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A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if
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non-negative and countably
If μ is a complex-valued countably additive Borel measure, μ is regular iff the non-negative countably additive measure | μ | is regular as defined above.
non-negative and additive
Use of the three primary colors is not sufficient to reproduce all colors ; only colors within the color triangle defined by the chromaticities of the primaries can be reproduced by additive mixing of non-negative amounts of those colors of light.
are additive, where n is a non-negative integer.
non-negative and measure
Technically, a measure is a function that assigns a non-negative real number or +∞ to ( certain ) subsets of a set X ( see Definition below ).
Finally, ψ is positive iff the measure μ is non-negative.
Here ' is Lebesgue measure and is a non-negative measurable function.
be a sequence of non-negative measurable functions on a measure space ( S, Σ, μ ).
Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
uses upper and lower variations to prove the Hahn – Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure.
non-negative and μ
If the highest weight is dominant and integral ( a weight μ is dominant and integral if μ satisfies the condition that is a non-negative integer for all i ), then the weight spectrum of the irreducible representation is invariant under the Weyl group for G, and the representation is integrable.
# If μ is non-negative and A ⊆ B, then μ ( A ) ≤ μ ( B ).
where φ is real, Q real and positive, Γ is the gamma function, the ω < sub > 1 </ sub > real and positive, and the μ < sub > i </ sub > complex with non-negative real part, so that the function
non-negative and on
( Expressed more technically, in each case the pair ( m, n ) decreases in the lexicographic order on pairs, which is a well-ordering, just like the ordering of single non-negative integers ; this means one cannot go down in the ordering infinitely many times in succession.
Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.
The unbounded knapsack problem ( UKP ) places no upper bound on the number of copies of each kind of item and can be formulated as above except for that the only restriction on is that it is a non-negative integer.
where n is the unit outward normal to B, and a is a non-negative function defined on B.
: If k is even then all the coefficients of the factors on the right ( considered as power series in T ) are non-negative ; this follows by writing
When k is even the coefficients of all its Euler factors are non-negative, so that each of the Euler factors has coefficients bounded by a constant times the coefficients of Z ( E < sup > k </ sup >, T ) and therefore converges on the same region and has no poles in this region.
The number l ( D ) is the quantity that is of primary interest: the dimension ( over C ) of the vector space of meromorphic functions h on the surface, such that all the coefficients of ( h ) + D are non-negative.
Thus, replacing E by K we may assume that f < sub > n </ sub > converge to f pointwise on E. Next, by the definition of the Lebesgue Integral, it is enough to show that if φ is any non-negative simple function less than or equal to f, then
In relativity, the Ricci curvature, which determines the collision properties of geodesics, is determined by the energy tensor, and its projection on light rays is equal to the null-projection of the energy-momentum tensor and is always non-negative.
Since all the eigenvalues are non-negative, the problem is convex and the maximum occurs on the edges of the domain, namely when and ( when the eigenvalues are ordered in decreasing magnitude ).
By induction on n it follows that the basis functions are non-negative for all values of and.
By the Krein extension theorem for positive linear functionals, there is a state f on A such that f ( z ) ≥ 0 for all non-negative z in A and f (− x * x ) < 0.
In mathematical analysis, the uniform norm assigns to real-or complex-valued bounded functions f defined on a set S the non-negative number
For instance, the usual ordering on the non-negative integers is a well-quasi-ordering, but the same ordering on the set of all integers is not, because it contains the infinite descending chain 0, − 1, − 2, − 3 ...
Using the Jensen's inequality on the definition of mutual information, we can show that I ( X ; Y ) is non-negative ; so consequently, H ( X ) ≥ H ( X | Y ).
Consider an integer N and a non-negative monotone decreasing function f defined on the unbounded interval < nowiki >
Operations on classes are carried out by combining these representatives and then reducing the result to its least non-negative residue.
There is a sequence of integrable non-negative functions φ < sub > n </ sub > with integral 1 on G such that λ ( g ) φ < sub > n </ sub > − φ < sub > n </ sub > tends to 0 in the weak topology on L < sup > 1 </ sup >( G ).
non-negative and locally
A dynamical system is the tuple, with a manifold ( locally a Banach space or Euclidean space ), the domain for time ( non-negative reals, the integers, ...) and f an evolution rule t → f < sup > t </ sup > ( with ) such that f < sup > t </ sup > is a diffeomorphism of the manifold to itself.
A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to the Euclidean space E < sup > n </ sup > ( or, equivalently, to some connected open subset of E < sup > n </ sup >).
non-negative and compact
: 3 ) For every compact set K ⊂ D there exists a constant C such that for every x ∈ K and every non-negative integer k the following bound holds:
Indeed if V is a compact neighborhood of the identity, let f < sub > V </ sub > be a non-negative continuous function supported in V such that
For every finite ( or compact ) subset F of G there is an integrable non-negative function φ with integral 1 such that λ ( g ) φ − φ is arbitrarily small in L < sup > 1 </ sup >( G ) for g in F.
In mathematics, the soul theorem is a theorem of Riemannian geometry that largely reduces the study of complete manifolds of non-negative sectional curvature to that of the compact case.
non-negative and Hausdorff
In mathematics, the Hausdorff dimension ( also known as the Hausdorff – Besicovitch dimension ) is an extended non-negative real number associated with any metric space.
* The Hausdorff dimension is an extended non-negative real number associated with any metric space that generalizes the notion of the dimension of a real vector space.
non-negative and space
A pseudometric space is a set together with a non-negative real-valued function ( called a pseudometric ) such that, for every,
For a non-negative integer k, the kth Betti number b < sub > k </ sub >( X ) of the space X is defined as the rank of the abelian group H < sub > k </ sub >( X ), the kth homology group of X. Equivalently, one can define it as the vector space dimension of H < sub > k </ sub >( X ; Q ), since the homology group in this case is a vector space over Q.
Recall that a density operator is a non-negative operator on a Hilbert space with unit trace.
Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions ( for the case of functions of one variable ) or on the space of integrable functions ( for the case of functions of several variables ).
A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system.
As with any RGB color space, for non-negative values of R, G, and B it is not possible to represent colors outside this triangle, which is well inside the range of colors visible to a human.
In convex geometry, a convex combination is a linear combination of points ( which can be vectors, scalars, or more generally points in an affine space ) where all coefficients are non-negative and sum up to 1.
Multiplication by a non-negative function on an L < sup > 2 </ sup > space is a non-negative self-adjoint operator.
The polar decomposition for matrices generalizes as follows: if A is a bounded linear operator then there is a unique factorization of A as a product A = UP where U is a partial isometry, P is a non-negative self-adjoint operator and the initial space of U is the closure of the range of P.
If X is a Banach space, a one-parameter semigroup of operators on X is a family of operators indexed on the non-negative real numbers
Suppose that X < sub > t </ sub > is a real-valued stochastic process defined on a probability space and with time index t ranging over the non-negative real numbers.
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