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https://en.wikipedia.org/wiki/H-derivative
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In mathematics, the H-derivative is a notion of derivative in the study of abstract Wiener spaces and the Malliavin calculus.
Definition
Let be an abstract Wiener space, and suppose that is differentiable. Then the Fréchet derivative is a map
;
i.e., for , is an element of , the dual space to .
Therefore, define the -derivative at by
,
a continuous linear map on .
Define the -gradient by
.
That is, if denotes the adjoint of , we have .
See also
Malliavin derivative
References
Generalizations of the derivative
Measure theory
Stochastic calculus
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https://en.wikipedia.org/wiki/Outline%20of%20trigonometry
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The following outline is provided as an overview of and topical guide to trigonometry:
Trigonometry – branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.
Basics
Geometry – mathematics concerned with questions of shape, size, the relative position of figures, and the properties of space. Geometry is used extensively in trigonometry.
Angle – the angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.
Ratio – a ratio indicates how many times one number contains another
Content of trigonometry
Trigonometry
Trigonometric functions
Trigonometric identities
Euler's formula
Scholars
Archimedes
Aristarchus
Aryabhata
Bhaskara I
Claudius Ptolemy
Euclid
Hipparchus
Madhava of Sangamagrama
Ptolemy
Pythagoras
Regiomontanus
History
Aristarchus's inequality
Bhaskara I's sine approximation formula
Greek astronomy
Indian astronomy
Jyā, koti-jyā and utkrama-jyā
Madhava's sine table
Ptolemy's table of chords
Rule of marteloio
Āryabhaṭa's sine table
Fields
Uses of trigonometry
Acoustics
Architecture
Astronomy
Biology
Cartography
Chemistry
Civil engineering
Computer graphics
Cryptography
Crystallography
Economics
Electrical engineering
Electronics
Game development
Geodesy
Mechanical engineering
Medical imaging
Meteorology
Music theory
Number theory
Oceanography
Optics
Pharmacy
Phonetics
Physical science
Probability theory
Seismology
Statistics
Surveying
Physics
Abbe sine condition
Greninger chart
Phasor
Snell's law
Astronomy
Equant
Parallax
Dialing scales
Chemistry
Greninger chart
Geography, geodesy, and land surveying
Hansen's problem
Snellius–Pothenot problem
Great-circle distance – how to find that distance if one knows the latitude and longitude.
Resection (orientation)
Vincenty's formulae
Geographic distance
Triangulation in three dimensions
Navigation
Haversine formula
Rule of marteloio
Engineering
Belt problem
Phase response
Phasor
Rake (angle)
Analog devices
Dialing scales
Gunter's quadrant
Gunter's scale
Protractor
Scale of chords
Calculus
Inverse trigonometric functions
List of integrals of trigonometric functions
List of integrals of inverse trigonometric functions
Regiomontanus' angle maximization problem
Tangent half-angle substitution
Trigonometric integral
Trigonometric substitution
Applications
Fourier transform
Wave equation
Other areas of mathematics
For examples of trigonometric functions as generating functions in combinatorics, see Alternating permutation.
Dirichlet kernel
Euler's formula
Exact trigonometric values
Exponential sum
Trigonometric integr
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https://en.wikipedia.org/wiki/Fiber%20%28mathematics%29
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In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context:
In naive set theory, the fiber of the element in the set under a map is the inverse image of the singleton under
In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.
Definitions
Fiber in naive set theory
Let be a function between sets.
The fiber of an element (or fiber over ) under the map is the set that is, the set of elements that get mapped to by the function. It is the preimage of the singleton (One usually takes in the image of to avoid being the empty set.)
The collection of all fibers for the function forms a partition of the domain The fiber containing an element is the set For example, the fibers of the projection map that sends to are the vertical lines, which form a partition of the plane.
If is a real-valued function of several real variables, the fibers of the function are the level sets of . If is also a continuous function and is in the image of the level set will typically be a curve in 2D, a surface in 3D, and, more generally, a hypersurface in the domain of
Fiber in algebraic geometry
In algebraic geometry, if is a morphism of schemes, the fiber of a point in is the fiber product of schemes
where is the residue field at
Fibers in topology
Every fiber of a local homeomorphism is a discrete subspace of its domain.
If is a continuous function and if (or more generally, if ) is a T1 space then every fiber is a closed subset of
A function between topological spaces is called if every fiber is a connected subspace of its domain. A function is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis.
A function between topological spaces is (sometimes) called a if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent competing definitions of "proper map" so it is advisable to always check how a particular author defines this term.
A continuous closed surjective function whose fibers are all compact is called a .
See also
Fibration
Fiber bundle
Fiber product
Preimage theorem
Zero set
Citations
References
Basic concepts in set theory
Mathematical relations
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https://en.wikipedia.org/wiki/Levku%C5%A1ka
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Levkuška () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
https://web.archive.org/web/20080111223415/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
Stub-Class Slovakia articles
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https://en.wikipedia.org/wiki/Luben%C3%ADk
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Lubeník () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
Lubenik.sk
https://web.archive.org/web/20070427022352/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Plosk%C3%A9%2C%20Rev%C3%BAca%20District
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Ploské () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Skere%C5%A1ovo
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Skerešovo () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Ra%C5%A1ice
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Rašice () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Otro%C4%8Dok
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Otročok () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Rybn%C3%ADk%2C%20Rev%C3%BAca%20District
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Rybník () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/S%C3%A1sa%2C%20Rev%C3%BAca%20District
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Sása () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Tur%C4%8Dok
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Turčok () is a village and municipality in Revúca District in the Banská Bystrica Region of Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Revúca District
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https://en.wikipedia.org/wiki/Vieska%20nad%20Blhom
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Vieska nad Blhom () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Strictly%20positive%20measure
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In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points".
Definition
Let be a Hausdorff topological space and let be a -algebra on that contains the topology (so that every open set is a measurable set, and is at least as fine as the Borel -algebra on ). Then a measure on is called strictly positive if every non-empty open subset of has strictly positive measure.
More concisely, is strictly positive if and only if for all such that
Examples
Counting measure on any set (with any topology) is strictly positive.
Dirac measure is usually not strictly positive unless the topology is particularly "coarse" (contains "few" sets). For example, on the real line with its usual Borel topology and -algebra is not strictly positive; however, if is equipped with the trivial topology then is strictly positive. This example illustrates the importance of the topology in determining strict positivity.
Gaussian measure on Euclidean space (with its Borel topology and -algebra) is strictly positive.
Wiener measure on the space of continuous paths in is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space.
Lebesgue measure on (with its Borel topology and -algebra) is strictly positive.
The trivial measure is never strictly positive, regardless of the space or the topology used, except when is empty.
Properties
If and are two measures on a measurable topological space with strictly positive and also absolutely continuous with respect to then is strictly positive as well. The proof is simple: let be an arbitrary open set; since is strictly positive, by absolute continuity, as well.
Hence, strict positivity is an invariant with respect to equivalence of measures.
See also
− a measure is strictly positive if and only if its support is the whole space.
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Andrei%20Okounkov
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Andrei Yuryevich Okounkov (, Andrej Okun'kov) (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at the University of California, Berkeley and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics. In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."
Education and career
He graduated with a B.S. in mathematics, summa cum laude, from Moscow State University in 1993 and received his doctorate, also at Moscow State, in 1995 under Alexandre Kirillov and Grigori Olshanski. He is a professor at Columbia University. He previously was a professor at Princeton University, where he was awarded a Packard Fellowship (2001), the European Mathematical Society Prize (2004), and the Fields Medal (2006); an assistant and associate professor at Berkeley, where he was awarded a Sloan Research Fellowship; and an instructor at the University of Chicago. He rejoined the faculty at Berkeley in the summer of 2022.
Work
He has worked on the representation theory of infinite symmetric groups, the statistics of plane partitions, and the quantum cohomology of the Hilbert scheme of points in the complex plane. Much of his work on Hilbert schemes was joint with Rahul Pandharipande.
Okounkov, along with Pandharipande, Nikita Nekrasov, and Davesh Maulik, has formulated well-known conjectures relating the Gromov–Witten invariants and Donaldson–Thomas invariants of threefolds.
In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."
In 2016, he became a fellow of the American Academy of Arts and Sciences.
See also
Newton–Okounkov body
References
External links
Andrei Okounkov home page at Columbia
Andrei Okounkov home page at Princeton
EMS Prize 2004 citation
Fields Medal citation
Andrei Okounkov's articles on the Arxiv
Daily Princetonian story
BBC story
21st-century Russian mathematicians
Fields Medalists
Moscow State University alumni
Princeton University faculty
Institute for Advanced Study visiting scholars
Columbia University faculty
University of California, Berkeley faculty
University of Chicago faculty
1969 births
Living people
Mathematicians from Moscow
Fellows of the American Academy of Arts and Sciences
Members of the United States National Academy of Sciences
Simons Investigator
Russian scientists
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https://en.wikipedia.org/wiki/Brian%20D.%20Ripley
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Brian David Ripley FRSE (born 29 April 1952) is a British statistician. From 1990, he was professor of applied statistics at the University of Oxford and is also a professorial fellow at St Peter's College. He retired August 2014 due to ill health.
Biography
Ripley has made contributions to the fields of spatial statistics and pattern recognition. His work on artificial neural networks in the 1990s helped to bring aspects of machine learning and data mining to the attention of statistical audiences. He emphasised the value of robust statistics in his books Modern Applied Statistics with S and Pattern Recognition and Neural Networks.
Ripley helped develop the S programming language and its implementations: S-PLUS and R. He co-authored two books based on S, Modern Applied Statistics with S and S Programming. From 2000 to 2021 he was one of the most active committers to the R core.
He was educated at the University of Cambridge, where he was awarded both the Smith's Prize (at the time awarded to the best graduate essay writer who had been undergraduate at Cambridge in that cohort) and the Rollo Davidson Prize. The university also awarded him the Adams Prize in 1987 for an essay entitled Statistical Inference for Spatial Processes, later published as a book. He served on the faculty of Imperial College, London from 1976 until 1983, at which point he moved to the University of Strathclyde.
Authored books
Ripley, B. D. (1981) Spatial Statistics. Wiley, 252pp. .
Ripley, B. D. (1983) Stochastic Simulation. Wiley, .
Ripley, B. D. (1996) Pattern Recognition and Neural Networks. Cambridge University Press. 403 pages. .
Venables, W. N. and Ripley, B. D. (2000) S Programming. Springer, 264pp. .
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S (Fourth Edition; previous editions published as Modern Applied Statistics with S-PLUS in 1994, 1997 & 1999). Springer, 462pp. .
References
External links
Brian Ripley's home page at the University of Oxford
1952 births
British statisticians
Fellows of the Royal Society of Edinburgh
Machine learning researchers
Artificial intelligence researchers
Living people
Fellows of St Peter's College, Oxford
R (programming language) people
Spatial statisticians
Computational statisticians
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https://en.wikipedia.org/wiki/Manzanares%20el%20Real
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Manzanares el Real is a 8,936 inhabitant town (2020 statistics from ine.es) in the northern area of the autonomous Community of Madrid. It is located at the foot of The Pedriza, a part of the Sierra de Guadarrama, and next to the embalse de Santillana (the Santillana reservoir).
Main sights
The New Castle of Manzanares el Real, the best conserved castle in the Community of Madrid. Construction commencing in 1475, it has been used in several motion pictures, most notably El Cid.
The Old Castle of Manzanares el Real is the ruin of a former fortress, also known as Plaza de Armas. Only two walls remain standing, now integrated into a garden complex. It was built in Mudejar style of granite with brick curbing.
Church of Nuestra Señora de las Nieves, founded in the early 14th century. It has a nave and two aisles, separated by arcades on stone columns. The nave, in Romanesque styles, ends into a pentagonal presbytery. The church has also a 16th-century Renaissance portico.
Hermitage of Nuestra Señora de la Peña Sacra.
The Town Square and the Town Hall Houses - The Square has always been, and remains the place for celebrations, where local events, celebrations, and social life take place. The Town Hall Houses are peculiar because although our municipality was the head of the County of El Real de Manzanares, they did not exist as such: they were the County jail. It has always preserved its portico, the balcony, and its railings, a construction that may have been commissioned by the Great Cardinal Mendoza in the 16th century.
Bus
There are three lines passing through the village, which are the following:
SE720: Colmenar Viejo - Manzanares el Real
720: Colmenar Viejo - Collado Villalba
724: Madrid (Plaza de Castilla) - Manzanares el Real - El Boalo
References
External links
Official website
The Castle of Manzanares el Real - History and photos
English Language tourist website
Municipalities in the Community of Madrid
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https://en.wikipedia.org/wiki/Lindel%C3%B6f%27s%20lemma
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In mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.
Statement of the lemma
Let the real line have its standard topology. Then every open subset of the real line is a countable union of open intervals.
Generalized Statement
Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover (Kelley 1955:49). This means that every second-countable space is also a Lindelöf space.
Proof of the generalized statement
Let be a countable basis of . Consider an open cover, . To get prepared for the following deduction, we define two sets for convenience, , .
A straight-forward but essential observation is that, which is from the definition of base. Therefore, we can get that,
where , and is therefore at most countable. Next, by construction, for each there is some such that . We can therefore write
completing the proof.
References
J.L. Kelley (1955), General Topology, van Nostrand.
M.A. Armstrong (1983), Basic Topology, Springer.
Covering lemmas
Lemmas
Topology
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https://en.wikipedia.org/wiki/Hilbert%20scheme
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In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
Hilbert scheme of projective space
The Hilbert scheme of classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of that are flat over . The closed subschemes of that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . The Hilbert scheme breaks up as a disjoint union of pieces corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial . Each of these pieces is projective over .
Construction as a determinantal variety
Grothendieck constructed the Hilbert scheme of -dimensional projective space as a subscheme of a Grassmannian defined by the vanishing of various determinants. Its fundamental property is that for a scheme , it represents the functor whose -valued points are the closed subschemes of that are flat over .
If is a subscheme of -dimensional projective space, then corresponds to a graded ideal of the polynomial ring in variables, with graded pieces . For sufficiently large all higher cohomology groups of with coefficients in vanish. Using the exact sequencewe have has dimension , where is the Hilbert polynomial of projective space. This can be shown by tensoring the exact sequence above by the locally flat sheaves , giving an exact sequence where the latter two terms have trivial cohomology, implying the triviality of the higher cohomology of . Note that we are using the equality of the Hilbert polynomial of a coherent sheaf with the Euler-characteristic of its sheaf cohomology groups.
Pick a sufficiently large value of . The -dimensional space is a subspace of the -dimensional space , so represents a point of the Grassmannian . This will give an embedding of the piece of the Hilbert scheme corresponding to the Hilbert polynomial into this Grassmannian.
It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map has rank at most for all positive , which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take .)
Properties
Universality
Given a closed subscheme over a field with Hilbert polynomial , the Hilbert scheme has a universal subscheme flat over such that
The fibers over closed point
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https://en.wikipedia.org/wiki/Hilbert%20series%20and%20Hilbert%20polynomial
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In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra.
These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes.
The typical situations where these notions are used are the following:
The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree.
The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree.
The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.
The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space.
The Hilbert polynomial and Hilbert series are important in computational algebraic geometry, as they are the easiest known way for computing the dimension and the degree of an algebraic variety defined by explicit polynomial equations. In addition, they provide useful invariants for families of algebraic varieties because a flat family has the same Hilbert polynomial over any closed point . This is used in the construction of the Hilbert scheme and Quot scheme.
Definitions and main properties
Consider a finitely generated graded commutative algebra over a field , which is finitely generated by elements of positive degree. This means that
and that .
The Hilbert function
maps the integer to the dimension of the -vector space . The Hilbert series, which is called Hilbert–Poincaré series in the more general setting of graded vector spaces, is the formal series
If is generated by homogeneous elements of positive degrees , then the sum of the Hilbert series is a rational fraction
where is a polynomial with integer coefficients.
If is generated by elements of degree 1 then the sum of the Hilbert series may be rewritten as
where is a polynomial with integer coefficients, and is the Krull dimension of .
In this case the series expansion of this rational fraction is
where
is the binomial coefficient for and is 0 otherwise.
If
the coefficient of in is thus
For the term of index in this sum is a polynomial in of degree with leading coefficient This shows that there exists a unique polynomial with rational coefficients which is equal to for large enough. This polynomial is the Hilbert polynomial, and has the form
The least such that for is called the Hilbert regularity. It may be lower than .
The Hilbert polynomial is a numerical polynomial, since the dimensions are integers, but the polynomial almost never has integer coefficients .
All these definitions may be extended to finitely generated graded modules over , with the only difference that a factor appe
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https://en.wikipedia.org/wiki/Equivalence%20%28measure%20theory%29
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In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero.
Definition
Let and be two measures on the measurable space and let
and
be the sets of -null sets and -null sets, respectively. Then the measure is said to be absolutely continuous in reference to if and only if This is denoted as
The two measures are called equivalent if and only if and which is denoted as That is, two measures are equivalent if they satisfy
Examples
On the real line
Define the two measures on the real line as
for all Borel sets Then and are equivalent, since all sets outside of have and measure zero, and a set inside is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.
Abstract measure space
Look at some measurable space and let be the counting measure, so
where is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is, So by the second definition, any other measure is equivalent to the counting measure if and only if it also has just the empty set as the only -null set.
Supporting measures
A measure is called a of a measure if is -finite and is equivalent to
References
Equivalence (mathematics)
Measure theory
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https://en.wikipedia.org/wiki/Serre%27s%20conjecture
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Serre's conjecture may refer to:
Quillen–Suslin theorem, formerly known as Serre's conjecture
Serre's conjecture II (algebra), concerning the Galois cohomology of linear algebraic groups
Serre's modularity conjecture, concerning Galois representations
Serre's multiplicity conjectures in commutative algebra
Ribet's theorem, formerly known as Serre's epsilon conjecture
See also
Jean-Pierre Serre
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https://en.wikipedia.org/wiki/Friedrichs%27s%20inequality
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In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the Lp norm of a function using Lp bounds on the weak derivatives of the function and the geometry of the domain, and can be used to show that certain norms on Sobolev spaces are equivalent. Friedrichs's inequality generalizes the Poincaré–Wirtinger inequality, which deals with the case k = 1.
Statement of the inequality
Let be a bounded subset of Euclidean space with diameter . Suppose that lies in the Sobolev space , i.e., and the trace of on the boundary is zero. Then
In the above
denotes the Lp norm;
α = (α1, ..., αn) is a multi-index with norm |α| = α1 + ... + αn;
Dαu is the mixed partial derivative
See also
Poincaré inequality
References
Sobolev spaces
Inequalities
Linear functionals
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https://en.wikipedia.org/wiki/Madrigueras
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Madrigueras is a municipality in Albacete, Castile-La Mancha, Spain. It has a population of 4,917 according to the official statistics by the National Statistics Institute of Spain (INE). The principal productions of this village are wine, knives and spatulas. Madrigueras is referred to as "Little China" due to the great number of inhabitants who use bikes for transportation.
References
External links
http://www.madrigueras.es Homepage of Madrigueras (in Spanish)
Municipalities of the Province of Albacete
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https://en.wikipedia.org/wiki/List%20of%20Everton%20F.C.%20records%20and%20statistics
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Everton Football Club is a professional association football club located in Liverpool. The club was formed in 1878, and was originally named as St Domingo FC. The club's first game was a 1–0 victory over Everton Church Club. In November 1879 the club was renamed to Everton FC.
In 1888, Everton were one of the twelve founding members of the English Football League. The club have played in the top-flight of English Football for a record 117 years, having missed only four top-flight seasons (1930–31, 1951–52, 1952–53, 1953–54).
Major competitions won by Everton F.C., records set by the club, associated managers and players will be included in the following list.
The player records section includes: appearances, goals scored, and clean sheets kept. Player and manager awards, transfer fees, club records (Wins, Draws, and Losses) are all included in the list, as well as several others.
Honours
Domestic
First Division:
Titles (9): 1890–91, 1914–15, 1927–28, 1931–32, 1938–39, 1962–63, 1969–70, 1984–85, 1986–87
Second Division:
Titles (1): 1930–31
FA Cup:
Titles (5): 1905–06, 1932–33, 1965–66, 1983–84, 1994–95
Football League Cup:
Runner-up (2): 1976–77, 1983–84
FA Charity Shield:
Titles (9): 1928, 1932, 1963, 1970, 1984, 1985, 1986 (shared), 1987, 1995
Full Members Cup:
Runner-up (2): 1989, 1991
Football League Super Cup:
Runner-up (1): 1985–86
European
European Cup Winners' Cup:
Winners: (1): 1984–85
Doubles
1984–85: League and European Cup Winners' Cup
Awards
1985 World Soccer Men's World Team of the Year
1985 France Football European Team of the Year
Player records
As of 31 October 2023
Appearances
Youngest Player (All Competitions): Thierry Small, 16 years and 176 days (vs Sheffield Wednesday, 24 January 2021)
Youngest Player in Europe: Jake Bidwell, 16 years and 271 days (vs BATE Borisov, 17 December 2009)
Oldest Player: Ted Sagar, 42 years and 281 days (vs Plymouth Argyle, 15 November 1952)
Most Appearances (All Competitions): Neville Southall, 751
Most League Appearances: Neville Southall, 578
Most FA Cup Appearances: Neville Southall, 70
Most League Cup Appearances: Neville Southall, 65
Most European Appearances: Tim Howard, 28
Most Substitute Appearances: Victor Anichebe, 95
All competitions appearances
All League appearances
FA Cup appearances
League Cup appearances
European appearances
Goalscorers
Most goals in a season – 60, Dixie Dean, (During the 1927–28 Season)
Most goals in a single match – 6, Jack Southworth (v. West Bromwich Albion, 30 December 1893)
Most goals in the League – 349, Dixie Dean
Most goals in the FA Cup – 28, Dixie Dean
Most goals in the League Cup – 19, Bob Latchford
Most goals in European competition – 8, Romelu Lukaku
Youngest goalscorer – James Vaughan, 16 yrs and 271 days (vs Crystal Palace, 10 April 2005) (Also a Premier League Record)
Oldest goalscorer – Wally Fielding, 38 yrs and 305 days (vs West Bromwich Albion F.C., 27 September 1958)
Top scorers (all competitions)
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https://en.wikipedia.org/wiki/Locally%20finite%20measure
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In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.
Definition
Let be a Hausdorff topological space and let be a -algebra on that contains the topology (so that every open set is a measurable set, and is at least as fine as the Borel -algebra on ). A measure/signed measure/complex measure defined on is called locally finite if, for every point of the space there is an open neighbourhood of such that the -measure of is finite.
In more condensed notation, is locally finite if and only if
Examples
Any probability measure on is locally finite, since it assigns unit measure to the whole space. Similarly, any measure that assigns finite measure to the whole space is locally finite.
Lebesgue measure on Euclidean space is locally finite.
By definition, any Radon measure is locally finite.
The counting measure is sometimes locally finite and sometimes not: the counting measure on the integers with their usual discrete topology is locally finite, but the counting measure on the real line with its usual Borel topology is not.
See also
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Comparison%20theorem
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In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.
Differential equations
In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property.
Chaplygin inequality
Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.
Sturm comparison theorem
Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.
Hille-Wintner comparison theorem
Riemannian geometry
In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.
Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
Toponogov's theorem
Myers's theorem
Hessian comparison theorem
Laplacian comparison theorem
Morse–Schoenberg comparison theorem
Berger comparison theorem, Rauch–Berger comparison theorem
Berger–Kazdan comparison theorem
Warner comparison theorem for lengths of N-Jacobi fields (N being a submanifold of a complete Riemannian manifold)
Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures
Lichnerowicz comparison theorem
Eigenvalue comparison theorem
Cheng's eigenvalue comparison theorem
See also: Comparison triangle
Other
Limit comparison theorem, about convergence of series
Comparison theorem for integrals, about convergence of integrals
Zeeman's comparison theorem, a technical tool from the theory of spectral sequences
References
Mathematical theorems
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https://en.wikipedia.org/wiki/Neporadza%2C%20Rimavsk%C3%A1%20Sobota%20District
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Neporadza () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Potok%2C%20Rimavsk%C3%A1%20Sobota%20District
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Potok () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Padarovce
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Padarovce (, also Balogpádár) is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
Municipalities in Slovakia where Hungarian is an official language
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https://en.wikipedia.org/wiki/Tachty
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Tachty () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Rumince
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Rumince () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Rakytn%C3%ADk
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Rakytník () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Valice
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Valice () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Vlky%C5%88a
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Vlkyňa () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Str%C3%A1nska
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Stránska () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Rimavsk%C3%A9%20Janovce
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Rimavské Janovce () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Transportation%20theory%20%28mathematics%29
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In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.
In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".
Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich. Consequently, the problem as it is stated is sometimes known as the Monge–Kantorovich transportation problem. The linear programming formulation of the transportation problem is also known as the Hitchcock–Koopmans transportation problem.
Motivation
Mines and factories
Suppose that we have a collection of m mines mining iron ore, and a collection of n factories which use the iron ore that the mines produce. Suppose for the sake of argument that these mines and factories form two disjoint subsets M and F of the Euclidean plane R2. Suppose also that we have a cost function c : R2 × R2 → [0, ∞), so that c(x, y) is the cost of transporting one shipment of iron from x to y. For simplicity, we ignore the time taken to do the transporting. We also assume that each mine can supply only one factory (no splitting of shipments) and that each factory requires precisely one shipment to be in operation (factories cannot work at half- or double-capacity). Having made the above assumptions, a transport plan is a bijection T : M → F.
In other words, each mine m ∈ M supplies precisely one target factory T(m) ∈ F and each factory is supplied by precisely one mine.
We wish to find the optimal transport plan, the plan T whose total cost
is the least of all possible transport plans from M to F. This motivating special case of the transportation problem is an instance of the assignment problem.
More specifically, it is equivalent to finding a minimum weight matching in a bipartite graph.
Moving books: the importance of the cost function
The following simple example illustrates the importance of the cost function in determining the optimal transport plan. Suppose that we have n books of equal width on a shelf (the real line), arranged in a single contiguous block. We wish to rearrange them into another contiguous block, but shifted one book-width to the right. Two obvious candidates for the optimal transport plan present themselves:
move all n books one book-width to the right ("many small moves");
move the left-most book n book-widths to the right and leave all other books fixed ("one big move").
If the cost function is proportional to Euclidean distance (c(x, y) = α|x − y|) then these two candidates are both optimal. If, on the other hand, we choose the strictly convex cost function proport
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https://en.wikipedia.org/wiki/List%20of%20towns%20and%20cities%20with%20100%2C000%20or%20more%20inhabitants
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By country name
A-B • C-D-E-F • G-H-I-J-K • L-M-N-O • P-Q-R-S • T-U-V-W-Y-Z
See also
List of largest cities
Notes
References
Sources
Population Density, United Nations Statistics Division, accessed 30 August 2010.
External links
Geopolis: research group, university of Paris-Diderot, France
Towns and cities with 100,000 or more inhabitants
100,000 or more inhabitants
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https://en.wikipedia.org/wiki/Nov%C3%A1%20Ba%C5%A1ta
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Nová Bašta () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Dennis%20DeTurck
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Dennis M. DeTurck (born July 15, 1954) is an American mathematician known for his work in partial differential equations and Riemannian geometry, in particular contributions to the theory of the Ricci flow and the prescribed Ricci curvature problem. He first used the DeTurck trick to give an alternative proof of the short time existence of the Ricci flow, which has found other uses since then.
Education
DeTurck received a B.S. (1976) from Drexel University. He received an M.A. (1978) and Ph.D. (1980) in mathematics from the University of Pennsylvania. His Ph.D. supervisor was Jerry Kazdan.
Career
DeTurck is currently Robert A. Fox Leadership Professor and Professor of Mathematics at the University of Pennsylvania, where he has been the Dean of the College of Arts and Sciences since 2005 and Faculty Director of Riepe College House. In 2002, DeTurck won the Haimo Award from the Mathematical Association of America for his teaching. Despite being recognized for excellence in teaching, he has been criticized for his belief that fractions are "as obsolete as Roman numerals" and suggesting that they not be taught to younger students.
In January 2012, he shared the Chauvenet Prize with three mathematical collaborators. In 2012, he became a fellow of the American Mathematical Society.
Selected publications
(explains the DeTurck trick; also see the improved version)
References
External links
Article with some career and biographical info
Dennis DeTurck's homepage
1954 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Mathematical analysts
University of Pennsylvania alumni
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
Fellows of the American Mathematical Society
Drexel University alumni
Differential geometers
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https://en.wikipedia.org/wiki/Susan%20Howson%20%28mathematician%29
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Susan Howson (born 1973) is a British mathematician whose research is in the fields of algebraic number theory and arithmetic geometry.
Education and career
Howson received her PhD in mathematics from the University of Cambridge in 1998 with thesis title Iwasawa Theory of Elliptic Curves for ρ-Adic Lie Extensions under the supervision of John H. Coates.
Howson has taught at MIT, University of Cambridge, University of Oxford, and University of Nottingham.
She then left academia and studied medicine in Southampton. After graduating she became a consultant in Child and Adolescent mental health, working in the NHS in Devon.
Recognition
In 2002, Howson won the Adams Prize for her work on number theory and elliptic curves. She was the first woman to win the prize in its 120-year history. In an interview, she indicated that the competitive and single-minded nature of higher mathematics is possibly part of what discourages women from pursuing it.
She also held a Royal Society Dorothy Hodgkin Research Fellowship.
References
External links
Woman joins Adams family March 2002
Living people
British women mathematicians
20th-century British mathematicians
21st-century British mathematicians
1973 births
Place of birth missing (living people)
20th-century women mathematicians
21st-century women mathematicians
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https://en.wikipedia.org/wiki/Edna%20Grossman
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Edna Grossman (born Edna Kalka) is an American mathematician. She was born in Germany, grew up in Brooklyn, New York, and graduated with a B.S. in mathematics from Brooklyn College. She earned her M.S. in mathematics from New York University's Courant Institute of Mathematical Sciences, where she also received her Ph.D. in mathematics in 1972; her thesis, supervised by Wilhelm Magnus, concerned the symmetries of free groups. Grossman worked for IBM, where she was part of the team that designed and analyzed the Data Encryption Standard. She is known for her development, along with Bryant Tuckerman, of the first slide attack in cryptanalysis.
References
20th-century American mathematicians
21st-century American mathematicians
American cryptographers
American women mathematicians
Group theorists
Brooklyn College alumni
Courant Institute of Mathematical Sciences alumni
Living people
Year of birth missing (living people)
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women
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https://en.wikipedia.org/wiki/Abdullah%20Sadiq
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Abdullah Sadiq (born 1940), is a Pakistani physicist and ICTP laureate who received the ICTP Prize in the honour of Nikolay Bogolyubov, in the fields of mathematics and solid state physics in 1987 for his contributions to scientific knowledge in the field of mathematics and statistical physics. He is the professor of physics and current dean of the department of physics of the Air University of the Pakistan Air Force (PAF).
Sadiq is also a renowned educationist of Pakistan with a specialisation in nuclear physics, solid-state physics, and computer programming. He has been a distinguished professor of nuclear physics and solid state physics in many universities of Pakistan.
Biography
He did his early education from Islamia Collegiate School. Therefore, after his matriculation from Islamia Collegiate school, Sadiq joined the Islamia College Peshawar in 1958. Influenced by Abdus Salam and his work, Sadiq studied for his double major in physics and mathematics, and learned the Zeeman effect, light interferences using the Pérot and Michelson interferometer. In 1962, Sadiq obtained his BSc in physics, and a minor in mathematics. In 1967, Abdullah Sadiq attended Peshawar University, where he joined the physics department as a graduate student, and taught courses in mathematics. In 1969, he received his MSc in physics under the supervision of physicist Abdul Majid Mian from the University of Peshawar. His mentor, Abdul Majid Mian, refused to recommend him for a job after his college degree and instead advised him to gain a doctorate in physics.
His mentor recommended him to go to the United States and gain a doctorate degree in material physics from the University of Illinois at Urbana-Champaign. He went to United States for higher studies. In 1971, he got his PhD in condensed matter physics under the supervision of Leo Kadanoff from the University of Illinois at Urbana-Champaign, US. Sadiq was a guest lecturer at International Centre for Theoretical Physics (ICTP).
Sadiq was a close friend of Russian theoretical physicist Nikolay Bogolyubov. Sadiq also worked as the rector of Ghulam Ishaq Khan Institute of Engineering Sciences and Technology in Topi, Pakistan. He retired in September 2007 and resided in his home town of Peshawar and focused on his school in RextinKore. Recently, he moved to Islamabad, where he joined Air University at Pakistan Air Force (PAF) as a nuclear physicist and also taught solid state physics and condensed matter physics there. He is also serving as a dean of physics department Air University.
Research in physics
Sadiq's keen research in statistical mechanics, fluid mechanics and superconductivity. Also, his research interests in condensed matter physics have included Ising models, correlated percolation and its relationship to spin glass transition and long-chain polymers. He has been active in the area of computer simulation of physical systems, and his current studies relate to the kinetics of irreversible chemical pro
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https://en.wikipedia.org/wiki/William%20Browder%20%28mathematician%29
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William Browder (born January 6, 1934) is an American mathematician, specializing in algebraic topology, differential topology and differential geometry. Browder was one of the pioneers with Sergei Novikov, Dennis Sullivan and C. T. C. Wall of the surgery theory method for classifying high-dimensional manifolds. He served as president of the American Mathematical Society until 1990.
Life and career
William Browder was born in New York City in 1934, the son of Raisa (née Berkmann), a Jewish Russian woman from Saint Petersburg, and American Communist Party leader Earl Browder, from Wichita, Kansas. His father had moved to the Soviet Union in 1927, where he met and married Raisa. Their sons Felix Browder and Andrew Browder (born 1931) were both born there. He attended local schools. He graduated from the Massachusetts Institute of Technology with a B.S. degree in 1954 and received his Ph.D. from Princeton University in 1958, with a dissertation entitled Homology of Loop Spaces, advised by John Coleman Moore.
Since 1964 Browder has been a professor at Princeton University; he was chair of the mathematics department at Princeton from 1971 to 1973. He was editor of the journal Annals of Mathematics from 1969 to 1981, and president of the American Mathematical Society from 1989 to 1991.
Browder was elected to the United States National Academy of Sciences in 1980, the American Academy of Arts and Sciences in 1984, and the Finnish Society of Sciences and Letters in 1990. In 1994 a conference was held at Princeton in celebration of his 60th birthday. In 2012 a conference was held at Princeton on the occasion of his retirement.
Selected bibliography
Books
"Surgery on Simply-Connected Manifolds", Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 65, Springer-Verlag, Berlin (1972)
"Algebraic Topology and Algebraic K-Theory", Princeton University Press, 1987,
Seminal papers
"Homotopy Type of Differentiable Manifolds", Proc. 1962 Aarhus Conference, published in Proc. 1993
Oberwolfach Novikov Conjecture Conference proceedings, London Mathematical Society Lecture Notes 226 (1995)
"The Kervaire invariant of framed manifolds and its generalization", Annals of Mathematics 90, 157–186 (1969)
See also
Assembly map
Exotic sphere
Kervaire invariant
Normal invariant
Signature (topology)
Surgery exact sequence
Earl Browder, father
Felix Browder, brother
Andrew Browder, brother
Bill Browder, nephew
Joshua Browder, grandnephew
References
External links
William Browder (AMS brief bio)
Browder, William, "My life in mathematics: How I became a mathematician and the milestones of my career" (2012 video)
1934 births
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Academy of Arts and Sciences
Living people
American people of Russian-Jewish descent
Massachusetts Institute of Technology School of Science alumni
Members of the United States National Academy of Sciences
Presidents of the American Mathematic
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https://en.wikipedia.org/wiki/Akshay%20Venkatesh
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Akshay Venkatesh (born 21 November 1981) is an Australian mathematician and a professor (since 15 August 2018) at the School of Mathematics at the Institute for Advanced Study. His research interests are in the fields of counting, equidistribution problems in automorphic forms and number theory, in particular representation theory, locally symmetric spaces, ergodic theory, and algebraic topology.
He was the first Australian to have won medals at both the International Physics Olympiad and International Mathematical Olympiad, which he did at the age of 12.
In 2018, he was awarded the Fields Medal for his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory. He is the second Australian and the second person of Indian descent to win the Fields Medal. He was on the Mathematical Sciences jury for the Infosys Prize in 2020.
Early years
Akshay Venkatesh was born in Delhi, India, and his family emigrated to Perth in Western Australia when he was two years old. He attended Scotch College. His mother, Svetha, is a computer science professor at Deakin University. A child prodigy, Akshay attended extracurricular training classes for gifted students in the state mathematical olympiad program, and in 1993, whilst aged only 11, he competed at the 24th International Physics Olympiad in Williamsburg, Virginia, winning a bronze medal. The following year, he switched his attention to mathematics and, after placing second in the Australian Mathematical Olympiad, he won a silver medal in the 6th Asian Pacific Mathematics Olympiad, before winning a bronze medal at the 1994 International Mathematical Olympiad held in Hong Kong. He completed his secondary education the same year, turning 13 before entering the University of Western Australia as its youngest ever student. Venkatesh completed the four-year course in three years and became, at 16, the youngest person to earn First Class Honours in pure mathematics from the university. He was awarded the J. A. Woods Memorial Prize as the most outstanding graduate of the year from the Faculties of Science, Engineering, Dentistry, or Medical Science. While at UWA he was also one of the founding members of the Honours Cricket Association.
Research career
Akshay commenced his PhD at Princeton University in 1998 under Peter Sarnak, which he completed in 2002, producing the thesis Limiting forms of the trace formula. He was supported by the Hackett Fellowship for postgraduate study. He was then awarded a postdoctoral position at the Massachusetts Institute of Technology, where he served as a C.L.E. Moore instructor. Venkatesh then held a Clay Research Fellowship from the Clay Mathematics Institute from 2004 to 2006, and was an associate professor at the Courant Institute of Mathematical Sciences at New York University. He was a member of the School of Mathematics at the Institute for Advanced Study (IAS) from 2005 to 2006. He became a full professor at Stanford University on 1
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https://en.wikipedia.org/wiki/Quasinorm
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In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by
for some
Definition
A on a vector space is a real-valued map on that satisfies the following conditions:
:
: for all and all scalars
there exists a real such that for all
If then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.
A is a quasi-seminorm that also satisfies:
Positive definite/: if satisfies then
A pair consisting of a vector space and an associated quasi-seminorm is called a .
If the quasi-seminorm is a quasinorm then it is also called a .
Multiplier
The infimum of all values of that satisfy condition (3) is called the of
The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition.
The term is sometimes used to describe a quasi-seminorm whose multiplier is equal to
A (respectively, a ) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is
Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
Topology
If is a quasinorm on then induces a vector topology on whose neighborhood basis at the origin is given by the sets:
as ranges over the positive integers.
A topological vector space with such a topology is called a or just a .
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a . Every Banach space is a quasi-Banach space, although not conversely.
Related definitions
A quasinormed space is called a if the vector space is an algebra and there is a constant such that
for all
A complete quasinormed algebra is called a .
Characterizations
A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.
Examples
Since every norm is a quasinorm, every normed space is also a quasinormed space.
spaces with
The spaces for are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology).
For the Lebesgue space is a complete metrizable TVS (an F-space) that is locally convex (in fact, its only convex open subsets are itself and the empty set) and the continuous linear functional on is the constant function .
In particular, the Hahn-Banach theorem does hold for when
See also
References
Linear algebra
Norms (mathematics)
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https://en.wikipedia.org/wiki/Petrovce%2C%20Rimavsk%C3%A1%20Sobota%20District
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Petrovce () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Star%C3%A1%20Ba%C5%A1ta
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Stará Bašta () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/Gromov%27s%20compactness%20theorem%20%28geometry%29
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In the mathematical field of metric geometry, Mikhael Gromov proved a fundamental compactness theorem for sequences of metric spaces. In the special case of Riemannian manifolds, the key assumption of his compactness theorem is automatically satisfied under an assumption on Ricci curvature. These theorems have been widely used in the fields of geometric group theory and Riemannian geometry.
Metric compactness theorem
The Gromov–Hausdorff distance defines a notion of distance between any two metric spaces, thereby setting up the concept of a sequence of metric spaces which converges to another metric space. This is known as Gromov–Hausdorff convergence. Gromov found a condition on a sequence of compact metric spaces which ensures that a subsequence converges to some metric space relative to the Gromov–Hausdorff distance:
Let be a sequence of compact metric spaces with uniformly bounded diameter. Suppose that for every positive number there is a natural number and, for every , the set can be covered by metric balls of radius . Then the sequence has a subsequence which converges relative to the Gromov–Hausdorff distance.
The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the Arzelà–Ascoli theorem in the theory of uniform convergence. Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to define the asymptotic cone of certain metric spaces. These techniques were later extended by Gromov and others, using the theory of ultrafilters.
Riemannian compactness theorem
Specializing to the setting of geodesically complete Riemannian manifolds with a fixed lower bound on the Ricci curvature, the crucial covering condition in Gromov's metric compactness theorem is automatically satisfied as a corollary of the Bishop–Gromov volume comparison theorem. As such, it follows that:
Consider a sequence of closed Riemannian manifolds with a uniform lower bound on the Ricci curvature and a uniform upper bound on the diameter. Then there is a subsequence which converges relative to the Gromov–Hausdorff distance.
The limit of a convergent subsequence may be a metric space without any smooth or Riemannian structure. This special case of the metric compactness theorem is significant in the field of Riemannian geometry, as it isolates the purely metric consequences of lower Ricci curvature bounds.
References
Sources.
Differential geometry
Theorems in Riemannian geometry
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https://en.wikipedia.org/wiki/Gromov%27s%20compactness%20theorem%20%28topology%29
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In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or (a finite tree of) "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory.
If the complex structures on the curves in the sequence do not vary, only bubbles can occur; nodes can occur only if the complex structures on the domain are allowed to vary. Usually, the energy bound is achieved by considering a symplectic manifold with compatible almost-complex structure as the target, and assuming that curves to lie in a fixed homology class in the target. This is because the energy of such a pseudoholomorphic curve is given by the integral of the target symplectic form over the curve, and thus by evaluating the cohomology class of that symplectic form on the homology class of the curve. The finiteness of the bubble tree follows from (positive) lower bounds on the energy contributed by a holomorphic sphere.
References
Symplectic topology
Compactness theorems
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https://en.wikipedia.org/wiki/Length%20%28disambiguation%29
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Length in its basic meaning is the long dimension of an object.
Length may also refer to:
Mathematics
Arc length, the distance between two points along a section of a curve.
Length of a sequence or tuple, the number of terms. (The length of an -tuple is )
Length of a module, in abstract algebra
Length of a polynomial, the sum of the magnitudes of the coefficients of a polynomial
Length of a vector, the size of a vector
Other uses
Length (phonetics), in phonetics
Vowel length, the perceived duration of a vowel sound
Geminate consonant, the articulation of a consonant for a longer period of time than that of a single instance
Line and length, the direction and point of bouncing on the pitch of a delivery in cricket
Horse length, the length of a horse in equestrianism
Length overall, the maximum length of a vessel's hull measured parallel to the waterline
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https://en.wikipedia.org/wiki/List%20of%20municipalities%20in%20Rio%20Grande%20do%20Norte
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Rio Grande do Norte () is a state located in the Northeast Region of Brazil. According to the 2010 Census conducted by the Brazilian Institute of Geography and Statistics (IBGE), Rio Grande do Norte has a population of 3,168,133 inhabitants over , which makes it the 16th largest state by population and the 22nd largest by area, out of 26 states. It is home to cities such as Natal, Mossoró, and São Gonçalo do Amarante.
The land that became Rio Grande do Norte was a donatário to João de Barros, the factor of the House of India and Mina, from John III of Portugal in 1535; prior to that, the Portuguese Crown owned the land. The French, who trafficked Brazil wood in the area, had a foothold on the land until the Portuguese expelled them in 1598. The Dutch took the land in 1634 as a part of Dutch Brazil and reigned until 1654, when they were defeated by the Portuguese. In 1701, Rio Grande do Norte joined the Captaincy of Pernambuco, and became a province in 1822 and a state of Brazil in 1889.
Rio Grande do Norte is divided into 167 municipalities, which are grouped into four mesoregions and 23 microregions. Of the 167 municipalities, Natal has the highest population, with 803,811 inhabitants, while Viçosa, with 1,618 inhabitants, has the lowest. The largest municipality by area is Mossoró, with an area of ; the smallest is Senador Georgino Avelino, named after the former Senator and Rio Grande do Norte Governor José Georgino Avelino, which covers an area of .
Municipalities
See also
Geography of Brazil
List of cities in Brazil
References
Rio Grande do Norte
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https://en.wikipedia.org/wiki/Tower%20rule
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The tower rule may refer to one of two rules in mathematics:
Law of total expectation, in probability and stochastic theory
a rule governing the degree of a field extension of a field extension in field theory
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https://en.wikipedia.org/wiki/Gnomon%20%28figure%29
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In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger figure of the same shape.
Building figurate numbers
Figurate numbers were a concern of Pythagorean mathematics, and Pythagoras is credited with the notion that these numbers are generated from a gnomon or basic unit. The gnomon is the piece which needs to be added to a figurate number to transform it to the next bigger one.
For example, the gnomon of the square number is the odd number, of the general form 2n + 1, n = 1, 2, 3, ... . The square of size 8 composed of gnomons looks like this:
To transform from the n-square (the square of size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to the end of each row (n elements), one to the end of each column (n elements), and a single one to the corner. For example, when transforming the 7-square to the 8-square, we add 15 elements; these adjunctions are the 8s in the above figure.
This gnomonic technique also provides a proof that the sum of the first n odd numbers is n2; the figure illustrates Applying the same technique to a multiplication table proves that each squared triangular number is a sum of cubes.
Isosceles triangles
In an acute isosceles triangle, it is possible to draw a similar but smaller triangle, one of whose sides is the base of the original triangle. The gnomon of these two similar triangles is the triangle remaining when the smaller of the two similar isosceles triangles is removed from the larger one. The gnomon is itself isosceles if and only if the ratio of the sides to the base of the original isosceles triangle, and the ratio of the base to the sides of the gnomon, is the golden ratio, in which case the acute isosceles triangle is the golden triangle and its gnomon is the golden gnomon.
Conversely, the acute golden triangle can be the gnomon of the obtuse golden triangle in an exceptional reciprocal exchange of roles
Metaphor and symbolism
A metaphor based around the geometry of a gnomon plays an important role in the literary analysis of James Joyce's Dubliners, involving both a play on words between "paralysis" and "parallelogram", and the geometric meaning of a gnomon as something fragmentary, diminished from its completed shape.
Gnomon shapes are also prominent in Arithmetic Composition I, an abstract painting by Theo van Doesburg.
There is also a very short geometric fairy tale illustrated by animations where gnomons play the role of invaders
See also
Theorem of the gnomon
References
Figurate numbers
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https://en.wikipedia.org/wiki/Gnomon%20%28disambiguation%29
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A gnomon is the part of a sundial that casts the shadow.
Gnomon may also refer to:
Gnomon (figure), in geometry, a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram
Gnomon (journal), a German language academic journal of classics
Gnomon (novel), a science fiction novel by Nick Harkaway
Gnomon, the difference between a pair of consecutive figurate numbers
Gnomon, one of the twenty-five fictional islands in the fantasy book series The Books of Abarat
Gnomon School of Visual Effects, a Hollywood-based university
See also
Gnomonic projection
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https://en.wikipedia.org/wiki/Sample-continuous%20process
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In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
Definition
Let (Ω, Σ, P) be a probability space. Let X : I × Ω → S be a stochastic process, where the index set I and state space S are both topological spaces. Then the process X is called sample-continuous (or almost surely continuous, or simply continuous) if the map X(ω) : I → S is continuous as a function of topological spaces for P-almost all ω in Ω.
In many examples, the index set I is an interval of time, [0, T] or [0, +∞), and the state space S is the real line or n-dimensional Euclidean space Rn.
Examples
Brownian motion (the Wiener process) on Euclidean space is sample-continuous.
For "nice" parameters of the equations, solutions to stochastic differential equations are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
The process X : [0, +∞) × Ω → R that makes equiprobable jumps up or down every unit time according to
is not sample-continuous. In fact, it is surely discontinuous.
Properties
For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.
See also
Continuous stochastic process
References
Stochastic processes
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https://en.wikipedia.org/wiki/Law%20%28stochastic%20processes%29
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In mathematics, the law of a stochastic process is the measure that the process induces on the collection of functions from the index set into the state space. The law encodes a lot of information about the process; in the case of a random walk, for example, the law is the probability distribution of the possible trajectories of the walk.
Definition
Let (Ω, F, P) be a probability space, T some index set, and (S, Σ) a measurable space. Let X : T × Ω → S be a stochastic process (so the map
is an (S, Σ)-measurable function for each t ∈ T). Let ST denote the collection of all functions from T into S. The process X (by way of currying) induces a function ΦX : Ω → ST, where
The law of the process X is then defined to be the pushforward measure
on ST.
Example
The law of standard Brownian motion is classical Wiener measure. (Indeed, many authors define Brownian motion to be a sample continuous process starting at the origin whose law is Wiener measure, and then proceed to derive the independence of increments and other properties from this definition; other authors prefer to work in the opposite direction.)
See also
Finite-dimensional distribution
Stochastic processes
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https://en.wikipedia.org/wiki/Kolmogorov%20extension%20theorem
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In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
Statement of the theorem
Let denote some interval (thought of as "time"), and let . For each and finite sequence of distinct times , let be a probability measure on Suppose that these measures satisfy two consistency conditions:
1. for all permutations of and measurable sets ,
2. for all measurable sets ,
Then there exists a probability space and a stochastic process such that
for all , and measurable sets , i.e. has as its finite-dimensional distributions relative to times .
In fact, it is always possible to take as the underlying probability space and to take for the canonical process . Therefore, an alternative way of stating Kolmogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure on with marginals for any finite collection of times . Kolmogorov's extension theorem applies when is uncountable, but the price to pay
for this level of generality is that the measure is only defined on the product σ-algebra of , which is not very rich.
Explanation of the conditions
The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process . Then the probability can be computed either as or as . Hence, for the finite-dimensional distributions to be consistent, it must hold that
.
The first condition generalizes this statement to hold for any number of time points , and any control sets .
Continuing the example, the second condition implies that . Also this is a trivial condition that will be satisfied by any consistent family of finite-dimensional distributions.
Implications of the theorem
Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.
The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process. The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Man
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https://en.wikipedia.org/wiki/Sublime%20number
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In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.
The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.
There are only two known sublime numbers: 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) . The second of these has 76 decimal digits:
6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.
References
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Cubic%20form
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In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic plane curve.
In , Boris Delone and Dmitry Faddeev showed that binary cubic forms with integer coefficients can be used to parametrize orders in cubic fields. Their work was generalized in to include all cubic rings (a is a ring that is isomorphic to Z3 as a Z-module), giving a discriminant-preserving bijection between orbits of a GL(2, Z)-action on the space of integral binary cubic forms and cubic rings up to isomorphism.
The classification of real cubic forms is linked to the classification of umbilical points of surfaces. The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus.
Examples
Cubic plane curve
Elliptic curve
Fermat cubic
Cubic 3-fold
Koras–Russell cubic threefold
Klein cubic threefold
Segre cubic
Notes
References
Multilinear algebra
Algebraic geometry
Algebraic varieties
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https://en.wikipedia.org/wiki/Italian%20society%20of%20economics%20demography%20and%20statistics
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The Italian society of economics demography and statistics (SIEDS, ) is a learned society aiming to further economic, demographic, and statistical studies and to establish active co-operation among professionals of the mentioned subjects in the field of social sciences and human behaviour. The society pursues this aim by:
organising seminars, congresses, or scientific meetings for analysing and discussing problems concerning its activity;
implementing scientific surveys, enquires and researches and promoting, training activities (courses, seminars, etc.);
publishing the Rivista italiana di economia demografia e statistica, a series of studies and monographs on specific items concerning the scientific interest of the society, as well as the proceedings of its congresses and seminars.
History
SIEDS was founded on June 29, 1939, at the initiative of Livio Livi and others. The first seat of the society was located in Florence. Initially named "Italian Society of Demography and Statistics", it originated in the Advisory Committee for the Population Study (), also founded by Livi. The first publications of the society were only the scientific meeting proceedings but, starting from January 1947, the society published its own scientific journal, the Rivista italiana di economia demografia e statistica. On 18 April 1950 the society decided to expand its fields of interest also to economics and obtained its current name. In 1950 the society's journal was renamed accordingly to Italian Review of Economics, Demography and Statistics. During the first meeting of the society, held at Istat (Rome, 28–29 May 1939), Livi underlined that at that time there were already 16 statistical societies in Europe as well as in North America, Argentina, Brazil, India, China, and Japan. Therefore, Italian academics thought that the time was arrived also for their own wide national society. During the scientific meeting held on December 28, 1940 the society already counted 122 ordinary members.
External links
Economics societies
Statistical societies
Demographics of Italy
Demographics organizations
Organizations established in 1939
1939 establishments in Italy
Organisations based in Rome
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https://en.wikipedia.org/wiki/Studen%C3%A1%2C%20Slovakia
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Studená () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/S%C3%BAtor
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Sútor () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/%C5%A0imonovce
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Šimonovce () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/%C5%A0irkovce
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Širkovce () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/%C5%A0panie%20Pole
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Španie Pole () is a village and municipality in the Rimavská Sobota District of the Banská Bystrica Region of southern Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Rimavská Sobota District
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https://en.wikipedia.org/wiki/United%20States%20Consumer%20Price%20Index
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The United States Consumer Price Index (CPI) is a set of various consumer price indices published monthly by the U.S. Bureau of Labor Statistics (BLS). The most commonly used are the CPI-U and the CPI-W, though many alternative versions exist. The CPI-U is the most popular measure of consumer inflation in the United States (though Social Security benefit payments are indexed to the CPI-W).
Methodology
Item coverage
The CPI measures the monthly price change of a basket of discretionary consumption goods whose price is borne by the consumer. There are eight major categories of items that are included in the CPI coverage; each includes both goods and services:
Food and beverages
Housing
Apparel
Transportation
Medical care
Recreation
Education and communication
Other goods and services
In line with this framework, the CPI excludes items such as life insurance, investment securities, financing costs, house prices (though the value of owned housing, distinct from a house price, is included in the CPI), as these are considered to be investment items, not consumption. Also excluded are income and property taxes, employer-provided benefits, and the portion of healthcare costs paid by the government or insurance plans, since these prices are not borne directly by consumers. However, sales and excise taxes, out-of-pocket healthcare costs, and health insurance premiums paid by the consumer (including Medicare Part B) are all included in the CPI, because consumers directly bear these costs. Finally, the prices of illegal goods such as marijuana are not measured, and so are also excluded. Some items, such as pleasure boats and pleasure aircraft, indeed belong in the scope of the CPI but are impractical to price; for these items, the price change is imputed to be the price change of a larger relevant category (eg. pleasure vehicles), while the weight is properly measured in the Consumer Expenditure Survey.
Population coverage and geographic sample
The CPI-U measures inflation as experienced by a representative household in a metropolitan statistical area. Rural (non-metropolitan) households, farm households, military members, and the institutionalized (eg. prisons or hospitals) are excluded from consideration; with this exclusion, the CPI-U covers about 93 percent of the US population. The items considered, prices collected, and the locations where the prices are collected are all designed to represent the spending habits of such households.
The BLS divides the urban population into Primary Sampling Units (PSUs), equivalent to core-based statistical areas from the 2010 United States census. Prices are measured in only 75 of these PSUs. 23 of these CBSAs are known as self-representing PSUs, whose measured price changes apply to only that PSU. Of these CBSAs, 21 are metropolitan statistical areas with a population greater than 2.5 million (such as the Detroit-Warren-Dearborn, MI Metropolitan Statistical Area), while the remaining two are Anchorag
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https://en.wikipedia.org/wiki/Basic%20pitch%20count%20estimator
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In baseball statistics, the basic pitch count estimator is a statistic used to estimate the number of pitches thrown by a pitcher where there is no pitch count data available. The formula was first derived by Tom Tango. The formula is , where PA refers to the number of plate appearances against the pitcher, SO to strikeouts and BB to base on balls.
See also
Pitch count
Batters faced by pitcher
References
Pitching statistics
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https://en.wikipedia.org/wiki/Rural%20Municipality%20of%20Piney
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The Rural Municipality of Piney is a rural municipality (RM) in southeastern Manitoba, Canada, along the border with Minnesota in the United States.
Geography
According to Statistics Canada, the municipality has a land area of 2,433.77 km2. It is bordered by the rural municipalities of Stuartburn, La Broquerie, and Reynolds, as well as the Buffalo Point 36 Indian reserve and an unorganized part of the province (Division No. 1, Unorganized, Manitoba).
The municipality borders Roseau County in the U.S. state of Minnesota. There are three international border crossings in Piney, the most of any Manitoba municipality: Pinecreek–Piney, Roseau–South Junction, and Warroad–Sprague Border Crossings.
A minority but large part of Sandilands Provincial Forest is located here, as is a small part of Northwest Angle Provincial Forest. Also, Cat Hills Provincial Forest and Wampum Provincial Forest are both entirely located here, but these two forests are relatively tiny in size.
Communities
Badger
Carrick
Menisino
Middlebro
Piney
St. Labre
Sandilands
South Junction
Sprague
Vassar
Wampum
Woodridge
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Piney had a population of 1,843 living in 804 of its 1,172 total private dwellings, a change of from its 2016 population of 1,726. With a land area of , it had a population density of in 2021.
In 2001, there were 690 housing units at an average density of 0.28/km2. 2.4% of the people in the municipality are members of a visible minority.
Of the 695 households, 24.5% had a couple (married or common law) with children, 38.1% had a couple without children, 26.6% had one person, and 10.1% had another household type. The average household size was 2.43 and the average family size was 3.48.
In the municipality the population was spread out, with 18.0% under the age of 15, 11.0% from 15 to 24, 22.3% from 25 to 44, 29.1% from 45 to 64, and 19.9% 65 or older. The median age was 44.2 years. For every 100 females there were 113.9 males. For every 100 females age 15 and over, there were 103.6 males.
The median household income was $32,237 and the median family income was $39,525. Males had a median income of $23,726 versus $19,268 for females. The per capita income for the municipality was $10,043.
Climate
Gallery
See also
Pinecreek–Piney Border Crossing
Roseau–South Junction Border Crossing
Warroad–Sprague Border Crossing
References
External links
Official website
Map of Piney R.M. at Statcan
Rural municipalities in Manitoba
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https://en.wikipedia.org/wiki/Fielden%20Professor%20of%20Pure%20Mathematics
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The Fielden Chair of Pure Mathematics is an endowed professorial position in the School of Mathematics, University of Manchester, England.
History
In 1870 Samuel Fielden, a wealthy mill owner from Todmorden, donated £150 to Owens College (as the Victoria University of Manchester was then called) for the teaching of evening classes and a further £3000 for the development of natural sciences at the college. From 1877 this supported the Fielden Lecturer, subsequently to become the Fielden Reader with the appointment of L. J. Mordell in 1922 and then Fielden Professor in 1923. Alex Wilkie FRS was appointed to the post in 2007.
Holders
Previous holders of the Fielden Chair (and lectureship) are:
A. T. Bentley (1876–1880) Lecturer in Pure Mathematics
J. E. A. Steggall (1880–1883) Lecturer in Pure Mathematics
R. F. Gwyther (1883–1907) Lecturer in Mathematics
F. T. Swanwick (1907–1912) Lecturer in Mathematics
H. R. Hasse (1912–1918) Lecturer in Mathematics
George Henry Livens (1920–1922) Lecturer in Mathematics
Louis Mordell (1923–1945)
Max Newman (1945–1964)
Frank Adams (1964–1971)
Ian G. Macdonald (1972–1976)
Norman Blackburn (1978–1994)
Mark Pollicott (1996–2004)
Alex Wilkie (2007–)
Related chairs
The other endowed chairs in mathematics at the University of Manchester are the Beyer Chair of Applied Mathematics, the Sir Horace Lamb Chair and the Richardson Chair of Applied Mathematics.
References
Professorships in mathematics
Mathematics education in the United Kingdom
Professorships at the University of Manchester
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https://en.wikipedia.org/wiki/Carl%20Benjamin%20Boyer
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Carl Benjamin Boyer (November 3, 1906 – April 26, 1976) was an American historian of sciences, and especially mathematics. Novelist David Foster Wallace called him the "Gibbon of math history". It has been written that he was one of few historians of mathematics of his time to "keep open links with contemporary history of science."
Life and career
Boyer was valedictorian of his high school class. He received a B.A. from Columbia College in 1928 and an M.A. in 1929. He received his Ph.D. in Mathematics from Columbia University in 1939. He was a full professor of Mathematics at the City University of New York's Brooklyn College from 1952 until his death, although he had begun tutoring and teaching at Brooklyn College in 1928.
Along with Carolyn Eisele of CUNY's Hunter College; C. Doris Hellman of the Pratt Institute, and later CUNY's Queens College; and Lynn Thorndike of Columbia University, Boyer was instrumental in the 1953 founding of the Metropolitan New York Section of the History of Science Society.
In 1954, Boyer was the recipient of a Guggenheim Fellowship to further his work in the history of science. In particular, the grant made reference to "the history of the theory of the rainbow".
Boyer wrote the books The History of the Calculus and Its Conceptual Development (1959), originally published as The Concepts of the Calculus (1939), History of Analytic Geometry (1956), The Rainbow: From Myth to Mathematics (1959), and A History of Mathematics (1968). He served as book-review editor of Scripta Mathematica.
Boyer died of a heart attack in New York City in 1976.
In 1978, Boyer's widow, the former Marjorie Duncan Nice, a professor of history, established the Carl B. Boyer Memorial Prize, to be awarded annually to a Columbia University undergraduate for the best essay on a scientific or mathematical topic.
References
Notes
Further reading
Boyer, Carl B. (August 30–September 6, 1950). Lecture: "The Foremost Textbook of Modern Times." International Congress of Mathematicians, Cambridge, Massachusetts. Retrieved on 2009-02-20.
Boyer, Carl B. (1949). The history of the calculus and its conceptual development Hafner Publishing Company, New York, ed. Dover 1959. Retrieved on 2010-03-30.
External links
1906 births
1976 deaths
20th-century American mathematicians
American historians of mathematics
20th-century American historians
20th-century American male writers
Columbia College (New York) alumni
Columbia Graduate School of Arts and Sciences alumni
Brooklyn College faculty
American male non-fiction writers
Educators from Pennsylvania
People from Lehigh County, Pennsylvania
Writers from Pennsylvania
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https://en.wikipedia.org/wiki/List%20of%20misnamed%20theorems
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This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, these items on this list illustrate Stigler's law of eponymy (which is not, of course, due to Stephen Stigler, who credits Robert K Merton).
== Applied mathematics ==
Benford's law. This was first stated in 1881 by Simon Newcomb, and rediscovered in 1938 by Frank Benford. The first rigorous formulation and proof seems to be due to Ted Hill in 1988.; see also the contribution by Persi Diaconis.
Bertrand's ballot theorem. This result concerning the probability that the winner of an election was ahead at each step of ballot counting was first published by W. A. Whitworth in 1878, but named after Joseph Louis François Bertrand who rediscovered it in 1887. A common proof uses André's reflection method, though the proof by Désiré André did not use any reflections.
Algebra
Burnside's lemma. This was stated and proved without attribution in Burnside's 1897 textbook, but it had previously been discussed by Augustin Cauchy, in 1845, and by Georg Frobenius in 1887.
Cayley–Hamilton theorem. The theorem was first proved in the easy special case of 2×2 matrices by Cayley, and later for the case of 4×4 matrices by Hamilton. But it was only proved in general by Frobenius in 1878.
Hölder's inequality. This inequality was first established by Leonard James Rogers, and published in 1888. Otto Hölder discovered it independently, and published it in 1889.
Marden's theorem. This theorem relating the location of the zeros of a complex cubic polynomial to the zeros of its derivative was named by Dan Kalman after Kalman read it in a 1966 book by Morris Marden, who had first written about it in 1945. But, as Marden had himself written, its original proof was by Jörg Siebeck in 1864.
Pólya enumeration theorem. This was proven in 1927 in a difficult paper by J. H. Redfield. Despite the prominence of the venue (the American Journal of Mathematics), the paper was overlooked. Eventually, the theorem was independently rediscovered in 1936 by George Pólya. Not until 1960 did Frank Harary unearth the much earlier paper by Redfield. See for historical and other information.
Analysis
Frobenius theorem. This fundamental theorem was stated and proved in 1840 by Feodor Deahna. Even though Frobenius cited Deahna's paper in his own 1875 paper, it became known after Frobenius, not Deahna. See for a historical review.
L'Hôpital's rule. This rule first appeared in l'Hôpital's book L'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes in 1696. The rule is believed to be the work of Johann Bernoulli since l'Hôpital, a nobleman, paid Bernoulli a retainer of 300 francs per year to keep him updated on developments in calculus and to solve problems he had. See L'Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes and
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https://en.wikipedia.org/wiki/Inner%20regular%20measure
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In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
Definition
Let (X, T) be a Hausdorff topological space and let Σ be a σ-algebra on X that contains the topology T (so that every open set is a measurable set, and Σ is at least as fine as the Borel σ-algebra on X). Then a measure μ on the measurable space (X, Σ) is called inner regular if, for every set A in Σ,
This property is sometimes referred to in words as "approximation from within by compact sets."
Some authors use the term tight as a synonym for inner regular. This use of the term is closely related to tightness of a family of measures, since a finite measure μ is inner regular if and only if, for all ε > 0, there is some compact subset K of X such that μ(X \ K) < ε. This is precisely the condition that the singleton collection of measures {μ} is tight.
Examples
When the real line R is given its usual Euclidean topology,
Lebesgue measure on R is inner regular; and
Gaussian measure (the normal distribution on R) is an inner regular probability measure.
However, if the topology on R is changed, then these measures can fail to be inner regular. For example, if R is given the lower limit topology (which generates the same σ-algebra as the Euclidean topology), then both of the above measures fail to be inner regular, because compact sets in that topology are necessarily countable, and hence of measure zero.
References
See also
Radon measure
Regular measure
Measures (measure theory)
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https://en.wikipedia.org/wiki/Ramanujan%27s%20congruences
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In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The mathematician Srinivasa Ramanujan discovered the congruences
This means that:
If a number is 4 more than a multiple of 5, i.e. it is in the sequence
4, 9, 14, 19, 24, 29, . . .
then the number of its partitions is a multiple of 5.
If a number is 5 more than a multiple of 7, i.e. it is in the sequence
5, 12, 19, 26, 33, 40, . . .
then the number of its partitions is a multiple of 7.
If a number is 6 more than a multiple of 11, i.e. it is in the sequence
6, 17, 28, 39, 50, 61, . . .
then the number of its partitions is a multiple of 11.
Background
In his 1919 paper, he proved the first two congruences using the following identities (using q-Pochhammer symbol notation):
He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these".
After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs the Eisenstein series.
In 1944, Freeman Dyson defined the rank function and conjectured the existence of a crank function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan found such a function, and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7 and 11.
In the 1960s, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences for small prime moduli. For example:
Extending the results of A. Atkin, Ken Ono in 2000 proved that there are such Ramanujan congruences modulo every integer coprime to 6. For example, his results give
Later Ken Ono conjectured that the elusive crank also satisfies exactly the same types of general congruences. This was proved by his Ph.D. student Karl Mahlburg in his 2005 paper Partition Congruences and the Andrews–Garvan–Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences Paper of the Year prize.
A conceptual explanation for Ramanujan's observation was finally discovered in January 2011 by considering the Hausdorff dimension of the following function in the l-adic topology:
It is seen to have dimension 0 only in the cases where ℓ = 5, 7 or 11 and since the partition function can be written as a linear combination of these functions this can be considered a formalization and proof of Ramanujan's observation.
In 2001, R.L. Weaver gave an effective algorithm for finding congruences of the partition function, and tabulated 76,065 congruences. This was extended in 2012 by F. Johansson to 22,474,608,014 congruences, one large example being
See also
Tau-function, for which there are other so-called Ramanujan congruences
Rank of a partition
Crank of a partition
References
External lin
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https://en.wikipedia.org/wiki/Wasserstein%20metric
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In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space . It is named after Leonid Vaseršteĭn.
Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on , the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781. Because of this analogy, the metric is known in computer science as the earth mover's distance.
The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after learning of it in the work of Leonid Vaseršteĭn on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by Leonid Kantorovich in The Mathematical Method of Production Planning and Organization (Russian original 1939) in the context of optimal transport planning of goods and materials. Some scholars thus encourage use of the terms "Kantorovich metric" and "Kantorovich distance". Most English-language publications use the German spelling "Wasserstein" (attributed to the name "Vaseršteĭn" () being of German origin).
Definition
Let be a metric space that is a Radon space. For , the Wasserstein -distance between two probability measures and on with finite -moments is
where is the set of all couplings of and ; is defined to be and corresponds to a supremum norm. A coupling is a joint probability measure on whose marginals are and on the first and second factors, respectively. That is,
Intuition and connection to optimal transport
One way to understand the above definition is to consider the optimal transport problem. That is, for a distribution of mass on a space , we wish to transport the mass in such a way that it is transformed into the distribution on the same space; transforming the 'pile of earth' to the pile . This problem only makes sense if the pile to be created has the same mass as the pile to be moved; therefore without loss of generality assume that and are probability distributions containing a total mass of 1. Assume also that there is given some cost function
that gives the cost of transporting a unit mass from the point to the point .
A transport plan to move into can be described by a function which gives the amount of mass to move from to . You can imagine the task as the need to move a pile of earth of shape to the hole in the ground of shape such that at the end, both the pile of earth and the hole in the ground completely vanish. In order for this plan to be meaningful, it must satisfy the following properties:
the amount of earth moved out of point must equal the amount that was there to begin with; that is,
and
the amount of earth moved into point must equal the depth of the hole that was there at the beginning; that is,
That is, that the total mas
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https://en.wikipedia.org/wiki/Philosophy%20of%20Arithmetic
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Philosophy of Arithmetic: Psychological and Logical Investigations () is an 1891 book about the philosophy of mathematics by the philosopher Edmund Husserl. Husserl's first published book, it is a synthesis of his studies in mathematics, under Karl Weierstrass, with his studies in philosophy and psychology, under Franz Brentano, to whom it is dedicated, and Carl Stumpf.
Structure
The Philosophy of Arithmetic constitutes the first volume of a work which Husserl intended to comprise two volumes, of which the second was never published. Comprehensively it would have encompassed four parts and an Appendix.
The first volume is divided in two parts, in the first of which Husserl purports to analyse the "Proper concepts of multiplicity, unity and amount" (Die eigentliche Begriffe von Vielheit, Einheit und Anzahl) and in the second "The symbolic amount-concepts and the logical sources of amount-arithmetic" (Die symbolischen Anzahlbegrife und die logischen Quellen der Anzahlen-Arithmetik).
Content
The basic issue of the book is a philosophical analysis of the concept of number, which is the most basic concept on which the entire edifice of arithmetic and mathematics can be founded. In order to proceed with this analysis, Husserl, following Brentano and Stumpf, uses the tools of psychology to look for the "origin and content" of the concept of number. He begins with the classical definition, already given by Euclid, Thomas Hobbes and Gottfried Wilhelm Leibniz, that "number is a multiplicity of unities" and then asks himself: what is multiplicity and what is unity? Anything that we can think of, anything we can present, can be considered at its most basic level to be "something". Multiplicity is then the "collective connection" of "something and something and something etc." In order to get a number instead of a mere quantity, we can also think of these featureless, abstract "somethings" as "ones" and then get "one and one and one etc." as basic definition of number in abstracto. However, these are just the proper numbers, i.e. number which we can conceive of properly, without the help of instruments or symbols. Psychologically we are limited to just the very first few numbers if we want to conceive of them properly, with higher numbers our short-term memory is not enough to think of them all together, but still as identical to themselves and different from all others. Husserl contends that as a result, we must proceed to the analysis of symbolically conceived numbers, which are in essence the numbers used in mathematics.
History
The book is a product of Husserl's years of study with Weierstrass (in Berlin) and his student Leo Königsberger (in Vienna) on the mathematical side and his studies with Brentano (in Vienna) and Stumpf (in Halle) on the psychological/philosophical side. The book is mostly based on his habilitationsschrift of 1887 "On the Concept of Number" (Über den Begriff der Zahl). Husserl also lectured on the concept of number between 18
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https://en.wikipedia.org/wiki/Varifold
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In mathematics, a varifold is, loosely speaking, a measure-theoretic generalization of the concept of a differentiable manifold, by replacing differentiability requirements with those provided by rectifiable sets, while maintaining the general algebraic structure usually seen in differential geometry. Varifolds generalize the idea of a rectifiable current, and are studied in geometric measure theory.
Historical note
Varifolds were first introduced by Laurence Chisholm Young in , under the name "generalized surfaces". Frederick J. Almgren Jr. slightly modified the definition in his mimeographed notes and coined the name varifold: he wanted to emphasize that these objects are substitutes for ordinary manifolds in problems of the calculus of variations. The modern approach to the theory was based on Almgren's notes and laid down by William K. Allard, in the paper .
Definition
Given an open subset of Euclidean space , an m-dimensional varifold on is defined as a Radon measure on the set
where is the Grassmannian of all m-dimensional linear subspaces of an n-dimensional vector space. The Grassmannian is used to allow the construction of analogs to differential forms as duals to vector fields in the approximate tangent space of the set .
The particular case of a rectifiable varifold is the data of a m-rectifiable set M (which is measurable with respect to the m-dimensional Hausdorff measure), and a density function defined on M, which is a positive function θ measurable and locally integrable with respect to the m-dimensional Hausdorff measure. It defines a Radon measure V on the Grassmannian bundle of ℝn
where
is the −dimensional Hausdorff measure
Rectifiable varifolds are weaker objects than locally rectifiable currents: they do not have any orientation. Replacing M with more regular sets, one easily see that differentiable submanifolds are particular cases of rectifiable manifolds.
Due to the lack of orientation, there is no boundary operator defined on the space of varifolds.
See also
Current
Geometric measure theory
Grassmannian
Plateau's problem
Radon measure
Notes
References
. This paper is also reproduced in .
.
.
.
.
. A set of mimeographed notes where Frederick J. Almgren Jr. introduces varifolds for the first time.
. The first widely circulated book describing the concept of a varifold. In chapter 4 is a section titled "A solution to the existence portion of Plateau's problem" but the stationary varifolds used in this section can only solve a greatly simplified version of the problem. For example, the only stationary varifolds containing the unit circle have support the unit disk. In 1968 Almgren used a combination of varifolds, integral currents, flat chains and Reifenberg's methods in an attempt to extend Reifenberg's celebrated 1960 paper to elliptic integrands. However, there are serious errors in his proof. A different approach to the Reifenberg problem for elliptic integrands has been recently provided by Harrison and
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https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner%20evolution
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In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter κ and a domain in the complex plane U, it gives a family of random curves in U, with κ controlling how much the curve turns. There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property.
It was discovered by as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendelin Werner in a series of joint papers.
Besides UST and LERW, the Schramm–Loewner evolution is conjectured or proven to describe the scaling limit of various stochastic processes in the plane, such as critical percolation, the critical Ising model, the double-dimer model, self-avoiding walks, and other critical statistical mechanics models that exhibit conformal invariance. The SLE curves are the scaling limits of interfaces and other non-self-intersecting random curves in these models. The main idea is that the conformal invariance and a certain Markov property inherent in such stochastic processes together make it possible to encode these planar curves into a one-dimensional Brownian motion running on the boundary of the domain (the driving function in Loewner's differential equation). This way, many important questions about the planar models can be translated into exercises in Itô calculus. Indeed, several mathematically non-rigorous predictions made by physicists using conformal field theory have been proven using this strategy.
The Loewner equation
If is a simply connected, open complex domain not equal to , and is a simple curve in starting on the boundary (a continuous function with on the boundary of and a subset of ), then for each , the complement
of is simply connected and therefore conformally isomorphic to by the Riemann mapping theorem. If is a suitable normalized isomorphism from to , then it satisfies a differential equation found by in his work on the Bieberbach conjecture.
Sometimes it is more convenient to use the inverse function of , which is a conformal mapping from to .
In Loewner's equation, , , and the boundary values at time are or
. The equation depends on a driving function taking values in the boundary of . If
is the unit disk and the curve is parameterized by "capacity", then Loewner's equation is
or
When is the upper half plane the Loewner equation differs from this by changes of variable and is
or
The driving function and the curve
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https://en.wikipedia.org/wiki/Pravica
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Pravica () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Vinica%2C%20Ve%C4%BEk%C3%BD%20Krt%C3%AD%C5%A1%20District
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Vinica () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.vinica.sk
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Vieska%2C%20Ve%C4%BEk%C3%BD%20Krt%C3%AD%C5%A1%20District
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Vieska () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Quiso
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Quiso may refer to:
A fictionary island in the novel Shardik
A Quasi-isomorphism in mathematics
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https://en.wikipedia.org/wiki/Random%20effects%20model
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In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model.
Contrast this to the biostatistics definitions, as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables).
Qualitative description
Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the model.
Two common assumptions can be made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual unobserved heterogeneity is uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables.
If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects model.
Simple example
Suppose m large elementary schools are chosen randomly from among thousands in a large country. Suppose also that n pupils of the same age are chosen randomly at each selected school. Their scores on a standard aptitude test are ascertained. Let Yij be the score of the jth pupil at the ith school. A simple way to model this variable is
where μ is the average test score for the entire population. In this model Ui is the school-specific random effect: it measures the difference between the average score at school i and the average score in the entire country. The term Wij is the individual-specific random effect, i.e., it's the deviation of the j-th pupil's score from the average for the i-th school.
The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups. For example:
where Sexij is the dummy variable for boys/girls and ParentsEducij records, say, the average education level of a child's parents. This is a mixed model, not a purely random effects model, as it introduces fixed-effects terms for Sex and Parents' Education.
Variance components
The variance of Yij is the sum of the variances τ2 and σ2 of Ui and Wij respectively.
Let
be the average, not of all scores at the ith school, but of those at the ith school that are included in the random sample. Let
be the grand average.
Let
be respectively the sum of squares due to differences within groups and the sum of squares du
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https://en.wikipedia.org/wiki/Mehdi%20Hasheminasab
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Seyyed Mehdi Hasheminasab (; born January 27, 1974) is a retired Iranian footballer.
Club career
He served his golden days in Persepolis and Esteghlal.
Club career statistics
International career
After a number of very good seasons with Persepolis, Hasheminasab was called up to the national team, earning his first cap versus Kuwait on February 15, 1999. He was also a member of the national team during the World Cup 2002 qualification campaign. Some consider him and a number of other players to be responsible for the poor atmosphere in the national team camp, and its eventual failure to qualify. In his career he achieved 28 caps and 2 goals.
References
External links
Mehdi Hasheminasab at PersianLeague.com
Mehdi Hasheminasab at TeamMelli.com
Quds Daily
Tebyan
1973 births
Living people
Footballers from Abadan, Iran
Iranian men's footballers
Iran men's international footballers
Iranian expatriate men's footballers
Expatriate men's footballers in Turkmenistan
FK Köpetdag Aşgabat players
Esteghlal F.C. players
Persepolis F.C. players
Siah Jamegan F.C. players
PAS Tehran F.C. players
Sanat Naft Abadan F.C. players
Payam Khorasan F.C. players
F.C. Aboomoslem players
Azadegan League players
2000 AFC Asian Cup players
Men's association football defenders
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https://en.wikipedia.org/wiki/Batting%20average%20on%20balls%20in%20play
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In baseball statistics, batting average on balls in play (abbreviated BABIP) is a measurement of how often batted balls result in hits, excluding home runs. It can be expressed as, "when you hit the ball and it’s not a home run, what’s your batting average?" The statistic is typically used to evaluate individual batters and individual pitchers.
Calculation
BABIP is computed per the following equation, where H is hits, HR is home runs, AB is at bats, K is strikeouts, and SF is sacrifice flies.
Effect
As compared to batting average, which is simply hits divided by at bats, BABIP excludes home runs and strikeouts from consideration while treating sacrifice flies as hitless at bats.
In Major League Baseball (MLB), .300 is considered an average BABIP. Various factors can impact BABIP, such as a player's home ballpark; for batters, being speedy enough to reach base on infield hits; or, for pitchers, the quality of their team's defense.
Usage
BABIP is commonly used as a red flag in sabermetric analysis, as a consistently high or low BABIP is hard to maintain—much more so for pitchers than hitters. Therefore, BABIP can be used to spot outlying seasons by pitchers. As with other statistical measures, those pitchers whose BABIPs are extremely high (bad) can often be expected to improve in the following season, and those pitchers whose BABIPs are extremely low (good) can often be expected to worsen in the following season.
While a pitcher's BABIP may vary from season to season, there are distinct differences between pitchers when looking at career BABIP figures.
See also
Defense independent pitching statistics
References
Batting statistics
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https://en.wikipedia.org/wiki/Catcher%27s%20ERA
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Catcher's ERA (CERA) in baseball statistics is the earned run average of the pitchers pitching when the catcher in question is catching. Its primary purpose is to measure a catcher's game-calling, rather than his effect on the opposing team's running game. Craig Wright first described the concept of CERA in his 1989 book The Diamond Appraised. With it, Wright developed a method of determining a catcher's effect on a team's pitching staff by comparing pitchers' performance when playing with different catchers.
Baseball Prospectus writer Keith Woolner has written that "catcher game-calling isn't a statistically significant skill" after doing statistical analysis of catcher performance. Sabermetrician Bill James also performed research into CERA, finding that while it is possible that catchers may have a significant effect on a pitching staff, there is too much yearly variation in CERA for it to be a reliable indicator of ability. James used simulations of catchers with assigned defensive values to directly compare CERAs, which influenced Woolner to perform similar simulations but instead using weighted events to calculate pitchers' runs per plate appearance. Through this, Woolner concluded that even if catchers do have an effect on pitchers' abilities to prevent runs, it is undetectable and thus has no practical usage. He also stated that "the hypothesis most consistent with the available facts appears to be that catchers do not have a significant effect on pitcher performance".
References
See also
Earned run
Earned run average
Catching statistics
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https://en.wikipedia.org/wiki/Canons%20of%20page%20construction
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The canons of page construction are historical reconstructions, based on careful measurement of extant books and what is known of the mathematics and engineering methods of the time, of manuscript-framework methods that may have been used in Medieval- or Renaissance-era book design to divide a page into pleasing proportions. Since their popularization in the 20th century, these canons have influenced modern-day book design in the ways that page proportions, margins and type areas (print spaces) of books are constructed.
The notion of canons, or laws of form, of book page construction was popularized by Jan Tschichold in the mid to late twentieth century, based on the work of J. A. van de Graaf, Raúl Rosarivo, Hans Kayser, and others. Tschichold wrote, "Though largely forgotten today, methods and rules upon which it is impossible to improve have been developed for centuries. To produce perfect books these rules have to be brought to life and applied." as cited in . Kayser's 1946 Ein harmonikaler Teilungskanon had earlier used the term canon in this context.
Typographers and book designers are influenced by these principles to this day in page layout, with variations related to the availability of standardized paper sizes, and the diverse types of commercially printed books.
Van de Graaf canon
The Van de Graaf canon is a historical reconstruction of a method that may have been used in book design to divide a page in pleasing proportions. This canon is also known as the "secret canon" used in many medieval manuscripts and incunabula.
The geometrical solution of the construction of Van de Graaf's canon, which works for any page width:height ratio, enables the book designer to position the type area in a specific area of the page. Using the canon, the proportions are maintained while creating pleasing and functional margins of size 1/9 and 2/9 of the page size. The resulting inner margin is one-half of the outer margin, and of proportions 2:3:4:6 (inner:top:outer:bottom) when the page proportion is 2:3 (more generally 1:R:2:2R for page proportion 1:R). This method was discovered by Van de Graaf, and used by Tschichold and other contemporary designers; they speculate that it may be older. The page proportions vary, but most commonly used is the 2:3 proportion. Tschichold writes "For purposes of better comparison I have based his figure on a page proportion of 2:3, which Van de Graaf does not use." In this canon the type area and page size are of same proportions, and the height of the type area equals the page width. This canon was popularized by Jan Tschichold in his book The Form of the Book.
Robert Bringhurst, in his The Elements of Typographic Style, asserts that the proportions that are useful for the shapes of pages are equally useful in shaping and positioning the textblock. This was often the case in medieval books, although later on in the Renaissance, typographers preferred to apply a more polyphonic page in which the proportions of pa
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https://en.wikipedia.org/wiki/Arthur%20Jaffe
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Arthur Michael Jaffe (; born December 22, 1937) is an American mathematical physicist at Harvard University, where in 1985 he succeeded George Mackey as the Landon T. Clay Professor of Mathematics and Theoretical Science.
Education and career
After graduating from Pelham Memorial High School in 1955, Jaffe attended Princeton University as an undergraduate obtaining a degree in chemistry in 1959, and later Clare College, Cambridge, as a Marshall Scholar, obtaining a degree in mathematics in 1961. He then returned to Princeton, obtaining a doctorate in physics in 1966 with Arthur Wightman. His whole career has been spent teaching mathematical physics and pursuing research at Harvard University. His 26 doctoral students include Joel Feldman, Ezra Getzler, and Clifford Taubes. He has had many post-doctoral collaborators, including Robert Schrader, Konrad Osterwalder, Juerg Froehlich, , Thomas Spencer, and Antti Kupiainen.
For several years Jaffe was president of the International Association of Mathematical Physics, and later of the American Mathematical Society. He chaired the Council of Scientific Society Presidents. He presently serves as chair of the board of the Dublin Institute for Advanced Studies, School of Theoretical Physics.
Jaffe conceived the idea of the Clay Mathematics Institute and its programs, including the employment of research fellows and the Millennium Prizes in mathematics. He served as a founding member, a founding member of the board, and the founding president of that organization.
Arthur Jaffe began as chief editor of Communications in Mathematical Physics in 1979 and served for 21 years until 2001. He is a distinguished visiting professor at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences.
Contributions
With James Glimm, he founded the subject called constructive quantum field theory. They established existence theorems for two- and three-dimensional examples of non-linear, relativistic quantum fields.
Awards and honors
Awarded the Dannie Heineman Prize for Mathematical Physics in 1980. In 2012 he became a fellow of the American Mathematical Society.
Personal life
Jaffe was married from 1971 to 1992 to Nora Frances Crow and they had one daughter, Margaret Collins, born in 1986. Jaffe was married to artist Sarah Robbins Warren from 1992 to 2002.
References
External links
Jaffe's website
Mathematical Picture Language Project at Harvard University
Dublin Institute for Advanced Study: Governing Boards
(lecture by Arthur Jaffe, 18 May 2016, Trinity College Dublin)
List of Past AMS Presidents (Jaffe is the 54th.)
20th-century American mathematicians
1937 births
Members of the United States National Academy of Sciences
21st-century American mathematicians
Quantum physicists
Harvard University faculty
Princeton University alumni
Marshall Scholars
Living people
Fellows of the Society for Industrial and Applied Mathematics
Members of the Royal Irish Academy
Presidents of t
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https://en.wikipedia.org/wiki/Such%C3%A9%20Brezovo
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Suché Brezovo () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Sidney%20Graham
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Sidney West Graham is a mathematician interested in analytic number theory and professor at Central Michigan University. He received his Ph.D., which was supervised by Hugh Montgomery, from the University of Michigan in 1977. In his Ph.D. thesis he lowered the upper bound for Linnik's constant to 36 and subsequently reduced the bound further to 20.
References
External links
Sidney West Graham page at Central Michigan University
20th-century American mathematicians
21st-century American mathematicians
Number theorists
University of Michigan alumni
Living people
Central Michigan University faculty
1950 births
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https://en.wikipedia.org/wiki/Tian%20Gang
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Tian Gang (; born November 24, 1958) is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis.
As of 2020, he is the Vice Chairman of the China Democratic League and the President of the Chinese Mathematical Society. From 2017 to 2019 he served as the Vice President of Peking University.
Biography
Tian was born in Nanjing, Jiangsu, China. He qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, under the supervision of Shing-Tung Yau.
In 1998, he was appointed as a Cheung Kong Scholar professor at Peking University. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was a professor of mathematics at the Massachusetts Institute of Technology from 1995 to 2006 (holding the chair of Simons Professor of Mathematics from 1996). His employment at Princeton started from 2003, and was later appointed the Higgins Professor of Mathematics. Starting 2005, he has been the director of the Beijing International Center for Mathematical Research (BICMR); from 2013 to 2017 he was the Dean of School of Mathematical Sciences at Peking University. He and John Milnor are Senior Scholars of the Clay Mathematics Institute (CMI). In 2011, Tian became director of the Sino-French Research Program in Mathematics at the Centre national de la recherche scientifique (CNRS) in Paris. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy.
Tian has served on many committees, including for the Abel Prize and the Leroy P. Steele Prize. He is a member of the editorial boards of many journals, including Advances in Mathematics and the Journal of Geometric Analysis. In the past he has been on the editorial boards of Annals of Mathematics and the Journal of the American Mathematical Society.
Among his awards and honors:
Sloan Research Fellowship (1991-1993)
Alan T. Waterman Award (1994)
Oswald Veblen Prize in Geometry (1996)
Elected to the Chinese Academy of Sciences (2001)
Elected to the American Academy of Arts and Sciences (2004)
Since at least 2013 he has been heavily involved in Chinese politics, serving as the Vice Chairman of the China Democratic League, the second most populous political party in China.
Mathematical contributions
The Kähler-Einstein problem
Tian is well-known for his contributions to Kähler geometry, and in particular to the study of Kähler-Einstein metrics. Shing-Tung Yau, in his renowned resolution of the Calabi conjecture, had settled the case of closed Kähler manifolds with nonpositive first Chern class. His work in applying the method of continuity showed
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https://en.wikipedia.org/wiki/Matti%20Jutila
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Matti Ilmari Jutila (born 1943) is a mathematician and a professor emeritus at the University of Turku. He researches in the field of analytic number theory.
Education and career
Jutila completed a doctorate at the University of Turku in 1970, with a dissertation related to Linnik's constant supervised by .
Jutila's work has repeatedly succeeded in lowering the upper bound for Linnik's constant. He is the author of a monograph, Lectures on a method in the theory of exponential sums (1987). He has been a member of the Finnish Academy of Science and Letters since 1982.
References
External links
1943 births
Living people
Finnish mathematicians
Number theorists
Members of the Finnish Academy of Science and Letters
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https://en.wikipedia.org/wiki/Slovensk%C3%A9%20%C4%8Earmoty
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Slovenské Ďarmoty () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
References
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Mu%C4%BEa
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Muľa () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
https://web.archive.org/web/20071217080336/http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/P%C3%B4tor
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Pôtor () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Pr%C3%ADbelce
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Príbelce () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Stredn%C3%A9%20Plachtince
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Stredné Plachtince () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Slovensk%C3%A9%20K%C4%BEa%C4%8Dany
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Slovenské Kľačany () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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https://en.wikipedia.org/wiki/Se%C4%8Dianky
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Sečianky () is a village and municipality in the Veľký Krtíš District of the Banská Bystrica Region of southern Slovakia.
External links
http://www.statistics.sk/mosmis/eng/run.html
Villages and municipalities in Veľký Krtíš District
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