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1.
We prove that ifX is a Polish space andF a face ofP(X) with the Baire property, thenF is either a meager or a co-meager subset ofP(X). As a consequence we show that for every abelian Polish groupX and every analytic Haar-null set Λ⊆X, the set of test measuresT(Λ) of Λ is either meager or co-meager. We characterize the non-locally-compact groups as the ones for which there exists
a closed Haar-null setF⊆X withT(F) meager, Moreover, we answer negatively a question of J. Mycielski by showing that for every non-locally-compact abelian Polish
group and every σ-compact subgroupG ofX there exists aG-invariantF
σ subset ofX which is neither prevalent nor Haar-null.
Research supported by a grant of EPEAEK program “Pythagoras”. 相似文献
2.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function
X ? *\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}}
. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient
Mw(X)/M0(X){\cal M}_{\omega}(X)/{\cal M}_0(X)
, for certain external subspaces
M0(X), Mw(X){\cal M}_0(X), {\cal M}_{\omega}(X)
of the hyperfinite dimensional Banach space
*\Bbb CX{}{^{\ast}{\Bbb C}}^X
, with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and
Mw(G)/M0(G){\cal M}_{\omega}(G)/{\cal M}_0(G)
are isometrically isomorphic as Banach algebras. 相似文献
3.
Alexander Šostak 《Journal of Mathematical Sciences》2007,144(5):4527-4534
Given an M-valued equality E: X×X → M on a set X, we extend it to the M-valued equality ε: L
X
× L
X
→ M on the L-powerset L
X
of X, where L is a complete sublattice of a GL-monoid M. As a result, we come to a category SET(M,L) whose objects are quadruples (X,E,L
X
, ε). This category serves as a ground category for the category L-TOP(M) of (L,M)-valued topological spaces and some of its subcategories, which are the main subject of this paper. In particular, as special
cases, we obtain here Chang-Goguen, Lowen, Kubiak-Šostak, and some other known categories related to fuzzy topology.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 237–247, 2005. 相似文献
4.
If X is a compact-covering image of a closed subspace of product of a σ-compact Polish space and a compact space, then Ck(X,M), the space of continuous maps of X into M with the compact-open topology, is stratifiable for any metric space M.If X is σ-compact Polish, K is compact and M metric then every point of Ck(X×K,M) has a closure-preserving local base, and hence this function space is M1. 相似文献
5.
SUDARSHAN RAJENDRA GURJAR 《Proceedings Mathematical Sciences》2011,121(2):165-170
Let X be a normal projective variety defined over an algebraically closed field k. Let |O
X
(1)| be a very ample invertible sheaf on X. Let G be an affine algebraic group defined over k. Let E
G
and F
G
be two principal G-bundles on X. Then there exists an integer n > > 0 (depending on E
G
and F
G
) such that if the restrictions of E
G
and F
G
to a curve C ∈ |O
X
(n)| are isomorphic, then they are isomorphic on all of X. 相似文献
6.
Let X be a locally compact topological space and (X, E, Xω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set Xω ⊆ X, such that all internal subsets of Xω are relatively compact in the induced topology and X is homeomorphic to the quotient Xω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function
. The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient
, for certain external subspaces
of the hyperfinite dimensional Banach space
, with the norm ‖f‖1 = ∑x ∈ X |f(x)|. If additionally X = G is a hyperfinite group, Xω = Gω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G0 of Gω, and G is isomorphic to the locally compact group Gω/G0, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and
are isometrically isomorphic as Banach algebras.
Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic. 相似文献
7.
Homogeneous graded metrics over split ℤ2-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul,
are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature
tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry:
It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the
Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with
κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only
over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesR
Δ′ (X,Y)2 = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection
Δ′ satisfies a specific set of equations. 0 Finally,
graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics
on Ω(M) are also discussed.
Partially supported by DGICYT grants #PB94-0972, and SAB94-0311; IVEI grant 95-031; CONACyT grant #3189-E9307. 相似文献
8.
Diana Caponetti 《Rendiconti del Circolo Matematico di Palermo》1991,40(2):241-250
LetE′ be the separable dual of a Banach spaceE. LetK be the class of all non-empty convex, weak*-compact subsets ofE′. In this paper we prove that Edgar’s inequality, given in [2], extends to
adapted sequences ofK-valued random variables. 相似文献
9.
In this note we prove that the game chromatic index χ
g
′(G) of a graph G of arboricity k is at most Δ + 3k − 1. This improves a bound obtained by Cai and Zhu [J. Graph Theory 36 (2001), 144–155] for k-degenerate graphs.
Tomasz Bartnicki: Research of the first author is supported by a PhD grant from Polish Ministry of Science and Higher Education
N201 2128 33.
Received: November 1, 2006. Final version received: December 22, 2007. 相似文献
10.
We define and study a class of summable processes, called additive summable processes, that is larger than the class used
by Dinculeanu and Brooks [D-B].
We relax the definition of a summable processesX:Ω×ℝ+→E⊂L(F, G) by asking for the associated measureI
X to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (I
X)
z
, forz∈(L
G
p
)*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the
integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ+→E⊂L(F, G) with integrable semivariation ifc
0 ∋G. 相似文献
11.
For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re
jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R
R(RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕
i=1
n
Mi is δ-coatomic if and only if each M
i (i=1,…, n) is δ-coatomic. 相似文献
12.
C. T. Tucker 《Periodica Mathematica Hungarica》1984,15(4):253-257
An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL.
A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism.
These properties were introduced by Schmidt in [5].
In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic
to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM.
An analogous theorem is derived for Riesz spaces with extension property (E2). 相似文献
13.
Elias Gabriel Minian 《Order》2010,27(2):213-224
The classical way to study a finite poset (X, ≤ ) using topology is by means of the simplicial complex Δ
X
of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84–95, 1952), we define the simplicial complexes K
X
and L
X
associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ
X
. We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K
X
and L
X
are topologically equivalent to the smaller complexes K′
X
, L′
X
induced by the relation <. More precisely, we prove that K
X
(resp. L
X
) simplicially collapses to K′
X
(resp. L′
X
). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y. 相似文献
14.
Let G be an outerplanar graph with maximum degree △. Let χ(G^2) and A(G) denote the chromatic number of the square and the L(2, 1)-labelling number of G, respectively. In this paper we prove the following results: (1) χ(G^2) = 7 if △= 6; (2) λ(G) ≤ △ +5 if △ ≥ 4, and ),(G)≤ 7 if △ = 3; and (3) there is an outerplanar graph G with △ = 4 such that )λ(G) = 7. These improve some known results on the distance two labelling of outerplanar graphs. 相似文献
15.
Joshua Rushton 《Journal of Theoretical Probability》2007,20(3):397-427
We establish a functional LIL for the maximal process M(t) :=sup 0≤s≤t
‖X(s)‖ of an ℝ
d
-valued α-stable Lévy process X, provided X(1) has density bounded away from zero over some neighborhood of the origin. We also provide a broad invariance result governing
a class independent-increment processes related to the domain of attraction of X(1). This breadth is particularly notable for two types of processes captured: First, it not only describes any partial sum
process built from iid summands in the domain of normal attraction of X(1), but also addresses those with arbitrary iid summands in the full domain of attraction (here we give a technical condition
necessary and sufficient for the partial sum process to share the exact LIL we prove for X). Second, it reveals that any Lévy process L such that L(1) satisfies the technical condition just mentioned will also share the LIL of X.
Supported in part by NSF Grant DMS 02-05034. 相似文献
16.
Haruko Okamura 《Graphs and Combinatorics》2005,21(4):503-514
Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {si, ti}⊆Xi⊆V(G) (i=1,2) and X1∩X2=∅. We here prove that if k is even and e(Xi)≤2k−1 (i=1,2), then there exist paths P1 and P2 such that Pi joins si and ti, V(Pi)⊆Xi (i=1,2) and G − E(P1∪P2) is (k−2)-edge-connected (for odd k, if e(X1)≤2k−2 and e(X2)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing
given edges. 相似文献
17.
R. Grant Woods 《Israel Journal of Mathematics》1971,9(2):250-256
We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 22ℵ0 pairwise disjoint, dense ℵ0-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ0-bounded. 相似文献
18.
Suppose X and Y are Polish spaces with non-atomic Borel probability measures μ and ν and suppose that T and S are ergodic measure-preserving homeomorphisms of (X, μ) and (Y, ν). Then there are invariant G
δ
subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a homeomorphism ϕ: X′ → Y′ which maps μ|X′ to ν|Y′ and maps T-orbits onto S-orbits. We also deal with the case where T and S preserve infinite invariant measures. 相似文献
19.
V. E. Slyusarchuk 《Ukrainian Mathematical Journal》1997,49(7):1089-1101
We establish conditions of asymptotic stability for all solutions of the equation X
n+1=F(X
n
), n≥0, in the Banach space E in the case where r(F′(x))<1 ∀ x ∈ E, r′(x) is the spectral radius of F′(x). An example of an equation with an unstable solution is given.
Ukrainian Academy of the Water Industry, Rovno. Translated from Ukrainskii Matematicheskii zhurnal, Vol. 49, No. 7, pp. 970–980.
July, 1997. 相似文献
20.
We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ : θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X(ω) by an increasing sequence of cylinder functions Lip(Ω) in order to prove that Cb(Ω) belongs to the completion of Lip(Ω) under the natural norm E[|·|]. 相似文献