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https://en.wikipedia.org/wiki/D%27Alembert%27s%20equation
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In mathematics, d'Alembert's equation is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as
where . After differentiating once, and rearranging we have
The above equation is linear. When , d'Alembert's equation is reduced to Clairaut's equation.
References
Eponymous equations of physics
Mathematical physics
Differential equations
Ordinary differential equations
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https://en.wikipedia.org/wiki/Interchange%20of%20limiting%20operations
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In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series.
Formulation
In symbols, the assumption
LM = ML,
where the left-hand side means that M is applied first, then L, and vice versa on the right-hand side, is not a valid equation between mathematical operators, under all circumstances and for all operands. An algebraist would say that the operations do not commute. The approach taken in analysis is somewhat different. Conclusions that assume limiting operations do 'commute' are called formal. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient conditions for the formal analysis to hold. This approach justifies, for example, the notion of uniform convergence. It is relatively rare for such sufficient conditions to be also necessary, so that a sharper piece of analysis may extend the domain of validity of formal results.
Professionally speaking, therefore, analysts push the envelope of techniques, and expand the meaning of well-behaved for a given context. G. H. Hardy wrote that "The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics". An opinion apparently not in favour of the piece-wise approach, but of leaving analysis at the level of heuristic, was that of Richard Courant.
Examples
Examples abound, one of the simplest being that for a double sequence am,n: it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. For example take
am,n = 2m − n
in which taking the limit first with respect to n gives 0, and with respect to m gives ∞.
Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators.
One of the major reasons why the Lebesgue integral is used is that theorems exist, such as the dominated convergence theorem, that give sufficient conditions under which integration and limit operation can be interchanged. Necessary and sufficient conditions for this interchange were discovered by Federico Cafiero.
List of related theorems
Interchange of limits:
Moore-Osgood theorem
Interchange of limit and infinite summation:
Tannery's theorem
Interchange of partial derivatives:
Schwarz's theorem
Interchange of integrals:
Fubini's theorem
Interchange of limit and integral:
Dominated convergence theorem
Vitali convergence theorem
Fichera convergence theorem
Cafiero convergence theorem
Fatou's lemma
Monotone converge
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https://en.wikipedia.org/wiki/Pullback%20attractor
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In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatment because random dynamical systems are necessarily non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense.
Set-up and motivation
Consider a random dynamical system on a complete separable metric space , where the noise is chosen from a probability space with base flow .
A naïve definition of an attractor for this random dynamical system would be to require that for any initial condition , as . This definition is far too limited, especially in dimensions higher than one. A more plausible definition, modelled on the idea of an omega-limit set, would be to say that a point lies in the attractor if and only if there exists an initial condition, , and there is a sequence of times such that
as .
This is not too far from a working definition. However, we have not yet considered the effect of the noise , which makes the system non-autonomous (i.e. it depends explicitly on time). For technical reasons, it becomes necessary to do the following: instead of looking seconds into the "future", and considering the limit as , one "rewinds" the noise seconds into the "past", and evolves the system through seconds using the same initial condition. That is, one is interested in the pullback limit
.
So, for example, in the pullback sense, the omega-limit set for a (possibly random) set is the random set
Equivalently, this may be written as
Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth.
Several examples of pullback attractors of non-autonomous dynamical systems are presented analytically and numerically.
Definition
The pullback attractor (or random global attractor) for a random dynamical system is a -almost surely unique random set such that
is a random compact set: is almost surely compact and is a -measurable function for every ;
is invariant: for all almost surely;
is attractive: for any deterministic bounded set ,
almost surely.
There is a slight abuse of notation in the above: the first use of "dist" refers to the Hausdorff semi-distance from a point to a set,
whereas the second use of "dist" refers to the Hausdorff semi-distance between two sets,
As noted in the previous section, in the absence of noise, this definition of attractor coincides with the deterministic definition of the attractor as the minimal compact invariant set that attracts all bounded deterministic sets.
Theorems relating omega-limit sets to attractors
The attractor as a union of omega-limit sets
If a random dynamical system has a compact random absor
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https://en.wikipedia.org/wiki/Pfister%20form
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In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms
for some nonzero elements a1, ..., an of F. (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.) An n-fold Pfister form can also be constructed inductively from an (n−1)-fold Pfister form q and a nonzero element a of F, as .
So the 1-fold and 2-fold Pfister forms look like:
.
For n ≤ 3, the n-fold Pfister forms are norm forms of composition algebras. In that case, two n-fold Pfister forms are isomorphic if and only if the corresponding composition algebras are isomorphic. In particular, this gives the classification of octonion algebras.
The n-fold Pfister forms additively generate the n-th power I n of the fundamental ideal of the Witt ring of F.
Characterizations
A quadratic form q over a field F is multiplicative if, for vectors of indeterminates x and y, we can write q(x).q(y) = q(z) for some vector z of rational functions in the x and y over F. Isotropic quadratic forms are multiplicative. For anisotropic quadratic forms, Pfister forms are multiplicative, and conversely.
For n-fold Pfister forms with n ≤ 3, this had been known since the 19th century; in that case z can be taken to be bilinear in x and y, by the properties of composition algebras. It was a remarkable discovery by Pfister that n-fold Pfister forms for all n are multiplicative in the more general sense here, involving rational functions. For example, he deduced that for any field F and any natural number n, the set of sums of 2n squares in F is closed under multiplication, using that
the quadratic form
is an n-fold Pfister form (namely, ).
Another striking feature of Pfister forms is that every isotropic Pfister form is in fact hyperbolic, that is, isomorphic to a direct sum of copies of the hyperbolic plane . This property also characterizes Pfister forms, as follows: If q is an anisotropic quadratic form over a field F, and if q becomes hyperbolic over every extension field E such that q becomes isotropic over E, then q is isomorphic to aφ for some nonzero a in F and some Pfister form φ over F.
Connection with K-theory
Let kn(F) be the n-th Milnor K-group modulo 2. There is a homomorphism from kn(F) to the quotient In/In+1 in the Witt ring of F, given by
where the image is an n-fold Pfister form. The homomorphism is surjective, since the Pfister forms additively generate In. One part of the Milnor conjecture, proved by Orlov, Vishik and Voevodsky, states that this homomorphism is in fact an isomorphism . That gives an explicit description of the abelian group In/In+1 by generators and relations. The other part of the Milnor conjecture, prove
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https://en.wikipedia.org/wiki/Norm%20form
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In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis e1, ..., en for L as a vector space over K, the form is given by
N(x1e1 + ... + xnen)
in variables x1, ..., xn.
In number theory norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation. For this application the field K is usually the rational number field, the field L is an algebraic number field, and the basis is taken of some order in the ring of integers OL of L.
See also
Trace form
References
Field (mathematics)
Diophantine equations
Homogeneous polynomials
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https://en.wikipedia.org/wiki/Naagarahaavu
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Naagarahaavu () is a 1972 Indian Kannada-language film directed by Puttanna Kanagal, based on T. R. Subba Rao's three novels Nagarahavu, Ondu Gandu Eradu Hennu and Sarpa Mathsara, and starring Vishnuvardhan, Aarathi , K. S. Ashwath and Shubha. The supporting cast features Leelavathi, M. Jayashree, M. N. Lakshmi Devi, Ambareesh, Shivaram, Dheerendra Gopal, Lokanath and Vajramuni. The film has a musical score by Vijaya Bhaskar. Cinematography was done by Chittibabu.
The film revolves around the protagonist's relationship with his teacher, Chamayya (K. S. Ashwath). Chamayya, who is childless treats Ramachari (Vishnuvardhan) as his son. He takes it upon himself to guide Ramachari on the right path despite Ramachari's legendary anger. He usually acts as the negotiator between his student and the people who have issues with Ramachari's behaviour. Ramachari is a young man whose anger is his weakness. He is difficult to reason with and has a great deal of pride. Chamayya is the only person who can convince him to do anything. His love interests are Alamelu (Aarathi) and Margaret (Shubha) who play pivotal parts in his life.
The film was released on 29 December 1972 to widespread critical acclaim and was a success in the box office and paved way for the stardom of Vishnuvardhan, Ambareesh and Aarathi who became leading actors in Kannada cinema. The character roles of Leelavathi, Dheerendra Gopal, Loknath, M. N. Lakshmi Devi were also critically acclaimed. The film won eight Karnataka State Film Awards for Second Best Film, Best Actor, Best Actress, Best Supporting Actor, Best Supporting Actress, Best Story, Best Screenplay and Best Dialogue. The film also won two Filmfare Awards South for Best Film – Kannada and a Special Award for excellent performance. This film was remade in Hindi as Zehreela Insaan, directed by Puttanna Kanagal himself and in Tamil as Raja Nagam and Kode Nagu in Telugu.
The film was re-released in its digitized version on 20 July 2018. This movie was digitalized by Balaji who is the brother of V. Ravichandran and son of N. Veeraswamy, who was the producer.
Plot details
Ramachari
The story revolves around a short-tempered, yet affable college student named Ramachari in the town of Chitradurga. The story begins with Ramachari being caught in class while trying to copy in an examination and being suspended by the college principal (Loknath). Humiliated and angry, Ramachari throws stones at the principal's house in the night and when the principal wakes up and comes out of his house, ties the half-naked principal to a pole and runs away.
Ramachari is the son of pious parents. His father doesn't like Ramachari because he is unpopular in the town as a ruffian. His mother is worried about his future. One person who cares for Ramachari is his primary school teacher, Chamayya (K.S.Ashwath) master. Ramachari finds the company of Chamayya and his wife Tunga (Leelavathi), more comforting than that of his parents. Ramachari has very great re
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https://en.wikipedia.org/wiki/Base%20flow%20%28random%20dynamical%20systems%29
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In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system.
Definition
In the definition of a random dynamical system, one is given a family of maps on a probability space . The measure-preserving dynamical system is known as the base flow of the random dynamical system. The maps are often known as shift maps since they "shift" time. The base flow is often ergodic.
The parameter may be chosen to run over
(a two-sided continuous-time dynamical system);
(a one-sided continuous-time dynamical system);
(a two-sided discrete-time dynamical system);
(a one-sided discrete-time dynamical system).
Each map is required
to be a -measurable function: for all ,
to preserve the measure : for all , .
Furthermore, as a family, the maps satisfy the relations
, the identity function on ;
for all and for which the three maps in this expression are defined. In particular, if exists.
In other words, the maps form a commutative monoid (in the cases and ) or a commutative group (in the cases and ).
Example
In the case of random dynamical system driven by a Wiener process , where is the two-sided classical Wiener space, the base flow would be given by
.
This can be read as saying that "starts the noise at time instead of time 0".
Random dynamical systems
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https://en.wikipedia.org/wiki/Norm%20variety
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In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to geometric objects V, having function fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups).
The formulation is that p is a given prime number, different from the characteristic of F, and a symbol is the class mod p of an element
of the n-th Milnor K-group. A field extension is said to split the symbol, if its image in the K-group for that field is 0.
The conditions on a norm variety V are that V is irreducible and a non-singular complete variety. Further it should have dimension d equal to
The key condition is in terms of the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number
should not be divisible by p2, it being known it is divisible by p.
Examples
These include (n = 2) cases of the Severi–Brauer variety and (p = 2) Pfister forms. There is an existence theorem in the general case (paper of Markus Rost cited).
References
External links
Paper by Rost
Algebraic varieties
K-theory
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https://en.wikipedia.org/wiki/Absorbing%20set%20%28random%20dynamical%20systems%29
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In mathematics, an absorbing set for a random dynamical system is a subset of the phase space. A dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.
The absorbing set eventually contains the image of any bounded set under the cocycle ("flow") of the random dynamical system. As with many concepts related to random dynamical systems, it is defined in the pullback sense.
Definition
Consider a random dynamical system φ on a complete separable metric space (X, d), where the noise is chosen from a probability space (Ω, Σ, P) with base flow θ : R × Ω → Ω. A random compact set K : Ω → 2X is said to be absorbing if, for all d-bounded deterministic sets B ⊆ X, there exists a (finite) random time τB : Ω → 0, +∞) such that
This is a definition in the pullback sense, as indicated by the use of the negative time shift θ−t.
See also
Glossary of areas of mathematics
Lists of mathematics topics
Mathematics Subject Classification
Outline of mathematics
References
(See footnote (e) on p. 104)
Random dynamical systems
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https://en.wikipedia.org/wiki/Discrete%20series%20representation
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In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.
Properties
If G is unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one (and hence all) matrix coefficient
with v, w non-zero vectors is square-integrable on G, with respect to Haar measure.
When G is unimodular, the discrete series representation has a formal dimension d, with the property that
for v, w, x, y in the representation. When G is compact this coincides with the dimension when the Haar measure on G is normalized so that G has measure 1.
Semisimple groups
classified the discrete series representations of connected semisimple groups G. In particular, such a group has discrete series representations if and only if it has the same rank as a maximal compact subgroup K. In other words, a maximal torus T in K must be a Cartan subgroup in G. (This result required that the center of G be finite, ruling out groups such as the simply connected cover of SL(2,R).) It applies in particular to special linear groups; of these only SL(2,R) has a discrete series (for this, see the representation theory of SL(2,R)).
Harish-Chandra's classification of the discrete series representations of a semisimple connected Lie group is given as follows. If L is the weight lattice of the maximal torus T, a sublattice of it where t is the Lie algebra of T, then there is a discrete series representation for every vector v of
L + ρ,
where ρ is the Weyl vector of G, that is not orthogonal to any root of G. Every discrete series representation occurs in this way. Two such vectors v correspond to the same discrete series representation if and only if they are conjugate under the Weyl group WK of the maximal compact subgroup K. If we fix a fundamental chamber for the Weyl group of K, then the discrete series representation are in 1:1 correspondence with the vectors of L + ρ in this Weyl chamber that are not orthogonal to any root of G. The infinitesimal character of the highest weight representation is given by v (mod the Weyl group WG of G) under the Harish-Chandra correspondence identifying infinitesimal characters of G with points of
t ⊗ C/WG.
So for each discrete series representation, there are exactly
|WG|/|WK|
discrete series representations with the same infinitesimal character.
Harish-Chandra went on to prove an analogue for these representations of the Weyl character formula. In the case where G is not compact, the representations have infinite dimension, and the notion of character is therefore more subtle to define since it is a Schwartz distribution (represented by a locally
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https://en.wikipedia.org/wiki/It%C3%B4%20isometry
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In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals.
Let denote the canonical real-valued Wiener process defined up to time , and let be a stochastic process that is adapted to the natural filtration of the Wiener process. Then
where denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space of square-integrable adapted processes to the space of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
and
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
for .
References
Stochastic calculus
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https://en.wikipedia.org/wiki/Functional%20square%20root
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In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying for all .
Notation
Notations expressing that is a functional square root of are and .
History
The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950.
The solutions of over (the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation. A particular solution is for . Babbage noted that for any given solution , its functional conjugate by an arbitrary invertible function is also a solution. In other words, the group of all invertible functions on the real line acts on the subset consisting of solutions to Babbage's functional equation by conjugation.
Solutions
A systematic procedure to produce arbitrary functional -roots (including arbitrary real, negative, and infinitesimal ) of functions relies on the solutions of Schröder's equation. Infinitely many trivial solutions exist when the domain of a root function f is allowed to be sufficiently larger than that of g.
Examples
is a functional square root of .
A functional square root of the th Chebyshev polynomial, , is , which in general is not a polynomial.
is a functional square root of .
[red curve]
[blue curve]
[orange curve]
[black curve above the orange curve]
[dashed curve]
(See. For the notation, see .)
See also
Iterated function
Function composition
Abel equation
Schröder's equation
Flow (mathematics)
Superfunction
Fractional calculus
Half-exponential function
References
Functional analysis
Functional equations
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https://en.wikipedia.org/wiki/Desargues%20configuration
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In geometry, the Desargues configuration is a configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues.
The Desargues configuration can be constructed in two dimensions from the points and lines occurring in Desargues's theorem, in three dimensions from five planes in general position, or in four dimensions from the 5-cell, the four-dimensional regular simplex. It has a large group of symmetries, taking any point to any other point and any line to any other line. It is also self-dual, meaning that if the points are replaced by lines and vice versa using projective duality, the same configuration results.
Graphs associated with the Desargues configuration include the Desargues graph (its graph of point-line incidences) and the Petersen graph (its graph of non-incident lines). The Desargues configuration is one of ten different configurations with ten points and lines, three points per line, and three lines per point, nine of which can be realized in the Euclidean plane.
Constructions
Two dimensions
Two triangles and are said to be in perspective centrally if the lines , , and meet in a common point, called the center of perspectivity. They are in perspective axially if the intersection points of the corresponding triangle sides, , , and all lie on a common line, the axis of perspectivity. Desargues's theorem in geometry states that these two conditions are equivalent: if two triangles are in perspective centrally then they must also be in perspective axially, and vice versa. When this happens, the ten points and ten lines of the two perspectivities (the six triangle vertices, three crossing points, and center of perspectivity, and the six triangle sides, three lines through corresponding pairs of vertices, and axis of perspectivity) together form an instance of the Desargues configuration.
Three dimensions
Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration. This construction is closely related to the property that every projective plane that can be embedded into a 3-dimensional projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron.
Four dimensions
The 5-cell or pentatope (a regular simplex in four dimensions) has five vertices, ten edges, ten triangular ridges (2-dimensional faces), and five tetrahedral facets; the edges and ridges touch each other in the same pattern as the Desargues configuration. Extend each of the edges of the 5-cell to the line that contains it (its affine hull), similarly extend each triangle of the 5-cell to the 2-dimensional plane that contains i
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https://en.wikipedia.org/wiki/Kay%20Toliver
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Kay Toliver is a teacher specialising in mathematics education.
Background
Kay Toliver was born and raised in East Harlem and the South Bronx. A product of the New York City public school system, she graduated from Harriet Beecher Stowe Junior High, Walton High School and Hunter College (AB 1967, MA 1971) with graduate work at the City College of New York in mathematics.
For more than 30 years, Kay Toliver taught mathematics and communication arts at P.S. 72/East Harlem Tech in Community School District 4. Prior to instructing seventh and eighth grade students, she taught grades one through six for 15 years.
"Becoming a teacher was the fulfillment of a childhood dream. My parents always stressed that education was the key to a better life. By becoming a teacher, I hoped to inspire African-American and Hispanic youths to realize their own dreams. I wanted to give something back to the communities I grew up in."
At East Harlem Tech, with the support of her principal, she established the "Challenger" program. The program, for grades 4-8, presents the basics of geometry and algebra in an integrated curriculum. This is a program for "gifted" students, but following her belief that all children can learn, she accepted students from all ability levels.
Teaching methods
The Math Fair
These events are similar to science fairs but involve students in creating and displaying projects relating to mathematics. Participants had to be able to explain thoroughly the mathematical theories and concepts behind their projects, which were placed on display at the school so that students from the lower grades could examine the older students' research. Students have created mathematic games such as "Dunking for Prime Numbers," "Fishing for Palindromes," and "Black Jack Geometry."
The Math Trail
Kay Toliver developed a lesson called the "Math Trail" to give students an appreciation for the community as well as an opportunity to see mathematics at work. To create a Math Trail, the class must first do some research on the history of the community. Then, they are instructed to plot a course, starting from the school building, that leads the class through the community and back to school, with stops along the way to visit several sites and create math problems about various real-life situations.
Teaching Honors
Presidential Awardee Secondary Mathematics, State and National levels
Reliance Award for Excellence in Education, Middle School
Outstanding Teacher for Mathematics Instruction, Disney American Teacher Awards
Fellow of FAME (Foundation for the Advancement of Mathematics Education)
Featured in the Peabody Award-winning PBS special, "Good Morning Miss Toliver," and the Peabody Award-winning classroom series, "The Eddie Files".
Outstanding Educator of the Year, National Conference on Diversity in the Scientific and Technological Workforce (National Science Foundation).
Kilby Award
Essence Award
References
External links
Kay Toliver Math Program
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https://en.wikipedia.org/wiki/Harmonic%20coordinates
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In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian metric on the manifold. They are useful in many problems of geometric analysis due to their regularity properties.
In two dimensions, certain harmonic coordinates known as isothermal coordinates have been studied since the early 1800s. Harmonic coordinates in higher dimensions were developed initially in the context of Lorentzian geometry and general relativity by Albert Einstein and Cornelius Lanczos (see harmonic coordinate condition). Following the work of Dennis DeTurck and Jerry Kazdan in 1981, they began to play a significant role in the geometric analysis literature, although Idzhad Sabitov and S.Z. Šefel had made the same discovery five years earlier.
Definition
Let be a Riemannian manifold of dimension . One says that a coordinate chart , defined on an open subset of , is harmonic if each individual coordinate function is a harmonic function on . That is, one requires that
where is the Laplace–Beltrami operator. Trivially, the coordinate system is harmonic if and only if, as a map , the coordinates are a harmonic map. A direct computation with the local definition of the Laplace-Beltrami operator shows that is a harmonic coordinate chart if and only if
in which are the Christoffel symbols of the given chart. Relative to a fixed "background" coordinate chart , one can view as a collection of functions on an open subset of Euclidean space. The metric tensor relative to is obtained from the metric tensor relative to by a local calculation having to do with the first derivatives of , and hence the Christoffel symbols relative to are calculated from second derivatives of . So both definitions of harmonic coordinates, as given above, have the qualitative character of having to do with second-order partial differential equations for the coordinate functions.
Using the definition of the Christoffel symbols, the above formula is equivalent to
Existence and basic theory
Harmonic coordinates always exist (locally), a result which follows easily from standard results on the existence and regularity of solutions of elliptic partial differential equations. In particular, the equation has a solution in some open set around any given point , such that and are both prescribed.
The basic regularity theorem concerning the metric in harmonic coordinates is that if the components of the metric are in the Hölder space when expressed in some coordinate chart, regardless of the smoothnness of the chart itself, then the transition function from that coordinate chart to any harmonic coordinate chart will be in the Hölder space . In particular this implies that the metric will also be in relative to harmonic coordinate charts.
As was first discovered by Cornelius Lanczos in 1922, relative to a harmonic coordinate chart, the Ricci curvature is given by
The fundamental aspect of this
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https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Hermann%20Weyl
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This is a list of topics named after Hermann Weyl, the influential German mathematician from the 20th century.
Mathematics and physics
Cartan–Weyl theory
Cartan–Weyl basis
Courant–Fischer–Weyl min-max principle
De Donder–Weyl theory
Hodge−Weyl decomposition
Majorana–Weyl spinor
Peter–Weyl theorem
Schur–Weyl duality
Weyl–Berry conjecture
Weyl–Groenewold product
Wigner–Weyl transform
Weyl algebra
Weyl almost periodic functions
Weyl anomaly
Weyl basis of the gamma matrices
Weyl chamber
Weyl character formula
Weyl denominator formula
Weyl dimension formula
Weyl–Kac character formula
Weyl curvature: see Weyl tensor
Weyl curvature hypothesis
Weyl dimension formula, a specialization of the character formula
Weyl distance function
Weyl equation, a relativistic wave equation
Weyl expansion
Weyl fermion
Weyl gauge
Weyl gravity
Weyl group
Length of a Weyl group element
Restricted Weyl group
Weyl integral
Weyl integration formula
Weyl law
Weyl metrics
Weyl module
Weyl notation
Weyl quantization
Weyl relations
Weyl scalar
Weyl semimetal
Weyl sequence
Weyl spinor
Weyl representation
Weyl sum, a type of exponential sum
Weyl symmetry: see Weyl transformation
Weyl tensor
Weyl transform
Weyl transformation
Weyl vector of a compact Lie group
Weyl–Brauer matrices
Weyl−Lewis−Papapetrou coordinates
Weyl–Schouten theorem
Weyl–von Neumann theorem
Weyl-squared theories
Weyl's axioms
Weyl's construction
Weyl's criterion
Weyl's criterion for essential spectrum
Weyl's criterion for equidistribution
Weyl's inequality
Weyl's inequality (number theory)
Weyl's infinitesimal geometry
Weyl's lemma: several results, for example;
Weyl's lemma on the "very weak" form of the Laplace equation
Weyl's lemma on hypoellipticity
Weyl's paradox (properly the Grelling–Nelson paradox)
Weyl's postulate
Weyl's theorem on complete reducibility
Weyl's tile argument
Weyl–Titchmarsh–Kodaira theory
Weyl's tube formula
Weyl's unitary trick
Other
Weyl (crater)
References
Lists of things named after mathematicians
Weyl
Weyl
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https://en.wikipedia.org/wiki/Semipermutable%20subgroup
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In mathematics, in algebra, in the realm of group theory, a subgroup of a finite group is said to be semipermutable if commutes with every subgroup whose order is relatively prime to that of .
Clearly, every permutable subgroup of a finite group is semipermutable. The converse, however, is not necessarily true.
External links
The Influence of semipermutable subgroups on the structure of finite groups
Subgroup properties
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https://en.wikipedia.org/wiki/Bundle%20map
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In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.
Bundle maps over a common base
Let and be fiber bundles over a space M. Then a bundle map from E to F over M is a continuous map such that . That is, the diagram
should commute. Equivalently, for any point x in M, maps the fiber of E over x to the fiber of F over x.
General morphisms of fiber bundles
Let πE:E→ M and πF:F→ N be fiber bundles over spaces M and N respectively. Then a continuous map is called a bundle map from E to F if there is a continuous map f:M→ N such that the diagram
commutes, that is, . In other words, is fiber-preserving, and f is the induced map on the space of fibers of E: since πE is surjective, f is uniquely determined by . For a given f, such a bundle map is said to be a bundle map covering f.
Relation between the two notions
It follows immediately from the definitions that a bundle map over M (in the first sense) is the same thing as a bundle map covering the identity map of M.
Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle. If πF:F→ N is a fiber bundle over N and f:M→ N is a continuous map, then the pullback of F by f is a fiber bundle f*F over M whose fiber over x is given by (f*F)x = Ff(x). It then follows that a bundle map from E to F covering f is the same thing as a bundle map from E to f*F over M.
Variants and generalizations
There are two kinds of variation of the general notion of a bundle map.
First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over a smooth manifold.
Second, one can consider fiber bundles with extra structure in their fibers, and restrict attention to bundle maps which preserve this structure. This leads, for example, to the notion of a (vector) bundle homomorphism between vector bundles, in which the fibers are vector spaces, and a bundle map φ is required to be a linear map on each fiber. In this case, such a bundle map φ (covering f) may also be viewed as a section of the vector bundle Hom(E,f*F) over M, whose fiber over x is the vector space Hom(Ex,Ff(x)) (also denoted L(Ex,Ff(x))) of linear maps from
Ex to Ff(x).
Fiber bundles
Theory of continuous functions
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https://en.wikipedia.org/wiki/Tannakian%20formalism
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In mathematics, a Tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K. A number of major applications of the theory have been made, or might be made in pursuit of some of the central conjectures of contemporary algebraic geometry and number theory.
The name is taken from Tadao Tannaka and Tannaka–Krein duality, a theory about compact groups G and their representation theory. The theory was developed first in the school of Alexander Grothendieck. It was later reconsidered by Pierre Deligne, and some simplifications made. The pattern of the theory is that of Grothendieck's Galois theory, which is a theory about finite permutation representations of groups G which are profinite groups.
The gist of the theory is that the fiber functor Φ of the Galois theory is replaced by a tensor functor T from C to K-Vect. The group of natural transformations of Φ to itself, which turns out to be a profinite group in the Galois theory, is replaced by the group (a priori only a monoid) of natural transformations of T into itself, that respect the tensor structure. This is by nature not an algebraic group, but an inverse limit of algebraic groups (pro-algebraic group).
Formal definition
A neutral Tannakian category is a rigid abelian tensor category, such that there exists a K-tensor functor to the category of finite dimensional K-vector spaces that is exact and faithful.
Applications
The construction is used in cases where a Hodge structure or l-adic representation is to be considered in the light of group representation theory. For example, the Mumford–Tate group and motivic Galois group are potentially to be recovered from one cohomology group or Galois module, by means of a mediating Tannakian category it generates.
Those areas of application are closely connected to the theory of motives. Another place in which Tannakian categories have been used is in connection with the Grothendieck–Katz p-curvature conjecture; in other words, in bounding monodromy groups.
The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G. This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with .
Extensions
has established partial Tannaka duality results in the situation where the category is R-linear, where R is no longer a field (as in classical Tannakian duality), but certain valuation rings.
References
Further reading
M. Larsen and R. Pink. Determining representations from invariant dimensions. Invent. math., 102:377–389, 1990.
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https://en.wikipedia.org/wiki/Demographics%20of%20Oslo
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The population of Oslo is monitored by Statistics Norway. As of 2022, the population of Oslo sat at 702,543.
Population
As of 2022, the population of Oslo sat at 702,543.
Origin
Immigrants and Norwegian-born to immigrant parents, by country background for Oslo, 2023
As of 2022, immigrants of non-Western origin and their children enumerated 164,824, and made up an estimated 24% of Oslo's population.
Immigrants of Western origin and their children enumerated 71,858, and made up an estimated 10% of the city's population. Immigrants made up a total of 35% of Oslo's population in 2022.
Number of immigrants
The current number of immigrants by country living in Oslo, Norway, as of 1 January 2020 are as follows;
Religion
Religiously, the residents of Oslo are in a majority-minority state with the largest group religious group being adherents to the Lutheran Church of Norway, but these do not make up the majority of residents. Irreligious people make up 28.9% of the population with the largest other religious group being Islam which makes up 9.5% of the city.
See also
Norwegian immigrant statistics
References
Oslo society
Oslo
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https://en.wikipedia.org/wiki/Andorran%20Federation%20of%20Ice%20Sports
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The Andorran Federation of Ice Sports (, FAEG) is the governing body of ice hockey, curling, and figure skating in Andorra.
Ice hockey statistics
52 players total
17 male players
24 junior players
11 female players
No referees
1 indoor rink
Not ranked in the world ranking
References
External links
Andorra – IIHF.com
Web oficial de la Federació Andorrana d'Esports de Gel
1995 establishments in Andorra
Ice hockey governing bodies in Europe
Federation
National members of the International Ice Hockey Federation
Ice hockey
Sports organizations established in 1995
National governing bodies for ice skating
National members of the International Skating Union
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https://en.wikipedia.org/wiki/History%20of%20education%20in%20the%20Indian%20subcontinent
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Education in the Indian subcontinent began with teaching of traditional elements such as Indian religions, Indian mathematics, Indian logic at early Hindu and Buddhist centres of learning such as ancient Takshashila (in modern-day Pakistan) and Nalanda (in India). Islamic education became ingrained with the establishment of Islamic empires in the Indian subcontinent in the Middle Ages while the coming of the Europeans later brought western education to colonial India.
Several Western-style universities were established during the period of British rule in the 19th century. A series of measures continuing throughout the early half of the 20th century ultimately laid the foundation of the educational system of the Republic of India, Pakistan and much of the Indian subcontinent.
Early history
Early education in India commenced under the supervision of a guru or prabhu. Initially, education was open to all and seen as one of the methods to achieve Moksha in those days, or enlightenment. As time progressed, due to a decentralised social structure, the education was imparted on the basis of varna and the related duties that one had to perform as a member of a specific caste. The Brahmans learned about scriptures and religion while the Kshatriya were educated in the various aspects of warfare. The Vaishya caste learned commerce and other specific vocational courses. The other caste Shudras, were men of working class and they were trained on skills to carry out these jobs. The earliest venues of education in India were often secluded from the main population. Students were expected to follow strict monastic guidelines prescribed by the guru and stay away from cities in ashrams. However, as population increased under the Gupta empire centres of urban learning became increasingly common and Cities such as Varanasi and the Buddhist centre at Nalanda became increasingly visible.
Education in India is a piece of education traditional form was closely related to religion. Among the Heterodox schools of belief were the Jain and Buddhist schools. Heterodox Buddhist education was more inclusive and aside of the monastic orders the Buddhist education centres were urban institutes of learning such as Taxila and Nalanda where grammar, medicine, philosophy, logic, metaphysics, arts and crafts etc. were also taught. Early Buddhist institutions of higher learning like Taxila and Nalanda continued to function well into the common era and were attended by students from China and Central Asia.
On the subject of education for the nobility Joseph Prabhu writes: "Outside the religious framework, kings and princes were educated in the arts and sciences related to government: politics (danda-nıti), economics (vartta), philosophy (anvıksiki), and historical traditions (itihasa). Here the authoritative source was Kautilya’s Arthashastra, often compared to Niccolò Machiavelli’s The Prince for its worldly outlook and political scheming." The Rigveda (c.1700-1000 BCE) mentions
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https://en.wikipedia.org/wiki/Yoram%20Moses
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Yoram Moses () is a Professor in the Electrical Engineering Department at the Technion - Israel Institute of Technology.
Yoram Moses received a B.Sc. in mathematics from the Hebrew University of Jerusalem in 1981, and a Ph.D. in Computer Science from Stanford University in 1986. Moses is a co-author of the book Reasoning About Knowledge, and is a winner of the 1997 Gödel Prize in theoretical computer science and the 2009 Dijkstra Prize in Distributed Computing.
His major research interests are distributed systems and reasoning about knowledge.
He is married to the computer scientist Yael Moses.
External links
Yoram Moses's homepage
Electrical engineering academics
Gödel Prize laureates
Dijkstra Prize laureates
Researchers in distributed computing
Hebrew University of Jerusalem alumni
Academic staff of Technion – Israel Institute of Technology
Stanford University alumni
Living people
Year of birth missing (living people)
Israeli electrical engineers
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https://en.wikipedia.org/wiki/F%C3%A1bio%20Pinto
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Fábio Nascimento Pinto (born 9 October 1980) is a Brazilian forme footballer who played as a forward.
Career statistics
Fábio Pinto played for several clubs in the Campeonato Brasileiro, including Sport Club Internacional, Grêmio Foot-Ball Porto Alegrense, Associação Desportiva São Caetano, Cruzeiro Esporte Clube and Guarani Futebol Clube. He also had a spell with Galatasaray S.K. in the Turkish Super Lig.
He played for Brazil at the 1997 FIFA U-17 World Championship in Egypt.
Honours
Club
Campeonato Gaúcho Juvenil: 1997
Campeonato Gaúcho Júnior: 1997
Copa São Paulo de Juniores: 1998
Campeonato Gaúcho: 2002
Campeonato Pernambucano: 2005
International
Brazil U-17
FIFA U-17 World Championship: 1997
Individual
Third-highest scorer at the 1997 FIFA U-17 World Championship
References
1980 births
Living people
Footballers from Santa Catarina (state)
Brazilian men's footballers
Cruzeiro Esporte Clube players
Associação Desportiva São Caetano players
Grêmio Foot-Ball Porto Alegrense players
Sport Club Internacional players
Coritiba Foot Ball Club players
Guarani FC players
La Liga players
Real Oviedo players
Galatasaray S.K. footballers
Pakhtakor Tashkent FK players
Uzbekistan Super League players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Turkey
Süper Lig players
Expatriate men's footballers in Turkey
Expatriate men's footballers in Spain
Expatriate men's footballers in Uzbekistan
Men's association football forwards
People from Itajaí
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https://en.wikipedia.org/wiki/Bradley%20Efron
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Bradley Efron (; born May 24, 1938) is an American statistician. Efron has been president of the American Statistical Association (2004) and of the Institute of Mathematical Statistics (1987–1988). He is a past editor (for theory and methods) of the Journal of the American Statistical Association, and he is the founding editor of the Annals of Applied Statistics. Efron is also the recipient of many awards (see below).
Efron is especially known for proposing the bootstrap resampling technique, which has had a major impact in the field of statistics and virtually every area of statistical application. The bootstrap was one of the first computer-intensive statistical techniques, replacing traditional algebraic derivations with data-based computer simulations.
Life and career
Efron was born in St. Paul, Minnesota in May 1938, the son of Russian Jewish immigrants Esther and Miles Efron. He attended the California Institute of Technology, graduating in mathematics in 1960. By his own admission he "had no talent for modern abstract math". His interest in statistics emerged after reading a Harald Cramér book cover to cover. Soon later, he arrived at Stanford in fall of 1960, earning his Ph.D., under the direction of Rupert Miller and Herbert Solomon, in the Department of Statistics. While at Stanford, he was suspended for six months for his involvement with the Stanford Chaparral'''s parody of Playboy magazine.
He is currently a professor of Statistics and Biostatistics at Stanford. At Stanford he has been the Chair of the Department of Statistics, Associate Dean of the School of Humanities and Sciences, Chairman of the University Advisory Board, Chair of the Faculty Senate, and co-director of the undergraduate-level Mathematical & Computational Science Program.
Efron holds the Max H. Stein endowed chair as Professor of Humanities and Sciences at Stanford.
He has made many important contributions to many areas of statistics. Efron's work has spanned both theoretical and applied topics, including empirical Bayes analysis (with Carl Morris), applications of differential geometry to statistical inference, the analysis of survival data, and inference for microarray gene expression data. He is the author of a classic monograph, The Jackknife, the Bootstrap and Other Resampling Plans (1982) and has also co-authored (with Robert Tibshirani) the text An Introduction to the Bootstrap (1994).
He created a set of intransitive dice called Efron's dice.
Awards
He has been given many honors, including a MacArthur Prize Fellowship, membership in the National Academy of Sciences and the American Academy of Arts and Sciences, fellowship in the Institute of Mathematical Statistics (IMS) and the American Statistical Association (ASA), the Lester R. Ford Award, the Wilks Medal, the Parzen Prize, and the Rao Prize, Fisher, Rietz, and Wald lecturer.
In 2005, he was awarded the National Medal of Science, the highest scientific honor by the United States, for his excep
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https://en.wikipedia.org/wiki/Cremona%20group
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In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the -dimensional projective space over a field It is denoted by
or or .
The Cremona group is naturally identified with the automorphism group of the field of the rational functions in indeterminates over , or in other words a pure transcendental extension of , with transcendence degree .
The projective general linear group of order , of projective transformations, is contained in the Cremona group of order . The two are equal only when or , in which case both the numerator and the denominator of a transformation must be linear.
The Cremona group in 2 dimensions
In two dimensions, Max Noether and Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with , though there was some controversy about whether their proofs were correct, and gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it.
showed that the Cremona group is not simple as an abstract group;
Blanc showed that it has no nontrivial normal subgroups that are also closed in a natural topology.
For the finite subgroups of the Cremona group see .
The Cremona group in higher dimensions
There is little known about the structure of the Cremona group in three dimensions and higher though many elements of it have been described. showed that it is (linearly) connected, answering a question of . There is no easy analogue of the Noether–Castelnouvo theorem as showed that the Cremona group in dimension at least 3 is not generated by its elements of degree bounded by any fixed integer.
De Jonquières groups
A De Jonquières group is a subgroup of a Cremona group of the following form . Pick a transcendence basis
for a field extension of . Then a De Jonquières group is the subgroup of automorphisms of mapping the subfield into itself for some . It has a normal subgroup given by the Cremona group of automorphisms of over the field , and the quotient group is the Cremona group of over the field . It can also be regarded as the group of birational automorphisms of the fiber bundle .
When and the De Jonquières group is the group of Cremona transformations fixing a pencil of lines through a given point, and is the semidirect product of
and .
References
Birational geometry
Group theory
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https://en.wikipedia.org/wiki/Semisimple%20operator
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In mathematics, a linear operator T : V → V on a vector space V is semisimple if every T-invariant subspace has a complementary T-invariant subspace. If T is a semisimple linear operator on V, then V is a semisimple representation of T. Equivalently, a linear operator is semisimple if its minimal polynomial is a product of distinct irreducible polynomials.
A linear operator on a finite dimensional vector space over an algebraically closed field is semisimple if and only if it is diagonalizable.
Over a perfect field, the Jordan–Chevalley decomposition expresses an endomorphism as a sum of a semisimple endomorphism s and a nilpotent endomorphism n such that both s and n are polynomials in x.
See also
Jordan–Chevalley decomposition
Notes
References
Linear algebra
Invariant subspaces
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https://en.wikipedia.org/wiki/Real%20algebraic%20geometry
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In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings).
Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets.
Terminology
Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory and real analytic geometry.
Examples: Real plane curves are examples of real algebraic sets and polyhedra are examples of semialgebraic sets. Real algebraic functions and Nash functions are examples of semialgebraic mappings. Piecewise polynomial mappings (see the Pierce–Birkhoff conjecture) are also semialgebraic mappings.
Computational real algebraic geometry is concerned with the algorithmic aspects of real algebraic (and semialgebraic) geometry. The main algorithm is cylindrical algebraic decomposition. It is used to cut semialgebraic sets into nice pieces and to compute their projections.
Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields) and their applications to the study of positive polynomials and sums-of-squares of polynomials. (See Hilbert's 17th problem and Krivine's Positivestellensatz.) The relation of real algebra to real algebraic geometry is similar to the relation of commutative algebra to complex algebraic geometry. Related fields are the theory of moment problems, convex optimization, the theory of quadratic forms, valuation theory and model theory.
Timeline of real algebra and real algebraic geometry
1826 Fourier's algorithm for systems of linear inequalities. Rediscovered by Lloyd Dines in 1919 and Theodore Motzkin in 1936.
1835 Sturm's theorem on real root counting
1856 Hermite's theorem on real root counting.
1876 Harnack's curve theorem. (This bound on the number of components was later extended to all Betti numbers of all real algebraic sets and all semialgebraic sets.)
1888 Hilbert's theorem on ternary quartics.
1900 Hilbert's problems (especially the 16th and the 17th problem)
1902 Farkas' lemma (Can be reformulated as linear positivstellensatz.)
1914 Annibale Comessatti showed that not every real algebraic surface is birational to RP2
1916 Fejér's conjecture about nonnegative tr
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https://en.wikipedia.org/wiki/Sweet%20Body%20of%20Bianca
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Sweet Body of Bianca () is a 1984 Italian comedy-mystery film directed by Nanni Moretti.
Plot
In Rome, Michele Apicella moves to a new apartment and starts a new job as mathematics teacher in the experimental Marilyn Monroe high school where most of the staff are, like him, eccentric. A solitary man, scrupulous about his work, one of his obsessions is the life of his new neighbours. He befriends a young couple, Maximilian and Aurora, but is deeply upset when he sees the girl with another man. She is found dead and the police inspector, thinking that Michele may know more than he reveals, puts him under surveillance.
An attractive new teacher, Bianca, arrives at the school and the two show interest in each other. She is living with a man, but decides to leave him and move in with Michele. While he is overjoyed to have the love of a beautiful and affectionate young woman, he is afraid that this perfection will not last and that like so many other couples he knows they will fall out.
One couple he is upset by are Ignazio and Maria who, despite his efforts to reconcile them, are breaking up. When they are both murdered, the police inspector arrests Michele as a suspect, but he is freed when Bianca gives him a false alibi. He then breaks with Bianca, telling her it is better to part while they are happy and, once on his own, his already fragile mental equilibrium crumbles. The film ends with his rambling confession to the patient inspector over how the dead neighbour and friends had disappointed him and upset his need for order in life.
Themes in Moretti's other works
"Michele Apicella" is the nom-de-plume used by Nanni Moretti for his roles in all his films (except The Mass Is Ended (1985)) until Palombella rossa of 1989. Another recurrent theme of Moretti's film, the Sacher torte, which is also present in Bianca.
Production
Songs in the film include "Insieme a te non ci sto più" (sung by Caterina Caselli), "Il cielo in una stanza" by Gino Paoli and "Scalo a Grado" by Franco Battiato.
The school psychologist is played by Luigi Moretti, Nanni's father.
In the Marilyn Monroe school, the traditional portraits of the President of the Italian Republic in the rooms are replaced by that of football goalkeeper Dino Zoff.
See also
List of Italian films of 1984
References
External links
1980s comedy mystery films
Films directed by Nanni Moretti
Films set in Rome
Italian comedy mystery films
1984 comedy films
1980s Italian films
1980s Italian-language films
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https://en.wikipedia.org/wiki/Ehrling%27s%20lemma
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In mathematics, Ehrling's lemma, also known as Lions' lemma, is a result concerning Banach spaces. It is often used in functional analysis to demonstrate the equivalence of certain norms on Sobolev spaces. It was named after Gunnar Ehrling.
Statement of the lemma
Let (X, ||·||X), (Y, ||·||Y) and (Z, ||·||Z) be three Banach spaces. Assume that:
X is compactly embedded in Y: i.e. X ⊆ Y and every ||·||X-bounded sequence in X has a subsequence that is ||·||Y-convergent; and
Y is continuously embedded in Z: i.e. Y ⊆ Z and there is a constant k so that ||y||Z ≤ k||y||Y for every y ∈ Y.
Then, for every ε > 0, there exists a constant C(ε) such that, for all x ∈ X,
Corollary (equivalent norms for Sobolev spaces)
Let Ω ⊂ Rn be open and bounded, and let k ∈ N. Suppose that the Sobolev space Hk(Ω) is compactly embedded in Hk−1(Ω). Then the following two norms on Hk(Ω) are equivalent:
and
For the subspace of Hk(Ω) consisting of those Sobolev functions with zero trace (those that are "zero on the boundary" of Ω), the L2 norm of u can be left out to yield another equivalent norm.
References
Notes
Bibliography
Banach spaces
Sobolev spaces
Lemmas in analysis
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https://en.wikipedia.org/wiki/Pink%20Book
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The Pink Book is an informal name for any of several books with pink covers. It may refer to:
The annual publication by the Office for National Statistics that details the United Kingdom's balance of payments
Epidemiology and Prevention of Vaccine-Preventable Diseases, a book published by the US Centers for Disease Control and Prevention (14th edition, 2021)
The member of the Coloured Book protocols family (1980–1992) that defined protocols for transport over Ethernet
See also
Black Book (disambiguation)
Blue book (disambiguation)
Green Book (disambiguation)
Orange Book (disambiguation)
Plum Book
White book (disambiguation)
Yellow Book (disambiguation)
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https://en.wikipedia.org/wiki/List%20of%20Leeds%20United%20F.C.%20records%20and%20statistics
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This article lists the records of Leeds United Football Club.
Honours and achievements
Domestic
League
First Division (level 1)
Champions: 1968–69, 1973–74, 1991–92
Runners-up: 1964–65, 1965–66, 1969–70, 1970–71, 1971–72
Second Division / Championship (level 2)
Champions: 1923–24, 1963–64, 1989–90, 2019–20
Runners-up: 1927–28, 1931–32, 1955–56
Play-off runners-up: 1987, 2006
League One (level 3)
Runners-up: 2009–10
Play-off runners-up: 2008
Cup
FA Cup
Winners: 1971–72
Runners-up: 1964–65, 1969–70, 1972–73
League Cup
Winners: 1967–68
Runners-up: 1995–96
FA Charity Shield
Winners: 1969, 1992
Runners-up: 1974
European
European Cup
Runners-up: 1974–75
European Cup Winners' Cup
Runners-up: 1972–73
Inter-Cities Fairs Cup
Winners: 1967–68, 1970–71
Runners-up: 1966–67
Inter-Cities Fairs Cup Trophy play-off
Runners-up: 1971
Record attendance
57,892 v Sunderland, FA Cup Rd. 5 replay, 15 March 1967
Record gate receipts
£1,230,000.00 Leeds United v Manchester United, Premier League 12 February 2023
Record victories
Overall: 10–0 v Lyn Oslo, European Cup Rd.1, 1st leg, 17 September 1969
League: 8–0 v Leicester City, Div. One, 7 April 1934
FA Cup: 8–1 v Crystal Palace, Rd.3, 11 January 1930
League Cup: 6–0 v Leicester City, Rd.3, 9 October 2001
League Trophy: 3–1 v Grimsby Town, Quarter-Final, 10 November 2009
Europe: 10–0 v Lyn Oslo, European Cup Rd.1, 1st leg, 17 September 1969
Home: 10–0 v Lyn Oslo, European Cup Rd.1, 1st leg, 17 September 1969
Away: 9–0 v Spora Luxembourg, Inter Cities Fairs Cup Rd.1, 1st leg, 3 October 1967
Record defeats
Overall: 1–8 v Stoke City, Div. One, 27 August 1934, 0–7 v Arsenal, Rd.2, 4 September 1979 & 0–7 v West Ham United, Rd.3, 7 November 1966 & 0–7 vs Manchester City, Rd.17, Premier League, 14 December 2021
League: 1–8 v Stoke City, Div. One, 27 August 1934
FA Cup: 2–7 v Middlesbrough, Rd.3, 2nd leg, 9 January 1946
League Cup: 0–7 v Arsenal, Rd.2, Sept 4 1979 & 0–7 v West Ham United, Rd.3, 7 November 1966
League Trophy: 2–4 v Rotherham United, Area 2nd Round, 8 October 2008
Europe: 0–4 v Lierse S.K., UEFA Cup, Rd.1, 2nd leg, 29 September 1971 & 0–4 v Barcelona, UEFA Champions League, 1st Group Stage, Matchday 1, 13 September 2000
Away: 1–8 v Stoke City, Div. One, 27 August 1934, 0–7 v Arsenal, Rd.2, 4 September 1979, 0–7 v West Ham United, Rd.3, 7 November 1966 & 0–7 v Manchester City, Premier League, Rd.17, 14 December 2021
Sequence records
Most League goals
Div. Two (98) 1927–28
Most League goals in a season
John Charles (43) Div. Two 1953–54
Best undefeated start to a season
(29) 25 August 1973 – 23 February 1974
Most matches undefeated
(34) 19 October 1968 – 30 August 1969
Most home matches undefeated
(39) 4 May 1968 – 28 March 1970
Most away matches undefeated
(17) 19 October 1968 – 30 August 1969
Longest run without a home win
(10) 16 January 1982 – 15 May 1982
Longest run without an away win
(26) 18 March 1939 – 30 August 1947
Most League wins (dates inclusive)
(9) 2
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https://en.wikipedia.org/wiki/Statistical%20semantics
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In linguistics, statistical semantics applies the methods of statistics to the problem of determining the meaning of words or phrases, ideally through unsupervised learning, to a degree of precision at least sufficient for the purpose of information retrieval.
History
The term statistical semantics was first used by Warren Weaver in his well-known paper on machine translation. He argued that word sense disambiguation for machine translation should be based on the co-occurrence frequency of the context words near a given target word. The underlying assumption that "a word is characterized by the company it keeps" was advocated by J.R. Firth. This assumption is known in linguistics as the distributional hypothesis. Emile Delavenay defined statistical semantics as the "statistical study of the meanings of words and their frequency and order of recurrence". "Furnas et al. 1983" is frequently cited as a foundational contribution to statistical semantics. An early success in the field was latent semantic analysis.
Applications
Research in statistical semantics has resulted in a wide variety of algorithms that use the distributional hypothesis to discover many aspects of semantics, by applying statistical techniques to large corpora:
Measuring the similarity in word meanings
Measuring the similarity in word relations
Modeling similarity-based generalization
Discovering words with a given relation
Classifying relations between words
Extracting keywords from documents
Measuring the cohesiveness of text
Discovering the different senses of words
Distinguishing the different senses of words
Subcognitive aspects of words
Distinguishing praise from criticism
Related fields
Statistical semantics focuses on the meanings of common words and the relations between common words, unlike text mining, which tends to focus on whole documents, document collections, or named entities (names of people, places, and organizations). Statistical semantics is a subfield of computational semantics, which is in turn a subfield of computational linguistics and natural language processing.
Many of the applications of statistical semantics (listed above) can also be addressed by lexicon-based algorithms, instead of the corpus-based algorithms of statistical semantics. One advantage of corpus-based algorithms is that they are typically not as labour-intensive as lexicon-based algorithms. Another advantage is that they are usually easier to adapt to new languages or noisier new text types from e.g. social media than lexicon-based algorithms are. However, the best performance on an application is often achieved by combining the two approaches.
See also
Co-occurrence
Computational linguistics
Information retrieval
Latent semantic analysis
Latent semantic indexing
Semantic analytics
Semantic similarity
Statistical natural language processing
Text corpus
Text mining
Web mining
References
Sources
Reprinted in
Applic
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https://en.wikipedia.org/wiki/STAR%20model
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In statistics, Smooth Transition Autoregressive (STAR) models are typically applied to time series data as an extension of autoregressive models, in order to allow for higher degree of flexibility in model parameters through a smooth transition.
Given a time series of data xt, the STAR model is a tool for understanding and, perhaps, predicting future values in this series, assuming that the behaviour of the series changes depending on the value of the transition variable. The transition might depend on the past values of the x series (similar to the SETAR models), or exogenous variables.
The model consists of 2 autoregressive (AR) parts linked by the transition function. The model is usually referred to as the STAR(p) models proceeded by the letter describing the transition function (see below) and p is the order of the autoregressive part. Most popular transition function include exponential function and first and second-order logistic functions. They give rise to Logistic STAR (LSTAR) and Exponential STAR (ESTAR) models.
Definition
AutoRegressive Models
Consider a simple AR(p) model for a time series yt
where:
for i=1,2,...,p are autoregressive coefficients, assumed to be constant over time;
stands for white-noise error term with constant variance.
written in a following vector form:
where:
is a column vector of variables;
is the vector of parameters :;
stands for white-noise error term with constant variance.
STAR as an Extension of the AutoRegressive Model
STAR models were introduced and comprehensively developed by Kung-sik Chan and Howell Tong in 1986 (esp. p. 187), in which the same acronym was used. It originally stands for Smooth Threshold AutoRegressive. For some background history, see Tong (2011, 2012). The models can be thought of in terms of extension of autoregressive models discussed above, allowing for changes in the model parameters according to the value of a transition variable zt. Chan and Tong (1986) rigorously proved that the family of STAR models includes the SETAR model as a limiting case by showing the uniform boundedness and equicontinuity with respect to the switching parameter. Without this proof, to say that STAR models nest the SETAR model lacks justification. Unfortunately, whether one should use a SETAR model or a STAR model for one's data has been a matter of subjective judgement, taste and inclination in much of the literature. Fortunately, the test procedure, based on David Cox's test of separate family of hypotheses and developed by Gao, Ling and Tong (2018, Statistica Sinica, volume 28, 2857-2883) is now available to address this issue. Such a test is important before adopting a STAR model because, among other issues, the parameter controlling its rate of switching is notoriously data-hungry.
Defined in this way, STAR model can be presented as follows:
where:
is a column vector of variables;
is the transition function bounded between 0 and 1.
Basic Structure
They can be understood as
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https://en.wikipedia.org/wiki/Greedy%20algorithm%20for%20Egyptian%20fractions
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In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as . As the name indicates, these representations have been used as long ago as ancient Egypt, but the first published systematic method for constructing such expansions was described in 1202 in the Liber Abaci of Leonardo of Pisa (Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible unit fraction that can be used in any representation of the remaining fraction.
Fibonacci actually lists several different methods for constructing Egyptian fraction representations. He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to .
The expansion produced by this method for a number is called the greedy Egyptian expansion, Sylvester expansion, or Fibonacci–Sylvester expansion of . However, the term Fibonacci expansion usually refers, not to this method, but to representation of integers as sums of Fibonacci numbers.
Algorithm and examples
Fibonacci's algorithm expands the fraction to be represented, by repeatedly performing the replacement
(simplifying the second term in this replacement as necessary). For instance:
in this expansion, the denominator 3 of the first unit fraction is the result of rounding up to the next larger integer, and the remaining fraction is the result of simplifying = . The denominator of the second unit fraction, 8, is the result of rounding up to the next larger integer, and the remaining fraction is what is left from after subtracting both and .
As each expansion step reduces the numerator of the remaining fraction to be expanded, this method always terminates with a finite expansion; however, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators. For instance, this method expands
while other methods lead to the much better expansion
suggests an even more badly-behaved example, . The greedy method leads to an expansion with ten terms, the last of which has over 500 digits in its denominator; however, has a much shorter non-greedy representation, .
Sylvester's sequence and closest approximation
Sylvester's sequence 2, 3, 7, 43, 1807, ... () can be viewed as generated by an infinite greedy expansion of this type for the n
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https://en.wikipedia.org/wiki/Continuous%20embedding
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In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems.
Definition
Let X and Y be two normed vector spaces, with norms ||·||X and ||·||Y respectively, such that X ⊆ Y. If the inclusion map (identity function)
is continuous, i.e. if there exists a constant C > 0 such that
for every x in X, then X is said to be continuously embedded in Y. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "X ↪ Y" means "X and Y are normed spaces with X continuously embedded in Y". This is a consistent use of notation from the point of view of the category of topological vector spaces, in which the morphisms ("arrows") are the continuous linear maps.
Examples
A finite-dimensional example of a continuous embedding is given by a natural embedding of the real line X = R into the plane Y = R2, where both spaces are given the Euclidean norm:
In this case, ||x||X = ||x||Y for every real number X. Clearly, the optimal choice of constant C is C = 1.
An infinite-dimensional example of a continuous embedding is given by the Rellich–Kondrachov theorem: let Ω ⊆ Rn be an open, bounded, Lipschitz domain, and let 1 ≤ p < n. Set
Then the Sobolev space W1,p(Ω; R) is continuously embedded in the Lp space Lp∗(Ω; R). In fact, for 1 ≤ q < p∗, this embedding is compact. The optimal constant C will depend upon the geometry of the domain Ω.
Infinite-dimensional spaces also offer examples of discontinuous embeddings. For example, consider
the space of continuous real-valued functions defined on the unit interval, but equip X with the L1 norm and Y with the supremum norm. For n ∈ N, let fn be the continuous, piecewise linear function given by
Then, for every n, ||fn||Y = ||fn||∞ = n, but
Hence, no constant C can be found such that ||fn||Y ≤ C||fn||X, and so the embedding of X into Y is discontinuous.
See also
Compact embedding
References
Functional analysis
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https://en.wikipedia.org/wiki/Kadammanitta%20Vasudevan%20Pillai
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Prof. Kadammanitta Vasudevan Pillai, is a Padayani exponent from Kerala, India. He is the former Vice Chairman of the Kerala Folklore Academy, a professor in mathematics, a writer and public speaker.
Early and professional life
Vasudevan Pillai was born to M. R. Ramakrishna Pillai (late) and Parukutty Amma (late), at a small village called Kadammanitta in Pathanamthitta district Kerala on 24 May 1947. His only sister is Omana Kumariamma. From childhood itself, he was interested in Padayani and Kerala folklore. He completed his pre-school and graduated from Catholicate College in Pathanamthitta. He did his MSc in Mathematics from Ravi Shankar University Raipur state of Chhattisgarh (then Madhya Pradesh) and was the first rank holder. He completed his Post Graduation with Gold medal. He later came back to Kerala and joined as lecturer in Mathematics in various colleges of NSS Management in Kerala. He worked in NSS College, Pandalam for most of the time, till his retirement as Head of Department in 2002.
He has served as the vice chairman of The Kerala Folklore Academy from 1997 to 2001.
As Padayani exponent and writer
Vasudevan Pillai has contributed greatly to the field of Kerala folklore. He has written books on Padayani and other traditional art forms.
:
1. Padayaniyile pala kolangal
2. Padayani
3. Padayaniyude jeevathalam
4. Padayani : Janakeya anushtaana nadakam (Mahatma Gandhi University textbook)
5. Padayani : Oru ithihasa nadakam
6. Padayani : The traditional epi theater (English, but not a translation)
7. Vamsheeya sangeetha shastram
8. Ethino musicology (English, but not a translation)
9. Apasaraka bimbangalude aasura gethiroopangal (study on Kadammanitta poetry)
10. Kadinjoo pottan (drama based on Kadammanitta poetry)
Awards
He is the recipient of State Awards for his literary and cultural contributions.
:
Kerala Sangeetha Nataka Akademi Award (1995)
Kerala Sahitya Akademi Award for Scholarly Literature (1996)
Kerala youth welfare board award in 1997
P. K Kalan puraskaram sponsored by Kerala folklore academy and kerala cultural department. The cash award of Rs 1 Lakh along with his contributions were used to form a trust to award every year Rs10000 to a veteran ashan(Guru/Master) of padayani.
Evur Anushtana Samithi - Raman Pillai Smaraka Kalarathna bahumathi. April 2023.
Padayani paramaacarya Kadaminitta Munjanat Narayanan Nair Aashan Smaraka Samithy Puraskaram - Padayani Acharya Bahumathi. 22nd April 2023.
His Guru
The guru to Kadammanitta Vasudevan Pillai is Kadammanitta Raman Nair, the father of famous poet Kadammanitta Ramakrishnan.
Other contributions
His bond with villages can be seen clearly in his works. His association with the great poet of Kerala Kadammanitta Ramakrishnan led to many valuable literary contributions.
The youth of Kadammanitta, under the leadership of Vasudevan Pillai, had formed Kadammanitta Gothra Kala Kalari through which this art form has been taught to many and is being passed over
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https://en.wikipedia.org/wiki/Trailing%20zero
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In mathematics, trailing zeros are a sequence of 0 in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.
Trailing zeros to the right of a decimal point, as in 12.340, don’t affect the value of a number and may be omitted if all that is of interest is its numerical value. This is true even if the zeros recur infinitely. For example, in pharmacy, trailing zeros are omitted from dose values to prevent misreading. However, trailing zeros may be useful for indicating the number of significant figures, for example in a measurement. In such a context, "simplifying" a number by removing trailing zeros would be incorrect.
The number of trailing zeros in a non-zero base-b integer n equals the exponent of the highest power of b that divides n. For example, 14000 has three trailing zeros and is therefore divisible by 1000 = 103, but not by 104. This property is useful when looking for small factors in integer factorization. Some computer architectures have a count trailing zeros operation in their instruction set for efficiently determining the number of trailing zero bits in a machine word.
Factorial
The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, is simply the multiplicity of the prime factor 5 in n!. This can be determined with this special case of de Polignac's formula:
where k must be chosen such that
more precisely
and denotes the floor function applied to a. For n = 0, 1, 2, ... this is
0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 6, ... .
For example, 53 > 32, and therefore 32! = 263130836933693530167218012160000000 ends in
zeros. If n < 5, the inequality is satisfied by k = 0; in that case the sum is empty, giving the answer 0.
The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero.
Defining
the following recurrence relation holds:
This can be used to simplify the computation of the terms of the summation, which can be stopped as soon as q i reaches zero. The condition is equivalent to
See also
Leading zero
Trailing digit
References
External links
Why are trailing fractional zeros important? for some examples of when trailing zeros are significant
Number of trailing zeros for any factorialPython program to calculate the number of trailing zeros for any factorial
Elementary arithmetic
0 (number)
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https://en.wikipedia.org/wiki/Kendall%20rank%20correlation%20coefficient
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In statistics, the Kendall rank correlation coefficient, commonly referred to as Kendall's τ coefficient (after the Greek letter τ, tau), is a statistic used to measure the ordinal association between two measured quantities. A τ test is a non-parametric hypothesis test for statistical dependence based on the τ coefficient. It is a measure of rank correlation: the similarity of the orderings of the data when ranked by each of the quantities. It is named after Maurice Kendall, who developed it in 1938, though Gustav Fechner had proposed a similar measure in the context of time series in 1897.
Intuitively, the Kendall correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully different for a correlation of −1) rank between the two variables.
Both Kendall's and Spearman's can be formulated as special cases of a more general correlation coefficient. Its notions of concordance and discordance also appear in other areas of statistics, like the Rand index in cluster analysis.
Definition
Let be a set of observations of the joint random variables X and Y, such that all the values of () and () are unique (ties are neglected for simplicity). Any pair of observations and , where , are said to be concordant if the sort order of and agrees: that is, if either both and holds or both and ; otherwise they are said to be discordant.
The Kendall τ coefficient is defined as:
where is the binomial coefficient for the number of ways to choose two items from n items.
Properties
The denominator is the total number of pair combinations, so the coefficient must be in the range −1 ≤ τ ≤ 1.
If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value 1.
If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value −1.
If X and Y are independent and not constant, then the expectation of the coefficient is zero.
An explicit expression for Kendall's rank coefficient is .
Hypothesis test
The Kendall rank coefficient is often used as a test statistic in a statistical hypothesis test to establish whether two variables may be regarded as statistically dependent. This test is non-parametric, as it does not rely on any assumptions on the distributions of X or Y or the distribution of (X,Y).
Under the null hypothesis of independence of X and Y, the sampling distribution of τ has an expected value of zero. The precise distribution cannot be characterized in terms of common distributions, but may be calculated exactly for small samples; for larger samples, it is common to use an approximation to the normal distribution, with mean zero and variance
.
Accounting for ties
A pair is said to be tied if an
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https://en.wikipedia.org/wiki/Rotation%20formalisms%20in%20three%20dimensions
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In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.
According to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom.
An example where rotation representation is used is in computer vision, where an automated observer needs to track a target. Consider a rigid body, with three orthogonal unit vectors fixed to its body (representing the three axes of the object's local coordinate system). The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.
Rotations and motions
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to. Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions. Any proper motion of the Euclidean space decomposes to a rotation around the origin and a translation. Whichever the order of their composition will be, the "pure" rotation component wouldn't change, uniquely determined by the complete motion.
One can also understand "pure" rotations as linear maps in a vector space equipped with Euclidean structure, not as maps of points of a corresponding affine space. In other words, a rotation formalism captures only the rotational part of a motion, that contains three degrees of freedom, and ignores the translational part, that contains another three.
When representing a rotation as numbers in a computer, some people prefer the quaternion representation or the axis+angle representation, because they avoid the gimbal lock that can occur with Euler rotations.
Formalism alternatives
Rotation matrix
The above-mentioned triad of unit vectors is also called a basis. Specifying the coordinates (components) of vectors of this basis in its current (rotated) position, in terms of the reference (non-rotated) coordinate axes, will completely describe the rotation. The three unit vectors, , and , that fo
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https://en.wikipedia.org/wiki/N%20%28disambiguation%29
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N is the fourteenth letter of the Latin alphabet.
N or n may also refer to:
Mathematics
, the set of natural numbers
N, the field norm
N for nullae, a rare Roman numeral for zero
n, the size of a statistical sample
Science
ATC code N Nervous system, a section of the Anatomical Therapeutic Chemical Classification System
Haplogroup N (mtDNA), a human mitochondrial DNA haplogroup
Haplogroup N (Y-DNA), a human Y-chromosome DNA haplogroup
N band, an atmospheric transmission window in the mid-infrared centred on 10 micrometres
N ray, a hypothesized form of radiation, found to be illusory
N., abbreviation of the Latin word nervus meaning nerve, used in anatomy, e.g. N. vagus
Quantities and units
N for Newton (unit), the SI derived unit of force
N or , a normal force in mechanics
Nitrogen, symbol N, a chemical element
N or Asn, the symbol for the common natural amino acid asparagine
N, the Normality (chemistry) or chemical concentration of a solution
N, the neutron number, the number of neutrons in a nuclide
N, in Brillouin zone, the center of a face of a body-centered cubic lattice
N, the physical quantity "rotation" in the International System of Quantities
n, for nano-, prefix in the SI system of units denoting a factor of 10−9
n, the optical refractive index of a material
n, the principal quantum number, the first of a set of quantum numbers of an atomic orbital
n, an electron density, the measure of the probability of an electron being present at a specific location
n, an amount of substance in chemical physics
n, the coordination number of a substance
n-, a lowercase prefix in chemistry denoting the straight-chain form of an open-chain compound in contrast to its branched isomer
N-, an uppercase prefix in chemistry denoting that the substituent is bonded to the nitrogen, as in amines
Transport
N (New York City Subway service), a service of the New York City Subway
N Judah, a Muni Metro line in San Francisco, California
N (Los Angeles Railway), a line operated by the Los Angeles Railway from 1920 to 1950
Tokyo Metro Namboku Line, a subway service operated by the Tokyo Metro, labeled
Nagahori Tsurumi-ryokuchi Line, a subway service operated by the Osaka Metro, labeled
Transilien Line N, a line of the Paris transport network
, the official West Japan Railway Company service symbol for the Akō Line.
Engineering
N, in geotechnical engineering, the result of a standard penetration test
N, in electrical systems, the connection to neutral
N, an ITU prefix allocated to the United States for radio and television stations
n (Poland), a digital TV platform
N battery or N cell, a standard size of dry cell battery
N connector, a threaded RF connector for joining coaxial cables
Entertainment
N. (novella), a 2008 short story by Stephen King
Near (Death Note) (alias N), a character in the manga series Death Note
N (video game), a 2005 computer game
N (Pokémon), the leader and king of crime syndicate Team Pla
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https://en.wikipedia.org/wiki/NSO
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NSO or Nso may refer to:
NSO
NATO Standardization Office
Netherlands Space Office
National Statistics Office (Philippines)
Nationale SIGINT Organisatie, the Netherlands
National Safeman's Organization of safe technicians, US
National Socialist Order, the rebranded name of Atomwaffen Division
National Solar Observatory, U.S.A.
National Symphony Orchestra (disambiguation), orchestras of several countries
Nigerian Security Organization, security forces 1976-1985
Nintendo Switch Online, online subscription service for the Nintendo Switch
NSO Group (Niv, Shalev, Omri), an Israeli phone spyware company
Non-qualified stock option
Non-Skating Official in roller derby
Nso
Nso people of Cameroon
Nso language (ISO 639-3 language code: lns)
Other uses
Northern Sotho language (ISO 639-2 language code: nso)
Scone Airport (IATA airport code: NSO), Upper Hunter Valley, New South Wales, Australia
Aerolíneas Sosa (ICAO airline code: NSO), Honduran airline
Nashipur Road railway station (train station code: NSO), West Bengal, India
See also
NS0 cell, a model cell line
Country Code Names Supporting Organization (ccNSO)
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https://en.wikipedia.org/wiki/Automatically%20Tuned%20Linear%20Algebra%20Software
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Automatically Tuned Linear Algebra Software (ATLAS) is a software library for linear algebra. It provides a mature open source implementation of BLAS APIs for C and Fortran77.
ATLAS is often recommended as a way to automatically generate an optimized BLAS library. While its performance often trails that of specialized libraries written for one specific hardware platform, it is often the first or even only optimized BLAS implementation available on new systems and is a large improvement over the generic BLAS available at Netlib. For this reason, ATLAS is sometimes used as a performance baseline for comparison with other products.
ATLAS runs on most Unix-like operating systems and on Microsoft Windows (using Cygwin). It is released under a BSD-style license without advertising clause, and many well-known mathematics applications including MATLAB, Mathematica, Scilab, SageMath, and some builds of GNU Octave may use it.
Functionality
ATLAS provides a full implementation of the BLAS APIs as well as some additional functions from LAPACK, a higher-level library built on top of BLAS. In BLAS, functionality is divided into three groups called levels 1, 2 and 3.
Level 1 contains vector operations of the form
as well as scalar dot products and vector norms, among other things.
Level 2 contains matrix-vector operations of the form
as well as solving for with being triangular, among other things.
Level 3 contains matrix-matrix operations such as the widely used General Matrix Multiply (GEMM) operation
as well as solving for triangular matrices , among other things.
Optimization approach
The optimization approach is called Automated Empirical Optimization of Software (AEOS), which identifies four fundamental approaches to computer assisted optimization of which ATLAS employs three:
Parameterization—searching over the parameter space of a function, used for blocking factor, cache edge, etc.
Multiple implementation—searching through various approaches to implementing the same function, e.g., for SSE support before intrinsics made them available in C code
Code generation—programs that write programs incorporating what knowledge they can about what will produce the best performance for the system
Optimization of the level 1 BLAS uses parameterization and multiple implementation
Every ATLAS level 1 BLAS function has its own kernel. Since it would be difficult to maintain thousands of cases in ATLAS there is little architecture specific optimization for Level 1 BLAS. Instead multiple implementation is relied upon to allow for compiler optimization to produce high performance implementation for the system.
Optimization of the level 2 BLAS uses parameterization and multiple implementation
With data and operations to perform the function is usually limited by bandwidth to memory, and thus there is not much opportunity for optimization
All routines in the ATLAS level 2 BLAS are built from two Level 2 BLAS kernels:
GEMV—matrix by vector mul
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https://en.wikipedia.org/wiki/Local%20martingale
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In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local martingale that is bounded from below is a supermartingale, and every local martingale that is bounded from above is a submartingale; however, in general a local martingale is not a martingale, because its expectation can be distorted by large values of small probability. In particular, a driftless diffusion process is a local martingale, but not necessarily a martingale.
Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem).
Definition
Let be a probability space; let be a filtration of ; let be an -adapted stochastic process on the set . Then is called an -local martingale if there exists a sequence of -stopping times such that
the are almost surely increasing: ;
the diverge almost surely: ;
the stopped process is an -martingale for every .
Examples
Example 1
Let Wt be the Wiener process and T = min{ t : Wt = −1 } the time of first hit of −1. The stopped process Wmin{ t, T } is a martingale; its expectation is 0 at all times, nevertheless its limit (as t → ∞) is equal to −1 almost surely (a kind of gambler's ruin). A time change leads to a process
The process is continuous almost surely; nevertheless, its expectation is discontinuous,
This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as if there is such t, otherwise . This sequence diverges almost surely, since for all k large enough (namely, for all k that exceed the maximal value of the process X). The process stopped at τk is a martingale.
Example 2
Let Wt be the Wiener process and ƒ a measurable function such that Then the following process is a martingale:
where
The Dirac delta function (strictly speaking, not a function), being used in place of leads to a process defined informally as and formally as
where
The process is continuous almost surely (since almost surely), nevertheless, its expectation is discontinuous,
This process is not a martingale. However, it is a local martingale. A localizing sequence may be chosen as
Example 3
Let be the complex-valued Wiener process, and
The process is continuous almost surely (since does not hit 1, almost surely), and is a local martingale, since the function is harmonic (on the complex plane without the point 1). A localizing sequence may be chosen as Nevertheless, the expectation of this process is non-constant; moreover,
as
which can be deduced from the fact that the mean value of over the circle tends to infinity as . (In fact, it is equal to for r ≥ 1 but to 0 for r ≤ 1).
Martingales via local martingales
Let be a local martingale. In order to prove that it is a martingale it is sufficient to prove that in L1 (as ) for every t, that
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https://en.wikipedia.org/wiki/Stopped%20process
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In mathematics, a stopped process is a stochastic process that is forced to assume the same value after a prescribed (possibly random) time.
Definition
Let
be a probability space;
be a measurable space;
be a stochastic process;
be a stopping time with respect to some filtration of .
Then the stopped process is defined for and by
Examples
Gambling
Consider a gambler playing roulette. Xt denotes the gambler's total holdings in the casino at time t ≥ 0, which may or may not be allowed to be negative, depending on whether or not the casino offers credit. Let Yt denote what the gambler's holdings would be if he/she could obtain unlimited credit (so Y can attain negative values).
Stopping at a deterministic time: suppose that the casino is prepared to lend the gambler unlimited credit, and that the gambler resolves to leave the game at a predetermined time T, regardless of the state of play. Then X is really the stopped process YT, since the gambler's account remains in the same state after leaving the game as it was in at the moment that the gambler left the game.
Stopping at a random time: suppose that the gambler has no other sources of revenue, and that the casino will not extend its customers credit. The gambler resolves to play until and unless he/she goes broke. Then the random time
is a stopping time for Y, and, since the gambler cannot continue to play after he/she has exhausted his/her resources, X is the stopped process Yτ.
Brownian motion
Let be a one-dimensional standard Brownian motion starting at zero.
Stopping at a deterministic time : if , then the stopped Brownian motion will evolve as per usual up until time , and thereafter will stay constant: i.e., for all .
Stopping at a random time: define a random stopping time by the first hitting time for the region :
Then the stopped Brownian motion will evolve as per usual up until the random time , and will thereafter be constant with value : i.e., for all .
See also
Killed process
References
Robert G. Gallager. Stochastic Processes: Theory for Applications. Cambridge University Press, Dec 12, 2013 pg. 450
Stochastic processes
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https://en.wikipedia.org/wiki/Louis%20Kauffman
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Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, mathematical physicist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He does research in topology, knot theory, topological quantum field theory, quantum information theory, and diagrammatic and categorical mathematics. He is best known for the introduction and development of the bracket polynomial and the Kauffman polynomial.
Biography
Kauffman was valedictorian of his graduating class at Norwood Norfolk Central High School in 1962. He received his B.S. at the Massachusetts Institute of Technology in 1966 and his Ph.D. in mathematics from Princeton University in 1972 (with William Browder as thesis advisor).
Kauffman has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institut des Hautes Études Scientifiques in Bures Sur Yevette, France, the Institut Henri Poincaré in Paris, France, the University of Bologna, Italy, the Federal University of Pernambuco in Recife, Brazil, and the Newton Institute in Cambridge England.
He is the founding editor and one of the managing editors of the Journal of Knot Theory and Its Ramifications, and editor of the World Scientific Book Series On Knots and Everything. He writes a column entitled Virtual Logic for the journal Cybernetics and Human Knowing
From 2005 to 2008 he was president of the American Society for Cybernetics. He plays
clarinet in the ChickenFat Klezmer Orchestra in Chicago.
Work
Kauffman's research interests are in the fields of cybernetics, topology, and mathematical physics. His work is primarily in the topics of knot theory and its connections with statistical mechanics, quantum theory, algebra, combinatorics, and foundations. In topology, he introduced and developed the bracket polynomial and Kauffman polynomial.
Bracket polynomial
In the mathematical field of knot theory, the bracket polynomial, also known as the Kauffman bracket, is a polynomial invariant of framed links. Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial. The bracket polynomial plays an important role in unifying the Jones polynomial with other quantum invariants. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to state sum invariants of 3-manifolds. Recently, the bracket polynomial formed the basis for Mikhail Khovanov's construction of a homology for knots and links, creating
a stronger invariant than the Jones polynomial and such that the graded Euler characteristic of the Khovanov homology is equal to the original
Jones polynomial. The generators for the chain complex of the Khovanov homology are states of the bracket polynomial decorated with elements
of
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https://en.wikipedia.org/wiki/List%20of%20Canadian%20census%20agglomerations%20by%20province%20or%20territory
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The tables below list Canada's 117 census agglomerations at the 2016 Census, as determined by Statistics Canada, up from 113 in the 2011 Census.
2016 changes
Statistics Canada's review of CMAs and CAs for the 2016 Census resulted in the addition of eight new CAs and the demotion of two CAs, and the promotion of two CAs to census metropolitan areas (CMAs).
New census agglomerations
Arnprior, Ontario
Carleton Place, Ontario
Gander, Newfoundland and Labrador
Nelson, British Columbia
Sainte-Marie, Quebec
Wasaga Beach, Ontario
Weyburn, Saskatchewan
Winkler, Manitoba
Promotion to census metropolitan areas
Belleville, Ontario
Lethbridge, Alberta
Retired census agglomerations
Amos, Quebec
Temiskaming Shores, Ontario
Lists
Alberta
Statistics Canada recognized fifteen census agglomerations within Alberta in the 2016 Census.
British Columbia
Statistics Canada recognized 22 census agglomerations within British Columbia in the 2016 Census.
Manitoba
Statistics Canada recognized five census agglomerations within Manitoba in the 2016 Census.
New Brunswick
Statistics Canada recognized five census agglomerations within New Brunswick in the 2016 Census.
Newfoundland and Labrador
Statistics Canada recognized four census agglomerations within Newfoundland and Labrador in the 2016 Census.
Northwest Territories
Statistics Canada recognized one census agglomeration within the Northwest Territories in the 2016 Census.
Nova Scotia
Statistics Canada recognized four census agglomerations within Nova Scotia in the 2016 Census.
Nunavut
Statistics Canada recognized no census agglomerations within Nunavut in the 2016 Census.
Ontario
Statistics Canada recognized 29 census agglomerations within Ontario in the 2016 Census.
Prince Edward Island
Statistics Canada recognized two census agglomerations within Prince Edward Island in the 2016 Census.
Quebec
Statistics Canada recognized 24 census agglomerations within Quebec in the 2016 Census.
Saskatchewan
Statistics Canada recognized eight census agglomerations within Saskatchewan in the 2016 Census.
Yukon
Statistics Canada recognized one census agglomeration within Yukon in the 2016 Census.
References
Census agglomerations
Statistics Canada
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https://en.wikipedia.org/wiki/Faqqua
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Faqqu'a () is a village on the northern West Bank, known for its cactus fruits, located along the Green Line on the Gilboa ridge. According to the Palestinian Central Bureau of Statistics, the town had a population of 3,490 inhabitants in mid-year 2006 and 4,410 in 2017, an exclusively Muslim population. The village belongs to the Jenin Governorate.
History
The village's history is rather unknown, although there are numerous findings that reveal a Roman or Byzantine presence. Roman coins have been found in the area and there are several sites that are believed to be burial grounds as well as remains of ancient olive oil production. It’s possible to find fragments of ancient pottery when simply wandering around the surrounding olive orchards. There is a common belief in local folklore that a Roman settlement once thrived nearby the current village.
Ottoman period
In 1838, Fuku'a was noted as one of a range of villages round a height, the other villages being named as Deir Abu Da'if, Beit Kad, Deir Ghuzal and Araneh. It was located in the Jenin district.
In 1870 Victor Guérin visited the village and noted that the village gave name to the mountain range. He further noted that the village had about 400 inhabitants, with houses of stone. There were several old cisterns cut into the rock, and some gardens bordered by cactus.
In 1870/1871 (1288 AH), an Ottoman census listed the village in the nahiya (sub-district) of Shafa al-Qibly.
In 1882, the PEF's Survey of Western Palestine described Fukua as "A large village on top of a spur. It gives its name to the Gilboa range, which is often called Jebel Fukua. It is surrounded by olive-gardens, and supplied by cisterns east and west of the village."
British Mandate period
In the 1922 census of Palestine, conducted by the British Mandate authorities, Faqu'a had a population of 553; all Muslims, increasing in the 1931 census to 663; still all Muslims, in a total of 153 houses.
In the 1945 statistics, Faqqu'a had a population of 880 Muslims, and the jurisdiction of the village was 30,179 dunams of land, according to an official land and population survey. Of this, 1,131 dunams were used for plantations and irrigable land, 8,440 dunams for cereals, while 22 dunams were built-up (urban) land.
Jordanian period
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Faqqua came under Jordanian rule.
The Jordanian census of 1961 found 1,099 inhabitants in Faqqu'a.
Post-1967
Since the Six-Day War in 1967, Faqqua has been under Israeli occupation.
Geography
The village is located in the most northeastern part of the West Bank, 11 km east of the city of Jenin, adjacent to the Green Line. Faqqu'a lies just below the ridge of the Gilboa hills, which locals eponymously call Jebel Faqqu'a, overlooking the fertile Jezreel Valley (Marj Ibn Amer in Arabic), the city of Jenin and other Palestinian villages.
The higher part of the range, which is located on the Israeli side, is now an area
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https://en.wikipedia.org/wiki/Path%20analysis
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Path Analysis may refer to:
Path analysis (statistics), a statistical method of testing cause/effect relationships
Path analysis (computing), a method for finding the trail that leads users to websites
Critical path method, an operations research technique
Main path analysis, a method for tracing the most significant citation chains in a citation network.
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https://en.wikipedia.org/wiki/Excitable
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Excitable may refer to:
a song on the 1987 Def Leppard album Hysteria
a hit song by the British band Amazulu
a cell that can respond to stimuli
See also
Excitable medium (mathematics / system analysis)
Cell excitability (biology)
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https://en.wikipedia.org/wiki/Numerical%20linear%20algebra
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Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.
Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences are as vast as the applications of continuous mathematics. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, and fluid dynamics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations. Noting the broad applications of numerical linear algebra, Lloyd N. Trefethen and David Bau, III argue that it is "as fundamental to the mathematical sciences as calculus and differential equations", even though it is a comparatively small field. Because many properties of matrices and vectors also apply to functions and operators, numerical linear algebra can also be viewed as a type of functional analysis which has a particular emphasis on practical algorithms.
Common problems in numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition, which can then be used to answer common linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central concern with developing algorithms that do not introduce errors when applied to real data on a finite precision computer is often achieved by iterative methods rather than direct ones.
History
Numerical linear algebra was developed by computer pioneers like John von Neumann, Alan Turing, James H. Wilkinson, Alston Scott Householder, George Forsythe, and Heinz Rutishauser, in order to apply the earliest computers to problems in continuous mathematics, such as ballistics problems and the solutions to systems of partial differential equations. The first serious attempt to minimize computer error in the application of algorithms to real data is John von Neumann and Herman Goldstine's wo
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https://en.wikipedia.org/wiki/Barrow%27s%20inequality
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In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow.
Statement
Let P be an arbitrary point inside the triangle ABC. From P and ABC, define U, V, and W as the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively. Then Barrow's inequality states that
with equality holding only in the case of an equilateral triangle and P is the center of the triangle.
Generalisation
Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices let be an inner point and the intersections of the angle bisectors of with the associated polygon sides , then the following inequality holds:
Here denotes the secant function. For the triangle case the inequality becomes Barrow's inequality due to .
History
Barrow's inequality strengthens the Erdős–Mordell inequality, which has identical form except with PU, PV, and PW replaced by the three distances of P from the triangle's sides. It is named after David Francis Barrow. Barrow's proof of this inequality was published in 1937, as his solution to a problem posed in the American Mathematical Monthly of proving the Erdős–Mordell inequality. This result was named "Barrow's inequality" as early as 1961.
A simpler proof was later given by Louis J. Mordell.
See also
Euler's theorem in geometry
List of triangle inequalities
References
External links
Hojoo Lee: Topics in Inequalities - Theorems and Techniques
Triangle inequalities
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https://en.wikipedia.org/wiki/K%C3%A4hler%E2%80%93Einstein%20metric
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In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.
The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds. This problem can be split up into three cases dependent on the sign of the first Chern class of the Kähler manifold:
When the first Chern class is negative, there is always a Kähler–Einstein metric, as Thierry Aubin and Shing-Tung Yau proved independently.
When the first Chern class is zero, there is always a Kähler–Einstein metric, as Yau proved in the Calabi conjecture. That leads to the name Calabi–Yau manifolds. He was awarded with the Fields Medal partly because of this work.
The third case, the positive or Fano case, remained a well-known open problem for many years. In this case, there is a non-trivial obstruction to existence. In 2012, Xiuxiong Chen, Simon Donaldson, and Song Sun proved that in this case existence is equivalent to an algebro-geometric criterion called K-stability. Their proof appeared in a series of articles in the Journal of the American Mathematical Society. A proof was produced independently by Gang Tian at the same time.
When first Chern class is not definite, or we have intermediate Kodaira dimension, then finding canonical metric remained as an open problem, which is called the algebrization conjecture via analytical minimal model program.
Definition
Einstein manifolds
Suppose is a Riemannian manifold. In physics the Einstein field equations are a set of partial differential equations on the metric tensor which describe how the manifold should curve due to the existence of mass or energy, a quantity encapsulated by the stress–energy tensor . In a vacuum where there is no mass or energy, that is , the Einstein Field Equations simplify. Namely, the Ricci curvature of is a symmetric -tensor, as is the metric itself, and the equations reduce to
where is the scalar curvature of . That is, the Ricci curvature becomes proportional to the metric. A Riemannian manifold satisfying the above equation is called an Einstein manifold.
Every two-dimensional Riemannian manifold is Einstein. It can be proven using the Bianchi identities that, in any larger dimension, the scalar curvature of any connected Einstein manifold must be constant. For this reason, the Einstein condition is often given as
for a real number
Kähler manifolds
When the Riemannian manifold is also a complex manifold, that is it comes with an integrable almost-complex structure , it is possible to ask for a compatibility between the metric structure and the complex structure . There are many equivalent ways of formulating this compatibility condition, and one succinct interpretation is to ask that
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https://en.wikipedia.org/wiki/VNBG
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VNBG may refer to:
Bajhang Airport (ICAO airport code)
von Neumann–Bernays–Gödel set theory
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https://en.wikipedia.org/wiki/Characteristic%20function%20%28convex%20analysis%29
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In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis.
Definition
Let be a set, and let be a subset of . The characteristic function of is the function
taking values in the extended real number line defined by
Relationship with the indicator function
Let denote the usual indicator function:
If one adopts the conventions that
for any , and , except ;
; and
;
then the indicator and characteristic functions are related by the equations
and
Subgradient
The subgradient of for a set is the tangent cone of that set in .
Bibliography
Convex analysis
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https://en.wikipedia.org/wiki/Homology%20manifold
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In mathematics, a homology manifold (or generalized manifold)
is a locally compact topological space X that looks locally like a topological manifold from the point of view of homology theory.
Definition
A homology G-manifold (without boundary) of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G-cohomological dimension such that for any x∈X, the homology groups
are trivial unless p=n, in which case they are isomorphic to G. Here H is some homology theory, usually singular homology. Homology manifolds are the same as homology Z-manifolds.
More generally, one can define homology manifolds with boundary, by allowing the local homology groups to vanish
at some points, which are of course called the boundary of the homology manifold. The boundary of an n-dimensional first-countable homology manifold is an n−1 dimensional homology manifold (without boundary).
Examples
Any topological manifold is a homology manifold.
An example of a homology manifold that is not a manifold is the suspension of a homology sphere that is not a sphere.
Properties
If X×Y is a topological manifold, then X and Y are homology manifolds.
References
W. J .R. Mitchell, "Defining the boundary of a homology manifold", Proceedings of the American Mathematical Society, Vol. 110, No. 2. (Oct., 1990), pp. 509-513.
Algebraic topology
Generalized manifolds
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https://en.wikipedia.org/wiki/Standardized%20coefficient
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In statistics, standardized (regression) coefficients, also called beta coefficients or beta weights, are the estimates resulting from a regression analysis where the underlying data have been standardized so that the variances of dependent and independent variables are equal to 1. Therefore, standardized coefficients are unitless and refer to how many standard deviations a dependent variable will change, per standard deviation increase in the predictor variable.
Usage
Standardization of the coefficient is usually done to answer the question of which of the independent variables have a greater effect on the dependent variable in a multiple regression analysis where the variables are measured in different units of measurement (for example, income measured in dollars and family size measured in number of individuals).
It may also be considered a general measure of effect size, quantifying the "magnitude" of the effect of one variable on another.
For simple linear regression with orthogonal predictors, the standardized regression coefficient equals the correlation between the independent and dependent variables.
Implementation
A regression carried out on original (unstandardized) variables produces unstandardized coefficients. A regression carried out on standardized variables produces standardized coefficients. Values for standardized and unstandardized coefficients can also be re-scaled to one another subsequent to either type of analysis.
Suppose that is the regression coefficient resulting from a linear regression (predicting by ). The standardized coefficient simply results as , where and are the (estimated) standard deviations of and , respectively.
Sometimes, standardization is done only without respect to the standard deviation of the regressor (the independent variable ).
Advantages and disadvantages
Standardized coefficients' advocates note that the coefficients are independent of the involved variables' units of measurement (i.e., standardized coefficients are unitless), which makes comparisons easy.
Critics voice concerns that such a standardization can be very misleading.
Due to the re-scaling based on sample standard deviations, any effect apparent in the standardized coefficient may be due to confounding with the particularities (especially: variability) of the involved data sample(s).
Also, the interpretation or meaning of a "one standard deviation change" in the regressor may vary markedly between non-normal distributions (e.g., when skewed, asymmetric or multimodal).
Terminology
Some statistical software packages like PSPP, SPSS and SYSTAT label the standardized regression coefficients as "Beta" while the unstandardized coefficients are labeled "B". Others, like DAP/SAS label them "Standardized Coefficient". Sometimes the unstandardized variables are also labeled as "b".
See also
Linear regression
Correlation coefficient
Effect size
Unit-weighted regression
References
Further reading
External links
Which Predictors
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https://en.wikipedia.org/wiki/Alan%20Weinstein
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Alan David Weinstein (17 June 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry.
Education and career
After attending Roslyn High School, Weinstein obtained a bachelor's degree at the Massachusetts Institute of Technology in 1964. His teachers included, among others, James Munkres, Gian-Carlo Rota, Irving Segal, and, for the first senior course of differential geometry, Sigurður Helgason.
He received a PhD at University of California, Berkeley in 1967 under the direction of Shiing-Shen Chern. His dissertation was entitled "The cut locus and conjugate locus of a Riemannian manifold".
He worked then at MIT on 1967 (as Moore instructor) and at Bonn University in 1968/69. In 1969 he returned to Berkeley as assistant professor and from 1976 he is full professor. During 1975/76 he visited IHES in Paris and during 1978/79 he was visiting professor at Rice University.
Weinstein was awarded in 1971 a Sloan Research Fellowship and in 1985 a Guggenheim Fellowship. In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki. In 1992 he was elected Fellow of the American Academy of Arts and Sciences and in 2012 Fellow of the American Mathematical Society. In 2003 he was awarded a honorary doctorate from Universiteit Utrecht.
Research
Weinstein's works cover many areas in differential geometry and mathematical physics, including Riemannian geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization.
Among his most important contributions, in 1971 he proved a tubular neighbourhood theorem for Lagrangians in symplectic manifolds.
In 1974 he worked with Jerrold Marsden on the theory of reduction for mechanical systems with symmetries, introducing the famous Marsden–Weinstein quotient.
In 1978 he formulated a celebrated conjecture on the existence of periodic orbits, which has been later proved in several particular cases and has led to many new developments in symplectic and contact geometry.
In 1981 he formulated a general principle, called symplectic creed, stating that "everything is a Lagrangian submanifold". Such insight has been constantly quoted as the source of inspiration for many results in symplectic geometry.
Building on the work of André Lichnerowicz, in a 1983 foundational paper Weinstein proved many results which laid the ground for the development of modern Poisson geometry. A further influential idea in this field was its introduction of symplectic groupoids.
He is author of more than 50 research papers in peer-reviewed journals and he has supervised 34 PhD students.
Books
Geometric Models for Noncommutative Algebras (with A. Cannas da Silva), Berkeley Mathematics Lecture Notes series, American Mathematical Society (1999)
Lectures on the Geometry of Quantization (with S. Bates), Berkeley Mathematics Lecture Notes series, American Mathematical So
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https://en.wikipedia.org/wiki/Vinko%20Dvo%C5%99%C3%A1k
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Vinko Dvořák (January 21, 1848 – May 6, 1922) was a Czech-Croatian physicist, professor and academician.
He studied mathematics and physics at the Charles University in Prague, and after graduating he became an assistant to professor Ernst Mach. After obtaining his doctorate in Prague in 1873/1874 he came to Zagreb (at the time also part of Austria-Hungary) and founded the Physics Cabinet at the Faculty of Philosophy in 1875.
Dvořák made many important discoveries in the field of experimental acoustics and optics, which are known as the Dvořák-Rayleigh current, the Dvořák acoustic repulsion, and the Dvořák circuit. His work on acoustic radiometers coincided with that of Lord Rayleigh.
He was the dean of the Faculty of Philosophy in 1881/82 and again in 1891/92 and the rector of the University of Zagreb in 1893/94.
Professor Dvorak made constant advancements in physics experimentation at the Faculty—in 1896 he obtained a Röntgen radiation device just six months after it was discovered.
He became a member of the Academy of Sciences and Arts in 1883 (associate) and 1887 (full member). He was also an associate member of the Czech Academy of Franz Joseph I, a member of the Société francaise de physique (French Physics Society) and the Paris Société internationale des électriciens, and a member of the Royal Czech Society of Sciences in Prague.
Dvořák retired in 1911.
References
1848 births
1922 deaths
19th-century Croatian people
19th-century Czech people
Croatian physicists
Czech physicists
Rectors of the University of Zagreb
Charles University alumni
Croatian people of Czech descent
Members of the Croatian Academy of Sciences and Arts
Croatian mountain climbers
Burials at Mirogoj Cemetery
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https://en.wikipedia.org/wiki/Kendall%27s%20notation
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In queueing theory, a discipline within the mathematical theory of probability, Kendall's notation (or sometimes Kendall notation) is the standard system used to describe and classify a queueing node. D. G. Kendall proposed describing queueing models using three factors written A/S/c in 1953 where A denotes the time between arrivals to the queue, S the service time distribution and c the number of service channels open at the node. It has since been extended to A/S/c/K/N/D where K is the capacity of the queue, N is the size of the population of jobs to be served, and D is the queueing discipline.
When the final three parameters are not specified (e.g. M/M/1 queue), it is assumed K = ∞, N = ∞ and D = FIFO.
First example: M/M/1 queue
A M/M/1 queue means that the time between arrivals is Markovian (M), i.e. the inter-arrival time follows an exponential distribution of parameter λ. The second M means that the service time is Markovian: it follows an exponential distribution of parameter μ. The last parameter is the number of service channel which one (1).
Description of the parameters
In this section, we describe the parameters A/S/c/K/N/D from left to right.
A: The arrival process
A code describing the arrival process. The codes used are:
S: The service time distribution
This gives the distribution of time of the service of a customer. Some common notations are:
c: The number of servers
The number of service channels (or servers). The M/M/1 queue has a single server and the M/M/c queue c servers.
K: The number of places in the queue
The capacity of queue, or the maximum number of customers allowed in the queue. When the number is at this maximum, further arrivals are turned away. If this number is omitted, the capacity is assumed to be unlimited, or infinite.
Note: This is sometimes denoted c + K where K is the buffer size, the number of places in the queue above the number of servers c.
N: The calling population
The size of calling source. The size of the population from which the customers come. A small population will significantly affect the effective arrival rate, because, as more customers are in system, there are fewer free customers available to arrive into the system. If this number is omitted, the population is assumed to be unlimited, or infinite.
D: The queue's discipline
The Service Discipline or Priority order that jobs in the queue, or waiting line, are served:
Note: An alternative notation practice is to record the queue discipline before the population and system capacity, with or without enclosing parenthesis. This does not normally cause confusion because the notation is different.
References
Mathematical notation
Single queueing nodes
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https://en.wikipedia.org/wiki/Dennis%20Sullivan
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Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University.
Sullivan was awarded the Wolf Prize in Mathematics in 2010 and the Abel Prize in 2022.
Early life and education
Sullivan was born in Port Huron, Michigan, on February 12, 1941. His family moved to Houston soon afterwards.
He entered Rice University to study chemical engineering but switched his major to mathematics in his second year after encountering a particularly motivating mathematical theorem. The change was prompted by a special case of the uniformization theorem, according to which, in his own words:
He received his Bachelor of Arts degree from Rice in 1963. He obtained his Doctor of Philosophy from Princeton University in 1966 with his thesis, Triangulating homotopy equivalences, under the supervision of William Browder.
Career
Sullivan worked at the University of Warwick on a NATO Fellowship from 1966 to 1967. He was a Miller Research Fellow at the University of California, Berkeley from 1967 to 1969 and then a Sloan Fellow at Massachusetts Institute of Technology from 1969 to 1973. He was a visiting scholar at the Institute for Advanced Study in 1967–1968, 1968–1970, and again in 1975.
Sullivan was an associate professor at Paris-Sud University from 1973 to 1974, and then became a permanent professor at the Institut des Hautes Études Scientifiques (IHÉS) in 1974. In 1981, he became the Albert Einstein Chair in Science (Mathematics) at the Graduate Center, City University of New York and reduced his duties at the IHÉS to a half-time appointment. He joined the mathematics faculty at Stony Brook University in 1996 and left the IHÉS the following year.
Sullivan was involved in the founding of the Simons Center for Geometry and Physics and is a member of its board of trustees.
Research
Topology
Geometric topology
Along with Browder and his other students, Sullivan was an early adopter of surgery theory, particularly for classifying high-dimensional manifolds. His thesis work was focused on the Hauptvermutung.
In an influential set of notes in 1970, Sullivan put forward the radical concept that, within homotopy theory, spaces could directly "be broken into boxes" (or localized), a procedure hitherto applied to the algebraic constructs made from them.
The Sullivan conjecture, proved in its original form by Haynes Miller, states that the classifying space BG of a finite group G is sufficiently different from any finite CW complex X, that it maps to such an X only 'with difficulty'; in a more formal statement, the space of all mappings BG to X, as pointed spaces and given the compact-open topology, is weakly contractible. Sullivan's conjecture was also first presented in his 1970 notes.
Sullivan and Daniel Quillen (independently
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https://en.wikipedia.org/wiki/Preload%20%28software%29
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preload is a free Linux program which runs as a daemon to record statistics about usage of files by more frequently-used programs. This information is then used to keep these files preloaded into memory. This results in faster application startup times as less data needs to be fetched from disk. preload is often paired with prelink. preload was written by Behdad Esfahbod and uses Markov chains in its implementation;
See also
Prefetching
Readahead
References
External links
Project homepage on SourceForge
"Analyzing and improving GNOME startup time" -(PDF)
Preloading and prebinding
Linux process- and task-management-related software
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https://en.wikipedia.org/wiki/Granville%20Sewell
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Edward Granville Sewell is an American mathematician, university professor, and intelligent design advocate. He is a professor of mathematics at the University of Texas, El Paso.
Education
Sewell received his PhD from Purdue University in 1972 and an M.S. in mechanical engineering 1977 from the University of Texas, Austin. His BS was from Harding College (now Harding University)
Contributions
Mathematics
Sewell's primary work is on the solution of differential equations. He published "The Numerical Solution of Ordinary and Partial Differential Equations, Third Edition," World Scientific Publishing, 2014 . His major development effort has been the equation solver PDE2D--A general-purpose PDE solver. Sewell similarly published: "Computational Methods of Linear Algebra, Third Edition," and "Solving Partial Differential Equation Applications with PDE2D".
Views on origins
Sewell is signatory to the Discovery Institute's "A Scientific Dissent from Darwinism" petition. In 2000 Sewell compared the lifelong development of his state of the art software program with Darwin's predictions. After positing modeling the early universe and predicting its evolution, Sewell concludes:
Clearly something extremely improbable has happened here on our planet, with the origin and development of life, and especially with the development of human consciousness and creativity. This is cited by the Discovery Institute as one of the "Peer-Reviewed & Peer-Edited Scientific Publications Supporting the Theory of Intelligent Design", a claim rejected by critics and the judge in the Dover trial. He also wrote an article in The American Spectator. In these articles he reiterates the view that evolution violates the second law of thermodynamics. Mathematician Jason Rosenhouse wrote a response in The Mathematical Intelligencer entitled "How Anti-evolutionists Abuse Mathematics" and "Does Evolution Have a Thermodynamics Problem?". Physicist Mark Perakh called Sewell's thermodynamics work "depressingly fallacious".
In 2010, Sewell published a collection of essays on origins: In The Beginning And Other Essays on Intelligent Design The Discovery Institute lists as one of the "Peer-Reviewed & Peer-Edited Scientific Publications Supporting the Theory of Intelligent Design" is a postscript to his 1985 book Analysis of a Finite Element Method: PDE/PROTRAN.
References
External links
Granville Sewell Mathematics Dept. University of Texas El Paso
PDE2D--A general-purpose PDE solver
"Computational Methods of Linear Algebra, 3rd Edition," Granville Sewell, World Scientific Publishing Company, 2014
Entropy, Disorder and Life, John Pieper, talkorigins Archive, Copyright © 2000, updated May 24, 2002
20th-century American mathematicians
21st-century American mathematicians
Intelligent design advocates
Living people
Purdue University College of Engineering alumni
Cockrell School of Engineering alumni
Harding University alumni
University of Texas at El Paso faculty
Year of birth m
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https://en.wikipedia.org/wiki/Fabinho%20%28footballer%2C%20born%201976%29
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Fábio de Jesus or simply Fabinho (born October 16, 1976, in Nova Iguaçu), is a Brazilian defensive midfielder.
Club statistics
Honours
Santos
Brazilian League Série A: 2004
Internacional
Copa Libertadores: 2006
FIFA Club World Championship: 2006
Fluminense
Brazilian Cup: 2007
References
External links
zerozero.pt
Guardian Stats Centre
globoesporte
fluminense.com
External links
1976 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
J1 League players
Sport Club Internacional players
Bonsucesso Futebol Clube players
Associação Atlética Ponte Preta players
Gamba Osaka players
CR Flamengo footballers
Shimizu S-Pulse players
Expatriate men's footballers in Japan
Santos FC players
Fluminense FC players
Sportspeople from Nova Iguaçu
Men's association football midfielders
Footballers from Rio de Janeiro (state)
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https://en.wikipedia.org/wiki/Guy%20Terjanian
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Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1966, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified had just been proved as the Ax-Kochen theorem.
In 1977, he proved that if p is an odd prime number, and the natural numbers x, y and z satisfy , then 2p must divide x or y.
See also
Ax–Kochen theorem
References
Further reading
math.unicaen.fr article Topic: Arithmetic & geometry
French people of Armenian descent
20th-century French mathematicians
Algebraists
Number theorists
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Ibsen%20Mart%C3%ADnez
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Ibsen Martínez (born 20 October 1951) is a columnist, mathematician, journalist, and playwright from Caracas, Venezuela. Ibsen is a graduate from Central University of Venezuela in pure mathematics.
Since 1995, he has written a weekly column for El Nacional. His writings have appeared in El Nuevo Herald, The New York Times, Letras Libres, and El País. Martinez has also written several plays for the theater.
Martínez also writes soap operas (called "telenovelas" in Latin America). One of them was "Por Estas Calles" (Along these streets) a television drama, in 1992.
References
External links
The Washington Post, PostGlobal Panelist
1951 births
Living people
People from Caracas
Venezuelan journalists
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https://en.wikipedia.org/wiki/Finham%20Park%20School
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Finham Park School is a secondary school and sixth form with academy status. It is situated on Green Lane in Finham, Coventry, England.
In September 2003, it became the first Mathematics and Computing College in Coventry. The Headteacher is Mr Chris Bishop, with Deputy Headteachers Ms Sarah Megeney and Mr Rob Morey. The previous headteacher had been Mr Mark Bailie who was head between 2009 and 2017 and is now CEO for the Finham Park Multi Academy Trust set up in 2015.
The school has 1,650 students across the five mandatory years and the two optional years of the sixth form. The student intake is from the Finham, Styvechale, Cheylesmore, Green Lane, Gibbet Hill and Fenside districts of the city, plus certain parts of Earlsdon.
History
Finham Park School opened in 1970, but construction of the final school buildings was not completed until late 1971. Before Finham Park School opened, pupils in the area went to a variety of Coventry schools.
A sixth-form block was opened in 2006 and the old sixth-form block became the Personalised Learning Center (PLC), used for Supportive Studies for the main school. The Sixth Form is now in a bigger, two-storey purpose built block (T-Block).
In 2005, Finham Park School became the first school in Coventry to offer the IB Diploma Programme, as an alternative to A Levels. Upper sixth form students, the first to take the IB diploma from the school, sat their IB examinations in May 2007; though the IB Diploma is no longer offered.
In 2007, Finham Park was one of the first schools in England to test out the 'Biometric Cashless system' when buying food and drink items from school. Students have their thumbprint scan converted into an 11 digit number, and can then pay by having their thumbprint scanned into a system. Students' parents can top-up their child's account by using ParentPay – an online payment system allowing payments for lunches, trips, book etc.
In 2012 the main school buildings were all painted blue.
In January 2014, the school opened a fitness suite for use by students during and after the school day as well as by staff before and after school. The suite was officially opened by Coventry born athlete, David Moorcroft OBE on Friday 17 January
In December 2022, Finham Park School's 'New E Block' was opened and is officially called 'Peter Burns Performing Arts Centre' this space provides extra classrooms, a large music suite, a large performing arts studio plus a large canteen for extra students.
Bishop Ullathorne RC School is close by, towards the A45. The school gates of the two schools are between 5 and 10 minutes walking distance.
Colleges
The school now has five colleges:
These college names are based on different subspecies of lion, some being extinct.
Each pupil belongs to one of the colleges and participates in intercollegiate events to earn points for their house.
These school houses were previously split into Newton, Ada, Whittle, and Galileo; but in 2008 the houses were changed as abo
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https://en.wikipedia.org/wiki/Sergey%20Stechkin
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Sergey Borisovich Stechkin () (6 September 1920 – 22 November 1995) was a prominent Soviet mathematician who worked in theory of functions (especially in approximation theory) and number theory.
Biography
Sergey Stechkin was born on 6 September 1920 in Moscow. His father (Boris Stechkin) was a Soviet turbojet engine designer, academician. His great uncle, N.Ye. Zhukovsky, was the founding father of modern aero- and hydrodynamics. His maternal grandfather, N.A. Shilov, was a notable chemist. His paternal grandfather was Sergey Solomin, a science fiction author.
Stechkin attended school 58 and then attempted to matriculate to Moscow State University. He was turned down, likely due to the fact that the Soviet regime viewed his father as a political dissident at the time. He matriculated to Gorky State University instead. A year later, he was nevertheless able to transfer to the Mechanics and Mathematics department at Moscow State University, where he studied mathematics and was a student of D. E. Menshov. Stechkin received his PhD in 1948 with a dissertation titled "On the order of best approximations of continuous functions".
Later he worked as a mathematician at the Steklov Institute of Mathematics in Moscow. He was the founder and first director of the department of the Institute in Yekaterinburg. Later this department became the Institute of Mechanics and Mathematics at the Ural branch of the Russian Academy of Sciences.
Stechkin founded, and, for more than 20 years, served as editor-in-chief for the mathematical journal “Mathematical Notes” ().
Stechkin served as professor of mathematics at Moscow State University and his honors include the Chebyshev Award of the Russian Academy of Sciences in 1993. He died in 1995 in Moscow from age-related chronic illness.
His contribution to mathematics include the generalization of Jackson's inequality for all spaces.
References
External links
1920 births
1995 deaths
20th-century Russian mathematicians
Soviet mathematicians
Moscow State University alumni
Academic staff of the Moscow Institute of Physics and Technology
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https://en.wikipedia.org/wiki/Taro%20Morishima
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was a Japanese mathematician specializing in algebra who attended University of Tokyo in Japan. Morishima published at least thirteen papers, including his work on Fermat's Last Theorem. and a collected works volume published in 1990 after his death. He also corresponded several times with American mathematician H. S. Vandiver.
Morishima's Theorem on FLT
Let m be a prime number not exceeding 31. Let p be prime, and let x, y, z be integers such that xp + yp + zp = 0. Assume that p does not divide the product xyz. Then, p² must divide mp − 1-1.
Review
Granville wrote that Morishima's proof could not be accepted.
References
External links
Collected papers at Queen's University
1903 births
1989 deaths
20th-century Japanese mathematicians
Algebraists
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https://en.wikipedia.org/wiki/Interactive%20Mathematics%20Program
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The Interactive Mathematics Program (IMP) is a four-year, problem-based mathematics curriculum for high schools. It was one of several curricula funded by the National Science Foundation and designed around the 1989 National Council of Teachers of Mathematics (NCTM) standards. The IMP books were authored by Dan Fendel and Diane Resek, professors of mathematics at San Francisco State University, and by Lynne Alper and Sherry Fraser. IMP was published by Key Curriculum Press in 1997 and sold in 2012 to It's About Time.
Curriculum
Designed in response to national reports pointing to the need for a major overhaul in mathematics education, the IMP curriculum is markedly different in structure, content, and pedagogy from courses more typically found in the high school sequence.
Each book of the curriculum is divided into five- to eight-week units, each having a central problem or theme. This larger problem is intended to serve as motivation for students to develop the underlying skills and concepts needed to solve it, through solving a variety of smaller related problems.
There is an emphasis on asking students to work together in collaborative groups.
It is hoped that communication skills will be developed; exercises aimed at this goal are embedded throughout the curriculum, through the use of group and whole class discussions, the use of writing to present and clarify mathematical solutions; in some IEP classes, formal oral presentations are required.
The IMP curriculum expects students to make nearly daily use of a scientific graphing calculator.
Controversy
Nearly every one of these distinctive characteristics has generated controversy and placed the IMP curriculum right in the middle of the “math wars,” the conflict between those that favor more traditional curricula in mathematics education and the supporters of the reform curricula that were largely an outgrowth of the 1989 NCTM standards.
IMP is among the reform curricula that have been heavily criticized by organizations such as Mathematically Correct. That organization's Internet site begins with a statement that “advocates of the new, fuzzy math” (focus) “on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement.” Former NCTM president Frank Allen states, “Trying to organize school mathematics around problem solving instead of using its own internal structure for that purpose … (is destroying) essential connections….”
Criticism often includes anecdotal evidence including stories of school districts that have decided to discontinue or supplement use of the IMP curriculum and of students who did not feel they had been prepared adequately for college. "Regular math is much better, it makes much more sense," says Aimee Lynn Stearns, a student at Taos High School in Taos, New Mexico.
On t
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https://en.wikipedia.org/wiki/De%20Branges%20space
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In mathematics, a de Branges space (sometimes written De Branges space) is a concept in functional analysis and is constructed from a de Branges function.
The concept is named after Louis de Branges who proved numerous results regarding these spaces, especially as Hilbert spaces, and used those results to prove the Bieberbach conjecture.
De Branges functions
A Hermite-Biehler function, also known as de Branges function is an entire function E from to that satisfies the inequality , for all z in the upper half of the complex plane .
Definition 1
Given a Hermite-Biehler function , the de Branges space is defined as the set of all entire functions F such that
where:
is the open upper half of the complex plane.
.
is the usual Hardy space on the open upper half plane.
Definition 2
A de Branges space can also be defined as all entire functions satisfying all of the following conditions:
Definition 3
There exists also an axiomatic description, useful in operator theory.
As Hilbert spaces
Given a de Branges space . Define the scalar product:
A de Branges space with such a scalar product can be proven to be a Hilbert space.
References
Operator theory
Hardy spaces
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https://en.wikipedia.org/wiki/Beta-binomial%20distribution
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In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support of non-negative integers arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
The beta-binomial is a one-dimensional version of the Dirichlet-multinomial distribution as the binomial and beta distributions are univariate versions of the multinomial and Dirichlet distributions respectively. The special case where α and β are integers is also known as the negative hypergeometric distribution.
Motivation and derivation
As a compound distribution
The Beta distribution is a conjugate distribution of the binomial distribution. This fact leads to an analytically tractable compound distribution where one can think of the parameter in the binomial distribution as being randomly drawn from a beta distribution.
Suppose we were interested in predicting the number of heads, in future trials. This is given by
Using the properties of the beta function, this can alternatively be written
Beta-binomial as an urn model
The beta-binomial distribution can also be motivated via an urn model for positive integer values of α and β, known as the Pólya urn model. Specifically, imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, then two black balls are returned to the urn. If this is repeated n times, then the probability of observing x red balls follows a beta-binomial distribution with parameters n, α and β.
If the random draws are with simple replacement (no balls over and above the observed ball are added to the urn), then the distribution follows a binomial distribution and if the random draws are made without replacement, the distribution follows a hypergeometric distribution.
Moments and properties
The first three raw moments are
and the kurtosis is
Letting we note, suggestively, that the mean can be written as
and the variance as
where . The parameter is known as the "intra class" or "intra cluster" correlation. It is this positive correlation which gives rise to overdispersion. Note that when , no information is available to distinguish between the beta and binomial variation, and the two models have equal variances.
Factorial Moments
The -th factorial moment of a Beta-binomial random variable is
.
Point estimates
Method of moments
The method of moments estimates can be gained by noting the first and second moments of the beta-binomial and setting those equal to the sampl
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https://en.wikipedia.org/wiki/Piedmont%20Governor%27s%20School%20for%20Mathematics%2C%20Science%2C%20and%20Technology
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The Piedmont Governor's School for Mathematics, Science, and Technology is one of Virginia's 18 state-initiated magnet Governor's Schools. It is a half-day school program where 11th and 12th grade students take advanced classes in the morning (receiving their remaining classes from their home high school.)
Four classes are to be taken at the gov. school and two or three more per semester at their base school.
Students at PGSMST have the opportunity to earn an associate degree through Danville Community College (Danville students) or Patrick Henry Community College (Martinsville students) while enrolled at the Governor's School.
The School does not have a classroom facility of its own. Instead, students travel to the Institute for Advanced Learning and Research (Danville site) or New College Institute (Martinsville site) to take courses offered by the program.
History
Before The Piedmont Governors School for Mathematics Science and Technology, it was commonly called the GSGET (Governors School for Global Economics and Technology.)
Mission
The mission of the Piedmont Governor's School for Mathematics, Science, and Technology is to provide a challenging, project-driven, research-based curriculum in a technology-infused environment. The school is designed for academically gifted, highly motivated, and high achieving eleventh (11th) and twelfth (12th) grade students from Danville City, Henry County, Martinsville City, and Pittsylvania County school divisions.
Upon Consideration for Gov. School...
Nominations of potential candidates may be received from a variety of sources including students, parents, school personnel, and community leaders. Multiple criteria will be used to create a pool of applicants from the nominations to insure that a student can not be eliminated by one measure. The criteria may include such items as grade-point averages, Standards of Learning End-of-Course test results, scores from standardized tests, and successful completion of specific courses in mathematics, science, and technology.
The selection process will utilize percentile rankings on standardized achievement and ability tests, such as the Matrix Analogies Test (MAT); writing samples; unweighted grade-point averages; and teacher recommendations. In the event of a tie between students’ scores, science and mathematics course grades, and achievement tests scores in those areas will be compared.
Participating school systems
As of 2008 the school has 125 students enrolled from its participating school systems:
Danville City
Henry County
Martinsville City
Pittsylvania County
Junior Curriculum
College Chemistry (CHM 111, 112)
College Biology (BIO 101, 102)
Advanced Calculus I (MTH 273)
Advanced Mathematical Analysis (MTH 166)
Research Methodology and Design (ENG 131)
English 11 IALR Site only (ENG 111, 112)
Applied Statistics (MTH 157)
Computer Applications and Concepts NCI Site only (ITE 115)
Senior Curriculum
College Physics (PHY 201, 202)
College Biology (BIO 10
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https://en.wikipedia.org/wiki/Franz%20Kugler
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Franz Kugler is the name of:
Franz Theodor Kugler (1808–1858), German art historian and poet
Franz Xaver Kugler (1862–1929), professor of mathematics, chemist, assyriologist, and Jesuit priest
Franz Xaver Kugler (Radler) from Munich who invented a beer shandy drink called Radler
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https://en.wikipedia.org/wiki/Hilkka%20Rantasepp%C3%A4-Helenius
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Hilkka Rantaseppä-Helenius (1925–1975) was a Finnish astronomer.
Rantaseppä-Helenius began studying mathematics in hopes of becoming a teacher. Finnish astronomer Yrjö Väisälä inspired her to become an astronomer instead. Helenius, as a daughter of a farmer, was among the lucky few astronomers that had the privilege of having an observatory at their own corral.
Rantaseppä-Helenius worked on observing minor planets. She worked as an assistant at Tuorla Observatory from 1956 to 1962. In 1962 she became an observer when a vacancy became available. She remained an observer until 1975. She was also involved in building the Kevola Observatory by Tähtitieteellis-optillinen seura (Astronomy-Optical Society) on her own property in 1963.
Rantaseppä-Helenius died at age 50 in an accident. The Florian asteroid 1530 Rantaseppä was named in her memory.
References
1925 births
1975 deaths
Finnish astronomers
Women astronomers
Astronomy-optics society
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https://en.wikipedia.org/wiki/Graduate%20Texts%20in%20Mathematics
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Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book.
The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.
List of books
Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., )
Measure and Category – A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., )
Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., )
A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., )
Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., )
Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, )
A Course in Arithmetic, Jean-Pierre Serre (1996, )
Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring, (1973, )
Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, )
A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, )
Functions of One Complex Variable I, John B. Conway (1978, 2nd ed., )
Advanced Mathematical Analysis, Richard Beals (1973, )
Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, 2nd ed., )
Stable Mappings and Their Singularities, Martin Golubitsky, Victor Guillemin, (1974, )
Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian, (1974, )
The Structure of Fields, David J. Winter, (1974, )
Random Processes, Murray Rosenblatt, (1974, )
Measure Theory, Paul R. Halmos (1974, )
A Hilbert Space Problem Book, Paul R. Halmos (1982, 2nd ed., )
Fibre Bundles, Dale Husemoller (1994, 3rd ed., )
Linear Algebraic Groups, James E. Humphreys (1975, )
An Algebraic Introduction to Mathematical Logic, Donald W. Barnes, John M. Mack (1975, )
Linear Algebra, Werner H. Greub (1975, )
Geometric Functional Analysis and Its Applications, Richard B. Holmes, (1975, )
Real and Abstract Analysis, Edwin Hewitt, Karl Stromberg (1975, )
Algebraic Theories, Ernest G. Manes, (1976, )
General Topology, John L. Kelley (1975, )
Commutative Algebra I, Oscar Zariski, Pierre Samuel (1975, )
Commutative Algebra II, Oscar Zariski, Pierre Samuel (1975, )
Lectures in Abstract Algebra I: Basic Concepts, Nathan Jacobson (1976, )
Lectures in Abstract Algebra II: Linear Algebra, Nathan Jacobson (1984, )
Lectures in Abstract Algebra III: Theory of Fields and Galois Theory, Nathan Jacobson (1976, )
Differential Topology, Morris W. Hirsch (1976, )
Principles of Random Walk, Frank Spitzer (1964, 2nd ed., )
Several Complex Variables and Banach Algebras, Herbert Alexander, John Wermer (1998, 3rd ed., )
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https://en.wikipedia.org/wiki/Cumulative%20hierarchy
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In mathematics, specifically set theory, a cumulative hierarchy is a family of sets indexed by ordinals such that
If is a limit ordinal, then
Some authors additionally require that or that .
The union of the sets of a cumulative hierarchy is often used as a model of set theory.
The phrase "the cumulative hierarchy" usually refers to the standard cumulative hierarchy of the von Neumann universe with introduced by .
Reflection principle
A cumulative hierarchy satisfies a form of the reflection principle: any formula in the language of set theory that holds in the union of the hierarchy also holds in some stages .
Examples
The von Neumann universe is built from a cumulative hierarchy .
The sets of the constructible universe form a cumulative hierarchy.
The Boolean-valued models constructed by forcing are built using a cumulative hierarchy.
The well founded sets in a model of set theory (possibly not satisfying the axiom of foundation) form a cumulative hierarchy whose union satisfies the axiom of foundation.
References
Set theory
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https://en.wikipedia.org/wiki/Technology%20College
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In the United Kingdom, a Technology College is a specialist school that specialises in design and technology, mathematics and science. Beginning in 1994, they were the first specialist schools that were not CTC colleges. In 2008, there were 598 Technology Colleges in England, of which 12 also specialised in another subject.
History
The Education Reform Act 1988 made technology mandatory, however the Conservative government were unable to afford the cost of funding schools to teach the subject. A first attempt at developing specialist schools to solve this issue, the City Technology College (CTC) programme between 1988 and 1993, had produced only 15 schools, despite an initial aim of 200. In response, Cyril Taylor, chairman of the City Technology Colleges Trust, proposed to allow pre-existing schools to become specialists in technology (CTCs were newly opened schools). This was expected to mitigate the programme's failure and allow the government to gradually pay for the subject of technology. The Major government launched the £25 million Technology Schools Initiative (TSI) afterwards. From 1991, secondary schools were granted additional funds as a reward for specialising in technology in order to improve the curricular provision of technical education. 89 local education authorities applied to join the TSI, with a number of schools individually applying in authorities that chose not to take part. Some authorities, namely those run by the Labour Party, refused to participate on political grounds (Labour had opposed technology schools). 222 schools had specialised in technology by 1993 (not including CTCs), with government plans to have these schools collaborate and share their resources with other secondaries.
The Conservative manifesto for the 1992 general election promised to "expand the initiative across the country", with the July 1992 education white paper Choice and Diversity: A new framework for schools reinforcing the initiative's goal of encouraging schools to specialise in technology after the Conservatives' victory. However, the focus was no longer on improving technical education. Instead the focus drifted to increasing diversity in the school system. In the same year, another education white paper Technology colleges: schools for the future was released. Like technology schools, new Technology Colleges specialising in mathematics, technology and science were to be established from already existing secondary schools in hopes of furthering the CTC programme's impact and adding diversity to the school system. The Technology Colleges programme was launched in 1993, allowing schools with voluntary aided and grant-maintained status to apply for Technology College status after raising £100,000 in private sponsorship. The first successful applicants were then designated with this status in 1994. The TSI was scrapped and the programme was opened up to all other state schools in November 1994. The programme evolved into the specialist school
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https://en.wikipedia.org/wiki/Elkhorn%20Valley%20Schools
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Elkhorn Valley Schools is located in Tilden, in the northeast section of the state of Nebraska, United States.
District statistics
The district is a Class 3 school and categorized as a C2 class size. The district houses approximately 300 students in a K-12 campus location. The staff consists of 33 teachers, 9 paraprofessionals, 2 administrators and 15 classified personnel.
Curriculum
The school supports a strong education foundation of scientific research based educational programs and curricula. The curricula support and comply with the expectations and guidelines of the Nebraska A.L.L.S.T.A.R. program. The district continuously improves its educational structure through the training and support of a six-year Reading First grant and program implementation. An inhouse preschool, funded through Title I federal grant monies, along with the traditional kindergarten through 12th grades, focus on improving reading skills, for their school improvement plan. Focus areas supporting the main improvement plan include the use and implementation of technology in the learning environment of all students and an increased emphasis on math skills and practice time in the classrooms. These targeted areas provide direction for students and teachers to improve reading, math, and technology skills that will meet 21st-century expectations.
School symbols
The school colors are green and white, and the mascot is the Falcon.
References
External links
School districts in Nebraska
Education in Antelope County, Nebraska
Education in Madison County, Nebraska
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https://en.wikipedia.org/wiki/Ambient%20isotopy
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In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let and be manifolds and and be embeddings of in . A continuous map
is defined to be an ambient isotopy taking to if is the identity map, each map is a homeomorphism from to itself, and . This implies that the orientation must be preserved by ambient isotopies. For example, two knots that are mirror images of each other are, in general, not equivalent.
See also
Isotopy
Regular homotopy
Regular isotopy
References
M. A. Armstrong, Basic Topology, Springer-Verlag, 1983
Sasho Kalajdzievski, An Illustrated Introduction to Topology and Homotopy, CRC Press, 2010, Chapter 10: Isotopy and Homotopy
Topology
Maps of manifolds
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https://en.wikipedia.org/wiki/Prime%20form
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In algebraic geometry, the Schottky–Klein prime form E(x,y) of a compact Riemann surface X depends on two elements x and y of X, and vanishes if and only if x = y. The prime form E is not quite a holomorphic function on X × X, but is a section of a holomorphic line bundle over this space. Prime forms were introduced by Friedrich Schottky and Felix Klein.
Prime forms can be used to construct meromorphic functions on X with given poles and zeros. If Σniai is a divisor linearly equivalent to 0, then ΠE(x,ai)ni is a meromorphic function with given poles and zeros.
See also
Fay's trisecant identity
References
Riemann surfaces
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https://en.wikipedia.org/wiki/Legendrian%20knot
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In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into which is tangent to the standard contact structure on It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a contact manifold that is always tangent to the contact hyperplane.
Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots. Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots. More sophisticated invariants have been constructed, including one constructed combinatorially by Chekanov and using holomorphic discs by Eliashberg. This Chekanov-Eliashberg invariant yields an invariant for loops of Legendrian knots by considering the monodromy of the loops. This has yielded noncontractible loops of Legendrian knots which are contractible in the space of all knots.
Any Legendrian knot may be perturbed to a transverse knot (a knot transverse to a contact structure) by pushing off in a direction transverse to the contact planes. The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.
References
.
External links
The Legendrian knot atlas
Knots and links
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https://en.wikipedia.org/wiki/List%20of%20South%20Sydney%20Rabbitohs%20players
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Following are lists of all rugby league footballers who have played first-grade for the South Sydney Rabbitohs Rugby League Football Club.
Players and statistics
Correct as of round 22 of the 2023 NRL season
Club Internationals – Australia
The following players have represented Australia whilst playing for South Sydney.
Tommy Anderson
Jim Armstrong
Alf Blair
Cec Blinkhorn
Ray Branighan
Tim Brasher
Arthur Butler
Billy Cann
Mark Carroll
Clive Churchill
Michael Cleary
Arthur Conlin
Damien Cook
Ron Coote
Les Cowie
Frank Curran
Steve Darmody
Les Davidson
Jim Davis
Denis Donoghue
Terry Fahey
Harry Finch
Bryan Fletcher
Dane Gagai
Herb Gilbert
Campbell Graham
Bob Grant
John Graves
Howard Hallett
Ernie Hammerton
Greg Hawick
Bob Honan
Greg Inglis
Brian James
Alex Johnston
Harry Kadwell
Clem Kennedy
John Kerwick
Jack Leveson
Eric Lewis
Jimmy Lisle
Bob McCarthy
Eddie McGrath
Paddy Maher
Latrell Mitchell
Ian Moir
Cameron Murray
Ray Norman
Alf O'Connor
Frank O'Connor
John O'Neill
Arthur Oxford
George Piggins
Denis Pittard
Bernie Purcell
Jack Rayner
Eddie Root
John Rosewell
Paul Sait
John Sattler
Eric Simms
Bill Spence
Gary Stevens
David Taylor
George Treweek
Dylan Walker
Elwyn Walters
Benny Wearing
Jack Why
Percy Williams
Note that Jim Morgan was selected as a reserve for Australia whilst as a player for South Sydney but did not actually take the field in the representative match.
Represented Australia before or after playing with South Sydney
The following players have represented Australia either before or after they played for South Sydney.
Royce Ayliffe
Peter Burge
Hugh Byrne
Darrel Chapman
Michael Crocker
Ron Crowe
Col Donohoe
Percy Fairall
Robbie Farah
Dane Gagai
Bob Gehrke
Brian Hambly
Shannon Hegarty
Terry Hill
Ray Hines
Johnny Hutchinson
Luke Keary
Matt King
Adam MacDougall
Ian Mackay
Mark McGaw
Keith Middleton
Jim Morgan
Adam Muir
Webby Neill
Rex Norman
Claud O'Donnell
Julian O'Neill
Bryan Orrock
Bill Owen
David Peachey
Russell Richardson
Ian Roberts
Craig Salvatori
Jim Serdaris
Glenn Stewart
Billy Thompson
Peter Tunks
Lote Tuqiri
Harry Wells
Craig Wing
Club Internationals – New Zealand
The following players have represented New Zealand whilst playing for South Sydney.
Roy Asotasi
David Kidwell
David Fa'alogo
Bryson Goodwin
Terry Hermansson
Issac Luke
Gene Ngamu
Jeremy Smith
Tyran Smith
Nigel Vagana1
Jason Williams
1 Represented New Zealand All Golds in 2007
Club Internationals – Other Countries
The following players have represented other rugby league playing nations (i.e. other than Australia or New Zealand) whilst playing for South Sydney.
Neccrom Areaiiti (Cook Islands)
Roy Asotasi (Samoa)
George Burgess (England)
Sam Burgess (England)
Tom Burgess (England)
Angelo Dymock (Tonga)
Robbie Farah (Lebanon)
Tere Glassie (Cook Islands)
Siliva Havili (Tonga)
Lachlan Ilias ([{Greece]])
Robert Jennings (Tonga)
Alex Johnston (Papua New Guinea)
K
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https://en.wikipedia.org/wiki/Ars%20Magna%20%28Cardano%20book%29
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The Ars Magna (The Great Art, 1545) is an important Latin-language book on algebra written by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnae, Sive de Regulis Algebraicis Liber Unus (Book number one about The Great Art, or The Rules of Algebra). There was a second edition in Cardano's lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the early Renaissance, together with Copernicus' De revolutionibus orbium coelestium and Vesalius' De humani corporis fabrica. The first editions of these three books were published within a two-year span (1543–1545).
History
In 1535 Niccolò Fontana Tartaglia became famous for having solved cubics of the form x3 + ax = b (with a,b > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, Pratica Arithmeticæ et mensurandi singularis (The Practice of Arithmetic and Simple Mensuration). That same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student Lodovico Ferrari found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano became aware of the fact that Scipione del Ferro had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.
Contents
The book, which is divided into forty chapters, contains the first published algebraic solution to cubic and quartic equations. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipione del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations.
Since at the time negative numbers were not generally acknowledged, knowing how to solve cubics of the form x3 + ax = b did not mean knowing how to solve cubics of the form x3 = ax + b (with a,b > 0), for instance. Besides, Cardano also explains how to reduce equations of the form x3 + ax2 + bx + c = 0 to cubic equations without a quadratic term, but, again, he has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII).
In Ars Magna the concept of multiple root appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is x3 = 12x + 16, of which −2 is a double root.
Ars Magna also contains the first occurrence of complex numbers (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative
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https://en.wikipedia.org/wiki/Minimum%20bounding%20rectangle
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In computational geometry, the minimum bounding rectangle (MBR), also known as bounding box (BBOX) or envelope, is an expression of the maximum extents of a two-dimensional object (e.g. point, line, polygon) or set of objects within its coordinate system; in other words , , , . The MBR is a 2-dimensional case of the minimum bounding box.
MBRs are frequently used as an indication of the general position of a geographic feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes.
The degree to which an "overlapping rectangles" query based on MBRs will be satisfactory (in other words, produce a low number of "false positive" hits) will depend on the extent to which individual spatial objects occupy (fill) their associated MBR. If the MBR is full or nearly so (for example, a mapsheet aligned with axes of latitude and longitude will normally entirely fill its associated MBR in the same coordinate space), then the "overlapping rectangles" test will be entirely reliable for that and similar spatial objects. On the other hand, if the MBR describes a dataset consisting of a diagonal line, or a small number of disjunct points (patchy data), then most of the MBR will be empty and an "overlapping rectangles" test will produce a high number of false positives. One system that attempts to deal with this problem, particularly for patchy data, is c-squares.
MBRs are also an essential prerequisite for the R-tree method of spatial indexing.
As spatial metadata
Owing to their simplicity of expression and ease of use for searching, MBRs (frequently as "bounding box" or "bounding coordinates") are also commonly included in relevant standards for geospatial metadata, i.e. metadata that describes spatial (geographic) objects; examples include DCMI Box as an extension to the Dublin Core metadata scheme, "Bounding Coordinates" in the (U.S.) FGDC metadata standard, and "Geographic Bounding Box" in the (2003–current) ISO 19115 Metadata Standard for geographic information (ISO/TC 211). It is also (as "boundingBox") an element in Geography Markup Language (GML), that is utilised by a range of Web Service specifications from the Open Geospatial Consortium (OGC). In the ISO 19107 Spatial Schema (ISO/TC 211), MBR appears as the datatype GM_Envelope that is returned by the envelope() operation on the root class GM_Object.
Web-accessible articles that deal further with the concept of the MBR include "Unlocking the Mysteries of the Bounding Box" by Douglas R. Caldwell, and "Geographic Database Search Interfaces and the Equatorial Cylindrical Equidistant Projection" by Ross S. Swick and Kenneth W. Knowles. The section on "searching" on the Geospatial Methods site is also well worth investigating. See also documentation for specific spatially enabled databases, e.g.
See also
Bounding parallelogram
C-squares
Darboux integral
Elongatedness
Geographic information system
Geospatial metadata
Largest empty rectangle, also known as maxima
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https://en.wikipedia.org/wiki/Constraint%20%28mathematics%29
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In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.
Example
The following is a simple optimization problem:
subject to
and
where denotes the vector (x1, x2).
In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions.
Without the constraints, the solution would be (0,0), where has the lowest value. But this solution does not satisfy the constraints. The solution of the constrained optimization problem stated above is , which is the point with the smallest value of that satisfies the two constraints.
Terminology
If an inequality constraint holds with equality at the optimal point, the constraint is said to be , as the point cannot be varied in the direction of the constraint even though doing so would improve the value of the objective function.
If an inequality constraint holds as a strict inequality at the optimal point (that is, does not hold with equality), the constraint is said to be , as the point could be varied in the direction of the constraint, although it would not be optimal to do so. Under certain conditions, as for example in convex optimization, if a constraint is non-binding, the optimization problem would have the same solution even in the absence of that constraint.
If a constraint is not satisfied at a given point, the point is said to be infeasible.
Hard and soft constraints
If the problem mandates that the constraints be satisfied, as in the above discussion, the constraints are sometimes referred to as hard constraints. However, in some problems, called flexible constraint satisfaction problems, it is preferred but not required that certain constraints be satisfied; such non-mandatory constraints are known as soft constraints. Soft constraints arise in, for example, preference-based planning. In a MAX-CSP problem, a number of constraints are allowed to be violated, and the quality of a solution is measured by the number of satisfied constraints.
Global constraints
Global constraints are constraints representing a specific relation on a number of variables, taken altogether. Some of them, such as the alldifferent constraint, can be rewritten as a conjunction of atomic constraints in a simpler language: the alldifferent constraint holds on n variables , and is satisfied if the variables take values which are pairwise different. It is semantically equivalent to the conjunction of inequaliti
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https://en.wikipedia.org/wiki/Lehmer%20mean
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In mathematics, the Lehmer mean of a tuple of positive real numbers, named after Derrick Henry Lehmer, is defined as:
The weighted Lehmer mean with respect to a tuple of positive weights is defined as:
The Lehmer mean is an alternative to power means
for interpolating between minimum and maximum via arithmetic mean and harmonic mean.
Properties
The derivative of is non-negative
thus this function is monotonic and the inequality
holds.
The derivative of the weighted Lehmer mean is:
Special cases
is the minimum of the elements of .
is the harmonic mean.
is the geometric mean of the two values and .
is the arithmetic mean.
is the contraharmonic mean.
is the maximum of the elements of . Sketch of a proof: Without loss of generality let be the values which equal the maximum. Then
Applications
Signal processing
Like a power mean, a Lehmer mean serves a non-linear moving average which is shifted towards small signal values for small and emphasizes big signal values for big . Given an efficient implementation of a moving arithmetic mean called you can implement a moving Lehmer mean according to the following Haskell code.
lehmerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
lehmerSmooth smooth p xs =
zipWith (/)
(smooth (map (**p) xs))
(smooth (map (**(p-1)) xs))
For big it can serve an envelope detector on a rectified signal.
For small it can serve an baseline detector on a mass spectrum.
Gonzalez and Woods call this a "contraharmonic mean filter" described for varying values of p (however, as above, the contraharmonic mean can refer to the specific case ). Their convention is to substitute p with the order of the filter Q:
Q=0 is the arithmetic mean. Positive Q can reduce pepper noise and negative Q can reduce salt noise.
See also
Mean
Power mean
Notes
External links
Lehmer Mean at MathWorld
Means
Articles with example Haskell code
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https://en.wikipedia.org/wiki/Stolarsky%20mean
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In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.
Definition
For two positive real numbers x, y the Stolarsky Mean is defined as:
Derivation
It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function at and , has the same slope as a line tangent to the graph at some point in the interval .
The Stolarsky mean is obtained by
when choosing .
Special cases
is the minimum.
is the geometric mean.
is the logarithmic mean. It can be obtained from the mean value theorem by choosing .
is the power mean with exponent .
is the identric mean. It can be obtained from the mean value theorem by choosing .
is the arithmetic mean.
is a connection to the quadratic mean and the geometric mean.
is the maximum.
Generalizations
One can generalize the mean to n + 1 variables by considering the mean value theorem for divided differences for the nth derivative.
One obtains
for .
See also
Mean
References
Means
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https://en.wikipedia.org/wiki/Integrated%20mathematics
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Integrated mathematics is the term used in the United States to describe the style of mathematics education which integrates many topics or strands of mathematics throughout each year of secondary school. Each math course in secondary school covers topics in algebra, geometry, trigonometry and functions. Nearly all countries throughout the world, except the United States, follow this type of curriculum.
In the United States, topics are usually integrated throughout elementary school up to the seventh or sometimes eighth grade. Beginning with high school level courses, topics are usually separated so that one year a student focuses entirely on algebra (if it was not already taken in the eighth grade), the next year entirely on geometry, then another year of algebra (sometimes with trigonometry), and later an optional fourth year of precalculus or calculus. Precalculus is the exception to the rule, as it usually integrates algebra, trigonometry, and geometry topics. Statistics may be integrated into all the courses or presented as a separate course.
New York State began using integrated math curricula in the 1980s, but recently returned to a traditional curriculum. A few other localities in the United States have also tried such integrated curricula, including Georgia, which mandated them in 2008 but subsequently made them optional. More recently, a few other states have mandated that all districts change to integrated curricula, including North Carolina, Illinois, West Virginia and Utah. Some districts in other states, including California, have either switched or are considering switching to an integrated curriculum.
Under the Common Core Standards adopted by most states in 2012, high school mathematics may be taught using either a traditional American approach or an integrated curriculum. The only difference would be the order in which the topics are taught. Supporters of using integrated curricula in the United States believe that students will be able to see the connections between algebra and geometry better in an integrated curriculum.
General mathematics is another term for a mathematics course organized around different branches of mathematics, with topics arranged according to the main objective of the course. When applied to primary education, the term general mathematics may encompass mathematical concepts more complex than basic arithmetic, like number notation, addition and multiplication tables, fractions and related operations, measurement units. When used in context of higher education, the term may encompass mathematical terminology and concepts, finding and applying appropriate techniques to solve routine problems, interpreting and representing practical information given in various forms, interpreting and using mathematical models, and constructing mathematical arguments to solve familiar and unfamiliar problems.
References
Mathematics education
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https://en.wikipedia.org/wiki/Molecular%20biophysics
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Molecular biophysics is a rapidly evolving interdisciplinary area of research that combines concepts in physics, chemistry, engineering, mathematics and biology. It seeks to understand biomolecular systems and explain biological function in terms of molecular structure, structural organization, and dynamic behaviour at various levels of complexity (from single molecules to supramolecular structures, viruses and small living systems). This discipline covers topics such as the measurement of molecular forces, molecular associations, allosteric interactions, Brownian motion, and cable theory. Additional areas of study can be found on Outline of Biophysics. The discipline has required development of specialized equipment and procedures capable of imaging and manipulating minute living structures, as well as novel experimental approaches.
Overview
Molecular biophysics typically addresses biological questions similar to those in biochemistry and molecular biology, seeking to find the physical underpinnings of biomolecular phenomena. Scientists in this field conduct research concerned with understanding the interactions between the various systems of a cell, including the interactions between DNA, RNA and protein biosynthesis, as well as how these interactions are regulated. A great variety of techniques are used to answer these questions.
Fluorescent imaging techniques, as well as electron microscopy, X-ray crystallography, NMR spectroscopy, atomic force microscopy (AFM) and small-angle scattering (SAS) both with X-rays and neutrons (SAXS/SANS) are often used to visualize structures of biological significance. Protein dynamics can be observed by neutron spin echo spectroscopy. Conformational change in structure can be measured using techniques such as dual polarisation interferometry, circular dichroism, SAXS and SANS. Direct manipulation of molecules using optical tweezers or AFM, can also be used to monitor biological events where forces and distances are at the nanoscale. Molecular biophysicists often consider complex biological events as systems of interacting entities which can be understood e.g. through statistical mechanics, thermodynamics and chemical kinetics. By drawing knowledge and experimental techniques from a wide variety of disciplines, biophysicists are often able to directly observe, model or even manipulate the structures and interactions of individual molecules or complexes of molecules.
Areas of Research
Computational biology
Computational biology involves the development and application of data-analytical and theoretical methods, mathematical modeling and computational simulation techniques to the study of biological, ecological, behavioral, and social systems. The field is broadly defined and includes foundations in biology, applied mathematics, statistics, biochemistry, chemistry, biophysics, molecular biology, genetics, genomics, computer science and evolution. Computational biology has become an important part of develo
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https://en.wikipedia.org/wiki/Switzerland%20at%20the%201904%20Summer%20Olympics
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According to the official statistics, one gymnast, Adolf Spinnler, and one wrestler, Gustav Thiefenthaler, from Switzerland competed at the 1904 Summer Olympics in St. Louis, United States.
But there were more athletes with Swiss roots at the Olympics: Andreas Kempf competed in three gymnastics events, finishing 8th in the combined three events finals. He arrived in the United States in 1902 and represented the Kansas City Turnverein. Kempf applied for naturalization as a US citizen in 1908, but was denied citizenship. Emil Schwegler competed for the United States in three gymnastics events, representing the St. Louis Schweizer Turnverein. Born in Switzerland, he was naturalized as a US citizen with his parents and then was a college student in Kansas City, Missouri. And Oscar Schwab was born in Paris to a Swiss mother and was adopted by her American husband. He competed for the US in the quarter mile cycling race. After the games he raced mostly in Europe and was Swiss sprint champion in 1907.
In fact, eighteen-year old Gustav Tiefenthaler was also born in Switzerland, moved to the United States with his family when he was a child and was naturalized as a US citizen with his parents. So he was "less swiss" than Andreas Kempf. Tiefenthaler represented the South Broadway Athletic Club of St. Louis. At the Olympics, Tiefenthaler had a single bout in the men’s freestyle light flyweight event and lose, but still earned a bronze medal.
Adolf Spinnler on the other hand was clearly European, but he was born in Switzerland, later moved to Esslingen am Neckar in the German Empire and represented a German sports club - so his affiliation to a national team is also disputed.
Medalists
Results by event
Athletics
Gymnastics
References
Official Olympic Reports
International Olympic Committee results database
External links
"Switzerland at the 1904 Olympics" at switzerlandusa.medium.com
Nations at the 1904 Summer Olympics
1904
Olympics
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https://en.wikipedia.org/wiki/Equilateral%20%28disambiguation%29
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Equilateral can refer to:
Equilateral polygon, in geometry
Equilateral triangle, in geometry
Equilateral dimension of a metric space, in mathematics
Equilateral triathlon, in which each leg would take an approximately equal time
See also
Venus Equilateral, a set of 13 science fiction short stories by George O. Smith
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https://en.wikipedia.org/wiki/Period%20%28algebraic%20geometry%29
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In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, such that the periods form a ring.
Maxim Kontsevich and Don Zagier gave a survey of periods and introduced some conjectures about them. Periods also arise in computing the integrals that arise from Feynman diagrams, and there has been intensive work trying to understand the connections.
Definition
A real number is a period if it is of the form
where is a polynomial and a rational function on with rational coefficients. A complex number is a period if its real and imaginary parts are periods.
An alternative definition allows and to be algebraic functions; this looks more general, but is equivalent. The coefficients of the rational functions and polynomials can also be generalised to algebraic numbers because irrational algebraic numbers are expressible in terms of areas of suitable domains.
In the other direction, can be restricted to be the constant function or , by replacing the integrand with an integral of over a region defined by a polynomial in additional variables. In other words, a (nonnegative) period is the volume of a region in defined by a polynomial inequality.
Examples
Besides the algebraic numbers, the following numbers are known to be periods:
The natural logarithm of any positive algebraic number a, which is
Elliptic integrals with rational arguments
All zeta constants (the Riemann zeta function of an integer) and multiple zeta values
Special values of hypergeometric functions at algebraic arguments
Γ(p/q)q for natural numbers p and q.
An example of a real number that is not a period is given by Chaitin's constant Ω. Any other non-computable number also gives an example of a real number that is not a period. Currently there are no natural examples of computable numbers that have been proved not to be periods, however it is possible to construct artificial examples. Plausible candidates for numbers that are not periods include e, 1/, and the Euler–Mascheroni constant γ.
Properties and motivation
The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable.
The set of all periods is countable, and all periods are computable, and in particular definable.
Conjectures
Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".
Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integra
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https://en.wikipedia.org/wiki/PCMB
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PCMB may refer to:
p-Chloromercuribenzoic acid
Plymouth-Canton Marching Band
PCMB (encoding), a mixed multi-byte character set
Physics chemistry maths and biology together
Pixiv complex Mladá Boleslav (in planning)
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https://en.wikipedia.org/wiki/Gustavus%20Simmons
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Gustavus J. Simmons (born 1930) is a retired cryptographer and former manager of the applied mathematics Department and Senior Fellow at Sandia National Laboratories. He worked primarily with authentication theory, developing cryptographic techniques for solving problems of mutual distrust and in devising protocols whose function could be trusted, even though some of the inputs or participants cannot be. Simmons was born in West Virginia and was named after his grandfather, a prohibition officer who was gunned down three years before Gustavus was born. He began his post-secondary education at Deep Springs College, and received his Ph.D in mathematics from the University of New Mexico, Albuquerque.
Simmons has published over 170 papers, many of which are devoted to asymmetric encryption techniques. His technical contributions include the development of subliminal channels which make it possible to conceal covert communications in digital signatures and the mathematical formulation of an authentication channel paralleling in many respects the secrecy channel formulated by Claude Shannon in 1948. In the 1980s, he helped found the International Association for Cryptologic Research (IACR). He is also the creator of the Ramsey/graph theory-based mathematical game Sim.
At Sandia, Simmons was primarily concerned with the command and control of nuclear weapons, where the objective is to separate possession of a weapon from the ability to autonomously use it, something which should only be possible on receipt of an authenticated order from the National Command Authority, in using authentication to make possible the verification of compliance with various arms control treaties, and in the cryptographic aspects of verifying adherence to the Comprehensive Test Ban Treaty for nuclear weapons. In a review of Contemporary Cryptology (see publications), Don Coppersmith summarized the problem:
Is the host substituting a false signal to mask the fact that it is continuing tests? Is the monitor really using the device to transmit other information than that allowed by the treaty? Who supplies the hardware? Can that person cheat?
Awards and recognition
In 1947 he was one of 40 finalists in the Westinghouse Science Talent Search. In 1986, Simmons was the recipient of the U.S. Department of Energy Ernest Orlando Lawrence Award. In 1991, he was awarded an honorary doctorate from Lund University for his work in authentication theory. In 1996 he was made an Honorary Fellow of the Institute of Combinatorics and its Applications. In 2005, he was elected an IACR Fellow, "for pioneering research in information integrity, information theory, and secure protocols and for substantial contributions to the formation of the IACR." He was invited to write the section on cryptology in the 16th edition of the Encyclopædia Britannica (1986) and to revise the section for the current edition. He was Rothschild Professor at the Isaac Newton Institute for Mathematical Sciences, Cambr
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https://en.wikipedia.org/wiki/Compact%20convergence
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In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology.
Definition
Let be a topological space and be a metric space. A sequence of functions
,
is said to converge compactly as to some function if, for every compact set ,
uniformly on as . This means that for all compact ,
Examples
If and with their usual topologies, with , then converges compactly to the constant function with value 0, but not uniformly.
If , and , then converges pointwise to the function that is zero on and one at , but the sequence does not converge compactly.
A very powerful tool for showing compact convergence is the Arzelà–Ascoli theorem. There are several versions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly bounded maps has a subsequence that converges compactly to some continuous map.
Properties
If uniformly, then compactly.
If is a compact space and compactly, then uniformly.
If is a locally compact space, then compactly if and only if locally uniformly.
If is a compactly generated space, compactly, and each is continuous, then is continuous.
See also
Modes of convergence (annotated index)
Montel's theorem
References
R. Remmert Theory of complex functions (1991 Springer) p. 95
Functional analysis
Convergence (mathematics)
Topology of function spaces
Topological spaces
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https://en.wikipedia.org/wiki/Normal%20convergence
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In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
History
The concept of normal convergence was first introduced by René Baire in 1908 in his book Leçons sur les théories générales de l'analyse.
Definition
Given a set S and functions (or to any normed vector space), the series
is called normally convergent if the series of uniform norms of the terms of the series converges, i.e.,
Distinctions
Normal convergence implies, but should not be confused with, uniform absolute convergence, i.e. uniform convergence of the series of nonnegative functions . To illustrate this, consider
Then the series is uniformly convergent (for any ε take n ≥ 1/ε), but the series of uniform norms is the harmonic series and thus diverges. An example using continuous functions can be made by replacing these functions with bump functions of height 1/n and width 1 centered at each natural number n.
As well, normal convergence of a series is different from norm-topology convergence, i.e. convergence of the partial sum sequence in the topology induced by the uniform norm. Normal convergence implies norm-topology convergence if and only if the space of functions under consideration is complete with respect to the uniform norm. (The converse does not hold even for complete function spaces: for example, consider the harmonic series as a sequence of constant functions).
Generalizations
Local normal convergence
A series can be called "locally normally convergent on X" if each point x in X has a neighborhood U such that the series of functions ƒn restricted to the domain U
is normally convergent, i.e. such that
where the norm is the supremum over the domain U.
Compact normal convergence
A series is said to be "normally convergent on compact subsets of X" or "compactly normally convergent on X" if for every compact subset K of X, the series of functions ƒn restricted to K
is normally convergent on K.
Note: if X is locally compact (even in the weakest sense), local normal convergence and compact normal convergence are equivalent.
Properties
Every normal convergent series is uniformly convergent, locally uniformly convergent, and compactly uniformly convergent. This is very important, since it assures that any re-arrangement of the series, any derivatives or integrals of the series, and sums and products with other convergent series will converge to the "correct" value.
If is normally convergent to , then any re-arrangement of the sequence also converges normally to the same ƒ. That is, for every bijection , is normally convergent to .
See also
Modes of convergence (annotated index)
References
Mathematical analysis
Convergence (mathematics)
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https://en.wikipedia.org/wiki/Baer%20ring
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In abstract algebra and functional analysis, Baer rings, Baer *-rings, Rickart rings, Rickart *-rings, and AW*-algebras are various attempts to give an algebraic analogue of von Neumann algebras, using axioms about annihilators of various sets.
Any von Neumann algebra is a Baer *-ring, and much of the theory of projections in von Neumann algebras can be extended to all Baer *-rings, For example, Baer *-rings can be divided into types I, II, and III in the same way as von Neumann algebras.
In the literature, left Rickart rings have also been termed left PP-rings. ("Principal implies projective": See definitions below.)
Definitions
An idempotent element of a ring is an element e which has the property that e2 = e.
The left annihilator of a set is
A (left) Rickart ring is a ring satisfying any of the following conditions:
the left annihilator of any single element of R is generated (as a left ideal) by an idempotent element.
(For unital rings) the left annihilator of any element is a direct summand of R.
All principal left ideals (ideals of the form Rx) are projective R modules.
A Baer ring has the following definitions:
The left annihilator of any subset of R is generated (as a left ideal) by an idempotent element.
(For unital rings) The left annihilator of any subset of R is a direct summand of R. For unital rings, replacing all occurrences of 'left' with 'right' yields an equivalent definition, that is to say, the definition is left-right symmetric.
In operator theory, the definitions are strengthened slightly by requiring the ring R to have an involution . Since this makes R isomorphic to its opposite ring Rop, the definition of Rickart *-ring is left-right symmetric.
A projection in a *-ring is an idempotent p that is self-adjoint ().
A Rickart *-ring is a *-ring such that left annihilator of any element is generated (as a left ideal) by a projection.
A Baer *-ring is a *-ring such that left annihilator of any subset is generated (as a left ideal) by a projection.
An AW*-algebra, introduced by , is a C*-algebra that is also a Baer *-ring.
Examples
Since the principal left ideals of a left hereditary ring or left semihereditary ring are projective, it is clear that both types are left Rickart rings. This includes von Neumann regular rings, which are left and right semihereditary. If a von Neumann regular ring R is also right or left self injective, then R is Baer.
Any semisimple ring is Baer, since all left and right ideals are summands in R, including the annihilators.
Any domain is Baer, since all annihilators are except for the annihilator of 0, which is R, and both and R are summands of R.
The ring of bounded linear operators on a Hilbert space are a Baer ring and is also a Baer *-ring with the involution * given by the adjoint.
von Neumann algebras are examples of all the different sorts of ring above.
Properties
The projections in a Rickart *-ring form a lattice, which is complete if the ring is a Baer *-ring.
See
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https://en.wikipedia.org/wiki/Zero%20object%20%28algebra%29
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In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as . One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).
Instances of the zero object include, but are not limited to the following:
As a group, the zero group or trivial group.
As a ring, the zero ring or trivial ring.
As an algebra over a field or algebra over a ring, the trivial algebra.
As a module (over a ring ), the zero module. The term trivial module is also used, although it may be ambiguous, as a trivial G-module is a G-module with a trivial action.
As a vector space (over a field ), the zero vector space, zero-dimensional vector space or just zero space.
These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.
In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as:
, where .
The most general of them, the zero module, is a finitely-generated module with an empty generating set.
For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, , because there are no non-zero elements. This structure is associative and commutative. A ring which has both an additive and multiplicative identity is trivial if and only if , since this equality implies that for all within ,
In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of depend on exact definition of the multiplicative identity; see below.
Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.
The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra.
The zero-dimensional is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.
Properties
The zero ring, zero module and zero vector space are the zero objects of, respectively, the category of pseudo-rings, the category of modules and the category of vector spaces. However, the zero ring is not a zero object in the category of rings, since there is no ring homomorphism of the zero ring in any other ring.
The zero object, by definition, must be a terminal object, which means that a morphism
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https://en.wikipedia.org/wiki/Mathematische%20Zeitschrift
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Mathematische Zeitschrift (German for Mathematical Journal) is a mathematical journal for pure and applied mathematics published by Springer Verlag.
It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Helmut Wielandt, and Olivier Debarre.
External links
Mathematics journals
Academic journals established in 1918
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