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1.
The Hausdorff Centred measure of the symmetry Cantor sets 总被引:1,自引:0,他引:1
Let 0<λ≤1/3,K (λ) be the attractor of an iterated function system { ψ1,ψ2 } on the line, where ψ1 (x ) =λx, ψ2(x)=1-λ+λx, x∈ [0,1]. We call K (λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K (λ). 相似文献
2.
Ruan Huojun Dai Meifeng Su Weiyi Dept. of Math. Zhejiang Univ. Hangzhou China. Dept. of Math. Jiangsu Univ. Zhenjiang China. Dept. of Math. Nanjing Univ. Nanjing China. 《高校应用数学学报(英文版)》2005,20(2):235-242
§ 1 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha… 相似文献
3.
A. F. Izé 《Annali di Matematica Pura ed Applicata》1973,96(1):21-39
Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(xt)=L(xt) and its perturbed equation(2) [(d/dx) D(xt)−G(t, xt)]= =L(xt)+F(t, xt) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying
where δ(t)=αd{F(t, ϕμ)+μG(t, ϕμ)+F(t, X0G(t, ϕμ))}, ϕμ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X0 are given constants.
Entrata in Redazione il 5 gennaio 1972. 相似文献
4.
Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x
k
): (ϕk(|x
k
|)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψk), we characterize the continuity of the superposition operators P
f
from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications
we consider superposition operators on some multiplier sequence spaces of Maddox type.
This research was supported by Estonian Science Foundation Grant 5376. 相似文献
5.
Extremal probabilities for Gaussian quadratic forms 总被引:1,自引:0,他引:1
Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf
Q
P(Q≤x)=P(χ2
n
/n≤x), where χ
n
2
is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z
1
,Z
2
are independent standard Gaussian rv's, then sup0≤λ≤1/2
P{λZ
1
2
+(1−λ)Z
2
2
≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic
quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.
Received: 24 June 2002 / Revised version: 26 January 2003
Published online: 15 April 2003
Research supported by NSA Grant MDA904-02-1-0091
Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10 相似文献
6.
Let (S)⊄L
2(S′(∔),μ)⊄(S)* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e
n
,n>-0) be the ONB ofL
2(∔) consisting of the eigenfunctions of the s.a. operator
. In this paper the Euler operator Δ
E
is defined as the sum
, where ∂
i
stands for the differential operatorD
e
i. It is shown that Δ
E
is the infinitesimal generator of the semigroup (T
t
), where (T
t
ϕ)(x)=ϕ(e
t
x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation:
Δ
Eϕ
=λϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out.
Supported by the National Natural Science Foundation of China. 相似文献
7.
Jürgen Weyer 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1983,34(5):728-745
Summary We study positive solutions of the nonlinear eigenvalue problemF(u)=G(u) with some monotone operatorsF andG. In particular, we consider the case of nonlinear elliptic differential equations of second order and chooseF(u)=–divA(x, gradu)+b(x,u) and G(u)=g (x,u). Positive solutions are obtained by the Picard iterationsu
0=0 andF(u
n+1)=G(u
n
).In order to get convergence of the sequenceu
n
,one has to study some comparison principles for the operatorF. Finally, the Picard iteration scheme allows a-priori estimates and bifurcation results for the admissible eigenvalue parameter.
Supported by the Deutscher Akademischer Austauschdienst (DAAD) and DICYT-University of Santiago de Chile. 相似文献
Zusammenfassung Für gewisse monotone OperatorenF andG untersuchen wir positive Lösungen des nichtlinearen EigenwertproblemsF(u)=G(u). Insbesondere betrachten wir nichtlineare elliptische Differentialgleichungen zweiter Ordnung und wählenF(u)=–divA(x, gradu)+b(x,u) sowieG(u)=g(x,u). Man erhält positive Lösungen durch das Picard-Iterationsverfahrenu 0=0 undF(u n+1)=G(u n ).Um die Konvergenz der Folgeu n nachzuweisen, benötigt man Vergleichsprinzipien fürF. Dann gestattet das Iterationsschema sogar a-priori Abschätzungen und Verzweigungsaussagen für die zulässigen Eigenwertparameter.
Supported by the Deutscher Akademischer Austauschdienst (DAAD) and DICYT-University of Santiago de Chile. 相似文献
8.
Mikhail A. Chebotar Wen-Fong Ke Pjek-Hwee Lee Ruibin Zhang 《Monatshefte für Mathematik》2006,162(1):91-101
Let R be a ring, A = M
n
(R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains
\frac12\frac{1}{2}
and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. 相似文献
9.
Mikhail A. Chebotar Wen-Fong Ke Pjek-Hwee Lee Ruibin Zhang 《Monatshefte für Mathematik》2006,149(2):91-101
Let R be a ring, A = M
n
(R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains
and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism.
The third author is Corresponding author. 相似文献
10.
We consider the existence and uniqueness of singular solutions for equations of the formu
1=div(|Du|p−2
Du)-φu), with initial datau(x, 0)=0 forx⇑0. The function ϕ is a nondecreasing real function such that ϕ(0)=0 andp>2.
Under a growth condition on ϕ(u) asu→∞, (H1), we prove that for everyc>0 there exists a singular solution such thatu(x, t)→cδ(x) ast→0. This solution is unique and is called a fundamental solution. Under additional conditions, (H2) and (H3), we show the
existence of very singular solutions, i.e. singular solutions such that ∫|x|≤r
u(x,t)dx→∞ ast→0. Finally, for functions ϕ which behave like a power for largeu we prove that the very singular solution is unique. This is our main result.
In the case ϕ(u)=u
q, 1≤q, there are fundamental solutions forq<p*=p-1+(p/N) and very singular solutions forp-1<q<p*. These ranges are optimal.
Dedicated to Professor Shmuel Agmon 相似文献
11.
The product ϕ
λ
(α,β)
(t1)ϕ
λ
(α,β)
(t2) of two Jacobi functions is expressed as an integral in terms of ϕ
λ
(α,β)
(t3) with explicit non-negative kernel, when α≧β≧−1/2. The resulting convolution structure for Jacobi function expansions is
studied. For special values of α and β the results are known from the theory of symmetric spaces. 相似文献
12.
Jorge J. Betancor Juan C. Fariña Teresa Martinez Lourdes Rodríguez-Mesa 《Arkiv f?r Matematik》2008,46(2):219-250
In this paper we investigate Riesz transforms R
μ
(k) of order k≥1 related to the Bessel operator Δμ
f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We
obtain that for every k≥1, R
μ
(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x
2μ+1
dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R
μ
(k) maps L
p
(ω) into itself and L
1(ω) into L
1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman. 相似文献
13.
Let X and Y be Polish spaces with non-atomic Borel measures μ and ν of full support. Suppose that T and S are ergodic non-singular homeomorphisms of (X, μ) and (Y, ν) with continuous Radon-Nikodym derivatives. Suppose that either they are both of type III
1 or that they are both of type III
λ, 0 < λ < 1 and, in the III
λ case, suppose in addition that both ‘topological asymptotic ranges’ (defined in the article) are log λ · ℤ. Then there exist invariant dense G
δ-subsets X′ ⊂ X and Y′ ⊂ Y of full measure and a non-singular homeomorphism ϕ: X′ → Y′ which is an orbit equivalence between T|
X′ and S|
Y′, that is ϕ{T
i
x} = {S
i
ϕx} for all x ∈ X′. Moreover, the Radon-Nikodym derivative dν ∘ ϕ/dμ is continuous on X′ and, letting S′ = ϕ
−1
Sϕ, we have T
x
= S′
n(x)
x and S′x = T
m(x)
x where n and m are continuous on X′. 相似文献
14.
Patrick S. Hagan 《Studies in Applied Mathematics》1981,64(1):57-88
We consider the class of equations ut=f(uxx, ux, u) under the restriction that for all a,b,c. We first consider this equation over the unbounded domain ? ∞ < x < + ∞, and we show that very nearly every bounded nonmonotonic solution of the form u(t, x)=?(x?ct) is unstable to all nonnegative and all nonpositive perturbations. We then extend these results to nonmonotonic plane wave solutions u(t, x, y)=?(x?ct) of ut = F(uxx, uxy, ux, uy, u). Finally, we consider the class of equations ut=f(uxx, ux, u) over the bounded domain 0 < x < 1 with the boundary conditions u(t, x)=A at x=0 and u(t, x)=B at x=1, and we find the stability of all steady solutions u(t, x)=?(x). 相似文献
15.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
16.
For distinct points x1,x2,…,xn in ℛ (the reals), letϕ[x1, x2,…,xn] denote the divided difference ofϕ. In this paper, we determine the general solutionϕ,g: ℛ → ℛ of the functional equationϕ[x1,x2,…,xn] =g(x1,+ x2 + … + xn) for distinct x1,x2,…, xn in ℛ without any regularity assumptions on the unknown functions. 相似文献
17.
In this paper, the problem of minimizing a functionf(x) subject to a constraint (x)=0 is considered. Here,f is a scalar,x ann-vector, and aq-vector, withq<n. The use of the augmented penalty function is explored in connection with theconjugate gradient-restoration algorithm. The augmented penalty functionW(x, ,k) is defined to be the linear combination of the augmented functionF(x, ) and the constraint errorP(x), where theq-vector is the Lagrange multiplier and the scalark is the penalty constant.The conjugate gradient-restoration algorithm includes a conjugate-gradient phase involvingn-q iterations and a restoration phase involving one iteration. In the conjugate-gradient phase, one tries to improve the value of the function, while avoiding excessive constraint violation. In the restoration phase, one reduces the constraint error, while avoiding excessive change in the value of the function.Concerning the conjugate-gradient phase, two classes of algorithms are considered: for algorithms of Class I, the Lagrange multiplier is determined so that the error in the optimum condition is minimized for givenx; for algorithms of Class II, the Lagrange multiplier is determined so that the constraint is satisfied to first order. For each class, two versions are studied. In version (), the penalty constant is held unchanged throughout the entire algorithm. In version (), the penalty constant is updated at the beginning of each conjugate-gradient phase so as to achieve certain desirable properties.Concerning the restoration phase, the minimum distance algorithm is employed. Since the use of the augmented penalty function automatically prevents excessive constraint violation, single-step restoration is considered.If the functionf(x) is quadratic and the constraint (x) is linear, all the previous algorithms are identical, that is, they produce the same sequence of points and converge to the solution in the same number of iterations. This number of iterations is at mostN*=n–q if the starting pointx
s is such that (x
s)=0 and at mostN*=1+n–q if the starting pointx
s is such that (x
s)0.In order to illustrate the theory, seven numerical examples are developed. The first example refers to a quadratic function and a linear constraint. The remaining examples refer to nonquadratic functions and nonlinear constraints. For the linear-quadratic example, all the algorithms behave identically, as predicted by the theory. For the nonlinear-nonquadratic examples, algorithms of Class II generally exhibit faster convergence than algorithms of Class I, and algorithms of type () generally exhibit faster convergence than algorithms of type ().This research was supported by the National Science Foundation, Grant No. GP-27271. 相似文献
18.
S. K. Chatterjea 《Annali dell'Universita di Ferrara》1961,10(1):13-16
Summary Defining the function Δn, 1,k;x(J) asΔn, 1,k;x(J)=J
n+1(x)−J
n(x)J
n+k+1(x) associated with the Bessel functionJ
n(x), we derive a series of products of Bessel functions for Δn, f, k, x (J). Whenk=1,k;x (J) becomes Turàn expression for Bessel functions. Some consequences have been pointed out.
Riassunto Definita la Δn, f, k, x (J) come Δn, f, k, x, (J)=J n+1(x)J n+k(x)-J n(n+k+1)(x) associata alla funzioneJ n(x) di Bessel, si ricava una serie di prodotti di funzioni di Bessel per Δn, f, k, x, (J). 3 Quandok=1, Δn, f, k, x, (J) diventa una espressione di Turàn per le funzioni di 2 Bessel, vengono inoltre indicate alcune altre conseguenze.相似文献
19.
This paper presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(ux) in the inhomogenenous quasi‐linear parabolic equation ut(x, t)=(k(ux)ux(x, t))x +F(u), with the Dirichlet boundary conditions u(0, t)=ψ0, u(1, t)=ψ1 and source function F(u). The main purpose of this paper is to investigate the distinguishability of the input–output mappings Φ[·]:??→C1[0, T], Ψ[·]:??→C1[0, T] via semigroup theory. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
20.
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