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https://en.wikipedia.org/wiki/Greg%20Lindsay
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Gregory John Lindsay AO (b. 1949) was until 2018 the Executive Director of the Australian think tank the Centre for Independent Studies (CIS), which he founded in 1976 when a young mathematics teacher in the western suburbs of Sydney. CIS has become influential in Australia and New Zealand.
Biography
Lindsay initially studied agricultural science at the University of Sydney, but found that this was not his real interest and instead obtained secondary teaching qualifications in mathematics at Sydney Teachers' College. A short four-year stint at Richmond High School coincided with further study at Macquarie University in philosophy culminating in graduating with a BA majoring in philosophy in 1977. The CIS's first public events were also held at Macquarie in October 1976 and in April 1977.
He was made an Officer of the Order of Australia (AO) in 2003 for his contribution to education and public debate.
In 2006 he was elected President of the Mont Pelerin Society for a two-year term at its general meeting in Guatemala. He was elected to the Council of Macquarie University for a term from 1 January 2008.
References
Officers of the Order of Australia
Living people
Year of birth missing (living people)
Australian libertarians
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https://en.wikipedia.org/wiki/Bernard%20Picinbono
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Bernard Picinbono is a French scientist born in 1933 in Algiers. His scientific work focuses on statistics and its applications in optics, electronics, signal processing and automation.
Biography
He did his secondary and higher education in Algiers and then in Paris where he obtained the agrégation de sciences physiques.
He was associate professor of physical sciences at the Algiers high school from 1956 to 1960 and then, after obtaining a doctorate in science, lecturer at the Faculty of Science in Algiers from 1960 to 1965. He was then appointed professor at the Orsay Faculty of Sciences. He was President of the University of Paris XI (now Paris-Saclay University) from 1970 to 1975, President of SupOptique (Institute of Theoretical and Applied Optics) from 1980 to 1990, and Director General of Supélec from 1990 to 1995. In the early 1980s, he was director of the master (DEA) in Automation and Signal Processing at the University of Paris XI and lectures at Supélec's signals and systems laboratory.
He is Professor Emeritus at the Paris-Saclay University and at CentraleSupélec.
Bernard Picinbono was President of Cimade from 1970 to 1983 and again from 1997 to 2002.
Awards
Member of the French Academy of sciences, elected correspondant in 1983.
Member of the French Academy of technologies.
Recipient in 1970 of the Blondel Medal awarded by the Electricity, Electronics and Information and Communication Technologies Society
Fellow of the IEEE for contributions to signal processing, adaptive detection and stochastic processes.
Officier of the Légion d'Honneur (July 2009)
Officier of the Ordre du Mérite
Commandeur des Palmes académiques (2008)
Publications
B. Picinbono. On Instantaneous Amplitude and Phase of Signals. IEEE Trans. Signal Process., 45:552-560, 1997. [bibtex-entry]
B. Picinbono and W. Martin. Représentation des signaux par amplitude et phase instantanées. Annales des Télécommunications, 38:179-190, 1983. [bibtex-entry]
References
French physicists
Members of the French Academy of Sciences
Academic staff of Paris-Sud University
Living people
1933 births
Paris-Saclay University people
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https://en.wikipedia.org/wiki/Handle%20decompositions%20of%203-manifolds
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In mathematics, a handle decomposition of a 3-manifold allows simplification of the original 3-manifold into pieces which are easier to study.
Heegaard splittings
An important method used to decompose into handlebodies is the Heegaard splitting, which gives us a decomposition in two handlebodies of equal genus.
Examples
As an example: lens spaces are orientable 3-spaces and allow decomposition into two solid tori, which are genus-one-handlebodies. The genus one non-orientable space is a space which is the union of two solid Klein bottles and corresponds to the twisted product of the 2-sphere and the 1-sphere: .
Orientability
Each orientable 3-manifold is the union of exactly two orientable handlebodies; meanwhile, each non-orientable one needs three orientable handlebodies.
Heegaard genus
The minimal genus of the glueing boundary determines what is known as the Heegaard genus. For non-orientable spaces an interesting invariant is the tri-genus.
References
J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel-Whitney surfaces and decompositions of 3-manifolds into handlebodies, Topology Appl. 60 (1994), 267-280.
J.C. Gómez Larrañaga, W. Heil, V.M. Núñez. Stiefel-Whitney surfaces and the trigenus of non-orientable 3-manifolds, Manuscripta Math. 100 (1999), 405-422.
3-manifolds
Topology
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https://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb%20algebra
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In statistical mechanics, the Temperley–Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb. It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.
Structure
Generators and relations
Let be a commutative ring and fix . The Temperley–Lieb algebra is the -algebra generated by the elements , subject to the Jones relations:
for all
for all
for all
for all such that
Using these relations, any product of generators can be brought to Jones' normal form:
where and are two strictly increasing sequences in . Elements of this type form a basis of the Temperley-Lieb algebra.
The dimensions of Temperley-Lieb algebras are Catalan numbers:
The Temperley–Lieb algebra is a subalgebra of the Brauer algebra , and therefore also of the partition algebra . The Temperley–Lieb algebra is semisimple for where is a known, finite set. For a given , all semisimple Temperley-Lieb algebras are isomorphic.
Diagram algebra
may be represented diagrammatically as the vector space over noncrossing pairings of points on two opposite sides of a rectangle with n points on each of the two sides.
The identity element is the diagram in which each point is connected to the one directly across the rectangle from it. The generator is the diagram in which the -th and -th point on the left side are connected to each other, similarly the two points opposite to these on the right side, and all other points are connected to the point directly across the rectangle.
The generators of are:
From left to right, the unit 1 and the generators , , , .
Multiplication on basis elements can be performed by concatenation: placing two rectangles side by side, and replacing any closed loops by a factor , for example :
× = = .
The Jones relations can be seen graphically:
=
=
=
The five basis elements of are the following:
.
From left to right, the unit 1, the generators , , and , .
Representations
Structure
For such that is semisimple, a complete set of simple modules is parametrized by integers with . The dimension of a simple module is written in terms of binomial coefficients as
A basis of the simple module is the set of monic noncrossing pairings from points on the left to points on the right. (Monic means that each point on the right is connected to a point on the left.) There is a natural bijection between , and the set of diagrams that generate : any such diagram can be cut into two elements of for some .
Then acts on by diagram concatenation from the left. (Concatenation can produce non-monic pairings, which have to be modded out.) The module may be called a standard module or link module.
If with a root of unity, may not be semisimple, and may not be irreducible:
If is reducible, then its quotient by its maximal proper submodule is irreducible.
Branching rules
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https://en.wikipedia.org/wiki/Yves%20Laszlo
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Yves Laszlo () is a French mathematician working in the University of Paris-Sud. He specializes in algebraic geometry.
Laszlo obtained his Ph.D. in 1988 from the University of Paris-Sud under the supervision of Arnaud Beauville.
He started the Fondation Mathématique Jacques Hadamard in 2011, and directed it until 2012.
The Beauville–Laszlo theorem on gluing sheaves together is named after Laszlo and Beauville, who published it in 1995.
References
External links
Laszlo's Web page
Year of birth missing (living people)
Living people
20th-century French mathematicians
21st-century French mathematicians
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https://en.wikipedia.org/wiki/Algebraic%20holography
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Algebraic holography, also sometimes called Rehren duality, is an attempt to understand the holographic principle of quantum gravity within the framework of algebraic quantum field theory, due to Karl-Henning Rehren. It is sometimes described as an alternative formulation of the AdS/CFT correspondence of string theory, but some string theorists reject this statement . The theories discussed in algebraic holography do not satisfy the usual holographic principle because their entropy follows a higher-dimensional power law.
Rehren's duality
The conformal boundary of an anti-de Sitter space (or its universal covering space) is the conformal Minkowski space (or its universal covering space) with one fewer dimension. Let's work with the universal covering spaces. In AQFT, a QFT in the conformal space is given by a conformally covariant net of C* algebras over the conformal space and the QFT in AdS is given a covariant net of C* algebras over AdS. Any two distinct null geodesic hypersurfaces of codimension 1 which intersect at more than just a point in AdS divides AdS into four distinct regions, two of which are spacelike. Any of the two spacelike regions is called a wedge. It's a geometrical fact that the conformal boundary of a wedge is a double cone in the conformal boundary and that any double cone in the conformal boundary is associated with a unique wedge. In other words, we have a one-to-one correspondence between double cones in CFT and wedges in AdS. It's easy to check that any CFT defined in terms of algebras over the double cones which satisfy the Haag–Kastler axioms also gives rise to a net over AdS which satisfies these axioms if we assume that the algebra associated with a wedge is the same as the algebra associated with its corresponding double cone and vice versa. This correspondence between AQFTs on both sides is called algebraic holography.
Unlike the usual AdS/CFT correspondence, the Rehren-dual theory on the AdS side does not appear to be a theory of quantum gravity as there is no apparent diffeomorphism covariance on the AdS side. Also, if the algebra associated with any double cone in AdS is nontrivial (i.e. contains more than just the identity), the corresponding CFT does not satisfy primitive causality. From this, we can conclude that the AdS Rehren-dual of any realistic CFT does not have any local degrees of freedom (wedges are noncompact).
Differences when compared to AdS/CFT
"In AdS/CFT, the boundary values of bulk fields are sources for operators of the boundary theory. In Rehren Duality, the boundary values of the bulk fields are the operators of the boundary theory.
"In AdS/CFT, the bulk theory is necessarily a gravitational one. The source for the conserved stress tensor of the boundary theory is the boundary value of the bulk metric tensor. In Rehren Duality, the bulk theory is an 'ordinary' (non-gravitational) QFT."
References
For a classical counterpart to Rehren duality see
Axiomatic quantum field theory
C
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https://en.wikipedia.org/wiki/Quaternionic%20projective%20space
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In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions Quaternionic projective space of dimension n is usually denoted by
and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line is homeomorphic to the 4-sphere.
In coordinates
Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written
where the are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion c; that is, we identify all the
.
In the language of group actions, is the orbit space of by the action of , the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside one may also regard as the orbit space of by the action of , the group of unit quaternions. The sphere then becomes a principal Sp(1)-bundle over :
This bundle is sometimes called a (generalized) Hopf fibration.
There is also a construction of by means of two-dimensional complex subspaces of , meaning that lies inside a complex Grassmannian.
Topology
Homotopy theory
The space , defined as the union of all finite 's under inclusion, is the classifying space BS3. The homotopy groups of are given by These groups are known to be very complex and in particular they are non-zero for infinitely many values of . However, we do have that
It follows that rationally, i.e. after localisation of a space, is an Eilenberg–Maclane space . That is (cf. the example K(Z,2)). See rational homotopy theory.
In general, has a cell structure with one cell in each dimension which is a multiple of 4, up to . Accordingly, its cohomology ring is , where is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that has infinite homotopy groups only in dimensions 4 and .
Differential geometry
carries a natural Riemannian metric analogous to the Fubini-Study metric on , with respect to which it is a compact quaternion-Kähler symmetric space with positive curvature.
Quaternionic projective space can be represented as the coset space
where is the compact symplectic group.
Characteristic classes
Since , its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial Stiefel–Whitney and Pontryagin classes. The total classes are given by the following formulas:
where is the generator of and is its reduction mod 2.
Special cases
Quaternionic projective line
The one-dimensional projective space over is called the "projective line" in generalization of the complex projective line. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the Möbius group to the quaternion conte
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https://en.wikipedia.org/wiki/Variation
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Variation or Variations may refer to:
Science and mathematics
Variation (astronomy), any perturbation of the mean motion or orbit of a planet or satellite, particularly of the moon
Genetic variation, the difference in DNA among individuals or the differences between populations
Human genetic variation, genetic differences in and among populations of humans
Magnetic variation, difference between magnetic north and true north, measured as an angle
p-variation in mathematical analysis, a family of seminorms of functions
Coefficient of variation in probability theory and statistics, a standardized measure of dispersion of a probability distribution or frequency distribution
Total variation in mathematical analysis, a way of quantifying the change in a function over a subset of or a measure space
Calculus of variations in mathematical analysis, a method of finding maxima and minima of functionals
Arts
Variation (ballet) or pas seul, solo dance or dance figure
Variations (ballet), a 1966 ballet by choreographer George Balanchine
Variations (film), a 1998 short film by Nathaniel Dorsky
Variations (journal), a journal of literature published by Peter Lang
Music
Variation (music), a formal technique where material is altered during repetition
Variations (Cage), a series of works by American avant-garde composer John Cage
Variations (musical), 1982 Australian musical by Nick Enright and Terence Clarke
Variations (Stravinsky), Igor Stravinsky's last orchestral composition written in 1963–64
Variation, album by Akina Nakamori
Les Variations, a French rock group
Variations (Andrew Lloyd Webber album), 1978
Variations (Eddie Rabbitt album), 1978
Other uses
Variation (game), modifications made to a game by a community of players (as opposed to a central authority)
Variation (game tree), a particular series of moves
Variation (linguistics), a linguistic characteristic of languages
Variation (horse), a British Thoroughbred
See also
Variability (disambiguation)
Change (disambiguation)
Variations on a Theme (disambiguation)
Rate of change (disambiguation)
Repetition (disambiguation)
Variant (disambiguation)
Calculus of variations, a field of mathematics which deals with functions of functions
Genetic diversity, total number of genetic characteristics in the genetic makeup of a species
Variance, expectation of deviation in probability theory or statistics
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https://en.wikipedia.org/wiki/Pair%20of%20pants%20%28mathematics%29
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In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants.
Pairs of pants are used as building blocks for compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional manifolds.
Pants and pants decomposition
Pants as topological surfaces
A pair of pants is any surface that is homeomorphic to a sphere with three holes, which formally is the result of removing from the sphere three open disks with pairwise disjoint closures. Thus a pair of pants is a compact surface of genus zero with three boundary components.
The Euler characteristic of a pair of pants is equal to −1, and the only other surface with this property is the punctured torus (a torus minus an open disk).
Pants decompositions
The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface of genus with boundary components to be , and for a non-connected surface take the sum over all components. Then the only surfaces with negative Euler characteristic and complexity zero are disjoint unions of pairs of pants. Furthermore, for any surface and any simple closed curve on which is not homotopic to a boundary component, the compact surface obtained by cutting along has a complexity that is strictly less than . In this sense, pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic.
By a recursion argument, this implies that for any surface there is a system of simple closed curves which cut the surface into pairs of pants. This is called a pants decomposition for the surface, and the curves are called the cuffs of the decomposition. This decomposition is not unique, but by quantifying the argument one sees that all pants decompositions of a given surface have the same number of curves, which is exactly the complexity. For connected surfaces a pants decomposition has exactly pants.
A collection of simple closed curves on a surface is a pants decomposition if and only if they are disjoint, no two of them are homotopic and none is homotopic to a boundary component, and the collection is maximal for these properties.
The pants complex
A given surface has infinitely many distinct pants decompositions (we understand two decompositions to be distinct when they are not homotopic). One way to try to understand the relations between all these decompositions is the pants complex associated to the surface. This is a graph with vertex set the pants decompositions of , and two vertices are joined if they are related by an
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https://en.wikipedia.org/wiki/Mid-range
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In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set:
The mid-range is closely related to the range, a measure of statistical dispersion defined as the difference between maximum and minimum values.
The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values.
The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest, because it ignores all intermediate points, and lacks robustness, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an L-estimator, it is simple to understand and compute.
Robustness
The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-robust statistic, having a breakdown point of 0, meaning that a single observation can change it arbitrarily. Further, it is highly influenced by outliers: increasing the sample maximum or decreasing the sample minimum by x changes the mid-range by while it changes the sample mean, which also has breakdown point of 0, by only It is thus of little use in practical statistics, unless outliers are already handled.
A trimmed midrange is known as a – the n% trimmed midrange is the average of the n% and (100−n)% percentiles, and is more robust, having a breakdown point of n%. In the middle of these is the midhinge, which is the 25% midsummary. The median can be interpreted as the fully trimmed (50%) mid-range; this accords with the convention that the median of an even number of points is the mean of the two middle points.
These trimmed midranges are also of interest as descriptive statistics or as L-estimators of central location or skewness: differences of midsummaries, such as midhinge minus the median, give measures of skewness at different points in the tail.
Efficiency
Despite its drawbacks, in some cases it is useful: the midrange is a highly efficient estimator of μ, given a small sample of a sufficiently platykurtic distribution, but it is inefficient for mesokurtic distributions, such as the normal.
For example, for a continuous uniform distribution with unknown maximum and minimum, the mid-range is the uniformly minimum-variance unbiased estimator (UMVU) estimator for the mean. The sample maximum and sample minimum, together with sample size, are a sufficient statistic for the population maximum and minimum – the distribution of other samples, conditional on a given maximum and minimum, is just the uniform distribution between the maximum and minimum and thus add no information. See German tank problem for
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https://en.wikipedia.org/wiki/Elbert%20Frank%20Cox
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Elbert Frank Cox (5 December 1895 – 28 November 1969) was an American mathematician. He was the first Black person in history to receive a PhD in mathematics, which he earned at Cornell University in 1925.
Early life
Cox was born in Evansville, Indiana to Johnson D. Cox, a Kentucky-born teacher active in the church, and Eugenia Talbot Cox. He grew up with his parents, maternal grandmother and two brothers in a racially mixed neighborhood; in 1900, in his block, there were three Black and five white families.
Cox went to a segregated college with inadequate resources. Cox was offered a scholarship to study violin at the Prague Conservatory of Music, but chose to pursue his interest in mathematics instead.
Education
Indiana University
Cox studied at Indiana University Bloomington. Besides mathematics, Cox also took courses in German, English, Latin, history, hygiene, chemistry, education, philosophy and physics. Cox's brother Avalon went to Indiana University as well. There were three other Black students in his class. He received his bachelor's degree in 1917, at a time when the transcript of every Black student had the word "" printed across it. He received A's on all his exams while at Indiana.
Between colleges
After he graduated in 1917, Cox joined the U.S. Army to fight in France during World War I from 1918 to 1919. After he was discharged from the Army, he began his career as a high school math tutor.
Cox returned to pursue a career in teaching, as an instructor of mathematics at a high school in Henderson, Kentucky. In the autumn of 1919, he was appointed as a professor in physics, chemistry and biology at Shaw University in Raleigh, North Carolina where he also became chairman of the Department of Natural Sciences. He would continue there until 1922.
Cornell University
In December 1921, he applied for a graduate scholarship at Cornell University, one of seven American universities with a doctoral program in mathematics. One of his references wrote a positive letter followed by another letter anticipating difficulties for him because he was a "colored man". Because Cornell's founder, Ezra Cornell, had been an early opponent of slavery, Cornell University was an appropriate place to study for an African American at the time. Cox was approved May 5, 1922, and enrolled in the autumn of the same year.
Cox was very successful at Cornell. Important to him was a young instructor, William Lloyd Garrison Williams, a co-founder of the Canadian Mathematical Congress who became chair of Cox's "special committee" in March 1923, and was his supervisor. Cox received the Erastus Brooks fellowship in Mathematics ($400 per year) in autumn 1924 and followed Williams to McGill University in Montreal. He moved back to Cornell in the spring semester of 1925, and finished his dissertation, The polynomial solutions of the difference equation of(x+1) + bf(x) = φ(x), in the summer of the same year. On September 26, 1925, he received his Ph.D. He was c
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https://en.wikipedia.org/wiki/Gap%20theorem
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See also Gap theorem (disambiguation) for other gap theorems in mathematics.
In computational complexity theory, the Gap Theorem, also known as the Borodin–Trakhtenbrot Gap Theorem, is a major theorem about the complexity of computable functions.
It essentially states that there are arbitrarily large computable gaps in the hierarchy of complexity classes. For any computable function that represents an increase in computational resources, one can find a resource bound such that the set of functions computable within the expanded resource bound is the same as the set computable within the original bound.
The theorem was proved independently by Boris Trakhtenbrot and Allan Borodin.
Although Trakhtenbrot's derivation preceded Borodin's by several years, it was not known nor recognized in the West until after Borodin's work was published.
Gap theorem
The general form of the theorem is as follows.
Suppose is an abstract (Blum) complexity measure. For any total computable function for which for every , there is a total computable function such that with respect to , the complexity classes with boundary functions and are identical.
The theorem can be proved by using the Blum axioms without any reference to a concrete computational model, so it applies to time, space, or any other reasonable complexity measure.
For the special case of time complexity, this can be stated more simply as:
for any total computable function such that for all , there exists a time bound such that .
Because the bound may be very large (and often will be nonconstructible) the Gap Theorem does not imply anything interesting for complexity classes such as P or NP, and it does not contradict the time hierarchy theorem or space hierarchy theorem.
See also
Blum's speedup theorem
References
Theorems in computational complexity theory
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https://en.wikipedia.org/wiki/Algebra%20i%20Logika
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Algebra i Logika (English: Algebra and Logic) is a peer-reviewed Russian mathematical journal founded in 1962 by Anatoly Ivanovich Malcev, published by the Siberian Fund for Algebra and Logic at Novosibirsk State University. An English translation of the journal is published by Springer-Verlag as Algebra and Logic since 1968. It published papers presented at the meetings of the "Algebra and Logic" seminar at the Novosibirsk State University. The journal is edited by academician Yury Yershov.
The journal is reviewed cover-to-cover in Mathematical Reviews and Zentralblatt MATH.
Abstracting and Indexing
Algebra i Logika is indexed and abstracted in the following databases:
According to the Journal Citation Reports, the journal had a 2020 impact factor of 0.753.
References
External links
Algebra i Logika website
Algebra and Logic website
Mathematics journals
Academic journals established in 1962
Novosibirsk State University
Magazines published in Novosibirsk
Russian-language journals
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https://en.wikipedia.org/wiki/G.%20Mike%20Reed
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George Michael ("Mike") Reed is an American computer scientist. He has contributed to theoretical computer science in general and CSP in particular.
Mike Reed has a doctorate in pure mathematics from Auburn University, United States, and a doctorate in computation from Oxford University, England. He has an interest in mathematical topology.
Reed was a Senior Research Associate at NASA Goddard Space Flight Center. From 1986 to 2005, he was at the Oxford University Computing Laboratory (now the Oxford University Department of Computer Science) in England where he was also a Fellow in Computation of St Edmund Hall, Oxford (1986–2005). In 2005, he became Director of UNU/IIST, Macau, part of the United Nations University.
References
External links
Year of birth missing (living people)
Living people
Auburn University alumni
Alumni of the University of Oxford
Members of the Department of Computer Science, University of Oxford
Fellows of St Edmund Hall, Oxford
Academic staff of United Nations University
American computer scientists
Formal methods people
Topologists
20th-century American mathematicians
21st-century American mathematicians
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https://en.wikipedia.org/wiki/%CE%98%20%28set%20theory%29
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In set theory, Θ (pronounced like the letter theta) is the least nonzero ordinal α such that there is no surjection from the reals onto α.
If the axiom of choice (AC) holds (or even if the reals can be wellordered), then Θ is simply , the cardinal successor of the cardinality of the continuum. However, Θ is often studied in contexts where the axiom of choice fails, such as models of the axiom of determinacy.
Θ is also the supremum of the lengths of all prewellorderings of the reals.
Proof of existence
It may not be obvious that it can be proven, without using AC, that there even exists a nonzero ordinal onto which there is no surjection from the reals (if there is such an ordinal, then there must be a least one because the ordinals are wellordered). However, suppose there were no such ordinal. Then to every ordinal α we could associate the set of all prewellorderings of the reals having length α. This would give an injection from the class of all ordinals into the set of all sets of orderings on the reals (which can to be seen to be a set via repeated application of the powerset axiom). Now the axiom of replacement shows that the class of all ordinals is in fact a set. But that is impossible, by the Burali-Forti paradox.
Cardinal numbers
Descriptive set theory
Determinacy
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https://en.wikipedia.org/wiki/AD%2B
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In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DC (the axiom of dependent choice for real numbers), states two things:
Every set of reals is ∞-Borel.
For any ordinal λ less than Θ, any subset A of ωω, and any continuous function π:λω→ωω, the preimage π−1[A] is determined. (Here λω is to be given the product topology, starting with the discrete topology on λ.)
The second clause by itself is referred to as ordinal determinacy.
See also
Axiom of projective determinacy
Axiom of real determinacy
Suslin's problem
Topological game
References
Axioms of set theory
Determinacy
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https://en.wikipedia.org/wiki/Classifying%20space%20for%20U%28n%29
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In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy.
This space with its universal fibration may be constructed as either
the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or,
the direct limit, with the induced topology, of Grassmannians of n planes.
Both constructions are detailed here.
Construction as an infinite Grassmannian
The total space EU(n) of the universal bundle is given by
Here, H denotes an infinite-dimensional complex Hilbert space, the ei are vectors in H, and is the Kronecker delta. The symbol is the inner product on H. Thus, we have that EU(n) is the space of orthonormal n-frames in H.
The group action of U(n) on this space is the natural one. The base space is then
and is the set of Grassmannian n-dimensional subspaces (or n-planes) in H. That is,
so that V is an n-dimensional vector space.
Case of line bundles
For n = 1, one has EU(1) = S∞, which is known to be a contractible space. The base space is then BU(1) = CP∞, the infinite-dimensional complex projective space. Thus, the set of isomorphism classes of circle bundles over a manifold M are in one-to-one correspondence with the homotopy classes of maps from M to CP∞.
One also has the relation that
that is, BU(1) is the infinite-dimensional projective unitary group. See that article for additional discussion and properties.
For a torus T, which is abstractly isomorphic to U(1) × ... × U(1), but need not have a chosen identification, one writes BT.
The topological K-theory K0(BT) is given by numerical polynomials; more details below.
Construction as an inductive limit
Let Fn(Ck) be the space of orthonormal families of n vectors in Ck and let Gn(Ck) be the Grassmannian of n-dimensional subvector spaces of Ck. The total space of the universal bundle can be taken to be the direct limit of the Fn(Ck) as k → ∞, while the base space is the direct limit of the Gn(Ck) as k → ∞.
Validity of the construction
In this section, we will define the topology on EU(n) and prove that EU(n) is indeed contractible.
The group U(n) acts freely on Fn(Ck) and the quotient is the Grassmannian Gn(Ck). The map
is a fibre bundle of fibre Fn−1(Ck−1). Thus because is trivial and because of the long exact sequence of the fibration, we have
whenever . By taking k big enough, precisely for , we can repeat the process and get
This last group is trivial for k > n + p. Let
be the direct limit of all the Fn(Ck) (with the induced topology). Let
be the direct limit of all the Gn(Ck) (with the induced topology).
Lemma: The group is trivial for all p ≥ 1.
Proof: Let γ : Sp → EU(n), since Sp is compact, there exists k such that γ(Sp) is included in Fn(Ck). By taking k big enough, we see that γ is homotopic, with respect to the base point, to the constant ma
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https://en.wikipedia.org/wiki/HNN%20extension
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In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper Embedding Theorems for Groups by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group G into another group G' , in such a way that two given isomorphic subgroups of G are conjugate (through a given isomorphism) in G' .
Construction
Let G be a group with presentation , and let be an isomorphism between two subgroups of G. Let t be a new symbol not in S, and define
The group is called the HNN extension of G relative to α. The original group G is called the base group for the construction, while the subgroups H and K are the associated subgroups. The new generator t is called the stable letter.
Key properties
Since the presentation for contains all the generators and relations from the presentation for G, there is a natural homomorphism, induced by the identification of generators, which takes G to . Higman, Neumann, and Neumann proved that this morphism is injective, that is, an embedding of G into . A consequence is that two isomorphic subgroups of a given group are always conjugate in some overgroup; the desire to show this was the original motivation for the construction.
Britton's Lemma
A key property of HNN-extensions is a normal form theorem known as Britton's Lemma. Let be as above and let w be the following product in :
Then Britton's Lemma can be stated as follows:
Britton's Lemma. If w = 1 in G∗α then
either and g0 = 1 in G
or and for some i ∈ {1, ..., n−1} one of the following holds:
εi = 1, εi+1 = −1, gi ∈ H,
εi = −1, εi+1 = 1, gi ∈ K.
In contrapositive terms, Britton's Lemma takes the following form:
Britton's Lemma (alternate form). If w is such that
either and g0 ≠ 1 ∈ G,
or and the product w does not contain substrings of the form tht−1, where h ∈ H and of the form t−1kt where k ∈ K,
then in .
Consequences of Britton's Lemma
Most basic properties of HNN-extensions follow from Britton's Lemma. These consequences include the following facts:
The natural homomorphism from G to is injective, so that we can think of as containing G as a subgroup.
Every element of finite order in is conjugate to an element of G.
Every finite subgroup of is conjugate to a finite subgroup of G.
If contains an element such that is contained in neither nor for any integer , then contains a subgroup isomorphic to a free group of rank two.
Applications and generalizations
Applied to algebraic topology, the HNN extension constructs the fundamental group of a topological space X that has been 'glued back' on itself by a mapping f : X → X (see e.g. Surface bundle over the circle). Thus, HNN extensions describe the fundamental group of a self-glued space in the same way that free products with amalgamation do for two spaces X and Y glued along a connected common subspace, as in the Seifert-van Kampen theorem. These two constructions allow the description of the fundamental group of any reaso
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https://en.wikipedia.org/wiki/Delta%20method
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In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.
History
The delta method was derived from propagation of error, and the idea behind was known in the early 20th century. Its statistical application can be traced as far back as 1928 by T. L. Kelley. A formal description of the method was presented by J. L. Doob in 1935. Robert Dorfman also described a version of it in 1938.
Univariate delta method
While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables satisfying
where θ and σ2 are finite valued constants and denotes convergence in distribution, then
for any function g satisfying the property that its first derivative, evaluated at , exists and is non-zero valued.
Proof in the univariate case
Demonstration of this result is fairly straightforward under the assumption that is continuous. To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):
where lies between and θ.
Note that since and , it must be that and since is continuous, applying the continuous mapping theorem yields
where denotes convergence in probability.
Rearranging the terms and multiplying by gives
Since
by assumption, it follows immediately from appeal to Slutsky's theorem that
This concludes the proof.
Proof with an explicit order of approximation
Alternatively, one can add one more step at the end, to obtain the order of approximation:
This suggests that the error in the approximation converges to 0 in probability.
Multivariate delta method
By definition, a consistent estimator B converges in probability to its true value β, and often a central limit theorem can be applied to obtain asymptotic normality:
where n is the number of observations and Σ is a (symmetric positive semi-definite) covariance matrix. Suppose we want to estimate the variance of a scalar-valued function h of the estimator B. Keeping only the first two terms of the Taylor series, and using vector notation for the gradient, we can estimate h(B) as
which implies the variance of h(B) is approximately
One can use the mean value theorem (for real-valued functions of many variables) to see that this does not rely on taking first order approximation.
The delta method therefore implies that
or in univariate terms,
Example: the binomial proportion
Suppose Xn is binomial with parameters and n. Since
we can apply the Delta method with to see
Hence, even though for any finite n, the variance of does not actually exist (since Xn can be zero), the asymptotic variance of does exist and is equal to
Note that since p>0, as , so with probability converging to one, is finite for large n.
Moreover, if and
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https://en.wikipedia.org/wiki/Link-state%20advertisement
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The link-state advertisement (LSA) is a basic communication means of the OSPF routing protocol for the Internet Protocol (IP). It communicates the router's local routing topology to all other local routers in the same OSPF area. OSPF is designed for scalability, so some LSAs are not flooded out on all interfaces, but only on those that belong to the appropriate area. In this way detailed information can be kept localized, while summary information is flooded to the rest of the network. The original IPv4-only OSPFv2 and the newer IPv6-compatible OSPFv3 have broadly similar LSA types.
Types
The LSA types defined in OSPF are as follows:
The opaque LSAs, types 9, 10, and 11, are designated for upgrades to OSPF for application-specific purposes. For example, OSPF-TE has traffic engineering extensions to be used by RSVP-TE in Multiprotocol Label Switching (MPLS). Opaque LSAs are used to flood link color and bandwidth information. Standard link-state database (LSDB) flooding mechanisms are used for distribution of opaque LSAs. Each of the three types has a different flooding scope.
For all types of LSAs, there are 20-byte LSA headers. One of the fields of the LSA header is the link-state ID.
Each router link is defined as one of four types: type 1, 2, 3, or 4. The LSA includes a link ID field that identifies, by the network number and mask, the object that this link connects to.
Depending on the type, the link ID has different meanings as shown in below table:
OSPFv2 for IPv4
As per Appendix-A.3.1 of RFC 2328, all OSPF packets start with a common LSA "24-byte header" as shown below.
For
Options
The Options field is present in:
Hello packets
Database Description packets
all the LSAs
The option field Indicative the feature supported by the source router. In Hello packet, a mismatch, will result in reject of neighbor. for LSA only packet that matches the destination routes willingness is forward.
Options (8 bits)
E-Bit: Indicative if area is AS-external capable, or STUBed.
x-bit: Set 0, used previously used by MOSPF
N/P-bit: Indicative if area is NSSA.
EA-bit: Indicative receive and forward External-Attributes-LSAs
DC-bit: Indicative router's handling of demand circuits, .
O-bit: Indicative router's willingness to receive and forward Opaque-LSAs
*: Reserved set 0
Database description DBD
Database description messages contain descriptions of the topology of the autonomous system or area. They convey the contents of the link-state database (LSDB) for the area from one router to another. Communicating a large LSDB may require several messages to be sent by having the sending device designated as a master device and sending messages in sequence, with the slave (recipient of the LSDB information) responding with acknowledgments.
Interface MTU (16 bits) the largest IP datagram that can be sent without fragmentation. In bytes.
flags(8 bits) 3 bits are defined.
I-Bit: Indicative this is the first packet in the sequence of Database Desc
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https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger%20determinant
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In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a -dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger.
The pairwise distance polynomials between n points in a real Euclidean space are Euclidean invariants that are associated via the Cayley-Menger relations. These relations served multiple purposes such as generalising Heron's Formula, computing the content of a n-dimensional simplex, and ultimately determining if any real symmetric matrix is a Euclidean distance matrix in the field of Distance geometry.
History
Karl Menger was a young geometry professor at the University of Vienna and Arthur Cayley was a British mathematician who specialized in algebraic geometry. Menger extended Cayley's algebraic excellence to propose a new axiom of metric spaces using the concepts of distance geometry and relation of congruence, known as the Cayley-Menger determinant. This ended up generalising one of the first discoveries in distance geometry, Heron's formula, which computes the area of a triangle given its side lengths.
Definition
Let be points in -dimensional Euclidean space, with . These points are the vertices of an n-dimensional simplex: a triangle when ; a tetrahedron when , and so on. Let be the Euclidean distances between vertices and . The content, i.e. the n-dimensional volume of this simplex, denoted by , can be expressed as a function of determinants of certain matrices, as follows:
This is the Cayley–Menger determinant. For it is a symmetric polynomial in the 's and is thus invariant under permutation of these quantities. This fails for but it is always invariant under permutation of the vertices.
Except for the final row and column of 1s, the matrix in the second form of this equation is a Euclidean distance matrix.
Special cases
2-Simplex
To reiterate, a simplex is an n-dimensional polytope and the convex hull of points which do not lie in any dimensional plane. Therefore, a 2-simplex occurs when and the simplex results in a triangle. Therefore, the formula for determining of a triangle is provided below:
As a result, the equation above presents the content of a 2-simplex (area of a planar triangle with side lengths , , and ) and it is a generalised form of Heron's Formula.
3-Simplex
Similarly, a 3-simplex occurs when and the simplex results in a tetrahedron. Therefore, the formula for determining of a tetrahedron is provided below:
As a result, the equation above presents the content of a 3-simplex, which is the volume of a tetrahedron where the edge between vertices and has length .
Proof
Let the column vectors be points in -dimensional Euclidean space. Starting with the volume formula
we note that the determinant is unchanged when we add an extra row and column to make an matrix,
where is the squar
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https://en.wikipedia.org/wiki/Theodosius%20of%20Bithynia
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Theodosius of Bithynia (; 2nd–1st century BC) was a Hellenistic astronomer and mathematician from Bithynia who wrote the Spherics, a treatise about spherical geometry, as well as several other books on mathematics and astronomy, of which two survive, On Habitations and On Days and Nights.
Life
Little is known about Theodosius' life. The Suda (10th-century Byzantine encyclopedia) mentions him writing a commentary on Archimedes' Method (late 3rd century BC), and Strabo's Geographica mentioned mathematicians Hipparchus ( – ) and "Theodosius and his sons" as among the residents of Bithynia distinguished for their learning. Later Vitruvius (1st century BC) mentioned a sundial invented by Theodosius. Thus Theodosius lived sometime after Archimedes and before Vitruvius, likely contemporaneously with or after Hipparchus, probably sometime between 200–50 BC.
Historically he was called Theodosius of Tripolis due to a confusing paragraph in the Suda which probably fused the entries about separate people named Theodosius, and was interpreted to mean that he came either from the Tripolis in Phoenicia or the one in Africa. Some sources claim he moved from Bithynia to Tripolis, or came from a hypothetical city called Tripolis in Bithynia.
Works
Theodosius' chief work, the Spherics ( ), provided the mathematics for spherical astronomy. Euclid's Phenomena and Autolycus's On the Moving Sphere, both dating from two centuries prior, make use of theorems proven in Spherics, so it has been speculated that they may have expected readers to be familiar with a treatise on elementary spherical geometry, perhaps by Eudoxus of Cnidus (4th century BC), on which the Spherics may have been based. However, no mention of this hypothetical earlier work or its author remains today.
The Spherics is reasonably complete, and remained the main reference on the subject at least until the time of Pappus of Alexandria (4th century AD). The work was translated into Arabic in the 10th century, and then into Latin in the early 16th century, but these versions were faulty. Francesco Maurolico translated the works later in the 16th century.
In addition to the Spherics, two other works by Theodosius have survived: On Habitations, describing the appearances of the heavens at different climes and different times of the year, and On Days and Nights, a study of the apparent motion of the Sun. Both were published in Latin in the 16th century. Theodosius was cited by Vitruvius as having invented a sundial suitable for any place on Earth.
Notes
References
Ancient Greek astronomers
Ancient Greek geometers
2nd-century BC Greek people
2nd-century BC mathematicians
2nd-century BC astronomers
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https://en.wikipedia.org/wiki/Traffic%20flow
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In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.
History
Attempts to produce a mathematical theory of traffic flow date back to the 1920s, when American Economist Frank Knight first produced an analysis of traffic equilibrium, which was refined into Wardrop's first and second principles of equilibrium in 1952.
Nonetheless, even with the advent of significant computer processing power, to date there has been no satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques. These models are then developed into traffic forecasts, and take account of proposed local or major changes, such as increased vehicle use, changes in land use or changes in mode of transport (with people moving from bus to train or car, for example), and to identify areas of congestion where the network needs to be adjusted.
Overview
Traffic behaves in a complex and nonlinear way, depending on the interactions of a large number of vehicles. Due to the individual reactions of human drivers, vehicles do not interact simply following the laws of mechanics, but rather display cluster formation and shock wave propagation, both forward and backward, depending on vehicle density. Some mathematical models of traffic flow use a vertical queue assumption, in which the vehicles along a congested link do not spill back along the length of the link.
In a free-flowing network, traffic flow theory refers to the traffic stream variables of speed, flow, and concentration. These relationships are mainly concerned with uninterrupted traffic flow, primarily found on freeways or expressways.
Flow conditions are considered "free" when less than 12 vehicles per mile per lane are on a road. "Stable" is sometimes described as 12–30 vehicles per mile per lane. As the density reaches the maximum mass flow rate (or flux) and exceeds the optimum density (above 30 vehicles per mile per lane), traffic flow becomes unstable, and even a minor incident can result in persistent stop-and-go driving conditions. A "breakdown" condition occurs when traffic becomes unstable and exceeds 67 vehicles per mile per lane. "Jam density" refers to extreme traffic density when traffic flow stops completely, usually in the range of 185–250 vehicles per mile per lane.
However, calculations about congested networks are more complex and rely more on empirical studies and extrapolations from actual road counts. Because these are often urban or suburban in nature, other factors (such as road-user safety and environmental considerations) also influence the optimum conditions.
Traff
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https://en.wikipedia.org/wiki/Isotonic%20regression
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In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.
Applications
Isotonic regression has applications in statistical inference. For example, one might use it to fit an isotonic curve to the means of some set of experimental results when an increase in those means according to some particular ordering is expected. A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing.
Another application is nonmetric multidimensional scaling, where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isotonic regression is used iteratively to fit ideal distances to preserve relative dissimilarity order.
Isotonic regression is also used in probabilistic classification to calibrate the predicted probabilities of supervised machine learning models.
Isotonic regression for the simply ordered case with univariate has been applied to estimating continuous dose-response relationships in fields such as anesthesiology and toxicology. Narrowly speaking, isotonic regression only provides point estimates at observed values of Estimation of the complete dose-response curve without any additional assumptions is usually done via linear interpolation between the point estimates.
Software for computing isotone (monotonic) regression has been developed for R, Stata, and Python.
Problem statement and algorithms
Let be a given set of observations, where the and the fall in some partially ordered set. For generality, each observation may be given a weight , although commonly for all .
Isotonic regression seeks a weighted least-squares fit for all , subject to the constraint that whenever . This gives the following quadratic program (QP) in the variables :
subject to
where specifies the partial ordering of the observed inputs (and may be regarded as the set of edges of some directed acyclic graph (dag) with vertices ). Problems of this form may be solved by generic quadratic programming techniques.
In the usual setting where the values fall in a totally ordered set such as , we may assume WLOG that the observations have been sorted so that , and take . In this case, a simple iterative algorithm for solving the quadratic program is the pool adjacent violators algorithm. Conversely, Best and Chakravarti studied the problem as an active set identification problem, and proposed a primal algorithm. These two algorithms can be seen as each other's dual, and both have a computational complexity of on already sorted data.
To complete the isotonic regression task, we may then choose any non-decr
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https://en.wikipedia.org/wiki/Cauchy%27s%20theorem%20%28group%20theory%29
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In mathematics, specifically group theory, Cauchy's theorem states that if is a finite group and is a prime number dividing the order of (the number of elements in ), then contains an element of order . That is, there is in such that is the smallest positive integer with = , where is the identity element of . It is named after Augustin-Louis Cauchy, who discovered it in 1845.
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of . Cauchy's theorem implies that for any prime divisor of the order of , there is a subgroup of whose order is —the cyclic group generated by the element in Cauchy's theorem.
Cauchy's theorem is generalized by Sylow's first theorem, which implies that if is the maximal power of dividing the order of , then has a subgroup of order (and using the fact that a -group is solvable, one can show that has subgroups of order for any less than or equal to ).
Statement and proof
Many texts prove the theorem with the use of strong induction and the class equation, though considerably less machinery is required to prove the theorem in the abelian case. One can also invoke group actions for the proof.
Proof 1
We first prove the special case that where is abelian, and then the general case; both proofs are by induction on = ||, and have as starting case = which is trivial because any non-identity element now has order . Suppose first that is abelian. Take any non-identity element , and let be the cyclic group it generates. If divides ||, then ||/ is an element of order . If does not divide ||, then it divides the order [:] of the quotient group /, which therefore contains an element of order by the inductive hypothesis. That element is a class for some in , and if is the order of in , then = in gives () = in /, so divides ; as before / is now an element of order in , completing the proof for the abelian case.
In the general case, let be the center of , which is an abelian subgroup. If divides ||, then contains an element of order by the case of abelian groups, and this element works for as well. So we may assume that does not divide the order of . Since does divide ||, and is the disjoint union of and of the conjugacy classes of non-central elements, there exists a conjugacy class of a non-central element whose size is not divisible by . But the class equation shows that size is [ : ()], so divides the order of the centralizer () of in , which is a proper subgroup because is not central. This subgroup contains an element of order by the inductive hypothesis, and we are done.
Proof 2
This proof uses the fact that for any action of a (cyclic) group of prime order , the only possible orbit sizes are 1 and , which is immediate from the orbit stabilizer theorem.
The set that our cyclic group shall act on is the set
of -tuples of elements of whose product (in order) gives the identity. Such a -tuple is uni
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https://en.wikipedia.org/wiki/Dusa%20McDuff
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Dusa McDuff FRS CorrFRSE (born 18 October 1945) is an English mathematician who works on symplectic geometry. She was the first recipient of the Ruth Lyttle Satter Prize in Mathematics, was a Noether Lecturer, and is a Fellow of the Royal Society. She is currently the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College.
Personal life and education
Margaret Dusa Waddington was born in London, England, on 18 October 1945 to Edinburgh architect Margaret Justin Blanco White, second wife of biologist Conrad Hal Waddington, her father. Her sister is the anthropologist Caroline Humphrey, and she has an elder half-brother C. Jake Waddington by her father's first marriage. Her mother was the daughter of Amber Reeves, the noted feminist, author and lover of H. G. Wells. McDuff grew up in Scotland where her father was Professor of Genetics at the University of Edinburgh. McDuff was educated at St George's School for Girls in Edinburgh and, although the standard was lower than at the corresponding boys' school, The Edinburgh Academy, McDuff had an exceptionally good mathematics teacher. She writes:
I always wanted to be a mathematician (apart from a time when I was eleven when I wanted to be a farmer's wife), and assumed that I would have a career, but I had no idea how to go about it: I didn't realize that the choices which one made about education were important and I had no idea that I might experience real difficulties and conflicts in reconciling the demands of a career with life as a woman.
Turning down a scholarship to the University of Cambridge to stay with her boyfriend in Scotland, she enrolled at the University of Edinburgh. She graduated with a BSc Hons in 1967, going on to Girton College, Cambridge as a doctoral student. Here, under the guidance of mathematician George A. Reid, McDuff worked on problems in functional analysis. She solved a problem on Von Neumann algebras, constructing infinitely many different factors of type II1, and published the work in the Annals of Mathematics.
After completing her doctorate in 1971 McDuff was appointed to a two-year Science Research Council Postdoctoral Fellowship at Cambridge. Following her husband, the literary translator David McDuff, she left for a six-month visit to Moscow. Her husband was studying the Russian Symbolist poet Innokenty Annensky. Though McDuff had no specific plans it turned out to be a profitable visit for her mathematically. There, she met Israel Gelfand in Moscow who gave her a deeper appreciation of mathematics. McDuff later wrote:
[My collaboration with him]... was not planned: it happened that his was the only name which came to mind when I had to fill out a form in the Inotdel office. The first thing that Gel'fand told me was that he was much more interested in the fact that my husband was studying the Russian Symbolist poet Innokenty Annensky than that I had found infinitely many type II-sub-one factors, but then he proceeded to open my eyes to the world of
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https://en.wikipedia.org/wiki/Calculus%20on%20Manifolds%20%28book%29
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Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates.
Description
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results:
The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin–Stokes theorem).
Reception
Calculus on Manifolds aims to present the topics of multivariable and vector calculus in the manner in which they are seen by a modern working mathematician, yet simply and selectively enough to be understood by undergraduate students whose previous coursework in mathematics comprises only one-variable calculus and introductory linear algebra. While Spivak's elementary treatment of modern mathematical tools is broadly successful—and this approach has made Calculus on Manifolds a standard introduction to the rigorous theory of multivariable calculus—the text is also well known for its laconic style, lack of motivating examples, and frequent omission of non-obvious steps and arguments. For example, in order to state and prove the generalized Stokes' theorem on chains, a profusion of unfamiliar concepts and constructions (e.g., tensor products, differential forms, tangent spaces, pullbacks, exterior derivatives, cube and chains) are introduced in quick succession within the span of 25 pages. Moreover, careful readers have noted a number of nontrivial oversights throughout the text, including missing hypotheses in theorems, inaccurately stated theorems, and proofs that fail to handle all cases.
Other textbooks
A more recent textbook which also covers these topics at an undergraduate level is the text Analysis on Manifolds by James Munkres (366 pp.). At more than twice the length of Calculus on Manifolds, Munkres's work presents a more careful and detailed treatment of the subject matter at a leisurely pace. Nevertheless, Munkres acknowledges the influence of Spivak's earlier text in the preface of Analysis on Manifolds.
Spivak's five-volume textbo
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https://en.wikipedia.org/wiki/Martingale%20central%20limit%20theorem
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In probability theory, the central limit theorem says that, under certain conditions, the sum of many independent identically-distributed random variables, when scaled appropriately, converges in distribution to a standard normal distribution. The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.
Statement
Here is a simple version of the martingale central limit theorem: Let
be a martingale with bounded increments; that is, suppose
and
almost surely for some fixed bound k and all t. Also assume that almost surely.
Define
and let
Then
converges in distribution to the normal distribution with mean 0 and variance 1 as . More explicitly,
The sum of variances must diverge to infinity
The statement of the above result implicitly assumes the variances sum to infinity, so the following holds with probability 1:
This ensures that with probability 1:
This condition is violated, for example, by a martingale that is defined to be zero almost surely for all time.
Intuition on the result
The result can be intuitively understood by writing the ratio as a summation:
The first term on the right-hand-side asymptotically converges to zero, while the second term is qualitatively similar to the summation formula for the central limit theorem in the simpler case of i.i.d. random variables. While the terms in the above expression are not necessarily i.i.d., they are uncorrelated and have zero mean. Indeed:
References
Many other variants on the martingale central limit theorem can be found in:
Note, however, that the proof of Theorem 5.4 in Hall & Heyde contains an error. For further discussion, see
Martingale theory
Central limit theorem
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https://en.wikipedia.org/wiki/Holomorph%20%28mathematics%29
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In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group that simultaneously contains (copies of) the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in a uniform context. In group theory, for a group , the holomorph of denoted can be described as a semidirect product or as a permutation group.
Hol(G) as a semidirect product
If is the automorphism group of then
where the multiplication is given by
[Eq. 1]
Typically, a semidirect product is given in the form where and are groups and is a homomorphism and where the multiplication of elements in the semidirect product is given as
which is well defined, since and therefore .
For the holomorph, and is the identity map, as such we suppress writing explicitly in the multiplication given in [Eq. 1] above.
For example,
the cyclic group of order 3
where
with the multiplication given by:
where the exponents of are taken mod 3 and those of mod 2.
Observe, for example
and this group is not abelian, as , so that is a non-abelian group of order 6, which, by basic group theory, must be isomorphic to the symmetric group .
Hol(G) as a permutation group
A group G acts naturally on itself by left and right multiplication, each giving rise to a homomorphism from G into the symmetric group on the underlying set of G. One homomorphism is defined as λ: G → Sym(G), (h) = g·h. That is, g is mapped to the permutation obtained by left-multiplying each element of G by g. Similarly, a second homomorphism ρ: G → Sym(G) is defined by (h) = h·g−1, where the inverse ensures that (k) = ((k)). These homomorphisms are called the left and right regular representations of G. Each homomorphism is injective, a fact referred to as Cayley's theorem.
For example, if G = C3 = {1, x, x2 } is a cyclic group of order three, then
(1) = x·1 = x,
(x) = x·x = x2, and
(x2) = x·x2 = 1,
so λ(x) takes (1, x, x2) to (x, x2, 1).
The image of λ is a subgroup of Sym(G) isomorphic to G, and its normalizer in Sym(G) is defined to be the holomorph N of G.
For each n in N and g in G, there is an h in G such that n· = ·n. If an element n of the holomorph fixes the identity of G, then for 1 in G, (n·)(1) = (·n)(1), but the left hand side is n(g), and the right side is h. In other words, if n in N fixes the identity of G, then for every g in G, n· = ·n. If g, h are elements of G, and n is an element of N fixing the identity of G, then applying this equality twice to n·· and once to the (equivalent) expression n· gives that n(g)·n(h) = n(g·h). That is, every element of N that fixes the identity of G is in fact an automorphism of G. Such an n normalizes , and the only that fixes the identity is λ(1). Setting A to be the stabilizer of the identity, the subgroup generated by A and is semidirect product with normal subgroup and complement A. Since is transit
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https://en.wikipedia.org/wiki/Holomorph
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Holomorph may refer to:
Mathematics
Holomorph (mathematics), a group which simultaneously contains (copies of) a group and its automorphism group
Holomorphic functions, the central object of study of complex analysis
Biology
Teleomorph, anamorph and holomorph, applying to portions of the life cycles of fungi in the phyla Ascomycota and Basidiomycota
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https://en.wikipedia.org/wiki/Exist
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Exist may refer to:
Existence
eXist, an open source database management system built on XML
Existential quantification, in logic and mathematics (symbolized by ∃, read "exists")
Energetic X-ray Survey Telescope, a proposed hard X-ray imaging all-sky deep survey mission
Exist (album), a studio album by Exo
Exists (band), formerly Exist, a Malaysian band
Exists (film), a 2014 horror film
XIST (gene) X inactive specific transcript, a gene which inactivates extra copies of X-chromosomes.
See also
Existentialism
Existence (disambiguation), for other meanings of "existence" and "existential"
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https://en.wikipedia.org/wiki/Spin%20structure
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In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry.
Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry.
Overview
In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M has a spin structure. This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes. Furthermore, if w2(M) = 0, then the set of the isomorphism classes of spin structures on M is acted upon freely and transitively by H1(M, Z2) . As the manifold M is assumed to be oriented, the first Stiefel–Whitney class w1(M) ∈ H1(M, Z2) of M vanishes too. (The Stiefel–Whitney classes wi(M) ∈ Hi(M, Z2) of a manifold M are defined to be the Stiefel–Whitney classes of its tangent bundle TM.)
The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP: P → M of spin frames over M and the spin representation of its structure group Spin(n) on the space of spinors Δn. The bundle S is called the spinor bundle for a given spin structure on M.
A precise definition of spin structure on manifold was possible only after the notion of fiber bundle had been introduced; André Haefliger (1956) found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold and Max Karoubi (1968) extended this result to the non-orientable pseudo-Riemannian case.
Spin structures on Riemannian manifolds
Definition
A spin structure on an orientable Riemannian manifold with an oriented vector bundle is an equivariant lift of the orthonormal frame bundle with respect to the double covering . In other words, a pair is a spin structure on the SO(n)-principal bundle when
a) is a principal Spin(n)-bundle over , and
b) is an equivariant 2-fold covering map such thatandfor all and .
Two spin structures and on the same oriented Riemannian manifold are called "equivalent" if there exists a Spin(n)-equivariant map such that
and for all and .
In this case and are two equivalent double coverings.
The definition of spin structure on as a spin structure on the principal bundle is due to André Haefliger (1956).
Obstruction
Haefliger found necessary and sufficient conditions for the existence of a spin structure on an oriented Riemannian manifold (M,g). The obstruction to having a spin structu
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https://en.wikipedia.org/wiki/Square%20planar%20molecular%20geometry
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The square planar molecular geometry in chemistry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds. As the name suggests, molecules of this geometry have their atoms positioned at the corners.
Examples
Numerous compounds adopt this geometry, examples being especially numerous for transition metal complexes. The noble gas compound xenon tetrafluoride adopts this structure as predicted by VSEPR theory. The geometry is prevalent for transition metal complexes with d8 configuration, which includes Rh(I), Ir(I), Pd(II), Pt(II), and Au(III). Notable examples include the anticancer drugs cisplatin, [PtCl2(NH3)2], and carboplatin. Many homogeneous catalysts are square planar in their resting state, such as Wilkinson's catalyst and Crabtree's catalyst. Other examples include Vaska's complex and Zeise's salt. Certain ligands (such as porphyrins) stabilize this geometry.
Splitting of d-orbitals
A general d-orbital splitting diagram for square planar (D4h) transition metal complexes can be derived from the general octahedral (Oh) splitting diagram, in which the dz2 and the dx2−y2 orbitals are degenerate and higher in energy than the degenerate set of dxy, dxz and dyz orbitals. When the two axial ligands are removed to generate a square planar geometry, the dz2 orbital is driven lower in energy as electron-electron repulsion with ligands on the z-axis is no longer present. However, for purely σ-donating ligands the dz2 orbital is still higher in energy than the dxy, dxz and dyz orbitals because of the torus shaped lobe of the dz2 orbital. It bears electron density on the x- and y-axes and therefore interacts with the filled ligand orbitals. The dxy, dxz and dyz orbitals are generally presented as degenerate but they have to split into two different energy levels with respect to the irreducible representations of the point group D4h. Their relative ordering depends on the nature of the particular complex. Furthermore, the splitting of d-orbitals is perturbed by π-donating ligands in contrast to octahedral complexes. In the square planar case strongly π-donating ligands can cause the dxz and dyz orbitals to be higher in energy than the dz2 orbital, whereas in the octahedral case π-donating ligands only affect the magnitude of the d-orbital splitting and the relative ordering of the orbitals is conserved.
See also
AXE method
Molecular geometry
References
External links
3D Chem – Chemistry, Structures, and 3D Molecules
IUMSC – Indiana University Molecular Structure Center
Interactive molecular examples for point groups
– Coordination numbers and complex ions
Molecular geometry
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https://en.wikipedia.org/wiki/Trend%20line
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Trend line can refer to:
A linear regression in statistics
The result of trend estimation in statistics
Trend line (technical analysis), a tool in technical analysis
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https://en.wikipedia.org/wiki/Alexander%27s%20trick
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Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
Statement
Two homeomorphisms of the n-dimensional ball which agree on the boundary sphere are isotopic.
More generally, two homeomorphisms of Dn that are isotopic on the boundary are isotopic.
Proof
Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.
If satisfies , then an isotopy connecting f to the identity is given by
Visually, the homeomorphism is 'straightened out' from the boundary, 'squeezing' down to the origin. William Thurston calls this "combing all the tangles to one point". In the original 2-page paper, J. W. Alexander explains that for each the transformation replicates at a different scale, on the disk of radius , thus as it is reasonable to expect that merges to the identity.
The subtlety is that at , "disappears": the germ at the origin "jumps" from an infinitely stretched version of to the identity. Each of the steps in the homotopy could be smoothed (smooth the transition), but the homotopy (the overall map) has a singularity at . This underlines that the Alexander trick is a PL construction, but not smooth.
General case: isotopic on boundary implies isotopic
If are two homeomorphisms that agree on , then is the identity on , so we have an isotopy from the identity to . The map is then an isotopy from to .
Radial extension
Some authors use the term Alexander trick for the statement that every homeomorphism of can be extended to a homeomorphism of the entire ball .
However, this is much easier to prove than the result discussed above: it is called radial extension (or coning) and is also true piecewise-linearly, but not smoothly.
Concretely, let be a homeomorphism, then
defines a homeomorphism of the ball.
Exotic spheres
The failure of smooth radial extension and the success of PL radial extension
yield exotic spheres via twisted spheres.
See also
Clutching construction
References
Geometric topology
Homeomorphisms
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https://en.wikipedia.org/wiki/Algebraic%20semantics%20%28mathematical%20logic%29
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In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic. For example, the modal logic S4 is characterized by the class of topological boolean algebras—that is, boolean algebras with an interior operator. Other modal logics are characterized by various other algebras with operators. The class of boolean algebras characterizes classical propositional logic, and the class of Heyting algebras propositional intuitionistic logic. MV-algebras are the algebraic semantics of Łukasiewicz logic.
See also
Algebraic semantics (computer science)
Lindenbaum–Tarski algebra
Further reading
(2nd published by ASL in 2009) open access at Project Euclid
Good introduction for readers with prior exposure to non-classical logics but without much background in order theory and/or universal algebra; the book covers these prerequisites at length. The book, however, has been criticized for poor and sometimes incorrect presentation of abstract algebraic logic results.
Mathematical logic
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https://en.wikipedia.org/wiki/Unifying%20Theories%20of%20Programming
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Unifying Theories of Programming (UTP) in computer science deals with program semantics. It shows how denotational semantics, operational semantics and algebraic semantics can be combined in a unified framework for the formal specification, design and implementation of programs and computer systems.
The book of this title by C.A.R. Hoare and He Jifeng was published in the Prentice Hall International Series in Computer Science in 1998 and is now freely available on the web.
Theories
The semantic foundation of the UTP is the first-order predicate calculus, augmented with fixed point constructs from second-order logic. Following the tradition of Eric Hehner, programs are predicates in the UTP, and there is no distinction between programs and specifications at the semantic level. In the words of Hoare:
A computer program is identified with the strongest predicate describing every relevant observation that can be made of the behaviour of a computer executing that program.
In UTP parlance, a theory is a model of a particular programming paradigm. A UTP theory is composed of three ingredients:
an alphabet, which is a set of variable names denoting the attributes of the paradigm that can be observed by an external entity;
a signature, which is the set of programming language constructs intrinsic to the paradigm; and
a collection of healthiness conditions, which define the space of programs that fit within the paradigm. These healthiness conditions are typically expressed as monotonic idempotent predicate transformers.
Program refinement is an important concept in the UTP. A program is refined by if and only if every observation that can be made of is also an observation of .
The definition of refinement is common across UTP theories:
where denotes the universal closure of all variables in the alphabet.
Relations
The most basic UTP theory is the alphabetised predicate calculus, which has no alphabet restrictions or healthiness conditions. The theory of relations is slightly more specialised, since a relation's alphabet may consist of only:
undecorated variables (), modelling an observation of the program at the start of its execution; and
primed variables (), modelling an observation of the program at a later stage of its execution.
Some common language constructs can be defined in the theory of relations as follows:
The skip statement, which does not alter the program state in any way, is modelled as the relational identity:
The assignment of value to a variable is modelled as setting to and keeping all other variables (denoted by ) constant:
The sequential composition of two programs is just relational composition of intermediate state:
Non-deterministic choice between programs is their greatest lower bound:
Conditional choice between programs is written using infix notation:
A semantics for recursion is given by the least fixed point of a monotonic predicate transformer :
References
Further reading
External links
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https://en.wikipedia.org/wiki/He%20Jifeng
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He Jifeng (, born August 1943) is a Chinese computer scientist.
He Jifeng graduated from the mathematics department of Fudan University in 1965. From 1965 to 1985, he was an instructor at East China Normal University. During 1980–81, he was a visiting scholar at Stanford University and the University of San Francisco in California, United States.
From 1984 to 1998, He Jifeng was a senior research fellow at the Programming Research Group in the Oxford University Computing Laboratory (now the Oxford University Department of Computer Science). He worked extensively on formal aspects of computing science. In particular, he worked with Prof. Sir Tony Hoare, latterly on Unifying Theories of Programming, resulting in a book of that name.
Since 1986, He Jifeng has been Professor of Computer Science at East China Normal University in Shanghai. In 1996, he also became Professor of Computer Science at Shanghai Jiao Tong University.
In 1998, he became a senior research fellow at the International Institute for Software Technology (UNU-IIST), United Nations University, based in Macau. He moved back to Shanghai in 2005.
He Jifeng's research interests include sound methods for the specification of computer systems, communications, applications, standards, and techniques for designing and implementing those specifications in software and/or hardware with high reliability.
In 2005, he was elected to the Chinese Academy of Sciences. In 2013, his 70th birthday was celebrated at East China Normal University with an international three-day Festschrift in association with the International Conference on Theoretical Aspects of Computing (ICTAC). Ten years later in 2023, his 80th birthday was celebrated at the Shanghai Science Hall with a hybrid international two-day Festschrift Symposium. Since 2019, he has been a Distinguished Professor at Tongji University in Shanghai.
Books
He Jifeng has written a number of computer science books, including:
He Jifeng, Provably Correct Systems: Modelling of Communication Languages and Design of Optimized Compilers. McGraw-Hill International Series in Software, 1995. .
C.A.R. Hoare and He Jifeng, Unified Theories of Programming. Prentice Hall International Series in Computer Science, 1998. .
Zhiming Liu and He Jifeng, Mathematical Frameworks for Component Software: Models for Analysis and Synthesis. World Scientific Publishing Company, Series on Component-Based Software Development, 2007. .
References
External links
Shanghai Huake Zhigu Artificial Intelligence Research Institute led by He Jifeng
He Jifeng homepage at ECNU, archived in 2012
1943 births
Living people
Chinese computer scientists
Chinese technology writers
Computer science writers
Academic staff of the East China Normal University
Educators from Shanghai
Formal methods people
Fudan University alumni
Members of the Department of Computer Science, University of Oxford
Members of the Chinese Academy of Sciences
Scientists from Shanghai
Academic staff of
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https://en.wikipedia.org/wiki/List%20of%20uniform%20polyhedra
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In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.
This list includes these:
all 75 nonprismatic uniform polyhedra;
a few representatives of the infinite sets of prisms and antiprisms;
one degenerate polyhedron, Skilling's figure with overlapping edges.
It was proven in that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide.
Not included are:
The uniform polyhedron compounds.
40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
The uniform tilings (infinite polyhedra)
11 Euclidean convex uniform tilings;
28 Euclidean nonconvex or apeirogonal uniform tilings;
Infinite number of uniform tilings in hyperbolic plane.
Any polygons or 4-polytopes
Indexing
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
[C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
[W] Wenninger, 1974, has 119 figures: 1–5 for the Platonic solids, 6–18 for the Archimedean solids, 19–66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67–119 for the nonconvex uniform polyhedra.
[K] Kaleido, 1993: The 80 figures were grouped by symmetry: 1–5 as representatives of the infinite families of prismatic forms with dihedral symmetry, 6–9 with tetrahedral symmetry, 10–26 with octahedral symmetry, 27–80 with icosahedral symmetry.
[U] Mathematica, 1993, follows the Kaleido series with the 5 prismatic forms moved to last, so that the nonprismatic forms become 1–75.
Names of polyhedra by number of sides
There are generic geometric names for the most common polyhedra. The 5 Platonic solids are called a tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron with 4, 6, 8, 12, and 20 sides respectively. The regular hexahedron is a cube.
Table of polyhedra
The convex forms are listed in order of degree of vertex configurations from 3 faces/vertex and up, and in increasing sides per face. This ordering allows topological similarities to be shown.
There are infinitely many prisms and antiprisms, one for each regular polygon; the ones up to the 12-gonal cases are listed.
Convex uniform polyhe
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https://en.wikipedia.org/wiki/Riemannian%20submersion
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In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.
Formal definition
Let (M, g) and (N, h) be two Riemannian manifolds and a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution is a sub-bundle of the tangent bundle of which depends both on the projection and on the metric .
Then, f is called a Riemannian submersion if and only if, for all , the vector space isomorphism is isometric, i.e., length-preserving.
Examples
An example of a Riemannian submersion arises when a Lie group acts isometrically, freely and properly on a Riemannian manifold .
The projection to the quotient space equipped with the quotient metric is a Riemannian submersion.
For example, component-wise multiplication on by the group of unit complex numbers yields the Hopf fibration.
Properties
The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O'Neill's formula, named for Barrett O'Neill:
where are orthonormal vector fields on , their horizontal lifts to , is the Lie bracket of vector fields and is the projection of the vector field to the vertical distribution.
In particular the lower bound for the sectional curvature of is at least as big as the lower bound for the sectional curvature of .
Generalizations and variations
Fiber bundle
Submetry
co-Lipschitz map
See also
Fibered manifold
Geometric topology
Manifold
Notes
References
.
Barrett O'Neill. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469.
Riemannian geometry
Maps of manifolds
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https://en.wikipedia.org/wiki/Busemann%20function
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In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature). They are named after Herbert Busemann, who introduced them; he gave an extensive treatment of the topic in his 1955 book "The geometry of geodesics".
Definition and elementary properties
Let be a metric space. A geodesic ray is a path which minimizes distance everywhere along its length. i.e., for all ,
Equivalently, a ray is an isometry from the "canonical ray" (the set equipped with the Euclidean metric) into the metric space X.
Given a ray γ, the Busemann function is defined by
Thus, when t is very large, the distance is approximately equal to . Given a ray γ, its Busemann function is always well-defined: indeed the right hand side above , tends pointwise to the left hand side on compacta, since is bounded above by and non-increasing since, if ,
It is immediate from the triangle inequality that
so that is uniformly continuous. More specifically, the above estimate above shows that
Busemann functions are Lipschitz functions with constant 1.
By Dini's theorem, the functions tend to uniformly on compact sets as t tends to infinity.
Example: Poincaré disk
Let be the unit disk in the complex plane with the Poincaré metric
Then, for and , the Busemann function is given by
where the term in brackets on the right hand side is the Poisson kernel for the unit disk and corresponds to the radial geodesic from the origin towards ,
. The computation of can be reduced to that of , since the metric is invariant under Möbius transformations in ; the geodesics through have the form where is the 1-parameter subgroup of ,
The formula above also completely determines the Busemann function by Möbius invariance.
Busemann functions on a Hadamard space
In a Hadamard space, where any two points are joined by a unique geodesic segment, the function is convex, i.e. convex on geodesic segments . Explicitly this means that if is the point which divides in the ratio , then . For fixed the function is convex and hence so are its translates; in particular, if is a geodesic ray in , then is convex. Since the Busemann function is the pointwise limit of ,
Busemann functions are convex on Hadamard spaces.
On a Hadamard space, the functions converge uniformly to uniformly on any bounded subset of .
Let . Since is parametrised by arclength, Alexandrov's first comparison theorem for Hadamard spaces implies that the function is convex. Hence for
Thus
so that
Letting t tend to ∞, it follows that
so convergence is uniform on bounded sets.
Note that the inequality above for (together with its proof) also holds for geodesic segments: if is a geodesic segment starting at and parametrised by arclength then
Next suppose that are points in a Hadamard space, and let be the geodesic through with and ,
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https://en.wikipedia.org/wiki/Conjugate%20points
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In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are locally length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) globally length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.
Definition
Suppose p and q are points on a Riemannian manifold, and is a geodesic that connects p and q. Then p and q are conjugate points along if there exists a non-zero Jacobi field along that vanishes at p and q.
Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on Jacobi fields). Therefore, if p and q are conjugate along , one can construct a family of geodesics that start at p and almost end at q. In particular,
if is the family of geodesics whose derivative in s at generates the Jacobi field J, then the end point
of the variation, namely , is the point q only up to first order in s. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.
Examples
On the sphere , antipodal points are conjugate.
On the real coordinate space , there are no conjugate points.
On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.
See also
Cut locus
Jacobi field
References
Riemannian geometry
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https://en.wikipedia.org/wiki/Hadamard%20space
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In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces.
A Hadamard space is defined to be a nonempty complete metric space such that, given any points and there exists a point such that for every point
The point is then the midpoint of and
In a Hilbert space, the above inequality is equality (with ), and in general an Hadamard space is said to be if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space.
The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in particular, it is contractible. Quite generally, if is a bounded subset of a metric space, then the center of the closed ball of the minimum radius containing it is called the circumcenter of Every bounded subset of a Hadamard space is contained in the smallest closed ball (which is the same as the closure of its convex hull). If is the group of isometries of a Hadamard space leaving invariant then fixes the circumcenter of (Bruhat–Tits fixed point theorem).
The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. The analog holds for a Hadamard space: a complete, connected metric space which is locally isometric to a Hadamard space has an Hadamard space as its universal cover. Its variant applies for non-positively curved orbifolds. (cf. Lurie.)
Examples of Hadamard spaces are Hilbert spaces, the Poincaré disc, complete real trees (for example, complete Bruhat–Tits building), -space with and and Hadamard manifolds, that is, complete simply-connected Riemannian manifolds of nonpositive sectional curvature. Important examples of Hadamard manifolds are simply connected nonpositively curved symmetric spaces.
Applications of Hadamard spaces are not restricted to geometry. In 1998, Dmitri Burago and Serge Ferleger used CAT(0) geometry to solve a problem in dynamical billiards: in a gas of hard balls, is there a uniform bound on the number of collisions? The solution begins by constructing a configuration space for the dynamical system, obtained by joining together copies of corresponding billiard table, which turns out to be an Hadamard space.
See also
References
Burago, Dmitri; Yuri Burago, and Sergei Ivanov. A Course in Metric Geometry. American Mathematical Society. (1984)
Jacob Lurie: Notes on the Theory of Hadamard Spaces
Alexander S., Kapovich V., Petrunin A. Notes on Alexandrov Geometry
Functional analysis
Geometric topology
Hilbert spaces
Metric spaces
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https://en.wikipedia.org/wiki/Five-dimensional%20space
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A five-dimensional space is a space with five dimensions. In mathematics, a sequence of N numbers can represent a location in an N-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions and the fourth dimension of time used in relativistic physics. Whether or not the universe is five-dimensional is a topic of debate.
Physics
Much of the early work on five-dimensional space was in an attempt to develop a theory that unifies the four fundamental interactions in nature: strong and weak nuclear forces, gravity and electromagnetism. German mathematician Theodor Kaluza and Swedish physicist Oskar Klein independently developed the Kaluza–Klein theory in 1921, which used the fifth dimension to unify gravity with electromagnetic force. Although their approaches were later found to be at least partially inaccurate, the concept provided a basis for further research over the past century.
To explain why this dimension would not be directly observable, Klein suggested that the fifth dimension would be rolled up into a tiny, compact loop on the order of 10 centimeters. Under his reasoning, he envisioned light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops. While not detectable, it would indirectly imply a connection between seemingly unrelated forces. The KaluzaKlein theory experienced a revival in the 1970s due to the emergence of superstring theory and supergravity: the concept that reality is composed of vibrating strands of energy, a postulate only mathematically viable in ten dimensions or more. Superstring theory then evolved into a more generalized approach known as M-theory. M-theory suggested a potentially observable extra dimension in addition to the ten essential dimensions which would allow for the existence of superstrings. The other 10 dimensions are compacted, or "rolled up", to a size below the subatomic level. The KaluzaKlein theory today is seen as essentially a gauge theory, with the gauge being the circle group.
The fifth dimension is difficult to directly observe, though the Large Hadron Collider provides an opportunity to record indirect evidence of its existence. Physicists theorize that collisions of subatomic particles in turn produce new particles as a result of the collision, including a graviton that escapes from the fourth dimension, or brane, leaking off into a five-dimensional bulk. M-theory would explain the weakness of gravity relative to the other fundamental forces of nature, as can be seen, for example, when using a magnet to lift a pin off a table—the magnet is able to overcome the gravitational pull of the entire earth with ease.
Mathematical approaches were developed in the early 20th century that viewed the fifth dimension as a theoretical construct. These theories make reference to Hilbert space, a concept that pos
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https://en.wikipedia.org/wiki/Tetrahemihexahedron
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In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).
The tetrahemihexahedron is the only non-prismatic uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/2 3 | 2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired and coinciding in space. (It can more intuitively be seen as two coinciding tetrahemihexahedra.)
The tetrahemihexahedron is a hemipolyhedron. The "hemi" part of the name means some of the faces form a group with half as many members as some regular polyhedron—here, three square faces form a group with half as many faces as the regular hexahedron, better known as the cube—hence hemihexahedron. Hemi faces are also oriented in the same direction as the regular polyhedron's faces. The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular.
The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. Visually, each square is divided into four right triangles, with two visible from each side.
Related surfaces
The tetrahemihexahedron is a non-orientable surface. It is unique as the only uniform polyhedron with an Euler characteristic of 1 and is hence a projective polyhedron, yielding a representation of the real projective plane very similar to the Roman surface.
Related polyhedra
The tetrahemihexahedron has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, but has three additional square faces passing through the centre of the polyhedron.
The dual figure of the tetrahemihexahedron is the tetrahemihexacron.
The tetrahemihexahedron is 2-covered by the cuboctahedron, which accordingly has the same abstract vertex figure (2 triangles and two squares: 3.4.3.4) and twice the vertices, edges, and faces. It has the same topology as the abstract polyhedron hemi-cuboctahedron.
The tetrahemihexahedron may also be constructed as a crossed triangular cuploid. All cuploids and their duals are topologically projective planes.
Tetrahemihexacron
The tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra.
Since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity; properly, on the real projective plane at infinity. In Magnus Wenninger's Dual Models, they are represented with intersecting prisms, each extending in both directions to the same vertex at infinity, in order to maintain symmetry. In practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger sugg
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https://en.wikipedia.org/wiki/Whitney%20disk
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In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.
Their existence and embeddedness is crucial in proving the cobordism theorem, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc. Casson handles are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological four-manifolds.
Pseudoholomorphic Whitney discs are counted by the differential in Lagrangian intersection Floer homology.
References
Geometric topology
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https://en.wikipedia.org/wiki/Leray%20cover
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In mathematics, a Leray cover(ing) is a cover of a topological space which allows for easy calculation of its cohomology. Such covers are named after Jean Leray.
Sheaf cohomology measures the extent to which a locally exact sequence on a fixed topological space, for instance the de Rham sequence, fails to be globally exact. Its definition, using derived functors, is reasonably natural, if technical. Moreover, important properties, such as the existence of a long exact sequence in cohomology corresponding to any short exact sequence of sheaves, follow directly from the definition. However, it is virtually impossible to calculate from the definition. On the other hand, Čech cohomology with respect to an open cover is well-suited to calculation, but of limited usefulness because it depends on the open cover chosen, not only on the sheaves and the space. By taking a direct limit of Čech cohomology over arbitrarily fine covers, we obtain a Čech cohomology theory that does not depend on the open cover chosen. In reasonable circumstances (for instance, if the topological space is paracompact), the derived-functor cohomology agrees with this Čech cohomology obtained by direct limits. However, like the derived functor cohomology, this cover-independent Čech cohomology is virtually impossible to calculate from the definition. The Leray condition on an open cover ensures that the cover in question is already "fine enough." The derived functor cohomology agrees with the Čech cohomology with respect to any Leray cover.
Let be an open cover of the topological space , and a sheaf on X. We say that is a Leray cover with respect to if, for every nonempty finite set of indices, and for all , we have that , in the derived functor cohomology. For example, if is a separated scheme, and is quasicoherent, then any cover of by open affine subschemes is a Leray cover.
References
Sheaf theory
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https://en.wikipedia.org/wiki/Ambient%20space%20%28mathematics%29
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In mathematics, especially in geometry and topology, an ambient space is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line may be studied in isolation —in which case the ambient space of is , or it may be studied as an object embedded in 2-dimensional Euclidean space —in which case the ambient space of is , or as an object embedded in 2-dimensional hyperbolic space —in which case the ambient space of is . To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is , but false if the ambient space is , because the geometric properties of are different from the geometric properties of . All spaces are subsets of their ambient space.
See also
Configuration space
Geometric space
Manifold and ambient manifold
Submanifolds and Hypersurfaces
Riemannian manifolds
Ricci curvature
Differential form
References
Further reading
Geometry
Topology
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https://en.wikipedia.org/wiki/Multi-armed%20bandit
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In probability theory and machine learning, the multi-armed bandit problem (sometimes called the K- or N-armed bandit problem) is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices in a way that maximizes their expected gain, when each choice's properties are only partially known at the time of allocation, and may become better understood as time passes or by allocating resources to the choice. This is a classic reinforcement learning problem that exemplifies the exploration–exploitation tradeoff dilemma. The name comes from imagining a gambler at a row of slot machines (sometimes known as "one-armed bandits"), who has to decide which machines to play, how many times to play each machine and in which order to play them, and whether to continue with the current machine or try a different machine. The multi-armed bandit problem also falls into the broad category of stochastic scheduling.
In the problem, each machine provides a random reward from a probability distribution specific to that machine, that is not known a-priori. The objective of the gambler is to maximize the sum of rewards earned through a sequence of lever pulls. The crucial tradeoff the gambler faces at each trial is between "exploitation" of the machine that has the highest expected payoff and "exploration" to get more information about the expected payoffs of the other machines. The trade-off between exploration and exploitation is also faced in machine learning. In practice, multi-armed bandits have been used to model problems such as managing research projects in a large organization, like a science foundation or a pharmaceutical company. In early versions of the problem, the gambler begins with no initial knowledge about the machines.
Herbert Robbins in 1952, realizing the importance of the problem, constructed convergent population selection strategies in "some aspects of the sequential design of experiments". A theorem, the Gittins index, first published by John C. Gittins, gives an optimal policy for maximizing the expected discounted reward.
Empirical motivation
The multi-armed bandit problem models an agent that simultaneously attempts to acquire new knowledge (called "exploration") and optimize their decisions based on existing knowledge (called "exploitation"). The agent attempts to balance these competing tasks in order to maximize their total value over the period of time considered. There are many practical applications of the bandit model, for example:
clinical trials investigating the effects of different experimental treatments while minimizing patient losses,
adaptive routing efforts for minimizing delays in a network,
financial portfolio design
In these practical examples, the problem requires balancing reward maximization based on the knowledge already acquired with attempting new actions to further increase knowledge. This is known as the exploitation vs. exploration tradeoff in machine learnin
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https://en.wikipedia.org/wiki/Caleb%20Gattegno
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Caleb Gattegno (1911–1988) was an Egyptian educator, psychologist, and mathematician. He is considered one of the most influential and prolific mathematics educators of the twentieth century. He is best known for introducing new approaches to teaching and learning mathematics (Visible & Tangible Math), foreign languages (The Silent Way) and reading (Words in Color). Gattegno also developed pedagogical materials for each of these approaches, and was the author of more than 120 books and hundreds of articles largely on the topics of education and human development.
Background
Gattegno was born November 11, 1911, in Alexandria, Egypt. His parents, Menachem Gattegno, a Spanish merchant, and his wife, Bchora, had nine children. Because of poverty, Gattegno and his siblings had to work starting from a young age. The future mathematician had no formal education until he started to learn on his own at the age of 14. He took external examinations when he was 20 years old and obtained a teaching license in physics and chemistry from the University of Marseille in Cairo.
He moved to England, where he became involved in teacher education and helped establish the Association of Teachers of Mathematics and the International Commission for the Study and Improvement of Mathematics Teaching. He taught at several universities including the University of Liverpool and the University of London.
Pedagogical approach
Gattegno's pedagogical approach is characterised by propositions based on the observation of human learning in many and varied situations. This is a description of three of these propositions. He was also influenced by the works of Jean Piaget and worked on introducing the implications of the latter's cognitive theory on education.
Learning and effort
Gattegno noticed that there is an "energy budget" for learning. Human beings have a highly developed sense of the economics of their own energy and are very sensitive to the cost involved in using it. It is therefore essential to teach in ways that are efficient in terms of the amount of energy spent by learners. To be able to quantitively determine whether one method was more efficient than another, he created a unit of measurement for the effort used to learn. He called that unit an ogden, and one can only say an ogden has been spent if the learning was done outside of ordinary functionings, and was retained. For example, learning one word in a foreign language costs one ogden, but if the word cannot be recalled, the ogden has not truly been spent. Gattegno's teaching materials and techniques were designed to be economical with ogdens, so that the greatest amount of information can be recalled with the least sense of effort. In 1970s, he collaborated with the film maker Joeseph Koenig. They produced one-minute television films that featured animation contents that presented: 1) raw data on how the English language works; 2) the language's spatial ordering; 3) the effects of transformations; and, 4) t
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https://en.wikipedia.org/wiki/IMSL%20Numerical%20Libraries
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IMSL (International Mathematics and Statistics Library) is a commercial collection of software libraries of numerical analysis functionality that are implemented in the computer programming languages C, Java, C#.NET, and Fortran. A Python interface is also available.
The IMSL Libraries were developed by Visual Numerics, which was acquired in 2009 by Rogue Wave Software, which was acquired in 2019 by Minneapolis, Minnesotabased application software developer Perforce.
Version history
The first IMSL Library for the Fortran language was released in 1970, followed by a C-language version originally called C/Base in 1991, a Java-language version in 2002 and the C#-language version in 2004.
Several recent product releases have involved making IMSL Library functions available from Python. These releases are Python wrappers to IMSL C Library functions (PyIMSL wrappers) and PyIMSL Studio, a prototyping and production application development environment based on Python and the IMSL C Library. The PyIMSL wrappers were first released in August 2008. PyIMSL Studio was introduced in February 2009. PyIMSL Studio is available for download at no charge for non-commercial use or for commercial evaluation.
Current versions:
IMSL C Library V 8.0 – November 2011
IMSL C# Library V 6.5.2 – November 2015 (end of life announced as end of 2020)
IMSL Fortran Library V 7.0 – October 2010
PyIMSL Studio V 1.5 – August 2009
PyIMSL wrappers V 1.5 – August 2009
JMSL Library V 6.1 – August 2010
Platform availability
The IMSL Numerical Libraries are supported on various operating systems, hardware and compilers.
Operating system support includes Unix, Linux, Mac OS and Microsoft Windows
Hardware support includes AMD, Intel, Apple Inc., Cray, Fujitsu, Hitachi, HP, IBM, NEC, SGI and Sun Microsystems
Compiler support includes Absoft, GCC, Intel, Microsoft, and Portland
See also
List of numerical-analysis software
List of numerical libraries
References
External links
The IMSL Numerical Libraries home page
Fortran libraries
Numerical libraries
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https://en.wikipedia.org/wiki/Noncentral%20chi%20distribution
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In probability theory and statistics, the noncentral chi distribution is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.
Definition
If are k independent, normally distributed random variables with means and variances , then the statistic
is distributed according to the noncentral chi distribution. The noncentral chi distribution has two parameters: which specifies the number of degrees of freedom (i.e. the number of ), and which is related to the mean of the random variables by:
Properties
Probability density function
The probability density function (pdf) is
where is a modified Bessel function of the first kind.
Raw moments
The first few raw moments are:
where is a Laguerre function. Note that the 2th moment is the same as the th moment of the noncentral chi-squared distribution with being replaced by .
Bivariate non-central chi distribution
Let , be a set of n independent and identically distributed bivariate normal random vectors with marginal distributions , correlation , and mean vector and covariance matrix
with positive definite. Define
Then the joint distribution of U, V is central or noncentral bivariate chi distribution with n degrees of freedom.
If either or both or the distribution is a noncentral bivariate chi distribution.
Related distributions
If is a random variable with the non-central chi distribution, the random variable will have the noncentral chi-squared distribution. Other related distributions may be seen there.
If is chi distributed: then is also non-central chi distributed: . In other words, the chi distribution is a special case of the non-central chi distribution (i.e., with a non-centrality parameter of zero).
A noncentral chi distribution with 2 degrees of freedom is equivalent to a Rice distribution with .
If X follows a noncentral chi distribution with 1 degree of freedom and noncentrality parameter λ, then σX follows a folded normal distribution whose parameters are equal to σλ and σ2 for any value of σ.
References
Continuous distributions
Noncentral distributions
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https://en.wikipedia.org/wiki/Affine%20involution
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In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space Rn. Such involutions are easy to characterize and they can be described geometrically.
Linear involutions
To give a linear involution is the same as giving an involutory matrix, a square matrix A such that
where I is the identity matrix.
It is a quick check that a square matrix D whose elements are all zero off the main diagonal and ±1 on the diagonal, that is, a signature matrix of the form
satisfies (1), i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying (1) are of the form
A=U −1DU,
where U is invertible and D is as above. That is to say, the matrix of any linear involution is of the form D up to a matrix similarity. Geometrically this means that any linear involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through the origin. (The term oblique reflection as used here includes ordinary reflections.)
One can easily verify that A represents a linear involution if and only if A has the form
A = ±(2P - I)
for a linear projection P.
Affine involutions
If A represents a linear involution, then x→A(x−b)+b is an affine involution. One can check that any affine involution in fact has this form. Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through n hyperplanes going through a point b.
Affine involutions can be categorized by the dimension of the affine space of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix D (see above), i.e., the dimension of the eigenspace for eigenvalue 1.
The affine involutions in 3D are:
the identity
the oblique reflection in respect to a plane
the oblique reflection in respect to a line
the reflection in respect to a point.
Isometric involutions
In the case that the eigenspace for eigenvalue 1 is the orthogonal complement of that for eigenvalue −1, i.e., every eigenvector with eigenvalue 1 is orthogonal to every eigenvector with eigenvalue −1, such an affine involution is an isometry. The two extreme cases for which this always applies are the identity function and inversion in a point.
The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a reflection, and in 3D a rotation about the line by 180°), inversion in a plane (in 3D and up; in 3D this is a reflection in a plane), inversion in a 3D space (in 3D: the identity), etc.
References
Affine geometry
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https://en.wikipedia.org/wiki/Triangular%20prism
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In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.
Equivalently, it is a polyhedron of which two faces are parallel, while the surface normals of the other three are in the same plane (which is not necessarily parallel to the base planes). These three faces are parallelograms. All cross-sections parallel to the base faces are the same triangle.
As a semiregular (or uniform) polyhedron
A right triangular prism is semiregular or, more generally, a uniform polyhedron if the base faces are equilateral triangles, and the other three faces are squares. It can be seen as a truncated trigonal hosohedron, represented by Schläfli symbol t{2,3}. Alternately it can be seen as the Cartesian product of a triangle and a line segment, and represented by the product, The dual of a triangular prism is a triangular bipyramid.
The symmetry group of a right 3-sided prism with triangular base is D3h of order 12. The rotation group is D3 of order 6. The symmetry group does not contain inversion.
Volume
The volume of any prism is the product of the area of the base and the distance between the two bases. In this case the base is a triangle so we simply need to compute the area of the triangle and multiply this by the length of the prism:
where is the length of one side of the triangle, is the length of an altitude drawn to that side, and is the distance between the triangular faces.
Truncated triangular prism
A truncated right triangular prism has one triangular face truncated (planed) at an oblique angle.
The volume of a truncated triangular prism with base area A and the three heights h1, h2, and h3 is determined by
Facetings
There are two full D3h symmetry facetings of a triangular prism, both with 6 isosceles triangle faces, one keeping the original top and bottom triangles, and one the original squares. Two lower C3v symmetry facetings have one base triangle, 3 lateral crossed square faces, and 3 isosceles triangle lateral faces.
Related polyhedra and tilings
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
Compounds
There are 4 uniform compounds of triangular prisms:
Compound of four triangular prisms, compound of eight triangular prisms, compound of ten triangular prisms, compound of twenty triangular prisms.
Honeycombs
There are 9 uniform honeycombs that include triangular prism cells:
Gyroelongated
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https://en.wikipedia.org/wiki/Pentagonal%20prism
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In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices.
As a semiregular (or uniform) polyhedron
If faces are all regular, the pentagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t{2,5}. Alternately it can be seen as the Cartesian product of a regular pentagon and a line segment, and represented by the product {5}×{}. The dual of a pentagonal prism is a pentagonal bipyramid.
The symmetry group of a right pentagonal prism is D5h of order 20. The rotation group is D5 of order 10.
Volume
The volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. For a uniform pentagonal prism with edges h the formula is
Use
Nonuniform pentagonal prisms called pentaprisms are also used in optics to rotate an image through a right angle without changing its chirality.
In 4-polytopes
It exists as cells of four nonprismatic uniform 4-polytopes in four dimensions:
Related polyhedra
External links
Pentagonal Prism Polyhedron Model -- works in your web browser
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Hexagonal%20prism
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In geometry, the hexagonal prism is a prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.
Since it has 8 faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces. Because of the ambiguity of the term octahedron and tilarity of the various eight-sided figures, the term is rarely used without clarification.
Before sharpening, many pencils take the shape of a long hexagonal prism.
As a semiregular (or uniform) polyhedron
If faces are all regular, the hexagonal prism is a semiregular polyhedron, more generally, a uniform polyhedron, and the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. It can be seen as a truncated hexagonal hosohedron, represented by Schläfli symbol t{2,6}. Alternately it can be seen as the Cartesian product of a regular hexagon and a line segment, and represented by the product {6}×{}. The dual of a hexagonal prism is a hexagonal bipyramid.
The symmetry group of a right hexagonal prism is D6h of order 24. The rotation group is D6 of order 12.
Volume
As in most prisms, the volume is found by taking the area of the base, with a side length of , and multiplying it by the height , giving the formula:
and its surface area can be
.
Symmetry
The topology of a uniform hexagonal prism can have geometric variations of lower symmetry, including:
As part of spatial tesselations
It exists as cells of four prismatic uniform convex honeycombs in 3 dimensions:
It also exists as cells of a number of four-dimensional uniform 4-polytopes, including:
Related polyhedra and tilings
This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
See also
References
External links
Uniform Honeycombs in 3-Space VRML models
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra Prisms and antiprisms
Hexagonal Prism Interactive Model -- works in your web browser
Prismatoid polyhedra
Space-filling polyhedra
Zonohedra
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https://en.wikipedia.org/wiki/Square%20antiprism
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In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube.
If all its faces are regular, it is a semiregular polyhedron or uniform polyhedron.
A nonuniform D4-symmetric variant is the cell of the noble square antiprismatic 72-cell.
Points on a sphere
When eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, the resulting shape corresponds to a square antiprism rather than a cube. Specific methods of distributing the points include, for example, the Thomson problem (minimizing the sum of all the reciprocals of distances between points), maximising the distance of each point to the nearest point, or minimising the sum of all reciprocals of squares of distances between points.
Molecules with square antiprismatic geometry
According to the VSEPR theory of molecular geometry in chemistry, which is based on the general principle of maximizing the distances between points, a square antiprism is the favoured geometry when eight pairs of electrons surround a central atom. One molecule with this geometry is the octafluoroxenate(VI) ion () in the salt nitrosonium octafluoroxenate(VI); however, the molecule is distorted away from the idealized square antiprism. Very few ions are cubical because such a shape would cause large repulsion between ligands; is one of the few examples.
In addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The S8 molecule has a structure based on the square antiprism, in which the eight atoms occupy the eight vertices of the antiprism, and the eight triangle-triangle edges of the antiprism correspond to single covalent bonds between sulfur atoms.
In architecture
The main building block of the One World Trade Center (at the site of the old World Trade Center destroyed on September 11, 2001) has the shape of an extremely tall tapering square antiprism. It is not a true antiprism because of its taper: the top square has half the area of the bottom one.
Topologically identical polyhedra
Twisted prism
A twisted prism can be made (clockwise or counterclockwise) with the same vertex arrangement. It can be seen as the convex form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices. It has half of the symmetry of the uniform solution: D4 order 4.
Crossed antiprism
A crossed square antiprism is a star polyhedron, topologically identical to the square antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.4, with one triangle retrograde. It has d4d symmetry, order 8.
Related polyhedra
Derived polyhedra
The gyroelongated square pyramid is a Johnson solid (specifically, J10) constructed by augmenting one a sq
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https://en.wikipedia.org/wiki/Pentagonal%20antiprism
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In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of ten triangles for a total of twelve faces. Hence, it is a non-regular dodecahedron.
Geometry
If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. It can also be considered as a parabidiminished icosahedron, a shape formed by removing two pentagonal pyramids from a regular icosahedron leaving two nonadjacent pentagonal faces; a related shape, the metabidiminished icosahedron (one of the Johnson solids), is likewise form from the icosahedron by removing two pyramids, but its pentagonal faces are adjacent to each other. The two pentagonal faces of either shape can be augmented with pyramids to form the icosahedron.
Relation to polytopes
The pentagonal antiprism occurs as a constituent element in some higher-dimensional polytopes. Two rings of ten pentagonal antiprisms each bound the hypersurface of the four-dimensional grand antiprism. If these antiprisms are augmented with pentagonal prism pyramids and linked with rings of five tetrahedra each, the 600-cell is obtained.
See also
The pentagonal antiprism can be truncated and alternated to form a snub antiprism:
Crossed antiprism
A crossed pentagonal antiprism is topologically identical to the pentagonal antiprism, although it can't be made uniform. The sides are isosceles triangles. It has d5d symmetry, order 10. Its vertex configuration is 3.3/2.3.5, with one triangle retrograde and its vertex arrangement is the same as a pentagonal prism.
External links
Pentagonal Antiprism: Interactive Polyhedron Model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A5
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Hexagonal%20antiprism
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In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 6-sided base, one usually considers the case where its copy is twisted by an angle . Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two bases and, connecting those bases, isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
Crossed antiprism
A crossed hexagonal antiprism is a star polyhedron, topologically identical to the convex hexagonal antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.6, with one triangle retrograde. It has D6d symmetry, order 24.
Related polyhedra
The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.
External links
Hexagonal Antiprism: Interactive Polyhedron model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A6
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Great%20dodecahedron
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In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex.
The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.
The great dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -pentagonal polytope faces of the core -polytope (pentagons for the great dodecahedron, and line segments for the pentagram) until the figure again closes.
Images
Related polyhedra
It shares the same edge arrangement as the convex regular icosahedron; the compound with both is the small complex icosidodecahedron.
If only the visible surface is considered, it has the same topology as a triakis icosahedron with concave pyramids rather than convex ones. The excavated dodecahedron can be seen as the same process applied to a regular dodecahedron, although this result is not regular.
A truncation process applied to the great dodecahedron produces a series of nonconvex uniform polyhedra. Truncating edges down to points produces the dodecadodecahedron as a rectified great dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the small stellated dodecahedron.
Usage
This shape was the basis for the Rubik's Cube-like Alexander's Star puzzle.
The great dodecahedron provides an easy mnemonic for the binary Golay code
See also
Compound of small stellated dodecahedron and great dodecahedron
References
External links
Uniform polyhedra and duals
Metal sculpture of Great Dodecahedron
Kepler–Poinsot polyhedra
Regular polyhedra
Polyhedral stellation
Toroidal polyhedra
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https://en.wikipedia.org/wiki/Pentagrammic%20prism
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In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.
It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex pentagonal prism.
It is the 78th model in the list of uniform polyhedra, as the first representative of uniform star prisms, along with the pentagrammic antiprism, which is the 79th model.
Geometry
It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U78 as a uniform polyhedron.
The triangle face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered a angel or exterior depending on how the interior is defined. One definition of the interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the diameter
Gallery
Pentagrammic dipyramid
In geometry, the pentagrammic dipyramid (or bipyramid) is first of the infinite set of face-transitive star dipyramids containing star polygon arrangement of edges. It has 10 intersecting isosceles triangle faces. It is topologically identical to the pentagonal dipyramid.
Each star dipyramid is the dual of a star polygon based uniform prism.
Related polyhedra
There are two pentagrammic trapezohedra (or deltohedra), being dual to the pentagrammic antiprism and pentagrammic crossed antiprism respectively, each having intersecting kite-shaped faces (convex or concave), and a total of 12 vertices:
References
External links
http://www.mathconsult.ch/showroom/unipoly/78.html
http://bulatov.org/polyhedra/uniform/u03.html
Paper model of pentagrammic prism
https://web.archive.org/web/20050313234702/http://www.math.technion.ac.il/~rl/kaleido/data/03.html
https://web.archive.org/web/20060211140715/http://www.ac-noumea.nc/maths/amc/polyhedr/no_conv5_.htm
Paper Model (net) Pentagrammic Prism
Prismatoid polyhedra
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Variational%20methods%20in%20general%20relativity
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Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity. The most commonly used tools are Lagrangians and Hamiltonians and are used to derive the Einstein field equations.
Lagrangian methods
The equations of motion in physical theories can often be derived from an object called the Lagrangian. In classical mechanics, this object is usually of the form, 'kinetic energy − potential energy'. In general, the Lagrangian is that function which when integrated over produces the Action functional.
David Hilbert gave an early and classic formulation of the equations in Einstein's general relativity. This used the functional now called the Einstein-Hilbert action.
See also
Palatini action
Plebanski action
MacDowell–Mansouri action
Freidel–Starodubtsev action
Mathematics of general relativity
Fermat's and energy variation principles in field theory
References
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https://en.wikipedia.org/wiki/Geometry%20%26%20Topology
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Geometry & Topology is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation.
It was founded in 1997 by a group of topologists who were dissatisfied with recent substantial rises in subscription prices of journals published by major publishing corporations. The aim was to set up a high-quality journal, capable of competing with existing journals, but with substantially lower subscription fees. The journal was open-access for its first ten years of existence and was available free to individual users, although institutions were required to pay modest subscription fees for both online access and for printed volumes. At present, an online subscription is required to view full-text PDF copies of articles in the most recent three volumes; articles older than that are open-access, at which point copies of the published articles are uploaded to the arXiv. A traditional printed version is also published, at present on an annual basis.
The journal has grown to be well respected in its field, and has in recent years published a number of important papers, in particular proofs of the Property P conjecture and the Birman conjecture.
References
Walter Neumann on the Success of Geometry & Topology, May 2010, Sciencewatch.com, Thomson Reuters
External links
Geometry & Topology
MSP Open Access Policy
Mathematics journals
Academic journals established in 1997
English-language journals
Mathematical Sciences Publishers academic journals
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https://en.wikipedia.org/wiki/Compensation%20of%20employees
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Compensation of employees (CE) is a statistical term used in national accounts, balance of payments statistics and sometimes in corporate accounts as well. It refers basically to the total gross (pre-tax) wages paid by employers to employees for work done in an accounting period, such as a quarter or a year.
However, in reality, the aggregate includes more than just gross wages, at least in national accounts and balance of payments statistics. The reason is that in these accounts, CE is defined as "the total remuneration, in cash or in kind, payable by an enterprise to an employee in return for work done by the latter during the accounting period". It represents effectively a total labour cost to an employer, paid from the gross revenues or the capital of an enterprise.
Compensation of employees is accounted for on an accrual basis; i.e., it is measured by the value of the remuneration in cash or in kind which an employee becomes entitled to receive from an employer in respect of work done, during the relevant accounting period – whether paid in advance, simultaneously, or in arrears of the work itself. This contrasts with other inputs to production, which are to be valued at the point when they are actually used.
For statistical purposes, the relationship of employer to employee exists, when there is an agreement, formal or informal, between an enterprise and a person, normally entered into voluntarily by both parties, whereby the person works for the enterprise, in return for remuneration in cash or in kind. The remuneration is normally based on either the time spent at work, or some other objective indicator of the amount of work done.
For social accounting purposes, CE is considered a component of the value of net output or value added (as factor income). The aim is not to measure income actually received by workers, but the value which labour contributes to net output along with other factors of production.
The underlying idea is that the value of net output equals the factor incomes that generate it. For this reason, some types of remuneration received by employees are either included or excluded, because they are regarded as either related or unrelated to production or to the value of new output.
In different countries, what is actually included and excluded in CE may differ somewhat. The reason is that the way in which workers are compensated for their labour may be somewhat different in different types of economies. For example, in some countries workers get substantial payments "in kind", in others they don't. Systems of social insurance also differ between countries, and some countries have little social insurance. One has to keep this in mind when comparing CE magnitudes for different countries.
A compensation system has to be aligned to the mission, vision, business strategy and organizational structure of a company to design the compensation plan in an efficient way to can achieve the goals. Businesses within the same organ
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https://en.wikipedia.org/wiki/Hexagonal%20tiling
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In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathematician John Conway called it a hextille.
The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling.
Applications
Hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular Weaire–Phelan structure is slightly better.
This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as carbon nanotubes. They have many potential applications, due to their high tensile strength and electrical properties. Silicene is similar.
Chicken wire consists of a hexagonal lattice (often not regular) of wires.
The hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest sphere packings in three dimensions. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure copper, amongst other materials, forms a face-centered cubic lattice.
Uniform colorings
There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions. The (h,k) represent the periodic repeat of one colored tile, counting hexagonal distances as h first, and k second. The same counting is used in the Goldberg polyhedra, with a notation {p+,3}h,k, and can be applied to hyperbolic tilings for p > 6.
The 3-color tiling is a tessellation generated by the order-3 permutohedrons.
Chamfered hexagonal tiling
A chamfered hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling.
Related tilings
The hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, and the triangular tiling:
The hexagonal tiling can be considered an elongated rhombic tiling, where each vertex of the rhombic tiling is stretched into a new edge. This is similar to the re
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https://en.wikipedia.org/wiki/Square%20tiling
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In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille.
The internal angle of the square is 90 degrees so four squares at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the hexagonal tiling.
Uniform colorings
There are 9 distinct uniform colorings of a square tiling. Naming the colors by indices on the 4 squares around a vertex: 1111, 1112(i), 1112(ii), 1122, 1123(i), 1123(ii), 1212, 1213, 1234. (i) cases have simple reflection symmetry, and (ii) glide reflection symmetry. Three can be seen in the same symmetry domain as reduced colorings: 1112i from 1213, 1123i from 1234, and 1112ii reduced from 1123ii.
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra and tilings, extending into the hyperbolic plane: {4,p}, p=3,4,5...
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
Wythoff constructions from square tiling
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular square tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three topologically distinct forms: square tiling, truncated square tiling, snub square tiling.
Topologically equivalent tilings
Other quadrilateral tilings can be made which are topologically equivalent to the square tiling (4 quads around every vertex).
Isohedral tilings have identical faces (face-transitivity) and vertex-transitivity, there are 18 variations, with 6 identified as triangles that do not connect edge-to-edge, or as quadrilateral with two collinear edges. Symmetry given assumes all faces are the same color.
Circle packing
The square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 4 other circles in the packing (kissing number). The packing density is π/4=78.54% coverage. There are 4 uniform colorings of the circle packings.
Related regular complex apeirogons
There are 3 regular complex apeirogons, sharing the vertices of the square tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.
See also
Checkerboard
List of regular polytopes
List of uniform tilings
Square lattice
Tilings of regular polygons
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Do
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https://en.wikipedia.org/wiki/Triangular%20tiling
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In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of
English mathematician John Conway called it a deltille, named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille.
It is one of three regular tilings of the plane. The other two are the square tiling and the hexagonal tiling.
Uniform colorings
There are 9 distinct uniform colorings of a triangular tiling. (Naming the colors by indices on the 6 triangles around a vertex: 111111, 111112, 111212, 111213, 111222, 112122, 121212, 121213, 121314) Three of them can be derived from others by repeating colors: 111212 and 111112 from 121213 by combining 1 and 3, while 111213 is reduced from 121314.
There is one class of Archimedean colorings, 111112, (marked with a *) which is not 1-uniform, containing alternate rows of triangles where every third is colored. The example shown is 2-uniform, but there are infinitely many such Archimedean colorings that can be created by arbitrary horizontal shifts of the rows.
A2 lattice and circle packings
The vertex arrangement of the triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplectic honeycomb.
The A lattice (also called A) can be constructed by the union of all three A2 lattices, and equivalent to the A2 lattice.
+ + = dual of =
The vertices of the triangular tiling are the centers of the densest possible circle packing. Every circle is in contact with 6 other circles in the packing (kissing number). The packing density is or 90.69%.
The voronoi cell of a triangular tiling is a hexagon, and so the voronoi tessellation, the hexagonal tiling, has a direct correspondence to the circle packings.
Geometric variations
Triangular tilings can be made with the equivalent {3,6} topology as the regular tiling (6 triangles around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 5 variations. Symmetry given assumes all faces are the same color.
Related polyhedra and tilings
The planar tilings are related to polyhedra. Putting fewer triangles on a vertex leaves a gap and allows it to be folded into a pyramid. These can be expanded to Platonic solids: five, four and three triangles on a vertex define an icosahedron, octahedron, and tetrahedron respectively.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.
It is also topologically related as a part of sequence of Catalan solids with face configuration Vn.6.6, and also cont
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https://en.wikipedia.org/wiki/Truncated%20hexagonal%20tiling
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In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.
Conway calls it a truncated hextille, constructed as a truncation operation applied to a hexagonal tiling (hextille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There is only one uniform coloring of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.)
Topologically identical tilings
The dodecagonal faces can be distorted into different geometries, such as:
Related polyhedra and tilings
Wythoff constructions from hexagonal and triangular tilings
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Symmetry mutations
This tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
Related 2-uniform tilings
Two 2-uniform tilings are related by dissected the dodecagons into a central hexagonal and 6 surrounding triangles and squares.
Circle packing
The truncated hexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number). This is the lowest density packing that can be created from a uniform tiling.
Triakis triangular tiling
The triakis triangular tiling is a tiling of the Euclidean plane. It is an equilateral triangular tiling with each triangle divided into three obtuse triangles (angles 30-30-120) from the center point. It is labeled by face configuration V3.12.12 because each isosceles triangle face has two types of vertices: one with 3 triangles, and two with 12 triangles.
Conway calls it a kisdeltille, constructed as a kis operation applied to a triangular tiling (deltille).
In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron.
It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex.
It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile.
Related duals to uniform tilings
It is one of 7 dual uniform tilings in hexagonal symmetry, including the regular d
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https://en.wikipedia.org/wiki/Truncated%20trihexagonal%20tiling
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In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.
Names
Uniform colorings
There is only one uniform coloring of a truncated trihexagonal tiling, with faces colored by polygon sides. A 2-uniform coloring has two colors of hexagons. 3-uniform colorings can have 3 colors of dodecagons or 3 colors of squares.
Related 2-uniform tilings
The truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects the hexagons into 6 triangles. The other two dissect the dodecagons into a central hexagon and surrounding triangles and square, in two different orientations.
Circle packing
The Truncated trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number).
Kisrhombille tiling
The kisrhombille tiling or 3-6 kisrhombille tiling is a tiling of the Euclidean plane. It is constructed by congruent 30-60-90 triangles with 4, 6, and 12 triangles meeting at each vertex.
Subdividing the faces of these tilings creates the kisrhombille tiling. (Compare the disdyakis hexa-, dodeca- and triacontahedron, three Catalan solids similar to this tiling.)
Construction from rhombille tiling
Conway calls it a kisrhombille for his kis vertex bisector operation applied to the rhombille tiling. More specifically it can be called a 3-6 kisrhombille, to distinguish it from other similar hyperbolic tilings, like 3-7 kisrhombille.
It can be seen as an equilateral hexagonal tiling with each hexagon divided into 12 triangles from the center point. (Alternately it can be seen as a bisected triangular tiling divided into 6 triangles, or as an infinite arrangement of lines in six parallel families.)
It is labeled V4.6.12 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 12 triangles.
Symmetry
The kisrhombille tiling triangles represent the fundamental domains of p6m, [6,3] (*632 orbifold notation) wallpaper group symmetry. There are a number of small index subgroups constructed from [6,3] by mirror removal and alternation. [1+,6,3] creates *333 symmetry, shown as red mirror lines. [6,3+] creates 3*3 symmetry. [6,3]+ is the rotational subgroup. The commutator subgroup is [1+,6,3+], which is 333 symmetry. A larger index 6 subgroup constructed as [6,3*], also becomes (*333), shown in blue mirror lines, and which has its own 333 rotational symmetry, index 12.
Related polyhedra and tilings
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling). Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges,
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https://en.wikipedia.org/wiki/Truncated%20square%20tiling
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In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.
Conway calls it a truncated quadrille, constructed as a truncation operation applied to a square tiling (quadrille).
Other names used for this pattern include Mediterranean tiling and octagonal tiling, which is often represented by smaller squares, and nonregular octagons which alternate long and short edges.
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices around a vertex (4.8.8): 122, 123.)
Circle packing
The truncated square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (kissing number).
Variations
One variations on this pattern, often called a Mediterranean pattern, is shown in stone tiles with smaller squares and diagonally aligned with the borders. Other variations stretch the squares or octagons.
The Pythagorean tiling alternates large and small squares, and may be seen as topologically identical to the truncated square tiling. The squares are rotated 45 degrees and octagons are distorted into squares with mid-edge vertices.
A weaving pattern also has the same topology, with octagons flattened rectangles.
Related polyhedra and tilings
The truncated square tiling is topologically related as a part of sequence of uniform polyhedra and tilings with vertex figures 4.2n.2n, extending into the hyperbolic plane:
The 3-dimensional bitruncated cubic honeycomb projected into the plane shows two copies of a truncated tiling. In the plane it can be represented by a compound tiling, or combined can be seen as a chamfered square tiling.
Wythoff constructions from square tiling
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, all 8 forms are distinct. However treating faces identically, there are only three unique topologically forms: square tiling, truncated square tiling, snub square tiling.
Related tilings in other symmetries
Tetrakis square tiling
The tetrakis square tiling is the tiling of the Euclidean plane dual to the truncated square tiling. It can be constructed square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of .
Conway calls it a kisquadrille, represented by a kis operation that adds a center point and t
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https://en.wikipedia.org/wiki/Oblique%20reflection
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In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations.
Consider a plane P in the three-dimensional Euclidean space. The usual reflection of a point A in space in respect to the plane P is another point B in space, such that the midpoint of the segment AB is in the plane, and AB is perpendicular to the plane. For an oblique reflection, one requires instead of perpendicularity that AB be parallel to a given reference line.
Formally, let there be a plane P in the three-dimensional space, and a line L in space not parallel to P. To obtain the oblique reflection of a point A in space in respect to the plane P, one draws through A a line parallel to L, and lets the oblique reflection of A be the point B on that line on the other side of the plane such that the midpoint of AB is in P. If the reference line L is perpendicular to the plane, one obtains the usual reflection.
For example, consider the plane P to be the xy plane, that is, the plane given by the equation z=0 in Cartesian coordinates. Let the direction of the reference line L be given by the vector (a, b, c), with c≠0 (that is, L is not parallel to P). The oblique reflection of a point (x, y, z) will then be
The concept of oblique reflection is easily generalizable to oblique reflection in respect to an affine hyperplane in Rn with a line again serving as a reference, or even more generally, oblique reflection in respect to a k-dimensional affine subspace, with a n−k-dimensional affine subspace serving as a reference. Back to three dimensions, one can then define oblique reflection in respect to a line, with a plane serving as a reference.
An oblique reflection is an affine transformation, and it is an involution, meaning that the reflection of the reflection of a point is the point itself.
References
Affine geometry
Functions and mappings
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https://en.wikipedia.org/wiki/Rhombitrihexagonal%20tiling
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In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.
John Conway calls it a rhombihexadeltille. It can be considered a cantellated by Norman Johnson's terminology or an expanded hexagonal tiling by Alicia Boole Stott's operational language.
There are three regular and eight semiregular tilings in the plane.
Uniform colorings
There is only one uniform coloring in a rhombitrihexagonal tiling. (Naming the colors by indices around a vertex (3.4.6.4): 1232.)
With edge-colorings there is a half symmetry form (3*3) orbifold notation. The hexagons can be considered as truncated triangles, t{3} with two types of edges. It has Coxeter diagram , Schläfli symbol s2{3,6}. The bicolored square can be distorted into isosceles trapezoids. In the limit, where the rectangles degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, .
Examples
Related tilings
There is one related 2-uniform tiling, having hexagons dissected into six triangles. The rhombitrihexagonal tiling is also related to the truncated trihexagonal tiling by replacing some of the hexagons and surrounding squares and triangles with dodecagons:
Circle packing
The rhombitrihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with four other circles in the packing (kissing number). The translational lattice domain (red rhombus) contains six distinct circles.
Wythoff construction
There are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are eight forms, seven topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Symmetry mutations
This tiling is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.
Deltoidal trihexagonal tiling
The deltoidal trihexagonal tiling is a dual of the semiregular tiling known as the rhombitrihexagonal tiling. Conway calls it a tetrille. The edges of this tiling can be formed by the intersection overlay of the regular triangular tiling and a hexagonal tiling. Each kite face of this tiling has angles 120°, 90°, 60° and 90°. It is one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
The deltoidal trihexagonal tiling is a dual of the semiregular tiling rhombitrihexagonal tiling. Its faces are deltoids or kites.
Related polyhedra and tilings
It is one of seven dual uniform tilings in hexagonal symmetry, including the regular duals.
This tiling has face t
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https://en.wikipedia.org/wiki/Trihexagonal%20tiling
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In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.
This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. The pattern has long been used in Japanese basketry, where it is called kagome. The Japanese term for this pattern has been taken up in physics, where it is called a kagome lattice. It occurs also in the crystal structures of certain minerals. Conway calls it a hexadeltille, combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille).
Kagome
Kagome () is a traditional Japanese woven bamboo pattern; its name is composed from the words kago, meaning "basket", and me, meaning "eye(s)", referring to the pattern of holes in a woven basket.
The kagome pattern is common in bamboo weaving in East Asia. In 2022, archaeologists found bamboo weaving remains at the Dongsunba ruins in Chongqing, China, 200 BC. After 2200 years, the kagome pattern is still clear.
It is a woven arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The woven process gives the Kagome a chiral wallpaper group symmetry, p6, (632).
Kagome lattice
The term kagome lattice was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji.
The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling.
Despite the name, these crossing points do not form a mathematical lattice.
A related three dimensional structure formed by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra, has been called a hyper-kagome lattice. It is represented by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an orthorhombic-kagome lattice. The trihexagonal prismatic honeycomb represents its edges and vertices.
Some minerals, namely jarosites and herbertsmithite, contain two-dimensional layers or three-dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. For instance, the spin arrangemen
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https://en.wikipedia.org/wiki/Majorization
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In mathematics, majorization is a preorder on vectors of real numbers. Let denote the -th largest element of the vector . Given , we say that weakly majorizes (or dominates) from below (or equivalently, we say that is weakly majorized (or dominated) by from below) denoted as if for all . If in addition , we say that majorizes (or dominates) , written as , or equivalently, we say that is majorized (or dominated) by . The order of the entries of the vectors or does not affect the majorization, e.g., the statement is simply equivalent to . As a consequence, majorization is not a partial order, since and do not imply , it only implies that the components of each vector are equal, but not necessarily in the same order.
The majorization partial order on finite dimensional vectors, described here, can be generalized to the Lorenz ordering, a partial order on distribution functions. For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. Various other generalizations of majorization are discussed in chapters 14 and 15 of.
The majorization preorder can be naturally extended to density matrices in the context of quantum information. In particular, exactly when (where denotes the state's spectrum).
Examples
(Strong) majorization: . For vectors with components
(Weak) majorization: . For vectors with components:
Geometry of majorization
For we have if and only if is in the convex hull of all vectors obtained by permuting the coordinates of .
Figure 1 displays the convex hull in 2D for the vector . Notice that the center of the convex hull, which is an interval in this case, is the vector . This is the "smallest" vector satisfying for this given vector .
Figure 2 shows the convex hull in 3D. The center of the convex hull, which is a 2D polygon in this case, is the "smallest" vector satisfying for this given vector .
Schur convexity
A function is said to be Schur convex when implies . Hence, Schur-convex functions translate the ordering of vectors to a standard ordering in . Similarly, is Schur concave when implies
An example of a Schur-convex function is the max function, . Schur convex functions are necessarily symmetric that the entries of it argument can be switched without modifying the value of the function. Therefore, linear functions, which are convex, are not Schur-convex unless they are symmetric. If a function is symmetric and convex, then it is Schur-convex.
Equivalent conditions
Each of the following statements is true if and only if :
for some doubly stochastic matrix . This is equivalent to saying can be represented as a convex combination of the permutations of ; one can verify that there exists such a convex representation using at most permutations of .
From we can produce by a finite sequence of "Robin Hood operations" where we repl
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https://en.wikipedia.org/wiki/Snub%20trihexagonal%20tiling
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In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille).
There are three regular and eight semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.
There is only one uniform coloring of a snub trihexagonal tiling. (Labeling the colors by numbers, "3.3.3.3.6" gives "11213".)
Circle packing
The snub trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number). The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling.
Related polyhedra and tilings
Symmetry mutations
This semiregular tiling is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.
6-fold pentille tiling
In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is one of the 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower. Each of its pentagonal faces has four 120° and one 60° angle.
It is the dual of the uniform snub trihexagonal tiling, and has rotational symmetries of orders 6-3-2 symmetry.
Variations
The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.
Related k-uniform and dual k-uniform tilings
There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles; for example, labeling F for V34.6, C for V32.4.3.4, B for V33.42, H for V36:
Fractalization
Replacing every V36 hexagon by a rhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.
Replacing every V36 hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 32.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V36 hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.42.6, and one vertex of 3.4.6.4.
In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since ther
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https://en.wikipedia.org/wiki/Elongated%20triangular%20tiling
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In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.
Conway calls it a isosnub quadrille.
There are 3 regular and 8 semiregular tilings in the plane. This tiling is similar to the snub square tiling which also has 3 triangles and two squares on a vertex, but in a different order.
Construction
It is also the only convex uniform tiling that can not be created as a Wythoff construction. It can be constructed as alternate layers of apeirogonal prisms and apeirogonal antiprisms.
Uniform colorings
There is one uniform colorings of an elongated triangular tiling. Two 2-uniform colorings have a single vertex figure, 11123, with two colors of squares, but are not 1-uniform, repeated either by reflection or glide reflection, or in general each row of squares can be shifted around independently. The 2-uniform tilings are also called Archimedean colorings. There are infinite variations of these Archimedean colorings by arbitrary shifts in the square row colorings.
Circle packing
The elongated triangular tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).
Related tilings
Sections of stacked triangles and squares can be combined into radial forms. This mixes two vertex configurations, 3.3.3.4.4 and 3.3.4.3.4 on the transitions. Twelve copies are needed to fill the plane with different center arrangements. The duals will mix in cairo pentagonal tiling pentagons.
Symmetry mutations
It is first in a series of symmetry mutations with hyperbolic uniform tilings with 2*n2 orbifold notation symmetry, vertex figure 4.n.4.3.3.3, and Coxeter diagram . Their duals have hexagonal faces in the hyperbolic plane, with face configuration V4.n.4.3.3.3.
There are four related 2-uniform tilings, mixing 2 or 3 rows of triangles or squares.
Prismatic pentagonal tiling
The prismatic pentagonal tiling is a dual uniform tiling in the Euclidean plane. It is one of 15 known isohedral pentagon tilings. It can be seen as a stretched hexagonal tiling with a set of parallel bisecting lines through the hexagons.
Conway calls it an . Each of its pentagonal faces has three 120° and two 90° angles.
It is related to the Cairo pentagonal tiling with face configuration V3.3.4.3.4.
Geometric variations
Monohedral pentagonal tiling type 6 has the same topology, but two edge lengths and a lower p2 (2222) wallpaper group symmetry:
Related 2-uniform dual tilings
There are four related 2-uniform dual tilings, mixing in rows of squares or hexagons (the prismatic pentagon is half-square half-hexagon).
See also
Tilings of regular polygons
Elongated triangular prismatic honeycomb
Gyroelongated triangular prismatic honeycomb
Notes
References
(Chapter 2.1:
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https://en.wikipedia.org/wiki/Snub%20square%20tiling
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In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. Its Schläfli symbol is s{4,4}.
Conway calls it a snub quadrille, constructed by a snub operation applied to a square tiling (quadrille).
There are 3 regular and 8 semiregular tilings in the plane.
Uniform colorings
There are two distinct uniform colorings of a snub square tiling. (Naming the colors by indices around a vertex (3.3.4.3.4): 11212, 11213.)
Circle packing
The snub square tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).
Wythoff construction
The snub square tiling can be constructed as a snub operation from the square tiling, or as an alternate truncation from the truncated square tiling.
An alternate truncation deletes every other vertex, creating a new triangular faces at the removed vertices, and reduces the original faces to half as many sides. In this case starting with a truncated square tiling with 2 octagons and 1 square per vertex, the octagon faces into squares, and the square faces degenerate into edges and 2 new triangles appear at the truncated vertices around the original square.
If the original tiling is made of regular faces the new triangles will be isosceles. Starting with octagons which alternate long and short edge lengths, derived from a regular dodecagon, will produce a snub tiling with perfect equilateral triangle faces.
Example:
Related tilings
Related k-uniform tilings
This tiling is related to the elongated triangular tiling which also has 3 triangles and two squares on a vertex, but in a different order, 3.3.3.4.4. The two vertex figures can be mixed in many k-uniform tilings.
Related topological series of polyhedra and tiling
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.
The snub square tiling is third in a series of snub polyhedra and tilings with vertex figure 3.3.n.3.n.
See also
List of uniform planar tilings
Snub (geometry)
Snub square prismatic honeycomb
Tilings of regular polygons
Elongated triangular tiling
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008,
(Chapter 2.1: Regular and uniform tilings, p. 58-65)
p38
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, , pp. 50–56, dual p. 115
External links
Euclidean tilings
Isogonal tilings
Semiregular tilings
Square tilings
Snub tilings
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https://en.wikipedia.org/wiki/Linearized%20gravity
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In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects of gravity when the gravitational field is weak. The usage of linearized gravity is integral to the study of gravitational waves and weak-field gravitational lensing.
Weak-field approximation
The Einstein field equation (EFE) describing the geometry of spacetime is given as (using natural units)
where is the Ricci tensor, is the Ricci scalar, is the energy–momentum tensor, and is the spacetime metric tensor that represent the solutions of the equation.
Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally nonlinear dependencies on the metric which render the prospect of finding exact solutions impractical in most systems. However, when describing particular systems for which the curvature of spacetime is small (meaning that terms in the EFE that are quadratic in do not significantly contribute to the equations of motion), one can model the solution of the field equations as being the Minkowski metric plus a small perturbation term . In other words:
In this regime, substituting the general metric for this perturbative approximation results in a simplified expression for the Ricci tensor:
where is the trace of the perturbation, denotes the partial derivative with respect to the coordinate of spacetime, and is the d'Alembert operator.
Together with the Ricci scalar,
the left side of the field equation reduces to
and thus the EFE is reduced to a linear, second order partial differential equation in terms of .
Gauge invariance
The process of decomposing the general spacetime into the Minkowski metric plus a perturbation term is not unique. This is due to the fact that different choices for coordinates may give different forms for . In order to capture this phenomenon, the application of gauge symmetry is introduced.
Gauge symmetries are a mathematical device for describing a system that does not change when the underlying coordinate system is "shifted" by an infinitesimal amount. So although the perturbation metric is not consistently defined between different coordinate systems, the overall system which it describes is.
To capture this formally, the non-uniqueness of the perturbation is represented as being a consequence of the diverse collection of diffeomorphisms on spacetime that leave sufficiently small. Therefore to continue, it is required that be defined in terms of a general set of diffeomorphisms then select the subset of these that preserve the small scale that is required by the weak-field approximation. One may thus define to denote an arbitrary diffeomorphism that maps the flat Minkowski spacetime to the more general spacetime represented by the metric . With this, the perturbation metric may be de
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https://en.wikipedia.org/wiki/Fusion%20rules
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In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of two-dimensional conformal field theory where the relevant group is generated by the Virasoro algebra, the relevant representations are the conformal families associated with a primary field and the tensor product is realized by operator product expansions. The fusion rules contain the information about the kind of families that appear on the right hand side of these OPEs, including the multiplicities.
More generally, integrable models in 2 dimensions which aren't conformal field theories are also described by fusion rules for their charges.
References
Conformal field theory
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https://en.wikipedia.org/wiki/Conformal%20family
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In theoretical physics, a conformal family is an irreducible representation of the Virasoro algebra. In most cases, it is uniquely determined by its primary field or the highest weight vector. The family contains all of its descendant fields.
References
See also
Conformal field theory
Conformal field theory
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https://en.wikipedia.org/wiki/Lorentz%20Eichstadt
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Lorentz Eichstadt (10 August 1596 – 8 June 1660) was a German mathematician and astronomer. He was a doctor of medicine in Szczecin in Pomerania and taught medicine and mathematics in Danzig.
The lunar crater Eichstadt is named after him.
References
External links
Lunar Republic: Craters. Retrieved October 8, 2005.
1596 births
1660 deaths
17th-century German mathematicians
17th-century German astronomers
17th-century German physicians
Scientists from Gdańsk
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https://en.wikipedia.org/wiki/Equipotential
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In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential function in -dimensional space is typically an ()-dimensional space. The del operator illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'.
An equipotential region of a scalar potential in three-dimensional space is often an equipotential surface (or potential isosurface), but it can also be a three-dimensional mathematical solid in space. The gradient of the scalar potential (and hence also its opposite, as in the case of a vector field with an associated potential field) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.
Electrical conductors offer an intuitive example. If a and b are any two points within or at the surface of a given conductor, and given there is no flow of charge being exchanged between the two points, then the potential difference is zero between the two points. Thus, an equipotential would contain both points a and b as they have the same potential. Extending this definition, an isopotential is the locus of all points that are of the same potential.
Gravity is perpendicular to the equipotential surfaces of the gravity potential, and in electrostatics and steady electric currents, the electric field (and hence the current, if any) is perpendicular to the equipotential surfaces of the electric potential (voltage).
In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity from the sphere (see shell theorem). In electrostatics, a conductor is a three-dimensional equipotential region. In the case of a hollow conductor (Faraday cage), the equipotential region includes the space inside.
A ball will not be accelerated left or right by the force of gravity if it is resting on a flat, horizontal surface, because it is an equipotential surface.
For the gravity of Earth, the corresponding geopotential isosurface (the equigeopotential) that best fits mean sea level is called the geoid.
See also
Potential flow
Potential gradient
Isopotential map
Scalar potential
References
External links
Electric Field Applet
Multivariable calculus
Mathematical physics
Potentials
de:Äquipotentialfläche
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https://en.wikipedia.org/wiki/Rhombille%20tiling
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In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.
Properties
The rhombille tiling can be seen as a subdivision of a hexagonal tiling with each hexagon divided into three rhombi meeting at the center point of the hexagon. This subdivision represents a regular compound tiling. It can also be seen as a subdivision of four hexagonal tilings with each hexagon divided into 12 rhombi.
The diagonals of each rhomb are in the ratio 1:.
This is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the face configuration for monohedral tilings it is denoted [3.6.3.6].
It is also one of 56 possible isohedral tilings by quadrilaterals, and one of only eight tilings of the plane in which every edge lies on a line of symmetry of the tiling.
It is possible to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way that two vertices are adjacent if and only if the corresponding lattice points are at unit distance from each other, and more strongly such that the number of edges in the shortest path between any two vertices of the tiling is the same as the Manhattan distance between the corresponding lattice points. Thus, the rhombille tiling can be viewed as an example of an infinite unit distance graph and partial cube.
Artistic and decorative applications
The rhombille tiling can be interpreted as an isometric projection view of a set of cubes in two different ways, forming a reversible figure related to the Necker Cube. In this context it is known as the "reversible cubes" illusion.
In the M. C. Escher artworks Metamorphosis I, Metamorphosis II, and Metamorphosis III Escher uses this interpretation of the tiling as a way of morphing between two- and three-dimensional forms. In another of his works, Cycle (1938), Escher played with the tension between the two-dimensionality and three-dimensionality of this tiling: in it he draws a building that has both large cubical blocks as architectural elements (drawn isometrically) and an upstairs patio tiled with the rhombille tiling. A human figure descends from the patio past the cubes, becoming more stylized and two-dimensional as he does so. These works involve only a single three-dimensional interpretation of the tiling, but in Convex and Concave Escher experiments with reversible figures more generally, and includes a depiction of the reversible cubes illusion on a flag within the scene.
The rhombille tiling is also used as a design for parquetry and for floor or wall tiling, sometimes with variations in the s
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https://en.wikipedia.org/wiki/Morse%20homology
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In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories.
Formal definition
Given any (compact) smooth manifold, let f be a Morse function and g a Riemannian metric on the manifold. (These are auxiliary; in the end, the Morse homology depends on neither.) The pair gives us a gradient vector field. We say that is Morse–Smale if the stable and unstable manifolds associated to all of the critical points of f intersect each other transversely.
For any such pair , it can be shown that the difference in index between any two critical points is equal to the dimension of the moduli space of gradient flows between those points. Thus there is a one-dimensional moduli space of flows between a critical point of index i and one of index . Each flow can be reparametrized by a one-dimensional translation in the domain. After modding out by these reparametrizations, the quotient space is zero-dimensional — that is, a collection of oriented points representing unparametrized flow lines.
A chain complex may then be defined as follows. The set of chains is the Z-module generated by the critical points. The differential d of the complex sends a critical point p of index i to a sum of index- critical points, with coefficients corresponding to the (signed) number of unparametrized flow lines from p to those index- critical points. The fact that the number of such flow lines is finite follows from the compactness of the moduli space.
The fact that this defines a chain complex (that is, that ) follows from an understanding of how the moduli spaces of gradient flows compactify. Namely, in the coefficient of an index- critical point q is the (signed) number of broken flows consisting of an index-1 flow from p to some critical point r of index and another index-1 flow from r to q. These broken flows exactly constitute the boundary of the moduli space of index-2 flows: The limit of any sequence of unbroken index-2 flows can be shown to be of this form, and all such broken flows arise as limits of unbroken index-2 flows. Unparametrized index-2 flows come in one-dimensional families, which compactify to compact one-manifolds with boundaries. The fact that the boundary of a compact one-manifold has signed count zero proves that .
Invariance of Morse homology
It can be shown that the homology of this complex is independent of the Morse–Smale pair (f, g) used to define it. A homotopy of pairs (ft, gt) that interpolates between any two given pairs (f0, g0) and (f1, g1) may always be defined. Either through bifurcation analysis or by using a continuation map to define a chain map from
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https://en.wikipedia.org/wiki/Cocycle
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In mathematics a cocycle is a closed cochain. Cocycles are used in algebraic topology to express obstructions (for example, to integrating a differential equation on a closed manifold). They are likewise used in group cohomology. In autonomous dynamical systems, cocycles are used to describe particular kinds of map, as in the Oseledets theorem.
Definition
Algebraic Topology
Let X be a CW complex and be the singular cochains with coboundary map . Then elements of are cocycles. Elements of are coboundaries. If is a cocycle, then , which means cocycles vanish on boundaries.
See also
Čech cohomology
Cocycle condition
References
Algebraic topology
Cohomology theories
Dynamical systems
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https://en.wikipedia.org/wiki/Pohlmeyer%20charge
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In theoretical physics Pohlmeyer charge, named for Klaus Pohlmeyer, is a conserved charge invariant under the Virasoro algebra or its generalization. It can be obtained by expanding the holonomies (generating functions)
with respect to the constant matrices T. The gauge field is defined as a combination of and its conjugate.
According to the logic of loop quantum gravity and algebraic quantum field theory, these charges are the right physical quantities that should be used for quantization. This logic is however incompatible with the standard and well-established methods of quantum field theory based on Fock space and perturbation theory.
Theoretical physics
Quantum field theory
Conformal field theory
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https://en.wikipedia.org/wiki/Lottery%20mathematics
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Lottery mathematics is used to calculate probabilities of winning or losing a lottery game. It is based primarily on combinatorics, particularly the twelvefold way and combinations without replacement.
Choosing 6 from 49
In a typical 6/49 game, each player chooses six distinct numbers from a range of 1-49. If the six numbers on a ticket match the numbers drawn by the lottery, the ticket holder is a jackpot winner—regardless of the order of the numbers. The probability of this happening is 1 in 13,983,816.
The chance of winning can be demonstrated as follows: The first number drawn has a 1 in 49 chance of matching. When the draw comes to the second number, there are now only 48 balls left in the bag, because the balls are drawn without replacement. So there is now a 1 in 48 chance of predicting this number.
Thus for each of the 49 ways of choosing the first number there are 48 different ways of choosing the second. This means that the probability of correctly predicting 2 numbers drawn from 49 in the correct order is calculated as 1 in 49 × 48. On drawing the third number there are only 47 ways of choosing the number; but we could have arrived at this point in any of 49 × 48 ways, so the chances of correctly predicting 3 numbers drawn from 49, again in the correct order, is 1 in 49 × 48 × 47. This continues until the sixth number has been drawn, giving the final calculation, 49 × 48 × 47 × 46 × 45 × 44, which can also be written as or 49 factorial divided by 43 factorial or FACT(49)/FACT(43) or simply PERM(49,6) .
608281864034267560872252163321295376887552831379210240000000000 / 60415263063373835637355132068513997507264512000000000 = 10068347520
This works out to 10,068,347,520, which is much bigger than the ~14 million stated above.
Perm(49,6)=10068347520 and 49 nPr 6 =10068347520.
However, the order of the 6 numbers is not significant for the payout. That is, if a ticket has the numbers 1, 2, 3, 4, 5, and 6, it wins as long as all the numbers 1 through 6 are drawn, no matter what order they come out in. Accordingly, given any combination of 6 numbers, there are 6 × 5 × 4 × 3 × 2 × 1 = 6! or 720 orders in which they can be drawn. Dividing 10,068,347,520 by 720 gives 13,983,816, also written as , or COMBIN(49,6) or 49 nCr 6 or more generally as
, where n is the number of alternatives and k is the number of choices. Further information is available at binomial coefficient and multinomial coefficient.
This function is called the combination function, COMBIN(n,k). For the rest of this article, we will use the notation . "Combination" means the group of numbers selected, irrespective of the order in which they are drawn. A combination of numbers is usually presented in ascending order. An eventual 7th drawn number, the reserve or bonus, is presented at the end.
An alternative method of calculating the odds is to note that the probability of the first ball corresponding to one of the six chosen is 6/49; the probability of the second ball
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https://en.wikipedia.org/wiki/Interleave%20sequence
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In mathematics, an interleave sequence is obtained by merging two sequences via an in shuffle.
Let be a set, and let and , be two sequences in The interleave sequence is defined to be the sequence . Formally, it is the sequence given by
Properties
The interleave sequence is convergent if and only if the sequences and are convergent and have the same limit.
Consider two real numbers a and b greater than zero and smaller than 1. One can interleave the sequences of digits of a and b, which will determine a third number c, also greater than zero and smaller than 1. In this way one obtains an injection from the square to the interval (0, 1). Different radixes give rise to different injections; the one for the binary numbers is called the Z-order curve or Morton code.
References
Real analysis
Sequences and series
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https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro%20theorem
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In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time.
The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
Statement of the theorem for the case
Let and be two sequences of real numbers. Assume that is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching , or strictly decreasing and approaching ) and the following limit exists:
Then, the limit
Statement of the theorem for the case
Let and be two sequences of real numbers. Assume now that and while is strictly decreasing. If
then
Proofs
Proof of the theorem for the case
Case 1: suppose strictly increasing and divergent to , and . By hypothesis, we have that for all there exists such that
which is to say
Since is strictly increasing, , and the following holds
.
Next we notice that
thus, by applying the above inequality to each of the terms in the square brackets, we obtain
Now, since as , there is an such that for all , and we can divide the two inequalities by for all
The two sequences (which are only defined for as there could be an such that )
are infinitesimal since and the numerator is a constant number, hence for all there exists , such that
therefore
which concludes the proof. The case with strictly decreasing and divergent to , and is similar.
Case 2: we assume strictly increasing and divergent to , and . Proceeding as before, for all there exists such that for all
Again, by applying the above inequality to each of the terms inside the square brackets we obtain
and
The sequence defined by
is infinitesimal, thus
combining this inequality with the previous one we conclude
The proofs of the other cases with strictly increasing or decreasing and approaching or respectively and all proceed in this same way.
Proof of the theorem for the case
Case 1: we first consider the case with and strictly decreasing. This time, for each , we can write
and for any such that for all we have
The two sequences
are infinitesimal since by hypothesis as , thus for all there are such that
thus, choosing appropriately (which is to say, taking the limit with respect to ) we obtain
which concludes the proof.
Case 2: we assume and strictly decreasing. For all there exists such that for all
Therefore, for each
The sequence
converges to (keeping fixed). Hence
such that
and, choosing conveniently, we conclude the proof
Applications and examples
The theorem concerning the case has a few notable consequences which are useful in the computation of limits.
Arithmetic mean
Let be a sequence of real numbers which converges to , define
then is strictly increasing and diverges to . We compute
therefore
Given any sequence of real numbers, suppose that
exis
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https://en.wikipedia.org/wiki/Square%20lattice
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In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their symmetry groups; its symmetry group in IUC notation as , Coxeter notation as , and orbifold notation as .
Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as the upright square lattice and diagonal square lattice; the latter is also called the centered square lattice. They differ by an angle of 45°. This is related to the fact that a square lattice can be partitioned into two square sub-lattices, as is evident in the colouring of a checkerboard.
Symmetry
The square lattice's symmetry category is wallpaper group . A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
An upright square lattice can be viewed as a diagonal square lattice with a mesh size that is √2 times as large, with the centers of the squares added. Correspondingly, after adding the centers of the squares of an upright square lattice one obtains a diagonal square lattice with a mesh size that is √2 times as small as that of the original lattice.
A pattern with 4-fold rotational symmetry has a square lattice of 4-fold rotocenters that is a factor √2 finer and diagonally oriented relative to the lattice of translational symmetry.
With respect to reflection axes there are three possibilities:
None. This is wallpaper group .
In four directions. This is wallpaper group .
In two perpendicular directions. This is wallpaper group . The points of intersection of the reflexion axes form a square grid which is as fine as, and oriented the same as, the square lattice of 4-fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection axes.
Crystal classes
The square lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
See also
Centered square number
Euclid's orchard
Gaussian integer
Hexagonal lattice
Quincunx
Square tiling
References
Euclidean geometry
Lattice points
Crystal systems
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https://en.wikipedia.org/wiki/Icosahedral%20symmetry
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In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron.
Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.
Description
Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron.
As point group
Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups.
Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups.
Presentations corresponding to the above are:
These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups.
The first presentation was given by William Rowan Hamilton in 1856, in his paper on icosian calculus.
Note that other presentations are possible, for instance as an alternating group (for I).
Visualizations
The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.
Group structure
Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.
The I is of order 60. The group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes (which inscribe in the dodecahedron), the compound of five octahedra, or either of the two compounds of five tetrahedra (which are enantiomorphs, and inscribe in the dodecahedron). The group contains 5 versions of Th with 20 versions of D3 (10 axes, 2 per axis), and 6 versions of D5.
The Ih has order 120. It has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the center corresponding to element (identity,-1), where Z2 is written multiplicatively.
Ih acts on the compound of five cubes and the compound of five oct
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https://en.wikipedia.org/wiki/Developable%20surface
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In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). Conversely, it is a surface which can be made by transforming a plane (i.e. "folding", "bending", "rolling", "cutting" and/or "gluing"). In three dimensions all developable surfaces are ruled surfaces (but not vice versa). There are developable surfaces in four-dimensional space which are not ruled.
The envelope of a single parameter family of planes is called a developable surface.
Particulars
The developable surfaces which can be realized in three-dimensional space include:
Cylinders and, more generally, the "generalized" cylinder; its cross-section may be any smooth curve
Cones and, more generally, conical surfaces; away from the apex
The oloid and the sphericon are members of a special family of solids that develop their entire surface when rolling down a flat plane.
Planes (trivially); which may be viewed as a cylinder whose cross-section is a line
Tangent developable surfaces; which are constructed by extending the tangent lines of a spatial curve.
The torus has a metric under which it is developable, which can be embedded into three-dimensional space by the Nash embedding theorem and has a simple representation in four dimensions as the Cartesian product of two circles: see Clifford torus.
Formally, in mathematics, a developable surface is a surface with zero Gaussian curvature. One consequence of this is that all "developable" surfaces embedded in 3D-space are ruled surfaces (though hyperboloids are examples of ruled surfaces which are not developable). Because of this, many developable surfaces can be visualised as the surface formed by moving a straight line in space. For example, a cone is formed by keeping one end-point of a line fixed whilst moving the other end-point in a circle.
Application
Developable surfaces have several practical applications.
Developable Mechanisms are mechanisms that conform to a developable surface and can exhibit motion (deploy) off the surface.
Many cartographic projections involve projecting the Earth to a developable surface and then "unrolling" the surface into a region on the plane.
Since developable surfaces may be constructed by bending a flat sheet, they are also important in manufacturing objects from sheet metal, cardboard, and plywood. An industry which uses developed surfaces extensively is shipbuilding.
Non-developable surface
Most smooth surfaces (and most surfaces in general) are not developable surfaces. Non-developable surfaces are variously referred to as having "double curvature", "doubly curved", "compound curvature", "non-zero Gaussian curvature", etc.
Some of the most often-used non-developable surfaces are:
Spheres are not developable surfaces under any metric as they cannot be unrolled onto a plane.
The helicoid i
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https://en.wikipedia.org/wiki/Judy%20A.%20Holdener
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Judy Holdener (née Newhauser) is an American mathematician and educator. She is a professor of mathematics at Kenyon College. She was born in 1965. Holdener's primary interest is in number theory. She discovered a simpler proof of the theorem of Touchard, which states that every perfect number is of the form 2k, 12k+1, or 36k+9.
Holdener earned her B.S. in mathematics at Kent State University and her M.S. and Ph.D. in mathematics at the University of Illinois at Urbana-Champaign. Holdener joined the faculty of Kenyon College in 1997, where she is currently the John B. McCoy Distinguished Teaching Chair.
The poem Euler's Daughter by award-winning South African poet Athol Williams is dedicated to Holdener in celebration of her love of mathematics and life.
References
.
.
External links
Biography Page at Kenyon College
Number theorists
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Kent State University alumni
University of Illinois alumni
Kenyon College faculty
Living people
1965 births
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women
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https://en.wikipedia.org/wiki/List%20of%20probability%20distributions
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Many probability distributions that are important in theory or applications have been given specific names.
Discrete distributions
With finite support
The Bernoulli distribution, which takes value 1 with probability p and value 0 with probability q = 1 − p.
The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
The binomial distribution, which describes the number of successes in a series of independent Yes/No experiments all with the same probability of success.
The beta-binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with heterogeneity in the success probability.
The degenerate distribution at x0, where X is certain to take the value x0. This does not look random, but it satisfies the definition of random variable. This is useful because it puts deterministic variables and random variables in the same formalism.
The discrete uniform distribution, where all elements of a finite set are equally likely. This is the theoretical distribution model for a balanced coin, an unbiased die, a casino roulette, or the first card of a well-shuffled deck.
The hypergeometric distribution, which describes the number of successes in the first m of a series of n consecutive Yes/No experiments, if the total number of successes is known. This distribution arises when there is no replacement.
The negative hypergeometric distribution, a distribution which describes the number of attempts needed to get the nth success in a series of Yes/No experiments without replacement.
The Poisson binomial distribution, which describes the number of successes in a series of independent Yes/No experiments with different success probabilities.
Fisher's noncentral hypergeometric distribution
Wallenius' noncentral hypergeometric distribution
Benford's law, which describes the frequency of the first digit of many naturally occurring data.
The ideal and robust soliton distributions.
Zipf's law or the Zipf distribution. A discrete power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
The Zipf–Mandelbrot law is a discrete power law distribution which is a generalization of the Zipf distribution.
With infinite support
The Cauchy distribution
The beta negative binomial distribution
The Boltzmann distribution, a discrete distribution important in statistical physics which describes the probabilities of the various discrete energy levels of a system in thermal equilibrium. It has a continuous analogue. Special cases include:
The Gibbs distribution
The Maxwell–Boltzmann distribution
The Borel distribution
The discrete phase-type distribution, a generalization of the geometric distribution which describes the first hit time of the absorbing state of a finite terminating Markov chain.
The extended negative binomial distribution
The generalized log-series distribution
The Gauss–Kuzmin distribution
The geometric di
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https://en.wikipedia.org/wiki/Parametric%20model
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In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters.
Definition
A statistical model is a collection of probability distributions on some sample space. We assume that the collection, , is indexed by some set . The set is called the parameter set or, more commonly, the parameter space. For each , let denote the corresponding member of the collection; so is a cumulative distribution function. Then a statistical model can be written as
The model is a parametric model if for some positive integer .
When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:
Examples
The Poisson family of distributions is parametrized by a single number :
where is the probability mass function. This family is an exponential family.
The normal family is parametrized by , where is a location parameter and is a scale parameter:
This parametrized family is both an exponential family and a location-scale family.
The Weibull translation model has a three-dimensional parameter :
The binomial model is parametrized by , where is a non-negative integer and is a probability (i.e. and ):
This example illustrates the definition for a model with some discrete parameters.
General remarks
A parametric model is called identifiable if the mapping is invertible, i.e. there are no two different parameter values and such that .
Comparisons with other classes of models
Parametric models are contrasted with the semi-parametric, semi-nonparametric, and non-parametric models, all of which consist of an infinite set of "parameters" for description. The distinction between these four classes is as follows:
in a "parametric" model all the parameters are in finite-dimensional parameter spaces;
a model is "non-parametric" if all the parameters are in infinite-dimensional parameter spaces;
a "semi-parametric" model contains finite-dimensional parameters of interest and infinite-dimensional nuisance parameters;
a "semi-nonparametric" model has both finite-dimensional and infinite-dimensional unknown parameters of interest.
Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous. It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval. This difficulty can be avoided by considering only "smooth" parametric models.
See also
Parametric family
Parametric statistics
Statistical model
Statistical model specification
Notes
Bibliography
Parametric statistics
Statistical models
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https://en.wikipedia.org/wiki/Quadratrix
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In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circle.
Quadratrix of Dinostratus
The quadratrix of Dinostratus (also called the quadratrix of Hippias) was well known to the ancient Greek geometers, and is mentioned by Proclus, who ascribes the invention of the curve to a contemporary of Socrates, probably Hippias of Elis. Dinostratus, a Greek geometer and disciple of Plato, discussed the curve, and showed how it effected a mechanical solution of squaring the circle. Pappus, in his Collections, treats its history, and gives two methods by which it can be generated.
Let a helix be drawn on a right circular cylinder; a screw surface is then obtained by drawing lines from every point of this spiral perpendicular to its axis. The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.
A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. Any plane section of the screw (plectoidal of Pappus) surface so obtained is the quadratrix.
Another construction is as follows. is a quadrant in which the line and the arc are divided into the same number of equal parts. Radii are drawn from the centre of the quadrant to the points of division of the arc, and these radii are intersected by the lines drawn parallel to and through the corresponding points on the radius . The locus of these intersections is the quadratrix.
Letting be the origin of the Cartesian coordinate system, be the point , units from the origin along the -axis, and be the point , units from the origin along the -axis, the curve itself can be expressed by the equation
Because the cotangent function is invariant under negation of its argument, and has a simple pole at each multiple of , the quadratrix has reflection symmetry across the -axis, and similarly has a pole for each value of of the form , for integer values of , except at where the pole in the cotangent is canceled by the factor of in the formula for the quadratrix. These poles partition the curve into a central portion flanked by infinite branches. The point where the curve crosses the -axis has ; therefore, if it were possible to accurately construct the curve, one could construct a line segment whose length is a rational multiple of , leading to a solution of the classical problem of squaring the circle. Since this is impossible with compass and straightedge, the quadratrix in turn cannot be constructed with compass and straightedge.
An accurate construction of the quadratrix would also allow the solution
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https://en.wikipedia.org/wiki/Walter%20Trump
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Walter Trump (born 1952 or 1953 ) is a German mathematician and retired high school teacher. He is known for his work in recreational mathematics.
He has made contributions working on both the square packing problem and the magic tile problem. In 1979 he discovered the optimal known packing of 11 equal squares in a larger square, and in 2003, along with Christian Boyer, developed the first known magic cube of order 5. In 2012, Trump et al. described a model for retention of liquid on random surfaces.
In 2014, he and Francis Gaspalou were able to calculate all 8 × 8 bimagic squares.
Until he retired in 2016, Trump worked as a teacher for mathematics and physics at the Gymnasium in Stein, Bavaria.
References
External links
Walter Trump's pages on magic series
Walter Trump's listings on the OEIS
Walter Trump's solutions for one of Martin Gardner's puzzles
Scientists from Bavaria
20th-century German mathematicians
Recreational mathematicians
Living people
Year of birth missing (living people)
21st-century German mathematicians
People from Fürth (district)
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https://en.wikipedia.org/wiki/159%20%28number%29
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159 (one hundred [and] fifty-nine) is a natural number following 158 and preceding 160.
In mathematics
159 is:
the sum of 3 consecutive prime numbers: 47 + 53 + 59.
a Woodall number.
equal to the sum of the squares of the digits of its own square in base 15.
Only 5 numbers (greater than 1) have this property in base 15, none in base 10.
written CLIX in Roman numeral, which spells a proper noun with multiple meanings.
Given 159, the Mertens function returns 0.
In astronomy
159 Aemilia is a large Main belt asteroid
NGC 159 is a galaxy in the constellation of Phoenix
The Saros number of the solar eclipse series which will begin on May 23, 2134 and end June 17, 3378. The duration of Saros series 159 is 1244.0 years, and it will contain 70 solar eclipses
The Saros number of the lunar eclipse series, which will begin on September 9, 2147 and end November 7, 3445. The duration of Saros series 159 is 1298.1 years, and it will contain 73 lunar eclipses
159P/LONEOS is a periodic comet in the Solar System
In geography
The state of Georgia has 159 counties
Sherwood No. 159, Saskatchewan is a rural municipality in Saskatchewan, Canada
In the military
Aero L-159 ALCA (Advanced Light Combat Aircraft) is a Czechoslovakian-built multi-role combat aircraft in service with the Czech Air Force
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy concrete barge during World War II
was a United States Navy concrete barge following World War II
was a United States Navy during World War II
was a United States Navy during World War II
In sports
In professional darts, 159 is the lowest score a player can achieve with no available checkout.
In transportation
The Alfa Romeo 159 compact executive car produced from 2005 to 2011
The Ferrari 159 S racecar
The Peugeot Type 159 was produced in 1919
The British Rail Class 159, a member of the Sprinter family), is a diesel multiple unit, produced from 1989 to 1993
London Buses route 159
TWA Flight 159, a Boeing 707, while on its takeoff roll from Greater Cincinnati Airport, passed Delta Flight 379, a DC-9 on the runway on November 6, 1967
In other fields
159 is also:
The year AD 159 or 159 BC
159 AH is a year in the Islamic calendar that corresponds to 775 – 776 CE
The atomic number of an element temporarily called Unpentennium.
The chemical element terbium has a stable Isotope of 159 nucleons
Financial Accounting Standards Board (FASB) Number 159, The Fair Value Option for Financial Assets and Financial Liabilities
159 can be dialled to contact a variety of banks in the United Kingdom directly. This has been implemented as a method of preventing financial scams.
See also
List of highways numbered 159
United Nations Security Council Resolution 159
United States Supreme Court cases, Volume 159
References
External links
Number Facts and Trivia: 159
The Number 159
VirtueScien
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https://en.wikipedia.org/wiki/195%20%28number%29
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195 (one hundred [and] ninety-five) is the natural number following 194 and preceding 196.
In mathematics
195 is:
the sum of eleven consecutive primes: 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37
the smallest number expressed as a sum of distinct squares in 16 different ways
a centered tetrahedral number
in the middle of a prime quadruplet (191, 193, 197, 199).
See also
195 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/Robust%20statistics
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Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly.
Introduction
Robust statistics seek to provide methods that emulate popular statistical methods, but are not unduly affected by outliers or other small departures from model assumptions. In statistics, classical estimation methods rely heavily on assumptions that are often not met in practice. In particular, it is often assumed that the data errors are normally distributed, at least approximately, or that the central limit theorem can be relied on to produce normally distributed estimates. Unfortunately, when there are outliers in the data, classical estimators often have very poor performance, when judged using the breakdown point and the influence function, described below.
The practical effect of problems seen in the influence function can be studied empirically by examining the sampling distribution of proposed estimators under a mixture model, where one mixes in a small amount (1–5% is often sufficient) of contamination. For instance, one may use a mixture of 95% a normal distribution, and 5% a normal distribution with the same mean but significantly higher standard deviation (representing outliers).
Robust parametric statistics can proceed in two ways:
by designing estimators so that a pre-selected behaviour of the influence function is achieved
by replacing estimators that are optimal under the assumption of a normal distribution with estimators that are optimal for, or at least derived for, other distributions: for example using the t-distribution with low degrees of freedom (high kurtosis; degrees of freedom between 4 and 6 have often been found to be useful in practice ) or with a mixture of two or more distributions.
Robust estimates have been studied for the following problems:
estimating location parameters
estimating scale parameters
estimating regression coefficients
estimation of model-states in models expressed in state-space form, for which the standard method is equivalent to a Kalman filter.
Definition
There are various definitions of a "robust statistic." Strictly speaking, a robust statistic is resistant to errors in the results, produced by deviations from assumptions (e.g., of normality). This means that if the assumptions are only approximately met, the robust estimator will still have a reasonable efficiency, and reaso
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