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https://en.wikipedia.org/wiki/Star%20refinement
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In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. A related concept is the notion of barycentric refinement.
Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness.
Definitions
The general definition makes sense for arbitrary coverings and does not require a topology. Let be a set and let be a covering of that is, Given a subset of the star of with respect to is the union of all the sets that intersect that is,
Given a point we write instead of
A covering of is a refinement of a covering of if every is contained in some The following are two special kinds of refinement. The covering is called a barycentric refinement of if for every the star is contained in some The covering is called a star refinement of if for every the star is contained in some
Properties and Examples
Every star refinement of a cover is a barycentric refinement of that cover. The converse is not true, but a barycentric refinement of a barycentric refinement is a star refinement.
Given a metric space let be the collection of all open balls of a fixed radius The collection is a barycentric refinement of and the collection is a star refinement of
See also
Notes
References
General topology
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https://en.wikipedia.org/wiki/Killing%20form
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In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show that Killing form has a close relationship to the semisimplicity of the Lie algebras.
History and name
The Killing form was essentially introduced into Lie algebra theory by in his thesis. In a historical survey of Lie theory, has described how the term "Killing form" first occurred in 1951 during one of his own reports for the Séminaire Bourbaki; it arose as a misnomer, since the form had previously been used by Lie theorists, without a name attached. Some other authors now employ the term "Cartan-Killing form". At the end of the 19th century, Killing had noted that the coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows that the Killing form (i.e. the degree 2 coefficient) is invariant, but he did not make much use of the fact. A basic result that Cartan made use of was Cartan's criterion, which states that the Killing form is non-degenerate if and only if the Lie algebra is a direct sum of simple Lie algebras.
Definition
Consider a Lie algebra over a field . Every element of defines the adjoint endomorphism (also written as ) of with the help of the Lie bracket, as
Now, supposing is of finite dimension, the trace of the composition of two such endomorphisms defines a symmetric bilinear form
with values in , the Killing form on .
Properties
The following properties follow as theorems from the above definition.
The Killing form is bilinear and symmetric.
The Killing form is an invariant form, as are all other forms obtained from Casimir operators. The derivation of Casimir operators vanishes; for the Killing form, this vanishing can be written as
where [ , ] is the Lie bracket.
If is a simple Lie algebra then any invariant symmetric bilinear form on is a scalar multiple of the Killing form.
The Killing form is also invariant under automorphisms of the algebra , that is,
for in .
The Cartan criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
The Killing form of a nilpotent Lie algebra is identically zero.
If are two ideals in a Lie algebra with zero intersection, then and are orthogonal subspaces with respect to the Killing form.
The orthogonal complement with respect to of an ideal is again an ideal.
If a given Lie algebra is a direct sum of its ideals , then the Killing form of is the direct sum of the Killing forms of the individual summands.
Matrix elements
Given a basis of the Lie algebra , the matrix elements of the Killing form are given by
Here
in Einstein summation notation, where the are the structure coefficients of the Lie algebra. The index functions as column index and the index as row i
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https://en.wikipedia.org/wiki/Double%20coset
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In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups. More precisely, let be a group, and let and be subgroups. Let act on by left multiplication and let act on by right multiplication. For each in , the -double coset of is the set
When , this is called the -double coset of . Equivalently, is the equivalence class of under the equivalence relation
if and only if there exist in and in such that .
The set of all -double cosets is denoted by
Properties
Suppose that is a group with subgroups and acting by left and right multiplication, respectively. The -double cosets of may be equivalently described as orbits for the product group acting on by . Many of the basic properties of double cosets follow immediately from the fact that they are orbits. However, because is a group and and are subgroups acting by multiplication, double cosets are more structured than orbits of arbitrary group actions, and they have additional properties that are false for more general actions.
Two double cosets and are either disjoint or identical.
is the disjoint union of its double cosets.
There is a one-to-one correspondence between the two double coset spaces and given by identifying with .
If , then . If , then .
A double coset is a union of right cosets of and left cosets of ; specifically,
The set of -double cosets is in bijection with the orbits , and also with the orbits under the mappings and respectively.
If is normal, then is a group, and the right action of on this group factors through the right action of . It follows that . Similarly, if is normal, then .
If is a normal subgroup of , then the -double cosets are in one-to-one correspondence with the left (and right) -cosets.
Consider as the union of a -orbit of right -cosets. The stabilizer of the right -coset with respect to the right action of is . Similarly, the stabilizer of the left -coset with respect to the left action of is .
It follows that the number of right cosets of contained in is the index and the number of left cosets of contained in is the index . Therefore
If , , and are finite, then it also follows that
Fix in , and let denote the double stabilizer }. Then the double stabilizer is a subgroup of .
Because is a group, for each in there is precisely one in such that , namely ; however, may not be in . Similarly, for each in there is precisely one in such that , but may not be in . The double stabilizer therefore has the descriptions
(Orbit–stabilizer theorem) There is a bijection between and under which corresponds to . It follows that if , , and are finite, then
(Cauchy–Frobenius lemma) Let denote the elements fixed by the action of . Then
In particular, if , , and are finite, then the number of double cosets equals the average number of points fixed per pair of group elements.
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https://en.wikipedia.org/wiki/Mertens%27%20theorems
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In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
In the following, let mean all primes not exceeding n.
First theorem
Mertens' first theorem is that
does not exceed 2 in absolute value for any . ()
Second theorem
Mertens' second theorem is
where M is the Meissel–Mertens constant (). More precisely, Mertens proves that the expression under the limit does not in absolute value exceed
for any .
Proof
The main step in the proof of Mertens' second theorem is
where the last equality needs which follows from .
Thus, we have proved that
.
Since the sum over prime powers with converges, this implies
.
A partial summation yields
.
Changes in sign
In a paper on the growth rate of the sum-of-divisors function published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference
changes sign infinitely often, and that in Mertens' 3rd theorem the difference
changes sign infinitely often. Robin's results are analogous to Littlewood's famous theorem that the difference π(x) − li(x) changes sign infinitely often. No analog of the Skewes number (an upper bound on the first natural number x for which π(x) > li(x)) is known in the case of Mertens' 2nd and 3rd theorems.
Relation to the prime number theorem
Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre", the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), and also that a more elaborate version was proved by Chebyshev in 1851. Note that, already in 1737, Euler knew the asymptotic behaviour of this sum.
Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem.
Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable.
Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields
whereas the prime number theorem (in its simplest form, without error estimate), can be shown to imply
In 1909 Edmund Landau, by using the best version of the prime number theorem then at his disposition, proved that
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https://en.wikipedia.org/wiki/Worm%20%28marketing%29
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The "Worm" is a market research analysis tool developed by the Roy Morgan statistics company (known than as Roy Morgan Research, who called it "The Reactor"), with the purpose of gauging an audience's reaction to some visual stimuli over some time period. The name "worm" describes its visual appearance – as a line graph snaking up or down, usually depicted on TV during live political debates
Background
Each member of the audience firstly fills out a questionnaire, used to describe the composition of the audience. Then, each member is given a control device (such as a dial or keypad) with which they select their feelings towards the vision or stimuli (for example, whether they regard the comments currently being made by a speaker favourably or unfavourably). This dial is checked centrally three times per second, and as the audience reacts differently over time, the collective feelings of the audience are gathered.
Australian Federal Elections
The "worm" has been used in the televised political debates in Australian federal elections, including those between then Australian prime minister John Howard and then-leader of the opposition Kevin Rudd in 2007 and between prime minister Julia Gillard and opposition leader Tony Abbott in 2010.
Internationally
In the first UK general election debate on 15 April 2010 between Labour Prime Minister Gordon Brown, Conservative leader David Cameron and Liberal Democrat leader Nick Clegg, "the worm" was used in certain segments. A study published in March 2011 suggests that the worm may influence voters.
In New Zealand, the worm has been controversially credited with increasing the support for United Future leader Peter Dunne in the 2002 election.
References
External links
The Reactor (Roy Morgan Research)
Market research
Political terminology in Australia
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https://en.wikipedia.org/wiki/Keldysh%20Institute%20of%20Applied%20Mathematics
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The Keldysh Institute of Applied Mathematics () is a research institute specializing in computational mathematics. It was established to solve computational tasks related to government programs of nuclear and fusion energy, space research and missile technology. The Institute is a part of the Department of Mathematical Sciences of the Russian Academy of Sciences. The main direction of activity of the institute is the use of computer technology to solve complex scientific and technical issues of practical importance. Since 2016, the development of mathematical and computational methods for biological research, as well as a direct solution to the problems of computational biology with the use of such methods, has also been included in the circle of scientific activities of the institute.
Scientific activity
Nuclear physics
Theoretical physicist Yakov Borisovich Zel'dovich headed one of the departments placed in charge of the theoretical aspects of the work on the creation of nuclear and thermonuclear weapons. Alexander Andreevich Samarskii performed the first realistic calculations of macrokinetics of chain reaction of a nuclear explosion, which led to the practical importance estimated power of nuclear weapons. In relation to nuclear energy, the institute was also involved in modelling of processes of neutron transport and nuclear reactions. In particular, E. Kuznetsov is known for his work on the theory of nuclear reactors.
Currently, such work in KIAM is continuing in the field of plasma physics and controlled thermonuclear fusion, which began under the leadership of S. P. Kurdyumov, , .
Cosmonautics
D. E. Okhotsimsky directed the works in the department created for research of the dynamics of space flight. In 1966, the department was reorganized into the Ballistic Centre. The Ballistic Centre was involved in the calculations of optimal orbit trajectory and actual corrections for all space flights, from uncrewed interplanetary and lunar vehicles to the crewed "Soyuz" and orbital station "Salyut" and "Mir". The institute took an active part in the creation of the Soviet space shuttle "Buran".
KIAM continues to engage in Russian space projects. Current research is connected with:
development of systems for management and navigation of space vehicles in real time with the use of global satellite navigation systems GPS and GLONASS;
exploration of prospects for further interplanetary missions using electric rocket engines;
participation in such projects as "RadioAstron" and Fobos-Grunt.
Mathematics and Mathematical Modelling
One of the greatest mathematicians of the twentieth century I. M. Gelfand was at the head of the Department of heat transmission before his departure for the United States in 1989. He was carrying out the fundamental works on functional analysis, algebra and topology. A. N. Tikhonov worked initially also in these areas of mathematics. However, Tikhonov is better known for his works of more applied orientation, such as m
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https://en.wikipedia.org/wiki/Steklov%20Institute%20of%20Mathematics
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Steklov Institute of Mathematics or Steklov Mathematical Institute () is a premier research institute based in Moscow, specialized in mathematics, and a part of the Russian Academy of Sciences. The institute is named after Vladimir Andreevich Steklov, who in 1919 founded the Institute of Physics and Mathematics in Leningrad. In 1934, this institute was split into separate parts for physics and mathematics, and the mathematical part became the Steklov Institute. At the same time, it was moved to Moscow. The first director of the Steklov Institute was Ivan Matveyevich Vinogradov. From 19611964, the institute's director was the notable mathematician Sergei Chernikov.
The old building of the Institute in Leningrad became its Department in Leningrad. Today, that department has become a separate institute, called the St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences or PDMI RAS, located in Saint Petersburg, Russia. The name St. Petersburg Department is misleading, however, because the St. Petersburg Department is now an independent institute. In 1966, the Moscow-based Keldysh Institute of Applied Mathematics (Russian: Институт прикладной математики им. М.В.Келдыша) split off from the Steklov Institute.
References
External links
Steklov Mathematical Institute
Petersburg Department of Steklov Institute of Mathematics
Mathematical institutes
Buildings and structures in Moscow
Education in Moscow
Research institutes in Russia
Universities and institutes established in the Soviet Union
Research institutes in the Soviet Union
Institutes of the Russian Academy of Sciences
1934 establishments in the Soviet Union
Research institutes established in 1934
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https://en.wikipedia.org/wiki/Derived%20category
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In mathematics, the derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences.
The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkable strides. The basic theory of Verdier was written down in his dissertation, published finally in 1996 in Astérisque (a summary had earlier appeared in SGA 4½). The axiomatics required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalization of localization of a ring. The original impulse to develop the "derived" formalism came from the need to find a suitable formulation of Grothendieck's coherent duality theory. Derived categories have since become indispensable also outside of algebraic geometry, for example in the formulation of the theory of D-modules and microlocal analysis. Recently derived categories have also become important in areas nearer to physics, such as D-branes and mirror symmetry.
Motivations
In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves in place of a single dualizing sheaf became apparent. In fact the Cohen–Macaulay ring condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.
Despite the level of abstraction, derived categories became accepted over the following decades, especially as a convenient setting for sheaf cohomology. Perhaps the biggest advance was the formulation of the Riemann–Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted the language of derived categories, and the subsequent history of D-modules was of a theory expressed in those terms.
A parallel development was the category of spectra in homotopy theory. The homot
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https://en.wikipedia.org/wiki/Ext
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Ext, ext or EXT may refer to:
Ext functor, used in the mathematical field of homological algebra
Ext (JavaScript library), a programming library used to build interactive web applications
Exeter Airport (IATA airport code), in Devon, England
Exeter St Thomas railway station (station code), in Exeter, England
Extended file system, a file system created for Linux
Exton station (Pennsylvania) (Amtrak station code), in Exton, Pennsylvania
Extremaduran language (ISO language code), spoken in Spain
Extremeroller, a former roller coaster at Worlds of Fun, Kansas City, Missouri
Cadillac Escalade EXT, a sport utility truck
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https://en.wikipedia.org/wiki/Combinatorial%20class
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In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size.
Counting sequences and isomorphism
The counting sequence of a combinatorial class is the sequence of the numbers of elements of size i for i = 0, 1, 2, ...; it may also be described as a generating function that has these numbers as its coefficients. The counting sequences of combinatorial classes are the main subject of study of enumerative combinatorics. Two combinatorial classes are said to be isomorphic if they have the same numbers of objects of each size, or equivalently, if their counting sequences are the same. Frequently, once two combinatorial classes are known to be isomorphic, a bijective proof of this equivalence is sought; such a proof may be interpreted as showing that the objects in the two isomorphic classes are cryptomorphic to each other.
For instance, the triangulations of regular polygons (with size given by the number of sides of the polygon, and a fixed choice of polygon to triangulate for each size) and the set of unrooted binary plane trees (up to graph isomorphism, with a fixed ordering of the leaves, and with size given by the number of leaves) are both counted by the Catalan numbers, so they form isomorphic combinatorial classes. A bijective isomorphism in this case is given by planar graph duality: a triangulation can be transformed bijectively into a tree with a leaf for each polygon edge, an internal node for each triangle, and an edge for each two (polygon edges?) or triangles that are adjacent to each other.
Analytic combinatorics
The theory of combinatorial species and its extension to analytic combinatorics provide a language for describing many important combinatorial classes, constructing new classes from combinations of previously defined ones, and automatically deriving their counting sequences. For example, two combinatorial classes may be combined by disjoint union, or by a Cartesian product construction in which the objects are ordered pairs of one object from each of two classes, and the size function is the sum of the sizes of each object in the pair. These operations respectively form the addition and multiplication operations of a semiring on the family of (isomorphism equivalence classes of) combinatorial classes, in which the zero object is the empty combinatorial class, and the unit is the class whose only object is the empty set.
Permutation patterns
In the study of permutation patterns, a combinatorial class of permutation classes, enumerated by permutation length, is called a Wilf class. The study of enumerations of specific permutation classes has turned up unexpected equivalences in counting sequences of seemingly unrelated permutation classes.
References
Combinatorics
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https://en.wikipedia.org/wiki/Friedrich%20Hirzebruch
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Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period."
Education
Hirzebruch was born in Hamm, Westphalia in 1927.
His father of the same name was a maths teacher.
Hirzebruch studied at the University of Münster from 1945–1950, with one year at ETH Zürich.
Career
Hirzebruch then held a position at Erlangen, followed by the years 1952–54 at the Institute for Advanced Study in Princeton, New Jersey. After one year at Princeton University 1955–56, he was made a professor at the University of Bonn, where he remained, becoming director of the Max-Planck-Institut für Mathematik in 1981.
More than 300 people gathered in celebration of his 80th birthday in Bonn in 2007.
The Hirzebruch–Riemann–Roch theorem (1954) for complex manifolds was a major advance and quickly became part of the mainstream developments around the classical Riemann–Roch theorem;
it was also a precursor of the Atiyah–Singer index theorem and Grothendieck's powerful generalisation.
Hirzebruch's book Neue topologische Methoden in der algebraischen Geometrie (1956) was a basic text for the 'new methods' of sheaf theory, in complex algebraic geometry.
He went on to write the foundational papers on topological K-theory with Michael Atiyah, and collaborated with Armand Borel on the theory of characteristic classes. In his later work he provided a detailed theory of Hilbert modular surfaces, with Don Zagier. He even found connections between the Dedekind sum in number theory and differential topology, one of the many discoveries found between these different fields. His work influenced a generation of prominent mathematicians like Kunihiko Kodaira, John Milnor, Borel, Atiyah, Raoul Bott and Jean-Pierre Serre.
In March 1945, Hirzebruch became a soldier, and in April, in the last weeks of Hitler's rule, he was taken prisoner by the British forces then invading Germany from the west. When a British soldier found that he was studying mathematics, he drove him home and released him, and told him to continue studying.
Hirzebruch is famous for organizing the Mathematische Arbeitstagung ("working meetings" in German) in Bonn University, beginning from 1957, and the first speakers include Atiyah, Jacques Tits, Alexander Grothendieck, Hans Grauert, Nicolaas Kuiper, and Hirzebruch himself. It allowed international cooperation among the mathematical world for the last 60 years and was a major source of developments in topology, geometry, group theory, number theory as well as mathematical physics in a few decades' time. He also established the Max Planck Institute for Mathematics at Bonn in 1980. The institute became the place for the Arbeitstagung and Hirzebruch was its director until 1995. The second Arbeitstagung began in 1993 and contin
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https://en.wikipedia.org/wiki/Salomon%20Bochner
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Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry.
Life
He was born into a Jewish family in Podgórze (near Kraków), then Austria-Hungary, now Poland. Fearful of a Russian invasion in Galicia at the beginning of World War I in 1914, his family moved to Germany, seeking greater security. Bochner was educated at a Berlin gymnasium (secondary school), and then at the University of Berlin. There, he was a student of Erhard Schmidt, writing a dissertation involving what would later be called the Bergman kernel. Shortly after this, he left the academy to help his family during the escalating inflation. After returning to mathematical research, he lectured at the University of Munich from 1924 to 1933. His academic career in Germany ended after the Nazis came to power in 1933, and he left for a position at Princeton University. He was a visiting scholar at the Institute for Advanced Study in 1945-48. He was appointed as Henry Burchard Fine Professor in 1959, retiring in 1968. Although he was seventy years old when he retired from Princeton, Bochner was appointed as Edgar Odell Lovett Professor of Mathematics at Rice University and went on to hold this chair until his death in 1982. He became Head of Department at Rice in 1969 and held this position until 1976. He died in Houston, Texas. He was an Orthodox Jew.
Mathematical work
In 1925 he started work in the area of almost periodic functions, simplifying the approach of Harald Bohr by use of compactness and approximate identity arguments. In 1933 he defined the Bochner integral, as it is now called, for vector-valued functions. Bochner's theorem on Fourier transforms appeared in a 1932 book. His techniques came into their own as Pontryagin duality and then the representation theory of locally compact groups developed in the following years.
Subsequently, he worked on multiple Fourier series, posing the question of the Bochner–Riesz means. This led to results on how the Fourier transform on Euclidean space behaves under rotations.
In differential geometry, Bochner's formula on curvature from 1946 was published. Joint work with Kentaro Yano (1912–1993) led to the 1953 book Curvature and Betti Numbers. It had consequences, for the Kodaira vanishing theory, representation theory, and spin manifolds. Bochner also worked on several complex variables (the Bochner–Martinelli formula and the book Several Complex Variables from 1948 with W. T. Martin).
Selected publications
2016 reprint
2013 reprint
2014 reprint
See also
Bochner almost periodic functions
Bochner–Kodaira–Nakano identity
Bochner Laplacian
Bochner measurable function
References
External links
National Academy of Sciences Biographical Memoir
1899 births
1982 deaths
20th-century Austrian mathematicians
20th-century American mathematicians
American Orthodox Jews
Austrian Orthodox Jews
Complex analysts
Differential
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https://en.wikipedia.org/wiki/Gram%20matrix
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In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product . If the vectors are the columns of matrix then the Gram matrix is in the general case that the vector coordinates are complex numbers, which simplifies to for the case that the vector coordinates are real numbers.
An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero.
It is named after Jørgen Pedersen Gram.
Examples
For finite-dimensional real vectors in with the usual Euclidean dot product, the Gram matrix is , where is a matrix whose columns are the vectors and is its transpose whose rows are the vectors . For complex vectors in , , where is the conjugate transpose of .
Given square-integrable functions on the interval , the Gram matrix is:
where is the complex conjugate of .
For any bilinear form on a finite-dimensional vector space over any field we can define a Gram matrix attached to a set of vectors by . The matrix will be symmetric if the bilinear form is symmetric.
Applications
In Riemannian geometry, given an embedded -dimensional Riemannian manifold and a parametrization for the volume form on induced by the embedding may be computed using the Gramian of the coordinate tangent vectors: This generalizes the classical surface integral of a parametrized surface for :
If the vectors are centered random variables, the Gramian is approximately proportional to the covariance matrix, with the scaling determined by the number of elements in the vector.
In quantum chemistry, the Gram matrix of a set of basis vectors is the overlap matrix.
In control theory (or more generally systems theory), the controllability Gramian and observability Gramian determine properties of a linear system.
Gramian matrices arise in covariance structure model fitting (see e.g., Jamshidian and Bentler, 1993, Applied Psychological Measurement, Volume 18, pp. 79–94).
In the finite element method, the Gram matrix arises from approximating a function from a finite dimensional space; the Gram matrix entries are then the inner products of the basis functions of the finite dimensional subspace.
In machine learning, kernel functions are often represented as Gram matrices. (Also see kernel PCA)
Since the Gram matrix over the reals is a symmetric matrix, it is diagonalizable and its eigenvalues are non-negative. The diagonalization of the Gram matrix is the singular value decomposition.
Properties
Positive-semidefiniteness
The Gram matrix is symmetric in the case the real product is real-valued; it is Hermitian in the general, complex case by definition of an inner product.
The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact
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https://en.wikipedia.org/wiki/Turtle%20Geometry
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Turtle Geometry is a college-level math text written by Hal Abelson and Andrea diSessa which aims to engage students in exploring mathematical properties visually via a simple programming language to maneuver the icon of a turtle trailing lines across a personal computer display.
See also
Turtle graphics
Turtle Geometry at MIT Press
Computer science books
1981 non-fiction books
MIT Press books
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https://en.wikipedia.org/wiki/Lindenbaum%E2%80%93Tarski%20algebra
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In mathematical logic, the Lindenbaum–Tarski algebra (or Lindenbaum algebra) of a logical theory T consists of the equivalence classes of sentences of the theory (i.e., the quotient, under the equivalence relation ~ defined such that p ~ q exactly when p and q are provably equivalent in T). That is, two sentences are equivalent if the theory T proves that each implies the other. The Lindenbaum–Tarski algebra is thus the quotient algebra obtained by factoring the algebra of formulas by this congruence relation.
The algebra is named for logicians Adolf Lindenbaum and Alfred Tarski.
Starting in the academic year 1926-1927, Lindenbaum pioneered his method in Jan Łukasiewicz's mathematical logic seminar, and the method was popularized and generalized in subsequent decades through work
by Tarski.
The Lindenbaum–Tarski algebra is considered the origin of the modern algebraic logic.
Operations
The operations in a Lindenbaum–Tarski algebra A are inherited from those in the underlying theory T. These typically include conjunction and disjunction, which are well-defined on the equivalence classes. When negation is also present in T, then A is a Boolean algebra, provided the logic is classical. If the theory T consists of the propositional tautologies, the Lindenbaum–Tarski algebra is the free Boolean algebra generated by the propositional variables.
Related algebras
Heyting algebras and interior algebras are the Lindenbaum–Tarski algebras for intuitionistic logic and the modal logic S4, respectively.
A logic for which Tarski's method is applicable, is called algebraizable. There are however a number of logics where this is not the case, for instance the modal logics S1, S2, or S3, which lack the rule of necessitation (⊢φ implying ⊢□φ), so ~ (defined above) is not a congruence (because ⊢φ→ψ does not imply ⊢□φ→□ψ). Another type of logic where Tarski's method is inapplicable is relevance logics, because given two theorems an implication from one to the other may not itself be a theorem in a relevance logic. The study of the algebraization process (and notion) as topic of interest by itself, not necessarily by Tarski's method, has led to the development of abstract algebraic logic.
See also
Algebraic semantics (mathematical logic)
Leibniz operator
List of Boolean algebra topics
References
Algebraic logic
Algebraic structures
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https://en.wikipedia.org/wiki/Interior%20algebra
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In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras.
Definition
An interior algebra is an algebraic structure with the signature
⟨S, ·, +, ′, 0, 1, I⟩
where
⟨S, ·, +, ′, 0, 1⟩
is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities:
xI ≤ x
xII = xI
(xy)I = xIyI
1I = 1
xI is called the interior of x.
The dual of the interior operator is the closure operator C defined by xC = ((x′)I)′. xC is called the closure of x. By the principle of duality, the closure operator satisfies the identities:
xC ≥ x
xCC = xC
(x + y)C = xC + yC
0C = 0
If the closure operator is taken as primitive, the interior operator can be defined as xI = ((x′)C)′. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form ⟨S, ·, +, ′, 0, 1, C⟩, where ⟨S, ·, +, ′, 0, 1⟩ is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form dual pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm following the work of Wim Blok.
Open and closed elements
Elements of an interior algebra satisfying the condition xI = x are called open. The complements of open elements are called closed and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements which are both open and closed are called clopen. 0 and 1 are clopen.
An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras which are the single element interior algebras characterized by the identity 0 = 1.
Morphisms of interior algebras
Homomorphisms
Interior algebras, by virtue of being algebraic structures, have homomorphisms. Given two interior algebras A and B, a map f : A → B is an interior algebra homomorphism if and only if f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures. Hence:
f(xI) = f(x)I;
f(xC) = f(x)C.
Topomorphisms
Topomorphisms are another important, and more general, class of morphisms between interior algebras. A map f : A → B is a topomo
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https://en.wikipedia.org/wiki/Norm%20%28mathematics%29
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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".
A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "" in the homogeneity axiom.
It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set.
Definition
Given a vector space over a subfield of the complex numbers a norm on is a real-valued function with the following properties, where denotes the usual absolute value of a scalar :
Subadditivity/Triangle inequality: for all
Absolute homogeneity: for all and all scalars
Positive definiteness/positiveness/: for all if then
Because property (2.) implies some authors replace property (3.) with the equivalent condition: for every if and only if
A seminorm on is a function that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if is a norm (or more generally, a seminorm) then and that also has the following property:
Non-negativity: for all
Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Although this article defined "" to be a synonym of "positive definite", some authors instead define "" to be a synonym of "non-negative"; these definitions are not equivalent.
Equivalent norms
Suppose that and are two norms (or seminorms) on a vector space Then and are called equivalent, if there exist two positive real constants and with such that for every vector
The relation " is equivalent to " is reflexive, symmetric ( implies ), and transitive and thus defines an equivalence relation on the set of all norms on
The norms and are equivalent if and only if they induce the same topology on Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.
Notation
If a norm is given on a vector space then the norm of a vector is usually denoted by enclosing it within double vertical lines:
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https://en.wikipedia.org/wiki/Dynamical%20systems%20theory
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Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.
This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.
This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.
Overview
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
An important goal is to describe the fixed points, or steady states of a given dynamical system; these are values of the variable that don't change over time. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point.
Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system.
Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos. The branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory.
History
The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical sys
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https://en.wikipedia.org/wiki/Moving%20average
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In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different selections of the full data set. It is also called a moving mean (MM) or rolling mean and is a type of finite impulse response filter. Variations include: simple, cumulative, or weighted forms (described below).
A moving average filter is sometimes called a boxcar filter, especially when followed by decimation.
Given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series. Then the subset is modified by "shifting forward"; that is, excluding the first number of the series and including the next value in the subset.
A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly. It is also used in economics to examine gross domestic product, employment or other macroeconomic time series. Mathematically, a moving average is a type of convolution and so it can be viewed as an example of a low-pass filter used in signal processing. When used with non-time series data, a moving average filters higher frequency components without any specific connection to time, although typically some kind of ordering is implied. Viewed simplistically it can be regarded as smoothing the data.
Simple moving average
In financial applications a simple moving average (SMA) is the unweighted mean of the previous data-points. However, in science and engineering, the mean is normally taken from an equal number of data on either side of a central value. This ensures that variations in the mean are aligned with the variations in the data rather than being shifted in time. An example of a simple equally weighted running mean is the mean over the last entries of a data-set containing entries. Let those data-points be . This could be closing prices of a stock. The mean over the last data-points (days in this example) is denoted as and calculated as:
When calculating the next mean with the same sampling width the range from to is considered. A new value comes into the sum and the oldest value drops out. This simplifies the calculations by reusing the previous mean .
This means that the moving average filter can be computed quite cheaply on real time data with a FIFO / circular buffer and only 3 arithmetic steps.
During the initial filling of the FIFO / circular buffer the sampling window is equal to the data-set size thus and the average calculation is performed as a cumulative moving average.
The period selected () depends on the type of movement of interest, such as short, intermediate, or long-term.
If the data used are not centered around the mean, a simple moving average lags b
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https://en.wikipedia.org/wiki/Harold%20Davenport
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Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.
Early life
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge. He became a research student of John Edensor Littlewood, working on the question of the distribution of quadratic residues.
First steps in research
The attack on the distribution question leads quickly to problems that are now seen to be special cases of those on local zeta-functions, for the particular case of some special hyperelliptic curves such as .
Bounds for the zeroes of the local zeta-function immediately imply bounds for sums , where χ is the Legendre symbol modulo a prime number p, and the sum is taken over a complete set of residues mod p.
In the light of this connection it was appropriate that, with a Trinity research fellowship, Davenport in 1932–1933 spent time in Marburg and Göttingen working with Helmut Hasse, an expert on the algebraic theory. This produced the work on the Hasse–Davenport relations for Gauss sums, and contact with Hans Heilbronn, with whom Davenport would later collaborate. In fact, as Davenport later admitted, his inherent prejudices against algebraic methods ("what can you do with algebra?") probably limited the amount he learned, in particular in the "new" algebraic geometry and Artin/Noether approach to abstract algebra.
Later career
He took an appointment at the University of Manchester in 1937, just at the time when Louis Mordell had recruited émigrés from continental Europe to build an outstanding department. He moved into the areas of diophantine approximation and geometry of numbers. These were fashionable, and complemented the technical expertise he had in the Hardy-Littlewood circle method; he was later, though, to let drop the comment that he wished he'd spent more time on the Riemann hypothesis.
He was President of the London Mathematical Society from 1957 to 1959. After professorial positions at the University of Wales and University College London, he was appointed to the Rouse Ball Chair of Mathematics in Cambridge in 1958. There he remained until his death, of lung cancer.
Personal life
Davenport married Anne Lofthouse, whom he met at the University College of North Wales at Bangor in 1944; they had two children, Richard and James, the latter going on to become Hebron and Medlock Professor of Information Technology at the University of Bath.
Influence
From about 1950, Davenport was the obvious leader of a "school", somewhat unusually in the context of British mathematics. The successor to the school of mathematical analysis of G. H. Hardy and J. E. Littlewood, it was also more narrowly devoted to number theory, and indeed to its analytic side, as had flourished in the 1930s. This implied problem-solving, and hard-analysis methods. The outstanding works of Klaus Roth an
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https://en.wikipedia.org/wiki/Louis%20J.%20Mordell
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Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Education
Mordell was educated at the University of Cambridge where he completed the Cambridge Mathematical Tripos as a student of St John's College, Cambridge, starting in 1906 after successfully passing the scholarship examination. He graduated as third wrangler in 1909.
Research
After graduating Mordell began independent research into particular diophantine equations: the question of integer points on the cubic curve, and special case of what is now called a Thue equation, the Mordell equation
y2 = x3 + k.
He took an appointment at Birkbeck College, London in 1913. During World War I he was involved in war work, but also produced one of his major results, proving in 1917 the multiplicative property of Srinivasa Ramanujan's tau-function. The proof was by means, in effect, of the Hecke operators, which had not yet been named after Erich Hecke; it was, in retrospect, one of the major advances in modular form theory, beyond its status as an odd corner of the theory of special functions.
In 1920, he took a teaching position in UMIST, becoming the Fielden Chair of Pure Mathematics at the University of Manchester in 1922 and Professor in 1923. There he developed a third area of interest within number theory, the geometry of numbers. His basic work on Mordell's theorem is from 1921 to 1922, as is the formulation of the Mordell conjecture. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1928 in Bologna and in 1932 in Zürich and a Plenary Speaker of the ICM in 1936 in Oslo.
He took British citizenship in 1929. In Manchester he also built up the department, offering posts to a number of outstanding mathematicians who had been forced from posts on the continent of Europe. He brought in Reinhold Baer, G. Billing, Paul Erdős, Chao Ko, Kurt Mahler, and Beniamino Segre. He also recruited J. A. Todd, Patrick du Val, Harold Davenport and Laurence Chisholm Young, and invited distinguished visitors.
In 1945, he returned to Cambridge as a Fellow of St. John's, when elected to the Sadleirian Chair, and became Head of Department. He officially retired in 1953. It was at this time that he had his only formal research students, of whom J. W. S. Cassels was one. His idea of supervising research was said to involve the suggestion that a proof of the transcendence of the Euler–Mascheroni constant was probably worth a doctorate. His book Diophantine Equations (1969) is based on lectures, and gives an idea of his discursive style. Mordell is said to have hated administrative duties.
Anecdote
While visiting the University of Calgary, the elderly Mordell attended the Number Theory seminars and would frequently fall asleep during them. According to a story by number theorist Richard K. Guy, the departmen
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https://en.wikipedia.org/wiki/Circumscribed%20sphere
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In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term circumcircle. As in the case of two-dimensional circumscribed circles (circumcircles), the radius of a sphere circumscribed around a polyhedron is called the circumradius of , and the center point of this sphere is called the circumcenter of .
Existence and optimality
When it exists, a circumscribed sphere need not be the smallest sphere containing the polyhedron; for instance, the tetrahedron formed by a vertex of a cube and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the convex hull of a subset of the vertices of the polyhedron.
In De solidorum elementis (circa 1630), René Descartes observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: some bipyramids, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a simple polyhedron has a circumscribed circle for each of its faces, it also has a circumscribed sphere.
Related concepts
The circumscribed sphere is the three-dimensional analogue of the circumscribed circle.
All regular polyhedra have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere. The circumscribed sphere (when it exists) is an example of a bounding sphere, a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it in linear time.
Other spheres defined for some but not all polyhedra include a midsphere, a sphere tangent to all edges of a polyhedron, and an inscribed sphere, a sphere tangent to all faces of a polyhedron. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric.
When the circumscribed sphere is the set of infinite limiting points of hyperbolic space, a polyhedron that it circumscribes is known as an ideal polyhedron.
Point on the circumscribed sphere
There are five convex regular polyhedra, known as the Platonic solids. All Platonic solids have circumscribed spheres. For an arbitrary point on the circumscribed sphere of each Platonic solid with number of the vertices , if are the distances to
the vertices , then
References
External links
Elementary geometry
Spheres
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https://en.wikipedia.org/wiki/Inscribed%20sphere
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In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and is dual to the dual polyhedron's circumsphere.
The radius of the sphere inscribed in a polyhedron P is called the inradius of P.
Interpretations
All regular polyhedra have inscribed spheres, but most irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of an insphere does not seem to have been properly defined and various interpretations of an insphere are to be found:
The sphere tangent to all faces (if one exists).
The sphere tangent to all face planes (if one exists).
The sphere tangent to a given set of faces (if one exists).
The largest sphere that can fit inside the polyhedron.
Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.
For example, the regular small stellated dodecahedron has a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron. Which is the insphere? Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere. Again, such authorities agree that the Archimedean polyhedra (having regular faces and equivalent vertices) have no inspheres while the Archimedean dual or Catalan polyhedra do have inspheres. But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra.
See also
Circumscribed sphere
Inscribed circle
Midsphere
Sphere packing
References
Coxeter, H.S.M. Regular Polytopes 3rd Edn. Dover (1973).
Cundy, H.M. and Rollett, A.P. Mathematical Models, 2nd Edn. OUP (1961).
External links
Elementary geometry
Polyhedra
Spheres
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https://en.wikipedia.org/wiki/Topological%20Boolean%20algebra
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Topological Boolean algebra may refer to:
In abstract algebra and mathematical logic, topological Boolean algebra is one of the many names that have been used for an interior algebra in the literature.
In the work of the mathematician R.S. Pierce, a topological Boolean algebra is a Boolean algebra equipped with both a closure operator and a derivative operator generalizing T1 topological spaces and may be considered to be a special case of interior algebras rather than synonymous with them.
In topological algebra — the study of topological spaces with algebraic structure, a topological Boolean algebra is a Boolean algebra endowed with a topological structure in which the operations of complement, join, and meet are continuous functions.
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https://en.wikipedia.org/wiki/John%20Selfridge
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John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics.
Education
Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin.
Career
Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University (NIU) from 1971 to 1991 (retirement), chairing the NIU Department of Mathematical Sciences 1972–1976 and 1986–1990.
He was executive editor of Mathematical Reviews from 1978 to 1986, overseeing the computerization of its operations. He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour.
Research
In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when k = 78,557, all numbers of the form k2n + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. Five years later, he and Sierpiński proposed the conjecture that 78,557 is the smallest Sierpinski number, and thus the answer to the Sierpinski problem. A distributed computing project called Seventeen or Bust is currently trying to prove this statement, only five of the original seventeen possibilities remain.
In 1964, Selfridge and Alexander Hurwitz proved that the 14th Fermat number was composite.
However, their proof did not provide a factor. It was not until 2010 that the first factor of the 14th Fermat number was found.
In 1975 John Brillhart, Derrick Henry Lehmer, and Selfridge developed a method of proving the primality of p given only partial factorizations of p − 1 and p + 1.
Together with Samuel Wagstaff they also all participated in the Cunningham project.
Together with Paul Erdős, Selfridge solved a 150-year-old problem, proving that the product of consecutive numbers is never a power. It took them many years to find the proof, and John made extensive use of computers, but the final version of the proof requires only a modest amount of computation, namely evaluating an easily computed function f(n) for 30,000 consecutive values of n. Selfridge suffered from writer's block and thanked "R. B. Eggleton for reorganizing and writing the paper in its final form".
Selfridge also developed the Selfridge–Conway discrete procedure for creating an envy-free cake-cutting among three people. Selfridge developed this in 1960, and John Conway independently discovered it in 1993. Neither of them ever published the result, but Richard Guy told many people Selfridge's solution in the 1960s, and it was eventually attributed to the two of them in a number of books and articles.
Selfridge's conjecture about Fermat numbers
Selfridge made the following conjecture about the Fermat numbers Fn = 22n + 1 . Let g(n) be the number of distinct prime factors of Fn . As to 2016, g(n) is known only up to n = 11, and it is monotonic. Selfridge conjectured that contrary to appearances, g(n) i
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https://en.wikipedia.org/wiki/Universal%20coefficient%20theorem
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In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its integral homology groups:
completely determine its homology groups with coefficients in , for any abelian group :
Here might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor.
For example it is common to take to be , so that coefficients are modulo 2. This becomes straightforward in the absence of 2-torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers of and the Betti numbers with coefficients in a field . These can differ, but only when the characteristic of is a prime number for which there is some -torsion in the homology.
Statement of the homology case
Consider the tensor product of modules . The theorem states there is a short exact sequence involving the Tor functor
Furthermore, this sequence splits, though not naturally. Here is the map induced by the bilinear map .
If the coefficient ring is , this is a special case of the Bockstein spectral sequence.
Universal coefficient theorem for cohomology
Let be a module over a principal ideal domain (e.g., or a field.)
There is also a universal coefficient theorem for cohomology involving the Ext functor, which asserts that there is a natural short exact sequence
As in the homology case, the sequence splits, though not naturally.
In fact, suppose
and define:
Then above is the canonical map:
An alternative point-of-view can be based on representing cohomology via Eilenberg–MacLane space where the map takes a homotopy class of maps from to to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a weak right adjoint to the homology functor.
Example: mod 2 cohomology of the real projective space
Let , the real projective space. We compute the singular cohomology of with coefficients in .
Knowing that the integer homology is given by:
We have , so that the above exact sequences yield
In fact the total cohomology ring structure is
Corollaries
A special case of the theorem is computing integral cohomology. For a finite CW complex , is finitely generated, and so we have the following decomposition.
where are the Betti numbers of and is the torsion part of . One may check that
and
This gives the following statement for integral cohomology:
For an orientable, closed, and connected -manifold, this corollary coupled with Poincaré duality gives that .
Universal coefficient spectral sequence
There is a generalization of the universal coefficient theorem for (co)homology with twisted coefficients.
For cohomology we have
Where is a r
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https://en.wikipedia.org/wiki/Molecular%20geometry
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Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom.
Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of molecule, i.e. they can be understood as approximately local and hence transferable properties.
Determination
The molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas electron diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances,
dihedral angles,
angles, and connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries (conformational isomerism) that are close in energy on the potential energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas.
The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking bond lengths of two joined atoms, bond angles of three connected atoms, and torsion angles (dihedral angles) of three consecutive bonds.
The influence of thermal excitation
Since the motions of the atoms in a molecule are determined by quantum mechanics, "motion" must be defined in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) In addition to translation and rotation, a third type of motion is molecular vibration, which corresponds to internal motions of the atoms such as bond stretching and bond angle variation. The molecular vibrations are harmonic (at least to good approximation), and the atoms oscillate about their equilibrium positions, even at the absolute zero of temperature. At absolute zero all atoms
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https://en.wikipedia.org/wiki/Peravia%20Province
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Peravia () is a province in the southern region of the Dominican Republic. Before January 1, 2002 it was included in what is the new San José de Ocoa province, and published statistics and maps generally relate it to the old, larger, Peravia.
It is named after the Peravia Valley. Along the Azua Province, Peravia is characterized by its dry climate and its dunes that surround the coast. One popular attraction is the Salinas beach, which recently has grown to be a popular tourist destination with a developed town that has shops and hotels.
Municipalities and municipal districts
The province as of June 20, 2006 is divided into the following municipalities (municipios) and municipal districts (distrito municipal - D.M.) within them:
Baní
Catalina (D.M.)
El Carretón (D.M.)
El Limonal (D.M.)
Paya (D.M.)
Villa Fundación (D.M.)
Matanzas
Sabana Buey (D.M.)
Villa Sombrero (D.M.)
Nizao
Pizarrete (D.M.)
Santana (D.M.)
The following is a sortable table of the municipalities and municipal districts with population figures as of the 2012 census. Urban population are those living in the seats (cabeceras literally heads) of municipalities or of municipal districts. Rural population are those living in the districts (Secciones literally sections) and neighborhoods (Parajes literally places) outside of them.
For comparison with the municipalities and municipal districts of other provinces see the list of municipalities and municipal districts of the Dominican Republic.
Geography
Peravia province has an area of 792.33 km2 (305.92 sq mi). It is located in the southern region, it borders the San José de Ocoa province to the north, to the east it borders San Cristóbal, to the west the province of Azua and to the south it has coasts on the Caribbean Sea. The most important rivers that cross the province are the Nizao, the Ocoa and the Baní.
The province is made up of two main regions, the Central Mountain Range and the Coastal Plain of the Caribbean. The Central Mountain Range, known in the region as the Sierra de Ocoa, extends to the north and west of the province, where the hills of La Barbacoa are located at 1,743 meters above sea level, Valdesia with 1,723 meters above sea level, Firme Rodríguez, Los Guayuyos and Los Naranjos, El Manaclar with 1,400 meters above sea level, where most of the repeaters of the country's telephone companies are located and the surveillance radar of the south of the country.
La Barbacoa was declared a Scientific Reserve for the conservation of hydro-graphic basins and sources of streams and source aquifers such as the Rio Nizao. In the south of the province, in Las Calderas Bay, is the Los Corbanitos beach, Las Dunas, Salinas de Puerto Hermoso beach and the Las Caldera Naval Base of the Dominican Navy. About 80% of the Province is dominated by a dry forest, especially in its southern zone, in the north there are different types of humid forests.
Economy
The province has a diverse agricultural industry, producing vegetab
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https://en.wikipedia.org/wiki/Frobenius%20method
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In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form
with and .
in the vicinity of the regular singular point .
One can divide by to obtain a differential equation of the form
which will not be solvable with regular power series methods if either or are not analytic at . The Frobenius method enables one to create a power series solution to such a differential equation, provided that p(z) and q(z) are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite).
History: Frobenius' Actual Contributions
Frobenius' contribution was not so much in all the possible forms of the series solutions involved (see below). These forms had all been established earlier, by Fuchs. The indicial polynomial (see below) and its role had also been established by Fuchs.
A first contribution by Frobenius to the theory was to show that - as regards a first, linearly independent solution, which then has the form of an analytical power series multiplied by an arbitrary power r of the independent variable (see below) - the coefficients of the generalized power series obey a recurrence relation, so that they can always be straightforwardly calculated.
A second contribution by Frobenius was to show that, in cases in which the roots of the indicial equation differ by an integer, the general form of the second linearly independent solution (see below) can be obtained by a procedure which is based on differentiation with respect to the parameter r, mentioned above.
A large part of Frobenius' 1873 publication was devoted to proofs of convergence of all the series involved in the solutions, as well as establishing the radii of convergence of these series.
Explanation of Frobenius Method: first, linearly independent solution
The method of Frobenius is to seek a power series solution of the form
Differentiating:
Substituting the above differentiation into our original ODE:
The expression
is known as the indicial polynomial, which is quadratic in r. The general definition of the indicial polynomial is the coefficient of the lowest power of z in the infinite series. In this case it happens to be that this is the rth coefficient but, it is possible for the lowest possible exponent to be r − 2, r − 1 or, something else depending on the given differential equation. This detail is important to keep in mind. In the process of synchronizing all the series of the differential equation to start at the same index value (which in the above expression is k = 1), one can end up with complicated expressions. However, in solving for the indicial roots attention is focused only on the coefficient of the lowest power of z.
Using this, the general expression of the coefficient of is
These coefficients must be zero, since they should be solutions of the differential equation, so
The series solution wit
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https://en.wikipedia.org/wiki/San%20Jos%C3%A9%20de%20Ocoa%20Province
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San José de Ocoa () is a province in the southern region of the Dominican Republic, and also the name of the province's capital city. It was split from Peravia on January 1, 2000. Published statistics and maps generally include this province in the old, larger, Peravia.
Municipalities and municipal districts
The province as of June 20, 2006 is divided into the following municipalities (municipios) and municipal districts (distrito municipal - D.M.) within them:
Rancho Arriba
Sabana Larga
San José de Ocoa
El Pinar (D.M.)
La Ciénaga (D.M.)
Nizao-Las Auyamas (D.M.)
The following is a sortable table of the municipalities and municipal districts with population figures as of the 2012 census. Urban population are those living in the seats (cabeceras literally heads) of municipalities or of municipal districts. Rural population are those living in the districts (Secciones literally sections) and neighborhoods (Parajes literally places) outside of them.
For comparison with the municipalities and municipal districts of other provinces see the list of municipalities and municipal districts of the Dominican Republic.
References
External links
Oficina Nacional de Estadística, Statistics Portal of the Dominican Republic
Oficina Nacional de Estadística, Maps with administrative division of the provinces of the Dominican Republic, downloadable in PDF format
Provinces of the Dominican Republic
States and territories established in 2000
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https://en.wikipedia.org/wiki/William%20George%20Horner
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William George Horner (9 June 1786 – 22 September 1837) was a British mathematician. Proficient in classics and mathematics, he was a schoolmaster, headmaster and schoolkeeper who wrote extensively on functional equations, number theory and approximation theory, but also on optics. His contribution to approximation theory is honoured in the designation Horner's method, in particular respect of a paper in Philosophical Transactions of the Royal Society of London for 1819. The modern invention of the zoetrope, under the name Daedaleum in 1834, has been attributed to him.
Horner died comparatively young, before the establishment of specialist, regular scientific periodicals. So, the way others have written about him has tended to diverge, sometimes markedly, from his own prolific, if dispersed, record of publications and the contemporary reception of them.
Family life
The eldest son of the Rev. William Horner, a Wesleyan minister, Horner was born in Bristol. He was educated at Kingswood School, a Wesleyan foundation near Bristol, and at the age of sixteen became an assistant master there. In four years he rose to be headmaster (1806), but left in 1809, setting up his own school, The Classical Seminary, at Grosvenor Place, Bath, which he kept until he died there 22 September 1837. He and his wife Sarah (1787?–1864) had six daughters and two sons.
Physical sciences, optics
Although Horner's article on the Dædalum (zoetrope) appeared in Philosophical Magazine only in January, 1834, he had published on Camera lucida as early as August, 1815.
Mathematics
Horner's name first appears in the list of solvers of the mathematical problems in The Ladies' Diary: or, Woman's Almanack for 1811, continuing in the successive annual issues until that for 1817. Up until the issue for 1816, he is listed as solving all but a few of the fifteen problems each year; several of his answers were printed, along with two problems he proposed. He also contributed to other departments of the Diary, not without distinction, reflecting the fact that he was known to be an all-rounder, competent in the classics as well as in mathematics. Horner was ever vigilant in his reading, as shown by his characteristic return to the Diary for 1821 in a discussion of the Prize Problem, where he reminds readers of an item in (Thomson's) Annals of Philosophy for 1817; several other problems in the Diary that year were solved by his youngest brother, Joseph.
His record in The Gentleman's Diary: or, Mathematical Repository for this period is similar, including one of two published modes of proof in the volume for 1815 of a problem posed the previous year by Thomas Scurr (d. 1836), now dubbed the Butterfly theorem. Leaving the headmastership of Kingswood School would have given him more time for this work, while the appearance of his name in these publications, which were favoured by a network of mathematics teachers, would have helped publicize his own school.
At this stage, Horner's efforts
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https://en.wikipedia.org/wiki/Vital%20statistics%20%28government%20records%29
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Vital statistics is accumulated data gathered on live births, deaths, migration, foetal deaths, marriages and divorces. The most common way of collecting information on these events is through civil registration, an administrative system used by governments to record vital events which occur in their populations. Efforts to improve the quality of vital statistics will therefore be closely related to the development of civil registration systems in countries. Civil registration followed the practice of churches keeping such records since the 19th century.
This article covers mainly the US, UK and Canada, with references to global standards.
Definitions
A vital statistics system is defined by the United Nations "as the total process of (a) collecting information by civil registration or enumeration on the frequency or occurrence of specified and defined vital events, as well as relevant characteristics of the events themselves and the person or persons concerned, and (b) compiling, processing, analyzing, evaluating, presenting, and disseminating these data in statistical form"
Civil registration is defined by the United Nations as the" continuous, permanent, compulsory, and universal recording of the occurrence and characteristics of vital events (live births, deaths, fetal deaths, marriages, and divorces) and other civil status events pertaining to the population as provided by decree, law or regulation, in accordance with the legal requirements in each country."
History
United Kingdom
Prior to the creation of the General Register Office (GRO) in 1837, there was no national system of civil registration in England and Wales. Baptisms, marriages and burials were recorded in parish registers maintained by Church of England (Anglican) clergy. However, with the great increase in nonconformity and the gradual relaxation of the laws against Catholics and other dissenters from the late 17th century, more and more baptisms, marriages and burials were going unrecorded in the registers of the Anglican Church.
The increasingly poor state of English parish registration led to numerous attempts to shore up the system in the 18th and early 19th centuries. The Marriage Act of 1753 attempted to prevent 'clandestine' marriages by imposing a standard form of entry for marriages, which had to be signed by both parties to the marriage and by witnesses. Additionally, except in the case of Jews and Quakers, legal marriages had to be carried out according to the rites of the Church of England. Sir George Rose's Parochial Registers Act of 1812 laid down that all events had to be entered on standard entries in bound volumes. It also declared that the church registers of Nonconformists were not admissible in court as evidence of births, marriages and deaths. Only those maintained by the clergy of the Church of England could be presented in court as legal documents, and this caused considerable hardship for Nonconformists. A number of proposals were presented to Parlia
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https://en.wikipedia.org/wiki/Irreducibility%20%28mathematics%29
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In mathematics, the concept of irreducibility is used in several ways.
A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field.
In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.
Absolutely irreducible is a term applied to mean irreducible, even after any finite extension of the field of coefficients. It applies in various situations, for example to irreducibility of a linear representation, or of an algebraic variety; where it means just the same as irreducible over an algebraic closure.
In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if such directed graph is strongly connected.) A detailed definition is given here.
Also, a Markov chain is irreducible if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.
In the theory of manifolds, an n-manifold is irreducible if any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over S1 and the twisted 2-sphere bundle over S1. See, for example, Prime decomposition (3-manifold).
A topological space is irreducible if it is not the union of two proper closed subsets. This notion is used in algebraic geometry, where spaces are equipped with the Zariski topology; it is not of much significance for Hausdorff spaces. See also irreducible component, algebraic variety.
In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.
A 3-mani
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https://en.wikipedia.org/wiki/Class%20function
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In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such functions play a basic role in representation theory.
Characters
The character of a linear representation of G over a field K is always a class function with values in K. The class functions form the center of the group ring K[G]. Here a class function f is identified with the element .
Inner products
The set of class functions of a group G with values in a field K form a K-vector space. If G is finite and the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by where |G| denotes the order of G and bar is conjugation in the field K. The set of irreducible characters of G forms an orthogonal basis, and if K is a splitting field for G, for instance if K is algebraically closed, then the irreducible characters form an orthonormal basis.
In the case of a compact group and K = C the field of complex numbers, the notion of Haar measure allows one to replace the finite sum above with an integral:
When K is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.
See also
Brauer's theorem on induced characters
References
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, Berlin, 1977.
Group theory
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https://en.wikipedia.org/wiki/Semisimple%20module
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In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.
For a group-theory analog of the same notion, see Semisimple representation.
Definition
A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
For a module M, the following are equivalent:
M is semisimple; i.e., a direct sum of irreducible modules.
M is the sum of its irreducible submodules.
Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that .
For the proof of the equivalences, see .
The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring of integers is not a semisimple module over itself, since the submodule is not a direct summand.
Semisimple is stronger than completely decomposable,
which is a direct sum of indecomposable submodules.
Let A be an algebra over a field K. Then a left module M over A is said to be absolutely semisimple if, for any field extension F of K, is a semisimple module over .
Properties
If M is semisimple and N is a submodule, then N and M/N are also semisimple.
An arbitrary direct sum of semisimple modules is semisimple.
A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero.
Endomorphism rings
A semisimple module M over a ring R can also be thought of as a ring homomorphism from R into the ring of abelian group endomorphisms of M. The image of this homomorphism is a semiprimitive ring, and every semiprimitive ring is isomorphic to such an image.
The endomorphism ring of a semisimple module is not only semiprimitive, but also von Neumann regular, .
Semisimple rings
A ring is said to be (left-)semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity.
A semisimple ring may be characterized in terms of homological algebra: namely, a ring R is semisimple if and only if any short exact sequence of left (or right) R-modules splits. That is, for a short exact sequence
there exists such that the composition is the identity. The map s is known as a section. From this it follows that
or in more exact te
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https://en.wikipedia.org/wiki/Convex%20conjugate
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In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality.
Definition
Let be a real topological vector space and let be the dual space to . Denote by
the canonical dual pairing, which is defined by
For a function taking values on the extended real number line, its is the function
whose value at is defined to be the supremum:
or, equivalently, in terms of the infimum:
This definition can be interpreted as an encoding of the convex hull of the function's epigraph in terms of its supporting hyperplanes.
Examples
For more examples, see .
The convex conjugate of an affine function is
The convex conjugate of a power function is
The convex conjugate of the absolute value function is
The convex conjugate of the exponential function is
The convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers.
Connection with expected shortfall (average value at risk)
See this article for example.
Let F denote a cumulative distribution function of a random variable X. Then (integrating by parts),
has the convex conjugate
Ordering
A particular interpretation has the transform
as this is a nondecreasing rearrangement of the initial function f; in particular, for f nondecreasing.
Properties
The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
Order reversing
Declare that if and only if for all Then convex-conjugation is order-reversing, which by definition means that if then
For a family of functions it follows from the fact that supremums may be interchanged that
and from the max–min inequality that
Biconjugate
The convex conjugate of a function is always lower semi-continuous. The biconjugate (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function with
For proper functions
if and only if is convex and lower semi-continuous, by the Fenchel–Moreau theorem.
Fenchel's inequality
For any function and its convex conjugate , Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every and
Furthermore, the equality holds only when .
The proof follows from the definition of convex conjugate:
Convexity
For two functions and and a number the convexity relation
holds. The operation is a convex mapping itself.
Infimal convolution
The infimal convolution (or
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https://en.wikipedia.org/wiki/Axiom%20of%20determinacy
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In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy.
Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. Steel and W. Hugh Woodin concluded a long line of research. Assuming the existence of some uncountable cardinal numbers analogous to , they proved the original conjecture of Mycielski and Steinhaus that AD is true in L(R).
Types of game that are determined
The axiom of determinacy refers to games of the following specific form:
Consider a subset A of the Baire space ωω of all infinite sequences of natural numbers. Two players, I and II, alternately pick natural numbers
n0, n1, n2, n3, ...
After infinitely many moves, a sequence is generated. Player I wins the game if and only if the sequence generated is an element of A. The axiom of determinacy is the statement that all such games are determined.
Not all games require the axiom of determinacy to prove them determined. If the set A is clopen, the game is essentially a finite game, and is therefore determined. Similarly, if A is a closed set, then the game is determined. It was shown in 1975 by Donald A. Martin that games whose winning set is a Borel set are determined. It follows from the existence of sufficiently large cardinals that all games with winning set a projective set are determined (see Projective determinacy), and that AD holds in L(R).
The axiom of determinacy implies that for every subspace X of the real numbers, the Banach–Mazur game BM(X) is determined (and therefore that every set of reals has the property of Baire).
Incompatibility of the axiom of determinacy with the axiom of choice
Under assumption of the axiom of choice, we present two separate constructions of counterexamples to the axiom of determinacy. It follows that the axiom of determinacy and the axiom of choice are incompatible.
Using well-ordering of the continuum
The set S1 of all first player strategies in an ω-game G has the same cardinality as the continuum. The same is true for the set S2 of all second player strategies. Let SG be the set of all possible sequences in G, and A be the subset of sequences of SG that make the first player win. Wi
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https://en.wikipedia.org/wiki/Bronis%C5%82aw%20Knaster
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Bronisław Knaster (22 May 1893 – 3 November 1980) was a Polish mathematician; from 1939 a university professor in Lwów and from 1945 in Wrocław.
He is known for his work in point-set topology and in particular for his discoveries in 1922 of the hereditarily indecomposable continuum or pseudo-arc and of the Knaster continuum, or buckethandle continuum. Together with his teacher Hugo Steinhaus and his colleague Stefan Banach, he also developed the last diminisher procedure for fair cake cutting.
Knaster received his Ph.D. degree from University of Warsaw in 1922. under the supervision
of Stefan Mazurkiewicz.
See also
Knaster–Tarski theorem
Knaster–Kuratowski fan
Knaster's condition
References
1893 births
1990 deaths
People from Warsaw Governorate
University of Paris alumni
Warsaw School of Mathematics
Topologists
Recipients of the State Award Badge (Poland)
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https://en.wikipedia.org/wiki/Rodrigues%27%20formula
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In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail.
Statement
Let be a sequence of orthogonal polynomials satisfying the orthogonality condition
where is a suitable weight function, is a constant depending on , and is the Kronecker delta. If the weight function satisfies the following differential equation (called Pearson's differential equation),
where is a polynomial with degree at most 1 and is a polynomial with degree at most 2 and, further, the limits
then it can be shown that satisfies a recurrence relation of the form,
for some constants . This relation is called Rodrigues' type formula, or just Rodrigues' formula.
The most known applications of Rodrigues' type formulas are the formulas for Legendre, Laguerre and Hermite polynomials:
Rodrigues stated his formula for Legendre polynomials :
Laguerre polynomials are usually denoted L0, L1, ..., and the Rodrigues formula can be written as
The Rodrigues formula for the Hermite polynomials can be written as
Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.
References
Orthogonal polynomials
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https://en.wikipedia.org/wiki/Olinde%20Rodrigues
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Benjamin Olinde Rodrigues (6 October 1795 – 17 December 1851), more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer. In mathematics Rodrigues is remembered for Rodrigues' rotation formula for vectors, the Rodrigues formula about series of orthogonal polynomials and the Euler–Rodrigues parameters.
Biography
Rodrigues was born into a well-to-do Sephardi Jewish family in Bordeaux. His family was of Portuguese-Jewish descent. He was awarded a doctorate in mathematics on 28 June 1815 by the University of Paris. His dissertation contains the result now called Rodrigues' formula.
After graduation, Rodrigues became a banker. A close associate of the Comte de Saint-Simon, Rodrigues continued, after Saint-Simon's death in 1825, to champion the older man's socialist ideals, a school of thought that came to be known as Saint-Simonianism. During this period, Rodrigues published writings on politics, social reform, and banking.
In 1840 he published a result on transformation groups, which applied Leonhard Euler's four squares formula, a precursor to the quaternions of William Rowan Hamilton, to the problem of representing rotations in space.
In 1846 Arthur Cayley acknowledged Euler's and Rodrigues' priority describing orthogonal transformations.
Rodrigues is credited as originating the idea of the artist as an avant-garde.
Publications
Mouvement de rotation d'un corps de révolution pesant, Paris, 1815
"Mémoire sur l'attraction des sphéroïdes", Correspondence Sur l'École Impériale Polytechnique, vol. 3, pp. 361–385, 1815
Théorie de la caisse hypothécaire, ou Examen du sort des emprunteurs, des porteurs d'obligations et des actionnaires de cet établissement, 1820
Appel : religion saint-simonienne, 1831
L'artiste, le savant et l'industriel: Dialogue, 1825
Réunion générale de la famille : séances des 19 et 21 novembre, 1831
Son premier écrit / Saint-Simon, 1832
Le disciple de Saint-Simon aux Saint-Simoniens et au public, 1832
Aux saint-simoniens, 13 février 1832 : bases de la loi morale proposées à l'acceptation des femmes, 1832
Olinde Rodrigues à M. Michel Chevalier, rédacteur du "Globe" : religion saint-simonienne, 1832
"Sur le nombre de manière de décomposer un Polygone en triangles au moyen de diagonales", Journal de Mathématiques Pures et Appliquées, vol. 3, pp. 547–548, 1838
"Sur le nombre de manière de d’effectuer un produit de n facteurs", Journal de Mathématiques Pures et Appliquées, vol. 3, p. 549, 1838
"Démonstration élémentaire et purement algébrique du développement d’un binome élevé à une puissance négative ou fractionnaire", Journal de Mathématiques Pures et Appliquées, vol. 3, pp. 550–551, 1838
"Note sur les inversions, ou dérangements produits dans les permutations", Journal de Mathématiques Pures et Appliquées, vol. 4, pp. 236–240, 1839
De l'organisation des banques à propos du projet de loi sur la Banque de France, 1840
"Des lois géométriques qui régissent les déplacements d'un
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https://en.wikipedia.org/wiki/Similarity%20measure
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In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. Though, in more broad terms, a similarity function may also satisfy metric axioms.
Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. In machine learning, common kernel functions such as the RBF kernel can be viewed as similarity functions.
Use of different similarity measure formulas
Different types of similarity measures exist for various types of objects, depending on the objects being compared. For each type of object there are various similarity measurement formulas.
Similarity between two data points
There are many various options available when it comes to finding similarity between two data points, some of which are a combination of other similarity methods. Some of the methods for similarity measures between two data points include Euclidean distance, Manhattan distance, Minkowski distance, and Chebyshev distance. The Euclidean distance formula is used to find the distance between two points on a plane, which is visualized in the image below. Manhattan distance is commonly used in GPS applications, as it can be used to find the shortest route between two addresses. When you generalize the Euclidean distance formula and Manhattan distance formula you are left with the Minkowski distance formula, which can be used in a wide variety of applications.
Euclidean distance
Manhattan distance
Minkowski distance
Chebyshev distance
Similarity between strings
For comparing strings, there are various measures of string similarity that can be used. Some of these methods include edit distance, Levenshtein distance, Hamming distance, and Jaro distance. The best-fit formula is dependent on the requirements of the application. For example, edit distance is frequently used for natural language processing applications and features, such as spell-checking. Jaro distance is commonly used in record linkage to compare first and last names to other sources.
Edit distance
Levenshtein distance
Lee distance
Hamming distance
Jaro distance
Similarity between two probability distributions
When comparing probability distributions the Mahalanobis distance formula, Bhattacharyya distance formulas, and the Hellinger distance formula are all very powerful and useful. The Mahalanobis distance formula is commonly used in statistical analysis. It measures the distance between two probability distributions that have different means and variances. This makes it useful for finding outliers across the datase
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https://en.wikipedia.org/wiki/Boost
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Boost, boosted or boosting may refer to:
Science, technology and mathematics
Boost, positive manifold pressure in turbocharged engines
Boost (C++ libraries), a set of free peer-reviewed portable C++ libraries
Boost (material), a material branded and used by Adidas in the midsoles of shoes.
Boost, a loose term for turbo or supercharger
Boost converter, an electrical circuit variation of a DC to DC converter, which increases (boosts) the voltage
Boosted fission weapon, a type of nuclear bomb that uses a small amount of fusion fuel to increase the rate, and thus yield, of a fission reaction
Boosting (machine learning), a supervised learning algorithm
Intel Turbo Boost, a technology that enables a processor to run above its base operating frequency
Jump start (vehicle), to start a vehicle
Lorentz boost, a type of Lorentz transformation
Arts, entertainment, and media
Fictional characters
Boost (Cars), a character from the Pixar franchise Cars
Boost (comics), a character from Marvel Comics
Films
Boost (film), a 2017 Canadian film directed by Darren Curtis
The Boost, a 1988 drama film directed by Harold Becker
Brands and enterprises
Boost (chocolate bar), a chocolate bar produced by Cadbury
Boost (drink), nutritional drinks brand made by Nestlé
Boost Energy, the Pay As You Go brand of OVO Energy
Boost!, American non-carbonated cola brand
Boost Drinks, British drinks company
Boost ETP, British independent boutique Exchange Traded Products provider
Boost Juice, a company in Australia
Boost Mobile (disambiguation), a brand of mobile phone services in Australia and the United States
Boosted (company), a defunct American manufacturer of electric skateboards
Lego Boost, a robotics Lego theme
Other uses
Boost, a slang term meaning steal or shoplift
Boosting (doping), a form of doping used by athletes with a spinal cord injury
See also
Booster (disambiguation)
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https://en.wikipedia.org/wiki/Vertex%20figure
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In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines across the connected faces, joining adjacent points around the face. When done, these lines form a complete circuit, i.e. a polygon, around the vertex. This polygon is the vertex figure.
More precise formal definitions can vary quite widely, according to circumstance. For example Coxeter (e.g. 1948, 1954) varies his definition as convenient for the current area of discussion. Most of the following definitions of a vertex figure apply equally well to infinite tilings or, by extension, to space-filling tessellation with polytope cells and other higher-dimensional polytopes.
As a flat slice
Make a slice through the corner of the polyhedron, cutting through all the edges connected to the vertex. The cut surface is the vertex figure. This is perhaps the most common approach, and the most easily understood. Different authors make the slice in different places. Wenninger (2003) cuts each edge a unit distance from the vertex, as does Coxeter (1948). For uniform polyhedra the Dorman Luke construction cuts each connected edge at its midpoint. Other authors make the cut through the vertex at the other end of each edge.
For an irregular polyhedron, cutting all edges incident to a given vertex at equal distances from the vertex may produce a figure that does not lie in a plane. A more general approach, valid for arbitrary convex polyhedra, is to make the cut along any plane which separates the given vertex from all the other vertices, but is otherwise arbitrary. This construction determines the combinatorial structure of the vertex figure, similar to a set of connected vertices (see below), but not its precise geometry; it may be generalized to convex polytopes in any dimension. However, for non-convex polyhedra, there may not exist a plane near the vertex that cuts all of the faces incident to the vertex.
As a spherical polygon
Cromwell (1999) forms the vertex figure by intersecting the polyhedron with a sphere centered at the vertex, small enough that it intersects only edges and faces incident to the vertex. This can be visualized as making a spherical cut or scoop, centered on the vertex. The cut surface or vertex figure is thus a spherical polygon marked on this sphere. One advantage of this method is that the shape of the vertex figure is fixed (up to the scale of the sphere), whereas the method of intersecting with a plane can produce different shapes depending on the angle of the plane. Additionally, this method works for non-convex polyhedra.
As the set of connected vertices
Many combinatorial and computational approaches (e.g. Skilling, 1975) treat a vertex figure as the ordered (or partially ordered) set of points of all the neighboring (connected via an edge) vertices to the given vertex.
Abstract de
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https://en.wikipedia.org/wiki/List%20of%20theorems%20called%20fundamental
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In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus. The names are mostly traditional, so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Some of these are classification theorems of objects which are mainly dealt with in the field. For instance, the fundamental theorem of curves describe classification of regular curves in space up to translation and rotation.
Likewise, the mathematical literature sometimes refers to the fundamental lemma of a field. The term lemma is conventionally used to denote a proven proposition which is used as a stepping stone to a larger result, rather than as a useful statement in-and-of itself.
Fundamental theorems of mathematical topics
Fundamental theorem of algebra
Fundamental theorem of algebraic K-theory
Fundamental theorem of arithmetic
Fundamental theorem of Boolean algebra
Fundamental theorem of calculus
Fundamental theorem of calculus for line integrals
Fundamental theorem of curves
Fundamental theorem of cyclic groups
Fundamental theorem of dynamical systems
Fundamental theorem of equivalence relations
Fundamental theorem of exterior calculus
Fundamental theorem of finitely generated abelian groups
Fundamental theorem of finitely generated modules over a principal ideal domain
Fundamental theorem of finite distributive lattices
Fundamental theorem of Galois theory
Fundamental theorem of geometric calculus
Fundamental theorem on homomorphisms
Fundamental theorem of ideal theory in number fields
Fundamental theorem of Lebesgue integral calculus
Fundamental theorem of linear algebra
Fundamental theorem of linear programming
Fundamental theorem of noncommutative algebra
Fundamental theorem of projective geometry
Fundamental theorem of random fields
Fundamental theorem of Riemannian geometry
Fundamental theorem of tessarine algebra
Fundamental theorem of symmetric polynomials
Fundamental theorem of topos theory
Fundamental theorem of ultraproducts
Fundamental theorem of vector analysis
Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of quadratic residues.
Applied or informally stated "fundamental theorems"
There are also a number of "fundamental theorems" that are not directly related to mathematics:
Fundamental theorem of arbitrage-free pricing
Fisher's fundamental theorem of natural selection
Fundamental theorems of welfare economics
Fundamental equations of thermodynamics
Fundamental theorem of poker
Holland's schema theorem, or the "fundamental theorem of genetic algorithms"
Glivenko–Cantelli theorem, or the "fundamental theorem of statistics"
Fundamental theorem of software engineering
Fundamental lemmata
Fundamental lemma of the calculus of variations
Fundamenta
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https://en.wikipedia.org/wiki/Analytic%20proof
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In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and which does not predominantly make use of algebraic or geometrical methods. The term was first used by Bernard Bolzano, who first provided a non-analytic proof of his intermediate value theorem and then, several years later provided a proof of the theorem that was free from intuitions concerning lines crossing each other at a point, and so he felt happy calling it analytic (Bolzano 1817).
Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter (Sebastik 2007). In proof theory, an analytic proof has come to mean a proof whose structure is simple in a special way, due to conditions on the kind of inferences that ensure none of them go beyond what is contained in the assumptions and what is demonstrated.
Structural proof theory
In proof theory, the notion of analytic proof provides the fundamental concept that brings out the similarities between a number of essentially distinct proof calculi, so defining the subfield of structural proof theory. There is no uncontroversial general definition of analytic proof, but for several proof calculi there is an accepted notion. For example:
In Gerhard Gentzen's natural deduction calculus the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule;
In Gentzen's sequent calculus the analytic proofs are those that do not use the cut rule.
However, it is possible to extend the inference rules of both calculi so that there are proofs that satisfy the condition but are not analytic. For example, a particularly tricky example of this is the analytic cut rule, used widely in the tableau method, which is a special case of the cut rule where the cut formula is a subformula of side formulae of the cut rule: a proof that contains an analytic cut is by virtue of that rule not analytic.
Furthermore, structural proof theories that are not analogous to Gentzen's theories have other notions of analytic proof. For example, the calculus of structures organises its inference rules into pairs, called the up fragment and the down fragment, and an analytic proof is one that only contains the down fragment.
See also
Proof-theoretic semantics
References
Bernard Bolzano (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In Abhandlungen der koniglichen bohmischen Gesellschaft der Wissenschaften Vol. V, pp.225-48.
Frank Pfenning (1984). Analytic and Non-analytic Proofs. In Proc. 7th International Conference on Automated Deduction.
Jan Šebestik (2007). Bolzano's Logic. Entry in the Stanford Encyclopedia of Philosophy.
Proof theory
Methods of proof
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https://en.wikipedia.org/wiki/Generalized%20singular%20value%20decomposition
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In linear algebra, the generalized singular value decomposition (GSVD) is the name of two different techniques based on the singular value decomposition (SVD). The two versions differ because one version decomposes two matrices (somewhat like the higher-order or tensor SVD) and the other version uses a set of constraints imposed on the left and right singular vectors of a single-matrix SVD.
First version: two-matrix decomposition
The generalized singular value decomposition (GSVD) is a matrix decomposition on a pair of matrices which generalizes the singular value decomposition. It was introduced by Van Loan in 1976 and later developed by Paige and Saunders, which is the version described here. In contrast to the SVD, the GSVD decomposes simultaneously a pair of matrices with the same number of columns. The SVD and the GSVD, as well as some other possible generalizations of the SVD, are extensively used in the study of the conditioning and regularization of linear systems with respect to quadratic semi-norms. In the following, let , or .
Definition
The generalized singular value decomposition of matrices and iswhere
is unitary,
is unitary,
is unitary,
is unitary,
is real diagonal with positive diagonal, and contains the non-zero singular values of in decreasing order,
,
is real non-negative block-diagonal, where with , , and ,
is real non-negative block-diagonal, where with , , and ,
,
,
,
.
We denote , , , and . While is diagonal, is not always diagonal, because of the leading rectangular zero matrix; instead is "bottom-right-diagonal".
Variations
There are many variations of the GSVD. These variations are related to the fact that it is always possible to multiply from the left by where is an arbitrary unitary matrix. We denote
, where is upper-triangular and invertible, and is unitary. Such matrices exist by RQ-decomposition.
. Then is invertible.
Here are some variations of the GSVD:
MATLAB (gsvd):
LAPACK (LA_GGSVD):
Simplified:
Generalized singular values
A generalized singular value of and is a pair such that
We have
By these properties we can show that the generalized singular values are exactly the pairs . We haveTherefore
This expression is zero exactly when and for some .
In, the generalized singular values are claimed to be those which solve . However, this claim only holds when , since otherwise the determinant is zero for every pair ; this can be seen by substituting above.
Generalized inverse
Define for any invertible matrix , for any zero matrix , and for any block-diagonal matrix. Then defineIt can be shown that as defined here is a generalized inverse of ; in particular a -inverse of . Since it does not in general satisfy , this is not the Moore–Penrose inverse; otherwise we could derive for any choice of matrices, which only holds for certain class of matrices.
Suppose , where and . This generalized inverse has the following properties:
Quotien
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https://en.wikipedia.org/wiki/Pendulum%20%28disambiguation%29
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A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity.
Pendulum may also refer to:
Devices
Pendulum (mathematics), the mathematical principles of a pendulum
Pendulum clock, a kind of clock that uses a pendulum to keep time
Pendulum car, an experimental tilting train
Foucault pendulum, a pendulum that demonstrates the Earth's rotation
Spherical pendulum
Spring pendulum
Conical pendulum
Centrifugal pendulum absorber, torsional vibration reduction by using a pendulum principle
For other types and uses of pendulums, see: :Category:Pendulums
Mackerras Pendulum, a model devised by Malcolm Mackerras to predict election outcomes
Pendulum (torture device), a device allegedly used by the Spanish Inquisition
Pendulum Instruments, a Swedish manufacturer of scientific instruments
Music
Pendulum (Australian band), an Australian electronic rock group formed in 2002
Pendulum (ambient band), an Australian house music group formed in 1994
Albums
Pendulum (Broadcast EP), and its title track
Pendulum (Creedence Clearwater Revival album)
Pendulum (Dave Liebman album), and its title track
Pendulum (Eberhard Weber album), and its title track
Pendulum (Lowen & Navarro album), and its title track
Pendulum (Tara Simmons EP), and its title track
The Pendulum, a comic book miniseries based on Insane Clown Posse's Dark Carnival universe, and associated songs and album
Songs
"Pendulum" (song), by FKA Twigs
"Pendulum", by Embodyment from the album The Narrow Scope of Things
"Pendulum", by Katy Perry from the album Witness
"Pendulum", by Phinehas from the album Thegodmachine
"Pendulums", by Sarah Harmer from the album All of Our Names
"Penduli Pendulum", by Bobbie Gentry from The Delta Sweete
"The Pendulum Song", song written by Al Hoffman and John Murray
Pendulum Music, a composition by Steve Reich involving microphones swinging above speakers like pendulums
Books and periodicals
Pendulum, by John Christopher, 1968
The Pendulum, by Annie S. Swan, 1972
Pendulum, by A. E. Van Vogt, 1978
Pendulum, by Adam Hamdy, 2017
The Pendulum, a student newspaper at Elon University
The Pendulum, a publication devoted to radiesthesia
Film
Pendulum (1969 film), an American neo noir film starring George Peppard
Pendulum, a 2001 film starring Rachel Hunter and James Russo
Pendulum (2014 film), an Indian neo noir
Pendulum (2023 film), a Malayalam-language fantasy thriller film
Other
Pendulum, a trade name for the preemergent herbicide pendimethalin
Pendulum, several types of Digimon virtual pets
Mackerras pendulum, an electoral tool for describing the swing required for a change of government
See also
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https://en.wikipedia.org/wiki/Weak%20equivalence
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In mathematics, weak equivalence may refer to:
Weak equivalence of categories
Weak equivalence (homotopy theory)
Weak equivalence (formal languages)
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https://en.wikipedia.org/wiki/Julio%20Garavito%20Armero
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Julio Garavito Armero (January 5, 1865 – March 11, 1920) was a Colombian astronomer.
Life
Born in Bogotá, he was a child prodigy in science and mathematics. He obtained his degrees as mathematician and civil engineer in the Universidad Nacional de Colombia (National university of Colombia). In 1892, he worked as the director of the Observatorio Astronómico Nacional (National Astronomical Observatory). His investigative works had been published in Los Anales de Ingeniería (The Annals of Engineering) since 1890, seven years before he took over editing the publication.
In his youth he studied at San Bartolomé high school, but in 1885 he had to interrupt his studies temporarily because of the civil wars which were affecting his home country. During the Thousand Days War, Garavito was part of a secret scientific society called El Círculo de los Nueve Puntos (the nine-point circle), where the condition for admission was to solve a problem about Euler's theorem. This group was active until Garavito's death. As an astronomer of the observatory, he did many useful scientific investigations such as calculating the latitude of Bogotá, studies about the comets which passed by the Earth between 1901 and 1910 (such as Comet Halley), and the 1916 solar eclipse (seen in the majority of Colombia).
But perhaps the most important were his studies about celestial mechanics, which finally turned into studies about lunar fluctuations and their influence on weather, floods, polar ice, and the Earthorbital acceleration (this was corroborated later). He worked also in other areas such as optics (this work was left unfinished at his death), and economics, by which he helped the country recover from the rough civil war. With this objective, he gave lectures and conferences in economics and the human factors which affected it, such as war or overpopulation.
He was later the director of the Chorographic Commission, created with the objectives of developing the Colombian railways and defining the frontier with Venezuela. He is believed to have questioned Albert Einstein's theory of relativity . He has been compared to two great scientists of the 19th century: José Celestino Mutis and Francisco José de Caldas.
Trivia
A crater on the Moon's far side is named Garavito after him. One of the most prestigious universities in Colombia is also named after him: Escuela Colombiana de Ingeniería (Colombian School of Engineering "Julio Garavito"), created in 1972, with a special emphasis in Applied Sciences and Engineering.
His face appears on the 20,000 colombian peso bill, with the Moon on the same side of the bill, and the Earth as viewed from the Moon's surface on the other side. Because of this, and the blue color of the bill, there is a local folk superstition that bringing offerings of blue candles and blue flowers to his grave in the Central Cemetery of Bogotá and praying there can help one to become wealthy.
References
Further reading
1865 births
1920 deaths
Col
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https://en.wikipedia.org/wiki/Gilbreath%27s%20conjecture
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Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.
Motivating arithmetic
Gilbreath observed a pattern while playing with the ordered sequence of prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
Computing the absolute value of the difference between term n + 1 and term n in this sequence yields the sequence
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, ...
If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the following five sequences in this list are
1, 0, 2, 2, 2, 2, 2, 2, 4, ...
1, 2, 0, 0, 0, 0, 0, 2, ...
1, 2, 0, 0, 0, 0, 2, ...
1, 2, 0, 0, 0, 2, ...
1, 2, 0, 0, 2, ...
What Gilbreath—and François Proth before him—noticed is that the first term in each series of differences appears to be 1.
The conjecture
Stating Gilbreath's observation formally is significantly easier to do after devising a notation for the sequences in the previous section. Toward this end, let denote the ordered sequence of prime numbers, and define each term in the sequence by
where is positive. Also, for each integer greater than 1, let the terms in be given by
Gilbreath's conjecture states that every term in the sequence for positive is equal to 1.
Verification and attempted proofs
, no valid proof of the conjecture has been published. As mentioned in the introduction, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. Andrew Odlyzko verified that is equal to 1 for in 1993, but the conjecture remains an open problem. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0s and 2s for the next n numbers. This implies that the next n rows begin with a 1.
Generalizations
In 1980, Martin Gardner published a conjecture by Hallard Croft that stated that the property of Gilbreath's conjecture (having a 1 in the first term of each difference sequence) should hold more generally for every sequence that begins with 2, subsequently contains only odd numbers, and has a sufficiently low bound on the gaps between consecutive elements in the sequence. This conjecture has also been repeated by later authors. However, it is false: for every initial subsequence of 2 and odd numbers, and every
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https://en.wikipedia.org/wiki/Closed%20convex%20function
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In mathematics, a function is said to be closed if for each , the sublevel set
is a closed set.
Equivalently, if the epigraph defined by
is closed, then the function is closed.
This definition is valid for any function, but most used for convex functions. A proper convex function is closed if and only if it is lower semi-continuous. For a convex function that is not proper, there is disagreement as to the definition of the closure of the function.
Properties
If is a continuous function and is closed, then is closed.
If is a continuous function and is open, then is closed if and only if it converges to along every sequence converging to a boundary point of .
A closed proper convex function f is the pointwise supremum of the collection of all affine functions h such that h ≤ f (called the affine minorants of f).
References
Convex analysis
Types of functions
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https://en.wikipedia.org/wiki/Bourbaki%E2%80%93Witt%20theorem
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In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and
such that
for all
then f has a fixed point. Such a function f is called inflationary or progressive.
Special case of a finite poset
If the poset X is finite then the statement of the theorem has a clear interpretation that leads to the proof. The sequence of successive iterates,
where x0 is any element of X, is monotone increasing. By the finiteness of X, it stabilizes:
for n sufficiently large.
It follows that x∞ is a fixed point of f.
Proof of the theorem
Pick some . Define a function K recursively on the ordinals as follows:
If is a limit ordinal, then by construction
is a chain in X. Define
This is now an increasing function from the ordinals into X. It cannot be strictly increasing, as if it were we would have an injective function from the ordinals into a set, violating Hartogs' lemma. Therefore the function must be eventually constant, so for some
that is,
So letting
we have our desired fixed point. Q.E.D.
Applications
The Bourbaki–Witt theorem has various important applications. One of the most common is in the proof that the axiom of choice implies Zorn's lemma. We first prove it for the case where X is chain complete and has no maximal element. Let g be a choice function on
Define a function
by
This is allowed as, by assumption, the set is non-empty. Then f(x) > x, so f is an inflationary function with no fixed point, contradicting the theorem.
This special case of Zorn's lemma is then used to prove the Hausdorff maximality principle, that every poset has a maximal chain, which is easily seen to be equivalent to Zorn's Lemma.
Bourbaki–Witt has other applications. In particular in computer science, it is used in the theory of computable functions.
It is also used to define recursive data types, e.g. linked lists, in domain theory.
References
Order theory
Fixed-point theorems
Theorems in the foundations of mathematics
Articles containing proofs
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https://en.wikipedia.org/wiki/Fixed-point%20theorem
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In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms.
In mathematical analysis
The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.
By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
For example, the cosine function is continuous in [−1,1] and maps it into [−1, 1], and thus must have a fixed point. This is clear when examining a sketched graph of the cosine function; the fixed point occurs where the cosine curve y = cos(x) intersects the line y = x. Numerically, the fixed point (known as the Dottie number) is approximately x = 0.73908513321516 (thus x = cos(x) for this value of x).
The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology is notable because it gives, in some sense, a way to count fixed points.
There are a number of generalisations to Banach fixed-point theorem and further; these are applied in PDE theory. See fixed-point theorems in infinite-dimensional spaces.
The collage theorem in fractal compression proves that, for many images, there exists a relatively small description of a function that, when iteratively applied to any starting image, rapidly converges on the desired image.
In algebra and discrete mathematics
The Knaster–Tarski theorem states that any order-preserving function on a complete lattice has a fixed point, and indeed a smallest fixed point. See also Bourbaki–Witt theorem.
The theorem has applications in abstract interpretation, a form of static program analysis.
A common theme in lambda calculus is to find fixed points of given lambda expressions. Every lambda expression has a fixed point, and a fixed-point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression. An important fixed-point combinator is the Y combinator used to give recursive definitions.
In denotational semantics of programming languages, a special case of the Knaster–Tarski theorem is used to establish the semantics of recursive definitions. While the fixed-point theorem is applied to the "same" function (from a logical point of view), the development of the theory is quite different.
The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem. These results are not equivalent theorems; the Knaster–Tarski theorem is a much stronger result than what is used in denotational semantics. However, in light of the Church–Turing thesis their intuitive meaning is the same: a recur
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https://en.wikipedia.org/wiki/Regge%20calculus
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In general relativity, Regge calculus is a formalism for producing simplicial approximations of spacetimes that are solutions to the Einstein field equation. The calculus was introduced by the Italian theoretician Tullio Regge in 1961.
Overview
The starting point for Regge's work is the fact that every four dimensional time orientable Lorentzian manifold admits a triangulation into simplices. Furthermore, the spacetime curvature can be expressed in terms of deficit angles associated with 2-faces where arrangements of 4-simplices meet. These 2-faces play the same role as the vertices where arrangements of triangles meet in a triangulation of a 2-manifold, which is easier to visualize. Here a vertex with a positive angular deficit represents a concentration of positive Gaussian curvature, whereas a vertex with a negative angular deficit represents a concentration of negative Gaussian curvature.
The deficit angles can be computed directly from the various edge lengths in the triangulation, which is equivalent to saying that the Riemann curvature tensor can be computed from the metric tensor of a Lorentzian manifold. Regge showed that the vacuum field equations can be reformulated as a restriction on these deficit angles. He then showed how this can be applied to evolve an initial spacelike hyperslice according to the vacuum field equation.
The result is that, starting with a triangulation of some spacelike hyperslice (which must itself satisfy a certain constraint equation), one can eventually obtain a simplicial approximation to a vacuum solution. This can be applied to difficult problems in numerical relativity such as simulating the collision of two black holes.
The elegant idea behind Regge calculus has motivated the construction of further generalizations of this idea. In particular, Regge calculus has been adapted to study quantum gravity.
See also
Numerical relativity
Quantum gravity
Euclidean quantum gravity
Piecewise linear manifold
Euclidean simplex
Path integral formulation
Lattice gauge theory
Wheeler–DeWitt equation
Mathematics of general relativity
Causal dynamical triangulation
Ricci calculus
Notes
References
See chapter 42.
Chapters 4 and 6.
Available (subscribers only) at "Classical and Quantum Gravity".
Available at .
eprint
Available at "Living Reviews of Relativity". See section 3.
Available (subscribers only) at "Classical and Quantum Gravity".
External links
Regge calculus on ScienceWorld
Mathematical methods in general relativity
Simplicial sets
Numerical analysis
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https://en.wikipedia.org/wiki/Coefficient%20of%20variation
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In probability theory and statistics, the coefficient of variation (CV), also known as Normalized Root-Mean-Square Deviation (NRMSD), Percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation to the mean (or its absolute value, , and often expressed as a percentage ("%RSD"). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R, by economists and investors in economic models, and in psychology/neuroscience.
Definition
The coefficient of variation (CV) is defined as the ratio of the standard deviation to the mean ,
It shows the extent of variability in relation to the mean of the population.
The coefficient of variation should be computed only for data measured on scales that have a meaningful zero (ratio scale) and hence allow relative comparison of two measurements (i.e., division of one measurement by the other). The coefficient of variation may not have any meaning for data on an interval scale. For example, most temperature scales (e.g., Celsius, Fahrenheit etc.) are interval scales with arbitrary zeros, so the computed coefficient of variation would be different depending on the scale used. On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale. In plain language, it is meaningful to say that 20 Kelvin is twice as hot as 10 Kelvin, but only in this scale with a true absolute zero. While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale. Only the Kelvin scale can be used to compute a valid coefficient of variability.
Measurements that are log-normally distributed exhibit stationary CV; in contrast, SD varies depending upon the expected value of measurements.
A more robust possibility is the quartile coefficient of dispersion, half the interquartile range divided by the average of the quartiles (the midhinge), .
In most cases, a CV is computed for a single independent variable (e.g., a single factory product) with numerous, repeated measures of a dependent variable (e.g., error in the production process). However, data that are linear or even logarithmically non-linear and include a continuous range for the independent variable with sparse measurements across each value (e.g., scatter-plot) may be amenable to single CV calculation using a maximum-likelihood estimation approach.
Examples
In the examples below, we will take the values given as randomly chosen from a larger population of values.
The data set [100, 100, 100] has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0
The data set [90, 100,
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https://en.wikipedia.org/wiki/Chain-complete%20partial%20order
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In mathematics, specifically order theory, a partially ordered set is chain-complete if every chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains.
Examples
Every complete lattice is chain-complete. Unlike complete lattices, chain-complete posets are relatively common. Examples include:
The set of all linearly independent subsets of a vector space V, ordered by inclusion.
The set of all partial functions on a set, ordered by restriction.
The set of all partial choice functions on a collection of non-empty sets, ordered by restriction.
The set of all prime ideals of a ring, ordered by inclusion.
The set of all consistent theories of a first-order language.
Properties
A poset is chain-complete if and only if it is a pointed dcpo. However, this equivalence requires the axiom of choice.
Zorn's lemma states that, if a poset has an upper bound for every chain, then it has a maximal element. Thus, it applies to chain-complete posets, but is more general in that it allows chains that have upper bounds but do not have least upper bounds.
Chain-complete posets also obey the Bourbaki–Witt theorem, a fixed point theorem stating that, if f is a function from a chain complete poset to itself with the property that f(x) ≥ x for all x, then f has a fixed point. This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.
By analogy with the Dedekind–MacNeille completion of a partially ordered set, every partially ordered set can be extended uniquely to a minimal chain-complete poset.
See also
Completeness (order theory)
References
Order theory
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https://en.wikipedia.org/wiki/Multidimensional%20analysis
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In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set.
Higher dimensions
In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets.
Software
Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.
See also
MultiDimensional eXpressions (MDX)
Multidimensional panel data
Multivariate statistics
Dimension (data warehouse)
Dimension tables
Data cube
References
Dimension reduction
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https://en.wikipedia.org/wiki/LAPACK
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LAPACK ("Linear Algebra Package") is a standard software library for numerical linear algebra. It provides routines for solving systems of linear equations and linear least squares, eigenvalue problems, and singular value decomposition. It also includes routines to implement the associated matrix factorizations such as LU, QR, Cholesky and Schur decomposition. LAPACK was originally written in FORTRAN 77, but moved to Fortran 90 in version 3.2 (2008). The routines handle both real and complex matrices in both single and double precision. LAPACK relies on an underlying BLAS implementation to provide efficient and portable computational building blocks for its routines.
LAPACK was designed as the successor to the linear equations and linear least-squares routines of LINPACK and the eigenvalue routines of EISPACK. LINPACK, written in the 1970s and 1980s, was designed to run on the then-modern vector computers with shared memory. LAPACK, in contrast, was designed to effectively exploit the caches on modern cache-based architectures and the instruction-level parallelism of modern superscalar processors, and thus can run orders of magnitude faster than LINPACK on such machines, given a well-tuned BLAS implementation. LAPACK has also been extended to run on distributed memory systems in later packages such as ScaLAPACK and PLAPACK.
Netlib LAPACK is licensed under a three-clause BSD style license, a permissive free software license with few restrictions.
Naming scheme
Subroutines in LAPACK have a naming convention which makes the identifiers very compact. This was necessary as the first Fortran standards only supported identifiers up to six characters long, so the names had to be shortened to fit into this limit.
A LAPACK subroutine name is in the form pmmaaa, where:
p is a one-letter code denoting the type of numerical constants used. S, D stand for real floating-point arithmetic respectively in single and double precision, while C and Z stand for complex arithmetic with respectively single and double precision. The newer version, LAPACK95, uses generic subroutines in order to overcome the need to explicitly specify the data type.
mm is a two-letter code denoting the kind of matrix expected by the algorithm. The codes for the different kind of matrices are reported below; the actual data are stored in a different format depending on the specific kind; e.g., when the code DI is given, the subroutine expects a vector of length n containing the elements on the diagonal, while when the code GE is given, the subroutine expects an array containing the entries of the matrix.
aaa is a one- to three-letter code describing the actual algorithm implemented in the subroutine, e.g. SV denotes a subroutine to solve linear system, while R denotes a rank-1 update.
For example, the subroutine to solve a linear system with a general (non-structured) matrix using real double-precision arithmetic is called DGESV.
Use with other programming languages and librarie
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https://en.wikipedia.org/wiki/Heilbronn%20triangle%20problem
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In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed in a given area, the smallest triangle area will be at most inversely proportional to the square of the number of points. His conjecture was proven false, but the asymptotic growth rate of the minimum triangle area remains unknown.
Definition
The Heilbronn triangle problem concerns the placement of points within a shape in the plane, such as the unit square or the unit disk, for a given Each triple of points form the three vertices of a triangle, and among these triangles, the problem concerns the smallest triangle, as measured by area. Different placements of points will have different smallest triangles, and the problem asks: how should points be placed to maximize the area of the smallest
More formally, the shape may be assumed to be a compact set in the plane, meaning that it stays within a bounded distance from the origin and that points are allowed to be placed on its boundary. In most work on this problem, is additionally a convex set of nonzero area. When three of the placed points lie on a line, they are considered as forming a degenerate triangle whose area is defined to be zero, so placements that maximize the smallest triangle will not have collinear triples of points. The assumption that the shape is compact implies that there exists an optimal placement of points, rather than only a sequence of placements approaching optimality. The number may be defined as the area of the smallest triangle in this optimal An example is shown in the figure, with six points in a unit square. These six points form different triangles, four of which are shaded in the figure. Six of these 20 triangles, with two of the shaded shapes, have area 1/8; the remaining 14 triangles have larger areas. This is the optimal placement of six points in a unit square: all other placements form at least one triangle with area 1/8 or smaller. Therefore,
Although researchers have studied the value of for specific shapes and specific small numbers of points, Heilbronn was concerned instead about its asymptotic behavior: if the shape is held fixed, but varies, how does the area of the smallest triangle vary That is, Heilbronn's question concerns the growth rate as a function For any two shapes the numbers and differ only by a constant factor, as any placement of points within can be scaled by an affine transformation to fit changing the minimum triangle area only by a constant. Therefore, in bounds on the growth rate of that omit the constant of proportionality of that growth, the choice of is irrelevant and the subscript may be
Heilbronn's conjecture and its disproof
Heilbronn conjectured prior to 1951 that the minimum triangle area always shrinks rapidly as a function —more specifically, inversely propo
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https://en.wikipedia.org/wiki/Hartley%20Rogers%20Jr.
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Hartley Rogers Jr. (July 6, 1926 – July 17, 2015) was an American mathematician who worked in computability theory, and was a professor in the Mathematics Department of the Massachusetts Institute of Technology.
Biography
Born in 1926 in Buffalo, New York, he studied under Alonzo Church at Princeton, and received his Ph.D. there in 1952. He served on the MIT faculty from 1956 until his death, July 17, 2015. He is survived by his wife, Dr. Adrianne E. Rogers, by his three children, Hartley R. Rogers, Campbell D.K. Rogers, and Caroline R. Broderick, and by his 10 grandchildren.
At MIT he had been involved in many scholarly extracurricular activities, including running SPUR (Summer Program in Undergraduate Research) for MIT undergraduates, overseeing the mathematics section of RSI (Research Science Institute) for advanced high school students, and coaching the MIT Putnam exam team for nearly two decades starting in 1990, including the years 2003 and 2004 when MIT won for the first time since 1979. He also ran a seminar called 18.S34: Mathematical Problem Solving for MIT freshmen.
Rogers is known within the MIT undergraduate community also for having developed a multivariable calculus course (18.022: Multivariable Calculus with Theory) with the explicit goal of providing a firm mathematical foundation for the study of physics. In 2005 he announced that he would no longer be teaching the course himself, but it is likely that it will continue to be taught in a similar manner in the future. He is remembered for his witty mathematical comments during lectures as well as his tradition of awarding Leibniz Cookies and Fig Newtons to top performers in his class.
An avid oarsman, he was most recently a member of the Cambridge Boat Club on the Charles River, Cambridge, Massachusetts. In his spare time, he served for many years as the Chaplain for the World Indoor Rowing Championships as part of the C.R.A.S.H.-B. Sprints Board of Directors.
Mathematical work
Rogers worked in mathematical logic, particularly recursion theory, and wrote the classic text Theory of Recursive Functions and Effective Computability. The Rogers equivalence theorem is named after him.
His doctoral students included Patrick Fischer, Louis Hodes, Carl Jockusch, Andrew Kahr, David Luckham, Rohit Parikh, David Park, and John Stillwell.
Rogers won the Lester R. Ford Award in 1965 for his expository article Information Theory.
Selected works
Hartley Rogers Jr., The Theory of Recursive Functions and Effective Computability, MIT Press, (paperback), (textbook)
References
External links
1926 births
2015 deaths
20th-century American mathematicians
21st-century American mathematicians
American logicians
Princeton University alumni
Massachusetts Institute of Technology School of Science faculty
Logicians
People from Winchester, Massachusetts
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https://en.wikipedia.org/wiki/Real%20projective%20space
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In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties
Construction
As with all projective spaces, RPn is formed by taking the quotient of under the equivalence relation for all real numbers . For all x in one can always find a λ such that λx has norm 1. There are precisely two such λ differing by sign.
Thus RPn can also be formed by identifying antipodal points of the unit n-sphere, Sn, in Rn+1.
One can further restrict to the upper hemisphere of Sn and merely identify antipodal points on the bounding equator. This shows that RPn is also equivalent to the closed n-dimensional disk, Dn, with antipodal points on the boundary, , identified.
Low-dimensional examples
RP1 is called the real projective line, which is topologically equivalent to a circle.
RP2 is called the real projective plane. This space cannot be embedded in R3. It can however be embedded in R4 and can be immersed in R3 (see here). The questions of embeddability and immersibility for projective n-space have been well-studied.
RP3 is (diffeomorphic to) SO(3), hence admits a group structure; the covering map S3 → RP3 is a map of groups Spin(3) → SO(3), where Spin(3) is a Lie group that is the universal cover of SO(3).
Topology
The antipodal map on the n-sphere (the map sending x to −x) generates a Z2 group action on Sn. As mentioned above, the orbit space for this action is RPn. This action is actually a covering space action giving Sn as a double cover of RPn. Since Sn is simply connected for n ≥ 2, it also serves as the universal cover in these cases. It follows that the fundamental group of RPn is Z2 when n > 1. (When n = 1 the fundamental group is Z due to the homeomorphism with S1). A generator for the fundamental group is the closed curve obtained by projecting any curve connecting antipodal points in Sn down to RPn.
The projective n-space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the n-sphere, a simply connected space. It is a double cover. The antipode map on Rp has sign , so it is orientation-preserving if and only if p is even. The orientation character is thus: the non-trivial loop in acts as on orientation, so RPn is orientable if and only if is even, i.e., n is odd.
The projective n-space is in fact diffeomorphic to the submanifold of R(n+1)2 consisting of all symmetric matrices of trace 1 that are also idempotent linear transformations.
Geometry of real projective spaces
Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry).
For the standard round metric, this has sectional curvature identically 1.
In the standard round me
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https://en.wikipedia.org/wiki/Upper%20topology
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In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton is the order section for each If is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets. However, not all up-sets must necessarily be open sets. The lower topology induced by the preorder is defined similarly in terms of the down-sets. The preorder inducing the upper topology is its specialization preorder, but the specialization preorder of the lower topology is opposite to the inducing preorder.
The real upper topology is most naturally defined on the upper-extended real line by the system of open sets. Similarly, the real lower topology is naturally defined on the lower real line A real function on a topological space is upper semi-continuous if and only if it is lower-continuous, i.e. is continuous with respect to the lower topology on the lower-extended line Similarly, a function into the upper real line is lower semi-continuous if and only if it is upper-continuous, i.e. is continuous with respect to the upper topology on
See also
References
General topology
Order theory
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https://en.wikipedia.org/wiki/Monadic%20Boolean%20algebra
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In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature
of type 〈2,2,1,0,0,1〉,
where 〈A, ·, +, ', 0, 1〉 is a Boolean algebra.
The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃):
is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as .
A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that . (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra A has signature , with 〈A, ·, +, ', 0, 1〉 a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities:
.
is the universal closure of x.
Discussion
Monadic Boolean algebras have an important connection to topology. If ∀ is interpreted as the interior operator of topology, (1)–(3) above plus the axiom ∀(∀x) = ∀x make up the axioms for an interior algebra. But ∀(∀x) = ∀x can be proved from (1)–(4). Moreover, an alternative axiomatization of monadic Boolean algebras consists of the (reinterpreted) axioms for an interior algebra, plus ∀(∀x)' = (∀x)' (Halmos 1962: 22). Hence monadic Boolean algebras are the semisimple interior/closure algebras such that:
The universal (dually, existential) quantifier interprets the interior (closure) operator;
All open (or closed) elements are also clopen.
A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(x∨∀y) = ∀x∨∀y (Halmos 1962: 21). This axiomatization obscures the connection to topology.
Monadic Boolean algebras form a variety. They are to monadic predicate logic what Boolean algebras are to propositional logic, and what polyadic algebras are to first-order logic. Paul Halmos discovered monadic Boolean algebras while working on polyadic algebras; Halmos (1962) reprints the relevant papers. Halmos and Givant (1998) includes an undergraduate treatment of monadic Boolean algebra.
Monadic Boolean algebras also have an important connection to modal logic. The modal logic S5, viewed as a theory in S4, is a model of monadic Boolean algebras in the same way that S4 is a model of interior algebra. Likewise, monadic Boolean algebras supply the algebraic semantics for S5. Hence S5-algebra is a synonym for monadic Boolean algebra.
See also
Clopen set
Cylindric algebra
Interior algebra
Kuratowski closure axioms
Łukasiewicz–Moisil algebra
Modal logic
Monadic logic
References
Paul Halmos, 1962. Algebraic Logic. New York: Chelsea.
------ and Steven Givant, 1998. Logic as Algebra. Mathematical Association of America.
Algebraic logic
Boolean algebra
Closure operators
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https://en.wikipedia.org/wiki/Preference%20relation
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The term preference relation is used to refer to orderings that describe human preferences for one thing over an other.
In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One specific variation of weak ordering, a total preorder (= a connected, reflexive and transitive relation), is also sometimes called a preference relation.
In computer science, machine learning algorithms are used to infer preferences, and the binary representation of the output of a preference learning algorithm is called a preference relation, regardless of whether it fits the weak ordering or semiorder mathematical models.
Preference relations are also widely used in economics; see preference (economics).
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https://en.wikipedia.org/wiki/3-manifold
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In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
Introduction
Definition
A topological space is a 3-manifold if it is a second-countable Hausdorff space and if every point in has a neighbourhood that is homeomorphic to Euclidean 3-space.
Mathematical theory of 3-manifolds
The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.
Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. This special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology.
A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case.
Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry. Using a geometry in addition to special surfaces is often fruitful.
The fundamental groups of 3-manifolds strongly reflect the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group theory and topological methods.
Invariants describing 3-manifolds
3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let be a 3-manifold and be its fundamental group, then a lot of information can be derived from them. For example, using Poincare duality and the Hurewicz theorem, we have the following homology groups:
where the last two groups are isomorphic to the group homology and cohomology of , respectively; that is,From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov tower there is a canonical m
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https://en.wikipedia.org/wiki/Field%20of%20sets
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In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.
Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over " is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
Definitions
A field of sets is a pair consisting of a set and a family of subsets of called an algebra over that has the following properties:
:
as an element:
Assuming that (1) holds, this condition (2) is equivalent to:
Any/all of the following equivalent conditions hold:
:
:
:
:
In other words, forms a subalgebra of the power set Boolean algebra of (with the same identity element ).
Many authors refer to itself as a field of sets.
Elements of are called points while elements of are called complexes and are said to be the admissible sets of
A field of sets is called a σ-field of sets and the algebra is called a σ-algebra if the following additional condition (4) is satisfied:
Any/both of the following equivalent conditions hold:
:
for all
:
for all
Fields of sets in the representation theory of Boolean algebras
Stone representation
For an arbitrary set its power set (or, somewhat pedantically, the pair of this set and its power set) is a field of sets. If is finite (namely, -element), then is finite (namely, -element). It appears that every finite field of sets (it means, with finite, while may be infinite) admits a representation of the form with finite ; it means a function that establishes a one-to-one correspondence between and via inverse image: where and (that is, ). One notable consequence: the number of complexes, if finite, is always of the form
To this end one chooses to be the set of all atoms of the given field of sets, and defines by whenever for a point and a complex that is an atom; the latter means that a nonempty subset of different from cannot be a complex.
In other words: the atoms are a partition of ; is the corresponding quotient set; and is the corresponding canonical surjection.
Similarly, every finite Boolean algebra can be represented as a power set – the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power set representation can be constructed more generally for any complete atomic Boolean algebra.
In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole pow
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https://en.wikipedia.org/wiki/Dirac%20measure
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In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
Definition
A Dirac measure is a measure on a set (with any -algebra of subsets of ) defined for a given and any (measurable) set by
where is the indicator function of .
The Dirac measure is a probability measure, and in terms of probability it represents the almost sure outcome in the sample space . We can also say that the measure is a single atom at ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. The Dirac measures are the extreme points of the convex set of probability measures on .
The name is a back-formation from the Dirac delta function; considered as a Schwartz distribution, for example on the real line, measures can be taken to be a special kind of distribution. The identity
which, in the form
is often taken to be part of the definition of the "delta function", holds as a theorem of Lebesgue integration.
Properties of the Dirac measure
Let denote the Dirac measure centred on some fixed point in some measurable space .
is a probability measure, and hence a finite measure.
Suppose that is a topological space and that is at least as fine as the Borel -algebra on .
is a strictly positive measure if and only if the topology is such that lies within every non-empty open set, e.g. in the case of the trivial topology .
Since is probability measure, it is also a locally finite measure.
If is a Hausdorff topological space with its Borel -algebra, then satisfies the condition to be an inner regular measure, since singleton sets such as are always compact. Hence, is also a Radon measure.
Assuming that the topology is fine enough that is closed, which is the case in most applications, the support of is . (Otherwise, is the closure of in .) Furthermore, is the only probability measure whose support is .
If is -dimensional Euclidean space with its usual -algebra and -dimensional Lebesgue measure , then is a singular measure with respect to : simply decompose as and and observe that .
The Dirac measure is a sigma-finite measure.
Generalizations
A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. More formally, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable set.
See also
Discrete measure
Dirac delta function
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Cuthill%E2%80%93McKee%20algorithm
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In numerical linear algebra, the Cuthill–McKee algorithm (CM), named after Elizabeth Cuthill and James McKee, is an algorithm to permute a sparse matrix that has a symmetric sparsity pattern into a band matrix form with a small bandwidth. The reverse Cuthill–McKee algorithm (RCM) due to Alan George and Joseph Liu is the same algorithm but with the resulting index numbers reversed. In practice this generally results in less fill-in than the CM ordering when Gaussian elimination is applied.
The Cuthill McKee algorithm is a variant of the standard breadth-first search
algorithm used in graph algorithms. It starts with a peripheral node and then
generates levels for until all nodes
are exhausted. The set is created from set
by listing all vertices adjacent to all nodes in . These
nodes are ordered according to predecessors and degree.
Algorithm
Given a symmetric matrix we visualize the matrix as the adjacency matrix of a graph. The Cuthill–McKee algorithm is then a relabeling of the vertices of the graph to reduce the bandwidth of the adjacency matrix.
The algorithm produces an ordered n-tuple of vertices which is the new order of the vertices.
First we choose a peripheral vertex (the vertex with the lowest degree) and set .
Then for we iterate the following steps while
Construct the adjacency set of (with the i-th component of ) and exclude the vertices we already have in
Sort ascending by minimum predecessor (the already-visited neighbor with the earliest position in R), and as a tiebreak ascending by vertex degree.
Append to the Result set .
In other words, number the vertices according to a particular level structure (computed by breadth-first search) where the vertices in each level are visited in order of their predecessor's numbering from lowest to highest. Where the predecessors are the same, vertices are distinguished by degree (again ordered from lowest to highest).
See also
Graph bandwidth
Sparse matrix
References
Cuthill–McKee documentation for the Boost C++ Libraries.
A detailed description of the Cuthill–McKee algorithm.
symrcm MATLAB's implementation of RCM.
reverse_cuthill_mckee RCM routine from SciPy written in Cython.
Matrix theory
Graph algorithms
Sparse matrices
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https://en.wikipedia.org/wiki/Orust
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Orust () is an island in western Sweden, and Sweden's third largest island. In 2014 Statistics Sweden declared it to instead be the fourth largest island, under a definition which adds artificial canals to the possible bodies of water surrounding an island. It has been noted that under this definition, all of Götaland would be the country's largest island, rendering Orust instead the fifth largest. The largest town on Orust is Henån, the municipal capital, where approximately 1,800 inhabitants live. Other communities, many of which are fishing villages, include Ellös, Edshultshall, Hälleviksstrand, Mollösund, Morlanda, Stocken, Svanesund, Svanvik and Varekil. Orust is home to approximately 15,160 inhabitants in the winter and many more in the summer. Its main industry is the shipyards, the two largest being Najadvarvet and Hallberg-Rassy.
International relations
Twin towns – Sister cities
Orust is twinned with:
Aalborg, Denmark
See also
Orust Municipality
Orust Eastern Hundred
Orust Western Hundred
Haga dolmen
References
External links
Orust Municipality – Official site
Islands of Västra Götaland County
Islands on the Swedish West Coast
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https://en.wikipedia.org/wiki/Lie%E2%80%93Kolchin%20theorem
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In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras.
It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and
a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that
That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation. This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all .
It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. In fact, this is another way to state the Lie–Kolchin theorem.
The result for Lie algebras was proved by and for algebraic groups was proved by .
The Borel fixed point theorem generalizes the Lie–Kolchin theorem.
Triangularization
Sometimes the theorem is also referred to as the Lie–Kolchin triangularization theorem because by induction it implies that with respect to a suitable basis of V the image has a triangular shape; in other words, the image group is conjugate in GL(n,K) (where n = dim V) to a subgroup of the group T of upper triangular matrices, the standard Borel subgroup of GL(n,K): the image is simultaneously triangularizable.
The theorem applies in particular to a Borel subgroup of a semisimple linear algebraic group G.
Counter-example
If the field K is not algebraically closed, the theorem can fail. The standard unit circle, viewed as the set of complex numbers of absolute value one is a one-dimensional commutative (and therefore solvable) linear algebraic group over the real numbers which has a two-dimensional representation into the special orthogonal group SO(2) without an invariant (real) line. Here the image of is the orthogonal matrix
References
Lie algebras
Representation theory of algebraic groups
Theorems in representation theory
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https://en.wikipedia.org/wiki/Gerhard%20Frey
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Gerhard Frey (; born 1 June 1944) is a German mathematician, known for his work in number theory. Following an original idea of Hellegouarch, he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to Wiles's proof of Fermat's Last Theorem.
Education and career
He studied mathematics and physics at the University of Tübingen, graduating in 1967. He continued his postgraduate studies at Heidelberg University, where he received his PhD in 1970, and his Habilitation in 1973. He was assistant professor at Heidelberg University from 1969–1973, professor at the University of Erlangen (1973–1975) and at Saarland University (1975–1990). Until 2009, he held a chair for number theory at the Institute for Experimental Mathematics at the University of Duisburg-Essen, campus Essen.
Frey was a visiting scientist at several universities and research institutions, including the Ohio State University, Harvard University, the University of California, Berkeley, the Mathematical Sciences Research Institute (MSRI), the Institute for Advanced Studies at Hebrew University of Jerusalem, and the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro.
Frey was also the co-editor of the journal .
Research contributions
His research areas are number theory and diophantine geometry, as well as applications to coding theory and cryptography.
In 1985, Frey pointed out a connection between Fermat's Last Theorem and the Taniyama-Shimura Conjecture, and this connection was made precise shortly thereafter by Jean-Pierre Serre who formulated a conjecture and showed that Taniyama-Shimura+ implies Fermat. Soon after, Kenneth Ribet proved enough of conjecture to deduce that the Taniyama-Shimura Conjecture implies Fermat's Last Theorem. This approach provided a framework for the subsequent successful attack on Fermat's Last Theorem by Andrew Wiles in the 1990s.
In 1998, Frey proposed the idea of Weil descent attack for elliptic curves over finite fields with composite degree. As a result of this attack, cryptographers lost their interest in these curves.
Awards and honors
Frey was awarded the Gauss medal of the Braunschweigische Wissenschaftliche Gesellschaft in 1996 for his work on Fermat's Last Theorem. Since 1998, he has been a member of the Göttingen Academy of Sciences.
In 2006, he received the Certicom ECC Visionary Award for his contributions to elliptic-curve cryptography.
See also
Fermat's Last Theorem
Elliptic curves
Trace zero cryptography
References
External links
Gerhard Frey's webpage at University Duisburg-Essen
20th-century German mathematicians
21st-century German mathematicians
Living people
1944 births
University of Tübingen alumni
Heidelberg University alumni
Academic staff of Heidelberg University
Ohio State University faculty
University of California, Berkeley faculty
Harvard University staff
Modern cryptographers
Number theorists
P
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https://en.wikipedia.org/wiki/T%C3%A4by
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Täby () was previously a trimunicipal locality, with 66,292 inhabitants in 2013. However, as from 2016, Statistics Sweden has amalgamated this locality with the Stockholm urban area. It is the seat of Täby Municipality in Stockholm County, Sweden. It was also partly located in Danderyd Municipality (the Enebyberg area) and a very small part in Sollentuna Municipality.
Täby kyrkby in the northern part of Täby Municipality forms on the other hand part of the Vallentuna urban area.
References
Municipal seats of Stockholm County
Swedish municipal seats
Populated places in Danderyd Municipality
Populated places in Sollentuna Municipality
Populated places in Täby Municipality
Uppland
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https://en.wikipedia.org/wiki/Variety%20%28universal%20algebra%29
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In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called finitary algebraic categories.
A covariety is the class of all coalgebraic structures of a given signature.
Terminology
A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common.
The term "variety of algebras" refers to algebras in the general sense of universal algebra; there is also a more specific sense of algebra, namely as algebra over a field, i.e. a vector space equipped with a bilinear multiplication.
Definition
A signature (in this context) is a set, whose elements are called operations, each of which is assigned a natural number (0, 1, 2,...) called its arity. Given a signature and a set , whose elements are called variables, a word is a finite rooted tree in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation has as many branches away from the root as the arity of . An equational law is a pair of such words; the axiom consisting of the words and is written as .
A theory consists of a signature, a set of variables, and a set of equational laws. Any theory gives a variety of algebras as follows. Given a theory , an algebra of consists of a set together with, for each operation of with arity , a function such that for each axiom and each assignment of elements of to the variables in that axiom, the equation holds that is given by applying the operations to the elements of as indicated by the trees defining and . The class of algebras of a given theory is called a variety of algebras.
Given two algebras of a theory , say and , a homomorphism is a function such that
for every operation of arity . Any theory gives a category where the objects are algebras of that theory and the morphisms are homomorphisms.
Examples
The class of all semigroups forms a variety of algebras of signature (2), meaning that a semigroup has a single binary operation. A sufficient defining equation is the associative law:
The class of groups forms a variety of algebras of signature (2,0,1), the three operations being respectively multiplication (binary), identity (nullary, a constant) and inversion (unary). The familiar axioms of associativity, identity
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https://en.wikipedia.org/wiki/Derivative%20algebra
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In mathematics:
In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topology and which provides algebraic semantics for the modal logic wK3.
In abstract algebra, the derivative algebra of a not-necessarily associative algebra A over a field F is the subalgebra of the algebra of linear endomorphisms of A consisting of the derivations.
In differential geometry a derivative algebra is a vector space with a product operation that has similar behaviour to the standard cross product of 3-vectors.
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https://en.wikipedia.org/wiki/Derivative%20algebra%20%28abstract%20algebra%29
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In abstract algebra, a derivative algebra is an algebraic structure of the signature
<A, ·, +, ', 0, 1, D>
where
<A, ·, +, ', 0, 1>
is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities:
0D = 0
xDD ≤ x + xD
(x + y)D = xD + yD.
xD is called the derivative of x. Derivative algebras provide an algebraic abstraction of the derived set operator in topology. They also play the same role for the modal logic wK4 = K + p∧?p → ??p that Boolean algebras play for ordinary propositional logic.
References
Esakia, L., Intuitionistic logic and modality via topology, Annals of Pure and Applied Logic, 127 (2004) 155-170
McKinsey, J.C.C. and Tarski, A., The Algebra of Topology, Annals of Mathematics, 45 (1944) 141-191
Algebras
Boolean algebra
Topology
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https://en.wikipedia.org/wiki/Burgers%27%20equation
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Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948.
For a given field and diffusion coefficient (or kinematic viscosity, as in the original fluid mechanical context) , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:
When the diffusion term is absent (i.e. ), Burgers' equation becomes the inviscid Burgers' equation:
which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the advective form of the Burgers' equation. The conservative form is found to be more useful in numerical integration
Terms
There are 4 parameters in Burgers' equation: and . In a system consisting of a moving viscous fluid with one spatial () and one temporal () dimension, e.g. a thin ideal pipe with fluid running through it, Burgers' equation describes the speed of the fluid at each location along the pipe as time progresses. The terms of the equation represent the following quantities:
: spatial coordinate
: temporal coordinate
: speed of fluid at the indicated spatial and temporal coordinates
: viscosity of fluid
The viscosity is a constant physical property of the fluid, and the other parameters represent the dynamics contingent on that viscosity.
Inviscid Burgers' equation
The inviscid Burgers' equation is a conservation equation, more generally a first order quasilinear hyperbolic equation. The solution to the equation and along with the initial condition
can be constructed by the method of characteristics. The characteristic equations are
Integration of the second equation tells us that is constant along the characteristic and integration of the first equation shows that the characteristics are straight lines, i.e.,
where is the point (or parameter) on the x-axis (t = 0) of the x-t plane from which the characteristic curve is drawn. Since at -axis is known from the initial condition and the fact that is unchanged as we move along the characteristic emanating from each point , we write on each characteristic. Therefore, the family of trajectories of characteristics parametrized by is
Thus, the solution is given by
This is an implicit relation that determines the solution of the inviscid Burgers' equation provided characteristics don't intersect. If the characteristics do intersect, then a classical solution to the PDE does not exist and leads to the formation of a shock wave. Whether characteristics can intersect or not depends on the initial condition. In fact, the breaking time before a shock wave can be formed is given by
Inviscid Burgers' equation for linear initial condition
Subrahm
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https://en.wikipedia.org/wiki/Donald%20Kingsbury
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Donald MacDonald Kingsbury (born 12 February 1929, in San Francisco) is an American–Canadian science fiction author. Kingsbury taught mathematics at McGill University, Montreal, from 1956 until his retirement in 1986.
Bibliography
Books
Courtship Rite. New York : Simon and Schuster, July 1982. . (Nominated for Hugo for Best Novel in 1983) (Compton Crook Award winner) (Prometheus Award Hall of Fame 2016 winner) Published in UK as Geta.
The Moon Goddess and the Son. New York : Baen Books, December 1986. . (Short version nominated for Hugo Award for Best Novella in 1980)
Psychohistorical Crisis. New York : Tor Books, December 2001. . (Winner, 2002 Prometheus Award)
The Finger Pointing Solward has been awaited ever since the publication of Courtship Rite. Kingsbury has never finished the story, noting as far back as September 1982 that he was still "polishing" it (see interview with Robert J. Sawyer) and as recently as his self-supplied Readercon biography in July 2006. Artist Donato Giancola placed a copy of the intended cover on his gallery page: this cover was used in 2016 for the Bradley P. Beaulieu collection In the Stars I'll Find You. In 1994, an excerpt was published as "The Cauldron".
Short fiction
"The Ghost Town", Astounding Science Fiction, June 1952.
"Shipwright", Analog, April 1978.
"To Bring in the Steel", Analog, July 1978.
"The Moon Goddess and the Son", Analog, December 1979.
"The Survivor", Man-Kzin Wars IV, September 1991.
"The Heroic Myth of Lieutenant Nora Argamentine", Man-Kzin Wars VI, July 1994.
"The Cauldron", Northern Stars: The Anthology of Canadian Science Fiction, September 1994.
"Historical Crisis", Far Futures, December 1995.
References
External links
Donald Kingsbury
Donald Kingsbury interviewed by Robert J. Sawyer
1929 births
Living people
Canadian science fiction writers
Writers from San Francisco
Canadian mathematicians
Kings
American expatriate academics
American expatriates in Canada
Canadian male novelists
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https://en.wikipedia.org/wiki/Positive%20set%20theory
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In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the positive formulas (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification).
Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain topology. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get generalized positive comprehension): the justification of the existential quantifier seems to require that the topology be compact.
Axioms
The set theory of Olivier Esser consists of the following axioms:
Extensionality
Positive comprehension
where is a positive formula. A positive formula uses only the logical constants but not .
Closure
where is a formula. That is, for every formula , the intersection of all sets which contain every such that exists. This is called the closure of and is written in any of the various ways that topological closures can be presented. This can be put more briefly if class language is allowed (any condition on sets defining a class as in NBG): for any class C there is a set which is the intersection of all sets which contain C as a subclass. This is a reasonable principle if the sets are understood as closed classes in a topology.
Infinity
The von Neumann ordinal exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal.
Interesting properties
The universal set is a proper set in this theory.
The sets of this theory are the collections of sets which are closed under a certain topology on the classes.
The theory can interpret ZFC (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory (Morse–Kelley set theory with the proper class ordinal a weakly compact cardinal).
Researchers
Isaac Malitz originally introduced Positive Set Theory in his 1976 PhD Thesis at UCLA
Alonzo Church was the chairman of the committee supervising the aforementioned thesis
Olivier Esser seems to be the most active in this field.
See also
New Foundations by Quine
References
Systems of set theory
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https://en.wikipedia.org/wiki/Jean%20Bourgain
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Jean Louis, baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics.
Biography
Bourgain received his PhD from the Vrije Universiteit Brussel in 1977. He was a faculty member at the University of Illinois, Urbana-Champaign and, from 1985 until 1995, professor at Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France, at the Institute for Advanced Study in Princeton, New Jersey from 1994 until 2018. He was an editor for the Annals of Mathematics. From 2012 to 2014, he was a visiting scholar at UC Berkeley.
His research work included several areas of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, analytic number theory, combinatorics, ergodic theory, partial differential equations and spectral theory, and later also group theory. In 2000, Bourgain connected the Kakeya problem to arithmetic combinatorics. As a researcher, he was the author or coauthor of more than 500 articles.
Bourgain was diagnosed with pancreatic cancer in late 2014. He died of it on 22 December 2018 at a hospital in Bonheiden, Belgium.
Awards and recognition
Bourgain received several awards during his career, the most notable being the Fields Medal in 1994.
In 2009 Bourgain was elected a foreign member of the Royal Swedish Academy of Sciences.
In 2010, he received the Shaw Prize in Mathematics.
In 2012, he and Terence Tao received the Crafoord Prize in Mathematics from the Royal Swedish Academy of Sciences.
In 2015, he was made a baron by king Philippe of Belgium.
In 2016, he received the 2017 Breakthrough Prize in Mathematics.
In 2017, he received the 2018 Leroy P. Steele Prizes.
Selected publications
Articles
(See Banach space and martingale.)
(See Sobolev space.)
(See Lindelöf hypothesis.)
Books
(Bourgain's research on nonlinear dispersive equations was, according to Carlos Kenig, "deep and influential".)
References
External links
MathSciNet: "Items authored by Bourgain, Jean."
1954 births
2018 deaths
Fields Medalists
Members of the French Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Foreign associates of the National Academy of Sciences
Functional analysts
Mathematical analysts
Institute for Advanced Study faculty
University of Illinois Urbana-Champaign faculty
20th-century Belgian mathematicians
Belgian mathematicians
Vrije Universiteit Brussel alumni
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https://en.wikipedia.org/wiki/Lw%C3%B3w%20School%20of%20Mathematics
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The Lwów school of mathematics () was a group of Polish mathematicians who worked in the interwar period in Lwów, Poland (since 1945 Lviv, Ukraine). The mathematicians often met at the famous Scottish Café to discuss mathematical problems, and published in the journal Studia Mathematica, founded in 1929. The school was renowned for its productivity and its extensive contributions to subjects such as point-set topology, set theory and functional analysis. The biographies and contributions of these mathematicians were documented in 1980 by their contemporary, Kazimierz Kuratowski in his book A Half Century of Polish Mathematics: Remembrances and Reflections.
Members
Notable members of the Lwów school of mathematics included:
Stefan Banach
Feliks Barański
Władysław Orlicz
Stanisław Saks
Hugo Steinhaus
Stanisław Mazur
Stanisław Ulam
Józef Schreier
Juliusz Schauder
Mark Kac
Antoni Łomnicki
Stefan Kaczmarz
Herman Auerbach
Włodzimierz Stożek
Stanisław Ruziewicz
Eustachy Żyliński
The end of the school
Many of the mathematicians, especially those of Jewish background, fled this southeastern part of Poland in 1941 when it became clear that it would be invaded by Germany. Few of the mathematicians survived World War II, but after the war a group including some of the original community carried on their work in western Poland's Wrocław, the successor city to prewar Lwów; see Polish population transfers (1944–1946). A number of the prewar mathematicians, prominent among them Stanisław Ulam, became famous for work done in the United States.
See also
Kraków School of Mathematics
Lwów–Warsaw school
Polish School of Mathematics
Scottish Café
Warsaw School of Mathematics
References
History of mathematics
History of education in Poland
History of Lviv
Science and technology in Poland
Lviv Polytechnic
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https://en.wikipedia.org/wiki/Martin%20J.%20Taylor
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Sir Martin John Taylor, FRS (born 18 February 1952) is a British mathematician and academic. He was Professor of Pure Mathematics at the School of Mathematics, University of Manchester and, prior to its formation and merger, UMIST where he was appointed to a chair after moving from Trinity College, Cambridge in 1986. He was elected Warden of Merton College, Oxford on 5 November 2009, took office on 2 October 2010 and retired in September 2018.
Early life and education
Taylor was born in Leicester in 1952 and educated at Wyggeston Grammar School. He gained a first class degree from Pembroke College, Oxford in 1973, and a Ph.D. from King's College London with a thesis entitled Galois module structure of the ring of integers of l-extensions in 1976 under the supervision of Albrecht Fröhlich.
Research
His early research concerned various properties and structures of algebraic numbers. In 1981 he proved the Fröhlich conjecture relating the symmetries of algebraic integers to the behaviour of certain analytic functions called Artin L-functions. In recent years his research has led him to study various aspects of arithmetic geometry: in particular, he and his collaborators have demonstrated how geometric properties of zeros of integral polynomials in many variables can be determined by the behaviour of associated L-functions.
Awards
Taylor was awarded the London Mathematical Society Whitehead Prize in 1982 and shared the Adams Prize in 1983. He was elected a Fellow of the Royal Society in 1996. He was President of the London Mathematical Society from 1998 to 2000 and in 2004 was appointed Physical Secretary and Vice-President of the Royal Society. Taylor was knighted in the 2009 New Year Honours. Taylor received an honorary Doctorate of Science from the University of East Anglia in July 2012.
Personal life
His hobbies include fly fishing and hill walking, and he is an enthusiastic supporter of Manchester United.
Notes
External links
Sir Martin Taylor's profile on the Merton College website
1952 births
Living people
Alumni of Pembroke College, Oxford
Alumni of King's College London
20th-century British mathematicians
21st-century British mathematicians
Number theorists
Fellows of the Royal Society
Fellows of Trinity College, Cambridge
Academics of the University of Manchester Institute of Science and Technology
Academics of the University of Manchester
Knights Bachelor
People educated at Wyggeston Grammar School for Boys
People from Bramhall
People from Leicester
Whitehead Prize winners
Wardens of Merton College, Oxford
Fellows of Merton College, Oxford
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https://en.wikipedia.org/wiki/Killing%20vector%20field
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In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object.
Definition
Specifically, a vector field X is a Killing field if the Lie derivative with respect to X of the metric g vanishes:
In terms of the Levi-Civita connection, this is
for all vectors Y and Z. In local coordinates, this amounts to the Killing equation
This condition is expressed in covariant form. Therefore, it is sufficient to establish it in a preferred coordinate system in order to have it hold in all coordinate systems.
Examples
Killing field on the circle
The vector field on a circle that points counterclockwise and has the same length at each point is a Killing vector field, since moving each point on the circle along this vector field simply rotates the circle.
Killing fields on the hyperbolic plane
A toy example for a Killing vector field is on the upper half-plane equipped with the Poincaré metric . The pair is typically called the hyperbolic plane and has Killing vector field (using standard coordinates). This should be intuitively clear since the covariant derivative transports the metric along an integral curve generated by the vector field (whose image is parallel to the x-axis).
Furthermore, the metric is independent of from which we can immediately conclude that is a Killing field using one of the results below in this article.
The isometry group of the upper half-plane model (or rather, the component connected to the identity) is (see Poincaré half-plane model), and the other two Killing fields may be derived from considering the action of the generators of on the upper half-plane. The other two generating Killing fields are dilatation and the special conformal transformation .
Killing fields on a 2-sphere
The Killing fields of the two-sphere , or more generally the -sphere should be obvious from ordinary intuition: spheres, having rotational symmetry, should possess Killing fields which generate rotations about any axis. That is, we expect to have symmetry under the action of the 3D rotation group SO(3). That is, by using the a priori knowledge that spheres can be embedded in Euclidean space, it is immediately possible to guess the form of the Killing fields. This is not possible in general, and so this example is of very limited educational value.
The conventional chart for the 2-sphere embedded in in Cartesian coordinates is given by
so that parametrises the height, and parametrises rotation about the -axis.
The pull back of the standard Cartesian met
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https://en.wikipedia.org/wiki/L%C3%A9vy%20process
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In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk.
The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Further important examples include the Gamma process, the Pascal process, and the Meixner process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths. All Lévy processes are additive processes.
Mathematical definition
A Lévy process is a stochastic process that satisfies the following properties:
almost surely;
Independence of increments: For any , are mutually independent;
Stationary increments: For any , is equal in distribution to
Continuity in probability: For any and it holds that
If is a Lévy process then one may construct a version of such that is almost surely right-continuous with left limits.
Properties
Independent increments
A continuous-time stochastic process assigns a random variable Xt to each point t ≥ 0 in time. In effect it is a random function of t. The increments of such a process are the differences Xs − Xt between its values at different times t < s. To call the increments of a process independent means that increments Xs − Xt and Xu − Xv are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent.
Stationary increments
To call the increments stationary means that the probability distribution of any increment Xt − Xs depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.
If is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s.
If is a Poisson process, the probability distribution of Xt − Xs is a Poisson distribution with expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.
If is a Cauchy process, the probability distribution of Xt − Xs is a Cauchy distribution with density .
Infinite divisibility
The distribution of a Lévy process has the property of infinite divisibility: given any integer n, the law of a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumptions 2 and 3. Conversely, for each
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https://en.wikipedia.org/wiki/Weyl%20tensor
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In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear expression in the Ricci tensor.
In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.
In dimensions 2 and 3 the Weyl curvature tensor vanishes identically. In dimensions ≥ 4, the Weyl curvature is generally nonzero. If the Weyl tensor vanishes in dimension ≥ 4, then the metric is locally conformally flat: there exists a local coordinate system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity.
Definition
The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a (0,4) valence tensor (by contracting with the metric). The (0,4) valence Weyl tensor is then
where n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor, s is the scalar curvature, and denotes the Kulkarni–Nomizu product of two symmetric (0,2) tensors:
In tensor component notation, this can be written as
The ordinary (1,3) valent Weyl tensor is then given by contracting the above with the inverse of the metric.
The decomposition () expresses the Riemann tensor as an orthogonal direct sum, in the sense that
This decomposition, known as the Ricci decomposition, expresses the Riemann curvature tensor into its irreducible components under the action of the orthogonal group. In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the special orthogonal group, the self-dual and antiself-dual parts C+ and C−.
The Weyl tensor can also be expresse
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https://en.wikipedia.org/wiki/Edgar%20de%20Wahl
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Edgar Alexei Robert von Wahl (Interlingue: ; 23 August 1867 – 9 March 1948) was a Baltic German mathematics and physics teacher who lived in Tallinn, Estonia. He is best known as the creator of Interlingue, an international auxiliary language that was known as Occidental throughout his life.
A Baltic German, De Wahl was born, raised and lived most of his life in the Russian Empire. Born in the territory of today's Ukraine, he spent his childhood in Tallinn and Saint Petersburg. He studied at the University of Saint Petersburg and at the Saint Petersburg Academy of Arts. During and after his studies he served in the Imperial Russian Navy. After leaving the navy in 1894 he lived permanently in Tallinn and worked there as a teacher. When most Baltic Germans left Estonia in 1939–1941, he decided to stay. He was arrested during the German occupation in 1943 and was placed in a psychiatric clinic because of alleged dementia. He stayed there until his death in 1948.
De Wahl was engaged with interlinguistics from an early age. He was first introduced to Volapük by his father's colleague Waldemar Rosenberger and even started to compose a lexicon of marine terminology for the language, before turning to Esperanto in 1888. After the failure of Reformed Esperanto in 1894, of which de Wahl had been a proponent, de Wahl started work to find an ideal form of an international language. In 1922 published a "key" to a new language, Occidental, and the first edition of the periodical Kosmoglott (later Cosmoglotta). developed the language over several decades on the advice of its speakers, but became isolated from the movement (then centred in Switzerland) from 1939 after the start of World War II.
Biography
Ancestry
Edgar was a member of the Päinurme line of the Wahl noble family. Edgar de Wahl's great-grandfather was Carl Gustav von Wahl, who acquired the Pajusi, Tapiku and Kavastu manors and was also the owner of Kaave Manor for a short time. Carl Gustav von Wahl had a total of 14 children from two marriages, from whom various Wahl lines descended. Of them, Edgar von Wahl's grandfather Alexei von Wahl, a civil servant who bought Päinurme manor in 1837, laid the foundation for the Päinurme line. In addition, he was a tenant at Taevere Manor, where Edgar's father Oskar von Wahl was born.
Childhood and youth
Edgar Alexei Robert (born von Wahl) was born on 23 August 1867 to Oskar von Wahl and Lydia Amalie Marie (married 1866 in Tallinn), in what is now Pervomaisk, Mykolaiv Oblast, in Ukraine (then in the Kherson Governorate, Russian Empire). The Wahl family had moved to Ukraine after his father, a railway engineer, started work on the Odesa–Balta–Kremenchuk–Kharkov railway in 1866. By 1869, the Wahl family had moved to Kremenchuk, where de Wahl's brother Arthur Johann Oskar was born in 1870. After that, the family moved to Tallinn, where Wahl's two sisters were born – Lydia Jenny Cornelia in 1871 and Harriet Marie Jenny in 1873. The family later moved t
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https://en.wikipedia.org/wiki/Change%20of%20base
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In mathematics, change of base can mean any of several things:
Changing numeral bases, such as converting from base 2 (binary) to base 10 (decimal). This is known as base conversion.
The logarithmic change-of-base formula, one of the logarithmic identities used frequently in algebra and calculus.
The method for changing between polynomial and normal bases, and similar transformations, for purposes of coding theory and cryptography.
Construction of the fiber product of schemes, in algebraic geometry.
See also
Change of basis
Base change (disambiguation)
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https://en.wikipedia.org/wiki/X0
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X0 may refer to:
Grammar
X0, denoting a sentence component
Zero-level projection, in X-bar theory
Head (linguistics), or nucleus
Science, technology and mathematics
SpaceShipOne flight 15P, a 2004 private spaceflight
X/0, division by zero
Turner syndrome, a disorder in which all or part of an X chromosome is absent
X0 sex-determination system, as found in some insects
Vehicles
X0, a smaller rigged version of an X1 (dinghy)
X0, running number for N700 Series Shinkansen prototype from 2014 to 2021
See also
XO (disambiguation)
X00, a popular DOS-based FOSSIL driver which was commonly used in the mid 1980s to the late 1990s
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https://en.wikipedia.org/wiki/Simplicial%20set
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In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber.
Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a "well-behaved" topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces.
Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects.
Motivation
A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations. This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.
To get back to actual topological spaces, there is a geometric realization functor which turns simplicial sets into compactly generated Hausdorff spaces. Most classical results on CW complexes in homotopy theory are generalized by analogous results for simplicial sets. While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes do not naturally exist.
Intuition
Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs. A simplicial set contains vertices (known as "0-simplices" in this context) and arrows ("1-simplices") between some of these vertices. Two vertices may be connected by several arrows, and directed loops that connect a vertex to itself are also allowed. Unlike directed multigraphs, simplicial sets may also contain higher simplices. A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices A, B, C and three arrows B → C, A → C and A → B. In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices). The vertices of the i-th face are the vertices of the n-simplex minus the
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https://en.wikipedia.org/wiki/Canonical%20basis
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In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta.
In a polynomial ring, it refers to its standard basis given by the monomials, .
For finite extension fields, it means the polynomial basis.
In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix , if the set is composed entirely of Jordan chains.
In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.
Representation theory
The canonical basis for the irreducible representations of a quantized enveloping algebra of
type and also for the plus part of that algebra was introduced by Lusztig by
two methods: an algebraic one (using a braid group action and PBW bases) and a topological one
(using intersection cohomology). Specializing the parameter to yields a canonical basis for the irreducible representations of the corresponding simple Lie algebra, which was
not known earlier. Specializing the parameter to yields something like a shadow of a basis. This shadow (but not the basis itself) for the case of irreducible representations
was considered independently by Kashiwara; it is sometimes called the crystal basis.
The definition of the canonical basis was extended to the Kac-Moody setting by Kashiwara (by an algebraic method) and by Lusztig (by a topological method).
There is a general concept underlying these bases:
Consider the ring of integral Laurent polynomials with its two subrings and the automorphism defined by .
A precanonical structure on a free -module consists of
A standard basis of ,
An interval finite partial order on , that is, is finite for all ,
A dualization operation, that is, a bijection of order two that is -semilinear and will be denoted by as well.
If a precanonical structure is given, then one can define the submodule of .
A canonical basis of the precanonical structure is then a -basis of that satisfies:
and
for all .
One can show that there exists at most one canonical basis for each precanonical structure. A sufficient condition for existence is that the polynomials defined by satisfy and .
A canonical basis induces an isomorphism from to .
Hecke algebras
Let be a Coxeter group. The corresponding Iwahori-Hecke algebra has the standard basis , the group is partially ordered by the Bruhat order which is interval finite and has a dualization operation defined by . This is a precanonical structure on that satisfies the sufficient condition above and the corresponding canonical basis of is the Kazhdan–Lusztig basis
with being the Kazhdan–Lusztig polynomials.
Linear algebra
If we are given an n × n matrix and wish to find a matrix in Jordan normal form, similar to , we are interested only in sets of linearly independent general
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https://en.wikipedia.org/wiki/Warsaw%20School%20%28mathematics%29
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Warsaw School of Mathematics is the name given to a group of mathematicians who worked at Warsaw, Poland, in the two decades between the World Wars, especially in the fields of logic, set theory, point-set topology and real analysis. They published in the journal Fundamenta Mathematicae, founded in 1920—one of the world's first specialist pure-mathematics journals. It was in this journal, in 1933, that Alfred Tarski—whose illustrious career would a few years later take him to the University of California, Berkeley—published his celebrated theorem on the undefinability of the notion of truth.
Notable members of the Warsaw School of Mathematics have included:
Wacław Sierpiński
Kazimierz Kuratowski
Edward Marczewski
Bronisław Knaster
Zygmunt Janiszewski
Stefan Mazurkiewicz
Stanisław Saks
Karol Borsuk
Roman Sikorski
Nachman Aronszajn
Samuel Eilenberg
Additionally, notable logicians of the Lwów–Warsaw School of Logic, working at Warsaw, have included:
Stanisław Leśniewski
Adolf Lindenbaum
Alfred Tarski
Jan Łukasiewicz
Andrzej Mostowski
Helena Rasiowa
Fourier analysis has been advanced at Warsaw by:
Aleksander Rajchman
Antoni Zygmund
Józef Marcinkiewicz
Otton M. Nikodym
Jerzy Spława-Neyman
See also
Polish School of Mathematics
Kraków School of Mathematics
Lwów School of Mathematics
Polish mathematics
History of education in Poland
History of mathematics
History of Warsaw
Science and technology in Poland
Warsaw School of Mathematics
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https://en.wikipedia.org/wiki/Krak%C3%B3w%20School%20of%20Mathematics
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The Kraków School of Mathematics () was a subgroup of the Polish School of Mathematics represented by mathematicians from the Kraków universities—Jagiellonian University, and the AGH University of Science and Technology–active during the interwar period (1918–1939). Their areas of study were primarily classical analysis, differential equations, and analytic functions.
The Kraków School of Differential Equations was founded by Tadeusz Ważewski, a student of Stanisław Zaremba, and was internationally appreciated after World War II.
The Kraków School of Analytic Functions was founded by Franciszek Leja. Other notable members included Kazimierz Żorawski, Władysław Ślebodziński, Stanisław Gołąb, and Czesław Olech.
See also
Polish School of Mathematics
Lwów School of Mathematics
Warsaw School of Mathematics
Polish Mathematical Society
Kraków School of Mathematics and Astrology
References
Polish mathematics
History of mathematics
History of education in Poland
20th century in Kraków
Jagiellonian University
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https://en.wikipedia.org/wiki/Krystyna%20Kuperberg
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Krystyna M. Kuperberg (born Krystyna M. Trybulec; 17 July 1944) is a Polish-American mathematician who currently works as a professor of mathematics at Auburn University, where she was formerly an Alumni Professor of Mathematics.
Early life and family
Her parents, Jan W. and Barbara H. Trybulec, were pharmacists and owned a pharmacy in Tarnów. Her older brother is Andrzej Trybulec. Her husband Włodzimierz Kuperberg and her son Greg Kuperberg are also mathematicians, while her daughter Anna Kuperberg is a photographer.
Education and career
After attending high school in Gdańsk, she entered the University of Warsaw in 1962, where she studied mathematics. Her first mathematics course was taught by Andrzej Mostowski; later she attended topology lectures of Karol Borsuk and became fascinated by topology.
After obtaining her undergraduate degree, Kuperberg began graduate studies at Warsaw under Borsuk, but stopped after earning a master's degree. She left Poland in 1969 with her young family to live in Sweden, then moved to the United States in 1972. She finished her Ph.D. in 1974, from Rice University, under the supervision of William Jaco. In the same year, both she and her husband were appointed to the faculty of Auburn University. From 1996 to 1998, Kuperberg served as an American Mathematical Society Council member at large.
Contributions
In 1987 she solved a problem of Bronisław Knaster concerning bi-homogeneity of continua. In the 1980s she became interested in fixed points and topological aspects of dynamical systems. In 1989 Kuperberg and Coke Reed solved a problem posed by Stan Ulam in the Scottish Book. The solution to that problem led to her 1993 work in which she constructed a smooth counterexample to the Seifert conjecture. She has since continued to work in dynamical systems.
Recognition
In 1995 Kuperberg received the Alfred Jurzykowski Prize from the Kościuszko Foundation. Her major lectures include an American Mathematical Society Plenary Lecture in March 1995, a Mathematical Association of America Plenary Lecture in January 1996, and an International Congress of Mathematicians invited talk in 1998. In 2012 she became a fellow of the American Mathematical Society.
Selected publications
References
Polish emigrants to the United States
American women mathematicians
Polish women mathematicians
20th-century American mathematicians
21st-century American mathematicians
20th-century Polish mathematicians
21st-century Polish mathematicians
Topologists
1944 births
Living people
People from Tarnów
University of Warsaw alumni
Rice University alumni
Auburn University faculty
Fellows of the American Mathematical Society
Dynamical systems theorists
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women
20th-century Polish women
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https://en.wikipedia.org/wiki/Wigner%20semicircle%20distribution
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The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0):
for −R ≤ x ≤ R, and f(x) = 0 if |x| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution.
The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R.
The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise.
General properties
The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution.
For positive integers n, the 2n-th moment of this distribution is
where X is any random variable with this distribution and Cn is the nth Catalan number
so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.)
Making the substitution into the defining equation for the moment generating function it can be seen that:
which can be solved (see Abramowitz and Stegun §9.6.18)
to yield:
where is the modified Bessel function. Similarly, the characteristic function is given by:
where is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving is zero.)
In the limit of approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.
Relation to free probability
In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely,
in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution.
Related distributions
Wigner (spherical) parabolic distribution
The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0):
for −R ≤ x ≤ R, and f(x) = 0 if |x| > R.
Example. The joint distribution is
Hence, the marginal PDF of the spherical (parametric) distribution is:
such that R=1
The characteristic function of a spherical distribution becomes the patter
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https://en.wikipedia.org/wiki/Stanis%C5%82aw%20Zaremba%20%28mathematician%29
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Stanisław Zaremba (3 October 1863 – 23 November 1942) was a Polish mathematician and engineer. His research in partial differential equations, applied mathematics and classical analysis, particularly on harmonic functions, gained him a wide recognition. He was one of the mathematicians who contributed to the success of the Polish School of Mathematics through his teaching and organizational skills as well as through his research. Apart from his research works, Zaremba wrote many university textbooks and monographies.
He was a professor of the Jagiellonian University (since 1900), member of Academy of Learning (since 1903), co-founder and president of the Polish Mathematical Society (1919), and the first editor of the Annales de la Société Polonaise de Mathématique.
He should not be confused with his son Stanisław Krystyn Zaremba, also a mathematician.
Biography
Zaremba was born on 3 October 1863 in Romanówka, present-day Ukraine. The son of an engineer, he was educated at a grammar school in Saint Petersburg and studied at the Institute of Technology of the same city obtaining is diploma in engineering in 1886. The same year he left Saint Petersburg and went to Paris to study mathematics: he received his degree from the Sorbonne in 1889. He stayed in France until 1900, when he joined the faculty at the Jagiellonian University in Kraków. His years in France enabled him to establish a strong bridge between Polish mathematicians and those in France.
He died on 23 November 1942 in Kraków, during the German occupation of Poland.
Work
Research activity
Selected publication
, translated in Russian as .
See also
Kraków School of Mathematics
Mixed boundary condition
Notes
References
.
.
.
, .
External links
19th-century Polish mathematicians
20th-century Polish mathematicians
Corresponding Members of the Russian Academy of Sciences (1917–1925)
Corresponding Members of the USSR Academy of Sciences
Members of the Lwów Scientific Society
Polish engineers
Mathematical analysts
University of Paris alumni
Academic staff of Jagiellonian University
1863 births
1942 deaths
Mathematicians from Austria-Hungary
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https://en.wikipedia.org/wiki/Wigner%20distribution
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Wigner distribution or Wigner function may refer to:
Wigner quasiprobability distribution (what is most commonly intended by term "Wigner function"): a quasiprobability distribution used in quantum physics, also known at the Wigner-Ville distribution
Wigner distribution function, used in signal processing, which is the time-frequency variant of the Wigner quasiprobability distribution
Modified Wigner distribution function, used in signal processing
Wigner semicircle distribution, a probability function used in mathematics
See also
Breit–Wigner distribution (disambiguation)
Wigner D-matrix, an irreducible representation of the rotation group SO(3)
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https://en.wikipedia.org/wiki/Kazimierz%20%C5%BBorawski
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Paulin Kazimierz Stefan Żorawski (June 22, 1866 – January 23, 1953) was a Polish mathematician. His work earned him an honored place in mathematics alongside such Polish mathematicians as Wojciech Brudzewski, Jan Brożek (Broscius), Nicolas Copernicus, Samuel Dickstein, Stefan Banach, Stefan Bergman, Marian Rejewski, Wacław Sierpiński, Stanisław Zaremba and Witold Hurewicz.
Żorawski's main interests were invariants of differential forms, integral invariants of Lie groups, differential geometry and fluid mechanics. His work in these disciplines was to prove important in other fields of mathematics and science, such as differential equations, geometry and physics (especially astrophysics and cosmology).
Biography
Kazimierz Żorawski was born in Szczurzyn near Ciechanów, in the Russian Empire, now in Poland, to Juliusz Bronisław Wiktor Żórawski and Kazimiera Żórawska. In 1884 he completed secondary school in Warsaw. From 1884 to 1888 he studied mathematics at the University of Warsaw. In 1889 he was selected to continue his mathematics studies on the strength of a paper on observations that he had made at the Warsaw Astronomical Observatory.
In the years that followed he studied the theory of conversion groups and analytical mechanics in Leipzig, and differential equations in Göttingen. In 1891 he was awarded a PhD (under M. Sophius Lie) in Leipzig for his thesis on the applications of group conversion theory to differential geometry. In 1892 he became a lecturer at the Polytechnic Higher School of Lwów where he taught mathematics and, in 1893, assumed the Chair of Mechanical Science.
In 1893, Żorawski received a doctorate in mathematics from Jagiellonian University in Kraków, and in 1895 he traveled to Berlin to study higher level geodesy. He later returned to Kraków where, he was named assistant professor and, in 1898, full professor of mathematics at Jagiellonian where he taught higher analysis, geometry (analytic, differential and projective), theory of algebraic curves and theory of singularities. In 1900 he was elected a member of the Academy of Learning (from 1919 Polish Academy of Learning) in Kraków.
In 1905, Żorawski became a Dean of the Faculty of philosophy at the Jagiellonian University in Kraków, and in 1910, he became an associate member of the Czech Academy of Sciences in Prague. In 1911, he became a president of the Societies of the Scientific Committee. Two years later he took part in the Organizational Committee of Academy of Maining in Kraków. From 1917 to 1918, he was a rector and from 1918 to 1919 vice-rector of the Jagiellonian University
In 1919, Żorawski settled in Warsaw where he became a full professor in mathematics at the Warsaw University of Technology, while at the same time teaching courses on the application of geometric analysis at the University of Warsaw. That same year he became a member of the Polish Society of Mathematics.
In 1920, Żorawski was elected to the Warsaw Society of Science and Letter
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https://en.wikipedia.org/wiki/Normal%20basis
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In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.
Normal basis theorem
Let be a Galois extension with Galois group . The classical normal basis theorem states that there is an element such that forms a basis of K, considered as a vector space over F. That is, any element can be written uniquely as for some elements
A normal basis contrasts with a primitive element basis of the form , where is an element whose minimal polynomial has degree .
Group representation point of view
A field extension with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules is of form for some . Since is a linear basis of F[G] over F, it follows easily that is bijective iff generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if is finite Galois extension, then as left -module. In terms of representations of G over F, this means that K is isomorphic to the regular representation.
Case of finite fields
For finite fields this can be stated as follows: Let denote the field of q elements, where is a prime power, and let denote its extension field of degree . Here the Galois group is with a cyclic group generated by the q-power Frobenius automorphism with Then there exists an element such that
is a basis of K over F.
Proof for finite fields
In case the Galois group is cyclic as above, generated by with the normal basis theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying ; then any distinct characters are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms thought of as mappings from the multiplicative group . Now as an F-vector space, so we may consider as an element of the matrix algebra Mn(F); since its powers are linearly independent (over K and a fortiori over F), its minimal polynomial must have degree at least n, i.e. it must be .
The second basic fact is the classification of finitely generated modules over a PID such as . Every such module M can be represented as , where may be chosen so that they are monic polynomials or zero and is a multiple of . is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case , in the
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https://en.wikipedia.org/wiki/Simplex%20category
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In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects.
Formal definition
The simplex category is usually denoted by . There are several equivalent descriptions of this category. can be described as the category of non-empty finite ordinals as objects, thought of as totally ordered sets, and (non-strictly) order-preserving functions as morphisms. The objects are commonly denoted (so that is the ordinal ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.)
A simplicial object is a presheaf on , that is a contravariant functor from to another category. For instance, simplicial sets are contravariant with the codomain category being the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from .
Augmented simplex category
The augmented simplex category, denoted by is the category of all finite ordinals and order-preserving maps, thus , where . Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category.
A contravariant functor defined on is called an augmented simplicial object and a covariant functor out of is called an augmented cosimplicial object; when the codomain category is the category of sets, for example, these are called augmented simplicial sets and augmented cosimplicial sets respectively.
The augmented simplex category, unlike the simplex category, admits a natural monoidal structure. The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal (the lack of a unit prevents this from qualifying as a monoidal structure on ). In fact, is the monoidal category freely generated by a single monoid object, given by with the unique possible unit and multiplication. This description is useful for understanding how any comonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from to the monoidal category containing the comonoid; by forgetting the augmentation we obtain a simplicial object. Similarly, this also illuminates the construction of simplicial objects from monads (and hence adjoint functors) since monads can be viewed as monoid objects in endofunctor categories.
See also
Simplicial category
PROP (category theory)
Abstract simplicial complex
References
External links
What's special about the Simplex category?
Algebraic topology
Homotopy theory
Categories in category theory
Free algebraic structures
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https://en.wikipedia.org/wiki/Eigenplane
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In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term eigenvector for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term eigenplane can be used to describe a two-dimensional plane (a 2-plane), such that the operation of a linear operator on a vector in the 2-plane always yields another vector in the same 2-plane.
A particular case that has been studied is that in which the linear operator is an isometry M of the hypersphere (written S3) represented within four-dimensional Euclidean space:
where s and t are four-dimensional column vectors and Λθ is a two-dimensional eigenrotation within the eigenplane.
In the usual eigenvector problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom to multiply by an arbitrary non-zero rotation.
This case is potentially physically interesting in the case that the shape of the universe is a multiply connected 3-manifold, since finding the angles of the eigenrotations of a candidate isometry for topological lensing is a way to falsify such hypotheses.
See also
Bivector
Plane of rotation
External links
possible relevance of eigenplanes in cosmology
GNU GPL software for calculating eigenplanes
Proof constructed by J M Shelley 2017
Linear algebra
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https://en.wikipedia.org/wiki/Primitive%20polynomial%20%28field%20theory%29
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In finite field theory, a branch of mathematics, a primitive polynomial is the minimal polynomial of a primitive element of the finite field . This means that a polynomial of degree with coefficients in is a primitive polynomial if it is monic and has a root in such that is the entire field . This implies that is a primitive ()-root of unity in .
Properties
Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible.
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by x. Over GF(2), is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by (it has 1 as a root).
An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n such that F(x) divides is .
Over GF(p) there are exactly primitive polynomials of degree m, where φ is Euler's totient function.
A primitive polynomial of degree m has m different roots in GF(pm), which all have order . This means that, if α is such a root, then and for .
The primitive polynomial F(x) of degree m of a primitive element α in GF(pm) has explicit form .
Usage
Field element representation
Primitive polynomials can be used to represent the elements of a finite field. If α in GF(pm) is a root of a primitive polynomial F(x), then the nonzero elements of GF(pm) are represented as successive powers of α:
This allows an economical representation in a computer of the nonzero elements of the finite field, by representing an element by the corresponding exponent of This representation makes multiplication easy, as it corresponds to addition of exponents modulo
Pseudo-random bit generation
Primitive polynomials over GF(2), the field with two elements, can be used for pseudorandom bit generation. In fact, every linear-feedback shift register with maximum cycle length (which is , where n is the length of the linear-feedback shift register) may be built from a primitive polynomial.
In general, for a primitive polynomial of degree m over GF(2), this process will generate pseudo-random bits before repeating the same sequence.
CRC codes
The cyclic redundancy check (CRC) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see Mathematics of CRC. Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of for a degree n primitive polynomial.
Primitive trinomials
A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms: . Their simplicity makes for particularly small and fast linear-feedback shift registers.
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