For example, there is a smooth projective 3-fold over 2 which has Kodaira dimension −∞ but is not separably uniruled.E. Sato, Tohoku Math. J. 45 (1993), 447-460. Theorem. It is not known whether every smooth Fano variety in positive characteristic is separably uniruled.
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Proposition 1. So uniruledness does not imply that the Kodaira dimension is −∞ in positive characteristic. A variety X is separably uniruled if there is a variety Y with a dominant separable rational map Y × P1 – → X which does not factor through the projection to Y. ("Separable" means that the derivative is surjective at some point; this would be automatic for a dominant rational map in characteristic zero.) A separably uniruled variety has Kodaira dimension −∞. The converse is true in dimension 2, but not in higher dimensions.
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The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. > A function f is said to be almost surely separably valued (or essentially > separably valued) if there exists a subset N ⊆ X with μ(N) = 0 such that f(X > \ N) ⊆ B is separable. > Theorem (Pettis, 1938). A function f : X → B defined on a measure space (X, > Σ, μ) and taking values in a Banach space B is (strongly) measurable (that > equals a.e.
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The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. > Function f is almost surely separably valued (or essentially separably > valued) if there exists a subset N ⊆ X with μ(N) = 0 such that f(X \ N) ⊆ B > is separable. > A function f : X -> B defined on a measure space (X, Σ, μ) and taking > values in a Banach space B is (strongly) measurable (with respect to Σ and > the Borel algebra on B) if and only if it is both weakly measurable and > almost surely separably valued. In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.
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When a prefix can be used both separably and inseparably, there are cases where the same verb can have different meanings depending on whether its prefix is separable or inseparable (an equivalent example in English would be take over and overtake).
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A Henselian local ring is called strictly Henselian if its residue field is separably closed. A field with valuation is said to be Henselian if its valuation ring is Henselian. A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.
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In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1 factor. There are an uncountable number of other factors of type II1. Connes proved that the infinite one is also unique.
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An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying this another way, K is contained in a separably- closed algebraic extension field. It is unique (up to isomorphism).
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Note that the roles of the scale parameter \sigma and the shape parameter \xi under Y \sim exGPD(\sigma, \xi) are separably interpretable, which may lead to a robust efficient estimation for the \xi than using the X \sim GPD(\sigma, \xi) . The roles of the two parameters are associated each other under X \sim GPD(\mu=0,\sigma, \xi) (at least up to the second central moment); see the formula of variance Var(X) wherein both parameters are participated.
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Other typical problems of the back- propagation algorithm are the speed of convergence and the possibility of ending up in a local minimum of the error function. Today there are practical methods that make back-propagation in multi-layer perceptrons the tool of choice for many machine learning tasks. One also can use a series of independent neural networks moderated by some intermediary, a similar behavior that happens in brain. These neurons can perform separably and handle a large task, and the results can be finally combined.
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The most basic example of a fundamental group is π1(Spec k), the fundamental group of a field k. Essentially by definition, the fundamental group of k can be shown to be isomorphic to the absolute Galois group Gal (ksep / k). More precisely, the choice of a geometric point of Spec (k) is equivalent to giving a separably closed extension field K, and the fundamental group with respect to that base point identifies with the Galois group Gal (K / k). This interpretation of the Galois group is known as Grothendieck's Galois theory.
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For fields that are not algebraically closed (or not separably closed), the absolute Galois group is fundamentally important: extending the case of finite Galois extensions outlined above, this group governs all finite separable extensions of . By elementary means, the group can be shown to be the Prüfer group, the profinite completion of . This statement subsumes the fact that the only algebraic extensions of are the fields for , and that the Galois groups of these finite extensions are given by :. A description in terms of generators and relations is also known for the Galois groups of -adic number fields (finite extensions of ).
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