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1.
For any positive real numbers A, B, and d satisfying the conditions
, d>2, we construct a Gabor orthonormal basis for L2(ℝ), such that the generating function g∈L2(ℝ) satisfies the condition:∫ℝ|g(x)|2(1+|x|
A
)/log
d
(2+|x|)dx < ∞ and
. 相似文献
2.
The wave equation, ∂
tt
u=Δu, in ℝ
n+1, considered with initial data u(x,0)=f∈H
s
(ℝ
n
) and u’(x,0)=0, has a solution which we denote by . We give almost sharp conditions under which and are bounded from H
s
(ℝ
n
) to L
q
(ℝ
n
). 相似文献
3.
Linghai ZHANG 《数学年刊B辑(英文版)》2008,29(2):179-198
Let u=u(x,t,uo)represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt+δux+γHuxx+βuxxx+f(u)x=αuxx,u(x,0)=uo(x), whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f(0)=0,|f(u)|→∞as |u|→∞,and f∈C^1(R),and there exist the following limits Lo=lim sup/u→o f(u)/u^3 and L∞=lim sup/u→∞ f(u)/u^5 Suppose that the initial function u0∈L^I(R)∩H^2(R).By using energy estimates,Fourier transform,Plancherel's identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv(xxt)+δvx+γHv(xx)+βv(xxx)=αv(xx),v(x,0)=vo(x), the author justifies the following limits(with sharp rates of decay) lim t→∞[(1+t)^(m+1/2)∫|uxm(x,t)|^2dx]=1/2π(π/2α)^(1/2)m!!/(4α)^m[∫R uo(x)dx]^2, if∫R uo(x)dx≠0, where 0!!=1,1!!=1 and m!!=1·3…(2m-3)…(2m-1).Moreover lim t→∞[(1+t)^(m+3/2)∫R|uxm(x,t)|^2dx]=1/2π(x/2α)^(1/2)(m+1)!!/(4α)^(m+1)[∫Rρo(x)dx]^2, if the initial function uo(x)=ρo′(x),for some functionρo∈C^1(R)∩L^1(R)and∫Rρo(x)dx≠0. 相似文献
4.
In this article, we use a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality associated to a para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type $\dot{F}^{\alpha,q}_{b,p}In this article, we use a discrete Calderón-type reproducing formula and Plancherel-P?lya-type inequality associated to a
para-accretive function to characterize the Triebel-Lizorkin spaces of para-accretive type
, which reduces to the classical Triebel-Lizorkin spaces when the para-accretive function is constant. Moreover, we give a
necessary and sufficient condition for the
boundedness of paraproduct operators. From this, we show that a generalized singular integral operator T with M
b
TM
b
∈WBP is bounded from
to
if and only if
and T
*
b=0 for
, where ε is the regularity exponent of the kernel of T.
Chin-Cheng Lin supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-008-021-MY3.
Kunchuan Wang supported by National Science Council, Republic of China under Grant #NSC 97-2115-M-259-009 and NCU Center for
Mathematics and Theoretic Physics. 相似文献
5.
Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several
conditions on u
0 to prove the existence of a unique strong solution with u(0) = u
0 in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to
prove the following optimal result with the weakest possible initial value condition and the largest possible solution class:
Given u
0, q, s as above and the Stokes operator A
2, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions. 相似文献
6.
This article mainly concerns retracts in polydisk, analytic varieties with the H
∞-extension property and the three-point Pick problem on
. Arising in the study of Nevanlinna-Pick interpolation on the bidisk, Agler and McCarthy recently discovered a remarkable
theorem which characterizes subsets in the bidisk with the polynomial extension property, and in this case, these subsets
are retracts. To study H
∞-extensions of holomorphic functions from subvarieties of polydisk, one naturally is concerned with retracts in polydisk.
Under certain mild assumptions, it is shown that subvarieties with H
∞-extension property are exactly retracts. Furthermore, we apply our argument to determine those retracts whose retractions
are unique. In particular, a retract in
having at least two different retractions is exactly a balanced disk. As an application, we give a sufficient condition of
the uniqueness of the solution for the three-point Pick problem on
.
相似文献
7.
Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums
, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form
, where
is a continuous function with
,
runs over
, the set of Farey fractions of order Q in the unit interval [0,1] and
are consecutive elements of
. We show that the limit lim
Q→∞
A
h
(Q) exists and is independent of h. 相似文献
8.
Reinhard Farwig Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321
Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain
W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u
0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u
0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ¥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with
2 < s < ¥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math
J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g.
u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls¢(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy
\frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}. 相似文献
9.
Alina Sîntămărian 《Numerical Algorithms》2007,46(2):141-151
The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).
相似文献
10.
A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor
frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional
Gabor frame multipliers. We prove that a L∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular
and
is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which
there is a function ∈ L∞(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame. 相似文献
11.
We determine the minimum length n
q
(k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n
q
(k, d) = g
q
(k, d) + 1 for when k is odd, for when k is even, and for .
This work was supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD). (KRF-2005-214-C00175).
This research has been partially supported by Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science
under Contract Number 17540129. 相似文献
12.
We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S
2 of the form where u is the polar angle on the sphere, is the ground state harmonic map, λ(t)=t
-1-ν, and is a radiative error with local energy going to zero as t→0. The number can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative
analysis.
Mathematics Subject Classification (1991) 35L05, 35Q75, 35P25 相似文献
13.
Let H be a locally compact group and K be a locally compact abelian group. Also let G=H×
τ
K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L
2(K)) on G defined by
is called the quasi regular representation. The properties of this representation in the case K=(ℝ
n
,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and
extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i)
, and (ii) the stabilizers H
ω
are compact for a.e.
; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.
相似文献
14.
In this paper we establish results on the existence of nontangential limits for weighted
-harmonic functions in the weighted Sobolev space
, for some q>1 and w in the Muckenhoupt A
q
class, where
is the unit ball in
. These results generalize the ones in Sect. 3 of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996, where the weight was identically equal to one. Weighted
-harmonic functions are weak solutions of the partial differential equation
where
for some fixed q∈(1,∞), where 0<α≤β<∞, and w(x) is a q-admissible weight as in Chap. 1 of Heinonen et al., Nonlinear Potential Theory, 2006.
Later, we apply these results to improve on results of Koskela et al., Trans. Am. Math. Soc. 348(2), 755–766, 1996 and Martio and Srebro, Math. Scand. 85, 49–70, 1999 on the existence of radial limits for bounded quasiregular mappings in the unit ball of
with some growth restriction on their multiplicity function.
相似文献
15.
P. M. Akhmet’ev 《Journal of Mathematical Sciences》2009,159(6):761-776
We present an approach to the Kervaire-invariant-one problem. The notion of the geometric (ℤ/2 ⨁ ℤ/2)-control of self-intersection of a skew-framed immersion and the notion of the (ℤ/2 ⨁ ℤ/4)-structure on the self-intersection manifold of a D
4-framed immersion are introduced. It is shown that a skew-framed immersion ↬ℝ
n
, 0 < q ≪ n (in the -range), admits a geometric (ℤ/2 ⨁ ℤ/2)-control if the characteristic class of the skew-framing of this immersion admits a retraction of order q, i.e., there exists a mapping such that this composition → ℝP∞ is the characteristic class of the skew-framing of f. Using the notion of (ℤ/2 ⨁ ℤ/2)-control, we prove that for a sufficiently large n, n = 2
l
− 2, an arbitrarily immersed D
4-framed manifold admits in the regular cobordism class (modulo odd torsion) an immersion with a (ℤ/2 ⨁ ℤ/4)-structure.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 17–41, 2007. 相似文献
16.
A central limit theorem for convex sets 总被引:4,自引:1,他引:3
B. Klartag 《Inventiones Mathematicae》2007,168(1):91-131
We show that there exists a sequence for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that
where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝn. Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the
identity matrix, we may assert that most unit vectors θ satisfy (*), with t0=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets. 相似文献
17.
We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval (?1,1) is concave on $(-\frac{1}{2},\frac{1}{2})We prove that the ground-state eigenfunction for symmetric stable processes of order α∈(0,2) killed upon leaving the interval
(−1,1) is concave on
. We call this property “mid-concavity”. A similar statement holds for rectangles in ℝd, d>1. These result follow from similar results for finite-dimensional distributions of Brownian motion and subordination.
Mathematics Subject Classification (2000) 30C45.
Rodrigo Ba?uelos: R. Ba?uelos was supported in part by NSF grant # 9700585-DMS.
Tadeusz Kulczycki: T. Kulczycki was supported by KBN grant 2 P03A 041 22 and RTN Harmonic Analysis and Related Problems, contract
HPRN-CT-2001-00273-HARP. 相似文献
18.
Partha Guha 《Acta Appl Math》2007,95(1):1-30
We consider the action of vector field Vect(S
1) on the space of an sl
n
- opers on S
1, i.e., a space of nth order differential operator . This action takes the sections of Ω
–(n–1)/2 to those of Ω
(n+1)/2, where Ω is the cotangent bundle on S
1. In this paper we study Euler–Poincaré (EP) flows on the space of sl
n
opers, in particular, we demonstrate explicitly EP flows on the space of third and fourth order differential operators (or
sl
3 and sl
4 opers) and its relation to Drienfeld–Sokolov, Hirota–Satsuma and other coupled KdV type systems. We also discuss the Boussinesq
equation associated with the third order operator. The solutions of the sl
n
oper defines an immersion in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to Δ
(n), we study the factorization of higher order operators and its compatibility with the action of Vect(S
1). We obtain the generalized Miura transformation and its connection to the modified Boussinesq equation for sl
3 oper. We also study the eigenvalue problem associated to sl
4 oper. We discuss flows on the special higher order differential operators for all u
i
= f(u,u
x
,u
xx
⋯) and its connection to KdV equation. Finally we explore a relation between projective vector field equation and generalized
Riccati equations.
相似文献
19.
In this paper, the boundedness of Toeplitz operator T
b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε
(ℝn) is discussed from L
p(ℝn) to L
q(ℝn),
, and from L
p(ℝn) to Triebel-Lizorkin space
. We also obtain the boundedness of generalized Toeplitz operator Θ
α0
b
from L
p(ℝn) to L
q(ℝn),
. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator
T
b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L
p(ℝn), 1 < p < ∞. 相似文献
20.
Yanping Chen Bolin Ma 《分析论及其应用》2007,23(2):112-128
Let (→b)=(b1,…,bm),bi∈Λβi(Rn),1≤i≤m,0<βi<β,0<β<1,[(→b),T]f(x)=∫Rn,(b1(x)-b1(y))…(bm(x)-bm(y)))K(x-y)f(y)dy where K is a Calder(o)n-Zygmund kernel.In this paper,we show that[(→b),T] is bounded from Lp (Rn) to Fβ,∞p(Rn),as well as[(→b,Iα)] from Lp(Rn) to Fβ,∞p(Rn),where 1/q=1/p-α/n. 相似文献