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https://en.wikipedia.org/wiki/Stable%20manifold
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In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.
Physical example
The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction. Imagining the rings to be sand or gravel particles ("dust") in orbit around Saturn, the tidal forces are such that any perturbations that push particles above or below the equatorial plane results in that particle feeling a restoring force, pushing it back into the plane. Particles effectively oscillate in a harmonic well, damped by collisions. The stable direction is perpendicular to the ring. The unstable direction is along any radius, where forces stretch and pull particles apart. Two particles that start very near each other in phase space will experience radial forces causing them to diverge, radially. These forces have a positive Lyapunov exponent; the trajectories lie on a hyperbolic manifold, and the movement of particles is essentially chaotic, wandering through the rings. The center manifold is tangential to the rings, with particles experiencing neither compression nor stretching. This allows second-order gravitational forces to dominate, and so particles can be entrained by moons or moonlets in the rings, phase locking to them. The gravitational forces of the moons effectively provide a regularly repeating small kick, each time around the orbit, akin to a kicked rotor, such as found in a phase-locked loop.
The discrete-time motion of particles in the ring can be approximated by the Poincaré map. The map effectively provides the transfer matrix of the system. The eigenvector associated with the largest eigenvalue of the matrix is the Frobenius–Perron eigenvector, which is also the invariant measure, i.e the actual density of the particles in the ring. All other eigenvectors of the transfer matrix have smaller eigenvalues, and correspond to decaying modes.
Definition
The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow.
Let be a topological space, and a homeomorphism. If is a fixed point for , the stable set of is defined by
and the unstable set of is defined by
Here, denotes the inverse of the function , i.e.
, where is the identity map on .
If is a periodic point of least period , then it is a fixed point of , and the stable and unstable sets of are defined by
and
Given a neighborhood of , the local stable and unstable sets of are defined by
and
If is metrizable, we can define the stable and unsta
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https://en.wikipedia.org/wiki/Weitzenb%C3%B6ck%20identity
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In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle, although the precise conditions under which such a formula exists are difficult to formulate. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis.
Riemannian geometry
In Riemannian geometry there are two notions of the Laplacian on differential forms over an oriented compact Riemannian manifold M. The first definition uses the divergence operator δ defined as the formal adjoint of the de Rham operator d:
where α is any p-form and β is any ()-form, and is the metric induced on the bundle of ()-forms. The usual form Laplacian is then given by
On the other hand, the Levi-Civita connection supplies a differential operator
where ΩpM is the bundle of p-forms. The Bochner Laplacian is given by
where is the adjoint of . This is also known as the connection or rough Laplacian.
The Weitzenböck formula then asserts that
where A is a linear operator of order zero involving only the curvature.
The precise form of A is given, up to an overall sign depending on curvature conventions, by
where
R is the Riemann curvature tensor,
Ric is the Ricci tensor,
is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form,
is the universal derivation inverse to θ on 1-forms.
Spin geometry
If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð2 on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator
As in the case of Riemannian manifolds, let . This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields:
where Sc is the scalar curvature. This result is also known as the Lichnerowicz formula.
Complex differential geometry
If M is a compact Kähler manifold, there is a Weitzenböck formula relating the -Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let
and
in a unitary frame at each point.
According to the Weitzenböck formula, if , then
where is an operator of order zero involving the curvature. Specifically, if in a unitary frame, then with k in the s-th place.
Other Weitzenböck identities
In conformal geometry there is a Weitzenböck formula relating a particular pair of differential operators defined on the tractor bundle. See Branson, T. and Gover, A.R., "Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature", Communications in Partial Differenti
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https://en.wikipedia.org/wiki/Edward%20G.%20Begle
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Edward Griffith Begle (November 27, 1914 – March 2, 1978) was a mathematician best known for his role as the director of the School Mathematics Study Group (SMSG), the primary group credited for developing what came to be known as The New Math. Begle was a topologist and a researcher in mathematics education who served as a member of the faculty of Stanford University, Princeton University, The University of Michigan, and Yale University. Begle was also elected as the secretary of the American Mathematical Society in 1951, and he held the position for 6 years.
Biography
Edward G. Begle was born November 27, 1914 in Saginaw, Michigan. Studying at the University of Michigan, Begle earned his A.B. in Mathematics in 1936 and his M.A. in 1938. Begle's early academic work was in the field of topology, which is where he earned his Ph.D. at Princeton, studying under Solomon Lefschetz in 1940. While Begle's contributions to the field of mathematical research are limited, among them is the first proof of the Vietoris theorem, which caused it to become commonly known as the Vietoris–Begle mapping theorem.
Begle departed Princeton a year after completing his doctorate to spend a year as a Fellow of the National Research Council, after which he joined the faculty of Yale in 1942. Begle's interest in mathematics education is apparent in his early mathematics texts, where the writing departs from the tradition at the time of writing textbooks addressed to accomplished mathematicians. Instead, Begle's introductory mathematics texts actually address freshman mathematicians, a revolutionary concept in teaching math. As Begle's stature increased as an educator within the field of mathematics, he gained notice within his field and was elected secretary of the American Mathematical Society in 1951.
In the wake of Sputnik in 1958, Begle gained the directorship of the School Mathematics Study Group, a post he would hold for 15 years. Under his leadership, the SMSG published numerous reports and studies, culminating in its series of books detailing the teaching revolution known as The New Math. It was in this capacity that in 1961, Begle took on an appointment as professor in the School of Education at Stanford as well as a courtesy appointment in the Stanford Department of Mathematics. Also in 1961, Begle was awarded with the Mathematical Association of America's highest honor: The Award for Distinguished Service to Mathematics.
At the time of his death in 1978, Begle was working on a compilation of the results of his 15-year tenure as the director of the SMSG and a culmination of his lifetime of experience in mathematics education. Published posthumously in 1979, Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature was listed by the National Council of Teachers of Mathematics as Begle's most influential work.
Bibliography
1951 Introductory calculus, with analytic geometry, Holt, Rinehart and Winston, revise
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https://en.wikipedia.org/wiki/Axiom%20A
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In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale. The importance of such systems is demonstrated by the chaotic hypothesis, which states that, 'for all practical purposes', a many-body thermostatted system is approximated by an Anosov system.
Definition
Let M be a smooth manifold with a diffeomorphism f: M→M. Then f is an axiom A diffeomorphism if
the following two conditions hold:
The nonwandering set of f, Ω(f), is a hyperbolic set and compact.
The set of periodic points of f is dense in Ω(f).
For surfaces, hyperbolicity of the nonwandering set implies the density of periodic points, but this is no longer true in higher dimensions. Nonetheless, axiom A diffeomorphisms are sometimes called hyperbolic diffeomorphisms, because the portion of M where the interesting dynamics occurs, namely, Ω(f), exhibits hyperbolic behavior.
Axiom A diffeomorphisms generalize Morse–Smale systems, which satisfy further restrictions (finitely many periodic points and transversality of stable and unstable submanifolds). Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.
Properties
Any Anosov diffeomorphism satisfies axiom A. In this case, the whole manifold M is hyperbolic (although it is an open question whether the non-wandering set Ω(f) constitutes the whole M).
Rufus Bowen showed that the non-wandering set Ω(f) of any axiom A diffeomorphism supports a Markov partition. Thus the restriction of f to a certain generic subset of Ω(f) is conjugated to a shift of finite type.
The density of the periodic points in the non-wandering set implies its local maximality: there exists an open neighborhood U of Ω(f) such that
Omega stability
An important property of Axiom A systems is their structural stability against small perturbations. That is, trajectories of the perturbed system remain in 1-1 topological correspondence with the unperturbed system. This property is important, in that it shows that Axiom A systems are not exceptional, but are in a sense 'robust'.
More precisely, for every C1-perturbation fε of f, its non-wandering set is formed by two compact, fε-invariant subsets Ω1 and Ω2. The first subset is homeomorphic to Ω(f) via a homeomorphism h which conjugates the restriction of f to Ω(f) with the restriction of fε to Ω1:
If Ω2 is empty then h is onto Ω(fε). If this is the case for every perturbation fε then f is called omega stable. A diffeomorphism f is omega stable if and only if it satisfies axiom A and the no-cycle condition (that an orbit, once having left an invariant subset, does not return).
See also
Ergodic flow
References
Ergodic theory
Diffeomorphisms
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https://en.wikipedia.org/wiki/Otto%20M.%20Nikodym
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Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician.
Education and career
Nikodym studied mathematics at the University of Lemberg (today's University of Lviv). Immediately after his graduation in 1911, he started his teaching job at a high school in Kraków where he remained until 1924. He eventually obtained his doctorate in 1925 from the University of Warsaw; he also spent an academic year (1926-1927) in Sorbonne. Nikodym taught at the Jagiellonian University in Kraków and University of Warsaw and at the Akademia Górnicza in Kraków in the years that followed. He moved to the United States in 1948 and joined the faculty of Kenyon College. He retired in 1966 and moved to Utica, New York, where he continued his research until retirement.
Personal life
Nikodym was born in 1887 in Demycze, a suburb of Zabłotów (in modern-day Ukraine), to a family with Polish, Czech, Italian and French roots. Orphaned at a young age, he was brought up by his maternal grandparents. In 1924, he married Stanisława Nikodym, the first Polish woman to obtain a PhD in mathematics.
Research works
Nikodym worked in a wide range of areas, but his best-known early work was his contribution to the development of the Lebesgue–Radon–Nikodym integral (see Radon–Nikodym theorem). His work in measure theory led him to an interest in abstract Boolean lattices. His work after coming to the United States centered on the theory of operators in Hilbert space, based on Boolean lattices, culminating in his The Mathematical Apparatus for Quantum-Theories. He was also interested in the teaching of mathematics.
See also
Nikodym set
Radon–Nikodym theorem
Radon–Nikodym property of a Banach space
List of Polish mathematicians
References
External links
MacTutor Entry
1887 births
1974 deaths
University of Lviv alumni
University of Warsaw alumni
University of Paris alumni
Academic staff of Jagiellonian University
Academic staff of the University of Warsaw
Polish emigrants to the United States
Nikodym, Otto Martin
Kenyon College faculty
20th-century Polish mathematicians
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https://en.wikipedia.org/wiki/Inhabited%20set
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In mathematics, a set is inhabited if there exists an element .
In classical mathematics, the property of being inhabited is equivalent to being non-empty. However, this equivalence is not valid in constructive or intuitionistic logic, and so this separate terminology is mostly used in the set theory of constructive mathematics.
Definition
In the formal language of first-order logic, set has the property of being if
Related definitions
A set has the property of being if , or equivalently . Here stands for the negation .
A set is if it is not empty, that is, if , or equivalently .
Theorems
Modus ponens implies , and taking any a false proposition for establishes that is always valid. Hence, any inhabited set is provably also non-empty.
Discussion
In constructive mathematics, the double-negation elimination principle is not automatically valid. In particular, an existence statement is generally stronger than its double-negated form. The latter merely expresses that the existence cannot be ruled out, in the strong sense that it cannot consistently be negated. In a constructive reading, in order for to hold for some formula , it is necessary for a specific value of satisfying to be constructed or known. Likewise, the negation of a universal quantified statement is in general weaker than an existential quantification of a negated statement. In turn, a set may be proven to be non-empty without one being able to prove it is inhabited.
Examples
Sets such as or are inhabited, as e.g. witnessed by . The set is empty and thus not inhabited. Naturally, the example section thus focuses on non-empty sets that are not provably inhabited.
It is easy to give examples for any simple set theoretical property, because logical statements can always be expressed as set theoretical ones, using an axiom of separation. For example, with a subset defined as , the proposition may always equivalently be stated as . The double-negated existence claim of an entity with a certain property can be expressed by stating that the set of entities with that property is non-empty.
Example relating to excluded middle
Define a subset via
Clearly and , and from the principle of non-contradiction one concludes . Further, and in turn
Already minimal logic proves , the double-negation for any excluded middle statement, which here is equivalent to . So by performing two contrapositions on the previous implication, one establishes . In words: It cannot consistently be ruled out that exactly one of the numbers and inhabits . In particular, the latter can be weakened to , saying is proven non-empty.
As example statements for , consider the consistency of the theory at hand, an infamous provenly theory-independent statement such as the continuum hypothesis, or, informally, an unknowable claim about the past or future. By design, these are chosen to be unprovable. A variant of this is to consider mathematiacal propositions that are merely not yet established
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https://en.wikipedia.org/wiki/Graded%20Lie%20algebra
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In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any parabolic Lie algebra is also a graded Lie algebra.
A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in the theory of Lie derivatives.
A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supersymmetric) setting, and then tensorizes to obtain the supersymmetric analog.
Still greater generalizations are possible to Lie algebras over a class of braided monoidal categories equipped with a coproduct and some notion of a gradation compatible with the braiding in the category. For hints in this direction, see Lie superalgebra#Category-theoretic definition.
Graded Lie algebras
In its most basic form, a graded Lie algebra is an ordinary Lie algebra , together with a gradation of vector spaces
such that the Lie bracket respects this gradation:
The universal enveloping algebra of a graded Lie algebra inherits the grading.
Examples
sl(2)
For example, the Lie algebra of trace-free 2 × 2 matrices is graded by the generators:
These satisfy the relations , , and . Hence with , , and , the decomposition presents as a graded Lie algebra.
Free Lie algebra
The free Lie algebra on a set X naturally has a grading, given by the minimum number of terms needed to generate the group element. This arises for example as the associated graded Lie algebra to the lower central series of a free group.
Generalizations
If is any commutative monoid, then the notion of a -graded Lie algebra generalizes that of an ordinary (-) graded Lie algebra so that the defining relations hold with the integers replaced by . In particular, any semisimple Lie algebra is graded by the root spaces of its adjoint representation.
Graded Lie superalgebras
A graded Lie superalgebra over a field k (not of characteristic 2) consists of a graded vector space E over k, along with a bilinear bracket operation
such that the following axioms are satisfied.
[-, -] respects the gradation of E:
(Symmetry) For all x in Ei and y in Ej,
(Jacobi identity) For all x in Ei, y in Ej, and z in Ek, (If k has characteristic 3, then the Jacobi identity must be supplemented with the condition for all x in Eodd.)
Note, for instance, that when E carries the triv
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https://en.wikipedia.org/wiki/CD-Cops
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CD-Cops was the first CD-ROM protection system to use the geometry of the CD-ROM media rather than a hidden "mark". It was invented in 1996 by Danish Link Data Security, known for its Cops Copylock key-diskette security used in the 1990s by Lotus 1-2-3.
Overview
As a copy (CD-R or CD-ROM) will have a different geometry, Data Position Measurement needs to be used for copies. The geometry is not known before CDs have been produced, therefore a CD-code expressing the layout of the CD-ROM must be entered the first time a user runs the protected software. Using a special production process in some cases the CD-code is embedded on the CD-ROM. CD-Cops is popular for encyclopaedias/dictionaries and business applications but not used as much for games.
DVD-Cops, based on the same principles, was the first DVD-ROM protection system, made in 1998.
References
Sources
Analysis
Report
cdmediaworld
PDF
article
External links
CD-Cops at linkdata.com
DVD-Cops at linkdata.com
Compact Disc and DVD copy protection
Digital rights management for macOS
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https://en.wikipedia.org/wiki/Universal%20quadratic%20form
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In mathematics, a universal quadratic form is a quadratic form over a ring that represents every element of the ring. A non-singular form over a field which represents zero non-trivially is universal.
Examples
Over the real numbers, the form x2 in one variable is not universal, as it cannot represent negative numbers: the two-variable form over R is universal.
Lagrange's four-square theorem states that every positive integer is the sum of four squares. Hence the form over Z is universal.
Over a finite field, any non-singular quadratic form of dimension 2 or more is universal.
Forms over the rational numbers
The Hasse–Minkowski theorem implies that a form is universal over Q if and only if it is universal over Qp for all p (where we include , letting Q∞ denote R). A form over R is universal if and only if it is not definite; a form over Qp is universal if it has dimension at least 4. One can conclude that all indefinite forms of dimension at least 4 over Q are universal.
See also
The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers.
References
Field (mathematics)
Quadratic forms
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https://en.wikipedia.org/wiki/Differentiation%20in%20Fr%C3%A9chet%20spaces
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In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between general topological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.
Mathematical details
Formally, the definition of differentiation is identical to the Gateaux derivative. Specifically, let and be Fréchet spaces, be an open set, and be a function. The directional derivative of in the direction is defined by
if the limit exists. One says that is continuously differentiable, or if the limit exists for all and the mapping
is a continuous map.
Higher order derivatives are defined inductively via
A function is said to be if It is or smooth if it is for every
Properties
Let and be Fréchet spaces. Suppose that is an open subset of is an open subset of and are a pair of functions. Then the following properties hold:
Fundamental theorem of calculus. If the line segment from to lies entirely within then
The chain rule. For all and
Linearity. is linear in More generally, if is then is multilinear in the 's.
Taylor's theorem with remainder. Suppose that the line segment between and lies entirely within If is then where the remainder term is given by
Commutativity of directional derivatives. If is then for every permutation σ of
The proofs of many of these properties rely fundamentally on the fact that it is possible to define the Riemann integral of continuous curves in a Fréchet space.
Smooth mappings
Surprisingly, a mapping between open subset of Fréchet spaces is smooth (infinitely often differentiable) if it maps smooth curves to smooth curves; see Convenient analysis.
Moreover, smooth curves in spaces of smooth functions are just smooth functions of one variable more.
Consequences in differential geometry
The existence of a chain rule allows for the definition of a manifold modeled on a Frèchet space: a Fréchet manifold. Furthermore, the linearity of the derivative implies that there is an analog of the tangent bundle for Fréchet manifolds.
Tame Fréchet spaces
Frequently the Fréchet spaces that arise in practical applications of the derivative enjoy an additional property: they are tame. Roughly speaking, a tame Fréchet space is one which is almost a Banach space. On tame spaces, it is possible to define a preferred class of mappings, known as tame maps. On the category of tame spaces
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https://en.wikipedia.org/wiki/Iterated%20binary%20operation
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In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the multiplication operation to the product operation. Other operations, e.g., the set-theoretic operations union and intersection, are also often iterated, but the iterations are not given separate names. In print, summation and product are represented by special symbols; but other iterated operators often are denoted by larger variants of the symbol for the ordinary binary operator. Thus, the iterations of the four operations mentioned above are denoted
and , respectively.
More generally, iteration of a binary function is generally denoted by a slash: iteration of over the sequence is denoted by , following the notation for reduce in Bird–Meertens formalism.
In general, there is more than one way to extend a binary operation to operate on finite sequences, depending on whether the operator is associative, and whether the operator has identity elements.
Definition
Denote by aj,k, with and , the finite sequence of length of elements of S, with members (ai), for . Note that if , the sequence is empty.
For , define a new function Fl on finite nonempty sequences of elements of S, where
Similarly, define
If f has a unique left identity e, the definition of Fl can be modified to operate on empty sequences by defining the value of Fl on an empty sequence to be e (the previous base case on sequences of length 1 becomes redundant). Similarly, Fr can be modified to operate on empty sequences if f has a unique right identity.
If f is associative, then Fl equals Fr, and we can simply write F. Moreover, if an identity element e exists, then it is unique (see Monoid).
If f is commutative and associative, then F can operate on any non-empty finite multiset by applying it to an arbitrary enumeration of the multiset. If f moreover has an identity element e, then this is defined to be the value of F on an empty multiset. If f is idempotent, then the above definitions can be extended to finite sets.
If S also is equipped with a metric or more generally with topology that is Hausdorff, so that the concept of a limit of a sequence is defined in S, then an infinite iteration on a countable sequence in S is defined exactly when the corresponding sequence of finite iterations converges. Thus, e.g., if a0, a1, a2, a3, … is an infinite sequence of real numbers, then the infinite product is defined, and equal to if and only if that limit exists.
Non-associative binary operation
The general, non-associative binary operation is given by a magma. The act of iterating on a non-associative binary operation may be represented as a binary tree.
Notation
Iterated binary operations are used to represent an operation that will be repeated over a set subject to some constra
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https://en.wikipedia.org/wiki/Refactorable%20number
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A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that . The first few refactorable numbers are listed in as
1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...
For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.
Properties
Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation has solutions only if is a refactorable number, where is the greatest common divisor function.
Let be the number of refactorable numbers which are at most . The problem of determining an asymptotic for is open. Spiro has proven that
There are still unsolved problems regarding refactorable numbers. Colton asked if there are there arbitrarily large such that both and are refactorable. Zelinsky wondered if there exists a refactorable number , does there necessarily exist such that is refactorable and .
History
First defined by Curtis Cooper and Robert E. Kennedy where they showed that the tau numbers have natural density zero, they were later rediscovered by Simon Colton using a computer program he had made which invents and judges definitions from a variety of areas of mathematics such as number theory and graph theory. Colton called such numbers "refactorable". While computer programs had discovered proofs before, this discovery was one of the first times that a computer program had discovered a new or previously obscure idea. Colton proved many results about refactorable numbers, showing that there were infinitely many and proving a variety of congruence restrictions on their distribution. Colton was only later alerted that Kennedy and Cooper had previously investigated the topic.
See also
Divisor function
References
Integer sequences
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https://en.wikipedia.org/wiki/Correlogram
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In the analysis of data, a correlogram is a chart of correlation statistics.
For example, in time series analysis, a plot of the sample autocorrelations versus (the time lags) is an autocorrelogram.
If cross-correlation is plotted, the result is called a cross-correlogram.
The correlogram is a commonly used tool for checking randomness in a data set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero.
In addition, correlograms are used in the model identification stage for Box–Jenkins autoregressive moving average time series models. Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness.
In multivariate analysis, correlation matrices shown as color-mapped images may also be called "correlograms" or "corrgrams".
Applications
The correlogram can help provide answers to the following questions:
Are the data random?
Is an observation related to an adjacent observation?
Is an observation related to an observation twice-removed? (etc.)
Is the observed time series white noise?
Is the observed time series sinusoidal?
Is the observed time series autoregressive?
What is an appropriate model for the observed time series?
Is the model
valid and sufficient?
Is the formula valid?
Importance
Randomness (along with fixed model, fixed variation, and fixed distribution) is one of the four assumptions that typically underlie all measurement processes. The randomness assumption is critically important for the following three reasons:
Most standard statistical tests depend on randomness. The validity of the test conclusions is directly linked to the validity of the randomness assumption.
Many commonly used statistical formulae depend on the randomness assumption, the most common formula being the formula for determining the standard error of the sample mean:
where s is the standard deviation of the data. Although heavily used, the results from using this formula are of no value unless the randomness assumption holds.
For univariate data, the default model is
If the data are not random, this model is incorrect and invalid, and the estimates for the parameters (such as the constant) become nonsensical and invalid.
Estimation of autocorrelations
The autocorrelation coefficient at lag h is given by
where ch is the autocovariance function
and c0 is the variance function
The resulting value of rh will range between −1 and +1.
Alternate estimate
Some sources may use the following formula for the autocovariance function:
Although this definition has less bias, the (1/N) formulation has some desirable statistical properties and is the form most commonly used in the statistics literature. See pages 20 and 49–50 in Chatfield for details.
I
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https://en.wikipedia.org/wiki/Monogon
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In geometry, a monogon, also known as a henagon, is a polygon with one edge and one vertex. It has Schläfli symbol {1}.
In Euclidean geometry
In Euclidean geometry a monogon is a degenerate polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon.
In spherical geometry
In spherical geometry, a monogon can be constructed as a vertex on a great circle (equator). This forms a dihedron, {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a hosohedron, {2,1} has two antipodal vertices at the poles, one 360° lune face, and one edge (meridian) between the two vertices.
See also
Digon
References
Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955
Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc.
Polygons by the number of sides
1 (number)
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https://en.wikipedia.org/wiki/Divisia%20monetary%20aggregates%20index
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In econometrics and official statistics, and particularly in banking, the Divisia monetary aggregates index is an index of money supply. It uses Divisia index methods.
Background
The monetary aggregates used by most central banks (notably the US Federal Reserve) are simple-sum indexes in which all monetary components are assigned the same weight:
in which is one of the monetary components of the monetary aggregate . The summation index implies that all monetary components contribute equally to the money total, and it views all components as dollar-for-dollar perfect substitutes. It has been argued that such an index does not weight such components in a way that properly summarizes the services of the quantities of money.
There have been many attempts at weighting monetary components within a simple-sum aggregate. An index can rigorously apply microeconomic- and aggregation-theoretic foundations in the construction of monetary aggregates. That approach to monetary aggregation was derived and advocated by William A. Barnett (1980) and has led to the construction of monetary aggregates based on Diewert's (1976) class of superlative quantity index numbers. The new aggregates are called the Divisia aggregates or Monetary Services Indexes. Salam Fayyad's 1986 PhD dissertation did early research with those aggregates using U.S. data.
This index is a discrete-time approximation with this definition:
Here, the growth rate of the aggregate is the weighted average of the growth rates of the component quantities. The discrete time Divisia weights are defined as the expenditure shares averaged over the two periods of the change
for , where
is the expenditure share of asset during period , and is the user cost of asset , derived by Barnett (1978),
Which is the opportunity cost of holding a dollar's worth of the th asset. In the last equation, is the market yield on the th asset, and is the yield available on a benchmark asset, held only to carry wealth between different time periods.
In the literature on aggregation and index number theory, the Divisia approach to monetary aggregation, , is widely viewed as a viable and theoretically appropriate alternative to the simple-sum approach. See, for example, International Monetary Fund (2008), Macroeconomic Dynamics (2009), and Journal of Econometrics (2011). The simple-sum approach, , which is still in use by some central banks, adds up imperfect substitutes, such as currency and non-negotiable certificates of deposit, without weights reflecting differences in their contributions to the economy's liquidity. A primary source of theory, applications, and data from the aggregation-theoretic approach to monetary aggregation is the Center for Financial Stability in New York City. More details regarding the Divisia approach to monetary aggregation are provided by Barnett, Fisher, and Serletis (1992), Barnett and Serletis (2000), and Serletis (2007). Divisia Monetary Aggregates are available for the Unit
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https://en.wikipedia.org/wiki/Unitary%20divisor
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In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.
The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931), who used the term block divisor.
Example
5 is a unitary divisor of 60, because 5 and have only 1 as a common factor.
On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2.
Sum of unitary divisors
The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n):
If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.
Properties
Number 1 is a unitary divisor of every natural number.
The number of unitary divisors of a number n is 2k, where k is the number of distinct prime factors of n.
This is because each integer N > 1 is the product of positive powers prp of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the prime divisors {p} of N,
of the prime powers prp for p ∈ S. If there are k prime factors, then there are exactly 2k subsets S, and the statement follows.
The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise.
Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is
Every divisor of n is unitary if and only if n is square-free.
Odd unitary divisors
The sum of the k-th powers of the odd unitary divisors is
It is also multiplicative, with Dirichlet generating function
Bi-unitary divisors
A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. This concept originates from D. Suryanarayana (1972). [The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag].
The number of bi-unitary divisors of n is a multiplicative function of n with average order where
A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.
OEIS sequences
References
Section B3.
Section 4.2
External links
Mathoverflow | Boolean ring of unitary divisors
Number theory
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https://en.wikipedia.org/wiki/Continuum%20function
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In mathematics, the continuum function is , i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality.
See also
Continuum hypothesis
Cardinality of the continuum
Beth number
Easton's theorem
Gimel function
Cardinal numbers
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https://en.wikipedia.org/wiki/Superfinishing
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Superfinishing, also known as micromachining, microfinishing, and short-stroke honing, is a metalworking process that improves surface finish and workpiece geometry. This is achieved by removing just the thin amorphous surface layer left by the last process with an abrasive stone or tape; this layer is usually about 1 μm in magnitude. Superfinishing, unlike polishing which produces a mirror finish, creates a cross-hatch pattern on the workpiece.
The superfinishing process was developed by the Chrysler Corporation in 1934.
Process
After a metal piece is ground to an initial finish, it is superfinished with a finer grit solid abrasive. The abrasive is oscillated or rotated while the workpiece is rotated in the opposite direction; these motions are what causes the cross-hatching. The geometry of the abrasive depends on the geometry of the workpiece surface; a stone (rectangular shape) is for cylindrical surfaces and cups and wheels are used for flat and spherical surfaces. A lubricant is used to minimize heat production, which can alter the metallurgical properties, and to carry away the swarf; kerosene is a common lubricant.
The abrasive cuts the surface of the workpiece in three phases. The first phase is when the abrasive first contacts the workpiece surface: dull grains of the abrasive fracture and fall away leaving a new sharp cutting surface. In the second phase the abrasive "self dresses" while most of the stock is being removed. Finally, the abrasive grains become dull as they work which improves the surface geometry.
The average rotational speed of abrasive wheel and/or workpiece is 1 to 15 surface m/min, with 6 to 14 m/min preferred; this is much slower compared to grinding speeds around 1800 to 3500 m/min. The pressure applied to the abrasive is very light, usually between , but can be as high as . Honing is usually and grinding is between . When a stone is used it is oscillated at 200 to 1000 cycles with an amplitude of .
Superfinishing can give a surface finish of 0.01 μm.
Types
There are three types superfinishing: Through-feed, plunge, and wheels.
Through-feed This type of superfinishing is used for cylindrical workpieces. The workpiece is rotated between two drive rollers, which also move the machine as well. Four to eight progressively finer abrasive stones are used to superfinish the workpiece. The stones contact the workpiece at a 90° angle and are oscillated axially. Examples of parts that would be produced by process include tapered rolls, piston pins, shock absorber rods, shafts, and needles.
Plunge This type is used to finish irregularly shaped surfaces. The workpiece is rotated while the abrasive plunges onto the desired surface.
Wheels Abrasive cups or wheels are used to superfinish flat and spherical surfaces. The wheel and workpiece are rotated in opposite directions, which creates the cross-hatching. If the two are parallel then the result if a flat finish, but if the wheel is tilted slightly a convex or conc
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https://en.wikipedia.org/wiki/Lamplighter%20group
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In mathematics, the lamplighter group L of group theory is the restricted wreath product
Introduction
The name of the group comes from viewing the group as acting on a doubly infinite sequence of street lamps each of which may be on or off, and a lamplighter standing at some lamp An equivalent description for this, called the base group of is
an infinite direct sum of copies of the cyclic group where corresponds to a light that is off and corresponds to a light that is on, and the direct sum is used to ensure that only finitely many lights are on at once. An element of gives the position of the lamplighter, and to encode which bulbs are illuminated.
There are two generators for the group: the generator t increments k, so that the lamplighter moves to the next lamp (t -1 decrements k), while the generator a means that the state of lamp lk is changed (from off to on or from on to off.) Group multiplication is done by "following" these operations.
We may assume that only finitely many lamps are lit at any time, since the action of any element of L changes at most finitely many lamps. The number of lamps lit is, however, unbounded. The group action is thus similar to the action of a Turing machine in two ways. The Turing machine has unbounded memory, but has only used a finite amount of memory at any given time. Moreover, the Turing machine's head is analogous to the lamplighter.
Presentation
The standard presentation for the lamplighter group arises from the wreath product structure
, which may be simplified to
.
The generators a and t are intrinsic to the group's notable growth rate, though they are sometimes replaced with a and at, changing the logarithm of the growth rate by at most a factor of 2.
This presentation is not finite (it has infinitely many relations). In fact there is no finite presentation for the lamplighter group, that is, it is not finitely presented.
Matrix representation
Allowing to be a formal variable, the lamplighter group is isomorphic to the group of matrices
where and ranges over all polynomials in
Using the presentations above, the isomorphism is given by
Generalizations
One can also define lamplighter groups , with , so that "lamps" may have more than just the option of "off" and "on." The classical lamplighter group is recovered when
References
Further reading
Volodymyr Nekrashevych, 2005, Self-Similar Groups, Mathematical Surveys and Monographs v. 117, American Mathematical Society, .
Solvable groups
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https://en.wikipedia.org/wiki/Strict%20differentiability
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In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both points used in the difference quotient to "move".
Basic definition
The simplest setting in which strict differentiability can be considered, is that of a real-valued function defined on an interval I of the real line.
The function f:I → R is said strictly differentiable in a point a ∈ I if
exists, where is to be considered as limit in , and of course requiring .
A strictly differentiable function is obviously differentiable, but the converse is wrong, as can be seen from the counter-example
One has however the equivalence of strict differentiability on an interval I, and being of differentiability class (i.e. continuously differentiable).
In analogy with the Fréchet derivative, the previous definition can be generalized to the case where R is replaced by a Banach space E (such as ), and requiring existence of a continuous linear map L such that
where is defined in a natural way on E × E.
Motivation from p-adic analysis
In the p-adic setting, the usual definition of the derivative fails to have certain desirable properties. For instance, it is possible for a function that is not locally constant to have zero derivative everywhere. An example of this is furnished by the function F: Zp → Zp, where Zp is the ring of p-adic integers, defined by
One checks that the derivative of F, according to usual definition of the derivative, exists and is zero everywhere, including at x = 0. That is, for any x in Zp,
Nevertheless F fails to be locally constant at the origin.
The problem with this function is that the difference quotients
do not approach zero for x and y close to zero. For example, taking x = pn − p2n and y = pn, we have
which does not approach zero. The definition of strict differentiability avoids this problem by imposing a condition directly on the difference quotients.
Definition in p-adic case
Let K be a complete extension of Qp (for example K = Cp), and let X be a subset of K with no isolated points. Then a function F : X → K is said to be strictly differentiable at x = a if the limit
exists.
References
Number theory
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https://en.wikipedia.org/wiki/Walter%20Rudin
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Walter Rudin (May 2, 1921 – May 20, 2010) was an Austrian-American mathematician and professor of Mathematics at the University of Wisconsin–Madison.
In addition to his contributions to complex and harmonic analysis, Rudin was known for his mathematical analysis textbooks: Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis. Rudin wrote Principles of Mathematical Analysis only two years after obtaining his Ph.D. from Duke University, while he was a C. L. E. Moore Instructor at MIT. Principles, acclaimed for its elegance and clarity, has since become a standard textbook for introductory real analysis courses in the United States.
Rudin's analysis textbooks have also been influential in mathematical education worldwide, having been translated into 13 languages, including Russian, Chinese, and Spanish.
Biography
Rudin was born into a Jewish family in Austria in 1921. He was enrolled for a period of time at a Swiss boarding school, the Institut auf dem Rosenberg, where he was part of a small program that prepared its students for entry to British universities. His family fled to France after the Anschluss in 1938.
When France surrendered to Germany in 1940, Rudin fled to England and served in the Royal Navy for the rest of World War II, after which he left for the United States. He obtained both his B.A. in 1947 and Ph.D. in 1949 from Duke University. After his Ph.D., he was a C.L.E. Moore instructor at MIT. He briefly taught at the University of Rochester before becoming a professor at the University of Wisconsin–Madison where he remained for 32 years. His research interests ranged from harmonic analysis to complex analysis.
In 1970 Rudin was an Invited Speaker at the International Congress of Mathematicians in Nice. He was awarded the Leroy P. Steele Prize for Mathematical Exposition in 1993 for authorship of the now classic analysis texts, Principles of Mathematical Analysis and Real and Complex Analysis. He received an honorary degree from the University of Vienna in 2006.
In 1953, he married fellow mathematician Mary Ellen Estill, known for her work in set-theoretic topology. The two resided in Madison, Wisconsin, in the eponymous Walter Rudin House, a home designed by architect Frank Lloyd Wright. They had four children.
Rudin died on May 20, 2010, after suffering from Parkinson's disease.
Selected publications
Ph.D. thesis
Selected research articles
"Totally real Klein bottles in
Books
Textbooks:
(1953; 3rd ed., 1976, 342 pp.)
(1966; 3rd ed., 1987, 416 pp.)
(1973; 2nd ed., 1991, 424 pp.)
Monographs:
(1962)
(1969)
Function Theory in the Unit Ball of . (1980)
Autobiography:
(1991)
Major awards
Steele Prize for Mathematical Exposition (1993)
See also
Helson–Kahane–Katznelson–Rudin theorem
Rudin–Shapiro sequence
Rudin's conjecture
References
External links
UW Mathematics Dept obituary
MathDL obituary
Photos of Rudin Residence
Walter B. Rudin, "Set Theory: An Offsprin
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https://en.wikipedia.org/wiki/Limit%20set
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In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.
Types
fixed points
periodic orbits
limit cycles
attractors
In general, limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact -limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.
Definition for iterated functions
Let be a metric space, and let be a continuous function. The -limit set of , denoted by , is the set of cluster points of the forward orbit of the iterated function . Hence, if and only if there is a strictly increasing sequence of natural numbers such that as . Another way to express this is
where denotes the closure of set . The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that
If is a homeomorphism (that is, a bicontinuous bijection), then the -limit set is defined in a similar fashion, but for the backward orbit; i.e. .
Both sets are -invariant, and if is compact, they are compact and nonempty.
Definition for flows
Given a real dynamical system with flow , a point , we call a point y an -limit point of if there exists a sequence in so that
.
For an orbit of , we say that is an -limit point of , if it is an -limit point of some point on the orbit.
Analogously we call an -limit point of if there exists a sequence in so that
.
For an orbit of , we say that is an -limit point of , if it is an -limit point of some point on the orbit.
The set of all -limit points (-limit points) for a given orbit is called -limit set (-limit set) for and denoted ().
If the -limit set (-limit set) is disjoint from the orbit , that is (), we call () a ω-limit cycle (α-limit cycle).
Alternatively the limit sets can be defined as
and
Examples
For any periodic orbit of a dynamical system,
For any fixed point of a dynamical system,
Properties
and are closed
if is compact then and are nonempty, compact and connected
and are -invariant, that is and
See also
Julia set
Stable set
Limit cycle
Periodic point
Non-wandering set
Kleinian group
References
Further reading
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https://en.wikipedia.org/wiki/Bar%20induction
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Bar induction is a reasoning principle used in intuitionistic mathematics, introduced by L. E. J. Brouwer. Bar induction's main use is the intuitionistic derivation of the fan theorem, a key result used in the derivation of the uniform continuity theorem.
It is also useful in giving constructive alternatives to other classical results.
The goal of the principle is to prove properties for all infinite sequences of natural numbers (called choice sequences in intuitionistic terminology), by inductively reducing them to properties of finite lists. Bar induction can also be used to prove properties about all choice sequences in a spread (a special kind of set).
Definition
Given a choice sequence , any finite sequence of elements of this sequence is called an initial segment of this choice sequence.
There are three forms of bar induction currently in the literature, each one places certain restrictions on a pair of predicates and the key differences are highlighted using bold font.
Decidable bar induction (BID)
Given two predicates and on finite sequences of natural numbers such that all of the following conditions hold:
every choice sequence contains at least one initial segment satisfying at some point (this is expressed by saying that is a bar);
is decidable (i.e. our bar is decidable);
every finite sequence satisfying also satisfies (so holds for every choice sequence beginning with the aforementioned finite sequence);
if all extensions of a finite sequence by one element satisfy , then that finite sequence also satisfies (this is sometimes referred to as being upward hereditary);
then we can conclude that holds for the empty sequence (i.e. A holds for all choice sequences starting with the empty sequence).
This principle of bar induction is favoured in the works of, A. S. Troelstra, S. C. Kleene and Albert Dragalin.
Thin bar induction (BIT)
Given two predicates and on finite sequences of natural numbers such that all of the following conditions hold:
every choice sequence contains a unique initial segment satisfying at some point (i.e. our bar is thin);
every finite sequence satisfying also satisfies ;
if all extensions of a finite sequence by one element satisfy , then that finite sequence also satisfies ;
then we can conclude that holds for the empty sequence.
This principle of bar induction is favoured in the works of Joan Moschovakis and is (intuitionistically) provably equivalent to decidable bar induction.
Monotonic bar induction (BIM)
Given two predicates and on finite sequences of natural numbers such that all of the following conditions hold:
every choice sequence contains at least one initial segment satisfying at some point;
once a finite sequence satisfies , then every possible extension of that finite sequence also satisfies (i.e. our bar is monotonic);
every finite sequence satisfying also satisfies ;
if all extensions of a finite sequence by one element satisfy , then that finite seq
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https://en.wikipedia.org/wiki/ITT
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ITT may refer to:
Communication
Infantry-Tank Telephone, a device allowing infantrymen to speak to the occupants of armoured vehicles.
Mathematics
Intuitionistic type theory, other name of Martin-Löf Type Theory
Intensional type theory
Business
ITT Inc. (formerly International Telephone & Telegraph), US
Invitation to tender for a contract
ITT Semiconductors
Education
ITT Technical Institute, US
Former Institute of Technology, Tallaght, Dublin, Ireland
Institute of Technology, Tralee, Ireland
Media
Cousin Itt, of the fictional Addams Family
"I.T.T (International Thief Thief)", a political screed about ITT Corp. by Fela Kuti
Medicine
Insulin tolerance test
Intention to treat analysis in medicine
Intermittent testicular torsion
Sport
Individual time trial in bicycle racing
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https://en.wikipedia.org/wiki/Stability%20theory
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In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance.
In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.
Overview in dynamical systems
Many parts of the qualitative theory of differential equations and dynamical systems deal with asymptotic properties of solutions and the trajectories—what happens with the system after a long period of time. The simplest kind of behavior is exhibited by equilibrium points, or fixed points, and by periodic orbits. If a particular orbit is well understood, it is natural to ask next whether a small change in the initial condition will lead to similar behavior. Stability theory addresses the following questions: Will a nearby orbit indefinitely stay close to a given orbit? Will it converge to the given orbit? In the former case, the orbit is called stable; in the latter case, it is called asymptotically stable and the given orbit is said to be attracting.
An equilibrium solution to an autonomous system of first order ordinary differential equations is called:
stable if for every (small) , there exists a such that every solution having initial conditions within distance i.e. of the equilibrium remains within distance i.e. for all .
asymptotically stable if it is stable and, in addition, there exists such that whenever then as .
Stability means that the trajectories do not change too much under small perturbations. The opposite situation, where a nearby orbit is getting repelled from the given orbit, is also of interest. In general, perturbing the initial state in some directions results in the trajectory asymptotically approaching the given one and in other directions to the trajectory getting away from it. There may also be directions for which the behavior of the perturbed orbit is more complicated (neither converging nor escaping completely), and then stability theory does not give sufficient information about the dynamics.
One of the key ideas in stability theory is that the quali
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https://en.wikipedia.org/wiki/Paul%20Epstein
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Paul Epstein (July 24, 1871 – August 11, 1939) was a German mathematician. He was known for his contributions to number theory, in particular the Epstein zeta function.
Epstein was born and brought up in Frankfurt, where his father was a professor. He received his PhD in 1895 from the University of Strasbourg. From 1895 to 1918 he was a Privatdozent at the University in Strasbourg, which at that time was part of the German Empire. At the end of World War I the city of Strasbourg reverted to France, and Epstein, being German, had to return to Frankfurt.
Epstein was appointed to a non-tenured post at the university and he lectured in Frankfurt from 1919. Later he was appointed professor at Frankfurt. However, after the Nazis came to power in Germany he lost his university position. Because of his age (68) he was unable to find a new position abroad, and finally committed suicide by barbital overdose at Dornbusch, fearing Gestapo torture because he was a Jew.
External links
1871 births
1939 suicides
1939 deaths
20th-century German mathematicians
Number theorists
German Jews who died in the Holocaust
Drug-related suicides
Suicides in Nazi Germany
Scientists from Frankfurt
University of Strasbourg alumni
Academic staff of the University of Strasbourg
Academic staff of Goethe University Frankfurt
Suicides by Jews during the Holocaust
Barbiturates-related deaths
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https://en.wikipedia.org/wiki/Stable%20manifold%20theorem
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In mathematics, especially in the study of dynamical systems and differential equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly states that the existence of a local diffeomorphism near a fixed point implies the existence of a local stable center manifold containing that fixed point. This manifold has dimension equal to the number of eigenvalues of the Jacobian matrix of the fixed point that are less than 1.
Stable manifold theorem
Let
be a smooth map with hyperbolic fixed point at . We denote by the stable set and by the unstable set of .
The theorem states that
is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of at .
is a smooth manifold and its tangent space has the same dimension as the unstable space of the linearization of at .
Accordingly is a stable manifold and is an unstable manifold.
See also
Center manifold theorem
Lyapunov exponent
Notes
References
External links
Dynamical systems
Theorems in dynamical systems
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https://en.wikipedia.org/wiki/280%20%28number%29
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280 (two hundred [and] eighty) is the natural number after 279 and before 281.
In mathematics
The denominator of the eighth harmonic number, 280 is an octagonal number. 280 is the smallest octagonal number that is a half of another octagonal number.
There are 280 plane trees with ten nodes.
As a consequence of this, 18 people around a round table can shake hands with each other in non-crossing ways, in 280 different ways (this includes rotations).
Integers from 281 to 289
281
282
283
284
285
286
287
288
289
References
Integers
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https://en.wikipedia.org/wiki/290%20%28number%29
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290 (two hundred [and] ninety) is the natural number following 289 and preceding 291.
In mathematics
The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes (67 + 71 + 73 + 79). The sum of the squares of the divisors of 17 is 290.
Not only is it a nontotient and a noncototient, it is also an untouchable number.
290 is the 16th member of the Mian–Chowla sequence; it can not be obtained as the sum of any two previous terms in the sequence.
See also the Bhargava–Hanke 290 theorem.
Integers from 291 to 299
291
292
293
294
295
296
296 = 23·37, a refactorable number, unique period in base 2, the number of regions formed by drawing the line segments connecting any two of the 12 perimeter points of an 2 times 4 grid of squares (illustration) , and the number of surface points on a 83 cube.
297
297 = 33·11, the number of integer partitions of 17, a decagonal number, and a Kaprekar number
298
298 = 2·149, is nontotient, noncototient, and the number of polynomial symmetric functions of matrix of order 6 under separate row and column permutations
299
299 = 13·23, a highly cototient number, a self number, and the twelfth cake number
References
Integers
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https://en.wikipedia.org/wiki/Karl%20Menninger%20%28mathematics%29
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Karl Menninger (October 6, 1898 – October 2, 1963) was a German teacher of and writer about mathematics. His major work was Zahlwort und Ziffer (1934,; English trans., Number Words and Number Symbols), about non-academic mathematics in much of the world. (The omission of Africa was rectified by Claudia Zaslavsky in her book Africa Counts.)
References
Dauben, Joseph Warren, and Christoph Scriba, eds. (2002), Writing the History of Mathematics, Birkhäuser, Basel, page 483.
Menninger, Karl (1934), Zahlwort und Ziffer. Revised edition (1958). Göttingen: Vandenhoeck and Ruprecht.
Menninger, Karl (1969), Number Words and Number Symbols. Cambridge, Mass.: The M.I.T. Press.
German historians of mathematics
Ethnomathematicians
20th-century German mathematicians
1898 births
1963 deaths
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https://en.wikipedia.org/wiki/Manhattan%20Center%20for%20Science%20and%20Mathematics
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Manhattan Center for Science and Mathematics (abbreviated as MCSM) is a public high school at East 116th Street between Pleasant Avenue and FDR Drive in East Harlem, within Upper Manhattan, New York City.
The school building, which was formerly Benjamin Franklin High School, was designated a New York City landmark by the New York City Landmarks Preservation Commission on May 29, 2018.
History
The precursor of MCSM in the same building, Benjamin Franklin High School opened in 1934 and was sited at 200 Pleasant Avenue, between 114th Street and 116th Street. A long-time principal there was pioneering educational theorist Leonard Covello, the city's first Italian-American principal.
The New York City Board of Education shuttered the school in June 1982 for performance issues and converted the building into a four-year high school, the Manhattan Center for Science and Mathematics, and a grade 6-8 middle school, the Isaac Newton Middle School for Math and Science, effective September 1982.
Description
Like all New York City high schools, admission is by application. Admission priority for Manhattan Center is given first to students attending the Isaac Newton Junior High School, which shares the campus with Manhattan Center; second to students residing in District 4; and then to other residents citywide.
The academic performance of this school is extremely high, as measured by New York State Regents Examinations scores, scholarship rates and a 95% graduation rate. MCSM is consistently one of the highest performing schools in the State of New York. In 2007, David Jimenez became the Principal. In 2009, MCSM graduated 97% of its students. Students graduating from Manhattan Center have attended top-notch colleges, including Ivy League colleges.
The curriculum includes Advanced Placement courses and special programs, and research and internship opportunities. The school offers AP World History, AP Art History, AP Macroeconomics, AP US Government, AP US History, AP English Language and Composition, AP English Literature and Composition, AP Calculus AB, AP Calculus BC, AP Physics C, AP Biology, AP Chemistry, AP Statistics, AP Computer Science Principles, AP Spanish Language and Culture and AP Spanish Literature and Culture. There are several honors and accelerated courses.
Manhattan Center partners with institutions of higher education, such as New York University, Columbia University, Cornell and CUNY, to offer courses in science, mathematics and humanities. Through these partnerships and Mount Sinai Hospital, Metropolitan Hospital Center, General Electric, Sponsors for Educational Opportunity, American Globe Theatre, and Manhattan Theatre Club there is a wide range of opportunities for one-on-one mentoring, internship experiences, academic enrichment and summer programs. They also provide support services, and a variety of sports, clubs and leadership activities for students.
Notable alumni
Franklin High:
Richie Adams (1980)
Walter Berry (1982)
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https://en.wikipedia.org/wiki/Rothenberg%20propriety
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In diatonic set theory, Rothenberg propriety is an important concept, lack of contradiction and ambiguity, in the general theory of musical scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it coherence.
"Rothenberg calls a scale 'strictly proper' if it possesses a generic ordering, 'proper' if it admits ambiguities but no contradictions, and 'improper' if it admits contradictions." A scale is strictly proper if all two step intervals are larger than any one step interval, all three step intervals are larger than any two step interval and so on. For instance with the diatonic scale, the one step intervals are the semitone (1) and tone (2), the two step intervals are the minor (3) and major (4) third, the three step intervals are the fourth (5) and tritone (6), the four step intervals are the fifth (7) and tritone (6), the five step intervals are the minor (8) and major (9) sixth, and the six step intervals are the minor (t) and major (e) seventh. So it's not strictly proper because the three step intervals and the four step intervals share an interval size (the tritone), causing ambiguity ("two [specific] intervals, that sound the same, map onto different codes [general intervals]"). Such a scale is just called "proper".
For example, the major pentatonic scale is strictly proper:
The pentatonic scales which are proper, but not strictly, are:
{0,1,4,6,8} (Lydian chord)
{0,2,4,6,8} (whole tone scale)
{0,1,4,6,9} (gamma chord)
{0,2,4,6,9} (dominant ninth chord)
{0,1,3,6,9} (dominant minor ninth chord)
The one strictly proper pentatonic scale:
{0,2,4,7,9} (major pentatonic)
The heptatonic scales which are proper, but not strictly, are:
{0,1,3,4,6,8,9} (harmonic minor scale)
{0,1,3,5,6,8,t} (diatonic scale)
{0,1,3,4,6,8,t} (Altered scale)
{0,1,2,4,6,8,t} (Major Neapolitan scale)
Propriety may also be considered as scales whose stability = 1, with stability defined as, "the ratio of the number of non-ambiguous undirected intervals...to the total number of undirected intervals," in which case the diatonic scale has a stability of .
The twelve equal scale is strictly proper as is any equal tempered scale because it has only one interval size for each number of steps Most tempered scales are proper too. As another example, the otonal harmonic fragment , , , is strictly proper, with the one step intervals varying in size from to , two step intervals vary from to , three step intervals from to .
Rothenberg hypothesizes that proper scales provide a point or frame of reference which aids perception ("stable gestalt") and that improper scales contradictions require a drone or ostinato to provide a point of reference.
An example of an improper scale is the Japanese Hirajōshi scale.
Its steps in semitones are 2, 1, 4, 1, 4. The single step intervals vary from the semitone from G to A to the major third from A to C.
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https://en.wikipedia.org/wiki/Recurrent%20point
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In mathematics, a recurrent point for a function f is a point that is in its own limit set by f. Any neighborhood containing the recurrent point will also contain (a countable number of) iterates of it as well.
Definition
Let be a Hausdorff space and a function. A point is said to be recurrent (for ) if , i.e. if belongs to its -limit set. This means that for each neighborhood of there exists such that .
The set of recurrent points of is often denoted and is called the recurrent set of . Its closure is called the Birkhoff center of , and appears in the work of George David Birkhoff on dynamical systems.
Every recurrent point is a nonwandering point, hence if is a homeomorphism and is compact, then is an invariant subset of the non-wandering set of (and may be a proper subset).
References
Limit sets
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https://en.wikipedia.org/wiki/Topological%20conjugacy
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In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.
To illustrate this directly: suppose that and are iterated functions, and there exists a homeomorphism such that
so that and are topologically conjugate. Then one must have
and so the iterated systems are topologically conjugate as well. Here, denotes function composition.
Definition
, and are continuous functions on topological spaces, and .
being topologically semiconjugate to means, by definition, that is a surjection such that .
and being topologically conjugate means, by definition, that they are topologically semiconjugate and is furthermore injective, then bijective, and its inverse is continuous too; i.e. is a homeomorphism; further, is termed a topological conjugation between and .
Flows
Similarly, on , and on are flows, with , and as above.
being topologically semiconjugate to means, by definition, that is a surjection such that , for each , .
and being topologically conjugate means, by definition, that they are topologically semiconjugate and is a homeomorphism.
Examples
The logistic map and the tent map are topologically conjugate.
The logistic map of unit height and the Bernoulli map are topologically conjugate.
For certain values in the parameter space, the Hénon map when restricted to its Julia set is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.
Discussion
Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring and to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of dynamical systems, since each class contains all functions which share the same dynamics from the topological viewpoint. For example, orbits of are mapped to homeomorphic orbits of through the conjugation. Writing makes this fact evident: . Speaking informally, topological conjugation is a "change of coordinates" in the topological sense.
However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps and to be topologically conjugate for each , which is requiring more than simply that orbits of be mapped to orbits of homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in into classes of flows sharing the same dynamics, again from the topological viewpoint.
Topological equivalence
We say that two flows and are topologically equivalent, if there is a
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https://en.wikipedia.org/wiki/Gyroelongated%20bipyramid
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In geometry, the gyroelongated bipyramids are an infinite set of polyhedra, constructed by elongating an bipyramid by inserting an antiprism between its congruent halves.
Forms
Two members of the set can be deltahedra, that is, constructed entirely of equilateral triangles: the gyroelongated square bipyramid, a Johnson solid, and the icosahedron, a Platonic solid. The gyroelongated triangular bipyramid can be made with equilateral triangles, but is not a deltahedron because it has coplanar faces, i.e. is not strictly convex. With pairs of triangles merged into rhombi, it can be seen as a trigonal trapezohedron. The other members can be constructed with isosceles triangles.
See also
Elongated bipyramid
Gyroelongated pyramid
Elongated pyramid
Diminished trapezohedron
External links
Conway Notation for Polyhedra Try: "knAn", where n=4,5,6... example "k5A5" is an icosahedron.
Pyramids and bipyramids
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https://en.wikipedia.org/wiki/Oskar%20Heil
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Oskar Heil (20 March 1908, in Langwieden – 15 May 1994, San Mateo, California) was a German electrical engineer and inventor. He studied physics, chemistry, mathematics, and music at the Georg-August University of Göttingen and was awarded his PhD in 1933, for his work on molecular spectroscopy.
Personal life
At the Georg-August University in Göttingen, Oskar Heil met Agnesa Arsenjewa (Агнесса Николаевна Арсеньева, 1901–1991), a promising young Russian physicist who also earned her PhD there. They married in Leningrad, the Soviet Union in 1934.
Together they moved to the United Kingdom to work in the Cavendish Laboratory, University of Cambridge. While on a trip to Italy, they co-wrote a pioneering paper on the generation of microwaves which was published in Germany in the Zeitschrift für Physik (i.e., Journal on Physics) in 1935. Agnesa subsequently returned to Russia to pursue this work further at the Leningrad Physico-Chemical Institute with her husband. However, he then returned to the UK alone; Agnesa, working in what had by then become a highly sensitive subject, was possibly not allowed to leave. Back in Britain, Oskar Heil worked for Standard Telephones and Cables.
At the onset of the Second World War he returned to Germany via Switzerland. During the war Heil worked on a microwave generator for the C. Lorenz AG in Berlin-Tempelhof.
In 1947 Heil was invited to the USA. After doing scientific work for Eitel McCullough and later the Varian Eimac division in San Carlos from 1955 until 1983, he founded his own company called Heil Scientific Labs Inc. in 1963 in Belmont, California. Agnesa remained in the Soviet Union until her death in 1991.
Microwave vacuum tube
Oskar Heil and Agnesa Arsenjewa-Heil in their pioneering paper developed the concept of the velocity-modulated tube, in which a beam of electrons could be made to form into "bunches" and thereby generate with reasonable efficiency radio waves of considerably higher frequency and power than were possible with conventional vacuum tubes/thermionic valves. This resulted in production of the "Heil tube", the first truly-practicable microwave generator, which slightly predated the (independent) invention of the klystron and subsequently the reflex klystron based on the same operating principle. These devices were a significant milestone in the development of microwave technology (particularly radar), and velocity-modulated tubes are still very much in use at the present day.
Field-effect transistor
Heil is mentioned as the inventor of an early transistor-like device (see also History of the transistor), based on several patents that were issued to him.
Erno Borbely states the following: "Field-effect transistors (FETs) have been around for a long time; in fact, they were invented, at least theoretically, before the bipolar transistors. The basic principle of the FET has been known since J.E. Lilienfeld’s US patent from 1930, and Oscar Heil described the possibility of controlling
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https://en.wikipedia.org/wiki/Spyridon%20Stais
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Spyridon Stais (, 1859–1932) was a Greek politician from the island of Kythera.
He studied physics and mathematics and served as a teacher in gymnasia (secondary schools) of Greece. He became active in politics in 1892, joining first the party of Charilaos Trikoupis and later (after Trikoupis’ death) the Modernist Party of Georgios Theotokis. He served as a member of parliament, as Minister for Education under prime minister Theotokis (in 1900 and again in 1903), as Minister of the Interior (1921–1922) under Dimitrios Gounaris and finally as general governor of Thessaloniki (1922) under Petros Protopapadakis.
In some recent publications dealing with the Antikythera Mechanism, the name of Spyridon Stais has been confused with that of the archaeologist Valerios Stais, the discoverer of that archaeological find.
References
1859 births
1932 deaths
Greek educators
People from Kythira
Ministers of the Interior of Greece
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https://en.wikipedia.org/wiki/Automaton%20%28disambiguation%29
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An automaton is a self-operating machine.
Automaton may also refer to:
An automaton, an abstract machine in mathematics, computer science, and automata theory, a mathematical model of computer hardware and software
In particular, a finite-state automaton, an automaton limited to a finite state space
Film and TV
Automatons (film), a 2006 film
Music
Automaton (album), Jamiroquai 2017
Automaton (song), a song by Jamiroquai 2017
"Automaton", a song by DJ Robotaki 2017
"Automaton", a song by English indie rock band The Rakes
See also
Automat (disambiguation)
Automata (disambiguation)
Automation (disambiguation)
Other uses
Automaton Media, a gaming website operated by Active Gaming Media.
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https://en.wikipedia.org/wiki/Expansive%20homeomorphism
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In mathematics, the notion of expansivity formalizes the notion of points moving away from one another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definition of positive expansivity, below, as well as the Schwarz–Ahlfors–Pick theorem demonstrate.
Definition
If is a metric space, a homeomorphism is said to be expansive if there is a constant
called the expansivity constant, such that for every pair of points in there is an integer such that
Note that in this definition, can be positive or negative, and so may be expansive in the forward or backward directions.
The space is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. if is any other metric generating the same topology as , and if is expansive in , then is expansive in (possibly with a different expansivity constant).
If
is a continuous map, we say that is positively expansive (or forward expansive) if there is a
such that, for any in , there is an such that .
Theorem of uniform expansivity
Given f an expansive homeomorphism of a compact metric space, the theorem of uniform expansivity states that for every and there is an such that for each pair of points of such that , there is an with such that
where is the expansivity constant of (proof).
Discussion
Positive expansivity is much stronger than expansivity. In fact, one can prove that if is compact and is a positively
expansive homeomorphism, then is finite (proof).
External links
Expansive dynamical systems on scholarpedia
Dynamical systems
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https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von%20Mises%20criterion
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In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function compared to a given empirical distribution function , or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as
In one-sample applications is the theoretical distribution and is the empirically observed distribution. Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case.
The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928–1930. The generalization to two samples is due to Anderson.
The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test (1933).
Cramér–von Mises test (one sample)
Let be the observed values, in increasing order. Then the statistic is
If this value is larger than the tabulated value, then the hypothesis that the data came from the distribution can be rejected.
Watson test
A modified version of the Cramér–von Mises test is the Watson test which uses the statistic U2, where
where
Cramér–von Mises test (two samples)
Let and be the observed values in the first and second sample respectively, in increasing order. Let be the ranks of the xs in the combined sample, and let be the ranks of the ys in the combined sample. Anderson shows that
where U is defined as
If the value of T is larger than the tabulated values, the hypothesis that the two samples come from the same distribution can be rejected. (Some books give critical values for U, which is more convenient, as it avoids the need to compute T via the expression above. The conclusion will be the same.)
The above assumes there are no duplicates in the , , and sequences. So is unique, and its rank is in the sorted list . If there are duplicates, and through are a run of identical values in the sorted list, then one common approach is the midrank method: assign each duplicate a "rank" of . In the above equations, in the expressions and , duplicates can modify all four variables , , , and .
References
Further reading
Statistical tests
Statistical distance
Nonparametric statistics
Normality tests
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https://en.wikipedia.org/wiki/Kronecker%20limit%20formula
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In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker.
First Kronecker limit formula
The (first) Kronecker limit formula states that
where
E(τ,s) is the real analytic Eisenstein series, given by
for Re(s) > 1, and by analytic continuation for other values of the complex number s.
γ is Euler–Mascheroni constant
τ = x + iy with y > 0.
, with q = e2π i τ is the Dedekind eta function.
So the Eisenstein series has a pole at s = 1 of residue π, and the (first) Kronecker limit formula gives the constant term of the Laurent series at this pole.
This formula has an interpretation in terms of the spectral geometry of the elliptic curve associated to the lattice : it says that the zeta-regularized determinant of the Laplace operator associated to the flat metric on is given by . This formula has been used in string theory for the one-loop computation in Polyakov's perturbative approach.
Second Kronecker limit formula
The second Kronecker limit formula states that
where
u and v are real and not both integers.
q = e2π i τ and qa = e2π i aτ
p = e2π i z and pa = e2π i az
for Re(s) > 1, and is defined by analytic continuation for other values of the complex number s.
See also
Herglotz–Zagier function
References
Serge Lang, Elliptic functions,
C. L. Siegel, Lectures on advanced analytic number theory, Tata institute 1961.
External links
Chapter0.pdf
Theorems in analytic number theory
Modular forms
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https://en.wikipedia.org/wiki/Moore%20plane
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In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii.
Definition
If is the (closed) upper half-plane , then a topology may be defined on by taking a local basis as follows:
Elements of the local basis at points with are the open discs in the plane which are small enough to lie within .
Elements of the local basis at points are sets where A is an open disc in the upper half-plane which is tangent to the x axis at p.
That is, the local basis is given by
Thus the subspace topology inherited by is the same as the subspace topology inherited from the standard topology of the Euclidean plane.
Properties
The Moore plane is separable, that is, it has a countable dense subset.
The Moore plane is a completely regular Hausdorff space (i.e. Tychonoff space), which is not normal.
The subspace of has, as its subspace topology, the discrete topology. Thus, the Moore plane shows that a subspace of a separable space need not be separable.
The Moore plane is first countable, but not second countable or Lindelöf.
The Moore plane is not locally compact.
The Moore plane is countably metacompact but not metacompact.
Proof that the Moore plane is not normal
The fact that this space is not normal can be established by the following counting argument (which is very similar to the argument that the Sorgenfrey plane is not normal):
On the one hand, the countable set of points with rational coordinates is dense in ; hence every continuous function is determined by its restriction to , so there can be at most many continuous real-valued functions on .
On the other hand, the real line is a closed discrete subspace of with many points. So there are many continuous functions from L to . Not all these functions can be extended to continuous functions on .
Hence is not normal, because by the Tietze extension theorem all continuous functions defined on a closed subspace of a normal space can be extended to a continuous function on the whole space.
In fact, if X is a separable topological space having an uncountable closed discrete subspace, X cannot be normal.
See also
Moore space (disambiguation)
Hedgehog space
References
Stephen Willard. General Topology, (1970) Addison-Wesley .
(Example 82)
Topological spaces
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https://en.wikipedia.org/wiki/GAUSS%20%28software%29
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GAUSS is a matrix programming language for mathematics and statistics, developed and marketed by Aptech Systems. Its primary purpose is the solution of numerical problems in statistics, econometrics, time-series, optimization and 2D- and 3D-visualization. It was first published in 1984 for MS-DOS and is available for Linux, macOS and Windows.
Examples
GAUSS has several Application Modules as well as functions in its Run-Time Library (i.e., functions that come with GAUSS without extra cost)
Qprog – Quadratic programming
SqpSolvemt – Sequential quadratic programming
QNewton - Quasi-Newton unconstrained optimization
EQsolve - Nonlinear equations solver
GAUSS Applications
A range of toolboxes are available for GAUSS at additional cost.
See also
List of numerical-analysis software
Comparison of numerical-analysis software
References
External links
International homepage
GAUSS Mailing List
Review of version 7.0
Some more links
Econometrics software
Mathematical optimization software
Numerical programming languages
Statistical programming languages
Proprietary commercial software for Linux
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https://en.wikipedia.org/wiki/Extension%20topology
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In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.
Extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form A ∪ Q, where A is an open set of X and Q is a subset of P.
The closed sets of X ∪ P are of the form B ∪ Q, where B is a closed set of X and Q is a subset of P.
For these reasons this topology is called the extension topology of X plus P, with which one extends to X ∪ P the open and the closed sets of X. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology. As a topological space, X ∪ P is homeomorphic to the topological sum of X and P, and X is a clopen subset of X ∪ P.
If Y is a topological space and R is a subset of Y, one might ask whether the extension topology of Y – R plus R is the same as the original topology of Y, and the answer is in general no.
Note the similarity of this extension topology construction and the Alexandroff one-point compactification, in which case, having a topological space X which one wishes to compactify by adding a point ∞ in infinity, one considers the closed sets of X ∪ {∞} to be the sets of the form K, where K is a closed compact set of X, or B ∪ {∞}, where B is a closed set of X.
Open extension topology
Let be a topological space and a set disjoint from . The open extension topology of plus is Let . Then is a topology in . The subspace topology of is the original topology of , i.e. , while the subspace topology of is the discrete topology, i.e. .
The closed sets in are . Note that is closed in and is open and dense in .
If Y a topological space and R is a subset of Y, one might ask whether the open extension topology of Y – R plus R is the same as the original topology of Y, and the answer is in general no.
Note that the open extension topology of is smaller than the extension topology of .
Assuming and are not empty to avoid trivialities, here are a few general properties of the open extension topology:
is dense in .
If is finite, is compact. So is a compactification of in that case.
is connected.
If has a single point, is ultraconnected.
For a set Z and a point p in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z – {p} plus p.
Closed extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of the form X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X.
For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ P the closed sets of X. As subsets of X ∪ P the subspace topology of
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https://en.wikipedia.org/wiki/Instituto%20Nacional%20de%20Estad%C3%ADstica
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Instituto Nacional de Estadística may refer to:
National Institute of Statistics and Census of Argentina ()
National Institute of Statistics of Bolivia ()
National Statistics Institute (Chile)
Instituto Nacional de Estadística y Censos de Costa Rica
, see
National Institute of Statistics (Guatemala) ()
Instituto Nacional de Estadística y Censos (Honduras), see
Instituto Nacional de Estadística, Geografía e Informática, Mexico
Instituto Nacional de Estadística y Censo – Panamá, see
, see
Instituto Nacional de Estadística e Informática, Peru
Instituto Nacional de Estatística (Portugal)
Instituto Nacional de Estadística (Spain)
, see
Statistics National Institute (Venezuela) ()
See also
National Institute of Statistics (disambiguation)
National Institute of Statistics and Census (disambiguation)
Instituto Nacional de Estatística (disambiguation), in various lusophone countries
National Institute of Statistics, Geography and Data Processing, a Mexican government agency
INE (disambiguation)
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https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics
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National Institute of Statistics may refer to:
National Institute of Statistics of Bolivia
National Institute of Statistics of Cambodia
National Institute of Statistics and Census of Costa Rica
National Institute of Statistics and Census of Nicaragua
National Institute of Statistics (Guatemala)
National Institute of Statistics (Italy)
National Institute of Statistics (Portugal)
National Institute of Statistics (Romania)
National Institute of Statistics (Spain)
National Institute of Statistics (Tunisia)
See also
List of national and international statistical services
National Statistics Institute (Chile)
National Institute of Statistics and Census (disambiguation)
Instituto Nacional de Estadística (disambiguation)
Instituto Nacional de Estadística e Informática, a Peruvian government agency
Instituto Nacional de Estadística y Geografía, a Mexican government agency
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https://en.wikipedia.org/wiki/Super%20Virasoro%20algebra
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In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory: the Ramond algebra (named after Pierre Ramond) and the Neveu–Schwarz algebra (named after André Neveu and John Henry Schwarz). Both algebras have N = 1 supersymmetry and an even part given by the Virasoro algebra. They describe the symmetries of a superstring in two different sectors, called the Ramond sector and the Neveu–Schwarz sector.
The N = 1 super Virasoro algebras
There are two minimal extensions of the Virasoro algebra with N = 1 supersymmetry: the Ramond algebra and the Neveu–Schwarz algebra. They are both Lie superalgebras whose even part is the Virasoro algebra: this Lie algebra has a basis consisting of a central element C and generators Lm (for integer m) satisfying
where is the Kronecker delta.
The odd part of the algebra has basis , where is either an integer (the Ramond case), or half an odd integer (the Neveu–Schwarz case). In both cases, is central in the superalgebra, and the additional graded brackets are given by
Note that this last bracket is an anticommutator, not a commutator, because both generators are odd.
The Ramond algebra has a presentation in terms of 2 generators and 5 conditions; and the Neveu—Schwarz algebra has a presentation in terms of 2 generators and 9 conditions.
Representations
The unitary highest weight representations of these algebras have a classification analogous to that for the Virasoro algebra, with a continuum of representations together with an infinite discrete series. The existence of these discrete series was conjectured by Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984). It was proven by Peter Goddard, Adrian Kent and David Olive (1986), using a supersymmetric generalisation of the coset construction or GKO construction.
Application to superstring theory
In superstring theory, the fermionic fields on the closed string may be either periodic or anti-periodic on the circle around the string. States in the "Ramond sector" admit one option (periodic conditions are referred to as Ramond boundary conditions), described by the Ramond algebra, while those in the "Neveu–Schwarz sector" admit the other (anti-periodic conditions are referred to as Neveu–Schwarz boundary conditions), described by the Neveu–Schwarz algebra.
For a fermionic field, the periodicity depends on the choice of coordinates on the worldsheet. In the w-frame, in which the worldsheet of a single string state is described as a long cylinder, states in the Neveu–Schwarz sector are anti-periodic and states in the Ramond sector are periodic. In the z-frame, in which the worldsheet of a single string state is described as an infinite punctured plane, the opposite is true.
The Neveu–Schwarz sector and Ramond sector are also defined in the open string and depend on the boundary conditions o
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https://en.wikipedia.org/wiki/Shrieker%20%28film%29
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Shrieker is a 1998 American horror film directed by David DeCoteau and produced by Charles Band.
Plot
Clark (Tanya Dempsey), a young Mathematics major at a University, thinks she's found the best deal for student housing: a group of squatters who live in an abandoned hospital secretly. The quirky residents let her into their community provided she follow the rules, including not telling anyone about her living arrangements. All seems wonderful, until she discovers that the reason that the hospital was abandoned was a series of murders in the 1940s by a strange "shrieking killer" who was never captured - and the discovery that someone who's living in the hospital is using occult means to bring back the demonic "Shrieker".
Cast
Tanya Dempsey as Clark
Parry Shen
Jenya Lano
Jason-Shane Scott
Jamie Gannon
Alison Cuffe
Roger Crowe
Chris Boyd
Brannon Gould
Rick Buono
Release
Shrieker was released on DVD by Wizard on March 3, 1998. It was later re-released in both 2003 and 2007 by Full Moon Home Video and Vision Films respectively.
Reception
The Tubi Tuesdays Podcast stated in Episode 13 that Shrieker is a confusing mess of a film, with the sub-plot of students squatting in an abandoned hospital made no sense.
References
External links
1998 films
1990s monster movies
American supernatural horror films
1998 horror films
Films directed by David DeCoteau
American monster movies
Squatting in film
1990s English-language films
1990s American films
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https://en.wikipedia.org/wiki/Lax%20pair
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In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems.
Definition
A Lax pair is a pair of matrices or operators dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation:
where is the commutator.
Often, as in the example below, depends on in a prescribed way, so this is a nonlinear equation for as a function of .
Isospectral property
It can then be shown that the eigenvalues and more generally the spectrum of L are independent of t. The matrices/operators L are said to be isospectral as varies.
The core observation is that the matrices are all similar by virtue of
where is the solution of the Cauchy problem
where I denotes the identity matrix. Note that if P(t) is skew-adjoint, U(t,s) will be unitary.
In other words, to solve the eigenvalue problem Lψ = λψ at time t, it is possible to solve the same problem at time 0 where L is generally known better, and to propagate the solution with the following formulas:
(no change in spectrum)
Through principal invariants
The result can also be shown using the invariants for any . These satisfy
due to the Lax equation, and since the characteristic polynomial can be written in terms of these traces, the spectrum is preserved by the flow.
Link with the inverse scattering method
The above property is the basis for the inverse scattering method. In this method, L and P act on a functional space (thus ψ = ψ(t,x)), and depend on an unknown function u(t,x) which is to be determined. It is generally assumed that u(0,x) is known, and that P does not depend on u in the scattering region where .
The method then takes the following form:
Compute the spectrum of , giving and ,
In the scattering region where is known, propagate in time by using with initial condition ,
Knowing in the scattering region, compute and/or .
Spectral curve
If the Lax matrix additionally depends on a complex parameter (as is the case for say sine-Gordon), the equation
defines an algebraic curve in with coordinates . By the isospectral property, this curve is preserved under time translation. This is the spectral curve. Such curves appear in the theory of Hitchin systems.
Zero-curvature representation
Any PDE which admits a Lax pair representation also admits a zero-curvature representation. In fact, the zero-curvature representation is more general and for other integrable PDEs, such as the sine-Gordon equation, the Lax pair refers to matrices that satisfy the zero-curvature equation rather than the Lax equation. Furthermore, the zero-curvature representation makes the link between integrable systems and geometry manifest, culminating in Ward's programme to formulate known integ
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https://en.wikipedia.org/wiki/Ideal%20triangle
|
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles. The vertices are sometimes called ideal vertices. All ideal triangles are congruent.
Properties
Ideal triangles have the following properties:
All ideal triangles are congruent to each other.
The interior angles of an ideal triangle are all zero.
An ideal triangle has infinite perimeter.
An ideal triangle is the largest possible triangle in hyperbolic geometry.
In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties:
Any ideal triangle has area π.
Distances in an ideal triangle
The inscribed circle to an ideal triangle has radius
.
The distance from any point in the triangle to the closest side of the triangle is less than or equal to the radius r above, with equality only for the center of the inscribed circle.
The inscribed circle meets the triangle in three points of tangency, forming an equilateral contact triangle with side length where is the golden ratio.
A circle with radius d around a point inside the triangle will meet or intersect at least two sides of the triangle.
The distance from any point on a side of the triangle to another side of the triangle is equal or less than , with equality only for the points of tangency described above.
a is also the altitude of the Schweikart triangle.
If the curvature is −K everywhere rather than −1, the areas above should be multiplied by 1/K and the lengths and distances should be multiplied by 1/.
Thin triangle condition
Because the ideal triangle is the largest possible triangle in hyperbolic geometry, the measures above are maxima possible for any hyperbolic triangle, this fact is important in the study of δ-hyperbolic space.
Models
In the Poincaré disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles.
In the Poincaré half-plane model, an ideal triangle is modeled by an arbelos, the figure between three mutually tangent semicircles.
In the Beltrami–Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary circle. Note that in the Beltrami-Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami-Klein model, unlike the Poincaré disk and half-plane models, is not conformal i.e. it does not preserve angles.
Real ideal triangle group
The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic to the free product of three order-two groups (Schwartz 2001).
References
Bibliography
Hyperbolic geometry
Types of triangles
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https://en.wikipedia.org/wiki/Miroslav%20Fiedler
|
Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory.
His article, "Algebraic Connectivity of Graphs", published in the Czechoslovak Math Journal in 1973, established the use of the eigenvalues of the Laplacian matrix of a graph to create tools for measuring algebraic connectivity in algebraic graph theory. Fiedler is honored by the Fiedler eigenvalue (the second smallest eigenvalue of the graph Laplacian), with its associated Fiedler eigenvector, as the names for the quantities that characterize algebraic connectivity. Since Fiedler's original contribution, this structure has become essential to large areas of research in network theory, flocking, distributed control, clustering, multi-robot applications and image segmentation.
References
External links
Home page at the Academy of Sciences of the Czech Republic.
1926 births
2015 deaths
Mathematicians from Prague
Czech mathematicians
Graph theorists
Recipients of Medal of Merit (Czech Republic)
Combinatorialists
Charles University alumni
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https://en.wikipedia.org/wiki/Excluded%20point%20topology
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In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set and p ∈ X. The collection
of subsets of X is then the excluded point topology on X. There are a variety of cases which are individually named:
If X has two points, it is called the Sierpiński space. This case is somewhat special and is handled separately.
If X is finite (with at least 3 points), the topology on X is called the finite excluded point topology
If X is countably infinite, the topology on X is called the countable excluded point topology
If X is uncountable, the topology on X is called the uncountable excluded point topology
A generalization is the open extension topology; if has the discrete topology, then the open extension topology on is the excluded point topology.
This topology is used to provide interesting examples and counterexamples.
Properties
Let be a space with the excluded point topology with special point
The space is compact, as the only neighborhood of is the whole space.
The topology is an Alexandrov topology. The smallest neighborhood of is the whole space the smallest neighborhood of a point is the singleton These smallest neighborhoods are compact. Their closures are respectively and which are also compact. So the space is locally relatively compact (each point admits a local base of relatively compact neighborhoods) and locally compact in the sense that each point has a local base of compact neighborhoods. But points do not admit a local base of closed compact neighborhoods.
The space is ultraconnected, as any nonempty closed set contains the point Therefore the space is also connected and path-connected.
See also
Finite topological space
Fort space
List of topologies
Particular point topology
References
Topological spaces
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https://en.wikipedia.org/wiki/Borel%20hierarchy
|
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called the rank of the Borel set. The Borel hierarchy is of particular interest in descriptive set theory.
One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important in measure theory and analysis.
Borel sets
The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well.
A short proof that the Borel algebra is well-defined proceeds by showing that the entire powerset of the space is closed under complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets of the space that have these closure properties. This proof does not give a simple procedure for determining whether a set is Borel. A motivation for the Borel hierarchy is to provide a more explicit characterization of the Borel sets.
Boldface Borel hierarchy
The Borel hierarchy or boldface Borel hierarchy on a space X consists of classes , , and for every countable ordinal greater than zero. Each of these classes consists of subsets of X. The classes are defined inductively from the following rules:
A set is in if and only if it is open.
A set is in if and only if its complement is in .
A set is in for if and only if there is a sequence of sets such that each is in for some and .
A set is in if and only if it is both in and in .
The motivation for the hierarchy is to follow the way in which a Borel set could be constructed from open sets using complementation and countable unions.
A Borel set is said to have finite rank if it is in for some finite ordinal ; otherwise it has infinite rank.
If then the hierarchy can be shown to have the following properties:
For every α, . Thus, once a set is in or , that set will be in all classes in the hierarchy corresponding to ordinals greater than α
. Moreover, a set is in this union if and only if it is Borel.
If is an uncountable Polish space, it can be shown that is not contained in for any , and thus the hierarchy does not collapse.
Borel sets of small rank
The classes of small rank are known by alternate names in classical descriptive set theory.
The sets are the open sets. The sets are the closed sets.
The sets are countable unions of closed sets, and are called Fσ sets. The sets are the dual class, and can be written as a countable intersection of open sets. These sets are called Gδ sets.
Lightface hierarchy
The lightface Borel hierarchy (also called the effective Borel hierarchypp.163-
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https://en.wikipedia.org/wiki/Line%E2%80%93sphere%20intersection
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In analytic geometry, a line and a sphere can intersect in three ways:
No intersection at all
Intersection in exactly one point
Intersection in two points.
Methods for distinguishing these cases, and determining the coordinates for the points in the latter cases, are useful in a number of circumstances. For example, it is a common calculation to perform during ray tracing.
Calculation using vectors in 3D
In vector notation, the equations are as follows:
Equation for a sphere
: points on the sphere
: center point
: radius of the sphere
Equation for a line starting at
: points on the line
: origin of the line
: distance from the origin of the line
: direction of line (a non-zero vector)
Searching for points that are on the line and on the sphere means combining the equations and solving for , involving the dot product of vectors:
Equations combined
Expanded and rearranged:
The form of a quadratic formula is now observable. (This quadratic equation is an instance of Joachimsthal's equation.)
where
Simplified
Note that in the specific case where is a unit vector, and thus , we can simplify this further to (writing instead of to indicate a unit vector):
If , then it is clear that no solutions exist, i.e. the line does not intersect the sphere (case 1).
If , then exactly one solution exists, i.e. the line just touches the sphere in one point (case 2).
If , two solutions exist, and thus the line touches the sphere in two points (case 3).
See also
Intersection_(geometry)#A_line_and_a_circle
Analytic geometry
Line–plane intersection
Plane–plane intersection
Plane–sphere intersection
References
Analytic geometry
Geometric algorithms
Geometric intersection
Spherical geometry
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https://en.wikipedia.org/wiki/List%20of%20localities%20in%20Northern%20Ireland%20by%20population
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This is a list of settlements in Northern Ireland by population. The fifty largest settlements are listed. This list has been compiled from data published by the Northern Ireland Statistics and Research Agency (NISRA), based on the 2011 Census and the 2021 Census, where available(*). Settlements with city status are shown in bold. Districts are local government districts as established in April 2015.
See also
List of settlements on the island of Ireland by population
List of places in Northern Ireland
List of urban areas in the Republic of Ireland
References
Settlements
Settlements
Northern Ireland
Northern Ireland
Localities
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https://en.wikipedia.org/wiki/J.%20Murdoch%20Henderson
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J. Murdoch Henderson (31 March 1902 – November 1970) was a Scottish fiddler, composer, and music critic.
John Murdoch Henderson was born in New Deer, Scotland, and became a mathematics teacher in Aberdeen. A childhood accident led to him breaking both wrists and hampered his playing. He took an interest in the interpretation of fiddle music and recorded much of the information he found. He published The Flowers of Scottish Melody in 1935, which contained 130 tunes, including 40 original contributions. The collection was reprinted by The Buchan Heritage Society in 1986. Later, he edited and published The Scottish Music Maker (1957), which preserved a number of melodies by James Scott Skinner that may otherwise have been lost (Alburger, 1983).
One of Henderson's greatest influences was James F. Dickie, a renowned fiddler from Old Deer. Dickie's son-in-law, James Duncan, was the founder of the Buchan Heritage Society, and was largely responsible for the republication of The Flowers of Scottish Melody. Two of Henderson's best-known compositions are named after Dickie: the reel James F. Dickie and the strathspey James F. Dickie's Delight.
References
Alburger, Mary Anne (1983), Scottish Fiddlers And Their Music, Victor Gollancz Ltd., .
External links
The John Murdoch Henderson Music Collection
1902 births
1970 deaths
Scottish composers
Scottish fiddlers
British male violinists
20th-century violinists
20th-century classical musicians
20th-century British composers
20th-century Scottish musicians
20th-century British male musicians
People from New Deer
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https://en.wikipedia.org/wiki/Compact%20closed%20category
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In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.
Symmetric compact closed category
A symmetric monoidal category is compact closed if every object has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and is denoted .
In a bit more detail, an object is called the dual of if it is equipped with two morphisms called the unit and the counit , satisfying the equations
and
where are the introduction of the unit on the left and right, respectively, and is the associator.
For clarity, we rewrite the above compositions diagrammatically. In order for to be compact closed, we need the following composites to equal :
and :
Definition
More generally, suppose is a monoidal category, not necessarily symmetric, such as in the case of a pregroup grammar. The above notion of having a dual for each object A is replaced by that of having both a left and a right adjoint, and , with a corresponding left unit , right unit , left counit , and right counit . These must satisfy the four yanking conditions, each of which are identities:
and
That is, in the general case, a compact closed category is both left and right-rigid, and biclosed.
Non-symmetric compact closed categories find applications in linguistics, in the area of categorial grammars and specifically in pregroup grammars, where the distinct left and right adjoints are required to capture word-order in sentences. In this context, compact closed monoidal categories are called (Lambek) pregroups.
Properties
Compact closed categories are a special case of monoidal closed categories, which in turn are a special case of closed categories.
Compact closed categories are precisely the symmetric autonomous categories. They are also *-autonomous.
Every compact closed category C admits a trace. Namely, for every morphism , one can define
which can be shown to be a proper trace. It helps to draw this diagrammatically:
Examples
The canonical example is the category FdVect with finite-dimensional vector spaces as objects and linear maps as morphisms. Here is the usual dual of the vector space .
The category of finite-dimensional representations of any group is also compact closed.
The category Vect, with all vector spaces as objects and linear maps as morphisms, is not compact closed; it is symmetric monoidal closed.
Simplex category
The simplex category can be used to construct an example of non-symmetric compact closed category. The si
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https://en.wikipedia.org/wiki/Polarization%20of%20an%20algebraic%20form
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In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal.
Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory.
The technique
The fundamental ideas are as follows. Let be a polynomial in variables Suppose that is homogeneous of degree which means that
Let be a collection of indeterminates with so that there are variables altogether. The polar form of is a polynomial
which is linear separately in each (that is, is multilinear), symmetric in the and such that
The polar form of is given by the following construction
In other words, is a constant multiple of the coefficient of in the expansion of
Examples
A quadratic example. Suppose that and is the quadratic form
Then the polarization of is a function in and given by
More generally, if is any quadratic form then the polarization of agrees with the conclusion of the polarization identity.
A cubic example. Let Then the polarization of is given by
Mathematical details and consequences
The polarization of a homogeneous polynomial of degree is valid over any commutative ring in which is a unit. In particular, it holds over any field of characteristic zero or whose characteristic is strictly greater than
The polarization isomorphism (by degree)
For simplicity, let be a field of characteristic zero and let be the polynomial ring in variables over Then is graded by degree, so that
The polarization of algebraic forms then induces an isomorphism of vector spaces in each degree
where is the -th symmetric power of the -dimensional space
These isomorphisms can be expressed independently of a basis as follows. If is a finite-dimensional vector space and is the ring of -valued polynomial functions on graded by homogeneous degree, then polarization yields an isomorphism
The algebraic isomorphism
Furthermore, the polarization is compatible with the algebraic structure on so that
where is the full symmetric algebra over
Remarks
For fields of positive characteristic the foregoing isomorphisms apply if the graded algebras are truncated at degree
There do exist generalizations when is an infinite dimensional topological vector space.
See also
References
Claudio Procesi (2007) Lie Groups: an approach through invariants and representations, Springer, .
Abstract algebra
Homogeneous polynomials
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https://en.wikipedia.org/wiki/Shapiro%20inequality
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In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.
Statement of the inequality
Suppose is a natural number and are positive numbers and:
is even and less than or equal to , or
is odd and less than or equal to .
Then the Shapiro inequality states that
where .
For greater values of the inequality does not hold and the strict lower bound is with .
The initial proofs of the inequality in the pivotal cases (Godunova and Levin, 1976) and (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for .
The value of was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound is given by , where the function is the convex hull of and . (That is, the region above the graph of is the convex hull of the union of the regions above the graphs of ' and .)
Interior local minima of the left-hand side are always (Nowosad, 1968).
Counter-examples for higher n
The first counter-example was found by Lighthill in 1956, for :
where is close to 0.
Then the left-hand side is equal to , thus lower than 10 when is small enough.
The following counter-example for is by Troesch (1985):
(Troesch, 1985)
References
They give an analytic proof of the formula for even , from which the result for all follows. They state as an open problem.
External links
Usenet discussion in 1999 (Dave Rusin's notes)
PlanetMath
Inequalities
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https://en.wikipedia.org/wiki/Substitution%20tiling
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In geometry, a tile substitution is a method for constructing highly ordered tilings. Most importantly, some tile substitutions generate aperiodic tilings, which are tilings whose prototiles do not admit any tiling with translational symmetry. The most famous of these are the Penrose tilings. Substitution tilings are special cases of finite subdivision rules, which do not require the tiles to be geometrically rigid.
Introduction
A tile substitution is described by a set of prototiles (tile shapes) , an expanding map and a dissection rule showing how to dissect the expanded prototiles to form copies of some prototiles . Intuitively, higher and higher iterations of tile substitution produce a tiling of the plane called a substitution tiling. Some substitution tilings are periodic, defined as having translational symmetry.
Every substitution tiling (up to mild conditions) can be "enforced by matching rules"—that is, there exist a set of marked tiles that can only form exactly the substitution tilings generated by the system. The tilings by these marked tiles are necessarily aperiodic.
A simple example that produces a periodic tiling has only one prototile, namely a square:
By iterating this tile substitution, larger and larger regions of the plane are covered with a square grid. A more sophisticated example with two prototiles is shown below, with the two steps of blowing up and dissecting merged into one step.
One may intuitively get an idea how this procedure yields a substitution tiling of the entire plane. A mathematically rigorous definition is given below. Substitution tilings are notably useful as ways of defining aperiodic tilings, which are objects of interest in many fields of mathematics, including automata theory, combinatorics, discrete geometry, dynamical systems, group theory, harmonic analysis and number theory, as well as crystallography and chemistry. In particular, the celebrated Penrose tiling is an example of an aperiodic substitution tiling.
History
In 1973 and 1974, Roger Penrose discovered a family of aperiodic tilings, now called Penrose tilings. The first description was given in terms of 'matching rules' treating the prototiles as jigsaw puzzle pieces. The proof that copies of these prototiles can be put together to form a tiling of the plane, but cannot do so periodically, uses a construction that can be cast as a substitution tiling of the prototiles. In 1977 Robert Ammann discovered a number of sets of aperiodic prototiles, i.e., prototiles with matching rules forcing nonperiodic tilings; in particular, he rediscovered Penrose's first example. This work gave an impact to scientists working in crystallography, eventually leading to the discovery of quasicrystals. In turn, the interest in quasicrystals led to the discovery of several well-ordered aperiodic tilings. Many of them can be easily described as substitution tilings.
Mathematical definition
We will consider regions in that are well-behaved, i
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https://en.wikipedia.org/wiki/Dual%20object
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In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space V∗ doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property.
A category in which each object has a dual is called autonomous or rigid. The category of finite-dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not.
Motivation
Let V be a finite-dimensional vector space over some field K. The standard notion of a dual vector space V∗ has the following property: for any K-vector spaces U and W there is an adjunction HomK(U ⊗ V,W) = HomK(U, V∗ ⊗ W), and this characterizes V∗ up to a unique isomorphism. This expression makes sense in any category with an appropriate replacement for the tensor product of vector spaces. For any monoidal category (C, ⊗) one may attempt to define a dual of an object V to be an object V∗ ∈ C with a natural isomorphism of bifunctors
HomC((–)1 ⊗ V, (–)2) → HomC((–)1, V∗ ⊗ (–)2)
For a well-behaved notion of duality, this map should be not only natural in the sense of category theory, but also respect the monoidal structure in some way. An actual definition of a dual object is thus more complicated.
In a closed monoidal category C, i.e. a monoidal category with an internal Hom functor, an alternative approach is to simulate the standard definition of a dual vector space as a space of functionals. For an object V ∈ C define V∗ to be , where 1C is the monoidal identity. In some cases, this object will be a dual object to V in a sense above, but in general it leads to a different theory.
Definition
Consider an object in a monoidal category . The object is called a left dual of if there exist two morphisms
, called the coevaluation, and , called the evaluation,
such that the following two diagrams commute:
The object is called the right dual of .
This definition is due to .
Left duals are canonically isomorphic when they exist, as are right duals. When C is braided (or symmetric), every left dual is also a right dual, and vice versa.
If we consider a monoidal category as a bicategory with one object, a dual pair is exactly an adjoint pair.
Examples
Consider a monoidal category (VectK, ⊗K) of vector spaces over a field K with the standard tensor product. A space V is dualizable if and only if it is finite-dimensional, and in this case the dual object V∗ coincides with the standard notion of a dual vector space.
Consider a monoidal category (ModR, ⊗R) of modules over a commutative ring R with the standard tensor product. A module M is dualizable if and
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https://en.wikipedia.org/wiki/Protein%20A
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Protein A is a 42 kDa surface protein originally found in the cell wall of the bacteria Staphylococcus aureus. It is encoded by the spa gene and its regulation is controlled by DNA topology, cellular osmolarity, and a two-component system called ArlS-ArlR. It has found use in biochemical research because of its ability to bind immunoglobulins. It is composed of five homologous Ig-binding domains that fold into a three-helix bundle. Each domain is able to bind proteins from many mammalian species, most notably IgGs. It binds the heavy chain within the Fc region of most immunoglobulins and also within the Fab region in the case of the human VH3 family. Through these interactions in serum, where IgG molecules are bound in the wrong orientation (in relation to normal antibody function), the bacteria disrupts opsonization and phagocytosis.
History
As a by-product of his work on type-specific staphylococcus antigens, Verwey reported in
1940 that a protein fraction prepared from extracts of these bacteria non-specifically precipitated rabbit antisera raised against different staphylococcus types. In 1958, Jensen confirmed Verwey’s finding and showed that rabbit pre-immunization sera as well as normal human sera bound to the active component in the staphylococcus extract; he designated this component Antigen A (because it was found in fraction A of the extract) but thought it was a polysaccharide. The misclassification of the protein was the result of faulty tests but it was not long thereafter (1962) that Löfkvist and Sjöquist corrected the error and confirmed that Antigen A was in fact a surface protein on the bacterial wall of certain strains of S. aureus. The Bergen group from Norway named the protein "Protein A" after the antigen fraction isolated by Jensen.
Protein A antibody binding
It has been shown via crystallographic refinement that the primary binding site for protein A is on the Fc region, between the CH2 and CH3 domains. In addition, protein A has been shown to bind human IgG molecules containing IgG F(ab')2 fragments from the human VH3 gene family.
Protein A can bind with strong affinity to the Fc portion of immunoglobulin of certain species as shown in the below table.
Other antibody binding proteins
In addition to protein A, other immunoglobulin-binding bacterial proteins such as Protein G, Protein A/G and Protein L are all commonly used to purify, immobilize or detect immunoglobulins.
Role in pathogenesis
As a pathogen, Staphylococcus aureus utilizes protein A, along with a host of other proteins and surface factors, to aid its survival and virulence. To this end, protein A plays a multifaceted role:
By binding the Fc portion of antibodies, protein A renders them inaccessible to the opsonins, thus impairing phagocytosis of the bacteria via immune cell attack.
Protein A facilitates the adherence of S. aureus to human von Willebrand factor (vWF)-coated surfaces, thus increasing the bacteria's infectiousness at the site of skin
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https://en.wikipedia.org/wiki/Real%20analytic%20Eisenstein%20series
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In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function.
There are many generalizations associated to more complicated groups.
Definition
The Eisenstein series E(z, s) for z = x + iy in the upper half-plane is defined by
for Re(s) > 1, and by analytic continuation for other values of the complex number s. The sum is over all pairs of coprime integers.
Warning: there are several other slightly different definitions. Some authors omit the factor of ½, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2s).
Properties
As a function on z
Viewed as a function of z, E(z,s) is a real-analytic eigenfunction of the Laplace operator on H with the eigenvalue s(s-1). In other words, it satisfies the elliptic partial differential equation
where
The function E(z, s) is invariant under the action of SL(2,Z) on z in the upper half plane by fractional linear transformations. Together with the previous property, this means that the Eisenstein series is a Maass form, a real-analytic analogue of a classical elliptic modular function.
Warning: E(z, s) is not a square-integrable function of z with respect to the invariant Riemannian metric on H.
As a function on s
The Eisenstein series converges for Re(s)>1, but can be analytically continued to a meromorphic function of s on the entire complex plane, with in the half-plane Re(s) 1/2 a unique pole of residue 3/π at s = 1 (for all z in H) and infinitely many poles in the strip 0 < Re(s) < 1/2 at where corresponds to a non-trivial zero of the Riemann zeta-function. The constant term of the pole at s = 1 is described by the Kronecker limit formula.
The modified function
satisfies the functional equation
analogous to the functional equation for the Riemann zeta function ζ(s).
Scalar product of two different Eisenstein series E(z, s) and E(z, t) is given by the Maass-Selberg relations.
Fourier expansion
The above properties of the real analytic Eisenstein series, i.e. the functional equation for E(z,s) and E*(z,s) using Laplacian on H, are shown from the fact that E(z,s) has a Fourier expansion:
where
and modified Bessel functions
Epstein zeta function
The Epstein zeta function ζQ(s) for a positive definite integral quadratic form Q(m, n) = cm2 + bmn +an2 is defined by
It is essentially a special case of the real analytic Eisenstein series for a special value of z, since
for
This zeta function was named after Paul Epstein.
Generalizations
The real analytic Eisenstein series E(z, s) is really the Eisenstein series associated to the discrete subgroup SL(2,Z) of SL(2,R). Selberg described generalizations to other discrete subgroups Γ of SL(2,R), and used these to study the representation of SL(2,R) on L2(SL(2,R)/Γ). Langlands extended Selberg's work to higher di
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https://en.wikipedia.org/wiki/Vasili%20Pronchishchev
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Vasili Vasilyevich Pronchishchev () (1702–) was a Russian explorer.
In 1718, Vasili Pronchishchev graduated from Moscow School of Mathematics and Navigation and was promoted to naval cadet. In 1733, he was promoted to the rank of lieutenant and appointed head of one of the units of the Second Kamchatka Expedition, the purpose of which was to map the shores of the Arctic Ocean from the mouth of the Lena to the mouth of the Yenisey.
In 1735, Vasili Pronchishchev went down the Lena River (from Yakutsk) on his sloop Yakutsk, doubled its delta, and stopped for wintering at the mouth of the Olenek River. Many members of the crew fell ill and died, mainly owing to scurvy. Despite the difficulties, in 1736, he reached the eastern shore of the Taymyr Peninsula and went north along its coastline. Finally Pronchishchev and his wife Maria (also referred to as Tatyana Feodorovna) succumbed to scurvy and died on the way back.
Despite the death toll, the expedition was successful regarding the fulfillment of its goals. During his journey, Vasili Pronchishchev discovered a number of islands off the northeastern coast of the Taymyr Peninsula (Faddey Islands, Komsomolskoy Pravdy Islands, Saint Peter Islands). His expedition was the first to accurately map the Lena River from Yakutsk to its estuary and the Laptev seacoast from the Lena's mouth to the Gulf of Faddey. Pronchishchev's wife Maria Pronchishcheva (died 12(23) September 1736), who took part in his expedition, is considered the first female polar explorer. After their deaths, both of them were interred at the mouth of the Olenek River.
Further information is now available from the Hakluit Society via a summary written by William Barr in July 2018, "The Arctic Detachments of the Russian Great Northern Expedition (1733-43) and their largely forgotten and even Clandestine Predecessors". On page 12 of the summary is shown information and maps on the Lena-Khatanga detachment led by Pronchishchev.
A part of the eastern coastline of the Taymyr Peninsula and a ridge between the mouths of the Olenek and Anabar Rivers bear Vasili Pronchishchev's name. The 1961-built Project 97A icebreaker Ledokol-1 was renamed Vasiliy Pronchishchev in 1996 after this pioneering Arctic explorer.
Maria Pronchishcheva Bay in the Laptev Sea is named after his wife Maria.
References
Historical data
Excavations at the burial site of the couple:
1702 births
1736 deaths
People from Ferzikovsky District
Imperial Russian Navy personnel
Explorers of the Arctic
Explorers from the Russian Empire
Explorers of Asia
Explorers of Siberia
Laptev Sea
18th-century people from the Russian Empire
Great Northern Expedition
Deaths from scurvy
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https://en.wikipedia.org/wiki/Career%20Guide%20to%20Industries
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The Career Guide to Industries was a publication of the United States Department of Labor's Bureau of Labor Statistics that included information about the nature of the industry, working conditions, training and education, earnings, and job outlook for workers in dozens of different industries. The Career Guide was released biennially with its companion publication the Occupational Outlook Handbook.
It is no longer an independent product and similar information is to be found in other publications, in particular: information about current and projected occupational employment within industries and information about current and projected industry employment for occupations.
The 2006-07 edition was released in December 2005 and included employment projections for the period 2004–2014. The 2010-11 edition printed by Claitors Publishing Division was released in August 2010.
References
External links
Economics publications
United States Department of Labor publications
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https://en.wikipedia.org/wiki/CLAS%20%28education%29
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CLAS was a test and given in California in the early 1990s. It was based on concepts of new standards such as whole language and reform mathematics. Instead of multiple choice tests with one correct answer, it used open written responses that were graded according to rubrics. Test takers would have to write about passages of literature that they were asked to read and relate the passage to their own experiences, or to explain how they found solutions to math problems that they were asked to solve. Such tests were thought to be fairer to students of all abilities.
The system debuted in 1993, when about 1 million students in fourth, eighth, and tenth grade took the exams, although only some of them were graded to save money. The system was originally nationally praised as an example of "'performance based' testing".
Failure rates among all groups, particularly minorities, was so high that it generated concern. It was terminated in 1995 by the governor after two years.
Minorities scored even lower than on standardized tests, huge numbers scored in the lowest categories, as open response questions with more than one answer proved to be even more difficult than multiple choice problems.
In September 1994, Pete Wilson vetoed a bill, introduced by Gary Hart, that would have continued CLAS for another five years and provided $24M in funding, and called on the California state legislature to enact another statewide testing program. According to Maureen DiMarco, Wilson vetoed this bill because it did not provide achievement scores for individual students, even though Wilson supported the CLAS exams overall.
Educators complained about mismanagement and problems with scoring the CLAS exams. Additionally, religious conservatives described some of the literary passages on the CLAS exams as being "anti-family" or "an invasion of students’ privacy". According to Debra Saunders, CLAS graders were told to give higher scores to students who answered a math problem about planting trees incorrectly but who wrote enthusiastic essays than to students who answered this problem correctly without writing an essay.
Maureen DiMarco testified to the California State Legislature in charge of the [CLAS] that no graders were allowed to give a "4" top score in mathematics in the first year. It was based on open responses scored holistically, so that the correct answer to how to share 5 apples among 4 people might be to give the 5th to a food bank.
It was replaced by STAR, which is a testing system based on traditional rigorous academic standards which largely discards the theory of outcome-based education which was widely rejected by the late 1990s in the United States.
References
Standardized tests in the United States
Education in California
Education reform
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https://en.wikipedia.org/wiki/Hyperbolic%20tree
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A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry.
Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level n is 2n, while the number of nodes for trees with more branching grows much more quickly. Drawing the tree as a node-link diagram thus requires exponential amounts of space to be displayed.
One approach is to use a hyperbolic tree, first introduced by Lamping et al. Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children.
Displaying a hyperbolic tree commonly utilizes the Poincaré disk model of hyperbolic geometry, though the Klein-Beltrami model can also be used. Both display the entire hyperbolic plane within a unit disk, making the entire tree visible at once. The unit disk gives a fish-eye lens view of the plane, giving more emphasis to nodes which are in focus and displaying nodes further out of focus closer to the boundary of the disk. Traversing the hyperbolic tree requires Möbius transformations of the space, bringing new nodes into focus and moving higher levels of the hierarchy out of view.
Hyperbolic trees were patented in the U.S. by Xerox in 1996, but the patent has since expired.
See also
Hyperbolic geometry
Binary tiling
Information visualization
Radial tree – is also circular, but uses linear geometry.
Tree (data structure)
Tree (graph theory)
References
External links
d3-hypertree – HTML5 Hyperbolic tree implementation, MIT licensed
Hyperbolic Tree of life – Open source tree of life visualisation using Open Tree of Life data set
The Green Tree of Life – Tree of life – University of California at Berkeley and Jepson Herbaria
Tree of life Similar to the above, but with pictures
RogueViz supports hyperbolic trees.
Hyperbolic geometry
Graph drawing
Trees (data structures)
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https://en.wikipedia.org/wiki/Mathematics%20education%20in%20Australia
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Mathematics education in Australia varies significantly between states, especially at the upper secondary level. While every school offers a state-based education systems, some may also offer the International Baccalaureate program.
Secondary
New South Wales
Higher School Certificate
The Higher School Certificate (HSC) in NSW contains a number of mathematics courses catering for a range of abilities. There are four courses offered by the NSW Education Standards Authority (NESA) for HSC Study:
Mathematics Standard 1 or 2: A basic mathematics course containing precalculus concepts; the course is heavily based on practical mathematics used in everyday life. While the more advanced courses include statistical topics, this is the only course which introduces normal distributions, standard deviations and z-scores. These topics are alluded to in more advanced courses though not formally considered.
Mathematics Advanced: An advanced level calculus-based course with detailed study in probability and statistics, trigonometry, curve sketching, and applications of calculus. It is the highest level non-extension mathematics course. The calculus is only a single variable in all of year 12 mathematics in NSW. Computational methods such as the trapezoidal rule are encountered for evaluating integrals. The course includes a brief foray into series and sequences, including an application to basic finance through the modelling of compound interest. The nature of lines, circles and parabolas as loci are investigated however these properties are not exploited by the plane geometry coursework. Quadratic equations are studied and students learn techniques to reduce special quintic and exponential equations to quadratics.
Mathematics Extension 1 (Must be studied concurrently with Mathematics Advanced): A more advanced course building on concepts in calculus, trigonometry, polynomials, basic combinatorics, vectors, and further statistics. Students learn the binomial theorem to extend their knowledge of probability, along with using circle geometry to prove a greater family of statements. The trigonometry component includes double-angle identities and factoring the addition of a sine and cosine function into a single sinusoid. In calculus, students are exposed to a greater variety of integration techniques such as substitution. Parametrization of planar curves is introduced, mainly focusing on lines, circles and parabolas. The plotting of cubic equations and solution of specific cases through polynomial long division and the remainder theorem enable a deeper understanding of polynomials.
Mathematics Extension 2 (Must be studied concurrently with Mathematics Advanced and Mathematics Extension 1): A highly advanced mathematics course containing an introduction to complex numbers, advanced calculus, motion, and further work with vectors. While NSW Mathematics curricula does not include matrix theory nor group theory, the geometric properties of complex numbers allude
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https://en.wikipedia.org/wiki/Indefinite%20inner%20product%20space
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In mathematics, in the field of functional analysis, an indefinite inner product space
is an infinite-dimensional complex vector space equipped with both an indefinite inner product
and a positive semi-definite inner product
where the metric operator is an endomorphism of obeying
The indefinite inner product space itself is not necessarily a Hilbert space; but the existence of a positive semi-definite inner product on implies that one can form a quotient space on which there is a positive definite inner product. Given a strong enough topology on this quotient space, it has the structure of a Hilbert space, and many objects of interest in typical applications fall into this quotient space.
An indefinite inner product space is called a Krein space (or -space) if is positive definite and possesses a majorant topology. Krein spaces are named in honor of the Soviet mathematician Mark Grigorievich Krein (3 April 1907 – 17 October 1989).
Inner products and the metric operator
Consider a complex vector space equipped with an indefinite hermitian form . In the theory of Krein spaces it is common to call such an hermitian form an indefinite inner product. The following subsets are defined in terms of the square norm induced by the indefinite inner product:
("neutral")
("positive")
("negative")
("non-negative")
("non-positive")
A subspace lying within is called a neutral subspace. Similarly, a subspace lying within () is called positive (negative) semi-definite, and a subspace lying within () is called positive (negative) definite. A subspace in any of the above categories may be called semi-definite, and any subspace that is not semi-definite is called indefinite.
Let our indefinite inner product space also be equipped with a decomposition into a pair of subspaces , called the fundamental decomposition, which respects the complex structure on . Hence the corresponding linear projection operators coincide with the identity on and annihilate , and they commute with multiplication by the of the complex structure. If this decomposition is such that and , then is called an indefinite inner product space; if , then is called a Krein space, subject to the existence of a majorant topology on (a locally convex topology where the inner product is jointly continuous).
The operator is called the (real phase) metric operator or fundamental symmetry, and may be used to define the Hilbert inner product :
On a Krein space, the Hilbert inner product is positive definite, giving the structure of a Hilbert space (under a suitable topology). Under the weaker constraint , some elements of the neutral subspace may still be neutral in the Hilbert inner product, but many are not. For instance, the subspaces are part of the neutral subspace of the Hilbert inner product, because an element obeys . But an element () which happens to lie in because will have a positive square norm under the Hilbert inner product.
We note that th
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https://en.wikipedia.org/wiki/S.%20Barry%20Cooper
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S. Barry Cooper (9 October 1943 – 26 October 2015) was an English mathematician and computability theorist. He was a professor of Pure Mathematics at the University of Leeds.
Early life and education
Cooper grew up in Bognor Regis and attended Chichester High School for Boys, during which time he played scrum-half for the under-15s England rugby team.
Cooper graduated from Jesus College, Oxford in 1966 and in 1970 received his PhD from the University of Leicester under the supervision of Reuben Goodstein and C.E.M. Yates, with a thesis entitled Degrees of Unsolvability.
Academic career
Cooper was appointed Lecturer in the School of Mathematics at the University of Leeds in 1969, where he remained for the rest of his career. He was promoted to Reader in Mathematical Logic in 1991 and to Professor of Pure Mathematics in 1996. In 2011, he was awarded an honorary doctorate at the University of Sofia "St. Kliment Ohridski".
His book Computability Theory made the technical research area accessible to a new generation of students. He was a leading mover of the return to basic questions of the kind considered by Alan Turing, and of interdisciplinary developments related to computability. He was President of the Association Computability in Europe, and Chair of the Turing Centenary Advisory Committee (TCAC), which co-ordinated the Alan Turing Year. The book Alan Turing: His Work and Impact, edited by Cooper and Jan van Leeuwen, won the Association of American Publishers' R. R. Hawkins Award.
Cooper was a member of the editorial board for The Rutherford Journal.
Interests
Cooper was a keen long-distance runner, and was also interested in jazz and improvised music, founding Leeds Jazz and being involved in the Termite Club.
In the 1970s, he was also a leading figure in the Chile Solidarity Campaign, welcoming Chilean refugees to Leeds.
Death
Cooper died on 26 October 2015 after a short illness.
Selected publications
S. B. Cooper, 2004. Computability Theory, Chapman & Hall/CRC.
S. B. Cooper; J. van Leeuwen (eds.), 2013. Alan Turing – His Work and Impact, New York: Elsevier,
S. B. Cooper, B. Löwe, A. Sorbi (eds.), 2008. New Computational Paradigms – Changing Conceptions of What is Computable, Springer.
References
External links
S. Barry Cooper homepage
S. Barry Cooper's Mathematics Genealogy Page
Computability in Europe homepage
The Alan Turing Centenary homepage
1943 births
2015 deaths
English logicians
20th-century English mathematicians
21st-century English mathematicians
Alumni of Jesus College, Oxford
Alumni of the University of Leicester
Academics of the University of Leeds
English philosophers
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https://en.wikipedia.org/wiki/Witt%20vector
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In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors over the finite field of order is isomorphic to , the ring of -adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers.
The main idea behind Witt vectors is instead of using the standard -adic expansionto represent an element in , we can instead consider an expansion using the Teichmüller characterwhich sends each element in the solution set of in to an element in the solution set of in . That is, we expand out elements in in terms of roots of unity instead of as profinite elements in . We can then express a -adic integer as an infinite sumwhich gives a Witt vectorThen, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give an additive and multiplicative structure such that induces a commutative ring morphism.
History
In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummer theory. Let be a field containing a primitive -th root of unity. Kummer theory classifies degree cyclic field extensions of . Such fields are in bijection with order cyclic groups , where corresponds to .
But suppose that has characteristic . The problem of studying degree extensions of , or more generally degree extensions, may appear superficially similar to Kummer theory. However, in this situation, cannot contain a primitive -th root of unity. Indeed, if is a -th root of unity in , then it satisfies . But consider the expression . By expanding using binomial coefficients we see that the operation of raising to the -th power, known here as the Frobenius homomorphism, introduces the factor to every coefficient except the first and the last, and so modulo these equations are the same. Therefore . Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.
The case where the characteristic divides the degree is today called Artin–Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two. This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree extensions of a field of characteristic were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form By repeating their construction, they described degree extensions. Abraham Adrian
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https://en.wikipedia.org/wiki/Genus%20of%20a%20multiplicative%20sequence
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In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.
Definition
A genus assigns a number to each manifold X such that
(where is the disjoint union);
;
if X is the boundary of a manifold with boundary.
The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms.
The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.
Example: If is the signature of the oriented manifold X, then is a genus from oriented manifolds to the ring of integers.
The genus associated to a formal power series
A sequence of polynomials in variables is called multiplicative if
implies that
If is a formal power series in z with constant term 1, we can define a multiplicative sequence
by
,
where is the kth elementary symmetric function of the indeterminates . (The variables will often in practice be Pontryagin classes.)
The genus of compact, connected, smooth, oriented manifolds corresponding to Q is given by
where the are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus . A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4k for positive integers k, implies that this gives a bijection between formal power series Q with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.
L genus
The L genus is the genus of the formal power series
where the numbers are the Bernoulli numbers. The first few values are:
(for further L-polynomials see or ). Now let M be a closed smooth oriented manifold of dimension 4n with Pontrjagin classes . Friedrich Hirzebruch showed that the L genus of M in dimension 4n evaluated on the fundamental class of , denoted , is equal to , the signature of M (i.e., the signature of the intersection form on the 2nth cohomology group of M):
.
This is now known as the Hirzebruch signature theorem (or sometimes the Hirzebruch index theorem).
The fact that is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional PL manifold with no smooth structure. Pontryagin numbers can also be defined for PL manifolds, an
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https://en.wikipedia.org/wiki/Conrad%20Dasypodius
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Conrad Dasypodius (1532 – 26 April 1600) was a professor of mathematics in Strasbourg, Alsace. He was born in Frauenfeld, Thurgau, Switzerland. His first name was also rendered as Konrad or Conradus or Cunradus, and his last name has been alternatively stated as Rauchfuss, Rauchfuß, and Hasenfratz. He was the son of Petrus Dasypodius (Peter Hasenfuss) (1490–1559, or Peter Hasenfratz), a humanist and lexicographer.
In 1564, Dasypodius edited various parts of the Elements of Euclid. In the preface, he says that for 26 years it had been the rule at his school that all who were promoted from the classes to public lectures should learn Book I of the Elements, but there were no longer any copies to be had so he was bringing out a new edition so as to maintain a good and fruitful regulation of his school.
In 1568, Dasypodius published a work about the heliocentric theory of Nicolaus Copernicus, Hypotyposes orbium coelestium congruentes cum tabulis Alfonsinis et Copernici seu etiam tabulis Prutenicis editae a Cunrado Dasypodio. It is unclear whether Dasypodius was a heliocentrist himself or rather followed the "Wittenberg interpretation."
Dasypodius designed an astronomical clock for the Strasbourg Cathedral; that clock was built in 1572-1574 by Isaac Habrecht and Josia Habrecht. This monumental clock represented the synthesis of the most advanced scientific knowledge of the era, in the domains of astronomy, mathematics, and physics. That mechanism remained in the Cathedral until 1842, when it was replaced by a clock built by Jean Baptiste Schwilgué.
Dasypodius translated writings of Hero of Alexandria from Greek into Latin: one source says it was Hero's Automata; but more likely it was the Mechanica.
Dasypodius died in Strasbourg.
Works
Euclidis Catoptrica (1557) link 1, link 2
Euclidis quindecim elementorum geometriae secundum (1564) link
Propositiones reliquorum librorum geometriae Euclidis (1564) link 1, link 2
(collaboration with Christianus Herlinus) Analysis geometriæ sex librórum Euclidis (1566), impr. J. Richelius, Strassburg, link
link
Eukleidu Stoicheiōn to Prōton (1570) link
Mathematicum, complectens praecepta (1570) link
Eukleidu Protaseis (1570) link
Euclidis elementorum liber primus (1571) link
Sphæricæ doctrinæ propositiones Græcæ et latinæ : Theodosi de sphæra libri III, De habitationibus liber, de Diebus et noctibus libri II. Autolici de sphæra mobili liber. De ortu et occasu stellarum libri II... (1572), impr. Christian Mylius, Strasbourg link
Lexicon seu dictionarium mathematicum (1573) (8 vol. 4).
Kalender oder Laaßbüchlein sampt der Schreibtafel, Mässen vnd Jarmärckren [!] auff das M.D.LXXIIII. Jar (1573) (we don't know if it has been written by Dasypodius)
Brevis et succincta descriptio Corporis luminosi, Quod Nunc Aliqvot Mensibvs Apparvit (1573)
Ein Richtiger vnd kurtzer Bericht über den WunderSternen/ oder besondern Cometen/ so nůn manche Monatszeit/ diß 72. vnd 73. Jar zů sonderem Warnungszeichen diser letzsten zeit
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https://en.wikipedia.org/wiki/Witt%20ring
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In mathematics, a Witt ring may be
A ring of Witt vectors
The Witt ring (forms), a ring structure on the Witt group of symmetric bilinear forms
See also Witt algebra, a Lie algebra.
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https://en.wikipedia.org/wiki/Witt%20group
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In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
Definition
Fix a field k of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector. Each class is represented by the core form of a Witt decomposition.
The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms. Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk : W(k) → Z/2Z is a homomorphism.
The elements of finite order in the Witt group have order a power of 2; the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k; it is generated by the Pfister forms with a non-zero sum of squares. If k is not formally real, then the Witt group is torsion, with exponent a power of 2. The height of the field k is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.
Ring structure
The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors.
To discuss the structure of this ring we assume that k is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms.
The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ring termed the fundamental ideal. The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering. The Witt ring is a Jacobson ring. It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group of k.
If k is not formally real, the fundamental ideal is the only prime ideal of W and consists precisely of the nilpotent elements; W is a local ring and has Krull dimension 0.
If k is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero; W has Krull dimension 1.
If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.
If k is an ordered field with pos
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https://en.wikipedia.org/wiki/Drinfeld%20module
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In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
Drinfeld modules were introduced by , who used them to prove the Langlands conjectures for GL2 of an algebraic function field in some special cases. He later invented shtukas and used shtukas of rank 2 to prove
the remaining cases of the Langlands conjectures for GL2. Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n.
"Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück”, meaning “piece, item, or unit". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind.
Drinfeld modules
The ring of additive polynomials
We let be a field of characteristic . The ring is defined to be the ring of noncommutative (or twisted) polynomials over , with the multiplication given by
The element can be thought of as a Frobenius element: in fact, is a left module over , with elements of acting as multiplication and acting as the Frobenius endomorphism of . The ring can also be thought of as the ring of all (absolutely) additive polynomials
in , where a polynomial is called additive if (as elements of ). The ring of additive polynomials is generated as an algebra over by the polynomial . The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.
Definition of Drinfeld modules
Let F be an algebraic function field with a finite field of constants and fix a place of F. Define A to be the ring of elements in F that are regular at every place except possibly . In particular, A is a Dedekind domain and it is discrete in F (with the topology induced by ). For example, we may take A to be the polynomial ring . Let L be a field equipped with a ring homomorphism .
A Drinfeld A-module over L is a ring homomorphism whose image is not contained in L, such that the composition of with coincides with .
The condition that the image of A is not in L is a non-degeneracy condition, put in to eliminate trivial cases, while the condition that gives the impression that a Drinfeld module is simply a deformation of the map .
As L{τ} can be thought of as endomorphisms of the additive group of L, a Drinfeld A-module can be regarded as an action of A on the additive group of L, or in other words as an A-module whose underlying additive group is the
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https://en.wikipedia.org/wiki/JTS%20Topology%20Suite
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JTS Topology Suite (Java Topology Suite) is an open-source Java software library that provides an object model for Euclidean planar linear geometry together with a set of fundamental geometric functions. JTS is primarily intended to be used as a core component of vector-based geomatics software such as geographical information systems. It can also be used as a general-purpose library providing algorithms in computational geometry.
JTS implements the geometry model and API defined in the OpenGIS Consortium Simple Features Specification for SQL.
JTS defines a standards-compliant geometry system for building spatial applications; examples include viewers, spatial query processors, and tools for performing data validation, cleaning and integration.
In addition to the Java library, the foundations of JTS and selected functions are maintained in a C++ port, for use in C-style linking on all major operating systems, in the form of the GEOS software library.
Up to JTS 1.14, and the GEOS port, are published under the GNU Lesser General Public License (LGPL).
With the LocationTech adoption future releases will be under the EPL/BSD licenses.
Scope
JTS provides the following functionality:
Geometry model
Geometry classes support modelling points, linestrings, polygons, and collections. Geometries are linear, in the sense that boundaries are implicitly defined by linear interpolation between vertices. Geometries are embedded in the 2-dimensional Euclidean plane. Geometry vertices may also carry a Z value.
User-defined precision models are supported for geometry coordinates. Computation is performed using algorithms which provide robust geometric computation under all precision models.
Geometric functions
Topological validity checking
Area and Distance functions
Spatial Predicates based on the Egenhofer DE-9IM model
Overlay functions (including intersection, difference, union, symmetric difference)
Buffer computation (including different cap and join types)
Convex hull
Geometric simplification including the Douglas–Peucker algorithm
Geometric densification
Linear referencing
Precision reduction
Delaunay triangulation and constrained Delaunay triangulation
Voronoi diagram generation
Smallest enclosing rectangle
Discrete Hausdorff distance
Spatial structures and algorithms
Robust line segment intersection
Efficient line arrangement intersection
Efficient point in polygon
Spatial index structures including quadtree and STR-tree
Planar graph structures and algorithms
I/O capabilities
Reading and writing of WKT, WKB and GML formats
History
Funding for the initial work on JTS was obtained in the Fall 2000 from GeoConnections and the Government of British Columbia, based on a proposal put forward by Mark Sondheim and David Skea. The work was carried out by Martin Davis (software design and lead developer) and Jonathan Aquino (developer), both of Vivid Solutions at the time. Since then JTS has been maintained as an independen
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https://en.wikipedia.org/wiki/Schmidt%20decomposition
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In linear algebra, the Schmidt decomposition (named after its originator Erhard Schmidt) refers to a particular way of expressing a vector in the tensor product of two inner product spaces. It has numerous applications in quantum information theory, for example in entanglement characterization and in state purification, and plasticity.
Theorem
Let and be Hilbert spaces of dimensions n and m respectively. Assume . For any vector in the tensor product , there exist orthonormal sets and such that , where the scalars are real, non-negative, and unique up to re-ordering.
Proof
The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases and . We can identify an elementary tensor with the matrix , where is the transpose of . A general element of the tensor product
can then be viewed as the n × m matrix
By the singular value decomposition, there exist an n × n unitary U, m × m unitary V, and a positive semidefinite diagonal m × m matrix Σ such that
Write where is n × m and we have
Let be the m column vectors of , the column vectors of , and the diagonal elements of Σ. The previous expression is then
Then
which proves the claim.
Some observations
Some properties of the Schmidt decomposition are of physical interest.
Spectrum of reduced states
Consider a vector of the tensor product
in the form of Schmidt decomposition
Form the rank 1 matrix . Then the partial trace of , with respect to either system A or B, is a diagonal matrix whose non-zero diagonal elements are . In other words, the Schmidt decomposition shows that the reduced states of on either subsystem have the same spectrum.
Schmidt rank and entanglement
The strictly positive values in the Schmidt decomposition of are its Schmidt coefficients, or Schmidt numbers. The total number of Schmidt coefficients of , counted with multiplicity, is called its Schmidt rank.
If can be expressed as a product
then is called a separable state. Otherwise, is said to be an entangled state. From the Schmidt decomposition, we can see that is entangled if and only if has Schmidt rank strictly greater than 1. Therefore, two subsystems that partition a pure state are entangled if and only if their reduced states are mixed states.
Von Neumann entropy
A consequence of the above comments is that, for pure states, the von Neumann entropy of the reduced states is a well-defined measure of entanglement. For the von Neumann entropy of both reduced states of is , and this is zero if and only if is a product state (not entangled).
Schmidt-rank vector
The Schmidt rank is defined for bipartite systems, namely quantum states
The concept of Schmidt rank can be extended to quantum systems made up of more than two subsystems.
Consider the tripartite quantum system:
There are three ways to reduce this to a bipartite system by performing the partial trace with respect to or
Each of the systems obtained is a b
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https://en.wikipedia.org/wiki/Cantamath
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Cantamath is a mathematics competition competed in Christchurch, Canterbury, New Zealand by years 6 to 10 students.
There are two sections, the Competition section and the Project section.
The sponsors of Cantamath are Casio, Trimble Navigation, Every Educaid, Mathletics and University of Canterbury.
Team Competition section
In the Team Competition section, each participating school sends in four selected student mathematicians per year level. The participants compete against other schools in the Christchurch Arena. It's a speed competition and takes 30 minutes. There are 20 questions for each team to complete, the aim being for each team to answer all questions the fastest. One of the four team members is a runner who runs to a judge to check if the answer to their current question is right. Each question is worth 5 points, allowing a maximum score of 100. A team can only attempt one question at a time and have to keep working on it until they get it right. Passing is allowed, but no points will be received for that question, as well as preventing the team from returning to that question.
The winning team gets a badge and a prize from Casio.
Project section
In the Project section, the student submits a project on a certain topic. Projects can be awarded with an Excellence or Highly Commended award, depending on their quality. There is also an Outstanding award for the best few projects in the display section.
The categories include:
Computer Generated Design (CGD): This consists of a picture made in programs. CGDs might include polygons, curves, and circles.
Publicity Motif: This is a design for the following year's poster. One poster will be chosen and will be the poster for that year.
Mathematical Poster: A poster which is based on the current year's theme.
Geometrical Design: This is a hand-done A4 paper project which should include geometric shapes and curves.
Mathematical Models: A 3D model. The design may be static or mobile.
Varies other exist, and may be found at the Cantamath site.
References
External links
Cantamath
Education in Canterbury, New Zealand
Mathematics competitions
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https://en.wikipedia.org/wiki/Albert%20Russell%20Nichols
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Albert Russell Nichols (1859–1933 ) was an English museum curator and zoologist who worked mainly in Ireland.
Nichols was educated at Clare College, Cambridge, graduating B.A. in mathematics as 16th wrangler in 1882. Nichols came from England to Dublin in 1883 as Assistant in the Museum of Science and Art (now the National Museum of Ireland). He worked on zoology, classifying and arranging the invertebrates throughout his forty-one years of service. He eventually became Keeper of the Natural History Division. Nichols took part in the Lord Bandon dredging expedition of 1886 with Haddon, sponsored by the Royal Irish Academy, and in the biological surveys of Lambay, Clare Island and Malahide.
He compiled or revised lists of echinoderms, marine Mollusca and birds of Ireland, issued by the Museum or by the Royal Irish Academy.
References
1859 births
1933 deaths
English curators
Irish zoologists
Alumni of Clare College, Cambridge
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https://en.wikipedia.org/wiki/Ted%20Lewis%20%28computer%20scientist%29
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Theodore Gyle (Ted) Lewis (born 1941) is an American computer scientist and mathematician, and professor at the Naval Postgraduate School.
Biography
Lewis received his BS in Mathematics and his PhD in computer Science. He started his career at the Oregon State University, where he became Professor of Computer Science and directed its Industry Research Center OACIS.
In 1993 he moved to the Naval Postgraduate School, where he was chairman of computer science for four years. In 1997 he moved to DaimlerChrysler Research and Technology, North America, Inc., where he served as president and CEO. After about three years he moved to the Eastman Kodak Company, where he directed the Digital Business Development division. In his retirement from industry in 2002, he became professor at the Naval Postgraduate School.
A columnist for IEEE Internet Computing, he has contributed pieces to Scientific American and Upside. He has served two stints of Editor-in-Chief, at IEEE Software from 1987 to 1990 and at Computer from 1993 to 1994.
Selected publications
Lewis has written or co-authored 30 books, including:
El-Rewini, Hesham, Theodore G. Lewis, and Hesham H. Ali. Task scheduling in parallel and distributed systems. Prentice-Hall, Inc., 1994.
Lewis, Theodore Gyle. The friction-free economy: Marketing strategies for a wired world. HarperBusiness, 1997.
El-Rewini, Hesham, and Ted G. Lewis. Distributed and parallel computing. Manning Publications Co., 1998.
Lewis, Ted G. Microsoft Rising: And Other Tales of the Silicon Valley. IEEE Computer Society Press, 1999.
Articles, a selection
Lewis, Theodore G., and William H. Payne. "Generalized feedback shift register pseudorandom number algorithm." Journal of the ACM 20.3 (1973): 456–468.
El-Rewini, Hesham, and Ted G. Lewis. "Scheduling parallel program tasks onto arbitrary target machines." Journal of parallel and Distributed Computing 9.2 (1990): 138–153.
References
External links
Ted Lewis's web site
Living people
Place of birth missing (living people)
1941 births
American computer scientists
Naval Postgraduate School faculty
Oregon State University faculty
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https://en.wikipedia.org/wiki/Strict%20%28disambiguation%29
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The term strict refers to relational operators in mathematics.
Strict may also refer to:
Strict, a function classification in programming languages - see Strict function
the strict pragma in the programming language Perl used to restrict unsafe constructs
See also
List of people known as the Strict
Strict histories (or executions) in scheduling
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https://en.wikipedia.org/wiki/Frobenius%20theorem
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There are several mathematical theorems named after Ferdinand Georg Frobenius. They include:
Frobenius theorem (differential topology) in differential geometry and topology for integrable subbundles
Frobenius theorem (real division algebras) in abstract algebra characterizing the finite-dimensional real division algebras
Frobenius reciprocity theorem in group representation theory describing the reciprocity relation between restricted and induced representations on a subgroup
Perron–Frobenius theorem in matrix theory concerning the eigenvalues and eigenvectors of a matrix with positive real coefficients
Frobenius's theorem (group theory) about the number of solutions of xn=1 in a group
Mathematics disambiguation pages
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https://en.wikipedia.org/wiki/Open%20book%20decomposition
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In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books have relevance to contact geometry, with a famous theorem of Emmanuel Giroux (given below) that shows that contact geometry can be studied from an entirely topological viewpoint.
Definition and construction
Definition. An open book decomposition of a 3-dimensional manifold M is a pair (B, π) where
B is an oriented link in M, called the binding of the open book;
π: M \ B → S1 is a fibration of the complement of B such that for each θ ∈ S1, π−1(θ) is the interior of a compact surface Σ ⊂ M whose boundary is B. The surface Σ is called the page of the open book.
This is the special case m = 3 of an open book decomposition of an m-dimensional manifold, for any m.
The definition for general m is similar, except that the surface with boundary (Σ, B) is replaced by an (m − 1)-manifold with boundary (P, ∂P). Equivalently, the open book decomposition can be thought of as a homeomorphism of M to the quotient space
where f:P → P is a self-homeomorphism preserving the boundary. This quotient space is called a relative mapping torus.
When Σ is an oriented compact surface with n boundary components and φ: Σ → Σ is a homeomorphism which is the identity near the boundary, we can construct an open book by first forming the mapping torus Σφ. Since φ is the identity on ∂Σ, ∂Σφ is the trivial circle bundle over a union of circles, that is, a union of tori; one torus for each boundary component. To complete the construction, solid tori are glued to fill in the boundary tori so that each circle S1 × {p} ⊂ S1×∂D2 is identified with the boundary of a page. In this case, the binding is the collection of n cores S1×{q} of the n solid tori glued into the mapping torus, for arbitrarily chosen q ∈ D2. It is known that any open book can be constructed this way. As the only information used in the construction is the surface and the homeomorphism, an alternate definition of open book is simply the pair (Σ, φ) with the construction understood. In short, an open book is a mapping torus with solid tori glued in so that the core circle of each torus runs parallel to the boundary of the fiber.
Each torus in ∂Σφ is fibered by circles parallel to the binding, each circle a boundary component of a page. One envisions a rolodex-looking structure for a neighborhood of the binding (that is, the solid torus glued to ∂Σφ)—the pages of the rolodex connect to pages of the open book and the center of the rolodex is the binding. Thus the term open book.
It is a 1972 theorem of Elmar Winkelnkemper that for m > 6, a simply-connected m-dimensional manifold has an open book decomposition if and only if it has signature 0. In 1977 Terry Lawson proved that for odd m > 6, every m-dimensional manifold has an open book decomposition, a result extended to 5-manifolds and manifolds with boundary by
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https://en.wikipedia.org/wiki/Andrzej%20Bia%C5%82ynicki-Birula
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Andrzej Białynicki-Birula (26 December 1935 – 19 April 2021) was a Polish mathematician, best known for his work on algebraic geometry. He was considered one of the pioneers of differential algebra. He was a member of the Polish Academy of Sciences.
Białynicki-Birula was born in Nowogrodek, Polish Republic, currently known as Navahrudak, West Belarus. His elder brother, , was born two years earlier and is a theoretical physicist and a fellow member of the Polish Academy of Sciences. He received his Ph.D. from the University of California, Berkeley in 1960. His thesis was written under the direction of Gerhard Hochschild. Since 1970, he was Professor of Mathematics at Warsaw University.
See also
List of Polish mathematicians
References
External links
Home page at Warsaw University
1935 births
2021 deaths
20th-century Polish mathematicians
21st-century Polish mathematicians
Algebraic geometers
University of California, Berkeley alumni
Academic staff of the University of Warsaw
Members of the Polish Academy of Sciences
People from Nowogródek Voivodeship (1919–1939)
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https://en.wikipedia.org/wiki/List%20of%20Juventus%20FC%20players
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This is a list of Juventus FC players who have earned 100 or more caps for Juventus.
For a list of notable Juventus players, major or minor, see Juventus FC players. For statistics and records see the statistics and records related article. For the list of Juventus players who played with the Italy national team during their careers at Juventus, see Juventus FC and the Italy national football team. For the current squad and its notable players, see the main Juventus FC article.
List of players
Players with 100 or more appearances for the club are listed in alphabetical order according to the date of their first-team official debut for the club. Appearances and goals are for all first-team competitive matches. Substitute appearances included. Bold denotes current Juventus players.
Statistics correct as of 4 June 2023.
Club captains
The role of captain in Italian football made its first appearance in the early 1920s.
List of Juventus players to have won all three major UEFA club competitions
The table below shows the Juventus players who have won all three major UEFA club competitions (chronological order).
Source: Players regarding European Club Cups, Record Sport Soccer Statistics Foundation, rsssf.com.
List of Juventus' players to have won all UEFA club competitions
The table below show the Juventus players who have won all UEFA club competitions (chronological order).
Source: Players regarding European Club Cups, Record Sport Soccer Statistics Foundation, rsssf.com.
See also
Lega Calcio Serie A-winning players
Juventus F.C. and the Italy national football team
Footnotes and references
Bibliography
External links
Champions of the past: History of Juventus FC players and managers (bianconerionline.com)
Statistic Area – All Juventus FC players since 1900 (juword.net)
Juventus Story: more than 100 years of Juve (juvecentus.too.it)
players
Juventus
Association football player non-biographical articles
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https://en.wikipedia.org/wiki/Contrast%20%28statistics%29
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In statistics, particularly in analysis of variance and linear regression, a contrast is a linear combination of variables (parameters or statistics) whose coefficients add up to zero, allowing comparison of different treatments.
Definitions
Let be a set of variables, either parameters or statistics, and be known constants. The quantity is a linear combination. It is called a contrast if Furthermore, two contrasts, and , are orthogonal if
Examples
Let us imagine that we are comparing four means, . The following table describes three possible contrasts:
The first contrast allows comparison of the first mean with the second, the second contrast allows comparison of the third mean with the fourth, and the third contrast allows comparison of the average of the first two means with the average of the last two.
In a balanced one-way analysis of variance, using orthogonal contrasts has the advantage of completely partitioning the treatment sum of squares into non-overlapping additive components that represent the variation due to each contrast. Consider the numbers above: each of the rows sums up to zero (hence they are contrasts). If we multiply each element of the first and second row and add those up, this again results in zero, thus the first and second contrast are orthogonal and so on.
Sets of contrast
Orthogonal contrasts are a set of contrasts in which, for any distinct pair, the sum of the cross-products of the coefficients is zero (assume sample sizes are equal). Although there are potentially infinite sets of orthogonal contrasts, within any given set there will always be a maximum of exactly k – 1 possible orthogonal contrasts (where k is the number of group means available).
Polynomial contrasts are a special set of orthogonal contrasts that test polynomial patterns in data with more than two means (e.g., linear, quadratic, cubic, quartic, etc.).
Orthonormal contrasts are orthogonal contrasts which satisfy the additional condition that, for each contrast, the sum squares of the coefficients add up to one.
Background
A contrast is defined as the sum of each group mean multiplied by a coefficient for each group (i.e., a signed number, cj). In equation form,
, where L is the weighted sum of group means, the cj coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and j represents the group means. Coefficients can be positive or negative, and fractions or whole numbers, depending on the comparison of interest. Linear contrasts are very useful and can be used to test complex hypotheses when used in conjunction with ANOVA or multiple regression. In essence, each contrast defines and tests for a particular pattern of differences among the means.
Contrasts should be constructed "to answer specific research questions", and do not necessarily have to be orthogonal.
A simple (not necessarily orthogonal) contrast is the difference between two means. A more complex contrast can test diffe
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https://en.wikipedia.org/wiki/Ramsey%20RESET%20test
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In statistics, the Ramsey Regression Equation Specification Error Test (RESET) test is a general specification test for the linear regression model. More specifically, it tests whether non-linear combinations of the explanatory variables help to explain the response variable. The intuition behind the test is that if non-linear combinations of the explanatory variables have any power in explaining the response variable, the model is misspecified in the sense that the data generating process might be better approximated by a polynomial or another non-linear functional form.
The test was developed by James B. Ramsey as part of his Ph.D. thesis at the University of Wisconsin–Madison in 1968, and later published in the Journal of the Royal Statistical Society in 1969.
Technical summary
Consider the model
The Ramsey test then tests whether has any power in explaining . This is executed by estimating the following linear regression
and then testing, by a means of a F-test whether through are zero. If the null-hypothesis that all coefficients are zero is rejected, then the model suffers from misspecification.
See also
Harvey–Collier test
References
Further reading
Statistical tests
Regression diagnostics
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https://en.wikipedia.org/wiki/Bretschneider%27s%20formula
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In geometry, Bretschneider's formula is a mathematical expression for the area of a general quadrilateral.
It works on both convex and concave quadrilaterals (but not crossed ones), whether it is cyclic or not.
History
The German mathematician Carl Anton Bretschneider discovered the formula in 1842. The formula was also derived in the same year by the German mathematician Karl Georg Christian von Staudt.
Formulation
Bretschneider's formula is expressed as:
Here, , , , are the sides of the quadrilateral, is the semiperimeter, and and are any two opposite angles, since as long as
Proof
Denote the area of the quadrilateral by . Then we have
Therefore
The law of cosines implies that
because both sides equal the square of the length of the diagonal . This can be rewritten as
Adding this to the above formula for yields
Note that: (a trigonometric identity true for all )
Following the same steps as in Brahmagupta's formula, this can be written as
Introducing the semiperimeter
the above becomes
and Bretschneider's formula follows after taking the square root of both sides:
The second form is given by using the cosine half-angle identity
yielding
Emmanuel García has used the generalized half angle formulas to give an alternative proof.
Related formulae
Bretschneider's formula generalizes Brahmagupta's formula for the area of a cyclic quadrilateral, which in turn generalizes Heron's formula for the area of a triangle.
The trigonometric adjustment in Bretschneider's formula for non-cyclicality of the quadrilateral can be rewritten non-trigonometrically in terms of the sides and the diagonals and to give
Notes
References & further reading
C. A. Bretschneider. Untersuchung der trigonometrischen Relationen des geradlinigen Viereckes. Archiv der Mathematik und Physik, Band 2, 1842, S. 225-261 ( online copy, German)
F. Strehlke: Zwei neue Sätze vom ebenen und sphärischen Viereck und Umkehrung des Ptolemaischen Lehrsatzes. Archiv der Mathematik und Physik, Band 2, 1842, S. 323-326 (online copy, German)
External links
Bretschneider's formula at proofwiki.org
Theorems about quadrilaterals
Area
Articles containing proofs
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https://en.wikipedia.org/wiki/K0
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K0 may refer to:
Spectral class K0, a star spectral class
the 1965 first model of the Honda CB450 motorbike
the Grothendieck group in abstract algebra
the Lateral earth pressure at rest
the neutral kaon, a strange meson with no charge in nuclear physics
K0 may refer to Khinchin's constant
K0 the order-zero graph
See also
KO (disambiguation)
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https://en.wikipedia.org/wiki/Kent%20distribution
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In directional statistics, the Kent distribution, also known as the 5-parameter Fisher–Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution on the unit sphere (2-sphere S2 in 3-space R3). It is the analogue on S2 of the bivariate normal distribution with an unconstrained covariance matrix. The Kent distribution was proposed by John T. Kent in 1982, and is used in geology as well as bioinformatics.
Definition
The probability density function of the Kent distribution is given by:
where is a three-dimensional unit vector, denotes the transpose of , and the normalizing constant is:
Where is the modified Bessel function and is the gamma function. Note that and , the normalizing constant of the Von Mises–Fisher distribution.
The parameter (with ) determines the concentration or spread of the distribution, while (with ) determines the ellipticity of the contours of equal probability. The higher the and parameters, the more concentrated and elliptical the distribution will be, respectively. Vector is the mean direction, and vectors are the major and minor axes. The latter two vectors determine the orientation of the equal probability contours on the sphere, while the first vector determines the common center of the contours. The 3×3 matrix must be orthogonal.
Generalization to higher dimensions
The Kent distribution can be easily generalized to spheres in higher dimensions. If is a point on the unit sphere in , then the density function of the -dimensional Kent distribution is proportional to
where and and the vectors are orthonormal. However, the normalization constant becomes very difficult to work with for .
See also
Directional statistics
Von Mises–Fisher distribution
Bivariate von Mises distribution
Von Mises distribution
Bingham distribution
References
Boomsma, W., Kent, J.T., Mardia, K.V., Taylor, C.C. & Hamelryck, T. (2006) Graphical models and directional statistics capture protein structure . In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Interdisciplinary Statistics and Bioinformatics, pp. 91–94. Leeds, Leeds University Press.
Hamelryck T, Kent JT, Krogh A (2006) Sampling Realistic Protein Conformations Using Local Structural Bias. PLoS Comput Biol 2(9): e131
Kent, J. T. (1982) The Fisher–Bingham distribution on the sphere., J. Royal. Stat. Soc., 44:71–80.
Kent, J. T., Hamelryck, T. (2005). Using the Fisher–Bingham distribution in stochastic models for protein structure . In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Quantitative Biology, Shape Analysis, and Wavelets, pp. 57–60. Leeds, Leeds University Press.
Mardia, K. V. M., Jupp, P. E. (2000) Directional Statistics (2nd edition), John Wiley and Sons Ltd.
Peel, D., Whiten, WJ., McLachlan, GJ. (2001) Fitting mixtures of Kent distributions to aid in joint set identification. J. Am. Stat. Ass., 96:56–63
Directional statistics
Continuous di
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https://en.wikipedia.org/wiki/Von%20Mises%E2%80%93Fisher%20distribution
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In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the -sphere in . If
the distribution reduces to the von Mises distribution on the circle.
Definition
The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector is given by:
where and
the normalization constant is equal to
where denotes the modified Bessel function of the first kind at order . If , the normalization constant reduces to
The parameters and are called the mean direction and concentration parameter, respectively. The greater the value of , the higher the concentration of the distribution around the mean direction . The distribution is unimodal for , and is uniform on the sphere for .
The von Mises–Fisher distribution for is also called the Fisher distribution.
It was first used to model the interaction of electric dipoles in an electric field. Other applications are found in geology, bioinformatics, and text mining.
Note on the normalization constant
In the textbook, Directional Statistics by Mardia and Jupp, the normalization constant given for the Von Mises Fisher probability density is apparently different from the one given here: . In that book, the normalization constant is specified as:
where is the gamma function. This is resolved by noting that Mardia and Jupp give the density "with respect to the uniform distribution", while the density here is specified in the usual way, with respect to Lebesgue measure. The density (w.r.t. Lebesgue measure) of the uniform distribution is the reciprocal of the surface area of the (p-1)-sphere, so that the uniform density function is given by the constant:
It then follows that:
While the value for was derived above via the surface area, the same result may be obtained by setting in the above formula for . This can be done by noting that the series expansion for divided by has but one non-zero term at . (To evaluate that term, one needs to use the definition .)
Support
The support of the Von Mises–Fisher distribution is the hypersphere, or more specifically, the -sphere, denoted as
This is a -dimensional manifold embedded in -dimensional Euclidean space, .
Relation to normal distribution
Starting from a normal distribution with isotropic covariance and mean of length , whose density function is:
the Von Mises–Fisher distribution is obtained by conditioning on . By expanding
and using the fact that the first two right-hand-side terms are fixed, the Von Mises-Fisher density, is recovered by recomputing the normalization constant by integrating over the unit sphere. If , we get the uniform distribution, with density .
More succinctly, the restriction of any isotropic multivariate normal density to the unit hypersphere, gives a Von Mises-Fisher density, up to normalization.
This construction can be generalized by starting with a normal distribution with a g
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https://en.wikipedia.org/wiki/Intersection%20theorem
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In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects and (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects and must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.
For example, Desargues' theorem can be stated using the following incidence structure:
Points:
Lines:
Incidences (in addition to obvious ones such as ):
The implication is then —that point is incident with line .
Famous examples
Desargues' theorem holds in a projective plane if and only if is the projective plane over some division ring (skewfield} — . The projective plane is then called desarguesian.
A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane satisfies the intersection theorem if and only if the division ring satisfies the rational identity.
Pappus's hexagon theorem holds in a desarguesian projective plane if and only if is a field; it corresponds to the identity .
Fano's axiom (which states a certain intersection does not happen) holds in if and only if has characteristic ; it corresponds to the identity .
References
Incidence geometry
Theorems in projective geometry
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https://en.wikipedia.org/wiki/Reach%20%28advertising%29
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In the application of statistics to advertising and media analysis, reach refers to the total number of different people or households exposed, at least once, to a medium during a given period. Reach should not be confused with the number of people who will actually be exposed to and consume the advertising, though. It is just the number of people who are exposed to the medium and therefore have an opportunity to see or hear the ad or commercial. Reach may be stated either as an absolute number, or as a fraction of a given population (for instance 'TV households', 'men' or 'those aged 25–35').
For any given viewer, they have been "reached" by the work if they have viewed it at all (or a specified amount) during the specified period. Multiple viewings by a single member of the audience in the cited period do not increase reach; however, media people use the term effective reach to describe the quality of exposure. Effective reach and reach are two different measurements for a target audience who receive a given message or ad.
Since reach is a time-dependent summary of aggregate audience behavior, reach figures are meaningless without a period associated with them: an example of a valid reach figure would be to state that "[example website] had a one-day reach of 1565 per million on 21 March 2004" (though unique users, an equivalent measure, would be a more typical metric for a website).
Reach of television channels is often expressed in the form of "x minute weekly reach" – that is, the number (or percentage) of viewers who watched the channel for at least x minutes in a given week.
For example, in the UK, BARB defines the reach of a television channel as the percentage of the population in private households who view a channel for more than 3 minutes in a given day or week. Similarly, for radio, RAJAR defines the weekly reach of a radio station as the number of people who tune into a radio station for at least 5 minutes (within at least one 15 min period) in a given week.
Reach is an important measure for the BBC, which is funded by a mandatory licence fee. It seeks to maximise its reach to ensure all licence fee payers are receiving value. Reach and frequency of exposure are also two of the most important statistics used in advertising management. When reach is multiplied by average frequency a composite measure called gross rating points (GRPs) is obtained. Reach can be calculated indirectly as: reach = GRPs / average frequency.
Reach Calculation
The following formulas are used to calculate the reach of a marketing campaign.
F=I/U
Where F is the frequency
I is the total number of impressions
U is the total number of unique users
See also
Social media reach
Television advertisement Vehicle Exposure of Media research
References
Audience measurement
Advertising
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https://en.wikipedia.org/wiki/Definite%20form
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Definite form may refer to:
Definite quadratic form in mathematics
Definiteness in linguistics
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https://en.wikipedia.org/wiki/Power%20automorphism
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In mathematics, in the realm of group theory, a power automorphism of a group is an automorphism that takes each subgroup of the group to within itself. It is worth noting that the power automorphism of an infinite group may not restrict to an automorphism on each subgroup. For instance, the automorphism on rational numbers that sends each number to its double is a power automorphism even though it does not restrict to an automorphism on each subgroup.
Alternatively, power automorphisms are characterized as automorphisms that send each element of the group to some power of that element. This explains the choice of the term power. The power automorphisms of a group form a subgroup of the whole automorphism group. This subgroup is denoted as where is the group.
A universal power automorphism is a power automorphism where the power to which each element is raised is the same. For instance, each element may go to its cube. Here are some facts about the powering index:
The powering index must be relatively prime to the order of each element. In particular, it must be relatively prime to the order of the group, if the group is finite.
If the group is abelian, any powering index works.
If the powering index 2 or -1 works, then the group is abelian.
The group of power automorphisms commutes with the group of inner automorphisms when viewed as subgroups of the automorphism group. Thus, in particular, power automorphisms that are also inner must arise as conjugations by elements in the second group of the upper central series.
References
Subgroup lattices of groups by Roland Schmidt (PDF file)
Group theory
Group automorphisms
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https://en.wikipedia.org/wiki/IA%20automorphism
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In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization. The abelianization of a group is its quotient by its commutator subgroup. An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself.
The IA automorphisms of a group form a normal subgroup of the automorphism group. Every inner automorphism is an IA automorphism.
See also
Torelli group
References
Group theory
Group automorphisms
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https://en.wikipedia.org/wiki/Quotientable%20automorphism
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In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group.
All family automorphisms are quotientable, and particularly, all class automorphisms and power automorphisms are. As well, all inner automorphisms are quotientable, and more generally, any automorphism defined by an algebraic formula is quotientable.
Group automorphisms
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https://en.wikipedia.org/wiki/Class%20automorphism
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In mathematics, in the realm of group theory, a class automorphism is an automorphism of a group that sends each element to within its conjugacy class. The class automorphisms form a subgroup of the automorphism group. Some facts:
Every inner automorphism is a class automorphism.
Every class automorphism is a family automorphism and a quotientable automorphism.
Under a quotient map, class automorphisms go to class automorphisms.
Every class automorphism is an IA automorphism, that is, it acts as identity on the abelianization.
Every class automorphism is a center-fixing automorphism, that is, it fixes all points in the center.
Normal subgroups are characterized as subgroups invariant under class automorphisms.
For infinite groups, an example of a class automorphism that is not inner is the following: take the finitary symmetric group on countably many elements and consider conjugation by an infinitary permutation. This conjugation defines an outer automorphism on the group of finitary permutations. However, for any specific finitary permutation, we can find a finitary permutation whose conjugation has the same effect as this infinitary permutation. This is essentially because the infinitary permutation takes permutations of finite supports to permutations of finite support.
For finite groups, the classical example is a group of order 32 obtained as the semidirect product of the cyclic ring on 8 elements, by its group of units acting via multiplication. Finding a class automorphism in the stability group that is not inner boils down to finding a cocycle for the action that is locally a coboundary but is not a global coboundary.
Group theory
Group automorphisms
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https://en.wikipedia.org/wiki/Stability%20group
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In mathematics, in the realm of group theory, the stability group of subnormal series is the group of automorphisms that act as identity on each quotient group.
Group theory
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