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generalized and definition
The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.
A more abstract definition, which is equivalent but more easily generalized to infinite-dimensional spaces, is to say that bras are linear functionals on kets, i. e. operators that input a ket and output a complex number.
Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.
Like the " dimension of image " characterization this can be generalized to a definition of the rank of a linear map: the rank of a linear map f from V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. While this definition does not suggest an efficient manner to compute the rank ( for which it is better to use one of the alternative definitions ), it does allow to easily understand many of the properties of the rank, for instance that the rank of the transpose of A is the same as that of A.
For generalized or extended thermodynamics, the definition of the quantity known as the entropy of a small local region is in terms beyond those of classical thermodynamics ; in particular, flow rates are admitted into the definition of the entropy of a small local region.
A suitable definition of the entropy of a small local region depends on the physically insightful and judicious selection of the independent local non-equilibrium macroscopic state variables, and different selections provide different generalized or extended thermodynamical accounts of one and the same given physical process.
With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces ; they may also be generalized further, to oriented matroids.
This generalized definition can then be used on any system, not only information technology system.
2 ( c ) of Directive No. 2004 / 83 / EC, essentially reproduces the narrow definition of refugee offered by the UN 1951 Convention ; nevertheless, by virtue of articles 2 ( e ) and 15 of the same Directive, persons who have fled a war-caused generalized violence are, at certain conditions, elegible for a complementary form of protection, called subsidiary protection.
Since the ordinary notion of consistency involves that of contradiction, which again involves negation, and since this function does not appear in general as a primitive in generalized set of postulates a new definition must be given ".
Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space R < sup > n </ sup >.
However, the word Kanto eventually became more of a generalized umbrella term than a precise definition of a musical genre.
The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges.
Then a C-valued presheaf on X is the same as a contravariant functor from O ( X ) to C. This definition can be generalized to the case when the source category is not of the form O ( X ) for any X ; see presheaf ( category theory ).
A matrix satisfying the first two conditions of the definition is known as a generalized inverse.
This choice is motivated by the fact that in generalized recursion theories, such as α-recursion theory, the definition corresponding to domains has been found to be more natural.
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations ( e. g. conic sections ).
The definition can be vastly generalized, for example to intersections along subvarieties instead of just at points, or to arbitrary complete varieties.
This definition can be generalized to the case in which is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid ( or even a group ) of transformations parametrized by ( or, or, or ), where each transformation satisfies the same requirements as above.
The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
In a Western context, Rosenau cites a British Secret Intelligence Service definition as " a generalized intention to ( emphasis added ) “ overthrow or undermine parliamentary democracy by political, industrial or violent means .” While insurgents do not necessarily use terror, it is hard to imagine any insurgency meeting its goals without undermining aspects of the legitimacy or power of the government or faction it opposes.
In the definition of a generalized BV algebra, one drops the second-order assumption for Δ.

generalized and implies
In particular, for, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means.
In 1997, Deshouillers, Effinger, te Riele and Zinoviev showed that the generalized Riemann hypothesis implies Goldbach's weak conjecture for all numbers.
When fluxes are included the supersymmetry condition instead implies that the compactification manifold be a generalized Calabi – Yau, a notion introduced by.
This implies uniqueness, in the sense that if there is a natural transformation between two generalized cohomology theories, which is an isomorphism for a one point space, then it is an isomorphism for all CW complexes.
For example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical ( in fact, ) non-measurable set of real numbers, all of which are independent of ZFC.
Lafforgue's theorem implies that the generalized Ramanujan conjecture is true for the general linear group over a global function field, by an argument due to.
The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic ; this implies that its Sylow subgroups are cyclic or generalized quaternion groups.

generalized and above
All the above approaches tend to be generalized to the form of integrated computational models of a synthetic / abstract intelligence, in order to be applied to the explanation and improvement of individual and social / organizational decision-making and reasoning.
The generalized relationship was already explained above.
Folding ( see above ) can generally be considered a generalized form of outlining.
In the second form given above, the phase is often generalized to, by replacing the wavenumber k with a wave vector that specifies the direction and wavenumber of a plane wave in 3-space, parameterized by position vector r. In that case, the wavenumber k, the magnitude of k, is still in the same relationship with wavelength as shown above, with v being interpreted as scalar speed in the direction of the wave vector.
Wang tiles can be generalized in various ways, all of which are also undecidable in the above sense.
The power series defined above is generalized by the idea of a germ.
Imagine that the action I defined above is invariant under small perturbations ( warpings ) of the time variable t and the generalized coordinates q ; in a notation commonly used in physics,
It doesn't matter if L depends on the time derivative of that generalized coordinate, since the Lagrangian independence of the coordinate always makes the above partial derivative zero.
This formulation expresses logistic regression as a type of generalized linear model, which predicts variables with various types of probability distributions by fitting a linear predictor function of the above form to some sort of arbitrary transformation of the expected value of the variable.
The above formula comes from using the generalized chain rule to obtain the total derivative — with respect to x — of both sides of F ( x, y ) = 0:
As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K ( X ) into the rational cohomology of X.
The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form.
This equation gives a sine wave for a single dimension, thus the generalized equation given above gives the amplitude of the wave at a position x at time t along a single line.
The example about free groups given above can be generalized to any type of algebra in the sense of a variety of algebras in universal algebra.
The index of the generalized coordinates is written here as a superscript (), not as a subscript as done above ().
The above concepts can be generalized to talk about space duality, where the terms " points " and " planes " are interchanged ( and lines remain lines ).
This may be explained by the generalized Trivers-Willard hypothesis, which states that parents who possess any heritable trait that increases males ’ reproductive success above females ’ will have more sons, and will have more daughters if they possess traits that increase females ’ reproductive success above males ’.
The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators.
If in the algebraic derivation above more than two polynomials are used, it results in the generalized Vandermonde's identity.
It is based on the fact that the vector space can be canonically decomposed into a direct sum of stable subspaces corresponding to the distinct irreducible factors P of the characteristic polynomial ( as stated by the lemme des noyaux ), where the characteristic polynomial of each summand is a power of the corresponding P. These summands can be further decomposed, non-canonically, as a direct sum of cyclic F-modules ( like is done for the Frobenius normal form above ), where the characteristic polynomial of each summand is still a ( generally smaller ) power of P. The primary rational canonical form is a block diagonal matrix corresponding to such a decomposition into cyclic modules, with a particular form called generalized Jordan block in the diagonal blocks, corresponding to a particular choice of a basis for the cyclic modules.
Wacquant notes that underclass status is imposed on urban blacks from outside and above them ( e. g., by journalists, politicians, and academics ), stating that “ underclass ” is a derogatory and " negative label that nobody claims or invokes except to pin it on to others .” And, although the underclass concepts is homogenizing, Wacquant argues that underclass imagery differentiates on gender lines, with the underclass male being depicted as a violent “ gang banger ,” a physical threat to public safety, and the underclass female being generalized as “ welfare mother ” ( also see welfare queen ), a “ moral assault on American values .”

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