共查询到20条相似文献,搜索用时 31 毫秒
1.
It is shown that for compact metric spaces ( X, d) the following statements are pairwise equivalent: “ X is Loeb”, “ X is separable”, “ X has a we ordered dense subset”, “ X is second countable”, and “ X has a dense set G = ∪{ Gn : n ∈ ω}, ∣ Gn∣ < ω, with lim n→∞ diam ( G n) = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF 0 , the Zermelo‐Fraenkel set theory without the axiom of regularity, and that the countable axiom of choice for families of finite sets CAC fin does not imply the statement “Compact metric spaces are separable”. 相似文献
2.
Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure \(\overline{\omega} : \mathscr{C}(S) \rightarrow S\) on a uniruled projective manifold \((X,\,{\cal K})\) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ? emanating from a general point x ∈ S, there exists an immersed neighborhood N? of ? which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ? X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ? 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ?*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ? X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ? ?2n, n ? 3, and when X0 ? X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W. 相似文献
3.
Let T = { T ( t)} t ∈? be a C0‐group on a Banach space X with generator A. Under what conditions the assumption σ ( A) = {0} implies that A = 0? This is called “ A = 0” problem. In this paper we present some results related to this problem. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
The principal goal of this article is to present two asymptotic solutions for the classical Graetz‐Nusselt problem. The method of lines (MOL) has been adopted for solving the governing partial differential energy equation in two independent variables in an asymptotic manner. Two temperature subfields are determined semianalytically: one for small x ( x → 0) and the other for large x ( x → ∞). Later, the two asymptotic mean Nusselt number subdistributions, Nu X→0( x) and Nu X→∞( x), blend themselves into a generalized correlation equation for the mean Nusselt number distribution Nu ( x) covering the entire x‐domain. The simplicity of the MOL procedure, combined with the high quality asymptotic mean Nusselt number subdistributions, provides an alternative methodology for solving the Graetz‐Nusselt problem without using higher level mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004. 相似文献
5.
In his long and illuminating paper [1] Joe Barback defined and showed to be non‐vacuous a class of infinite regressive isols he has termed “complete y torre” (CT) isols. These particular isols a enjoy a property that Barback has since labelled combinatoriality. In [2], he provides a list of properties characterizing the combinatoria isols. In Section 2 of our paper, we extend this list of characterizations to include the fact that an infinite regressive isol X is combinatorial if and only if its associated Dekker semiring D ( X) satisfies all those Π 2 sentences of the anguage LN for isol theory that are true in the set ω of natural numbers. (Moreover, with X combinatorial, the interpretations in D( X)of the various function and relation symbols of LN via the “lifting ” to D( X) of their Σ 1 definitions in ω coincide with their interpretations via isolic extension.) We also note in Section 2 that Π 2( L)‐correctness, for semirings D( X), cannot be improved to Π 3( L)‐correctness, no matter how many additional properties we succeed in attaching to a combinatoria isol; there is a fixed ( L) sentence that blocks such extension. (Here L is the usual basic first‐order language for arithmetic.) In Section 3, we provide a proof of the existence of combinatorial isols that does not involve verification of the extremely strong properties that characterize Barback's CT isols. 相似文献
6.
Summary Let X be a smooth, projective variety defined over a number field and let CH 2 ( X) denote the Chow group of codimension two cycles modulo rational equivalence. We show that if the cohomology group H
2(X,O x) vanishes then the torsion subgroup of CH 2 ( X) is a finite group. This result covers all previous results in this direction. The hypothesis H
2(X,O x)=0 is used to lift line bundles.
Oblatum 17-IX-1990 相似文献
7.
In this paper, we study nonautonomous Cauchy problems ( NCP) { u̇(t) = A(t)u(t)u(s) = x ∈ X for a family of linear operators ( A( t)) t∈I on some Banach space X by means of evolution semigroups. In particular, we characterize “stability” in the so called “hyperbolic case” on the level of evolution semigroups and derive a product formula for the solutions of ( NCP). Moreover, in Section 4 we connect the “hyperbolic” and the “parabolic” case by showing, that integrals ∫ ts A(τ) dτ always define generators. This yields another product formula. 相似文献
8.
Let {( Xi, Ti): i ∈ I } be a family of compact spaces and let X be their Tychonoff product. ??( X) denotes the family of all basic non‐trivial closed subsets of X and ?? R( X) denotes the family of all closed subsets H = V × Π Xi of X, where V is a non‐trivial closed subset of Π Xi and QH is a finite non‐empty subset of I. We show: (i) Every filterbase ?? ? ?? R( X) extends to a ?? R( X)‐ultrafilter ? if and only if every family H ? ??( X) with the finite intersection property (fip for abbreviation) extends to a maximal ??( X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ?? R( X) extends to a ?? R( X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {( Xi, Ti): i ∈ I } of compact spaces, every filterbase ?? ? ?? R( Y), Y = Π i ∈IYi, extends to a ?? R( Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {( Xi, Ti): i ∈ ω } of compact spaces, every countable filterbase ?? ? ?? R( X), X = Π i ∈ωXi, extends to a ?? R( X)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {( Xi, Ti): i ∈ ω } of compact topological spaces, every countable family ?? ? ??( X) with the fip extends to a maximal ??( X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Let ( X1, Y1), ( X2, Y2), …, be two-dimensional random vectors which are independent and distributed as ( X, Y). For 0< p<1, let ξ( px) be the conditional pth quantile of Ygiven X= x; that is, ξ( px)=inf{ y : P( YyX= x) p}. We consider the problem of estimating ξ( px) from the data ( X1, Y1), ( X2, Y2), …, ( Xn, Yn). In this paper, a new kernel estimator of ξ( px) is proposed. The asymptotic normality and a law of the iterated logarithm are obtained. 相似文献
10.
Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1 , and their (2 k+ 1) th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in ( 1 ) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [ 1 ]. Various other nonlinear operator and numerical methods for ( 1 ) are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in ( 1 ), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values { p= pl( n)} l≥0 of the absorption exponents is discussed. 相似文献
11.
In this paper we relate the generator property of an operator A with (abstract) generalized Wentzell boundary conditions on a Banach space X and its associated (abstract) Dirichlet‐to‐Neumann operator N acting on a “boundary” space . Our approach is based on similarity transformations and perturbation arguments and allows to split A into an operator A00 with Dirichlet‐type boundary conditions on a space X0 of states having “zero trace” and the operator N. If A00 generates an analytic semigroup, we obtain under a weak Hille–Yosida type condition that A generates an analytic semigroup on X if and only if N does so on . Here we assume that the (abstract) “trace” operator is bounded that is typically satisfied if X is a space of continuous functions. Concrete applications are made to various second order differential operators. 相似文献
12.
Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M( x ) of limit measures of x . It is defined as the set of accumulation points of the sequence of the discrete measures induced by x . Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = ( xn) n∈ℕ ∈ Xℕ and every a I there corresponds a subsequence denoted by a x . We investigate the set M( a x ) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension. 相似文献
13.
Let G be a closed subgroup of S∞ and X be a Polish G ‐space. To every x ∈ X we associate an admissible set A x and show how questions about X which involve Baire category can be formalized in A x . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Let X be a Banach space. We show that each m : ? \ {0} → L ( X ) satisfying the Mikhlin condition sup x ≠0(‖ m ( x )‖ + ‖ xm ′( x )‖) < ∞ defines a Fourier multiplier on B s p,q (?; X ) if and only if 1 < p < ∞ and X is isomorphic to a Hilbert space; each bounded measurable function m : ? → L ( X ) having a uniformly bounded variation on dyadic intervals defines a Fourier multiplier on B s p,q (?; X ) if and only if 1 < p < ∞ and X is a UMD space. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
15.
A metric space X = U
i-0
x
X is constructed such that X
o={ x
o} consists of a single point x
o
, X
i
, i=0, 1, 2, … are disjoint and closed, X
i
, i=1, 2, … are open, ind X
i
=0 for i=0, 1, … and ind X=1. The above space (proved to be, in some sense, most simple) shows also that the dimension ind of a metric space can be
raised by adjoining of a single point, a fact proved recently by E.K. Van Douwen and by T. Przymusiński. Some maximality property
of the family { X; Ind X=0} is proved and conditions implying P-ind= P-Ind are given.
This is part of a research thesis at the Technion, Israel Institute of Technology, towards an M.Sc. degree, directed by Professor
M. Reichaw. 相似文献
16.
By using Bernstein‐type inequality we define analogs of spaces of entire functions of exponential type in Lp ( X), 1 ≤ p ≤ ∞, where X is a symmetric space of non‐compact. We give estimates of Lp ‐norms, 1 ≤ p ≤ ∞, of such functions (the Nikolskii‐type inequalities) and also prove the Lp ‐Plancherel–Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of “measurements” in a stable way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
Let X be a closed subset of a topological space F; let a(·) be a continuous map from X into X; let { x
i} be a sequence generated iteratively by a(·) from x
0 in X, i.e., x
i+1 = a( x
i), i=0, 1, 2, ...; and let Q( x
0) be the cluster point set of { x
i}. In this paper, we prove that, if there exists a point z in Q( x
0) such that (i) z is isolated with respect to Q( x
0), (ii) z is a periodic point of a(·) of period p, and (iii) z possesses a sequentially compact neighborhood, then (iv) Q( x
0) contains p points, (v) the sequence { x
i} is contained in a sequentially compact set, and (vi) every point in Q( x
0) possesses properties (i) and (ii). The application of the preceding results to the case F= E
n leads to the following: (vii) if Q( x
0) contains one and only one point, then { x
i} converges; (viii) if Q( x
0) contains a finite number of points, then { x
i} is bounded; and (ix) if Q( x
0) contains p points, then every point in Q( x
0) is a periodic point of a(·) of period p. 相似文献
18.
We will prove that some so‐called union theorems (see [2]) are equivalent in ZF 0 to statements about the transitive closure of relations. The special case of “bounded” union theorems dealing with κ‐hereditary sets yields equivalents to statements about the transitive closure of κ‐narrow relations. The instance κ = ω1 (i. e., hereditarily countable sets) yields an equivalent to Howard‐Rubin's Form 172 (the transitive closure Tc( x) of every hereditarily countable set x is countable). In particular, the countable union theorem (Howard‐Rubin's Form 31) and, a fortiori, the axiom of countable choice imply Form 172. 相似文献
19.
In this paper we consider nonlinear ill-posed problems F( x) = y 0, where x and y 0 are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces. 相似文献
20.
A a set-valued optimization problem min
C
F( x), x ∈ X
0, is considered, where X
0 ⊂ X, X and Y are normed spaces, F: X
0 ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs ( x
0, y
0), y
0 ∈ F( x
0), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between
p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini
directional derivative first order necessary conditions and sufficient conditions a pair ( x
0, y
0) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done. 相似文献
|