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Page "Economics" ¶ 49
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generalizes and developed
While the example below is developed for the one-dimensional case, many of the arguments, and almost all of the notation, generalizes easily to any number of dimensions.

generalizes and such
Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.
The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements ( wherein the last statement asserts the truth / falsity of the first statement ), provided there are an odd number of statements asserting the falsity of their successor ; the following is a three-sentence version, with each statement asserting the falsity of its successor:
The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.
For this reason, the notion of a Noetherian ring generalizes such rings as.
It generalizes an exchanging sort, such as insertion or bubble sort, by starting the comparison and exchange of elements with elements that are far apart before finishing with neighboring elements.
such that it generalizes length, and the angle θ between two vectors a and b by
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold ( such as a surface in space ) which takes as input a pair of tangent vectors v and w and produces a real number ( scalar ) g ( v, w ) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
This definition generalizes in a straightforward manner to a discrete signal with an infinite number of values such as a signal sampled at discrete times:
The definition of the power spectral density generalizes in a straightforward manner to finite time-series with such as a signal sampled at discrete times for a total measurement period.
* Two-point tensor, which generalizes this notion to tensors that have indices not just in the primal and dual space but in other vector spaces ( such as other tangent spaces on the same manifold ).
Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure.
The homotopy principle generalizes such results as Smale's proof of sphere eversion.
This example not only generalizes the previous example, but includes many measures on non-locally compact spaces, such as Wiener measure on the space of real-valued continuous functions on the interval.
As such, the following summary only generalizes the actual sequence of events which take place.
The HOMFLY polynomial is one such invariant and it generalizes two polynomials previously discovered, the Alexander polynomial and the Jones polynomial both of which can be obtained by appropriate substitutions from HOMFLY.
While the use of " call / cc " is unnecessary for a linear collection, such as, the code generalizes to any collection that can be traversed.
But unlike the constraint-based gauge fixing procedures above, the R < sub > ξ </ sub > gauge generalizes well to non-abelian gauge groups such as the SU ( 3 ) of QCD.

generalizes and supply
This criticism of the application of the model of supply and demand generalizes, particularly to all markets for factors of production.

generalizes and model
Another model, which generalizes Gilbert's random graph model, is the random dot-product model.
The autoregressive fractionally integrated moving average ( ARFIMA ) model generalizes the former three.
The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
The approach generalizes to a model with multiple explanatory variables.
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct.

generalizes and allows
More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.
This perspective is generalized by the linear canonical transformation, which generalizes the fractional Fourier transform and allows linear transforms of the time-frequency domain other than rotation.
It allows for redundancy and generalizations, because the language user generalizes over recurring experiences of use.
Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints.

generalizes and for
This generalizes an analogous fact for normal operators.
Compactness generalizes many important properties of closed and bounded intervals in the real line ; that is, intervals of the form for real numbers and.
Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case.
The above development generalizes in a more-or-less straightforward fashion to general principal G-bundles for some arbitrary Lie group G taking the place of U ( 1 ).
For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.
This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely-generated pro-p groups.
Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.
Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors ; note that for tensors of order greater than 2 ( matrices are order 2 tensors ), rank is very hard to compute, unlike for matrices.
The notion of paracompactness generalizes ordinary compactness ; a key motivation for the notion of paracompactness is that it is a sufficient condition for the existence of partitions of unity.
Taylor's theorem also generalizes to multivariate and vector valued functions on any dimensions n and m. This generalization of Taylor's theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations.
This generalizes the notion of geodesic for Riemannian manifolds.
Thinking of curvature as a measure, rather than as a function, Descartes ' theorem is Gauss – Bonnet where the curvature is a discrete measure, and Gauss – Bonnet for measures generalizes both Gauss – Bonnet for smooth manifolds and Descartes ' theorem.
Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero.
The coefficient of determination generalizes the correlation coefficient for relationships beyond simple linear regression.

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