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1.
We study self adjoint operators of the form?H
ω = H
0 + ∑λω(n) <δ
n
,·>δ
n
,?where the δ
n
’s are a family of orthonormal vectors and the λω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general
structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ
n
and δ
m
are not completely orthogonal, then the restrictions of H
ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of
such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must
be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases.
Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000 相似文献
2.
Constance van Eeden 《Annals of the Institute of Statistical Mathematics》1985,37(1):461-472
Summary Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX
1, …,X
n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh
n0 for the scale factorh
n=Kn−α is given byh
n0=K0n−1/5 withK
0 depending onf and the kernel; they also showed that MISE(h
n0)=O(n−4/5) and Epanechnikov [1] found the optimal kernel.
In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness
of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h
n0)=O(n−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h
n0)=O(n−1/2). For this last case the optimal kernel is shown to be the double exponential density.
Supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nr. A 3114 and by the Gouvernement
du Québec, Programme de formation de chercheurs et d'action concertée. 相似文献
3.
Irina Krasikova Miguel Martín Javier Merí Vladimir Mykhaylyuk Mikhail Popov 《Central European Journal of Mathematics》2009,7(4):683-693
It is known that there is a continuous linear functional on L
∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L
∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L
∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L
∞(μ) to c
0(Γ) is narrow while not every such an operator is AM-compact. 相似文献
4.
Idealization of a decomposition theorem 总被引:1,自引:1,他引:0
In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A
I-sets and A
I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A
I-continuous.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
5.
Nicolas Eisen 《Arkiv f?r Matematik》2012,50(1):69-87
We give a holomorphic extension result for continuous CR functions on a non-generic CR submanifold N of ℂ
n
to complex transversal wedges with edges containing N. We show that given any v∈ℂ
n
∖(T
p
N+iT
p
N), there exists a wedge of direction v whose edge contains a neighborhood of p in N, such that any continuous CR function defined locally near p extends holomorphically to that wedge. 相似文献
6.
Joel H. Shapiro 《Israel Journal of Mathematics》1978,29(2-3):248-264
LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL
E
P
(G), the closure inL
p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL
E
p
(G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem,
and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL
P (T), where T is the unit circle; and they show that theH
p spaces of the “big disc algebra” have one-dimensional dual. 相似文献
7.
Christopher Boyd 《Monatshefte für Mathematik》2000,18(4):177-188
We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ b locally Asplund we show that the space of n-homogeneous polynomials on (E′ b )′ b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. 相似文献
8.
Christopher Boyd 《Monatshefte für Mathematik》2000,130(3):177-188
We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′
b
locally Asplund we show that the space of n-homogeneous polynomials on (E′
b
)′
b
is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.
(Received 24 March 1999; in final form 14 February 2000) 相似文献
9.
Christophe Breuil 《Israel Journal of Mathematics》2011,182(1):349-382
Let ρ be a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ
p
/ℚ
p
) over ̅F
p
. The modulo p Langlands correspondence for GL2(ℚ
p
) defined in [5], as realized in [9], can be reformulated as a quite simple recipee giving back the (φ, Γ)-module of the dual of ρ starting from the “Diamond diagram” associated to ρ. Let F be a finite unramified extension of ℚ
p
and ρ a 2-dimensional continuous semi-simple generic representation of Gal(̅ℚ
p
/F) over ̅F
p
. When one formally extends this recipee to the Diamond diagrams associated to ρ in [6], we show that one essentially finds the (φ, Γ)-module of the tensor induction from F to ℚ
p
of the dual of ρ. 相似文献
10.
A. Yu. Pilipenko 《Ukrainian Mathematical Journal》2006,58(12):1891-1903
Let ϕt(x), x ∈ ℝ+ be a value taken at time t ≥ 0 by a solution of a stochastic equation with normal reflection from a hyperplane starting at initial time from x. We characterize the absolutely continuous (with respect to Lebesgue measure) component and the singular component of a stochastic
measure-valued process μt = μ ○ ϕ
t
−1
that is the image of a certain absolutely continuous measure μ under random mapping ϕt(·). We prove that the restriction of the Hausdorff measure H
d−1 to the support of the singular component is σ-finite and give sufficient conditions guaranteeing that the singular component
is absolutely continuous with respect to H
d−1.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1663–1673, December, 2006. 相似文献
11.
Paolo Albano 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(2):273-281
In an open bounded set Ω, we consider the distance function from ∂Ω associated to a Riemannian metric with C
1,1 coefficients. Assuming that Ω is convex near a boundary point x
0, we show that the distance function is differentiable at x
0 if and only if there exists the tangent space to ∂Ω at x
0. Furthermore, if the distance function is not differentiable at x
0 then there exists a Lipschitz continuous curve, with initial point at x
0, such that the distance function is not differentiable along such a curve.
相似文献
12.
13.
We prove that a Markov operatorT onL
1 has an invariant density if and only if there exists a densityf that satisfies lim sup
n→∞‖T
n
f − f‖ < 2. Using this result, we show that a Frobenius-Perron operatorP is mean ergodic if and only if there exists a densityw such that lim sup
n→∞ ‖P
n
f − w‖<2 for every densityf. Corresponding results hold for strongly continuous semigroups. 相似文献
14.
We consider the collision dynamics produced by three beads with masses (m
1, m
2, m
3) sliding without friction on a ring, where the masses are scaled so that m
1 = 1/ɛ, m
2 = 1, m
3 = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these
cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of
collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e.
0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics
are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses
from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ
2 ∼ αɛ
β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ
2 ∼ log(ɛ
β). 相似文献
15.
V. D. Zalizko 《Ukrainian Mathematical Journal》2007,59(1):28-44
The Jackson inequality
relates the value of the best uniform approximation E
n
(f) of a continuous 2π-periodic function f: ℝ → ℝ by trigonometric polynomials of degree ≤ n − 1 to its third modulus of continuity ω
3(f, t). In the present paper, we show that this inequality is true if continuous 2π-periodic functions that change their convexity
on [−π, π) only at every point of a fixed finite set consisting of an even number of points are approximated by polynomials
coconvex to them.
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 29–43, January, 2007. 相似文献
16.
E. A. Barabanov 《Differential Equations》2009,45(8):1087-1104
We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise
continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values
for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness
set of some family if and only if it is a G
δσ
-set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the
same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness
sets are the G
δσ
-sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness
set is an arbitrary open subset of the real line. 相似文献
17.
We show here that any two finite state irreducible Markov chains of the same entropy are finitarily Kakutani equivalent. By
this we mean they are orbit equivalent by an invertible measure preserving mapping that is almost continuous and monotone
in time when restricted to some cylinder set. Smorodinsky and Keane have shown that any two irreducible Markov chains of equal
entropy and period are finitarily isomorphic. Hence, all that is necessary to obtain our result is to show that for every
entropy h > 0 and period p ∈ ℕ there exists two irreducible Markov chains σ
1, σ
2 both of entropy h, where
(1) σ
1 is mixing
(2) ς
2 has period p and
(3) σ1 and σ
2 are finitarily Kakutani equivalent. 相似文献
18.
A. A. Murach 《Ukrainian Mathematical Journal》2009,61(3):467-477
We study a linear system of pseudodifferential equations uniformly elliptic in Petrovskii’s sense in the Hilbert scale of
H?rmander functional spaces defined in ℝ
n
. An a priori estimate is proved for the solution of the system and its interior smoothness in this scale of spaces is investigated.
As an application, we establish a sufficient condition for the existence of continuous bounded derivatives of the solution. 相似文献
19.
Dorin Bucur Ioan R. Ionescu 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2006,57(6):1042-1056
We consider an eigenvalue problem associated to the antiplane shearing on a system of collinear faults under a slip-dependent
friction law. Firstly we consider a periodic system of faults in the whole plane. We prove that the first eigenvalues/eigenfunctions
of different physical periodicity are all equal and that the other eigenvalues converge to this first common eigenvalue as
their physical period becomes indefinitely large. Secondly we consider a large scale fault system composed on a small scale
collinear faults periodically disposed. If β0* is the first eigenvalue of the periodic problem in the whole plane, we prove that the first eigenvalue of the microscopic
problem behaves as β0*/∈ when ∈→ 0 regardless the geometry of the domain (here ∈ is the scale quotient). The geophysical implications of this result
is that the macroscopic critical slip Dc scales with Dc∈/∈ (here Dc∈ is the small scale critical slip). 相似文献
20.
Let A and B be the generators of strongly continuous semigroups (S(t))
t \geq 0
and (T(t))
t \geq 0
, respectively. Denote by Δ (t) = T(t) -S(t) . We show that if Δ(t) is norm continuous for t>0 and R(λ,B)-R(λ,A) is compact for λ ∈ ρ(A)\cap ρ(B) , then Δ(t) is compact. The converse is true if the perturbing operator is of Miyadera-Voigt-type. A characterization of norm continuity
of Δ(t) in terms of the resolvents of the generators is given in Hilbert spaces. 相似文献