its life cycle. At its "birth," a shape can be instantiated through one of three methods: a model, a mathematical representation, or a scan. After a shape is born, it can be analyzed and edited repeatedly in a cycle. This usually involves acquiring different measurements, such as the distances between the points of the shape, the smoothness of the shape, or its Euler characteristic. Editing may involve denoising, deforming, or performing rigid transformations. At the final stage of the shape's "life," it is consumed. This can mean it is consumed by a viewer as a rendered asset in a game or movie, for instance. The end of a shape's life can also be defined by a decision about the shape, like whether or not it satisfies some criteria. Or it can even be fabricated in the real world, through a method such as 3D printing or laser cutting.
Discrete Representation of a Shape
Like any other shape, the shapes used in geometry processing have properties pertaining to their geometry and topology. The geometry of a shape concerns the position of the shape's points in space, tangents, normals, and curvature. It also includes the dimension in which the shape lives (ex. or ). The topology of a shape is a collection of properties that do not change even after smooth transformations have been applied to the shape. It concerns dimensions such as the number of holes and boundaries, as well as the orientability of the shape. One example of a non-orientable shape is the Mobius strip.
In computers, everything must be discretized. Shapes in geometry processing are usually represented as triangle meshes, which can be seen as a graph. Each node in the graph is a v
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https://en.wikipedia.org/wiki/Dirichlet%20beta%20function
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In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.
Definition
The Dirichlet beta function is defined as
or, equivalently,
In each case, it is assumed that Re(s) > 0.
Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:
Another equivalent definition, in terms of the Lerch transcendent, is:
which is once again valid for all complex values of s.
The Dirichlet beta function can also be written in terms of the polylogarithm function:
Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function
but this formula is only valid at positive integer values of .
Euler product formula
It is also the simplest example of a series non-directly related to which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.
At least for Re(s) ≥ 1:
where are the primes of the form (5,13,17,...) and are the primes of the form (3,7,11,...). This can be written compactly as
Functional equation
The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It is given by
where Γ(s) is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842 (see Blagouchine, 2014).
Special values
Some special values include:
where G represents Catalan's constant, and
where in the above is an example of the polygamma function.
Hence, the function vanishes for all odd negative integral values of the argument.
For every positive integer k:
where is the Euler zigzag number.
Also it was derived by Malmsten in 1842 (see Blagouchine, 2014) that
There are zeros at -1; -3; -5; -7 etc.
See also
Hurwitz zeta function
Dirichlet eta function
Polylogarithm
References
J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
Zeta and L-functions
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https://en.wikipedia.org/wiki/Term%20algebra
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In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra and anarchic algebra.
From a category theory perspective, a term algebra is the initial object for the category of all X-generated algebras of the same signature, and this object, unique up to isomorphism, is called an initial algebra; it generates by homomorphic projection all algebras in the category.
A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming, which is (absolutely freely) defined starting from the set of constants and function symbols in a set of clauses. That is, the Herbrand universe consists of all ground terms: terms that have no variables in them.
An atomic formula or atom is commonly defined as a predicate applied to a tuple of terms; a ground atom is then a predicate in which only ground terms appear. The Herbrand base is the set of all ground atoms that can be formed from predicate symbols in the original set of clauses and terms in its Herbrand universe. These two concepts are named after Jacques Herbrand.
Term algebras also play a role in the semantics of abstract data types, where an abstract data type declaration provides the signature of a multi-sorted algebraic structure and the term algebra is a concrete model of the abstract declaration.
Universal algebra
A type is a set of function symbols, with each having an associated arity (i.e. number of inputs). For any non-negative integer , let denote the function symbols in of arity . A constant is a function symbol of arity 0.
Let be a type, and let be a non-empty set of symbols, representing the variable symbols. (For simplicity, assume and are disjoint.) Then the set of terms of type over is the set of all well-formed strings that can be constructed using the variable symbols of and the constants and operations of . Formally, is the smallest set such that:
— each variable symbol from is a term in , and so is each constant symbol from .
For all and for all function symbols and terms , we have the string — given terms , the application of an -ary function symbol to them represents again a term.
The term algebra of type over is, in summary, the algebra of type that maps each expression to its string representation. Formally, is defined as follows:
The domain of is .
For each nullary function in , is defined as the string .
For all and for each n-ary function in and elements in the domain, is defined as the string .
A term algebra is called absolutely free because for any algebra of type , and for any function , extends to a unique homomorphism , which simply evaluates each term to its corresponding value . Formally, for each :
If , t
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https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s%20constant
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In mathematics, Apéry's constant is the sum of the reciprocals of the positive cubes. That is, it is defined as the number
where is the Riemann zeta function. It has an approximate value of
.
The constant is named after Roger Apéry. It arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient, which appear occasionally in physics, for instance, when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.
Irrational number
was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.
Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for ,
by the Legendre polynomials.
In particular, van der Poorten's article chronicles this approach by noting that
where , are the Legendre polynomials, and the subsequences are integers or almost integers.
It is still not known whether Apéry's constant is transcendental.
Series representations
Classical
In addition to the fundamental series:
Leonhard Euler gave the series representation:
in 1772, which was subsequently rediscovered several times.
Fast convergence
Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of . Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "Known digits").
The following series representation was found by A. A. Markov in 1890, rediscovered by Hjortnaes in 1953, and rediscovered once more and widely advertised by Apéry in 1979:
The following series representation gives (asymptotically) 1.43 new correct decimal places per term:
The following series representation gives (asymptotically) 3.01 new correct decimal places per term:
The following series representation gives (asymptotically) 5.04 new correct decimal places per term:
It has been used to calculate Apéry's constant with several million correct decimal places.
The following series representation gives (asymptotically) 3.92 new correct decimal places per term:
Digit by digit
In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained in nearly linear time and logarithmic space.
Thue-Morse sequence
The following representation was found by Tóth in 2022:
where is the term of the Thue-Morse sequence. In fact, this is a special case of the following formula (valid for all with real part greater than ):
Others
The follo
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https://en.wikipedia.org/wiki/Covariance%20and%20correlation
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In probability theory and statistics, the mathematical concepts of covariance and correlation are very similar. Both describe the degree to which two random variables or sets of random variables tend to deviate from their expected values in similar ways.
If X and Y are two random variables, with means (expected values) μX and μY and standard deviations σX and σY, respectively, then their covariance and correlation are as follows:
covariance
correlation
so that
where E is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables.
If Y always takes on the same values as X, we have the covariance of a variable with itself (i.e. ), which is called the variance and is more commonly denoted as the square of the standard deviation. The correlation of a variable with itself is always 1 (except in the degenerate case where the two variances are zero because X always takes on the same single value, in which case the correlation does not exist since its computation would involve division by 0). More generally, the correlation between two variables is 1 (or –1) if one of them always takes on a value that is given exactly by a linear function of the other with respectively a positive (or negative) slope.
Although the values of the theoretical covariances and correlations are linked in the above way, the probability distributions of sample estimates of these quantities are not linked in any simple way and they generally need to be treated separately.
Multiple random variables
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one. Likewise, the correlations can be placed in a correlation matrix.
Time series analysis
In the case of a time series which is stationary in the wide sense, both the means and variances are constant over time (E(Xn+m) = E(Xn) = μX and var(Xn+m) = var(Xn) and likewise for the variable Y). In this case the cross-covariance and cross-correlation are functions of the time difference:
cross-covariance
cross-correlation
If Y is the same variable as X, the above expressions are called the autocovariance and autocorrelation:
autocovariance
autocorrelation
References
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https://en.wikipedia.org/wiki/Circular%20symmetry
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In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself.
Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the special orthogonal group SO(2), and unitary group U(1). Reflective circular symmetry is isomorphic with the orthogonal group O(2).
Two dimensions
A 2-dimensional object with circular symmetry would consist of concentric circles and annular domains.
Rotational circular symmetry has all cyclic symmetry, Zn as subgroup symmetries. Reflective circular symmetry has all dihedral symmetry, Dihn as subgroup symmetries.
Three dimensions
In 3-dimensions, a surface or solid of revolution has circular symmetry around an axis, also called cylindrical symmetry or axial symmetry. An example is a right circular cone. Circular symmetry in 3 dimensions has all pyramidal symmetry, Cnv as subgroups.
A double-cone, bicone, cylinder, toroid and spheroid have circular symmetry, and in addition have a bilateral symmetry perpendular to the axis of system (or half cylindrical symmetry). These reflective circular symmetries have all discrete prismatic symmetries, Dnh as subgroups.
Four dimensions
In four dimensions, an object can have circular symmetry, on two orthogonal axis planes, or duocylindrical symmetry. For example, the duocylinder and Clifford torus have circular symmetry in two orthogonal axes. A spherinder has spherical symmetry in one 3-space, and circular symmetry in the orthogonal direction.
Spherical symmetry
An analogous 3-dimensional equivalent term is spherical symmetry.
Rotational spherical symmetry is isomorphic with the rotation group SO(3), and can be parametrized by the Davenport chained rotations pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D point groups as subgroups. Reflectional spherical symmetry is isomorphic with the orthogonal group O(3) and has the 3-dimensional discrete point groups as subgroups.
A scalar field has spherical symmetry if it depends on the distance to the origin only, such as the potential of a central force. A vector field has spherical symmetry if it is in radially inward or outward direction with a magnitude and orientation (inward/outward) depending on the distance to the origin only, such as a central force.
See also
Isotropy
Rotational symmetry
Particle in a spherically symmetric potential
Gauss's theorem
References
Symmetry
Rotation
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https://en.wikipedia.org/wiki/Solenoid%20%28mathematics%29
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This page discusses a class of topological groups. For the wrapped loop of wire, see Solenoid.
In mathematics, a solenoid is a compact connected topological space (i.e. a continuum) that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms
where each is a circle and fi is the map that uniformly wraps the circle for times () around the circle . This construction can be carried out geometrically in the three-dimensional Euclidean space R3. A solenoid is a one-dimensional homogeneous indecomposable continuum that has the structure of a compact topological group.
Solenoids were first introduced by Vietoris for the case, and by van Dantzig the case, where is fixed. Such a solenoid arises as a one-dimensional expanding attractor, or Smale–Williams attractor, and forms an important example in the theory of hyperbolic dynamical systems.
Construction
Geometric construction and the Smale–Williams attractor
Each solenoid may be constructed as the intersection of a nested system of embedded solid tori in R3.
Fix a sequence of natural numbers {ni}, ni ≥ 2. Let T0 = S1 × D be a solid torus. For each i ≥ 0, choose a solid torus Ti+1 that is wrapped longitudinally ni times inside the solid torus Ti. Then their intersection
is homeomorphic to the solenoid constructed as the inverse limit of the system of circles with the maps determined by the sequence {ni}.
Here is a variant of this construction isolated by Stephen Smale as an example of an expanding attractor in the theory of smooth dynamical systems. Denote the angular coordinate on the circle S1 by t (it is defined mod 2π) and consider the complex coordinate z on the two-dimensional unit disk D. Let f be the map of the solid torus T = S1 × D into itself given by the explicit formula
This map is a smooth embedding of T into itself that preserves the foliation by meridional disks (the constants 1/2 and 1/4 are somewhat arbitrary, but it is essential that 1/4 < 1/2 and 1/4 + 1/2 < 1). If T is imagined as a rubber tube, the map f stretches it in the longitudinal direction, contracts each meridional disk, and wraps the deformed tube twice inside T with twisting, but without self-intersections. The hyperbolic set Λ of the discrete dynamical system (T, f) is the intersection of the sequence of nested solid tori described above, where Ti is the image of T under the ith iteration of the map f. This set is a one-dimensional (in the sense of topological dimension) attractor, and the dynamics of f on Λ has the following interesting properties:
meridional disks are the stable manifolds, each of which intersects Λ over a Cantor set
periodic points of f are dense in Λ
the map f is topologically transitive on Λ
General theory of solenoids and expanding attractors, not necessarily one-dimensional, was developed by R. F. Williams and involves a projective system of infinitely many copies of a compact branched manifold in place of the circle, to
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https://en.wikipedia.org/wiki/Blum%20integer
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In mathematics, a natural number n is a Blum integer if is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form , for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are
21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ...
The integers were named for computer scientist Manuel Blum.
Properties
Given a Blum integer, Qn the set of all quadratic residues modulo n and coprime to n and . Then:
a has four square roots modulo n, exactly one of which is also in Qn
The unique square root of a in Qn is called the principal square root of a modulo n
The function f : Qn → Qn defined by f(x) = x2 mod n is a permutation. The inverse function of f is: f(x) = .
For every Blum integer n, −1 has a Jacobi symbol mod n of +1, although −1 is not a quadratic residue of n:
History
Before modern factoring algorithms, such as MPQS and NFS, were developed, it was thought to be useful to select Blum integers as RSA moduli. This is no longer regarded as a useful precaution, since MPQS and NFS are able to factor Blum integers with the same ease as RSA moduli constructed from randomly selected primes.
References
Integer sequences
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https://en.wikipedia.org/wiki/Bishop%20Stopford%27s%20School
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Bishop Stopford's School, commonly known as Bishop Stopford's, or (simply) just Bishop's, is a voluntary aided co-educational secondary school specialising in mathematics, computing and engineering, with a sixth form. It is a London Diocesan Church of England school with worship in a relatively High Church Anglo-Catholic tradition. It is in Brick Lane, Enfield, near Enfield Highway, Greater London, England.
Overview
Bishop Stopford's has about 920 pupils aged 11 to 19. In 2004 the school received an award for mathematics and computing and in 2008 engineering specialist status.
Key Stage 3
At Key Stage 3 pupils follow the same subjects for years 7–9. All pupils start to take French in Year 7.
GCSE
In Year 9 pupils can choose what subjects they wish to take for their GCSEs. All pupils take maths, science, English language, English literature, religious education, and physical education.
Sixth form
Entry to the Sixth Form is subject to a satisfactory report from the Year 11 Head of House and an interview with the Head of the Sixth Form or other relevant teacher. In the sixth form, pupils again choose what they wish to study. There are two routes which they may take. Pupils may take a 1-year BTEC course in either OCR business studies or BTEC art and design, or AS/A2 levels. The conditions for taking AS/A2 Levels are:
a minimum of 5 A* to C grades at GCSE level in a suitable combination of subjects, and C grades or better in English Language, Literature, and Maths.
a recommendation from the appropriate head of department.
History
After almost a century of attempts by the Church to found a church secondary school in Enfield, Bishop Stopford's was founded on St. Polycarp's Day 1967 and opened its doors to its first pupils on 7 September 1967. Its founder was the then Bishop of London, the Right Reverend Robert Wright Stopford. The school was founded to provide an Anglican church school for the children of Enfield, who at that time had several Church primary schools but no Church secondary school. The school was established in the buildings of the old Suffolk's Secondary Modern School.
The school hit the headlines in February 1990 when three rottweiler dogs escaped from a nearby property and entered the school premises and attacked and injured several pupils. The incident became known as the 'St. Valentine's Day Massacre' among pupils at the time, and was a contributing factor in the introduction of the Dangerous Dogs Act (1991). Former pupil, comedian Russell Kane described the experience in his 2019 memoir, Son of a Silverback.
The former Heads of Bishop Stopford's have been Dr Geoffrey Roberts B.A. PhD F.R.S.A, JP from 1967 to 1988 d. 2005, Brian Robin Pickard M.A. from 1988 to 2001 d. 2015, Mrs Bridget Sarah Evans from 2001 to 2008, ( Mrs E. Kohler was acting Head from May 2008 - July 2009) and Jim Owen from 2009 to 2012.
Mrs Tammy Day (Current Deputy Head /Senior Mistress) was appointed as Acting Head for a term until Mr Paul Woods ass
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https://en.wikipedia.org/wiki/Cycles%20and%20fixed%20points
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In mathematics, the cycles of a permutation of a finite set S correspond bijectively to the orbits of the subgroup generated by acting on S. These orbits are subsets of S that can be written as , such that
for , and .
The corresponding cycle of is written as ( c1 c2 ... cn ); this expression is not unique since c1 can be chosen to be any element of the orbit.
The size of the orbit is called the length of the corresponding cycle; when , the single element in the orbit is called a fixed point of the permutation.
A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order. For example, let
be a permutation that maps 1 to 2, 6 to 8, etc. Then one may write
= ( 1 2 4 3 ) ( 5 ) ( 6 8 ) (7) = (7) ( 1 2 4 3 ) ( 6 8 ) ( 5 ) = ( 4 3 1 2 ) ( 8 6 ) ( 5 ) (7) = ...
Here 5 and 7 are fixed points of , since (5) = 5 and (7)=7. It is typical, but not necessary, to not write the cycles of length one in such an expression. Thus, = (1 2 4 3)(6 8), would be an appropriate way to express this permutation.
There are different ways to write a permutation as a list of its cycles, but the number of cycles and their contents are given by the partition of S into orbits, and these are therefore the same for all such expressions.
Counting permutations by number of cycles
The unsigned Stirling number of the first kind, s(k, j) counts the number of permutations of k elements with exactly j disjoint cycles.
Properties
(1) For every k > 0 :
(2) For every k > 0 :
(3) For every k > j > 1,
Reasons for properties
(1) There is only one way to construct a permutation of k elements with k cycles: Every cycle must have length 1 so every element must be a fixed point.
(2.a) Every cycle of length k may be written as permutation of the number 1 to k; there are k! of these permutations.
(2.b) There are k different ways to write a given cycle of length k, e.g. ( 1 2 4 3 ) = ( 2 4 3 1 ) = ( 4 3 1 2 ) = ( 3 1 2 4 ).
(2.c) Finally:
(3) There are two different ways to construct a permutation of k elements with j cycles:
(3.a) If we want element k to be a fixed point we may choose one of the permutations with elements and cycles and add element k as a new cycle of length 1.
(3.b) If we want element k not to be a fixed point we may choose one of the permutations with elements and j cycles and insert element k in an existing cycle in front of one of the elements.
Some values
Counting permutations by number of fixed points
The value counts the number of permutations of k elements with exactly j fixed points. For the main article on this topic, see rencontres numbers.
Properties
(1) For every j < 0 or j > k :
(2) f(0, 0) = 1.
(3) For every k > 1 and k ≥ j ≥ 0,
Reasons for properties
(3) There are three different methods to construct a permutation of k elements with j fixed points:
(3.a) We may choose one of the permutations wit
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https://en.wikipedia.org/wiki/Perfect%20set%20property
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In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set.
As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum.
The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set. In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable open set.
The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets. However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every analytic set has the perfect set property. It follows from the existence of sufficiently large cardinals that every projective set has the perfect set property.
References
Descriptive set theory
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https://en.wikipedia.org/wiki/Somos%27%20quadratic%20recurrence%20constant
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In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number
This can be easily re-written into the far more quickly converging product representation
which can then be compactly represented in infinite product form by:
The constant σ arises when studying the asymptotic behaviour of the sequence
with first few terms 1, 1, 2, 12, 576, 1658880, ... . This sequence can be shown to have asymptotic behaviour as follows:
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent:
where ln is the natural logarithm and (z, s, q) is the Lerch transcendent.
Finally,
.
Notes
References
Steven R. Finch, Mathematical Constants (2003), Cambridge University Press, p. 446. .
Jesus Guillera and Jonathan Sondow, "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", Ramanujan Journal 16 (2008), 247–270 (Provides an integral and a series representation).
Mathematical constants
Infinite products
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https://en.wikipedia.org/wiki/Bartlett%27s%20test
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In statistics, Bartlett's test, named after Maurice Stevenson Bartlett, is used to test homoscedasticity, that is, if multiple samples are from populations with equal variances. Some statistical tests, such as the analysis of variance, assume that variances are equal across groups or samples, which can be verified with Bartlett's test.
In a Bartlett test, we construct the null and alternative hypothesis. For this purpose several test procedures have been devised. The test procedure due to M.S.E (Mean Square Error/Estimator) Bartlett test is represented here. This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (k − 1) degrees of freedom, where k is the number of random samples, which may vary in size and are each drawn from independent normal distributions.
Bartlett's test is sensitive to departures from normality. That is, if the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. Levene's test and the Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.
Specification
Bartlett's test is used to test the null hypothesis, H0 that all k population variances are equal against the alternative that at least two are different.
If there are k samples with sizes and sample variances then Bartlett's test statistic is
where and is the pooled estimate for the variance.
The test statistic has approximately a distribution. Thus, the null hypothesis is rejected if (where is the upper tail critical value for the distribution).
Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the distribution better (Bartlett, 1937).
Notes
The test statistics may be written in some sources with logarithms of base 10 as:
See also
Box's M test
Levene's test
Kaiser–Meyer–Olkin test
References
External links
NIST page on Bartlett's test
Analysis of variance
Statistical tests
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https://en.wikipedia.org/wiki/William%20Gemmell%20Cochran
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William Gemmell Cochran (15 July 1909 – 29 March 1980) was a prominent statistician. He was born in Scotland but spent most of his life in the United States.
Cochran studied mathematics at the University of Glasgow and the University of Cambridge. He worked at Rothamsted Experimental Station from 1934 to 1939, when he moved to the United States. There he helped establish several departments of statistics. His longest spell in any one university was at Harvard, which he joined in 1957 and from which he retired in 1976.
Writings
Cochran wrote many articles and books. His books became standard texts:
Experimental Designs (with Gertrude Mary Cox) 1950
Statistical Methods Applied to Experiments in Agriculture and Biology by George W. Snedecor (Cochran contributed from the fifth (1956) edition)
Planning and Analysis of Observational Studies (edited by Lincoln E. Moses and Frederick Mosteller) 1983.
References
External links
Brief biography
ASA biography
Morris Hansen and Frederick Mosteller (1987) William Gemmell Cochran NAS Biographical Memoirs V.56
Morris Hansen and Frederick Mosteller, "William Gemmell Cochran", Biographical Memoirs of the National Academy of Sciences (1987)
"Designing Clinical Trials" (1961; Evaluation of Drug Therapy)
1909 births
1980 deaths
American statisticians
British statisticians
Harvard University faculty
20th-century Scottish mathematicians
Presidents of the American Statistical Association
Presidents of the Institute of Mathematical Statistics
Presidents of the International Statistical Institute
Rothamsted statisticians
Fellows of the American Statistical Association
Survey methodologists
People from Rutherglen
Members of the United States National Academy of Sciences
Alumni of the University of Glasgow
Alumni of the University of Cambridge
Mathematical statisticians
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https://en.wikipedia.org/wiki/Australian%20Science%20and%20Mathematics%20School
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The Australian Science and Mathematics School (ASMS) is a coeducational public senior high school for Years 10–12 located on the Sturt campus of Flinders University in Bedford Park, a southern suburb of Adelaide, the capital of South Australia. As the school is unzoned, it attracts students from all across the Adelaide metropolitan area as well as some regional and interstate locations, in addition to international students. The goal of the school is to prepare its students for university, particularly in the fields of mathematics and science. The ASMS is unconventional in its approach to education, emphasising a love of learning in both students and teaching staff; students are given the freedom to take control of their own education. ASMS aims to make students aware of their own learning and for them to become self-directed in the way they complete academic tasks.
Overview
The Australian Science and Mathematics School was opened in 2003 and has a total of around 380 students. As the school is designed to provide an adult environment for senior school students, there is no school uniform policy, which promotes a variety of culture and social styles and structures. A key feature of the ASMS is the productive relationship between the school and the Flinders University, on which the campus is located; the ASMS shares many resources with the university, including the library, cafeteria, student services, transport, recreational areas and car parks, in addition to booked access to lecture theatres and specialist science and support facilities. Furthermore, students at the ASMS in collaboration with Flinders University's Science and Technology Enterprise Partnership (STEP) may be involved in research projects in the business, industry and university sectors.
Curriculum
The ASMS is a specialist science and mathematics school, however it offers a comprehensive curriculum which covers all learning areas necessary for students to achieve their South Australian Certificate of Education (SACE) qualification.
The Year 10/11 curriculum is organised into interdisciplinary Central Studies; these alternate every year to ensure that a student will not do the same subjects twice. The Year 12 curriculum consists of standard SACE Stage 2 subjects, such as the various Mathematics, Science, English and Humanities subjects.
In 2022, the 10/11 students took part in the first "Field Trip Week", which involved activities both outside and inside the school premises. The goal of this week was to extend the students' knowledge in their current Central Studies.
Adventure Space/Passion Project
The ASMS also provides special activities for Year 10 and 11 students in the form of Adventure Space. Examples of these include Dance, Cryptography, Robotics, Aviation, Australian Space Design Competition, Paramedical Pathways, Electronics, Creative Writing, and Palaeontology. While not assessed, they do provide an opportunity to interact with university life, as well as an opportu
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https://en.wikipedia.org/wiki/Augmentation%20ideal
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In algebra, an augmentation ideal is an ideal that can be defined in any group ring.
If G is a group and R a commutative ring, there is a ring homomorphism , called the augmentation map, from the group ring to , defined by taking a (finite) sum to (Here and .) In less formal terms, for any element , for any elements and , and is then extended to a homomorphism of R-modules in the obvious way.
The augmentation ideal is the kernel of and is therefore a two-sided ideal in R[G].
is generated by the differences of group elements. Equivalently, it is also generated by , which is a basis as a free R-module.
For R and G as above, the group ring R[G] is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra.
The augmentation ideal plays a basic role in group cohomology, amongst other applications.
Examples of quotients by the augmentation ideal
Let G a group and the group ring over the integers. Let I denote the augmentation ideal of . Then the quotient is isomorphic to the abelianization of G, defined as the quotient of G by its commutator subgroup.
A complex representation V of a group G is a - module. The coinvariants of V can then be described as the quotient of V by IV, where I is the augmentation ideal in .
Another class of examples of augmentation ideal can be the kernel of the counit of any Hopf algebra.
Notes
References
Dummit and Foote, Abstract Algebra
Ideals (ring theory)
Hopf algebras
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https://en.wikipedia.org/wiki/Join%20Java
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Join Java is a programming language based on the join-pattern that extends the standard Java programming language with the join semantics of the join-calculus. It was written at the University of South Australia within the Reconfigurable Computing Lab by Dr. Von Itzstein.
Language characteristics
The Join Java extension introduces three new language constructs:
Join methods
Asynchronous methods
Order class modifiers for determining the order that patterns are matched
Concurrency in most popular programming languages is implemented using constructs such as semaphores and monitors. Libraries are emerging (such as the Java concurrency library JSR-166) that provide higher-level concurrency semantics. communicating sequential processes (CSP), Calculus of Communicating Systems (CCS) and Pi have higher-level synchronization behaviours defined implicitly through the composition of events at the interfaces of concurrent processes. Join calculus, in contrast, has explicit synchronization based on a localized conjunction of events defined as reduction rules. Join semantics try to provide explicit expressions of synchronization without breaching the object-oriented idea of modularization, including dynamic creation and destruction of processes and channels.
The Join Java language can express virtually all published concurrency patterns without explicit recourse to low-level monitor calls. In general, Join Java programs are more concise than their Java equivalents. The overhead introduced in Join Java by the higher-level expressions derived from the Join calculus is manageable. The synchronization expressions associated with monitors (wait and notify) which are normally located in the body of methods can be replaced by Join Java expressions (the Join methods) which form part of the method signature.
Join methods
A Join method is defined by two or more Join fragments. A Join method will
execute once all the fragments of the Join pattern have been called.
If the return type is a standard Java type then the leading fragment will
block the caller until the Join pattern is complete and the method has
executed. If the return type is of type signal then the
leading fragment will return immediately. All trailing fragments are
asynchronous so will not block the caller.
Example:
class JoinExample {
int fragment1() & fragment2(int x) {
//will return value of x
//to caller of fragment1
return x;
}
}
Ordering modifiers
Join fragments can be repeated in multiple Join patterns so there can be
a case when multiple Join patterns are completed when a fragment is called.
Such a case could occur in the example below if B(), C() and D() then A() are
called. The final A() fragment completes three of the patterns so there are
three possible methods that may be called. The ordered class
modifier is used here to determine which Join method will be called.
The default and when using the unordered class modifier is
to pick one of the m
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https://en.wikipedia.org/wiki/J%C3%BAlio%20C%C3%A9sar%20de%20Mello%20e%20Souza
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Júlio César de Mello e Souza (Rio de Janeiro, May 6, 1895 – Recife, June 18, 1974), was a Brazilian writer and mathematics teacher. He was well known in Brazil and abroad for his books on recreational mathematics, most of them published under the pen names of Malba Tahan and Breno de Alencar Bianco.
He wrote 69 novels and 51 books of mathematics and other subjects, with over than two million books sold by 1995. His most famous work, The Man Who Counted, saw its 54th printing in 2001.
Júlio César's most popular books, including The Man Who Counted, are collections of mathematical problems, puzzles, curiosities, and embedded in tales inspired by the Arabian Nights. He thoroughly researched his subject matters — not only the mathematics, but also the history, geography, and culture of the Islamic Empire which was the backdrop and connecting thread of his books. Yet Júlio César's travels outside Brazil were limited to short visits to Buenos Aires, Montevideo, and Lisbon: he never set foot in the deserts and cities which he so vividly described in his books.
Júlio César was very critical of the educational methods used in Brazilian classrooms, especially for mathematics. "The mathematics teacher is a sadist," he claimed, "who loves to make everything as complicated as possible." In education, he was decades ahead of his time, and his proposals are still more praised than implemented today.
For his books, Júlio César received a prize by the prestigious Brazilian Literary Academy and was made a member of the Pernambuco Literary Academy. The Malba Tahan Institute was founded in 2004 at Queluz to preserve his legacy. The State Legislature of Rio de Janeiro determined his birthday, May 6, to be commemorated as the Mathematician's Day.
Early life
Júlio César was born in Rio de Janeiro but spent most of his childhood in Queluz, a small rural town in the State of São Paulo. His father, João de Deus de Mello e Souza, was a civil servant with limited salary and eight (some reports say nine) children to support.
In 1905 he was sent with his older brother, João Batista, to Rio de Janeiro to attend preparatory classes for admission to the prestigious Colégio Militar do Rio de Janeiro, where he studied from 1906 to 1909, and later at Colégio Pedro II.
As a student, Júlio César was not academically successful. In a 1905 letter to their parents, João Batista tells that little Júlio "is bad at writing, and a failure in mathematics". His grade reports at Colégio Pedro II show that he once failed an Algebra exam, and barely passed one on Arithmetic. He later attributed these results to the teaching practices of the time, based on "the detestable method of salivation".
However, he did show signs of his originality and non-conventional approaches in other ways. As a child in Queluz, he used to keep frogs as pets, and at one point he had some 50 animals in his yard. One of them, nicknamed "Monsignor", would follow him through the town. As an adult, he kept up
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https://en.wikipedia.org/wiki/Stirling%20numbers%20of%20the%20second%20kind
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In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or . Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions. They are named after James Stirling.
The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the second kind. Identities linking the two kinds appear in the article on Stirling numbers.
Definition
The Stirling numbers of the second kind, written or or with other notations, count the number of ways to partition a set of labelled objects into nonempty unlabelled subsets. Equivalently, they count the number of different equivalence relations with precisely equivalence classes that can be defined on an element set. In fact, there is a bijection between the set of partitions and the set of equivalence relations on a given set. Obviously,
for n ≥ 0, and for n ≥ 1,
as the only way to partition an n-element set into n parts is to put each element of the set into its own part, and the only way to partition a nonempty set into one part is to put all of the elements in the same part. Unlike Stirling numbers of the first kind, they can be calculated using a one-sum formula:
The Stirling numbers of the second kind may also be characterized as the numbers that arise when one expresses powers of an indeterminate x in terms of the falling factorials
(In particular, (x)0 = 1 because it is an empty product.)
In other words
Notation
Various notations have been used for Stirling numbers of the second kind. The brace notation was used by Imanuel Marx and Antonio Salmeri in 1962 for variants of these numbers.<ref>Antonio Salmeri, Introduzione alla teoria dei coefficienti fattoriali, Giornale di Matematiche di Battaglini 90 (1962), pp. 44–54.</ref> This led Knuth to use it, as shown here, in the first volume of The Art of Computer Programming (1968).Donald E. Knuth, Fundamental Algorithms, Reading, Mass.: Addison–Wesley, 1968. According to the third edition of The Art of Computer Programming, this notation was also used earlier by Jovan Karamata in 1935.Jovan Karamata, Théorèmes sur la sommabilité exponentielle et d'autres sommabilités s'y rattachant, Mathematica (Cluj) 9 (1935), pp, 164–178. The notation S(n, k) was used by Richard Stanley in his book Enumerative Combinatorics and also, much earlier, by many other writers.
The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources.
Relation to Bell numbers
Since the Stirling number counts set partitions of an n-element set into k parts, the sum
over all values of k is the total number of partitions of a set with n members. This number is known as the nth Bell nu
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https://en.wikipedia.org/wiki/Stirling%20numbers%20of%20the%20first%20kind
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In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one).
The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers.
Definitions
Stirling numbers of the first kind are the coefficients in the expansion of the falling factorial
into powers of the variable :
For example, , leading to the values , , and .
Subsequently, it was discovered that the absolute values of these numbers are equal to the number of permutations of certain kinds. These absolute values, which are known as unsigned Stirling numbers of the first kind, are often denoted or . They may be defined directly to be the number of permutations of elements with disjoint cycles. For example, of the permutations of three elements, there is one permutation with three cycles (the identity permutation, given in one-line notation by or in cycle notation by ), three permutations with two cycles (, , and ) and two permutations with one cycle ( and ). Thus, , and . These can be seen to agree with the previous calculation of for .
It was observed by Alfréd Rényi that the unsigned Stirling number also count the number
of permutations of size with left-to-right maxima.
The unsigned Stirling numbers may also be defined algebraically, as the coefficients of the rising factorial:
.
The notations used on this page for Stirling numbers are not universal, and may conflict with notations in other sources. (The square bracket notation is also common notation for the Gaussian coefficients.)
Definition by permutation
can be defined as the number of permutations on elements with cycles.
The image at right shows that : the symmetric group on 4 objects has 3 permutations of the form
(having 2 orbits, each of size 2),
and 8 permutations of the form
(having 1 orbit of size 3 and 1 orbit of size 1).
These numbers can be calculated by considering the orbits as conjugancy classes (last bullet point).
Signs
The signs of the (signed) Stirling numbers of the first kind are predictable and depend on the parity of . In particular,
Recurrence relation
The unsigned Stirling numbers of the first kind can be calculated by the recurrence relation
for , with the initial conditions
for .
It follows immediately that the (signed) Stirling numbers of the first kind satisfy the recurrence
.
Table of values
Below is a triangular array of unsigned values for the Stirling numbers of the first kind, similar in form to Pascal's triangle. These values are easy to generate using the recurrence relation in the previous section.
Properties
Simple identities
U
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https://en.wikipedia.org/wiki/Fr%C3%B6licher%20space
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In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.
Definition
A Frölicher space consists of a non-empty set X together with a subset C of Hom(R, X) called the set of smooth curves, and a subset F of Hom(X, R) called the set of smooth real functions, such that for each real function
f : X → R
in F and each curve
c : R → X
in C, the following axioms are satisfied:
f in F if and only if for each γ in C, f . γ in C∞(R, R)
c in C if and only if for each φ in F, φ . c in C∞(R, R)
Let A and B be two Frölicher spaces. A map
m : A → B
is called smooth if for each smooth curve c in CA, m.c is in CB. Furthermore, the space of all such smooth maps has itself the structure of a Frölicher space. The smooth functions on
C∞(A, B)
are the images of
References
, section 23
Smooth functions
Structures on manifolds
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https://en.wikipedia.org/wiki/List%20of%20computer%20algebra%20systems
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The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language. A CAS may include a user interface and graphics capability; and to be effective may require a large library of algorithms, efficient data structures and a fast kernel.
General
These computer algebra systems are sometimes combined with "front end" programs that provide a better user interface, such as the general-purpose GNU TeXmacs.
Functionality
Below is a summary of significantly developed symbolic functionality in each of the systems.
via SymPy
<li> via qepcad optional package
Those which do not "edit equations" may have a GUI, plotting, ASCII graphic formulae and math font printing. The ability to generate plaintext files is also a sought-after feature because it allows a work to be understood by people who do not have a computer algebra system installed.
Operating system support
The software can run under their respective operating systems natively without emulation. Some systems must be compiled first using an appropriate compiler for the source language and target platform. For some platforms, only older releases of the software may be available.
Graphing calculators
Some graphing calculators have CAS features.
See also
:Category:Computer algebra systems
Comparison of numerical-analysis software
Comparison of statistical packages
List of information graphics software
List of numerical-analysis software
List of numerical libraries
List of statistical software
Mathematical software
Web-based simulation
References
External links
Comparisons of mathematical software
Mathematics-related lists
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https://en.wikipedia.org/wiki/Universally%20measurable%20set
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In mathematics, a subset of a Polish space is universally measurable if it is measurable with respect to every complete probability measure on that measures all Borel subsets of . In particular, a universally measurable set of reals is necessarily Lebesgue measurable (see below).
Every analytic set is universally measurable. It follows from projective determinacy, which in turn follows from sufficient large cardinals, that every projective set is universally measurable.
Finiteness condition
The condition that the measure be a probability measure; that is, that the measure of itself be 1, is less restrictive than it may appear. For example, Lebesgue measure on the reals is not a probability measure, yet every universally measurable set is Lebesgue measurable. To see this, divide the real line into countably many intervals of length 1; say, N0=[0,1), N1=[1,2), N2=[-1,0), N3=[2,3), N4=[-2,-1), and so on. Now letting μ be Lebesgue measure, define a new measure ν by
Then easily ν is a probability measure on the reals, and a set is ν-measurable if and only if it is Lebesgue measurable. More generally a universally measurable set must be measurable with respect to every sigma-finite measure that measures all Borel sets.
Example contrasting with Lebesgue measurability
Suppose is a subset of Cantor space ; that is, is a set of infinite sequences of zeroes and ones. By putting a binary point before such a sequence, the sequence can be viewed as a real number between 0 and 1 (inclusive), with some unimportant ambiguity. Thus we can think of as a subset of the interval [0,1], and evaluate its Lebesgue measure, if that is defined. That value is sometimes called the coin-flipping measure of , because it is the probability of producing a sequence of heads and tails that is an element of upon flipping a fair coin infinitely many times.
Now it follows from the axiom of choice that there are some such without a well-defined Lebesgue measure (or coin-flipping measure). That is, for such an , the probability that the sequence of flips of a fair coin will wind up in is not well-defined. This is a pathological property of that says that is "very complicated" or "ill-behaved".
From such a set , form a new set by performing the following operation on each sequence in : Intersperse a 0 at every even position in the sequence, moving the other bits to make room. Although is not intuitively any "simpler" or "better-behaved" than , the probability that the sequence of flips of a fair coin will be in is well-defined. Indeed, to be in , the coin must come up tails on every even-numbered flip, which happens with probability zero.
However is not universally measurable. To see that, we can test it against a biased coin that always comes up tails on even-numbered flips, and is fair on odd-numbered flips. For a set of sequences to be universally measurable, an arbitrarily biased coin may be used (even one that can "remember" the sequence of flip
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https://en.wikipedia.org/wiki/Distribution%20ensemble
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In cryptography, a distribution ensemble or probability ensemble is a family of distributions or random variables where is a (countable) index set, and each is a random variable, or probability distribution. Often and it is required that each have a certain property for n sufficiently large.
For example, a uniform ensemble is a distribution ensemble where each is uniformly distributed over strings of length n. In fact, many applications of probability ensembles implicitly assume that the probability spaces for the random variables all coincide in this way, so every probability ensemble is also a stochastic process.
See also
Provable security
Statistically close
Pseudorandom ensemble
Computational indistinguishability
References
Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. . Fragments available at the author's web site.
Theory of cryptography
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https://en.wikipedia.org/wiki/Basic%20hypergeometric%20series
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In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series.
A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of successive terms is a rational function of qn, then the series is called a basic hypergeometric series. The number q is called the base.
The basic hypergeometric series was first considered by . It becomes the hypergeometric series in the limit when base .
Definition
There are two forms of basic hypergeometric series, the unilateral basic hypergeometric series φ, and the more general bilateral basic hypergeometric series ψ.
The unilateral basic hypergeometric series is defined as
where
and
is the q-shifted factorial.
The most important special case is when j = k + 1, when it becomes
This series is called balanced if a1 ... ak + 1 = b1 ...bkq.
This series is called well poised if a1q = a2b1 = ... = ak + 1bk, and very well poised if in addition a2 = −a3 = qa11/2.
The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since
holds ().
The bilateral basic hypergeometric series, corresponding to the bilateral hypergeometric series, is defined as
The most important special case is when j = k, when it becomes
The unilateral series can be obtained as a special case of the bilateral one by setting one of the b variables equal to q, at least when none of the a variables is a power of q, as all the terms with n < 0 then vanish.
Simple series
Some simple series expressions include
and
and
The q-binomial theorem
The q-binomial theorem (first published in 1811 by Heinrich August Rothe) states that
which follows by repeatedly applying the identity
The special case of a = 0 is closely related to the q-exponential.
Cauchy binomial theorem
Cauchy binomial theorem is a special case of the q-binomial theorem.
Ramanujan's identity
Srinivasa Ramanujan gave the identity
valid for |q| < 1 and |b/a| < |z| < 1. Similar identities for have been given by Bailey. Such identities can be understood to be generalizations of the Jacobi triple product theorem, which can be written using q-series as
Ken Ono gives a related formal power series
Watson's contour integral
As an analogue of the Barnes integral for the hypergeometric series, Watson showed that
where the poles of lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for r+1φr. This contour integral gives an analytic continuation of the basic hypergeometric function in z.
Matrix version
The basic hypergeometric matrix function can be defined as follows:
The ratio test shows that this matrix function is absolutely convergent.
See also
Dixon's identity
Rogers–Ramanujan identities
Notes
References
W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathem
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https://en.wikipedia.org/wiki/Richard%20Rusczyk
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Richard Rusczyk (; ; born September 21, 1971) is the founder and chief executive officer of Art of Problem Solving Inc. (as well as the website, which serves as a mathematics forum and place to hold online classes) and a co-author of the Art of Problem Solving textbooks. Rusczyk was a national Mathcounts participant in 1985, and he won the USA Math Olympiad (USAMO) in 1989. He is one of the co-creators of the Mandelbrot Competition, and the director of the USA Mathematical Talent Search (USAMTS). He also founded the San Diego Math Circle.
Early life
Richard Rusczyk was born in Idaho Falls, Idaho in 1971. He signed up for the MathCounts program when he was in middle school. As a high schooler, Rusczyk was a part of his high school math team and took part in the American Mathematics Competitions. Rusczyk would later go on to attend Princeton University, which he graduated from in 1993.
Art of Problem Solving
In 1994, Rusczyk and Sandor Lehoczky wrote the Art of Problem Solving books, designed to prepare students for mathematical competitions by teaching them concepts and problem-solving methods rarely taught in school. These books lent their name to the company he founded in 2003.
After working for four years as a bond trader for D. E. Shaw & Co., Rusczyk created the Art of Problem Solving website, which provides resources for middle and high school students to develop their mathematics and problem-solving abilities. These include real-time competitions to solve math problems and online tools to learn how to solve problems with increasing difficulty as well as math forums. As of May 26, 2021, there have been 709,491 students, 1,322,594 topics, and a total of 15,182,054 posts on the site. Rusczyk has also published the Art of Problem Solving series of books aimed at a similar audience. The site also provides fee-based online mathematics classes, which range from Prealgebra to Group Theory and Calculus. Additionally, Art of Problem Solving offers Python programming classes and Olympiad level classes in Mathematics, Physics, Chemistry and Computer Science. They are collectively named WOOT.
Rusczyk founded and serves on the board of the nonprofit Art of Problem Solving Initiative, which manages the United States of America Mathematical Talent Search (USAMTS) and finances numerous local math initiatives around the United States. In 2012, Rusczyk won the Mathcounts distinguished alumnus award. In 2014, Rusczyk won the Paul Erdős Award from the World Federation of National Mathematics Competitions. Art of Problem Solving also has a vast community of over 500,000 math, computer science, and physics enthusiasts.
In 2020, Mathcounts was canceled due to the COVID-19 pandemic. In its stead, Art of Problem Solving hosted the online Mathcounts Week. Art of Problem Solving also hosted the second round of the American Invitational Mathematics Examination as the American Online Invitational Mathematics Examination. Art of Problem Solving also hosted the Am
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https://en.wikipedia.org/wiki/Infinity-Borel%20set
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In set theory, a subset of a Polish space is ∞-Borel if it
can be obtained by starting with the open subsets of , and transfinitely iterating the operations of complementation and wellordered union. This concept is usually considered without the assumption of the axiom of choice, which means that the ∞-Borel sets may fail to be closed under wellordered union; see below.
Formal definition
We define the set of ∞-Borel codes and the interpretation function below. A ∞-Borel set is a subset of which is in the image of the interpretation function .
The set of ∞-Borel codes is an inductive type generated by functions , and for each ; the interpretation function is defined inductively as , and . Here denotes the Hartogs number of : a sufficiently large ordinal such that there is no injection from to . Restricting to unions of length below doesn't affect the possible unions (as any union of length can be replaced by one of length by removing duplicates), but ensures that the ∞-Borel codes form a set, not a proper class.
This can be phrased more set-theoretically as a definition by transfinite recursion as follows:
For every open subset , the ordered pair is an ∞-Borel code; its interpretation is .
If is an ∞-Borel code, then the ordered pair is also an ∞-Borel code; its interpretation is the complement of , that is, .
If is a length-α sequence of ∞-Borel codes for some ordinal α < Ξ (that is, if for every β<α, is an ∞-Borel code), then the ordered pair is an ∞-Borel code; its interpretation is .
The axiom of choice implies that every set can be wellordered, and therefore that every subset of every Polish space is ∞-Borel. Therefore, the notion is interesting only in contexts where AC does not hold (or is not known to hold). Unfortunately, without the axiom of choice, it is not clear that the ∞-Borel sets are closed under wellordered union. This is because, given a wellordered union of ∞-Borel sets, each of the individual sets may have many ∞-Borel codes, and there may be no way to choose one code for each of the sets, with which to form the code for the union.
The assumption that every set of reals is ∞-Borel is part of AD+, an extension of the axiom of determinacy studied by Woodin.
Incorrect definition
It is very tempting to read the informal description at the top of this article as claiming that the ∞-Borel sets are the smallest class of subsets of containing all the open sets and closed under complementation and wellordered union. That is, one might wish to dispense with the ∞-Borel codes altogether and try a definition like this:
For each ordinal α define by transfinite recursion Bα as follows:
B0 is the collection of all open subsets of .
For a given even ordinal α, Bα+1 is the union of Bα with the set of all complements of sets in Bα.
For a given even ordinal α, Bα+2 is the set of all wellordered unions of sets in Bα+1.
For a given limit ordinal λ, Bλ is the union of all Bα for α<λ
Bβ equals BΞ for every β
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https://en.wikipedia.org/wiki/Octahedral%20molecular%20geometry
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In chemistry, octahedral molecular geometry, also called square bipyramidal, describes the shape of compounds with six atoms or groups of atoms or ligands symmetrically arranged around a central atom, defining the vertices of an octahedron. The octahedron has eight faces, hence the prefix octa. The octahedron is one of the Platonic solids, although octahedral molecules typically have an atom in their centre and no bonds between the ligand atoms. A perfect octahedron belongs to the point group Oh. Examples of octahedral compounds are sulfur hexafluoride SF6 and molybdenum hexacarbonyl Mo(CO)6. The term "octahedral" is used somewhat loosely by chemists, focusing on the geometry of the bonds to the central atom and not considering differences among the ligands themselves. For example, , which is not octahedral in the mathematical sense due to the orientation of the bonds, is referred to as octahedral.
The concept of octahedral coordination geometry was developed by Alfred Werner to explain the stoichiometries and isomerism in coordination compounds. His insight allowed chemists to rationalize the number of isomers of coordination compounds. Octahedral transition-metal complexes containing amines and simple anions are often referred to as Werner-type complexes.
Isomerism in octahedral complexes
When two or more types of ligands (La, Lb, ...) are coordinated to an octahedral metal centre (M), the complex can exist as isomers. The naming system for these isomers depends upon the number and arrangement of different ligands.
cis and trans
For MLL, two isomers exist. These isomers of MLL are cis, if the Lb ligands are mutually adjacent, and trans, if the Lb groups are situated 180° to each other. It was the analysis of such complexes that led Alfred Werner to the 1913 Nobel Prize–winning postulation of octahedral complexes.
Facial and meridional isomers
For MLL, two isomers are possible - a facial isomer (fac) in which each set of three identical ligands occupies one face of the octahedron surrounding the metal atom, so that any two of these three ligands are mutually cis, and a meridional isomer (mer) in which each set of three identical ligands occupies a plane passing through the metal atom.
Δ vs Λ isomers
Complexes with three bidentate ligands or two cis bidentate ligands can exist as enantiomeric pairs. Examples are shown below.
Other
For MLLL, a total of five geometric isomers and six stereoisomers are possible.
One isomer in which all three pairs of identical ligands are trans
Three isomers in which one pair of identical ligands (La or Lb or Lc) is trans while the other two pairs of ligands are mutually cis.
Two enantiomeric pair in which all three pairs of identical ligands are cis. These are equivalent to the Δ vs Λ isomers mentioned above.
The number of possible isomers can reach 30 for an octahedral complex with six different ligands (in contrast, only two stereoisomers are possible for a tetrahedral complex with four differe
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https://en.wikipedia.org/wiki/List%20of%20properties%20of%20sets%20of%20reals
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This article lists some properties of sets of real numbers. The general study of these concepts forms descriptive set theory, which has a rather different emphasis from general topology.
Definability properties
Borel set
Analytic set
C-measurable set
Projective set
Inductive set
Infinity-Borel set
Suslin set
Homogeneously Suslin set
Weakly homogeneously Suslin set
Set of uniqueness
Regularity properties
Property of Baire
Lebesgue measurable
Universally measurable set
Perfect set property
Universally Baire set
Largeness and smallness properties
Meager set
Comeager set - A comeager set is one whose complement is meager.
Null set
Conull set
Dense set
Nowhere dense set
Real numbers
Real numbers
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https://en.wikipedia.org/wiki/Exponential%20object
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In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law.
Definition
Let be a category, let and be objects of , and let have all binary products with . An object together with a morphism is an exponential object if for any object and morphism there is a unique morphism (called the transpose of ) such that the following diagram commutes:
This assignment of a unique to each establishes an isomorphism (bijection) of hom-sets,
If exists for all objects in , then the functor defined on objects by and on arrows by , is a right adjoint to the product functor . For this reason, the morphisms and are sometimes called exponential adjoints of one another.
Equational definition
Alternatively, the exponential object may be defined through equations:
Existence of is guaranteed by existence of the operation .
Commutativity of the diagrams above is guaranteed by the equality .
Uniqueness of is guaranteed by the equality .
Universal property
The exponential is given by a universal morphism from the product functor to the object . This universal morphism consists of an object and a morphism .
Examples
In the category of sets, an exponential object is the set of all functions . The map is just the evaluation map, which sends the pair to . For any map the map is the curried form of :
A Heyting algebra is just a bounded lattice that has all exponential objects. Heyting implication, , is an alternative notation for . The above adjunction results translate to implication () being right adjoint to meet (). This adjunction can be written as , or more fully as:
In the category of topological spaces, the exponential object exists provided that is a locally compact Hausdorff space. In that case, the space is the set of all continuous functions from to together with the compact-open topology. The evaluation map is the same as in the category of sets; it is continuous with the above topology. If is not locally compact Hausdorff, the exponential object may not exist (the space still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to be cartesian closed.
However, the category of locally compact topological spaces is not cartesian closed either, since need not be locally compact for locally compact spaces and . A cartesian closed category of spaces is, for example, given by the full subcategory spanned by the compactly generated Hausdorff spaces.
In functional programming languages, the morphism is often called , and the syntax is often written . The morphism here must not to be confused with the eval function in
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https://en.wikipedia.org/wiki/Association%20scheme
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The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and the theory of error-correcting codes. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
Definition
An n-class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R0, R1, ..., Rn which satisfy:
; it is called the identity relation.
Defining , if R in S, then R* in S.
If , the number of such that and is a constant depending on , , but not on the particular choice of and .
An association scheme is commutative if for all , and . Most authors assume this property.
A symmetric association scheme is one in which each is a symmetric relation. That is:
if (x, y) ∈ Ri, then (y, x) ∈ Ri. (Or equivalently, R* = R.)
Every symmetric association scheme is commutative.
Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.
Two points x and y are called i th associates if . The definition states that if x and y are i th associates then so are y and x. Every pair of points are i th associates for exactly one . Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points which are both i th associates of and j th associates of is a constant .
Graph interpretation and adjacency matrices
A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has vertices, one for each point of , and the edge joining vertices and is labeled if and are th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled having the other edges labeled and is a constant , depending on but not on the choice of the base. In particular, each vertex is incident with exactly edges labeled ; is the valency of the relation . There are also loops labeled at each vertex , corresponding to .
The relations are described by their adjacency matrices. is the adjacency matrix of for and is a v × v matrix with rows and columns labeled by the points of .
The definition of a symmetric association scheme is equivalent to saying that the are v × v (0,1)-matrices which satisfy
I. is symmetric,
II. (the all-ones matrix),
III. ,
IV. .
The (x, y)-th entry of the left side of (IV) is the number of paths of length two between x and y with labels i and j in the graph. Note that the rows and columns of contain 's:
Terminology
The numbers are called the parameters of the scheme. They
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https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Italy
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In the NUTS (Nomenclature of Territorial Units for Statistics) codes of Italy (IT), the three levels are:
NUTS codes
The following codes have been discontinued:
ITC45 (Milano) was split into ITC4C and ITC4D.
ITD (Northeast Italy) became ITH.
ITE (Central Italy) became ITI.
ITF41 (Foggia) and ITF42 (Bari) were split into ITF46, ITF47, and ITF48.
ITG21 (Sassari), ITG22 (Nuoro), ITG23 (Oristano), and ITG24 (Cagliari) were split into the current divisions of ITG2.
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of Italy can be downloaded here: ''
See also
Subdivisions of Italy
ISO 3166-2 codes of Italy
FIPS region codes of Italy
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
ITALIA - NUTS level 2
ITALIA - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Provinces of Italy, Statoids.com
Italy
Nuts
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https://en.wikipedia.org/wiki/Mathematical%20Tripos
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The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the university.
Origin
In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of Cambridge. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination". From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problems set for solution. By way of example, in 1854, the Tripos consisted of 16 papers spread over eight days, totaling 44.5 hours. The total number of questions was 211. It was divided into two parts, with Part I (the first three days) covering more elementary topics.
The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123, the lowest wrangler around 1500 and the lowest scoring candidate obtaining honours (the wooden spoon) 237; about 100 candidates were awarded honours. The 300-odd candidates below that level did not earn honours and were known as poll men. The questions for the 1841 examination may be found within Cambridge University Magazine (pages 191–208).
Influence
According to the study Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick during this period the style of teaching and study required for the successful preparation of students had a wide influence:
on the development of 'mixed mathematics' (a precursor of later applied mathematics, descriptive geometry and mathematical physics, with emphasis on algebraic manipulative mastery)
on mathematical education
as vocational training for fields such as astronomy
in the reception of new physical theories, particularly in electromagnetism as expounded by James Clerk Maxwell
Since Cambridge students did a lot of rote learning called "bookwork", it was noted by Augustus De Morgan and repeated by Andrew Warwick that authors of Cambridge textbooks skipped known material. In consequence, "non-Cambridge readers ... found the arguments impossible to follow."
From the 1820s to the 1840s, analytic topics such as elliptical integrals were introduced to the curriculum. Under William Whewell, the Tripos' scope changed to one of 'mixed mathematics', with the inclusion of topics from physics such as electricity, heat and magnetism. Students would have to study intensely to perform routine problems rapidly.
Early history
The early history is of the gradual replacement during the middle of the eighteenth century of a traditional method of oral examination by written papers, with a simultaneous switch in emphasis from Latin disputation to mathematical questions. That is, all degree candidates were e
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https://en.wikipedia.org/wiki/Fermionic%20field
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In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.
The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.
Basic properties
Free (non-interacting) fermionic fields obey canonical anticommutation relations; i.e., involve the anticommutators {a, b} = ab + ba, rather than the commutators [a, b] = ab − ba of bosonic or standard quantum mechanics. Those relations also hold for interacting fermionic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states.
It is these anticommutation relations that imply Fermi–Dirac statistics for the field quanta. They also result in the Pauli exclusion principle: two fermionic particles cannot occupy the same state at the same time.
Dirac fields
The prominent example of a spin-1/2 fermion field is the Dirac field (named after Paul Dirac), and denoted by . The equation of motion for a free spin 1/2 particle is the Dirac equation,
where are gamma matrices and is the mass. The simplest possible solutions to this equation are plane wave solutions, and . These plane wave solutions form a basis for the Fourier components of , allowing for the general expansion of the wave function as follows,
u and v are spinors, labelled by spin, s and spinor indices . For the electron, a spin 1/2 particle, s = +1/2 or s=−1/2. The energy factor is the result of having a Lorentz invariant integration measure. In second quantization, is promoted to an operator, so the coefficients of its Fourier modes must be operators too. Hence, and are operators. The properties of these operators can be discerned from the properties of the field. and obey the anticommutation relations:
We impose an anticommutator relation (as opposed to a commutation relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for and , the anticommutation relations for the coefficients can be computed.
In a manner analogous to non-relativistic annihilation and creation operators and their commutators, these algebras lead to the physical interpretation that creates a fermion of momentum p and spin s, and creates an antifermion of momentum q and spin r. The gene
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https://en.wikipedia.org/wiki/Generalized%20arithmetic%20progression
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In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by multiple common differences. For example, the sequence is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3 or 5, thus allowing multiple common differences to generate it.
A semilinear set generalizes this idea to multiple dimensions -- it is a set of vectors of integers, rather than a set of integers.
Finite generalized arithmetic progression
A finite generalized arithmetic progression, or sometimes just generalized arithmetic progression (GAP), of dimension d is defined to be a set of the form
where . The product is called the size of the generalized arithmetic progression; the cardinality of the set can differ from the size if some elements of the set have multiple representations. If the cardinality equals the size, the progression is called proper. Generalized arithmetic progressions can be thought of as a projection of a higher dimensional grid into . This projection is injective if and only if the generalized arithmetic progression is proper.
Semilinear sets
Formally, an arithmetic progression of is an infinite sequence of the form , where and are fixed vectors in , called the initial vector and common difference respectively. A subset of is said to be linear if it is of the form
where is some integer and are fixed vectors in . A subset of is said to be semilinear if it is a finite union of linear sets.
The semilinear sets are exactly the sets definable in Presburger arithmetic.
See also
Freiman's theorem
References
Algebra
Combinatorics
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https://en.wikipedia.org/wiki/Zone%20d%27%C3%A9tudes%20et%20d%27am%C3%A9nagement%20du%20territoire
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In 1967 the (French National Institute for Statistics and Economic Studies, INSEE), together with the French Commissariat général and DATAR () declared the nominal division of France into eight large regions. These were named (Research and National Development Zones) or ZEAT.
Until 2016, the ZEAT corresponded to the first level in the European Union Nomenclature of Territorial Units for Statistics (NUTS 1).
External links
ZEAT
ZEAT
Types of geographical division
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https://en.wikipedia.org/wiki/Mathematics%20%28producer%29
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Ronald Maurice Bean, better known professionally as Mathematics (also known as Allah Mathematics) (born October 21, 1971), is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects. He designed the Wu-Tang Clan logo.
Biography
Born and raised in Jamaica, Queens, New York, Mathematics was introduced to hip hop by his brother who used to bring home recordings of the genre's pioneers like Grandmaster Flash & The Furious Five, Treacherous Three and Cold Crush Brothers. He began his career in 1987 DJing block parties and park jams in Baisley Projects, going by the name Supreme Cut Master. In 1988, he became the full-time DJ for experienced rapper Victor C, doing countless shows in clubs and colleges in New York City.
In 1990, Mathematics linked up with GZA/Genius, who would soon become one of the Wu-Tang Clan's founding members, but at the time was struggling to build a career on the Cold Chillin' label. This partnership earned Mathematics a spot on his first official tour, The Cold Chillin Blizzard Tour (with popular acts such as Biz Markie, Big Daddy Kane, Kool G. Rap & DJ Polo and Marley Marl).
GZA left Cold Chillin after his first album, Words from the Genius, did not achieve the sales target that was anticipated. He and Mathematics took to the road again, but this time with the help of GZA's cousins, RZA and Ol' Dirty Bastard. These three soon became the founding members of Wu-Tang Clan, then known as All In Together Now. The group soon dissolved, however, and the trio set their minds on creating the Wu group. During the group's inception, Mathematics used his experience as a graffiti artist to design a logo for the up-and-coming crew, as well as various other logos and designs the Wu-Tang's artists would use. In the years to come, he became a Wu-Element under the guidance of RZA.
Mathematics' first real exposure to production came late one night when he attended a session where he assisted RZA, his mentor, in constructing a beat from nothing. The track eventually developed into "Ice Cream" on Raekwon's Only Built 4 Cuban Linx album. RZA inspired Mathematics to follow the Wu-Tang, giving him advice on the nuances of hip hop production over the coming years. In 1996, Mathematics began record producing, in Staten Island, New York (Shaolin) and at home, in the P-Funk City, Plainfield, New Jersey, between heavily scheduled tour dates and rigorous road travelling with his father's gospel group, The Soul Seekers.
His first track, "Fast Life" featuring Ghostface Killah and the American football star Andre Rison, was included in the NFL Jams compilation album. Though this track faded into obscurity somewhat, it led to several more collaborations between Mathematics and Ghostface; Mathematics also began to produce for many other Wu-Tang members and affiliates, including several tracks on GZA's second album Beneath The Surface as well as Method Man's Tical 2000: Judgement Day, Inspectah Deck's Uncontrolled Substance and Met
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https://en.wikipedia.org/wiki/Continuity
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Continuity or continuous may refer to:
Mathematics
Continuity (mathematics), the opposing concept to discreteness; common examples include
Continuous probability distribution or random variable in probability and statistics
Continuous game, a generalization of games used in game theory
Law of continuity, a heuristic principle of Gottfried Leibniz
Continuous function, in particular:
Continuity (topology), a generalization to functions between topological spaces
Scott continuity, for functions between posets
Continuity (set theory), for functions between ordinals
Continuity (category theory), for functors
Graph continuity, for payoff functions in game theory
Continuity theorem may refer to one of two results:
Lévy's continuity theorem, on random variables
Kolmogorov continuity theorem, on stochastic processes
In geometry:
Parametric continuity, for parametrised curves
Geometric continuity, a concept primarily applied to the conic sections and related shapes
In probability theory
Continuous stochastic process
Science
Continuity equations applicable to conservation of mass, energy, momentum, electric charge and other conserved quantities
Continuity test for an unbroken electrical path in an electronic circuit or connector
In materials science:
a colloidal system, consists of a dispersed phase evenly intermixed with a continuous phase
a continuous wave, an electromagnetic wave of constant amplitude and frequency
Entertainment
Continuity (broadcasting), messages played by broadcasters between programs
Continuity editing, a form of film editing that combines closely related shots into a sequence highlighting plot points or consistencies
Continuity (fiction), consistency of plot elements, such as characterization, location, and costuming, within a work of fiction (this is a mass noun)
Continuity (setting), one of several similar but distinct fictional universes in a broad franchise of related works (this is a count noun)
"Continuity" or continuity script, the precursor to a film screenplay
Other uses
Continuity (Apple), a set of features introduced by Apple
Continuity of operations (disambiguation)
Continuous and progressive aspects in linguistics
Business continuity
Health care continuity
Continuity in architecture (part of complementary architecture)
See also
Continuum (disambiguation)
Contiguity (disambiguation)
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https://en.wikipedia.org/wiki/Frobenius%20pseudoprime
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In number theory, a Frobenius pseudoprime is a pseudoprime, whose definition was inspired by the quadratic Frobenius test described by Jon Grantham in a 1998 preprint and published in 2000. Frobenius pseudoprimes can be defined with respect to polynomials of degree at least 2, but they have been most extensively studied in the case of quadratic polynomials.
Frobenius pseudoprimes w.r.t. quadratic polynomials
Definition of Frobenius pseudoprimes with respect to a monic quadratic polynomial , where the discriminant is not a square, can be expressed in terms of Lucas sequences and as follows.
A composite number n is a Frobenius pseudoprime if and only if
and
where is the Jacobi symbol.
When condition (2) is satisfied, condition (3) becomes equivalent to
Therefore, Frobenius pseudoprime can be equivalently defined by conditions (1-2) and (3), or by conditions (1-2) and (3′).
Since conditions (2) and (3) hold for all primes which satisfy the simple condition (1), they can be used as a probable prime test. (If condition (1) fails, either the greatest common divisor is less than , in which case it is a non-trivial factor and is composite, or the GCD equals , in which case you should try different parameters and which are not multiples of .)
Relations to other pseudoprimes
Every Frobenius pseudoprime is also
a Lucas pseudoprime with parameters , since it is defined by conditions (1) and (2);
a Dickson pseudoprime with parameters , since it is defined by conditions (1) and (3');
a Fermat pseudoprime base when .
Converse of none of these statements is true, making the Frobenius pseudoprimes a proper subset of each of the sets of Lucas pseudoprimes and Dickson pseudoprimes with parameters , and Fermat pseudoprimes base when .
Furthermore, it follows that for the same parameters , a composite number is a Frobenius pseudoprime if and only if it is both Lucas and Dickson pseudoprime. In other words, for every fixed pair of parameters , the set of Frobenius pseudoprimes equals the intersection of the sets of Lucas and Dickson pseudoprimes.
While each Frobenius pseudoprime is a Lucas pseudoprime, it is not necessarily a strong Lucas pseudoprime. For example, 6721 is the first Frobenius pseudoprime for , which is not a strong Lucas pseudoprime.
Every Frobenius pseudoprime to is also a restricted Perrin pseudoprime. Analogous statements hold for other cubic polynomials of the form .
Examples
Frobenius pseudoprimes with respect to the Fibonacci polynomial are determined in terms of the Fibonacci numbers and Lucas numbers . Such Frobenius pseudoprimes form the sequence:
4181, 5777, 6721, 10877, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 75077, 90061, 96049, 97921, 100127, 113573, 118441, 146611, 161027, 162133, 163081, 186961, 197209, 219781, 231703, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 330929, 399001, 430127, 433621, 438751, 489601, ... .
While 323 is the first Lucas pseudoprime with resp
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https://en.wikipedia.org/wiki/L%28R%29
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In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals.
Construction
It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructible universe), by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals.
Assumptions
In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannot show even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiom of choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, given only that the von Neumann universe, V, also satisfies that axiom.
Results
Given the assumptions above, some additional results of the theory are:
Every projective set of reals – and therefore every analytic set and every Borel set of reals – is an element of L(R).
Every set of reals in L(R) is Lebesgue measurable (in fact, universally measurable) and has the property of Baire and the perfect set property.
L(R) does not satisfy the axiom of uniformization or the axiom of real determinacy.
R#, the sharp of the set of all reals, has the smallest Wadge degree of any set of reals not contained in L(R).
While not every relation on the reals in L(R) has a uniformization in L(R), every such relation does have a uniformization in L(R#).
Given any (set-size) generic extension V[G] of V, L(R) is an elementary submodel of L(R) as calculated in V[G]. Thus the theory of L(R) cannot be changed by forcing.
L(R) satisfies AD+.
References
Inner model theory
Determinacy
Descriptive set theory
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