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1.
Werner Georg Nowak 《Monatshefte für Mathematik》2002,137(3):227-238
This article is concerned with sums 𝒮(t) = ∑
n
ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C
4-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫
T−Λ
T+Λ
(𝒮(t))2dt for any Λ = Λ(T) growing faster than log T.
Received April 30, 2001; in revised form February 15, 2002
RID="a"
ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday 相似文献
2.
The scattering problem is studied, which is described by the equation (-Δ
x
+q(x,x/ɛ)−E)ψ = f(x), where ψ = ψ (x,ɛ) ∈ ℂ, x ℂ ℝ
d
, ɛ > 0, E > 0, the function q(x,y) is periodic with respect to y, and the function f is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior
as ɛ → O is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown
that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential
$
\hat q(x) = \frac{1}
{{\left| \Omega \right|}}\int_\Omega {q(x,y)dy} .
$
\hat q(x) = \frac{1}
{{\left| \Omega \right|}}\int_\Omega {q(x,y)dy} .
相似文献
3.
M. F. Gamal' 《Journal of Mathematical Sciences》1998,92(1):3589-3596
Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H∞. It is known that ϕ+BH∞ is a weak* generator of the algebra H∞/BH∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak* generator of the algebra l∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists
B=B1·…·BN, where N ∈ℕ, and B1, …, BN are Blaschke products satisfying condition (C), there exists a function ϕ∈H∞ such that ϕ(Λ) is a weak* generator of the algebra l∞, and ϕ+BH∞ is not a weak* generator of the algebra H∞/BH∞. Bibliography: 12 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85.
Translated by M. F. Gamal'. 相似文献
4.
F. A. Shamoyan 《Journal of Mathematical Sciences》2006,139(2):6491-6495
Let ϕ(r) = (ϕ1(r1), …, ϕn(rn)) be a vector-valued function on R
+
n
. A necessary and sufficient condition is obtained under which any function f ∈, H∞ (D
n
), f(z) ≠ 0, z ∈, D
n
, is cyclic in the corresponding weighted space Lp(ϕ), where D
n
is the unit polydisk in C
n. Bibliography: 13 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 226–234. 相似文献
5.
We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw
*-lower semicontinuous function ϕ defined on aw
*-compact convex setC in a dual Banach spaceX
* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw
*−H
σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets.
Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second
author. 相似文献
6.
T. V. Dudnikova 《Theoretical and Mathematical Physics》2011,169(3):1668-1682
We study the dynamics of lattice systems in ℤd, d ≥ 1. We assume that the initial data are random functions. We introduce the family of initial measures {μ0ɛ, ɛ > 0}. The measures μ0ɛ are assumed to be locally homogeneous or “slowly changing” under spatial shifts of the order o(ɛ−1
) and inhomogeneous under shifts of the order ɛ−1
. Moreover, correlations of the measures μ0ɛ decrease uniformly in ɛ at large distances. For all τ ∈ ℝ \ 0, r ∈ ℝd, and κ > 0, we consider distributions of a random solution at the instants t = τ/ɛκ at points close to [r/ɛ] ∈ ℤd. Our main goal is to study the asymptotic behavior of these distributions as ɛ → 0 and to derive the limit hydrodynamic equations of the Euler and Navier-Stokes type. 相似文献
7.
Robert S. Strichartz 《Journal of Geometric Analysis》1991,1(3):269-289
Let μ be a measure on ℝn that satisfies the estimate μ(B
r(x))≤cr
α for allx ∈ ℝn and allr ≤ 1 (B
r(x) denotes the ball of radius r centered atx. Let ϕ
j,k
(ɛ)
(x)=2
nj2ϕ(ɛ)(2
j
x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤn, and ∈ ∈E, a finite set, and letP
j
(T)=Σɛ,k
<T,ϕ
j,k
(ɛ)
>ϕ
j,k
(ɛ)
denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls
p(dμ) for 1 ≤p ≤ ∞, we show thatP
j(f dμ) ∈ Lp(dx) and ||P
j
(fdμ)||L
p(dx)≤c2
j((n-α)/p′))||f||L
p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P
j
(fdμ)||L
p(dx) under more restrictive hypotheses.
Communicated by Guido Weiss 相似文献
8.
Summary. We study the 2D Ising model in a rectangular box Λ
L
of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑
t∈ΛL
σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m
* is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using
the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature
representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation.
We then study the Gibbs measure conditioned by {|∑
t∈ΛL
σ(t) −m|Λ
L
||≤|Λ
L
|L
−
c
}, with 0<c<1/4 and −m
*<m<m
*. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric
type.
Received: 17 October 1996 / In revised form: 7 March 1997 相似文献
9.
We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume
that for any submodular function f: ?→R on a distributive lattice ?⊆2
V
with ?,V∈? and f(?)=0 and for any vector x∈R
V
where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z
1,Z
2,···,Z
k
of V such that f(Z
1)>f(Z
2)>···>f(Z
k
)=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient
membership algorithms.
Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001 相似文献
10.
In this paper, a natural R
+
n+1
extension of singular integrals, i.e.,T
κ:f→K*φ
t
*t with K a standard C-Z kernel and ϕ usual one, is investigated. One of the main results is: Let (dμ, udx) ∈C1 and u-Mw, w∈A∞, then Tk is of type (Lp(udx), Lp(dμ)). As a related topic, a maximal operator
is proved to be of type
, where
, provided (dμ, udx) ∈C1 and u∈ A∞.
Supported by National Science Foundation of China 相似文献
11.
We prove that a convex functionf ∈ L
p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω
3
ϕ
(f,1/n)p where ω
3
ϕ
(f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω
3
ϕ
(f,1/n)p, and the impossibility of having such estimates involving ω4. We also give similar estimates for the approximation off by convexC
0 andC
1 piecewise quadratics as well as convexC
2 piecewise cubic polynomials.
Communicated by Dietrich Braess 相似文献
12.
Suppose that(T
t
)t>0 is aC
0 semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ
−1(Jx)={x}, whereJ is the duality mapping fromX toP(X
*). If |<T
t
x,f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible.
If the hypothesisJ
−1(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ
−1(Jx) is weakly compact, then if |<T
t
x, f>|→1, whent→+∞ for somef∈X
*, ‖f‖≤1, there existsy∈J
−1(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ1 orL
1. 相似文献
13.
Summary Let Γ=〈g
1〉*〈g
2〉*...*〈g
n
〉*... be a free product of cyclic groups with generators {g
i
}, andC
r
*
(Γ,℘
Λ) be the C*-algebra generated by the reduced group C*-algebraC
r
*
Γ and a set of projectionsP
gL associated with a subset Λ of {g
i
}. We prove the following: (1)C
r
*
(Γ,℘
Λ) is *-isomorphic to the reduced cross product
for certain Hausdorff compact spaceX
Λ constructed from Γ and its boundary ∂Γ. (2)C
r
*
(Γ,℘
Λ) is either a purely infinite, simple C*-algebra or an extension of a purely infinite, simple C*-altebra, depending on the
pair (Γ, Λ). (3)C
r
*
(Г,℘
Λ) is nuclear if and only if the subgroup ΓΛ generated by {g
i
}/Λ is amenable.
Partially supported by RMC grant 45/290/603 from the University of Newcastle
Partially supported by NSF grant DMS-9225076 and a Taft travel grant from the University of Cincinnati 相似文献
14.
Résumé Soit X un processus gaussien stationnaire non dérivable. Nous étudions le nombre de passages en zéro du processus régularisé par
convolution. Sous des hypothèses peu restrictives sur X, cette variable convenablement normalisée, converge au sens de L
2 quand la taille du filtre tend vers zéro. Lorsque X admet un temps local continu, la limite obtenue est le temps local.
Summary Let {X(t)} be a stationary non differentiable Gaussian process and let ϕɛ(u=ɛ−1 ϕ(u/ɛ) be an approximate identity. Setting X ɛ(t)=X*ϕɛ(t) and letting N ɛ(T) be the number of zeros of X ɛ in the interval [0, T] it is shown that under weak technical conditions there are constants C(ɛ) so that C(ɛ) N ɛ(T) converges in L 2 as ɛ→0. When X admits a continuous local time, the limit is the local time L(0, T) at zero of X(t).相似文献 15.
A. M. Vershik 《Journal of Mathematical Sciences》2011,176(1):1-6
The paper studies the region of values of the system {c
2, c
3, f(z
1), f′(z
1)},where z
1 is an arbitrary fixed point of the disk |z| < 1; f ∈ T,and the class T consists of all the functions f(z) = z + c
2
z
2 + c
3z3 + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z
1) in the subclass of functions f ∈ T with prescribed values c
2, c
3, and f(z
1) is determined. Bibliography: 10 titles. 相似文献
16.
W. M. Ruess 《Semigroup Forum》1995,51(1):335-341
For aC
0-contraction semigroup (S(t))
t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e
iλt
x, t≥0, for some λ∈ℝ, provided lim
t→∞
|<S(t)x,x
*
>|=|<x,x
*
>| for allx
*∈X*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context. 相似文献
17.
Sandra Cerrai 《Probability Theory and Related Fields》1999,113(1):85-114
In the present paper we consider the transition semigroup P
t
related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in
the Banach space of continuous functions , where ⊂ℝ
d
is a bounded open set. In L
2() the only result proved is the strong Feller property, for d=1. Here we are able to prove that if f∈C
∞(ℝ) and d≤3, then for any and t>0. An important application is to the study of the ergodic properties of the system. These results are also of interest for
some problem in stochastic control.
Received: 20 August 1997 / Revised version: 27 May 1998 相似文献
18.
Zhen-long Chen San-yang Liu 《应用数学学报(英文版)》2005,21(4):623-636
Let φ be a Hausdorff measure function and A be an infinite increasing sequence of positive integers. The Hausdorff-type measure φ - mA associated to φ and A is studied. Let X(t)(t ∈ R^N) be certain Gaussian random fields in R^d. We give the exact Hausdorff measure of the graph set GrX([0, 1]N), and evaluate the exact φ - mA measure of the image and graph set of X(t). A necessary and sufficient condition on the sequence A is given so that the usual Hausdorff measure function for X([0, 1] ^N) and GrX([0, 1]^N) are still the correct measure functions. If the sequence A increases faster, then some smaller measure functions will give positive and finite ( φ A)-Hausdorff measure for X([0, 1]^N) and GrX([0, 1]N). 相似文献
19.
T. Shulman 《Journal of Mathematical Sciences》2009,159(6):894-903
We consider maps defined on a real space Asa of all self-adjoint elements of a C*-algebra A commuting with the conjugation by unitaries: F(u* au) = u* F(a)u for any a ∈ A
sa, u ∈ (A). In the case where A is a full matrix algebra, there is a functional realization of these maps (in terms of multivariable functions) and analytical
properties of these maps can be expressed in terms of corresponding functions. In the present work, these results are generalized
to the class of uniformly hyperfinite C*-algebras and to the algebra of all compact operators in a Hilbert space.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 213–227, 2007. 相似文献
20.
M. Filali 《Semigroup Forum》1994,48(1):163-168
LetG be a discrete abelian group,Ĝ the character group ofG, andl
∞(G)* the conjugate ofl
∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l
∞(G)* such that γμ=0 whenever γ is an annihilator ofC
0(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl
∞(G)*, and each of them generates a left ideal ofl
∞(G)* that contains no minimal left ideal. 相似文献
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