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https://en.wikipedia.org/wiki/Kali%20S.%20Banerjee
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Kali S. Banerjee (September 17, 1914 – April 9, 2002) was a math and statistics expert, and a professor of statistics at the University of Delaware.
He was born in Dhaka, (now in Bangladesh) in 1914. He earned his bachelor's degree in mathematics and his master's and doctoral degrees in statistics from the University of Calcutta.
In 1962, Kali S. Banerjee moved to the United States and he was naturalized in 1974, and joined as a faculty of statistics at University of Delaware. Before joining the University of Delaware, he taught at Cornell University and at Kansas State University
Dr. Banerjee received the university's excellence in teaching award in 1972 and was named a fellow of Royal Statistical Society at London in 1975. He wrote many books such as The Cost Of Learning Index. He wrote about 15 study books on Statistics, about 13 books on Economics, in all writing about 40 books through his life.
He and his wife raised two children, a daughter, Swapna and a son, Deb.
External links
In Memoriam Kali S. Banerjee - University of Delaware
1914 births
2002 deaths
Indian emigrants to the United States
University of Delaware faculty
Bengali mathematicians
American Hindus
University of Calcutta alumni
20th-century Bengalis
21st-century Bengalis
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https://en.wikipedia.org/wiki/Mumford%20conjecture
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There are several conjectures in mathematics by David Mumford.
Mumford's conjecture about reductive groups, now called Haboush's theorem.
The Mumford conjecture on the cohomology of the stable mapping class group, proved by Ib Madsen and Michael Weiss.
The Manin-Mumford conjecture about Jacobians of curves, proved by Michel Raynaud.
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https://en.wikipedia.org/wiki/Green%20formula
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In mathematics, Green formula may refer to:
Green's theorem in integral calculus
Green's identities in vector calculus
Green's function in differential equations
the Green formula for the Green measure in stochastic analysis
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https://en.wikipedia.org/wiki/Top-coded
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In econometrics and statistics, a top-coded data observation is one for which data points whose values are above an upper bound are censored.
Survey data are often topcoded before release to the public to preserve the anonymity of respondents. For example, if a survey answer reported a respondent with self-identified wealth of $79 billion, it would not be anonymous because people would know there is a good chance the respondent was Bill Gates. Top-coding may be also applied to prevent possibly-erroneous outliers from being published.
Bottom-coding is analogous, e.g. if amounts below zero are reported as zero. Top-coding occurs for data recorded in groups, e.g. if age ranges are reported in these groups: 0-20, 21-50, 50-99, 100-and-up. Here we only know how many people have ages above 100, not their distribution. Producers of survey data sometimes release the average of the censored amounts to help users impute unbiased estimates of the top group.
Example: Top-coding of income at $30,000
Top-coding is a general problem for analysis of public use data sets. Top-coding in the Current Population Survey makes it hard to estimate measures of income inequality since the shape of the distribution of high incomes is blocked. To help overcome this problem, CPS provides the mean value of top-coded values.
The practice of top-coding, or capping the reported maximum value on tax returns to protect the earner's anonymity, complicates the analysis of the distribution of wealth in the United States.
Implications for ordinary least squares estimation
If the lower bound of the top-coded group is used as a regressor value (30000 in the example above), OLS is biased and inconsistent since the regressor's highest values are reported with a systematic error.
The top-coded observations can be omitted from the regression entirely. Provided there are no systematic differences between the omitted group and the included groups, OLS is consistent and unbiased.
The Tobit procedure is robust to top coding, and gives unbiased estimates.
See also
Tobit model
Heckit model
Truncated data
Censoring (statistics)
Further reading
Jenkins, S. P., Burkhauser, R. V., Feng, S., & Larrimore, J. (2009). Measuring inequality using censored data: a multiple imputation approach, ISER Working Paper Series 2009-04, Institute for Social and Economic Research.
References
Statistical data coding
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https://en.wikipedia.org/wiki/Notation%20for%20differentiation
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In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below.
Leibniz's notation
The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between dependent and independent variables and . Leibniz's notation makes this relationship explicit by writing the derivative as
Furthermore, the derivative of at is therefore written
Higher derivatives are written as
This is a suggestive notational device that comes from formal manipulations of symbols, as in,
The value of the derivative of at a point may be expressed in two ways using Leibniz's notation:
.
Leibniz's notation allows one to specify the variable for differentiation (in the denominator). This is especially helpful when considering partial derivatives. It also makes the chain rule easy to remember and recognize:
Leibniz's notation for differentiation does not require assigning a meaning to symbols such as or (known as differentials) on their own, and some authors do not attempt to assign these symbols meaning. Leibniz treated these symbols as infinitesimals. Later authors have assigned them other meanings, such as infinitesimals in non-standard analysis, or exterior derivatives. Commonly, is left undefined or equated with , while is assigned a meaning in terms of , via the equation
which may also be written, e.g.
(see below). Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the coefficient of ).
Some authors and journals set the differential symbol in roman type instead of italic: . The ISO/IEC 80000 scientific style guide recommends this style.
Leibniz's notation for antidifferentiation
Leibniz introduced the integral symbol in Analyseos tetragonisticae pars secunda and Methodi tangentium inversae exempla (both from 1675). It is now the standard symbol for integration.
Lagrange's notation
One of the most common modern notations for differentiation is named after Joseph Louis Lagrange, even though it was actually invented by Euler and just popularized by the former. In Lagrange's notation, a prime mark denotes a derivative. If f is a function, then its derivative evaluated at x is written
.
It first appeared in print in 1749.
Higher derivatives are indicated using additional prime marks, as in for the second derivative and for the third derivative. The use of repeated prime marks eventually becomes unwieldy. Some authors contin
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https://en.wikipedia.org/wiki/Positive-definite%20kernel
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In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Since then, positive-definite functions and their various analogues and generalizations have arisen in diverse parts of mathematics. They occur naturally in Fourier analysis, probability theory, operator theory, complex function-theory, moment problems, integral equations, boundary-value problems for partial differential equations, machine learning, embedding problem, information theory, and other areas.
Definition
Let be a nonempty set, sometimes referred to as the index set. A symmetric function is called a positive-definite (p.d.) kernel on if
holds for any , given .
In probability theory, a distinction is sometimes made between positive-definite kernels, for which equality in (1.1) implies , and positive semi-definite (p.s.d.) kernels, which do not impose this condition. Note that this is equivalent to requiring that any finite matrix constructed by pairwise evaluation, , has either entirely positive (p.d.) or nonnegative (p.s.d.) eigenvalues.
In mathematical literature, kernels are usually complex valued functions, but in this article we assume real-valued functions, which is the common practice in applications of p.d. kernels.
Some general properties
For a family of p.d. kernels
The conical sum is p.d., given
The product is p.d., given
The limit is p.d. if the limit exists.
If is a sequence of sets, and a sequence of p.d. kernels, then both and are p.d. kernels on .
Let . Then the restriction of to is also a p.d. kernel.
Examples of p.d. kernels
Common examples of p.d. kernels defined on Euclidean space include:
Linear kernel: .
Polynomial kernel: .
Gaussian kernel (RBF kernel): .
Laplacian kernel: .
Abel kernel: .
Kernel generating Sobolev spaces : , where is the Bessel function of the third kind.
Kernel generating Paley–Wiener space: .
If is a Hilbert space, then its corresponding inner product is a p.d. kernel. Indeed, we have
Kernels defined on and histograms: Histograms are frequently encountered in applications of real-life problems. Most observations are usually available under the form of nonnegative vectors of counts, which, if normalized, yield histograms of frequencies. It has been shown that the following family of squared metrics, respectively Jensen divergence, the -square, Total Variation, and two variations of the Hellinger distance:can be used to define p.d. kernels using the following formula
History
Positive-definite kernels, as defined in (1.1), appeared first in 1909 in a paper on integral equations by James Mercer. Several other authors made use of this concept in the following two decades, but none of them explicitly used kernels , i.e. p.d. functions (indeed M. Mathias and S. Bochner seem
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https://en.wikipedia.org/wiki/Projective%20cover
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In the branch of abstract mathematics called category theory, a projective cover of an object X is in a sense the best approximation of X by a projective object P. Projective covers are the dual of injective envelopes.
Definition
Let be a category and X an object in . A projective cover is a pair (P,p), with P a projective object in and p a superfluous epimorphism in Hom(P, X).
If R is a ring, then in the category of R-modules, a superfluous epimorphism is then an epimorphism such that the kernel of p is a superfluous submodule of P.
Properties
Projective covers and their superfluous epimorphisms, when they exist, are unique up to isomorphism. The isomorphism need not be unique, however, since the projective property is not a full fledged universal property.
The main effect of p having a superfluous kernel is the following: if N is any proper submodule of P, then . Informally speaking, this shows the superfluous kernel causes P to cover M optimally, that is, no submodule of P would suffice. This does not depend upon the projectivity of P: it is true of all superfluous epimorphisms.
If (P,p) is a projective cover of M, and P' is another projective module with an epimorphism , then there is a split epimorphism α from P' to P such that
Unlike injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have projective covers. A ring R is called left (right) perfect if every left (right) R-module has a projective cover in R-Mod (Mod-R).
A ring is called semiperfect if every finitely generated left (right) R-module has a projective cover in R-Mod (Mod-R). "Semiperfect" is a left-right symmetric property.
A ring is called lift/rad if idempotents lift from R/J to R, where J is the Jacobson radical of R. The property of being lift/rad can be characterized in terms of projective covers: R is lift/rad if and only if direct summands of the R module R/J (as a right or left module) have projective covers.
Examples
In the category of R modules:
If M is already a projective module, then the identity map from M to M is a superfluous epimorphism (its kernel being zero). Hence, projective modules always have projective covers.
If J(R)=0, then a module M has a projective cover if and only if M is already projective.
In the case that a module M is simple, then it is necessarily the top of its projective cover, if it exists.
The injective envelope for a module always exists, however over certain rings modules may not have projective covers. For example, the natural map from Z onto Z/2Z is not a projective cover of the Z-module Z/2Z (which in fact has no projective cover). The class of rings which provides all of its right modules with projective covers is the class of right perfect rings.
Any R-module M has a flat cover, which is equal to the projective cover if R has a projective cover.
See also
Projective resolution
References
Category theory
Homolo
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https://en.wikipedia.org/wiki/Topological%20modular%20forms
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In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.
The spectrum of topological modular forms is constructed as the global sections of a sheaf of E-infinity ring spectra on the moduli stack of (generalized) elliptic curves. This theory has relations to the theory of modular forms in number theory, the homotopy groups of spheres, and conjectural index theories on loop spaces of manifolds. tmf was first constructed by Michael Hopkins and Haynes Miller; many of the computations can be found in preprints and articles by Paul Goerss, Hopkins, Mark Mahowald, Miller, Charles Rezk, and Tilman Bauer.
Construction
The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf Otop ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve.
A second construction, due to Jacob Lurie, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted. These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne-Mumford compactification of this moduli stack, in which elliptic curves with nodal singularities are included. TMF is the spectrum that results from the global sections over the moduli stack of smooth curves, and tmf is the spectrum arising as the global sections of the Deligne–Mumford compactification.
TMF is a periodic version of the connective tmf. While the ring spectra used to construct TMF are periodic with period 2, T
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https://en.wikipedia.org/wiki/6-demicube
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In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,33,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
(±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
As a configuration
This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Images
Related polytopes
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
Skew icosahedron
Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Stra
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https://en.wikipedia.org/wiki/7-cube
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In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.
It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
Related polytopes
The 7-cube is 7th in a series of hypercube:
The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces.
As a configuration
This configuration matrix represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Cartesian coordinates
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.
Projections
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary: hypercube Garrett Jones
Rotation of 7D-Cube www.4d-screen.de
7-polytopes
Articles containing video clips
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https://en.wikipedia.org/wiki/7-demicube
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In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM7 for a 7-dimensional half measure polytope.
Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or {3,34,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:
(±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
As a configuration
This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Related polytopes
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1)
External links
Multi-dimensional Glossary
7-polytopes
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https://en.wikipedia.org/wiki/8-demicube
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In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM8 for an 8-dimensional half measure polytope.
Coxeter named this polytope as 151 from its Coxeter diagram, with a ring on
one of the 1-length branches, and Schläfli symbol or {3,35,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8-demicube centered at the origin are alternate halves of the 8-cube:
(±1,±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Related polytopes and honeycombs
This polytope is the vertex figure for the uniform tessellation, 251 with Coxeter-Dynkin diagram:
Images
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1)
External links
Multi-dimensional Glossary
8-polytopes
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https://en.wikipedia.org/wiki/9-demicube
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In geometry, a demienneract or 9-demicube is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM9 for a 9-dimensional half measure polytope.
Coxeter named this polytope as 161 from its Coxeter diagram, with a ring on
one of the 1-length branches, and Schläfli symbol or {3,36,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demienneract centered at the origin are alternate halves of the enneract:
(±1,±1,±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1)
External links
Multi-dimensional Glossary
9-polytopes
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https://en.wikipedia.org/wiki/8-cube
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In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7) with -1 < xi < 1.
As a configuration
This configuration matrix represents the 8-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Projections
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.
Related polytopes
The 8-cube is 8th in an infinite series of hypercube:
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary: hypercube Garrett Jones
8-polytopes
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https://en.wikipedia.org/wiki/9-cube
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In geometry, a 9-cube is a nine-dimensional hypercube with 512 vertices, 2304 edges, 4608 square faces, 5376 cubic cells, 4032 tesseract 4-faces, 2016 5-cube 5-faces, 672 6-cube 6-faces, 144 7-cube 7-faces, and 18 8-cube 8-faces.
It can be named by its Schläfli symbol {4,37}, being composed of three 8-cubes around each 7-face. It is also called an enneract, a portmanteau of tesseract (the 4-cube) and enne for nine (dimensions) in Greek. It can also be called a regular octadeca-9-tope or octadecayotton, as a nine-dimensional polytope constructed with 18 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of a 9-cube can be called a 9-orthoplex, and is a part of the infinite family of cross-polytopes.
Cartesian coordinates
Cartesian coordinates for the vertices of a 9-cube centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7, x8) with −1 < xi < 1.
Projections
Images
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the 9-cube, creates another uniform polytope, called a 9-demicube, (part of an infinite family called demihypercubes), which has 18 8-demicube and 256 8-simplex facets.
Notes
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary: hypercube Garrett Jones
9-polytopes
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https://en.wikipedia.org/wiki/7-orthoplex
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In geometry, a 7-orthoplex, or 7-cross polytope, is a regular 7-polytope with 14 vertices, 84 edges, 280 triangle faces, 560 tetrahedron cells, 672 5-cells 4-faces, 448 5-faces, and 128 6-faces.
It has two constructed forms, the first being regular with Schläfli symbol {35,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,31,1} or Coxeter symbol 411.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 7-hypercube, or hepteract.
Alternate names
Heptacross, derived from combining the family name cross polytope with hept for seven (dimensions) in Greek.
Hecatonicosoctaexon as a 128-facetted 7-polytope (polyexon).
As a configuration
This configuration matrix represents the 7-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Images
Construction
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or [4,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or [34,1,1] symmetry group. A lowest symmetry construction is based on a dual of a 7-orthotope, called a 7-fusil.
Cartesian coordinates
Cartesian coordinates for the vertices of a 7-orthoplex, centered at the origin are
(±1,0,0,0,0,0,0), (0,±1,0,0,0,0,0), (0,0,±1,0,0,0,0), (0,0,0,±1,0,0,0), (0,0,0,0,±1,0,0), (0,0,0,0,0,±1,0), (0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
See also
Rectified 7-orthoplex
Truncated 7-orthoplex
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes
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https://en.wikipedia.org/wiki/6-simplex
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In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
Alternate names
It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.
As a configuration
This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.
Coordinates
The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:
The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:
(0,0,0,0,0,0,1)
This construction is based on facets of the 7-orthoplex.
Images
Related uniform 6-polytopes
The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
Notes
References
Coxeter, H.S.M.:
(Paper 22)
(Paper 23)
(Paper 24)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes
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https://en.wikipedia.org/wiki/7-simplex
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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.
Alternate names
It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca.
As a configuration
This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.
Symmetry
There are many lower symmetry constructions of the 7-simplex.
Some are expressed as join partitions of two or more lower simplexes. The symmetry order of each join is the product of the symmetry order of the elements, and raised further if identical elements can be interchanged.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular octaexon having edge length 2 are:
More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.
Images
Orthographic projections
Related polytopes
This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram:
This polytope is one of 71 uniform 7-polytopes with A7 symmetry.
Notes
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes
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https://en.wikipedia.org/wiki/8-orthoplex
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In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.
It has two constructive forms, the first being regular with Schläfli symbol {36,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,31,1} or Coxeter symbol 511.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.
Alternate names
Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)
As a configuration
This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.
Construction
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8-cube, centered at the origin are
(±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),
(0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
8-polytopes
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https://en.wikipedia.org/wiki/Octonion%20algebra
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In mathematics, an octonion algebra or Cayley algebra over a field F is a composition algebra over F that has dimension 8 over F. In other words, it is a 8-dimensional unital non-associative algebra A over F with a non-degenerate quadratic form N (called the norm form) such that
for all x and y in A.
The most well-known example of an octonion algebra is the classical octonions, which are an octonion algebra over R, the field of real numbers. The split-octonions also form an octonion algebra over R. Up to R-algebra isomorphism, these are the only octonion algebras over the reals. The algebra of bioctonions is the octonion algebra over the complex numbers C.
The octonion algebra for N is a division algebra if and only if the form N is anisotropic. A split octonion algebra is one for which the quadratic form N is isotropic (i.e., there exists a non-zero vector x with N(x) = 0). Up to F-algebra isomorphism, there is a unique split octonion algebra over any field F. When F is algebraically closed or a finite field, these are the only octonion algebras over F.
Octonion algebras are always non-associative. They are, however, alternative algebras, alternativity being a weaker form of associativity. Moreover, the Moufang identities hold in any octonion algebra. It follows that the invertible elements in any octonion algebra form a Moufang loop, as do the elements of unit norm.
The construction of general octonion algebras over an arbitrary field k was described by Leonard Dickson in his book Algebren und ihre Zahlentheorie (1927) (Seite 264) and repeated by Max Zorn. The product depends on selection of a γ from k. Given q and Q from a quaternion algebra over k, the octonion is written q + Qe. Another octonion may be written r + Re. Then with * denoting the conjugation in the quaternion algebra, their product is
Zorn’s German language description of this Cayley–Dickson construction contributed to the persistent use of this eponym describing the construction of composition algebras.
Cohl Furey has proposed that octonion algebras can be utilized in an attempt to reconcile components of the standard model.
Classification
It is a theorem of Adolf Hurwitz that the F-isomorphism classes of the norm form are in one-to-one correspondence with the isomorphism classes of octonion F-algebras. Moreover, the possible norm forms are exactly the Pfister 3-forms over F.
Since any two octonion F-algebras become isomorphic over the algebraic closure of F, one can apply the ideas of non-abelian Galois cohomology. In particular, by using the fact that the automorphism group of the split octonions is the split algebraic group G2, one sees the correspondence of isomorphism classes of octonion F-algebras with isomorphism classes of G2-torsors over F. These isomorphism classes form the non-abelian Galois cohomology set .
References
External links
Composition algebras
Algebra
Non-associative algebras
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https://en.wikipedia.org/wiki/Isothermal%20coordinates
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In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form
where is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.)
Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold.
By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally flat if and only if its Weyl tensor vanishes.
Isothermal coordinates on surfaces
In 1822, Carl Friedrich Gauss proved the existence of isothermal coordinates on an arbitrary surface with a real-analytic Riemannian metric, following earlier results of
Joseph Lagrange in the special case of surfaces of revolution. The construction used by Gauss made use of the Cauchy–Kowalevski theorem, so that his method is fundamentally restricted to the real-analytic context. Following innovations in the theory of two-dimensional partial differential equations by Arthur Korn, Leon Lichtenstein found in 1916 the general existence of isothermal coordinates for Riemannian metrics of lower regularity, including smooth metrics and even Hölder continuous metrics.
Given a Riemannian metric on a two-dimensional manifold, the transition function between isothermal coordinate charts, which is a map between open subsets of , is necessarily angle-preserving. The angle-preserving property together with orientation-preservation is one characterization (among many) of holomorphic functions, and so an oriented coordinate atlas consisting of isothermal coordinate charts may be viewed as a holomorphic coordinate atlas. This demonstrates that a Riemannian metric and an orientation on a two-dimensional manifold combine to induce the structure of a Riemann surface (i.e. a one-dimensional complex manifold). Furthermore, given an oriented surface, two Riemannian metrics induce the same holomorphic atlas if and only if they are conformal to one another. For this reason, the study of Riemann surfaces is identical to the study of conformal classes of Riemannian metrics on oriented surfaces.
By the 1950s, expositions of the ideas of Korn and Lichtenstein were put into the language of complex derivatives and the Beltrami equation by Lipman Bers and Shiing-shen Chern, among others. In this context, it is natural to investigate the existence of generalized solutions, which satisfy the relevant partial differential equations but are no longer interpretable as coor
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https://en.wikipedia.org/wiki/Lie%20bracket%20of%20vector%20fields
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In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted .
Conceptually, the Lie bracket is the derivative of Y along the flow generated by X, and is sometimes denoted ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X.
The Lie bracket is an R-bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra.
The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems.
Definitions
There are three conceptually different but equivalent approaches to defining the Lie bracket:
Vector fields as derivations
Each smooth vector field on a manifold M may be regarded as a differential operator acting on smooth functions (where and of class ) when we define to be another function whose value at a point is the directional derivative of f at p in the direction X(p). In this way, each smooth vector field X becomes a derivation on C∞(M). Furthermore, any derivation on C∞(M) arises from a unique smooth vector field X.
In general, the commutator of any two derivations and is again a derivation, where denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:
Flows and limits
Let be the flow associated with the vector field X, and let D denote the tangent map derivative operator. Then the Lie bracket of X and Y at the point can be defined as the Lie derivative:
This also measures the failure of the flow in the successive directions to return to the point x:
In coordinates
Though the above definitions of Lie bracket are intrinsic (independent of the choice of coordinates on the manifold M), in practice one often wants to compute the bracket in terms of a specific coordinate system . We write for the associated local basis of the tangent bundle, so that general vector fields can be written and for smooth functions . Then the Lie bracket can be computed as:
If M is (an open subset of) Rn, then the vector fields X and Y can be written as smooth maps of the form and , and the Lie bracket is given by:
where and are Jacobian matrices ( and respectively using index notation) multiplying the column vectors X and Y.
Properties
The Lie bracket of vector fields equips the real vector space of all vector fields on M (i.e., smooth sections of the tangent bundle ) with the structure of a Lie algebra, which means [ • , • ] is a map with:
R-bilinearity
Anti-symmetry,
Jacobi identity,
An immediate consequence of the second property is that for any .
Furtherm
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https://en.wikipedia.org/wiki/Dirac%20algebra
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In mathematical physics, the Dirac algebra is the Clifford algebra . This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- particles with a matrix representation of the gamma matrices, which represent the generators of the algebra.
The gamma matrices are a set of four matrices with entries in , that is, elements of that satisfy
where by convention, an identity matrix has been suppressed on the right-hand side. The numbers are the components of the Minkowski metric.
For this article we fix the signature to be mostly minus, that is, .
The Dirac algebra is then the linear span of the identity, the gamma matrices as well as any linearly independent products of the gamma matrices. This forms a finite-dimensional algebra over the field or , with dimension .
Basis for the algebra
The algebra has a basis
where in each expression, each greek index is increasing as we move to the right. In particular, there is no repeated index in the expressions. By dimension counting, the dimension of the algebra is 16.
The algebra can be generated by taking products of the alone: the identity arises as
while the others are explicitly products of the .
These elements span the space generated by . We conclude that we really do have a basis of the Clifford algebra generated by the
Quadratic powers and Lorentz algebra
For the theory in this section, there are many choices of conventions found in the literature, often corresponding to factors of . For clarity, here we will choose conventions to minimise the number of numerical factors needed, but may lead to generators being anti-Hermitian rather than Hermitian.
There is another common way to write the quadratic subspace of the Clifford algebra:
with . Note .
There is another way to write this which holds even when :
This form can be used to show that the form a representation of the Lorentz algebra (with real conventions)
Physics conventions
It is common convention in physics to include a factor of , so that Hermitian conjugation (where transposing is done with respect to the spacetime greek indices) gives a 'Hermitian matrix' of sigma generators
only of which are non-zero due to antisymmetry of the bracket, span the six-dimensional representation space of the tensor -representation of the Lorentz algebra inside . Moreover, they have the commutation relations of the Lie algebra,
and hence constitute a representation of the Lorentz algebra (in addition to spanning a representation space) sitting inside the spin representation.
Spin(1, 3)
The exponential map for matrices is well defined. The satisfy the Lorentz algebra, and turn out to exponentiate to a representation of the spin group of the Lorentz group (strictly, the future-directed part connected to the identity). The are then the spin generators of this representation.
We emphasize that is itself a matrix, not the components of a matrix. Its components as
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https://en.wikipedia.org/wiki/Elliptic%20cohomology
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In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
History and motivation
Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning -actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to the K-theory of the free loop space.
Definitions and constructions
Call a cohomology theory even periodic if for i odd and there is an invertible element . These theories possess a complex orientation, which gives a formal group law. A particularly rich source for formal group laws are elliptic curves. A cohomology theory with
is called elliptic if it is even periodic and its formal group law is isomorphic to a formal group law of an elliptic curve over . The usual construction of such elliptic cohomology theories uses the Landweber exact functor theorem. If the formal group law of is Landweber exact, one can define an elliptic cohomology theory (on finite complexes) by
Franke has identified the condition needed to fulfill Landweber exactness:
needs to be flat over
There is no irreducible component of , where the fiber is supersingular for every
These conditions can be checked in many cases related to elliptic genera. Moreover, the conditions are fulfilled in the universal case in the sense that the map from the moduli stack of elliptic curves to the moduli stack of formal groups
is flat. This gives then a presheaf of cohomology theoriesover the site of affine schemes flat over the moduli stack of elliptic curves. The desire to get a universal elliptic cohomology theory by taking global sections has led to the construction of the topological modular formspg 20as the homotopy limit of this presheaf over the previous site.
See also
Spectral algebraic geometry
Intermediate Jacobian
Chromatic homotopy theory
References
.
.
.
.
.
Founding articles
Elliptic cohomology - Graeme Segal
Extensions to Calabi-Yau manifolds
K3 Spectra
Constructing explicit K3 spectra
The Elliptic curves in gauge theory, string theory, and cohomology
Cohomology theories
Elliptic curves
Modular forms
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https://en.wikipedia.org/wiki/Hodge%20structure
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In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989).
Hodge structures
Definition of Hodge structures
A pure Hodge structure of integer weight n consists of an abelian group and a decomposition of its complexification H into a direct sum of complex subspaces , where , with the property that the complex conjugate of is :
An equivalent definition is obtained by replacing the direct sum decomposition of H by the Hodge filtration, a finite decreasing filtration of H by complex subspaces subject to the condition
The relation between these two descriptions is given as follows:
For example, if X is a compact Kähler manifold, is the n-th cohomology group of X with integer coefficients, then is its n-th cohomology group with complex coefficients and Hodge theory provides the decomposition of H into a direct sum as above, so that these data define a pure Hodge structure of weight n. On the other hand, the Hodge–de Rham spectral sequence supplies with the decreasing filtration by as in the second definition.
For applications in algebraic geometry, namely, classification of complex projective varieties by their periods, the set of all Hodge structures of weight n on is too big. Using the Riemann bilinear relations, in this case called Hodge Riemann bilinear relations, it can be substantially simplified. A polarized Hodge structure of weight n consists of a Hodge structure and a non-degenerate integer bilinear form Q on (polarization), which is extended to H by linearity, and satisfying the conditions:
In terms of the Hodge filtration, these conditions imply that
where C is the Weil operator on H, given by on .
Yet another definition of a Hodge structure is based on the equivalence between the -grading on a complex vector space and the action of the circle group U(1). In this definition, an action of the multiplicative group of complex numbers viewed as a two-dimensional real algebraic torus, is given on H. This action must have the property that a real number a acts by an. The subspace is the subspace on which acts as multiplication by
A-Hodge structure
In the theory of motives, it becomes important to allow more general coefficients for the cohomology. The definition of a Hodge structure is modified by fixing a Noetherian subring A of the field of real numbers, for which is a field. Then a pure Hodge A-structure of weight n is d
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https://en.wikipedia.org/wiki/Reduced%20residue%20system
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In mathematics, a subset R of the integers is called a reduced residue system modulo n if:
gcd(r, n) = 1 for each r in R,
R contains φ(n) elements,
no two elements of R are congruent modulo n.
Here φ denotes Euler's totient function.
A reduced residue system modulo n can be formed from a complete residue system modulo n by removing all integers not relatively prime to n. For example, a complete residue system modulo 12 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is {1, 5, 7, 11}. The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are:
{13,17,19,23}
{−11,−7,−5,−1}
{−7,−13,13,31}
{35,43,53,61}
Facts
If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n with n > 2, then .
Every number in a reduced residue system modulo n is a generator for the additive group of integers modulo n.
If {r1, r2, ... , rφ(n)} is a reduced residue system modulo n, and a is an integer such that gcd(a, n) = 1, then {ar1, ar2, ... , arφ(n)} is also a reduced residue system modulo n.
See also
Complete residue system modulo m
Multiplicative group of integers modulo n
Congruence relation
Euler's totient function
Greatest common divisor
Least residue system modulo m
Modular arithmetic
Number theory
Residue number system
Notes
References
External links
Residue systems at PlanetMath
Reduced residue system at MathWorld
Modular arithmetic
Elementary number theory
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https://en.wikipedia.org/wiki/List%20of%20York%20City%20F.C.%20records%20and%20statistics
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York City Football Club is a professional association football club based in York, North Yorkshire, England. The club was founded in 1922 and was elected to the Midland League, which the team played in until 1929 when they were elected to the Football League. The highest level of the English football league system the team has reached is the second tier, spending two seasons in the Second Division during the 1970s. The club lost its Football League status following relegation to the Football Conference in 2004, but regained it eight years later with victory in the 2012 Conference Premier play-off final.
This list encompasses the major honours won by York City, and records set by the club, its players and its managers. The player records section itemises the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by York players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Fulfordgate, Bootham Crescent and the York Community Stadium are also included.
All figures are correct as of the match played on 21 May 2022, the final match of York's 2021–22 season.
Honours
York City have won one major honour in the Football League, when winning the 1983–84 Fourth Division championship. With 101 points, York became the first club in the Football League to reach a three-figure points total. They have achieved promotion on seven other occasions, most recently in the 2021–22 season, when winning the National League North play-offs. York won their first domestic cup competition in the 2011–12 season, beating Newport County in the 2012 FA Trophy Final.
York's honours and achievements include the following:
The Football League
Third Division (level 3)
Promotion: 1973–74
Fourth Division / Third Division (level 4)
Champions: 1983–84
Promotion: 1958–59, 1964–65, 1970–71, 1992–93
Football Conference/National League
Conference Premier (level 5)
Promotion: 2011–12
National League North (level 6)
Promotion: 2021–22
Domestic cup competition
FA Trophy
Winners: 2011–12, 2016–17
Finalists: 2008–09
Player records
Appearances
Youngest first-team player: Reg Stockill, 15 years 281 days (against Wigan Borough, Third Division North, 29 August 1929).
Oldest first-team player: Paul Musselwhite, 43 years 127 days (against Forest Green Rovers, Conference Premier, 28 April 2012).
Most appearances
Competitive matches only, appearances as substitute in brackets.
Goalscorers
Most goals in a season: 56, by Jimmy Cowie in 1928–29.
Most league goals in a season: 49, by Jimmy Cowie in the Midland League, 1928–29.
Most goals in a match:
6, by Jimmy Cowie against Stockton, FA Cup, 29 September 1928.
6, by Jimmy Cowie against Worksop Town, Midland League, 23 February 1929.
Top goalscorers
Competitive matches only. Matches played (including as substitute) appear in brackets.
International caps
This section refers only to caps earned while a
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https://en.wikipedia.org/wiki/Well%20intervention
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A well intervention, or well work, is any operation carried out on an oil or gas well during, or at the end of, its productive life that alters the state of the well or well geometry, provides well diagnostics, or manages the production of the well.
Types of well intervention
Pumping
Pumping is the simplest form of intervention as it does not involve putting hardware into the well itself. Frequently it simply involves rigging up to the kill wing valve on the Christmas tree and pumping in a fluid determined necessary for the particular well.
Wellhead and Christmas tree maintenance
The complexity of wellhead and Christmas tree maintenance can vary depending on the condition of the wellheads. Scheduled annual maintenance may simply involve greasing and pressure testing the valve on the hardware. Sometimes the downhole safety valve is pressure tested as well.
Slickline
Slickline operations may be used for fishing, gauge cutting, setting or removing plugs, deploying or removing wireline retrievable valves and memory logging.
Braided line
Braided line is more complex than slickline due to the need for a grease injection system in the rigup to ensure the blowout preventer (BOP) can seal around the braided contours of the wire. It also requires an additional shear-seal BOP as a tertiary barrier, as the upper master valve on the Christmas tree can only cut slickline. Braided line includes both the core-less variety used for heaving fishing and electric-line used for well logging and perforating.
Coiled tubing
Coiled tubing is used when it is desired to pump chemicals directly to the bottom of the well, such as in a circulating operation or a chemical wash. It can also be used for tasks normally done by wireline if the deviation in the well is too severe for gravity to lower the toolstring and circumstances prevent the use of a wireline tractor.
Snubbing
Snubbing, also known as hydraulic workover, involves forcing a string of pipe into the well against wellbore pressure to perform the required tasks. The rigup is larger than for coiled tubing and the pipe more rigid.
Workover
In some older wells, changing reservoir conditions or deteriorating condition of the completion may necessitate pulling it out to replace it with a fresh completion.
Subsea well intervention
Subsea well intervention offers many challenges and requires much planning. The cost of subsea intervention has in the past inhibited the intervention but in the current economic climate it is much more viable. These interventions are commonly executed from light/medium intervention vessels, or mobile offshore drilling units (MODU) for the heavier interventions such as snubbing and workover drilling rigs. Light interventions are generally performed with the well live, and usually involve adjustments of things such as valves; while heavy interventions are generally performed with the well shut down, and may be used to replace parts such as tubing strings or pumps, or
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https://en.wikipedia.org/wiki/Gordon%20Royle
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Gordon F. Royle is a professor at the School of Mathematics and Statistics at The University of Western Australia.
Royle is the co-author (with Chris Godsil) of the book Algebraic Graph Theory (Springer Verlag, 2001, ).
Royle is also known for his research into the mathematics of Sudoku and his search for the Sudoku puzzle with the smallest number of entries that has a unique solution.
Royle earned his Ph.D. in 1987 from the University of Western Australia under the supervision of Cheryl Praeger and Brendan McKay.
References
Living people
Australian mathematicians
Graph theorists
University of Western Australia alumni
Academic staff of the University of Western Australia
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Replication%20%28statistics%29
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In engineering, science, and statistics, replication is the repetition of an experimental condition so that the variability associated with the phenomenon can be estimated. ASTM, in standard E1847, defines replication as "... the repetition of the set of all the treatment combinations to be compared in an experiment. Each of the repetitions is called a replicate."
Replication is not the same as repeated measurements of the same item: they are dealt with differently in statistical experimental design and data analysis.
For proper sampling, a process or batch of products should be in reasonable statistical control; inherent random variation is present but variation due to assignable (special) causes is not. Evaluation or testing of a single item does not allow for item-to-item variation and may not represent the batch or process. Replication is needed to account for this variation among items and treatments.
Example
As an example, consider a continuous process which produces items. Batches of items are then processed or treated. Finally, tests or measurements are conducted. Several options might be available to obtain ten test values. Some possibilities are:
One finished and treated item might be measured repeatedly to obtain ten test results. Only one item was measured so there is no replication. The repeated measurements help identify observational error.
Ten finished and treated items might be taken from a batch and each measured once. This is not full replication because the ten samples are not random and not representative of the continuous nor batch processing.
Five items are taken from the continuous process based on sound statistical sampling. These are processed in a batch and tested twice each. This includes replication of initial samples but does not allow for batch-to-batch variation in processing. The repeated tests on each provide some measure and control of testing error.
Five items are taken from the continuous process based on sound statistical sampling. These are processed in five different batches and tested twice each. This plan includes proper replication of initial samples and also includes batch-to-batch variation. The repeated tests on each provide some measure and control of testing error.
Each option would call for different data analysis methods and yield different conclusions.
See also
Degrees of freedom (statistics)
Design of experiments
Pseudoreplication
Sample size
Statistical ensemble
Statistical process control
Test method
Bibliography
ASTM E122-07 Standard Practice for Calculating Sample Size to Estimate, With Specified Precision, the Average for a Characteristic of a Lot or Process
"Engineering Statistics Handbook", NIST/SEMATEK
Pyzdek, T, "Quality Engineering Handbook", 2003, .
Godfrey, A. B., "Juran's Quality Handbook", 1999, .
Design of experiments
Sampling (statistics)
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https://en.wikipedia.org/wiki/Differentiation%20rules
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This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Elementary rules of differentiation
Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers (C).
Constant term rule
For any value of , where , if is the constant function given by , then .
Proof
Let and . By the definition of the derivative,
This shows that the derivative of any constant function is 0.
Intuitive (geometric) explanation
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and it's angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
Differentiation is linear
For any functions and and any real numbers and , the derivative of the function with respect to is:
In Leibniz's notation this is written as:
Special cases include:
The constant factor rule
The sum rule
The difference rule
The product rule
For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is
In Leibniz's notation this is written
The chain rule
The derivative of the function is
In Leibniz's notation, this is written as:
often abridged to
Focusing on the notion of maps, and the differential being a map , this is written in a more concise way as:
The inverse function rule
If the function has an inverse function , meaning that and then
In Leibniz notation, this is written as
Power laws, polynomials, quotients, and reciprocals
The polynomial or elementary power rule
If , for any real number then
When this becomes the special case that if then
Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.
The reciprocal rule
The derivative of for any (nonvanishing) function is:
wherever is non-zero.
In Leibniz's notation, this is written
The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.
The quotient rule
If and are functions, then:
wherever is nonzero.
This can be derived from the product rule and the reciprocal rule.
Generalized power rule
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions and ,
wherever both sides are well defined.
Special cases
If , then when is any non-zero real number and is positive.
The reciprocal rule may be derived as the special case where .
Derivatives of exponential and logarithmic functions
the equation above is true for all , but the derivative for yields a complex number.
the equation above is also true for all , but yields a c
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https://en.wikipedia.org/wiki/De%20Bruijn%20index
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In mathematical logic, the De Bruijn index is a tool invented by the Dutch mathematician Nicolaas Govert de Bruijn for representing terms of lambda calculus without naming the bound variables. Terms written using these indices are invariant with respect to α-conversion, so the check for α-equivalence is the same as that for syntactic equality. Each De Bruijn index is a natural number that represents an occurrence of a variable in a λ-term, and denotes the number of binders that are in scope between that occurrence and its corresponding binder. The following are some examples:
The term λx. λy. x, sometimes called the K combinator, is written as λ λ 2 with De Bruijn indices. The binder for the occurrence x is the second λ in scope.
The term λx. λy. λz. x z (y z) (the S combinator), with De Bruijn indices, is λ λ λ 3 1 (2 1).
The term λz. (λy. y (λx. x)) (λx. z x) is λ (λ 1 (λ 1)) (λ 2 1). See the following illustration, where the binders are coloured and the references are shown with arrows.
De Bruijn indices are commonly used in higher-order reasoning systems such as automated theorem provers and logic programming systems.
Formal definition
Formally, λ-terms (M, N, ...) written using De Bruijn indices have the following syntax (parentheses allowed freely):
M, N, ... ::= n | M N | λ M
where n—natural numbers greater than 0—are the variables. A variable n is bound if it is in the scope of at least n binders (λ); otherwise it is free. The binding site for a variable n is the nth binder it is in the scope of, starting from the innermost binder.
The most primitive operation on λ-terms is substitution: replacing free variables in a term with other terms. In the β-reduction (λ M) N, for example, we must
find the instances of the variables n1, n2, ..., nk in M that are bound by the λ in λ M,
decrement the free variables of M to match the removal of the outer λ-binder, and
replace n1, n2, ..., nk with N, suitably incrementing the free variables occurring in N each time, to match the number of λ-binders, under which the corresponding variable occurs when N substitutes for one of the ni.
To illustrate, consider the application
(λ λ 4 2 (λ 1 3)) (λ 5 1)
which might correspond to the following term written in the usual notation
(λx. λy. z x (λu. u x)) (λx. w x).
After step 1, we obtain the term λ 4 □ (λ 1 □), where the variables that are destined for substitution are replaced with boxes. Step 2 decrements the free variables, giving λ 3 □ (λ 1 □). Finally, in step 3, we replace the boxes with the argument, namely λ 5 1; the first box is under one binder, so we replace it with λ 6 1 (which is λ 5 1 with the free variables increased by 1); the second is under two binders, so we replace it with λ 7 1. The final result is λ 3 (λ 6 1) (λ 1 (λ 7 1)).
Formally, a substitution is an unbounded list of terms, written M1.M2..., where Mi is the replacement for the ith free variable. The increasing operation in step 3 is sometimes called shift and written ↑k
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https://en.wikipedia.org/wiki/Katrin%20Wehrheim
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Katrin Wehrheim (born 1974) is an associate professor of mathematics at the University of California, Berkeley. Wehrheim's research centers around symplectic topology and gauge theory, and they are known for work on pseudoholomorphic quilts. With Dusa McDuff, they have challenged the foundational rigor of a classic proof in symplectic geometry.
Education and career
After attending school in Hamburg and studying at the University of Hamburg until 1995 and Imperial College until 1996, Wehrheim went to ETH Zürich for graduate studies. After almost dropping out to become an Olympic rower, Wehrheim completed her PhD in 2002, under the joint supervision of Dusa McDuff and Dietmar Salamon.
Wehrheim was an instructor at Princeton University and member of the Institute for Advanced Study before taking a tenure track position at the Massachusetts Institute of Technology in 2005. While they were at MIT, Wehrheim—who is openly gay—co-headed the 2008 Celebration of Women in Mathematics conference. Since 2013, Wehrheim has been teaching mathematics at the University of California, Berkeley.
Awards and honors
Wehrheim's PhD thesis in mathematics Anti-Self-Dual Instantons with Lagrangian Boundary Conditions won the 2002 ETH medal. In 2010, Wehrheim received the Presidential Career Award PECASE from Barack Obama in a ceremony at the White House. In 2012, Wehrheim became a fellow of the American Mathematical Society.
References
External links
Home page
1974 births
Living people
21st-century German mathematicians
German women mathematicians
University of Hamburg alumni
Alumni of Imperial College London
ETH Zurich alumni
Academic staff of ETH Zurich
University of California, Berkeley faculty
Princeton University faculty
Massachusetts Institute of Technology faculty
Fellows of the American Mathematical Society
21st-century women mathematicians
21st-century German women
21st-century German LGBT people
LGBT mathematicians
German lesbians
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https://en.wikipedia.org/wiki/De%20Bruijn%20notation
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In mathematical logic, the De Bruijn notation is a syntax for terms in the λ calculus invented by the Dutch mathematician Nicolaas Govert de Bruijn. It can be seen as a reversal of the usual syntax for the λ calculus where the argument in an application is placed next to its corresponding binder in the function instead of after the latter's body.
Formal definition
Terms () in the De Bruijn notation are either variables (), or have one of two wagon prefixes. The abstractor wagon, written , corresponds to the usual λ-binder of the λ calculus, and the applicator wagon, written , corresponds to the argument in an application in the λ calculus.
Terms in the traditional syntax can be converted to the De Bruijn notation by defining an inductive function for which:
All operations on λ-terms commute with respect to the translation. For example, the usual β-reduction,
in the De Bruijn notation is, predictably,
A feature of this notation is that abstractor and applicator wagons of β-redexes are paired like parentheses. For example, consider the stages in the β-reduction of the term , where the redexes are underlined:
Thus, if one views the applicator as an open paren ('(') and the abstractor as a close bracket (']'), then the pattern in the above term is '((](]]'. De Bruijn called an applicator and its corresponding abstractor in this interpretation partners, and wagons without partners bachelors. A sequence of wagons, which he called a segment, is well balanced if all its wagons are partnered.
Advantages of the De Bruijn notation
In a well balanced segment, the partnered wagons may be moved around arbitrarily and, as long as parity is not destroyed, the meaning of the term stays the same. For example, in the above example, the applicator can be brought to its abstractor , or the abstractor to the applicator. In fact, all commutatives and permutative conversions on lambda terms may be described simply in terms of parity-preserving reorderings of partnered wagons. One thus obtains a generalised conversion primitive for λ-terms in the De Bruijn notation.
Several properties of λ-terms that are difficult to state and prove using the traditional notation are easily expressed in the De Bruijn notation. For example, in a type-theoretic setting, one can easily compute the canonical class of types for a term in a typing context, and restate the type checking problem to one of verifying that the checked type is a member of this class. De Bruijn notation has also been shown to be useful in calculi for explicit substitution in pure type systems.
See also
Mathematical notation
References
Lambda calculus
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https://en.wikipedia.org/wiki/Beyer%20Professor%20of%20Applied%20Mathematics
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The Beyer Chair of Applied Mathematics is an endowed professorial position in the Department of Mathematics, University of Manchester, England. The endowment came from the will of the celebrated locomotive designer and founder of locomotive builder Beyer, Peacock & Company, Charles Frederick Beyer. He was the university's largest single donor.
The first appointment in 1881 was of Arthur Schuster who held the position until 1888. After Schuster’s departure, the chair of Mathematics to which Horace Lamb had been appointed in 1885 became the Beyer Professorship of Mathematics and remained so until Lamb’s retirement in 1920. At this point an existing chair, of Mathematics and Natural Philosophy to which Sydney Chapman had been appointed in 1919, was renamed the Beyer Professorship of Mathematics and Natural Philosophy. After Chapman’s resignation, the Beyer title was applied to the chair of Applied Mathematics. There was no incumbent between 1937-1945.
Most of the holders of the post were elected as Fellows of the Royal Society, an honour bestowed on a small minority of UK mathematics professors. Lamb, Champman, Milne and Goldstein all received the Smith's Prize and indication of early career promise.
The other endowed chairs in mathematics at the University of Manchester are the Richardson Chair of Applied Mathematics, and the Fielden Chair of Pure Mathematics as well as the named Sir Horace Lamb Chair.
Beyer Professors
1881–1888 Arthur Schuster
1888–1920 Horace Lamb
1920–1924 Sydney Chapman, Beyer Professor of Mathematics and Natural Philosophy
1924–1928 Edward Arthur Milne
1929–1937 Douglas Hartree
1945–1950 Sydney Goldstein
1950–1959 James Lighthill
1961–1990 Fritz Ursell
1991–1996 Philip Hall
1996–2017 David Abrahams
2017– Pending appointment
References
Professorships in mathematics
Professorships at the University of Manchester
Mathematics education in the United Kingdom
1881 establishments in England
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https://en.wikipedia.org/wiki/David%20Abrahams%20%28mathematician%29
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Ian David Abrahams (born 15 January 1958) is an English mathematician and held the Beyer Professor of Applied Mathematics at the University of Manchester, 2008–2016. From 2014 to 2016 he was Director of the International Centre for Mathematical Sciences in Edinburgh and in October 2016 he succeeded John Toland as Director of the Isaac Newton Institute for Mathematical Sciences, and N M Rothschild and Sons Professor of Mathematics, in Cambridge. He was President 2007–2009, of the Institute of Mathematics and its Applications. In 2017 he was awarded the IMA/LMS David Crighton Medal for services to mathematics.
Education
Born in Manchester, Abrahams was the son of Harry Abrahams and of Leila Abrahams.
He completed his BSc in aeronautical engineering in 1979 and PhD (and DIC) in applied mathematics in 1982, both at Imperial College London. There he won two scholarships and the Finsbury Medal for top undergraduate. For his PhD he was supervised by Frank Leppington for a thesis entitled The scattering of sound by finite thin elastic plates and cavities.
In the
same year, he moved to Manchester on a 1-year contract. This was the beginning of a collaboration with GR Wickham. First, they developed some general techniques for solving matrix Wiener–Hopf problems and this gave the solution to a basic problem of diffraction theory, namely, scattering by two parallel, semi-infinite, staggered plates. Motivated by the problems of austenitic steel welds, they went on to develop a theory for wave propagation in certain inhomogeneous anisotropic solids. They also gave asymptotic solutions for scattering by small defects in an elastic half-space making use of a certain expansion of the half-space Green's function.
More recently Abrahams has found aspects of the Wiener-Hopf technique that impinge on finance and probability. This has led to developments, for example in relating Wiener-Hopf factorisation to Spitzer's identity and other important results within probability theory.
Personal life
In 2004, Abrahams married Penelope Lawrence Warwick with whom he has one daughter and two step sons.
Abrahams's leisure interests include motorcycling and he owns a 1977 Triumph Bonneville T140V, as well as a 1000cc Moto Guzzi.
References
External links
Archived home page at University of Manchester
Living people
Alumni of Imperial College London
Academics of the University of Manchester
20th-century English mathematicians
21st-century English mathematicians
David Crighton medalists
1958 births
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https://en.wikipedia.org/wiki/Institute%20for%20Mathematics%20and%20its%20Applications
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The Institute for Mathematics and its Applications located at the University of Minnesota is an organization established in 1982 by the National Science Foundation (NSF) of the United States.
Objectives
The primary mission of the IMA is to increase the impact of mathematics by fostering interdisciplinary research and linking mathematics and scientific and technological problems from other disciplines and industry.
Activities
The IMA hosts long-term visitors, funds postdoctoral research positions, and holds several conferences annually. The NSF has granted the IMA $19.5 million over the period 2005–2010, the largest single mathematics grant the NSF has ever awarded.
Prize in Mathematics and its Applications
The IMA annually awards a prize to a mathematician who has received their PhD within the last 10 years. This award recognizes an individual who has made a transformative impact on the mathematical sciences and their applications.
2019: Jacob Bedrossian
2018: Anders C. Hansen
2017: Jianfeng Lu
2016: Rachel Ward and Deanna Needell
2015: Jonathan Weare
2014: David F. Anderson
Organization with similar names
Sharing a very similar name and the acronym IMA, it should not be confused with the Institute of Mathematics and its Applications, a professional body for mathematicians in the UK.
References
IMA Celebrates 20 Prodigious Years, SIAM News, October 31, 2003.
Notes
External links
Institute for Mathematics and its Applications
Mathematical institutes
University of Minnesota
Research institutes established in 1982
Research institutes in Minnesota
National Science Foundation mathematical sciences institutes
1982 establishments in Minnesota
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https://en.wikipedia.org/wiki/Hyperhomology
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In homological algebra, the hyperhomology or hypercohomology () is a generalization of (co)homology functors which takes as input not objects in an abelian category but instead chain complexes of objects, so objects in . It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor .
Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a derived functor between derived categories.
Motivation
One of the motivations for hypercohomology comes from the fact that there isn't an obvious generalization of cohomological long exact sequences associated to short exact sequencesi.e. there is an associated long exact sequenceIt turns out hypercohomology gives techniques for constructing a similar cohomological associated long exact sequence from an arbitrary long exact sequencesince its inputs are given by chain complexes instead of just objects from an abelian category. We can turn this chain complex into a distinguished triangle (using the language of triangulated categories on a derived category)which we denote byThen, taking derived global sections gives a long exact sequence, which is a long exact sequence of hypercohomology groups.
Definition
We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on.
Suppose that A is an abelian category with enough injectives and F a left exact functor to another abelian category B.
If C is a complex of objects of A bounded on the left, the hypercohomology
Hi(C)
of C (for an integer i) is
calculated as follows:
Take a quasi-isomorphism Φ : C → I, here I is a complex of injective elements of A.
The hypercohomology Hi(C) of C is then the cohomology Hi(F(I)) of the complex F(I).
The hypercohomology of C is independent of the choice of the quasi-isomorphism, up to unique isomorphisms.
The hypercohomology can also be defined using derived categories: the hypercohomology of C is just the cohomology of RF(C) considered as an element of the derived category of B.
For complexes that vanish for negative indices, the hypercohomology can be defined as the derived functors of H0 = FH0 = H0F.
The hypercohomology spectral sequences
There are two hypercohomology spectral sequences; one with E2 term
and the other with E1 term
and E2 term
both converging to the hypercohomology
,
where RjF is a right derived functor of F.
Applications
One application of hypercohomology spectral sequences are in the study of gerbes. Recall that rank n vector bundles on a space can be classified as the Cech-cohomology group . The main idea behind gerbes is to extend this idea cohomologically, so
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https://en.wikipedia.org/wiki/Geometric%20median
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In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the 1-median, spatial median, Euclidean minisum point, or Torricelli point.
The geometric median is an important estimator of location in statistics, where it is also known as the L1 estimator (after the L1 norm). It is also a standard problem in facility location, where it models the problem of locating a facility to minimize the cost of transportation.
The more general k-median problem asks for the location of k cluster centers minimizing the sum of distances from each sample point to its nearest center.
If the point is generalized into a line or a curve, the best-fitting solution is found via least absolute deviations.
The special case of the problem for three points in the plane (that is, = 3 and = 2 in the definition below) is sometimes also known as Fermat's problem; it arises in the construction of minimal Steiner trees, and was originally posed as a problem by Pierre de Fermat and solved by Evangelista Torricelli. Its solution is now known as the Fermat point of the triangle formed by the three sample points. The geometric median may in turn be generalized to the problem of minimizing the sum of weighted distances, known as the Weber problem after Alfred Weber's discussion of the problem in his 1909 book on facility location. Some sources instead call Weber's problem the Fermat–Weber problem, but others use this name for the unweighted geometric median problem.
provides a survey of the geometric median problem. See for generalizations of the problem to non-discrete point sets.
Definition
Formally, for a given set of m points with each , the geometric median is defined as
Here, arg min means the value of the argument which minimizes the sum. In this case, it is the point in n dimensional Euclidean space from where the sum of all Euclidean distances to the 's is minimum.
Properties
For the 1-dimensional case, the geometric median coincides with the median. This is because the univariate median also minimizes the sum of distances from the points. (More precisely, if the points are p1, …, pn, in that order, the geometric median is the middle point if n is odd, but is not uniquely determined if n is even, when it can be any point in the line segment between the two middling points and .)
The geometric median is unique whenever the points are not collinear.
The geometric median is equivariant for Euclidean similarity transformations, including translation and rotation. This means that one would get the same result either by transforming the geometric median, or by applying the same transformation to the sample data and finding the geometric median of the transformed data. This property
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https://en.wikipedia.org/wiki/Fermat%E2%80%93Weber%20problem
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In mathematics, statistics, and operations research, the Fermat–Weber problem is either of two closely related problems:
Geometric median, the problem of finding a point minimizing the sum of distances from given points
Weber problem, the problem of finding a point minimizing the sum of weighted distances from given (point, weight) pairs
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https://en.wikipedia.org/wiki/Negafibonacci%20coding
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In mathematics, negafibonacci coding is a universal code which encodes nonzero integers into binary code words. It is similar to Fibonacci coding, except that it allows both positive and negative integers to be represented. All codes end with "11" and have no "11" before the end.
Encoding method
To encode a nonzero integer X:
Calculate the largest (or smallest) encodeable number with N bits by summing the odd (or even) negafibonacci numbers from 1 to N.
When it is determined that N bits is just enough to contain X, subtract the Nth negafibonacci number from X, keeping track of the remainder, and put a one in the Nth bit of the output.
Working downward from the Nth bit to the first one, compare each of the corresponding negafibonacci numbers to the remainder. Subtract it from the remainder if the absolute value of the difference is less, AND if the next higher bit does not already have a one in it. A one is placed in the appropriate bit if the subtraction is made, or a zero if not.
Put a one in the N+1th bit to finish.
To decode a token in the code, remove the last "1", assign the remaining bits the values 1, −1, 2, −3, 5, −8, 13... (the negafibonacci numbers), and add the "1" bits.
Negafibonacci representation
Negafibonacci coding is closely related to negafibonacci representation, a positional numeral system sometimes used by mathematicians. The negafibonacci code for a particular nonzero integer is exactly that of the integer's negafibonacci representation, except with the order of its digits reversed and an additional "1" appended to the end. The negafibonacci code for all negative numbers has an odd number of digits, while those of all positive numbers have an even number of digits.
Table
The code for the integers from −11 to 11 is given below.
See also
Fibonacci numbers
Golden ratio base
Zeckendorf's theorem
References
Works cited
In the pre-publication draft of section 7.1.3 see in particular pp. 36–39.
Non-standard positional numeral systems
Lossless compression algorithms
Fibonacci numbers
fr:Codage de Fibonacci
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https://en.wikipedia.org/wiki/Shapiro%20polynomials
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In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are:
where the second sequence, indicated by Q, is said to be complementary to the first sequence, indicated by P.
Construction
The Shapiro polynomials Pn(z) may be constructed from the Golay–Rudin–Shapiro sequence an, which equals 1 if the number of pairs of consecutive ones in the binary expansion of n is even, and −1 otherwise. Thus a0 = 1, a1 = 1, a2 = 1, a3 = −1, etc.
The first Shapiro Pn(z) is the partial sum of order 2n − 1 (where n = 0, 1, 2, ...) of the power series
f(z) := a0 + a1 z + a2 z2 + ...
The Golay–Rudin–Shapiro sequence {an} has a fractal-like structure – for example, an = a2n – which implies that the subsequence (a0, a2, a4, ...) replicates the original sequence {an}. This in turn leads to remarkable
functional equations satisfied by f(z).
The second or complementary Shapiro polynomials Qn(z) may be defined in terms of this sequence, or by the relation Qn(z) = (1-)nz2n-1Pn(-1/z), or by the recursions
Properties
The sequence of complementary polynomials Qn corresponding to the Pn is uniquely characterized by the following properties:
(i) Qn is of degree 2n − 1;
(ii) the coefficients of Qn are all 1 or −1, and its constant term equals 1; and
(iii) the identity |Pn(z)|2 + |Qn(z)|2 = 2(n + 1) holds on the unit circle, where the complex variable z has absolute value one.
The most interesting property of the {Pn} is that the absolute value of Pn(z) is bounded on the unit circle by the square root of 2(n + 1), which is on the order
of the L2 norm of Pn. Polynomials with coefficients from the set {−1, 1} whose maximum modulus on the unit circle is close to their mean modulus are useful for various applications in communication theory (e.g., antenna design and data compression). Property (iii) shows that (P, Q) form a Golay pair.
These polynomials have further properties:
See also
Littlewood polynomials
Notes
References
Chapter 4.
Fourier analysis
Digital signal processing
Polynomials
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https://en.wikipedia.org/wiki/Realization%20%28probability%29
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In probability and statistics, a realization, observation, or observed value, of a random variable is the value that is actually observed (what actually happened). The random variable itself is the process dictating how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "empirical", as in empirical distribution function or empirical probability.
Conventionally, to avoid confusion, upper case letters denote random variables; the corresponding lower case letters denote their realizations.
Formal definition
In more formal probability theory, a random variable is a function X defined from a sample space Ω to a measurable space called the state space. If an element in Ω is mapped to an element in state space by X, then that element in state space is a realization. Elements of the sample space can be thought of as all the different possibilities that could happen; while a realization (an element of the state space) can be thought of as the value X attains when one of the possibilities did happen. Probability is a mapping that assigns numbers between zero and one to certain subsets of the sample space, namely the measurable subsets, known here as events. Subsets of the sample space that contain only one element are called elementary events. The value of the random variable (that is, the function) X at a point ω ∈ Ω,
is called a realization of X.
See also
Errors and residuals
Outcome (probability)
Random variate
Raw data
Notes
References
Statistical concepts
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https://en.wikipedia.org/wiki/Macdonald%20polynomials
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In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable t can be replaced by several different variables t=(t1,...,tk), one for each of the k orbits of roots in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
Definition
First fix some notation:
R is a finite root system in a real vector space V.
R+ is a choice of positive roots, to which corresponds a positive Weyl chamber.
W is the Weyl group of R.
Q is the root lattice of R (the lattice spanned by the roots).
P is the weight lattice of R (containing Q).
An ordering on the weights: if and only if is a nonnegative linear combination of simple roots.
P+ is the set of dominant weights: the elements of P in the positive Weyl chamber.
ρ is the Weyl vector: half the sum of the positive roots; this is a special element of P+ in the interior of the positive Weyl chamber.
F is a field of characteristic 0, usually the rational numbers.
A = F(P) is the group algebra of P, with a basis of elements written eλ for λ ∈ P.
If f = eλ, then f means e−λ, and this is extended by linearity to the whole group algebra.
mμ = Σλ ∈ Wμeλ is an orbit sum; these elements form a basis for the subalgebra AW of elements fixed by W.
, the infinite q-Pochhammer symbol.
is the inner product of two elements of A, at least when t is a positive integer power of q.
The Macdonald polynomials Pλ for λ ∈ P+ are uniquely defined by the following two conditions:
where uλμ is a rational function of q and t with uλλ = 1;
Pλ and Pμ are orthogonal if λ < μ.
In other words, the Macdonald polynomials are obtained by orthogonalizing the obvious basis for AW. The existence of polynomials with these properties is easy to show (for any inner product). A key property of the Macdonald polynomials is that they are orthogonal: 〈Pλ, Pμ〉 = 0 whenever λ ≠ μ. This is not a trivial consequence of the definition because P+ is not totally ordered, and so has plenty of elements that are incomparable. Thus one must check that the corresponding polynomials are still or
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https://en.wikipedia.org/wiki/Acyclic%20space
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In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point.
In other words, using the idea of reduced homology,
It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc
or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface."
The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic.
If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.
Examples
Acyclic spaces occur in topology, where they can be used to construct other, more interesting topological spaces.
For instance, if one removes a single point from a manifold M which is a homology sphere, one gets such a space. The homotopy groups of an acyclic space X do not vanish in general, because the fundamental group need not be trivial. For example, the punctured Poincaré homology sphere is an acyclic, 3-dimensional manifold which is not contractible.
This gives a repertoire of examples, since the first homology group is the abelianization of the fundamental group. With every perfect group G one can associate a (canonical, terminal) acyclic space, whose fundamental group is a central extension of the given group G.
The homotopy groups of these associated acyclic spaces are closely related to Quillen's plus construction on the classifying space BG.
Acyclic groups
An acyclic group is a group G whose classifying space BG is acyclic; in other words, all its (reduced) homology groups vanish, i.e., , for all . Every acyclic group is thus a perfect group, meaning its first homology group vanishes: , and in fact, a superperfect group, meaning the first two homology groups vanish: . The converse is not true: the binary icosahedral group is superperfect (hence perfect) but not acyclic.
See also
Aspherical space
References
External links
Algebraic topology
Homology theory
Homotopy theory
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https://en.wikipedia.org/wiki/Affine%20Hecke%20algebra
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In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.
Definition
Let be a Euclidean space of a finite dimension and an affine root system on . An affine Hecke algebra is a certain associative algebra that deforms the group algebra of the Weyl group of (the affine Weyl group). It is usually denoted by , where is multiplicity function that plays the role of deformation parameter. For the affine Hecke algebra indeed reduces to .
Generalizations
Ivan Cherednik introduced generalizations of affine Hecke algebras, the so-called double affine Hecke algebra (usually referred to as DAHA). Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials (building on work of Eric Opdam). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ equations.
References
Algebras
Representation theory
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https://en.wikipedia.org/wiki/Free%20lattice
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In mathematics, in the area of order theory, a free lattice is the free object corresponding to a lattice. As free objects, they have the universal property.
Formal definition
Because the concept of a lattice can be axiomatised in terms of two operations and satisfying certain identities, the category of all lattices constitute a variety (universal algebra), and thus there exist (by general principles of universal algebra) free objects within this category: lattices where only those relations hold which follow from the general axioms.
These free lattices may be characterised using the relevant universal property. Concretely, free lattice is a functor from sets to lattices, assigning to each set the free lattice equipped with a set map assigning to each the corresponding element . The universal property of these is that there for any map from to some arbitrary lattice exists a unique lattice homomorphism satisfying , or as a commutative diagram:
The functor is left adjoint to the forgetful functor from lattices to their underlying sets.
It is frequently possible to prove things about the free lattice directly using the universal property, but such arguments tend to be rather abstract, so a concrete construction provides a valuable alternative presentation.
Semilattices
In the case of semilattices, an explicit construction of the free semilattice is straightforward to give; this helps illustrate several features of the definition by way of universal property. Concretely, the free semilattice may be realised as the set of all finite nonempty subsets of , with ordinary set union as the join operation . The map maps elements of to singleton sets, i.e., for all . For any semilattice and any set map , the corresponding universal morphism is given by
where denotes the semilattice operation in .
This form of is forced by the universal property: any can be written as a finite union of elements on the form for some , the equality in the universal property says , and finally the homomorphism status of implies for all . Any extension of to infinite subsets of (if there even is one) need however not be uniquely determined by these conditions, so there cannot in be any elements corresponding to infinite subsets of .
Lower semilattices
It is similarly possible to define a free functor for lower semilattices, but the combination fails to produce the free lattice in several ways, because treats as just a set:
the join operation is not extended to the new elements of ,
the existing partial order on is not respected; views and as unrelated, not understanding it should make .
The actual structure of the free lattice is considerably more intricate than that of the free semilattice.
Word problem
The word problem for free lattices has some interesting aspects. Consider the case of bounded lattices, i.e. algebraic structures with the two binary operations ∨ and ∧ and the two constants (nullary operations) 0 and 1. The se
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https://en.wikipedia.org/wiki/Dror%20Bar-Natan
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Dror Bar-Natan (; born January 30, 1966) is a professor at the University of Toronto Department of Mathematics, Canada. His main research interests include knot theory, finite type invariants, and Khovanov homology.
Education
Bar-Natan earned his B.Sc. in mathematics at Tel Aviv University in 1984. After performing his military service as a teacher, he went to study at Princeton University in 1987. He obtained his Ph.D. in mathematics from Princeton in 1991, under the direction of physicist Edward Witten.
Professorship
After holding a Benjamin Peirce Assistant Professorship at Harvard University for four years from 1991–95, he returned to Israel, and became Associate Professor at the Hebrew University of Jerusalem. He moved to the University of Toronto in 2002, and was promoted to Full Professor in 2006.
Personal life
Bar-Natan holds US, Israeli, and Canadian citizenship, and currently resides in Canada. Bar-Natan originally refused to take the Canadian citizenship oath because it would require him to swear allegiance to royalty. He later decided to become a citizen but publicly announced his intention to renounce the oath immediately after becoming a citizen, which he did so in front of the presiding judge at his citizenship ceremony on November 30, 2015. From his marriage to mathematician Yael Karshon he has two sons, Assaf and Itai.
Research
In 1999, Bar-Natan collaborated on a paper with the goal of mathematically refuting claims made in The Bible Code by Michael Drosnin that hidden messages could be deciphered from within the bible. In particular, the paper demonstrated that practically any "code" could be found within the Bible, thereby debunking Drosnin's "discovery" of specific codes. This work is outside the main scope of his academic interests, although he is known for it because of the popularity of The Bible Code.
Academically, Bar-Natan has made significant contributions to the formalization of Khovanov homology.
Bar-Natan was a member of the Editorial Board for the journal Compositio Mathematica for 10 years, until 2010.
Selected publication
References
External links
Citizenship disavowal website maintained by Bar-Natan
20th-century Israeli mathematicians
21st-century Israeli mathematicians
Topologists
Academic staff of the University of Toronto
Tel Aviv University alumni
Princeton University alumni
Harvard University Department of Mathematics faculty
Harvard University faculty
Canadian mathematicians
1966 births
Living people
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https://en.wikipedia.org/wiki/Coreset
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In computational geometry, a coreset is a small set of points that approximates the shape of a larger point set, in the sense that applying some geometric measure to the two sets (such as their minimum bounding box volume) results in approximately equal numbers. Many natural geometric optimization problems have coresets that approximate an optimal solution to within a factor of , that can be found quickly (in linear time or near-linear time), and that have size bounded by a function of independent of the input size, where is an arbitrary positive number. When this is the case, one obtains a linear-time or near-linear time approximation scheme, based on the idea of finding a coreset and then applying an exact optimization algorithm to the coreset. Regardless of how slow the exact optimization algorithm is, for any fixed choice of , the running time of this approximation scheme will be plus the time to find the coreset.
References
Computational geometry
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https://en.wikipedia.org/wiki/Sharadchandra%20Shankar%20Shrikhande
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Sharadchandra Shankar Shrikhande (19 October 1917 – 21 April 2020) was an Indian mathematician with notable achievements in combinatorial mathematics. He was notable for his breakthrough work along with R. C. Bose and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal latin squares of order 4n + 2 for any n. Shrikhande's specialty was combinatorics, and statistical designs. Shrikhande graph is used in statistical designs.
Life, education and career
He was the fifth of ten siblings. His father worked at a flour mill. He completed his B.Sc. from Government Science College, Nagpur and went for further studies at the Indian Statistical Institute. He then briefly worked as a lecturer at the Government Science College, Nagpur.
Shrikhande received a Ph.D. in the year 1950 from the University of North Carolina at Chapel Hill under the supervision of Raj Chandra Bose. Shrikhande taught at various universities in the USA and in India. Shrikhande was a professor of mathematics at Banaras Hindu University, Banaras, and the founding head of the department of mathematics, University of Mumbai and the founding director of the Center of Advanced Study in Mathematics, Mumbai until he retired in 1978. He was a fellow of the Indian National Science Academy, the Indian Academy of Sciences and the Institute of Mathematical Statistics, USA.
In 1988, his wife Shakuntala passed away and he moved to the United States. Shrikhande returned to India in 2009. He turned 100 in October 2017 and died in April 2020 at the age of 102.
His son Mohan Shrikhande is a professor of combinatorial mathematics at Central Michigan University in Mt. Pleasant, Michigan.
References
External links
1917 births
2020 deaths
20th-century Indian mathematicians
Fellows of the Indian Academy of Sciences
Indian centenarians
Indian combinatorialists
Indian statisticians
Latin squares
Men centenarians
Scientists from Madhya Pradesh
University of North Carolina at Chapel Hill alumni
University of North Carolina at Chapel Hill faculty
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https://en.wikipedia.org/wiki/LLT%20polynomial
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In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions.
J. Haglund, M. Haiman, N. Loehr (2005) showed how to expand Macdonald polynomials in terms of LLT polynomials. Ian Grojnowski and Mark Haiman (2007, preprint) proved a positivity conjecture for LLT polynomials that combined with the previous result implies the Macdonald positivity conjecture for Macdonald polynomials, and extended the definition of LLT polynomials to arbitrary finite root systems.
References
I. Grojnowski, M. Haiman, Affine algebras and positivity (preprint available here)
J. Haglund, M. Haiman, N. Loehr A Combinatorial Formula for Macdonald Polynomials J. Amer. Math. Soc. 18 (2005), no. 3, 735–761
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties J. Math. Phys. 38 (1997), no. 2, 1041–1068.
Symmetric functions
Algebraic geometry
Algebraic combinatorics
Q-analogs
Polynomials
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https://en.wikipedia.org/wiki/Free-by-cyclic%20group
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In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group is said to be free-by-cyclic if it has a free normal subgroup such that the quotient group is cyclic. In other words, is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). Usually, we assume is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if is an automorphism of , the semidirect product is a free-by-cyclic group.
An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms represent the same outer automorphism, that is, for some inner automorphism , the free-by-cyclic groups and are isomorphic.
Examples
The class of free-by-cyclic groups contains various groups as follow:
A free-by-cyclic group is hyperbolic if and only if the attaching map is atoroidal.
Some free-by-cyclic groups are hyperbolic relative to free-abelian subgroups. More generally, all free-by-cyclic groups are hyperbolic relative to a collection of subgroups that are free-by-cyclic for an automorphism of polynomial growth.
Notably, there is a non-CAT(0) free-by-cyclic group.
References
A. Martino and E. Ventura (2004), The Conjugacy Problem for Free-by-Cyclic Groups . Preprint from the Centre de Recerca Matemàtica, Barcelona, Catalonia, Spain.
Feighn, Mark; Handel, Michael Mapping tori of free group automorphisms are coherent, Ann. Math., Volume 149 (1999) no. 3
Ghosh, P. (2023). Relative hyperbolicity of free-by-cyclic extensions. Compositio Mathematica, 159(1), 153-183.
F. Dahmani and R. Li, Relative hyperbolicity for automorphisms of free products and free groups, Journal of Topology and AnalysisVol. 14, No. 01, pp. 55-92 (2022)
Infinite group theory
Properties of groups
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https://en.wikipedia.org/wiki/Virtually
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In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup such that H has property P.
Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.
This terminology is also used when P is just another group. That is, if G and H are groups then G is virtually H if G has a subgroup K of finite index in G such that K is isomorphic to H.
In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are commensurable.
Examples
Virtually abelian
The following groups are virtually abelian.
Any abelian group.
Any semidirect product where N is abelian and H is finite. (For example, any generalized dihedral group.)
Any semidirect product where N is finite and H is abelian.
Any finite group (since the trivial subgroup is abelian).
Virtually nilpotent
Any group that is virtually abelian.
Any nilpotent group.
Any semidirect product where N is nilpotent and H is finite.
Any semidirect product where N is finite and H is nilpotent.
Gromov's theorem says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
Virtually polycyclic
Virtually free
Any free group.
Any virtually cyclic group.
Any semidirect product where N is free and H is finite.
Any semidirect product where N is finite and H is free.
Any free product , where H and K are both finite. (For example, the modular group .)
It follows from Stalling's theorem that any torsion-free virtually free group is free.
Others
The free group on 2 generators is virtually for any as a consequence of the Nielsen–Schreier theorem and the Schreier index formula.
The group is virtually connected as has index 2 in it.
References
Group theory
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https://en.wikipedia.org/wiki/Velgo%C5%A1ti
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Velgošti () is a village in the municipality of Ohrid, North Macedonia. It has a primary school called Živko Čingo dedicated to the author born there.
Demographics
According to the statistics of the Bulgarian ethnographer Vasil Kanchov from 1900, 1220 inhabitants lived in Velgošti, 1190 Bulgarian Exarchists and 30 Muslim Bulgarians.
As of the 2021 census, Velgošti had 3,141 residents with the following ethnic composition:
Macedonians 2,673
Persons for whom data are taken from administrative sources 394
Others 66
Vlachs 8
According to the 2002 census, the village had a total of 3,060 inhabitants. Ethnic groups in the village include:
Macedonians 3,002
Serbs 8
Aromanians 10
Others 40
References
Villages in Ohrid Municipality
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https://en.wikipedia.org/wiki/Empirical%20probability
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In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e., by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation.
Given an event in a sample space, the relative frequency of is the ratio being the number of outcomes in which the event occurs, and being the total number of outcomes of the experiment.
In statistical terms, the empirical probability is an estimator or estimate of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain assumptions are made for the prior distribution of the probability. If a trial yields more information, the empirical probability can be improved on by adopting further assumptions in the form of a statistical model: if such a model is fitted, it can be used to derive an estimate of the probability of the specified event
Advantages and disadvantages
Advantages
An advantage of estimating probabilities using empirical probabilities is that this procedure is relatively free of assumptions.
For example, consider estimating the probability among a population of men that they satisfy two conditions:
that they are over 6 feet in height.
that they prefer strawberry jam to raspberry jam.
A direct estimate could be found by counting the number of men who satisfy both conditions to give the empirical probability of the combined condition. An alternative estimate could be found by multiplying the proportion of men who are over 6 feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.
Disadvantages
A disadvantage in using empirical probabilities arises in estimating probabilities which are either very close to zero, or very close to one. In these cases very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. Here statistical models can help, depending on the context, and in general one can hope that such models would provide improvements in accuracy compared to empirical probabilities, provided that the assumptions involved actually do hold.
For example, consider estimating the probability that the lowest of the daily-maximum temperatures at a site in February in any one year is less than zero degrees Celsius. A record of such temperatures in past years could be used to estimate this probability. A model-based alternative would be to select a family of probability distributions and fit it to the
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https://en.wikipedia.org/wiki/Section%2051%28xi%29%20of%20the%20Constitution%20of%20Australia
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Section 51(xi) of the Constitution of Australia, a subsection of section 51, grants the Commonwealth the power to make laws for "census and statistics".
Background
The first version of the Constitution included a census power. Its inclusion was not controversial. It can be seen as a class of "nationhood powers" which reflected basic powers that a "nation" was viewed with possessing (similar nationhood powers would include the currency power, the weights and measures power, and the postal power).
Australian colonies had collected statistics from settlement. The first simultaneous census was held across Australia in 1881 as part of the Census of the British Empire.
In December 1905 the Commonwealth Government passed the Census and Statistics Act 1905.
The first Commonwealth Census after federation was held in 1911 (although a simultaneous state census was held in 1901).
The Australian Bureau of Statistics is the Commonwealth agency responsible for census and statistics.
Related Constitution sections
Section 24 says the number of members in the House of Representatives per state will be based on the based on quotas based on population, which will be based on "the latest statistics of the Commonwealth" (s. 24(i) and s. 24(ii)). Section 24 evinces a clear intention that the Commonwealth would use section 51(xi) to conduct census and collect information, rather than leaving the matter to the states.
Section 127 stated that "in reckoning the numbers of the people ... aboriginal natives should not be counted." This section was removed by a referendum to amend the constitution that was held in 1967.
Section 127 did not use the word "census" or "statistics" – the language of s. 51(xi). On a purposive approach, the debates at the Constitutional Conventions showed the clear purpose of section 127 was to limit section 24. Section 127 operated to prevent the number of Aboriginal Australians being used in the calculations for the number of members of the House of Representatives. Section 127 was quite a narrow provision, in that it did not use the word "statistics". Accordingly, section 51(xi) still allowed the Commonwealth had the power to collect statistics on Aboriginal people which it did, according population numbers.
References
Full text of Section 51 of the Constitution of Australia on Austlii
Full text of Section 126 and the deleted Section 127 of the Constitution of Australia on Austlii
Dr Helen Irving,"The Census, Constitution and Australian Democracy" (Australian Bureau of Statistics)
Ellen Percy Kraly, The Annual Censuses of Aborigines, 1925-1944: Technical Imperative, Social Demography, or Social Control? (Population Association of America, 2007)
Australian constitutional law
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https://en.wikipedia.org/wiki/Department%20of%20Mathematics%2C%20University%20of%20Manchester
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The Department of Mathematics at the University of Manchester is one of the largest unified mathematics departments in the United Kingdom, with over 90 academic staff and an undergraduate intake of roughly 400 students per year (including students studying mathematics with a minor in another subject) and approximately 200 postgraduate students in total.
The School of Mathematics was formed in 2004 by the merger of the mathematics departments of University of Manchester Institute of Science and Technology (UMIST) and the Victoria University of Manchester (VUM). In July 2007 the department moved into a purpose-designed building─the first three floors of the Alan Turing Building─on Upper Brook Street. In a Faculty restructure in 2019 the School of Mathematics reverted to the Department of Mathematics. It is one of five Departments that make up the School of Natural Sciences, which together with the School of Engineering now constitutes the Faculty of Science and Engineering at Manchester.
Organization
The current head of the department is Andrew Hazel. The department is divided into three groups: Pure Mathematics (Head: Charles Eaton), Applied Mathematics (Head: David Sylvester), and Probability and Statistics (Head: Korbinian Strimmer). The director of research is William Parnell.
The Manchester Institute for Mathematical Sciences (MIMS) is a unit of the department focusing on the organising of mathematical colloquia and conferences, and research visitors. MIMS is headed by Nick Higham. Other high-profile mathematicians at Manchester include Martin Taylor and Jeff Paris.
Since its formation, the department has made some influential appointments including the topologist Viktor Buchstaber and model theorist Alex Wilkie. Numerical analyst Jack Dongarra, one of the authors of LINPACK, was appointed in 2007 as Turing Fellow. In the autumn of 2007, Albert Shiryaev was appointed to a 20% chair. Shiryaev is known for his work on probability theory (he was a student of Andrey Kolmogorov) and for his work on financial mathematics.
Research
As might be expected from its size (about 30 academic staff in Probability & Statistics, 30 in Pure Mathematics and 45 in Applied Mathematics), the department has a wide range of research interests, including the traditionally pure areas of algebra, analysis, noncommutative geometry, ergodic theory, mathematical logic, number theory, geometry and topology; and the more applied dynamical system, fluid dynamics, solid mechanics, inverse problems, mathematical finance, wave propagation and scattering. The department also has a strong tradition in numerical analysis and well established groups in Probability theory, and Mathematical statistics.
Manchester mathematicians have a long tradition of applying mathematics to industrial problems. Nowadays this involves not only the traditional applications in engineering and the physical sciences, but also in the life sciences and the financial sector. Some of the recent indust
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https://en.wikipedia.org/wiki/Tangential%20and%20normal%20components
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In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector. Similarly, a vector at a point on a surface can be broken down the same way.
More generally, given a submanifold N of a manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to N.
Formal definition
Surface
More formally, let be a surface, and be a point on the surface. Let be a vector at Then one can write uniquely as a sum
where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other.
To calculate the tangential and normal components, consider a unit normal to the surface, that is, a unit vector perpendicular to at Then,
and thus
where "" denotes the dot product. Another formula for the tangential component is
where "" denotes the cross product.
Note that these formulas do not depend on the particular unit normal used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).
Submanifold
More generally, given a submanifold N of a manifold M and a point , we get a short exact sequence involving the tangent spaces:
The quotient space is a generalized space of normal vectors.
If M is a Riemannian manifold, the above sequence splits, and the tangent space of M at p decomposes as a direct sum of the component tangent to N and the component normal to N:
Thus every tangent vector splits as where and .
Computations
Suppose N is given by non-degenerate equations.
If N is given explicitly, via parametric equations (such as a parametric curve), then the derivative gives a spanning set for the tangent bundle (it is a basis if and only if the parametrization is an immersion).
If N is given implicitly (as in the above description of a surface, (or more generally as) a hypersurface) as a level set or intersection of level surfaces for , then the gradients of span the normal space.
In both cases, we can again compute using the dot product; the cross product is special to 3 dimensions however.
Applications
Lagrange multipliers: constrained critical points are where the tangential component of the total derivative vanish.
Surface normal
Frenet–Serret formulas
References
Differential geometry
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https://en.wikipedia.org/wiki/Mixed%20Hodge%20module
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In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module together with a perverse sheaf such that the functor from the Riemann–Hilbert correspondence sends to . This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves.
Abstract structure
Before going into the nitty gritty details of defining Mixed hodge modules, which is quite elaborate, it is useful to get a sense of what the category of Mixed Hodge modules actually provides. Given a complex algebraic variety there is an abelian category pg 339 with the following functorial properties
There is a faithful functor called the rationalization functor. This gives the underlying rational perverse sheaf of a mixed Hodge module.
There is a faithful functor sending a mixed Hodge module to its underlying D-module
These functors behave well with respect to the Riemann-Hilbert correspondence , meaning for every mixed Hodge module there is an isomorphism .
In addition, there are the following categorical properties
The category of mixed Hodge modules over a point is isomorphic to the category of Mixed hodge structures,
Every object in admits a weight filtration such that every morphism in preserves the weight filtration strictly, the associated graded objects are semi-simple, and in the category of mixed Hodge modules over a point, this corresponds to the weight filtration of a Mixed hodge structure.
There is a dualizing functor lifting the Verdier dualizing functor in which is an involution on .
For a morphism of algebraic varieties, the associated six functors on and have the following properties
don't increase the weights of a complex of mixed Hodge modules.
don't decrease the weights of a complex of mixed Hodge modules.
Relation between derived categories
The derived category of mixed Hodge modules is intimately related to the derived category of constructuctible sheaves equivalent to the derived category of perverse sheaves. This is because of how the rationalization functor is compatible with the cohomology functor of a complex of mixed Hodge modules. When taking the rationalization, there is an isomorphismfor the middle perversity . Notepg 310 this is the function sending , which differs from the case of pseudomanifolds where the perversity is a function where . Recall this is defined as taking th
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https://en.wikipedia.org/wiki/Shimura%20variety
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In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.
Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. In 1979, Robert Langlands remarked that Shimura varieties form a natural realm of examples for which equivalence between motivic and automorphic L-functions postulated in the Langlands program can be tested. Automorphic forms realized in the cohomology of a Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attaching Galois representations to them.
Definition
Shimura datum
Let S = ResC/R Gm be the Weil restriction of the multiplicative group from complex numbers to real numbers. It is a real algebraic group, whose group of R-points, S(R), is C* and group of C-points is C*×C*. A Shimura datum is a pair (G, X) consisting of a (connected) reductive algebraic group G defined over the field Q of rational numbers and a G(R)-conjugacy class X of homomorphisms h: S → GR satisfying the following axioms:
For any h in X, only weights (0,0), (1,−1), (−1,1) may occur in gC, i.e. the complexified Lie algebra of G decomposes into a direct sum
where for any z ∈ S, h(z) acts trivially on the first summand and via (respectively, ) on the second (respectively, third) summand.
The adjoint action of h(i) induces a Cartan involution on the adjoint group of GR.
The adjoint group of GR does not admit a factor H defined over Q such that the projection of h on H is trivial.
It follows from these axioms that X has a unique structure of a complex manifold (possibly, disconnected) such that for every representation ρ: GR → GL(V), the family (V, ρ ⋅ h) is a holomorphic family of Hodge structures; moreover, it forms a variation of Hodge structure, and X is a finite disjoint union of hermitian symmetric domains.
Shimura variety
Let Aƒ be the ring of finite adeles of Q. For every sufficiently small compact open subgroup K of G(Aƒ), the double coset space
is a finite disjoint union of locally symmetric varieties of the form , where the plus superscript indicates a connected component. The varieties ShK(G,X) are complex algebraic varieties and they form an inverse system over all sufficiently small co
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https://en.wikipedia.org/wiki/Spherical%20polyhedron
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In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron.
History
The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period (the New Stone Age).
During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) wrote the first serious study of spherical polyhedra.
Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.
In the middle of the 20th Century, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).
Examples
All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
Improper cases
Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as {2,n}, and dihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
Relation to tilings of the projective plane
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra (tessellations of the real projective plane) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra:
Hemi-cube, {4,3}/2
Hemi-octahedron, {3,4}/2
Hemi-dodecahedron, {5,3}/2
Hemi-icosahedron, {3,5}/2
Hemi-dihedron, {2p,2}/2, p>=1
Hemi-hosohedron, {2,2p}/2, p>=1
See also
Spherical geometry
Spherical trigonometry
Polyhedron
Projective polyhedron
Toroidal polyhedron
Conway polyhedron notation
References
Further reading
Polyhedra
Tessellation
Spheres
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https://en.wikipedia.org/wiki/Extravagant%20number
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In number theory, an extravagant number (also known as a wasteful number) is a natural number in a given number base that has fewer digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 4 = 22, 6 = 2×3, 8 = 23, and 9 = 32 are extravagant numbers .
There are infinitely many extravagant numbers in every base.
Mathematical definition
Let be a number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation
where is the p-adic valuation of , and is an extravagant number in base if
See also
Equidigital number
Frugal number
Notes
References
R.G.E. Pinch (1998), Economical Numbers.
Chris Caldwell, The Prime Glossary: extravagant number at The Prime Pages.
Base-dependent integer sequences
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https://en.wikipedia.org/wiki/Equidigital%20number
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In number theory, an equidigital number is a natural number in a given number base that has the same number of digits as the number of digits in its prime factorization in the given number base, including exponents but excluding exponents equal to 1. For example, in base 10, 1, 2, 3, 5, 7, and 10 (2 × 5) are equidigital numbers . All prime numbers are equidigital numbers in any base.
A number that is either equidigital or frugal is said to be economical.
Mathematical definition
Let be the number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation
where is the p-adic valuation of , and is an equidigital number in base if
Properties
Every prime number is equidigital. This also proves that there are infinitely many equidigital numbers.
See also
Extravagant number
Frugal number
Smith number
Notes
References
R.G.E. Pinch (1998), Economical Numbers.
Integer sequences
Base-dependent integer sequences
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https://en.wikipedia.org/wiki/Frugal%20number
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In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 = 35, and 256 = 28 are frugal numbers . The first frugal number which is not a prime power is 1029 = 3 × 73. In base 2, thirty-two is a frugal number, since 32 = 25 is written in base 2 as 100000 = 10101.
The term economical number has been used for a frugal number, but also for a number which is either frugal or equidigital.
Mathematical definition
Let be a number base, and let be the number of digits in a natural number for base . A natural number has the prime factorisation
where is the p-adic valuation of , and is an frugal number in base if
See also
Equidigital number
Extravagant number
Notes
References
R.G.E. Pinch (1998), Economical Numbers
Base-dependent integer sequences
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https://en.wikipedia.org/wiki/Lagrange%20multipliers%20on%20Banach%20spaces
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In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems. The method is a generalization of the classical method of Lagrange multipliers as used to find extrema of a function of finitely many variables.
The Lagrange multiplier theorem for Banach spaces
Let X and Y be real Banach spaces. Let U be an open subset of X and let f : U → R be a continuously differentiable function. Let g : U → Y be another continuously differentiable function, the constraint: the objective is to find the extremal points (maxima or minima) of f subject to the constraint that g is zero.
Suppose that u0 is a constrained extremum of f, i.e. an extremum of f on
Suppose also that the Fréchet derivative Dg(u0) : X → Y of g at u0 is a surjective linear map. Then there exists a Lagrange multiplier λ : Y → R in Y∗, the dual space to Y, such that
Since Df(u0) is an element of the dual space X∗, equation (L) can also be written as
where (Dg(u0))∗(λ) is the pullback of λ by Dg(u0), i.e. the action of the adjoint map (Dg(u0))∗ on λ, as defined by
Connection to the finite-dimensional case
In the case that X and Y are both finite-dimensional (i.e. linearly isomorphic to Rm and Rn for some natural numbers m and n) then writing out equation (L) in matrix form shows that λ is the usual Lagrange multiplier vector; in the case n = 1, λ is the usual Lagrange multiplier, a real number.
Application
In many optimization problems, one seeks to minimize a functional defined on an infinite-dimensional space such as a Banach space.
Consider, for example, the Sobolev space and the functional given by
Without any constraint, the minimum value of f would be 0, attained by u0(x) = 0 for all x between −1 and +1. One could also consider the constrained optimization problem, to minimize f among all those u ∈ X such that the mean value of u is +1. In terms of the above theorem, the constraint g would be given by
However this problem can be solved as in the finite dimensional case since the Lagrange multiplier is only a scalar.
See also
Pontryagin's minimum principle, Hamiltonian method in calculus of variations
References
(See Section 4.14, pp.270–271.)
Calculus of variations
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https://en.wikipedia.org/wiki/Boldface%20%28disambiguation%29
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Boldface may refer to:
A variety of emphasis (typography)
Boldface pointclass, a concept in descriptive set theory in mathematics
See also
Bold (disambiguation)
Bald face (disambiguation)
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https://en.wikipedia.org/wiki/Superegg
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In geometry, a superegg is a solid of revolution obtained by rotating an elongated superellipse with exponent greater than 2 around its longest axis. It is a special case of superellipsoid.
Unlike an elongated ellipsoid, an elongated superegg can stand upright on a flat surface, or on top of another superegg. This is due to its curvature being zero at the tips. The shape was popularized by Danish poet and scientist Piet Hein (1905–1996). Supereggs of various materials, including brass, were sold as novelties or "executive toys" in the 1960s.
Mathematical description
The superegg is a superellipsoid whose horizontal cross-sections are circles. It is defined by the inequality
where R is the horizontal radius at the "equator" (the widest part), and h is one half of the height. The exponent p determines the degree of flattening at the tips and equator. Hein's choice was p = 2.5 (the same one he used for the Sergels Torg roundabout), and R/h = 3/4.
The definition can be changed to have an equality rather than an inequality; this changes the superegg to being a surface of revolution rather than a solid.
Volume
The volume of a superegg can be derived via squigonometry, a generalization of trigonometry to squircles. It is related to the gamma function:
See also
Egg of Columbus
References
Algebraic curves
Surfaces
Office toys
Educational toys
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https://en.wikipedia.org/wiki/Standard%20probability%20space
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In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.
The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. Rokhlin showed that the unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, yet can be effectively substituted for many of these in probability theory. The dimension of the unit interval is not an obstacle, as was clear already to Norbert Wiener. He constructed the Wiener process (also called Brownian motion) in the form of a measurable map from the unit interval to the space of continuous functions.
Short history
The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940. For modernized presentations see , , and .
Nowadays standard probability spaces may be (and often are) treated in the framework of descriptive set theory, via standard Borel spaces, see for example . This approach is based on the isomorphism theorem for standard Borel spaces . An alternate approach of Rokhlin, based on measure theory, neglects null sets, in contrast to descriptive set theory.
Standard probability spaces are used routinely in ergodic theory.
Definition
One of several well-known equivalent definitions of the standardness is given below, after some preparations. All probability spaces are assumed to be complete.
Isomorphism
An isomorphism between two probability spaces , is an invertible map such that and both are (measurable and) measure preserving maps.
Two probability spaces are isomorphic if there exists an isomorphism between them.
Isomorphism modulo zero
Two probability spaces , are isomorphic if there exist null sets , such that the probability spaces , are isomorphic (being endowed naturally with sigma-fields and probability measures).
Standard probability space
A probability space is standard, if it is isomorphic to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination (disjoint union) of both.
See , , and . See also , and . In the measure is assumed finite, not necessarily probabilistic. In atoms are not allowed.
Examples of non-standard probability spaces
A naive white noise
The space of all functions may be thought of as the product of a continuum of copies of the real line . One may endow with a probability measure, say, the standard normal distribution , and treat the space of functions as the product of a continuum of identical probability spaces . The product measure is a probability measure on . Naively it might seem that describes white noise.
However, the integral of a white noise function from 0 to 1
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https://en.wikipedia.org/wiki/Paradoxes%20of%20set%20theory
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This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.
Basics
Cardinal numbers
Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. These sets have in common the cardinal number |N| = (aleph-nought), a number greater than every natural number.
Cardinal numbers can be defined as follows. Define two sets to have the same size by: there exists a bijection between the two sets (a one-to-one correspondence between the elements). Then a cardinal number is, by definition, a class consisting of all sets of the same size. To have the same size is an equivalence relation, and the cardinal numbers are the equivalence classes.
Ordinal numbers
Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order. The order of a well-ordered set is described by an ordinal number. For instance, 3 is the ordinal number of the set {0, 1, 2} with the usual order 0 < 1 < 2; and ω is the ordinal number of the set of all natural numbers ordered the usual way. Neglecting the order, we are left with the cardinal number |N| = |ω| = .
Ordinal numbers can be defined with the same method used for cardinal numbers. Define two well-ordered sets to have the same order type by: there exists a bijection between the two sets respecting the order: smaller elements are mapped to smaller elements. Then an ordinal number is, by definition, a class consisting of all well-ordered sets of the same order type. To have the same order type is an equivalence relation on the class of well-ordered sets, and the ordinal numbers are the equivalence classes.
Two sets of the same order type have the same cardinality. The converse is not true in general for infinite sets: it is possible to impose different well-orderings on the set of natural numbers that give rise to different ordinal numbers.
There is a natural ordering on the ordinals, which is itself a well-ordering. Given any ordinal α, one can consider the set of all ordinals less than α. This set turns out to have ordinal number α. This observation is used for a different way of introducing the ordi
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https://en.wikipedia.org/wiki/Tasmanian%20year%20book
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Tasmanian year book was the annual review of statistics collected for Tasmania.
It was a companion volume to Walch's Tasmanian Almanac bound in the same colour red cloth - and produced between 1967 and 2000.
It was issued by the Commonwealth Bureau of Census and Statistics Tasmanian Office, later known as the Australian Bureau of Statistics office.
It had regular special articles in each edition which were considered definitive in their writing and approach.
Special articles
No.11 (1977)
Townsley, W.A. The Tasmanian main line railway company (originally presented in 1956 to the Tasmanian Historical Research Association) pp. 6 – 22. along with photographs
Publishing details
No. 1 (1967)-no. 27 (2000) - Hobart, Tas. : Commonwealth Bureau of Census and Statistics, Tasmanian Office, 1967-2000. ISSN 0082-2116 (1987 not published)
External links
www.abs.gov.au
History of Tasmania
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https://en.wikipedia.org/wiki/Spherical%20mean
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In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.
Definition
Consider an open set U in the Euclidean space Rn and a continuous function u defined on U with real or complex values. Let x be a point in U and r > 0 be such that the closed ball B(x, r) of center x and radius r is contained in U. The spherical mean over the sphere of radius r centered at x is defined as
where ∂B(x, r) is the (n − 1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and ωn−1(r) is the "surface area" of this (n − 1)-sphere.
Equivalently, the spherical mean is given by
where ωn−1 is the area of the (n − 1)-sphere of radius 1.
The spherical mean is often denoted as
The spherical mean is also defined for Riemannian manifolds in a natural manner.
Properties and uses
From the continuity of it follows that the function is continuous, and that its limit as is
Spherical means can be used to solve the Cauchy problem for the wave equation in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in (for odd ) to the wave equation in , and then using d'Alembert's formula. The expression itself is presented in wave equation article.
If is an open set in and is a C2 function defined on , then is harmonic if and only if for all in and all such that the closed ball is contained in one has This result can be used to prove the maximum principle for harmonic functions.
References
External links
Partial differential equations
Means
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https://en.wikipedia.org/wiki/Phase%20space%20method
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In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations.
The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is transformed back into the original setting.
The phase space method is used widely in physics. It can be applied, for example, to find traveling wave solutions of reaction–diffusion systems.
See also
Reaction–diffusion system
Fisher's equation
References
Partial differential equations
Dynamical systems
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https://en.wikipedia.org/wiki/Semifield
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In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed.
Overview
The term semifield has two conflicting meanings, both of which include fields as a special case.
In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication. In other words, a semifield is a set S with two operations + (addition) and · (multiplication), such that
(S,+) is an abelian group,
multiplication is distributive on both the left and right,
there exists a multiplicative identity element, and
division is always possible: for every a and every nonzero b in S, there exist unique x and y in S for which b·x = a and y·b = a.
Note in particular that the multiplication is not assumed to be commutative or associative. A semifield that is associative is a division ring, and one that is both associative and commutative is a field. A semifield by this definition is a special case of a quasifield. If S is finite, the last axiom in the definition above can be replaced with the assumption that there are no zero divisors, so that a·b = 0 implies that a = 0 or b = 0. Note that due to the lack of associativity, the last axiom is not equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings.
In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse. These objects are also called proper semifields. A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group. However, the pair (S,+) is only a semigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative.
Primitivity of semifields
A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right (resp. left) principal powers of w.
Examples
We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples.
Positive rational numbers with the usual addition and multiplication form a commutative semifield.
This can be extended by an absorbing 0.
Positive real numbers with the usual addition and multiplication form a commutative semifield.
This can be extended by an absorbing
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https://en.wikipedia.org/wiki/Classification%20of%20Fatou%20components
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In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou.
Rational case
If f is a rational function
defined in the extended complex plane, and if it is a nonlinear function (degree > 1)
then for a periodic component of the Fatou set, exactly one of the following holds:
contains an attracting periodic point
is parabolic
is a Siegel disc: a simply connected Fatou component on which f(z) is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle.
is a Herman ring: a double connected Fatou component (an annulus) on which f(z) is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle.
Attracting periodic point
The components of the map contain the attracting points that are the solutions to . This is because the map is the one to use for finding solutions to the equation by Newton–Raphson formula. The solutions must naturally be attracting fixed points.
Herman ring
The map
and t = 0.6151732... will produce a Herman ring. It is shown by Shishikura that the degree of such map must be at least 3, as in this example.
More than one type of component
If degree d is greater than 2 then there is more than one critical point and then can be more than one type of component
Transcendental case
Baker domain
In case of transcendental functions there is another type of periodic Fatou components, called Baker domain: these are "domains on which the iterates tend to an essential singularity (not possible for polynomials and rational functions)" one example of such a function is:
Wandering domain
Transcendental maps may have wandering domains: these are Fatou components that are not eventually periodic.
See also
No-wandering-domain theorem
Montel's theorem
John Domains
Basins of attraction
References
Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993.
Alan F. Beardon Iteration of Rational Functions, Springer 1991.
Fractals
Limit sets
Theorems in complex analysis
Complex dynamics
Theorems in dynamical systems
Mathematical classification systems
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https://en.wikipedia.org/wiki/SOCR%20%28disambiguation%29
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SOCR is an acronym that can refer to:
Statistics Online Computational Resource
Seattle Office for Civil Rights
State Operated Community Residence
Stand-alone optical character reader
Special Operational Capability Report
Special Operations Craft – Riverine (SOC-R)
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https://en.wikipedia.org/wiki/Geoff%20Smith%20%28mathematician%29
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Geoffrey Charles Smith, MBE (born 1953) is a British mathematician. He is Senior Lecturer in Mathematics at the University of Bath (where he works in group theory) and current professor in residence at Wells Cathedral School.
He was educated at Trinity School in Croydon, and attended Keble College, Oxford, the University of Warwick, and the University of Manchester, where he gained a Ph.D. in group theory in 1983.
Smith was the leader of the United Kingdom team at the International Mathematical Olympiad between 2002 and 2010, a longer continuous period than any other person. He returned to the position as leader of the British Mathematical Olympiad from 2013.
Smith oversaw a quantitative increase in training: annual events in Bath (moving to The Queen's College, Oxford, from 2009), at Oundle School, in Hungary, at Trinity College, Cambridge, and immediately prior to the IMO itself. He also thrice won the IMO Golden Microphone, awarded to the national team leader who makes the most speeches to the IMO Jury. In 2010, he was elected to the IMO Advisory Board for a four-year period. Smith was elected as the chair of the International Mathematical Olympiad for the term of 2014-2018 and was re-elected in 2018.
Smith also prepared UK teams for the Romanian Masters in Mathematics tournament (which they won in 2008), and for participation as guests at the annual Balkan Mathematical Olympiad.
As well as group theory, he is also interested in Euclidean geometry. He often collaborates with Christopher Bradley and David Monk, and has published several papers on Forum Geometricorum, the online geometry journal.
In June 2011, Smith was awarded an MBE for services to education following his contributions toward organising Royal Institution Maths Masterclasses.
References
External links
Virtual Geoff Smith
Geoff Smith on Midweek, 28 January 2004
20th-century English mathematicians
21st-century English mathematicians
Group theorists
Academics of the University of Bath
Alumni of the University of Manchester
1953 births
Living people
Alumni of Keble College, Oxford
Members of the Order of the British Empire
Teachers of Oundle School
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https://en.wikipedia.org/wiki/Herbert%20Solomon
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Herbert Solomon (March 13, 1919 – September 20, 2004) was an American statistician. He was a professor emeritus of statistics at Stanford University and co-founder of the university's statistics department. Born in Harlem to Jewish-Russian immigrant parents, he attended DeWitt Clinton High School and later earned a bachelor's degree from the City College of New York in 1940 and a master's degree from Columbia University in 1941. His studies were interrupted by World War II, during which he was a member of the Statistical Research Group at Columbia. After the war, he would continue his doctoral studies at Stanford, and earned his doctorate in 1950. After serving in the Office of Naval Research from 1948 to 1952, he returned to Columbia as a professor, and taught there from 1952 to 1959. While on sabbatical, he returned to Stanford, where he would teach for the remainder of his life.
In 1954 he was named a Fellow of the American Statistical Association.
References
External links
1919 births
2004 deaths
American statisticians
City College of New York alumni
Columbia University alumni
Stanford University alumni
Stanford University faculty
Columbia University faculty
Fellows of the American Statistical Association
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https://en.wikipedia.org/wiki/Hemicube%20%28geometry%29
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In abstract geometry, a hemicube is an abstract, regular polyhedron, containing half the faces of a cube.
Realization
It can be realized as a projective polyhedron (a tessellation of the real projective plane by three quadrilaterals), which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing the hemisphere into three equal parts.
It has three square faces, six edges, and four vertices. It has an unexpected property that every face is in contact with every other face on two edges, and every face contains all the vertices, which gives an example of an abstract polytope whose faces are not determined by their vertex sets.
From the point of view of graph theory the skeleton is a tetrahedral graph, an embedding of K4 (the complete graph with four vertices) on a projective plane.
The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the hemicube is a quotient of the cube, while the vertices of the demicube are a subset of the vertices of the cube.
Related polytopes
The hemicube is the Petrie dual to the regular tetrahedron, with the four vertices, six edges of the tetrahedron, and three Petrie polygon quadrilateral faces. The faces can be seen as red, green, and blue edge colorings in the tetrahedral graph:
See also
hemi-octahedron
hemi-dodecahedron
hemi-icosahedron
Footnotes
References
External links
The hemicube
Projective polyhedra
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https://en.wikipedia.org/wiki/Petros%20Protopapadakis
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Petros Protopapadakis (; 1854–1922) was a politician and Prime Minister of Greece from May to September 1922.
Life and work
Born in 1860 in Apeiranthos, Naxos, Protopapadakis studied mathematics and engineering in Paris but was keenly interested in politics. He was a professor at the Scholi Evelpidon, the military academy of Greece.
Protopadakis was elected to the Hellenic Parliament in 1902 as a member of the conservative Nationalist Party. He later joined the People's Party and served as Minister of Economy and later, in the government of Dimitrios Gounaris, he was the Justice Minister (1921–22). In 1922, during the ill-fated Greco-Turkish War, Protopapadakis was asked to form a government by King Constantine when Gounaris resigned after almost losing a vote of confidence. Protopapadakis became Prime Minister and Gounaris the Justice Minister. Protopapadakis remained in his position for a little more than 3 months, as he was overthrown by a military coup d'état.
Death
Protopapadakis was executed in the Trial of the Six proceedings at Goudi on November 1922, along with the other five most senior members of his government.
See also
History of Modern Greece
References
19th-century births
1922 deaths
20th-century prime ministers of Greece
People from Naxos
Prime Ministers of Greece
Greek people of the Greco-Turkish War (1919–1922)
People's Party (Greece) politicians
People executed for treason against Greece
People executed by Greece by firing squad
Executed prime ministers
Finance ministers of Greece
Leaders ousted by a coup
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https://en.wikipedia.org/wiki/Computational%20mathematics
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Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.
A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra.
Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants.
Areas of computational mathematics
Computational mathematics emerged as a distinct part of applied mathematics by the early 1950s. Currently, computational mathematics can refer to or include:
Computational science, also known as scientific computation or computational engineering
Solving mathematical problems by computer simulation as opposed to analytic methods of applied mathematics
Numerical methods used in scientific computation, for example numerical linear algebra and numerical solution of partial differential equations
Stochastic methods, such as Monte Carlo methods and other representations of uncertainty in scientific computation
The mathematics of scientific computation, in particular numerical analysis, the theory of numerical methods
Computational complexity
Computer algebra and computer algebra systems
Computer-assisted research in various areas of mathematics, such as logic (automated theorem proving), discrete mathematics, combinatorics, number theory, and computational algebraic topology
Cryptography and computer security, which involve, in particular, research on primality testing, factorization, elliptic curves, and mathematics of blockchain
Computational linguistics, the use of mathematical and computer techniques in natural languages
Computational algebraic geometry
Computational group theory
Computational geometry
Computational number theory
Computational topology
Computational statistics
Algorithmic information theory
Algorithmic game theory
Mathematical economics, the use of mathematics in economics, finance and, to certain extents, of accounting.
Experimental mathematics
See also
References
Further reading
External links
Foundations of Computational Mathematics, a non-profit organization
International Journal of Computer Discovered Mathematics
Applied mathematics
Computational science
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https://en.wikipedia.org/wiki/Mikhail%20Khovanov
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Mikhail Khovanov (; born 1972) is a Russian-American professor of mathematics at Columbia University who works on representation theory, knot theory, and algebraic topology. He is known for introducing Khovanov homology for links, which was one of the first examples of categorification.
Education and career
Khovanov graduated from Moscow State School 57 mathematical class in 1988. He earned a PhD in mathematics from Yale University in 1997, where he studied under Igor Frenkel.
Khovanov was a faculty member at UC Davis before moving to Columbia University.
He is a half-brother of Tanya Khovanova.
References
External links
Khovanov's faculty page at Columbia.
List of Khovanov's publications.
1972 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Topologists
Columbia University faculty
Yale University alumni
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https://en.wikipedia.org/wiki/Roy%20Batchelor
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Roy A. Batchelor (born 23 March 1947) is Professor Emeritus in Political Economy and Statistics in Bayes Business School (formerly Cass), City, University of London.
Educated at Allan Glen's School and Glasgow University, Roy worked as a government scientist and economist; then at the UK National Institute of Economic and Social Research. He joined City University in 1977, and has since been active there in research, teaching and academic administration, this including spells as Head of Banking and Finance Department, Director of the Bayes (formerly Cass) Executive MBA programme in Dubai, and of the Executive MBA in London.
Professor Batchelor’s research has focussed on economic and financial market forecasting, and the interpretation and use of consumer and business survey data. He has published widely in these fields, often in the International Journal of Forecasting, and its sister practitioner journal, Foresight. In 2008 Professor Batchelor was elected Honorary Fellow of the International Institute of Forecasters, and he has since served as an elected Director of the IIF.
In parallel with his academic work, Professor Batchelor has been active in professional training and consultancy with business and governmental organisations around the world. He has held many visiting academic appointments, and is a Fellow and Research Professor at the ifo Institute for Economic Research in Munich.
Batchelor supervised the PhD thesis of Richard Ramyar, a former director of the United Kingdom Society of Technical Analysts. This claimed to debunk Fibonacci ratio technical analysis in the US equity market. This work was described in multiple business news outlets, and he has also received press coverage for his other work on finance.
References
External links
City University biography
Linkedin
Early Chess and Chess Problems
1947 births
Living people
English economists
Academics of City, University of London
Academics of Bayes Business School
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https://en.wikipedia.org/wiki/Jyri%20Marttinen
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Jyri Marttinen (born September 1, 1982) is a Finnish ice hockey defenceman.
Career statistics
Regular season and playoffs
International
References
External links
1982 births
Drakkars de Caen players
Finnish ice hockey defencemen
GKS Katowice (ice hockey) players
JYP Jyväskylä players
Living people
Lukko players
Malmö Redhawks players
Lahti Pelicans players
Skellefteå AIK players
Timrå IK players
Porin Ässät (men's ice hockey) players
Calgary Flames draft picks
HC 07 Detva players
Finnish expatriate ice hockey players in Slovakia
Finnish expatriate ice hockey players in Sweden
Finnish expatriate ice hockey players in Poland
Finnish expatriate ice hockey players in France
Finnish expatriate ice hockey players in Romania
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https://en.wikipedia.org/wiki/Nystr%C3%B6m%20method
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In mathematics numerical analysis, the Nyström method or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into discrete intervals; quadrature or numerical integration determines the weights and locations of representative points for the integral.
The problem becomes a system of linear equations with equations and unknowns, and the underlying function is implicitly represented by an interpolation using the chosen quadrature rule. This discrete problem may be ill-conditioned, depending on the original problem and the chosen quadrature rule.
Since the linear equations require operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.
Discretization of the integral
Standard quadrature methods seek to represent an integral as a weighed sum in the following manner:
where are the weights of the quadrature rule, and points are the abscissas.
Example
Applying this to the inhomogeneous Fredholm equation of the second kind
,
results in
.
See also
Boundary element method
References
Bibliography
Leonard M. Delves & Joan E. Walsh (eds): Numerical Solution of Integral Equations, Clarendon, Oxford, 1974.
Hans-Jürgen Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations, Springer, New York, 1985.
Integral equations
Numerical analysis
Numerical integration (quadrature)
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https://en.wikipedia.org/wiki/Nahm%20equations
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In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.
Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. They can also be viewed as a dimensional reduction of the anti-self-dual Yang-Mills equations . Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by , , and .
Equations
Let be three matrix-valued meromorphic functions of a complex variable . The Nahm equations are a system of matrix differential equations
together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form
More generally, instead of considering by matrices, one can consider Nahm's equations with values in a Lie algebra .
Additional conditions
The variable is restricted to the open interval , and the following conditions are imposed:
can be continued to a meromorphic function of in a neighborhood of the closed interval , analytic outside of and , and with simple poles at and ; and
At the poles, the residues of form an irreducible representation of the group SU(2).
Nahm–Hitchin description of monopoles
There is a natural equivalence between
the monopoles of charge for the group , modulo gauge transformations, and
the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of by the group .
Lax representation
The Nahm equations can be written in the Lax form as follows. Set
then the system of Nahm equations is equivalent to the Lax equation
As an immediate corollary, we obtain that the spectrum of the matrix does not depend on . Therefore, the characteristic equation
which determines the so-called spectral curve in the twistor space is invariant under the flow in .
See also
Bogomolny equation
Yang–Mills–Higgs equations
References
External links
Islands project – a wiki about the Nahm equations and related topics
Differential equations
Eponymous equations of physics
Mathematical physics
Integrable systems
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https://en.wikipedia.org/wiki/Inverse%20problem%20for%20Lagrangian%20mechanics
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In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can arise as the Euler–Lagrange equations for some Lagrangian function.
There has been a great deal of activity in the study of this problem since the early 20th century. A notable advance in this field was a 1941 paper by the American mathematician Jesse Douglas, in which he provided necessary and sufficient conditions for the problem to have a solution; these conditions are now known as the Helmholtz conditions, after the German physicist Hermann von Helmholtz.
Background and statement of the problem
The usual set-up of Lagrangian mechanics on n-dimensional Euclidean space Rn is as follows. Consider a differentiable path u : [0, T] → Rn. The action of the path u, denoted S(u), is given by
where L is a function of time, position and velocity known as the Lagrangian. The principle of least action states that, given an initial state x0 and a final state x1 in Rn, the trajectory that the system determined by L will actually follow must be a minimizer of the action functional S satisfying the boundary conditions u(0) = x0, u(T) = x1. Furthermore, the critical points (and hence minimizers) of S must satisfy the Euler–Lagrange equations for S:
where the upper indices i denote the components of u = (u1, ..., un).
In the classical case
the Euler–Lagrange equations are the second-order ordinary differential equations better known as Newton's laws of motion:
The inverse problem of Lagrangian mechanics is as follows: given a system of second-order ordinary differential equations
that holds for times 0 ≤ t ≤ T, does there exist a Lagrangian L : [0, T] × Rn × Rn → R for which these ordinary differential equations (E) are the Euler–Lagrange equations? In general, this problem is posed not on Euclidean space Rn, but on an n-dimensional manifold M, and the Lagrangian is a function L : [0, T] × TM → R, where TM denotes the tangent bundle of M.
Douglas' theorem and the Helmholtz conditions
To simplify the notation, let
and define a collection of n2 functions Φji by
Theorem. (Douglas 1941) There exists a Lagrangian L : [0, T] × TM → R such that the equations (E) are its Euler–Lagrange equations if and only if there exists a non-singular symmetric matrix g with entries gij depending on both u and v satisfying the following three Helmholtz conditions:
(The Einstein summation convention is in use for the repeated indices.)
Applying Douglas' theorem
At first glance, solving the Helmholtz equations (H1)–(H3) seems to be an extremely difficult task. Condition (H1) is the easiest to solve: it is always possible to find a g that satisfies (H1), and it alone will not imply that the Lagrangian is singular. Equation (H2) is a system of ordinary differential equations: the usual theorems on the existence and uniqueness of solutions to ordinary differential equations imply that it is, in principle, possible to solv
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https://en.wikipedia.org/wiki/Visual%20calculus
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Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner called "aha! solutions" or Roger Nelsen a proof without words.
Description
Mamikon devised his method in 1959 while an undergraduate, first applying it to a well-known geometry problem: find the area of a ring (annulus), given the length of a chord tangent to the inner circumference. Perhaps surprisingly, no additional information is needed; the solution does not depend on the ring's inner and outer dimensions.
The traditional approach involves algebra and application of the Pythagorean theorem. Mamikon's method, however, envisions an alternate construction of the ring: first the inner circle alone is drawn, then a constant-length tangent is made to travel along its circumference, "sweeping out" the ring as it goes.
Now if all the (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—and sweeping out a ring around a circle of zero radius is indistinguishable from simply rotating a line segment about one of its endpoints and sweeping out a disk.
Mamikon's insight was to recognize the equivalence of the two constructions; and because they are equivalent, they yield equal areas. Moreover, so long as it is given that the tangent length is constant, the two starting curves need not be circular—a finding not easily proven by more traditional geometric methods. This yields Mamikon's theorem:
The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.
Applications
Area of a cycloid
The area of a cycloid can be calculated by considering the area between it and an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius then this circle also has radius and area . The area of the rectangle is . Therefore the area of the cycloid is : it is 3 times the area of the generating circle.
The tangent cluster can be seen to be a circle because the cycloid is generated by a circle and the tangent to the cycloid will be at right angle to the line from the generating point to the rolling point. Thus the tangent and the line to the contact point form a right-angled triangle in the generating circle. This means that clustered together the tangents will describe the shape of the generating circle.
See also
Cavalieri's principle
Hodograph – This is a related construct that maps the velocity of a point using a polar diagram.
The Method of Mechanical Theorems
Pappus's centroid theorem
Plan
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https://en.wikipedia.org/wiki/Paul%20Malliavin
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Paul Malliavin (; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis.
He is known for the Malliavin calculus, an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem.
He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences from 1979 to 2010.
Personal life
Malliavin was the son of René Malliavin, also known as Michel Dacier, a political writer and journalist, and Madeleine Delavenne, a physician. On 27 April 1965 he married Marie-Paule Brameret, who was also a mathematician and with whom he published several mathematical papers. They had two children.
Scientific contributions
Malliavin's early work was in harmonic analysis, where he derived important results on the spectral synthesis problem, providing definitive answers to fundamental questions in this field, including a complete characterization of 'band-limited' functions whose Fourier transform has compact support, known as the Beurling-Malliavin theorem.
In stochastic analysis, Malliavin is known for his work on the stochastic calculus of variation, now known as the Malliavin calculus, a mathematical theory which has found many applications in Monte Carlo simulation and mathematical finance.
As stated by Stroock and Yor: "Like Norbert Wiener, Paul Malliavin came to probability theory from harmonic analysis, and, like Wiener, his analytic origins were apparent in everything he did there."
Malliavin introduced a differential operator on Wiener space, now called the Malliavin derivative, and derived an integration by parts formula for Wiener functionals. Using this integration by parts formula, Malliavin initiated a probabilistic approach to Hörmander's theorem for hypo-elliptic operators and gave a condition for the existence of smooth densities for Wiener functionals in terms of their Malliavin covariance matrix.
Selected publications
La quasi-analyticité généralisée sur un intervalle borné, Annales scientifiques de l’École Normale Supérieure 3e série 72, 1955, pp. 93–110
Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts, Publications Mathématiques de l’IHÉS 2, 1959, pp. 61–68
Calcul symbolique et sous-algèbres de L1(G), Bulletin de la Société Mathématique de France 87, 1959, pp. 181–186, suite, pp. 187–190
with Lee A. Rubel: On small entire functions of exponential type with given zeros, Bulletin de la Société Mathématique de France 89, 1961, pp. 175–206
Spectre des fonctions moyenne-périodiques. Totalité d’une suite d’exponentielles sur un segment, Séminaire Lelong. Analyse 3 Exposé No. 11, 1961
Un théorème taubérien relié aux estimations de valeurs propres, Séminaire Jean Leray, 1962–1963, pp. 224–231
Géométrie riemannienne stochastique, Séminaire Jean Leray 2 Exposé No. 1, 1973–1974
Geometrie differentielle stochastique, Presses de l’Universit
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https://en.wikipedia.org/wiki/Cerf%20theory
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In mathematics, at the junction of singularity theory and differential topology, Cerf theory is the study of families of smooth real-valued functions
on a smooth manifold , their generic singularities and the topology of the subspaces these singularities define, as subspaces of the function space. The theory is named after Jean Cerf, who initiated it in the late 1960s.
An example
Marston Morse proved that, provided is compact, any smooth function can be approximated by a Morse function. Thus, for many purposes, one can replace arbitrary functions on by Morse functions.
As a next step, one could ask, 'if you have a one-parameter family of functions which start and end at Morse functions, can you assume the whole family is Morse?' In general, the answer is no. Consider, for example, the one-parameter family of functions on given by
At time , it has no critical points, but at time , it is a Morse function with two critical points at .
Cerf showed that a one-parameter family of functions between two Morse functions can be approximated by one that is Morse at all but finitely many degenerate times. The degeneracies involve a birth/death transition of critical points, as in the above example when, at , an index 0 and index 1 critical point are created as increases.
A stratification of an infinite-dimensional space
Returning to the general case where is a compact manifold, let denote the space of Morse functions on , and the space of real-valued smooth functions on . Morse proved that is an open and dense subset in the topology.
For the purposes of intuition, here is an analogy. Think of the Morse functions as the top-dimensional open stratum in a stratification of (we make no claim that such a stratification exists, but suppose one does). Notice that in stratified spaces, the co-dimension 0 open stratum is open and dense. For notational purposes, reverse the conventions for indexing the stratifications in a stratified space, and index the open strata not by their dimension, but by their co-dimension. This is convenient since is infinite-dimensional if is not a finite set. By assumption, the open co-dimension 0 stratum of is , i.e.: . In a stratified space , frequently is disconnected. The essential property of the co-dimension 1 stratum is that any path in which starts and ends in can be approximated by a path that intersects transversely in finitely many points, and does not intersect for any .
Thus Cerf theory is the study of the positive co-dimensional strata of , i.e.: for . In the case of
,
only for is the function not Morse, and
has a cubic degenerate critical point corresponding to the birth/death transition.
A single time parameter, statement of theorem
The Morse Theorem asserts that if is a Morse function, then near a critical point it is conjugate to a function of the form
where .
Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum.
Precisely, if is a on
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https://en.wikipedia.org/wiki/Largest%20cities%20in%20Rio%20Grande%20do%20Sul%20by%20population
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Largest cities in the state of Rio Grande do Sul, Brazil by population, in descending order:
References
"Cidades@", Brazilian Institute of Geography and Statistics, Accessed on 2007-03-20.
Rio Grande do Sul
Rio Grande do Sul
de:Liste der Gemeinden in Rio Grande do Sul
pt:Anexo:Lista de municípios do Rio Grande do Sul por população
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https://en.wikipedia.org/wiki/Eigenvalue%20perturbation
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In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues . This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system.
This type of analysis was popularized by Lord Rayleigh, in his investigation of harmonic vibrations of a string perturbed by small inhomogeneities.
The derivations in this article are essentially self-contained and can be found in many texts on numerical linear algebra or numerical functional analysis.
This article is focused on the case of the perturbation of a simple eigenvalue (see in
multiplicity of eigenvalues).
Why generalized eigenvalues?
In the entry applications of eigenvalues and eigenvectors we find numerous scientific fields in which eigenvalues are used to obtain solutions. Generalized eigenvalue problems are less widespread but are a key in the study of vibrations.
They are useful when we use the Galerkin method or Rayleigh-Ritz method to find approximate
solutions of partial differential equations modeling vibrations of structures such as strings and plates; the paper of Courant (1943)
is fundamental. The Finite element method is a widespread particular case.
In classical mechanics, we may find generalized eigenvalues when we look for vibrations of multiple degrees of freedom systems close to equilibrium; the kinetic energy provides the mass matrix , the potential strain energy provides the rigidity matrix .
To get details, for example see the first section of this article of Weinstein (1941, in French)
With both methods, we obtain a system of differential equations or Matrix differential equation
with the mass matrix , the damping matrix and the rigidity matrix . If we neglect the damping effect, we use , we can look for a solution of the following form ; we obtain that and are solution of the generalized eigenvalue problem
Setting of perturbation for a generalized eigenvalue problem
Suppose we have solutions to the generalized eigenvalue problem,
where and are matrices. That is, we know the eigenvalues and eigenvectors for . It is also required that the eigenvalues are distinct.
Now suppose we want to change the matrices by a small amount. That is, we want to find the eigenvalues and eigenvectors of
where
with the perturbations and much smaller than and respectively. Then we expect the new eigenvalues and eigenvectors to be similar to the original, plus small perturbations:
Steps
We assume that the matrices are symmetric and positive definite, and assume we have scaled the eigenvectors such that
where is the Kronecker delta.
Now we want to solve the equation
In this article we restrict the study to first order perturbation.
First order expansion of the equation
Substituting in (1), we get
which expands to
Canceling from (0) () leaves
Removing the higher-order terms, this simpl
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https://en.wikipedia.org/wiki/L%C2%B2%20cohomology
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In mathematics, L2 cohomology is a cohomology theory for smooth non-compact manifolds M with Riemannian metric. It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms. The notion of square-integrability makes sense because the metric on M gives rise to a norm on differential forms and a volume form.
L2 cohomology, which grew in part out of L2 d-bar estimates from the 1960s, was studied cohomologically, independently by Steven Zucker (1978) and Jeff Cheeger (1979). It is closely related to intersection cohomology; indeed, the results in the preceding cited works can be expressed in terms of intersection cohomology.
Another such result is the Zucker conjecture, which states that for a Hermitian locally symmetric variety the L2 cohomology is isomorphic to the intersection cohomology (with the middle perversity) of its Baily–Borel compactification (Zucker 1982). This was proved in different ways by Eduard Looijenga (1988) and by Leslie Saper and Mark Stern (1990).
See also
Dirichlet form
Dirichlet principle
Riemannian manifold
References
Mark Goresky, L2 cohomology is intersection cohomology
Frances Kirwan, Jonathan Woolf An Introduction to Intersection Homology Theory,, chapter 6
Cohomology theories
Differential geometry
Differential topology
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https://en.wikipedia.org/wiki/GW2
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GW2 may refer to:
Gears of War 2, a science-fiction third-person shooter
Geometry Wars: Retro Evolved², a multidirectional shooter video game created by Bizarre Creations
Guild Wars 2, a massively multiplayer online role-playing game by ArenaNet
Iraq War of 2003, or Gulf War 2
Plants vs. Zombies: Garden Warfare 2, a 2016 video game by PopCap
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https://en.wikipedia.org/wiki/Elliptic%20boundary%20value%20problem
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In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distribution of heat in a room several hours after the heating is turned on.
Differential equations describe a large class of natural phenomena, from the heat equation describing the evolution of heat in (for instance) a metal plate, to the Navier-Stokes equation describing the movement of fluids, including Einstein's equations describing the physical universe in a relativistic way. Although all these equations are boundary value problems, they are further subdivided into categories. This is necessary because each category must be analyzed using different techniques. The present article deals with the category of boundary value problems known as linear elliptic problems.
Boundary value problems and partial differential equations specify relations between two or more quantities. For instance, in the heat equation, the rate of change of temperature at a point is related to the difference of temperature between that point and the nearby points so that, over time, the heat flows from hotter points to cooler points. Boundary value problems can involve space, time and other quantities such as temperature, velocity, pressure, magnetic field, etc.
Some problems do not involve time. For instance, if one hangs a clothesline between the house and a tree, then in the absence of wind, the clothesline will not move and will adopt a gentle hanging curved shape known as the catenary. This curved shape can be computed as the solution of a differential equation relating position, tension, angle and gravity, but since the shape does not change over time, there is no time variable.
Elliptic boundary value problems are a class of problems which do not involve the time variable, and instead only depend on space variables.
The main example
In two dimensions, let be the coordinates. We will use the notation for the first and second partial derivatives of with respect to , and a similar notation for . We will use the symbols and for the partial differential operators in and . The second partial derivatives will be denoted and . We also define the gradient , the Laplace operator and the divergence . Note from the definitions that .
The main example for boundary value problems is the Laplace operator,
where is a region in the plane and is the boundary of that region. The function is known data and the solution is what must be computed. This example has the same essential properties as all other elliptic boundary value problems.
The solution can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like , if this metal plate has its boundary adjacent to ice (which is kept at zero degrees, thus the Dirichlet boundary condition.) The function represents the intensity of heat generation at
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https://en.wikipedia.org/wiki/Optimal%20facility%20location
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The study of facility location problems (FLP), also known as location analysis, is a branch of operations research and computational geometry concerned with the optimal placement of facilities to minimize transportation costs while considering factors like avoiding placing hazardous materials near housing, and competitors' facilities. The techniques also apply to cluster analysis.
Minimum facility location
A simple facility location problem is the Weber problem, in which a single facility is to be placed, with the only optimization criterion being the minimization of the weighted sum of distances from a given set of point sites. More complex problems considered in this discipline include the placement of multiple facilities, constraints on the locations of facilities, and more complex optimization criteria.
In a basic formulation, the facility location problem consists of a set of potential facility sites L where a facility can be opened, and a set of demand points D that must be serviced. The goal is to pick a subset F of facilities to open, to minimize the sum of distances from each demand point to its nearest facility, plus the sum of opening costs of the facilities.
The facility location problem on general graphs is NP-hard to solve optimally, by reduction from (for example) the set cover problem. A number of approximation algorithms have been developed for the facility location problem and many of its variants.
Without assumptions on the set of distances between clients and sites (in particular, without assuming that the distances satisfy the triangle inequality), the problem is known as non-metric facility location and can be approximated to within a factor O(log n). This factor is tight, via an approximation-preserving reduction from the set cover problem.
If we assume distances between clients and sites are undirected and satisfy the triangle inequality, we are talking about a metric facility location (MFL) problem. The MFL is still NP-hard and hard to approximate within factor better than 1.463. The currently best known approximation algorithm achieves approximation ratio of 1.488.
Minimax facility location
The minimax facility location problem seeks a location which minimizes the maximum distance to the sites, where the distance from one point to the sites is the distance from the point to its nearest site. A formal definition is as follows:
Given a point set P ⊂ ℝd, find a point set S ⊂ ℝd, |S| = k, so that maxp ∈ P(minq ∈ S(d(p, q)) ) is minimized.
In the case of the Euclidean metric for k = 1, it is known as the smallest enclosing sphere problem or 1-center problem. Its study traced at least to the year of 1860. see smallest enclosing circle and bounding sphere for more details.
NP hardness
It has been proven that exact solution of k-center problem is NP hard.
Approximation to the problem was found to be also NP hard when the error is small. The error level in the approximation algorithm is measured as an approximation fact
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https://en.wikipedia.org/wiki/Schubert%20variety
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In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, of -dimensional subspaces of a vector space , usually with singular points. Like the Grassmannian, it is a kind of moduli space, whose elements satisfy conditions giving lower bounds to the dimensions of the intersections of its elements , with the elements of a specified complete flag. Here may be a vector space over an arbitrary field, but most commonly this taken to be either the real or the complex numbers.
A typical example is the set of -dimensional subspaces of a 4-dimensional space that intersect a fixed (reference) 2-dimensional subspace nontrivially.
Over the real number field, this can be pictured in usual xyz-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of , we obtain an open subset X° ⊂ X. This is isomorphic to the set of all lines L (not necessarily through the origin) which meet the x-axis. Each such line L corresponds to a point of X°, and continuously moving L in space (while keeping contact with the x-axis) corresponds to a curve in X°. Since there are three degrees of freedom in moving L (moving the point on the x-axis, rotating, and tilting), X is a three-dimensional real algebraic variety. However, when L is equal to the x-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes L a singular point of X.
More generally, a Schubert variety in is defined by specifying the minimal dimension of intersection of a -dimensional subspace with each of the spaces in a fixed reference complete flag , where . (In the example above, this would mean requiring certain intersections of the line L with the x-axis and the xy-plane.)
In even greater generality, given a semisimple algebraic group with a Borel subgroup and a standard parabolic subgroup , it is known that the homogeneous space , which is an example of a flag variety, consists of finitely many -orbits, which may be parametrized by certain elements of the Weyl group . The closure of the -orbit associated to an element is denoted and is called a Schubert variety in . The classical case corresponds to , with , the th maximal parabolic subgroup of , so that is the Grassmannian of -planes in .
Significance
Schubert varieties form one of the most important and best studied classes of singular algebraic varieties. A certain measure of singularity of Schubert varieties is provided by Kazhdan–Lusztig polynomials, which encode their local Goresky–MacPherson intersection cohomology.
The algebras of regular functions on Schubert varieties have deep significance in algebraic combinatorics and are examples of algebras with a straightening law. (Co)homology of the Grassmannian, and more generally, of more general flag varieties, has a basis consisting of the (co)homology classes of Schubert varieties, or Schubert cycles. The study of the intersection theo
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https://en.wikipedia.org/wiki/Standard%20conjectures%20on%20algebraic%20cycles
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In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his construction of pure motives gave an abelian category that is semisimple. Moreover, as he pointed out, the standard conjectures also imply the hardest part of the Weil conjectures, namely the "Riemann hypothesis" conjecture that remained open at the end of the 1960s and was proved later by Pierre Deligne; for details on the link between Weil and standard conjectures, see . The standard conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures, other methods have been found to prove such results unconditionally.
The classical formulations of the standard conjectures involve a fixed Weil cohomology theory . All of the conjectures deal with "algebraic" cohomology classes, which means a morphism on the cohomology of a smooth projective variety
induced by an algebraic cycle with rational coefficients on the product via the cycle class map, which is part of the structure of a Weil cohomology theory.
Conjecture A is equivalent to Conjecture B (see , p. 196), and so is not listed.
Lefschetz type Standard Conjecture (Conjecture B)
One of the axioms of a Weil theory is the so-called hard Lefschetz theorem (or axiom):
Begin with a fixed smooth hyperplane section
,
where is a given smooth projective variety in the ambient projective space and is a hyperplane. Then for , the Lefschetz operator
,
which is defined by intersecting cohomology classes with , gives an isomorphism
.
Now, for define:
The conjecture states that the Lefschetz operator () is induced by an algebraic cycle.
Künneth type Standard Conjecture (Conjecture C)
It is conjectured that the projectors
are algebraic, i.e. induced by a cycle with rational coefficients. This implies that the motive of any smooth projective variety (and more generally, every pure motive) decomposes as
The motives and can always be split off as direct summands. The conjecture therefore immediately holds for curves. It was proved for surfaces by .
have used the Weil conjectures to show the conjecture for algebraic varieties defined over finite fields, in arbitrary dimension.
proved the Künneth decomposition for abelian varieties A.
refined this result by exhibiting a functorial Künneth decomposition of the Chow motive of A such that the n-multiplication on the abelian variety acts as on the i-th summand .
proved the Künneth decomposition for the Hilbert scheme of points in a smooth surface.
Conjecture D (numerical equivalence vs. homological equivalence)
Conjecture D states that numerical and homological equivalence agree. (It implies in particular the latter does not depend on the choice of the Weil cohomolog
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https://en.wikipedia.org/wiki/Differentially%20closed%20field
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In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by . Differentially closed fields are the analogues
for differential equations of algebraically closed fields for polynomial equations.
The theory of differentially closed fields
We recall that a differential field is a field equipped with a derivation operator. Let K be a differential field with derivation operator ∂.
A differential polynomial in x is a polynomial in the formal expressions x, ∂x, ∂2x, ... with coefficients in K.
The order of a non-zero differential polynomial in x is the largest n such that ∂nx occurs in it, or −1 if the differential polynomial is a constant.
The separant Sf of a differential polynomial of order n≥0 is the derivative of f with respect to ∂nx.
The field of constants of K is the subfield of elements a with ∂a=0.
In a differential field K of nonzero characteristic p, all pth powers are constants. It follows that neither K nor its field of constants is perfect, unless ∂ is trivial. A field K with derivation ∂ is called differentially perfect if it is either of characteristic 0, or of characteristic p and every constant is a pth power of an element of K.
A differentially closed field is a differentially perfect differential field K such that if f and g are differential polynomials such that Sf≠ 0 and g≠0 and f has order greater than that of g, then there is some x in K with f(x)=0 and g(x)≠0. (Some authors add the condition that K has characteristic 0, in which case Sf is automatically non-zero, and K is automatically perfect.)
DCFp is the theory of differentially closed fields of characteristic p (where p is 0 or a prime).
Taking g=1 and f any ordinary separable polynomial shows that any differentially closed field is separably closed. In characteristic 0 this implies that it is algebraically closed, but in characteristic p>0 differentially closed fields are never algebraically closed.
Unlike the complex numbers in the theory of algebraically closed fields, there is no natural example of a differentially closed field.
Any differentially perfect field K has a differential closure, a prime model extension, which is differentially closed. Shelah showed that the differential closure is unique up to isomorphism over K. Shelah also showed that the prime differentially closed field of characteristic 0 (the differential closure of the rationals) is not minimal; this was a rather surprising result, as it is not what one would expect by analogy with algebraically closed fields.
The theory of DCFp is complete and model complete (for p=0 this was shown by Robinson, and for p>0 by ).
The theory DCFp is the model companion of the theory of differential fields of characteristic p. It is the model completion of the theory of differentially perfect fields of characteristic p if one adds to the lan
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https://en.wikipedia.org/wiki/Harald%20Ganzinger
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Harald Ganzinger (31 October 1950, Werneck – 3 June 2004, Saarbrücken) was a German computer scientist who together with Leo Bachmair developed the superposition calculus, which is (as of 2007) used in most of the state-of-the-art automated theorem provers for first-order logic.
He received his Ph.D. from the Technical University of Munich in 1978. Before 1991 he was a Professor of Computer Science at University of Dortmund. Then he joined the Max Planck Institute for Computer Science in Saarbrücken shortly after it was founded in 1991. Until 2004 he was the Director of the Programming Logics department of the Max Planck Institute for Computer Science and honorary professor at Saarland University. His research group created the SPASS automated theorem prover.
He received the Herbrand Award in 2004 (posthumous) for his important contributions to automated theorem proving.
References
Rewrite-Based Equational Theorem Proving with Selection and Simplification, Leo Bachmair and Harald Ganzinger, Journal of Logic and Computation 3(4), 1994.
External links
Personal Homepage of Harald Ganzinger — Version of Dec.7th, 2013 saved at archive.org
1950 births
2004 deaths
Automated theorem proving
German computer scientists
Technical University of Munich alumni
Academic staff of the Technical University of Dortmund
Max Planck Institute for Informatics
Max Planck Institute directors
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https://en.wikipedia.org/wiki/Index%20group
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In operator theory, a branch of mathematics, every Banach algebra can be associated with a group called its abstract index group.
Definition
Let A be a Banach algebra and G the group of invertible elements in A. The set G is open and a topological group. Consider the identity component
G0,
or in other words the connected component containing the identity 1 of A; G0 is a normal subgroup of G. The quotient group
ΛA = G/G0
is the abstract index group of A. Because G0, being the component of an open set, is both open and closed in G, the index group is a discrete group.
Examples
Let L(H) be the Banach algebra of bounded operators on a Hilbert space. The set of invertible elements in L(H) is path connected. Therefore, ΛL(H) is the trivial group.
Let T denote the unit circle in the complex plane. The algebra C(T) of continuous functions from T to the complex numbers is a Banach algebra, with the topology of uniform convergence. A function in C(T) is invertible (meaning that it has a pointwise multiplicative inverse, not that it is an invertible function) if it does not map any element of T to zero. The group G0 consists of elements homotopic, in G, to the identity in G, the constant function 1. One can choose the functions fn(z) = zn as representatives in G of distinct homotopy classes of maps T→T. Thus the index group ΛC(T) is the set of homotopy classes, indexed by the winding number of its members. Thus ΛC(T) is isomorphic to the fundamental group of T. It is a countable discrete group.
The Calkin algebra K is the quotient C*-algebra of L(H) with respect to the compact operators. Suppose π is the quotient map. By Atkinson's theorem, an invertible elements in K is of the form π(T) where T is a Fredholm operators. The index group ΛK is again a countable discrete group. In fact, ΛK is isomorphic to the additive group of integers Z, via the Fredholm index. In other words, for Fredholm operators, the two notions of index coincide.
References
Zhu, Kehe (1993). An Introduction to Operator Algebras, CRC Press, Boca Raton, LA,
Operator theory
Banach algebras
Discrete groups
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