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1.
We consider the following singularly perturbed boundary-value problem:
on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u
0(x) asε → 0 uniformly inx ε [0, 1], whereu
0(x) εC
∞ [0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx
0 ε (0, 1) such thata(x
0)=0,a′(x
0) > 0,a(x) < 0 for 0 ≤x <x
0, anda(x) > 0 forx
0 <x ≤ 1, wherea(x)=f′
v(x,u
0(x),u′
0(x)).
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 520–524, April, 2000. 相似文献
2.
Yuexu Zhao 《Bulletin of the Brazilian Mathematical Society》2006,37(3):377-391
Let X1, X2, ... be i.i.d. random variables with EX1 = 0 and positive, finite variance σ2, and set Sn = X1 + ... + Xn. For any α > −1, β > −1/2 and for κn(ε) a function of ε and n such that κn(ε) log log n → λ as n ↑ ∞ and
, we prove that
*Supported by the Natural Science Foundation of Department of Education of Zhejiang Province (Grant No. 20060237 and 20050494). 相似文献
3.
Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn). 相似文献
4.
A. A. Kotsiolis 《Journal of Mathematical Sciences》1997,83(2):233-243
In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following
two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:
and the following two dissipative ε-approximations for the equations of motion of the Kelvin-Voight fluids:
satisfying the free surface conditions on the boundary ϖΩ of a domain Ω⊂R3:
.
The free term f(x, t) in systems (1)–(4) is assumed to be t-periodic with period T. It is shown that as ε→0, the classical
t-periodic solutions (with period T) of Eqs. (1)–(4) satisfying the free surface conditions (5) converge to the classicat
t-periodic solutions (with period T) of the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya and to the
equations of motion of the Kelvin-Voight fruids, respectively, satisfying the boundary condition (5).
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 109–124.
Translated by N. S. Zabavnikova. 相似文献
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5.
In this paper we deal with the limit behaviour of the bounded solutions uε of quasi-linear equations of the form
of Ω with Dirichlet boundary conditions on σΩ. The map a=a(x,ϕ) is periodic in x, monotone in ϕ, and satisfies suitable coerciveness
and growth conditions. The function H=H(x,s,ϕ) is assumed to be periodic in x, continuous in [s,ϕ] and to grow at most like
|ξ|p. Under these assumptions on a and H we prove that there exists a function H0=H0(s,ϕ) with the same behaviour of H, such that, up to a subsequence, (uε) converges to a solution u of the homogenized problem -div(b(Du)) + γ|u|p-2u = H0(u,Du) + h(x) on Ω, where b depends only on a and has analogous qualitative properties. 相似文献
6.
T. V. Malovichko 《Ukrainian Mathematical Journal》2008,60(11):1789-1802
We consider the solution x
ε of the equation
where W is a Wiener sheet on . In the case where φε
2 converges to pδ(⋅ −a
1) + qδ(⋅ −a
2), i.e., the limit function describing the influence of a random medium is singular at more than one point, we establish the
weak convergence of (x
ε (u
1,⋅), …, x
ε (u
d
, ⋅)) as ε → 0+ to (X(u
1,⋅), …, X(u
d
, ⋅)), where X is the Arratia flow.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1529–1538, November, 2008. 相似文献
7.
K. N. Venkataraman K. Suresh Chandra 《Annals of the Institute of Statistical Mathematics》1984,36(1):101-118
Summary LetX(t) be a linear autoregressively generated explosive time series, with autoregressive coefficientsb
1,…,bq, and a constant termb
0, and an error term
; a0=1. Where ε(t),t≧1 are independent, Eε(t)=0, and Eε
2(t)=σ2 is positive and finite. In this paper two categories of
-consisent and asymptotically singularly normal estimators are proposed for (b
1,…,bq, b0) thus settling an open problem since the publication of the paper (Venkataraman [5]). Based on these estimators several additional
limit theorems based on estimated error residuals are proved. The parameter-free limit theorems of Spectral and Quenouille
types of this paper serve as asymptotic goodness of fit tests for the model generatingX(t). 相似文献
8.
A general result on precise asymptotics for linear processes of positively associated sequences 总被引:2,自引:0,他引:2
Let {εt; t ∈ Z^+} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2+1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^+} is a sequence of real numbers satisfying ∑j=0^∞|aj|〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E|ε1|^2+δ′〈 for some δ′〉0 and μ(n)=O(n^-ρ) for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}. 相似文献
9.
An Inverse Problem for Maxwell’s Equations in Anisotropic Media 总被引:3,自引:0,他引:3
The authors consider Maxwell's equations for an isomagnctic anisotropic and inhomogeneous medium in two dimensions, and discuss an inverse problem of determining the permittivity tensor (ε1ε2ε2ε3) and the permeabilityμin the constitutive relations from a finite number of lateral boundary measurements. Applying a Carlcman estimate, the authors prove an estimate of the Lipschitz type for stability, provided thatε1,ε2,ε3,μsatisfy some a priori conditions. 相似文献
10.
Considering the positive d-dimensional lattice point Z
+
d
(d ≥ 2) with partial ordering ≤, let {X
k: k ∈ Z
+
d
} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $
S_n = \sum\limits_{k \leqslant n} {X_k }
$
S_n = \sum\limits_{k \leqslant n} {X_k }
, n ∈ Z
+
d
. Let σ
i
2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ
2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
$
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)
d−2(log log ‖X‖)
b+4] < ∞ implies $
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
$
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
, where Γ(·) is the Gamma function and $
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
$
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
. 相似文献
11.
Achiya Dax 《BIT Numerical Mathematics》1997,37(3):600-622
This paper presents a proximal point algorithm for solving discretel
∞ approximation problems of the form minimize ∥Ax−b∥∞. Let ε∞ be a preassigned positive constant and let ε
l
,l = 0,1,2,... be a sequence of positive real numbers such that 0 < ε
l
< ε∞. Then, starting from an arbitrary pointz
0, the proposed method generates a sequence of points z
l
,l= 0,1,2,..., via the rule
. One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within
a finite number of iterations. The smaller are the numbers ε
l
the smaller is the number of iterations. In fact, if ε
0
is sufficiently small then z1 solves the original minimax problem.
The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax
problem of the form minimize
where ∈ is a given positive constant. It is shown that the dual of this problem has the form maximize
, and ify solves the dual thenx=A
T
y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design
and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate
the feasibility of our ideas. 相似文献
12.
Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes 总被引:1,自引:0,他引:1
Yun Xia LI Li Xin ZHANG 《数学学报(英文版)》2006,22(1):143-156
In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai+kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞) is an absolutely solutely summable sequence of real numbers. 相似文献
13.
D. V. Goryashin 《Moscow University Mathematics Bulletin》2011,66(3):125-128
For the number N(x) of solutions to the equation aq − bc = 1 in positive integers a, b, c and square-free numbers q satisfying the condition aq ≤ x the asymptotic formula
$N\left( x \right) = \sum\limits_{n \leqslant x} {2^{\omega \left( n \right)} \tau \left( {n - 1} \right) = \xi _0 x\ln ^2 x + \xi _1 x\ln x + \xi _2 x + O\left( {x^{{5 \mathord{\left/
{\vphantom {5 {6 + \varepsilon }}} \right.
\kern-\nulldelimiterspace} {6 + \varepsilon }}} } \right)}$N\left( x \right) = \sum\limits_{n \leqslant x} {2^{\omega \left( n \right)} \tau \left( {n - 1} \right) = \xi _0 x\ln ^2 x + \xi _1 x\ln x + \xi _2 x + O\left( {x^{{5 \mathord{\left/
{\vphantom {5 {6 + \varepsilon }}} \right.
\kern-\nulldelimiterspace} {6 + \varepsilon }}} } \right)} 相似文献
14.
T. A. Suslina 《Functional Analysis and Its Applications》2012,46(3):234-238
In a bounded domain O ⊂ ℝd with C 1,1 boundary a matrix elliptic second-order operator A D,ɛ with Dirichlet boundary condition is studied. The coefficients of this operator are periodic and depend on x/ɛ, where ɛ s 0 is a small parameter. The sharp-order error estimate $
\left\| {A_{D,\varepsilon }^{ - 1} - \left( {A_D^0 } \right)^{ - 1} } \right\|\left. {L_2 \to L_2 \leqslant C\varepsilon } \right|
$
\left\| {A_{D,\varepsilon }^{ - 1} - \left( {A_D^0 } \right)^{ - 1} } \right\|\left. {L_2 \to L_2 \leqslant C\varepsilon } \right|
相似文献
15.
G. Bouchitte 《Annali dell'Universita di Ferrara》1987,33(1):113-156
Résumé Partant d’un résultat abstrait de représentation intègrale pour une fonctionnelle convexe faiblement s.c.i. sur [M
b
(Ω)]
d
(obtenu grace à [1]); on établit un résultat de convergence pour une suite de fonctionnelles du type
μ
n
εM
b
+
(Ω),f
n
: Ω×R
d
→−∞, +∞] inté
Quelques exemples motivés par des applications à la mécanique sont ensuite traités.
Riassunto Partendo da un risultato astratto di rappresentazione integrale per un funzionale convesso debolmente s.c.i. su [M b (Ω)] d (ottenuto grazie a [1]), si dimostra un risultato di convergenza per una successione di funzionali del tipo con μ n εM b + (Ω),f n : Ω×R d →−∞, +∞] integrando normale convesso. Vengono inoltre trattati alcuni esempi motivati da applicazioni alla Meccanica.相似文献 16.
Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables 总被引:1,自引:0,他引:1
Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞. 相似文献
17.
B. I. Peleshenko 《Ukrainian Mathematical Journal》2000,52(7):1134-1140
We consider the integral convolution operators
\varepsilon } {k\left( {x - y} \right)f\left( y \right)dy}$$
" align="middle" border="0">
defined on spaces of functions of several real variables. For the kernels k(x) satisfying the Hörmander condition, we establish necessary and sufficient conditions under which the operators {T
} are uniformly bounded from Lorentz spaces into Marcinkiewicz spaces. 相似文献
18.
For positive integersn, m and realp≥1, let
Upper and lower bounds for this quantity are derived, extending results of Brown and Spencer forB
1(n,n), corresponding to the Gale-Berlekamp switching problem. For a Minkowski spaceM of dimensionm, define
a quantity investigated by Dvoretzky and Rogers. 相似文献
19.
Marina GHISI Massimo GOBBINO 《数学学报(英文版)》2006,22(4):1161-1170
We consider the Cauchy problem εu^″ε + δu′ε + Auε = 0, uε(0) = uo, u′ε(0) = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D(A). We study the convergence of (uε) to the solution of the limit problem ,δu' + Au = 0, u(0) = u0.
For initial data (u0, u1) ∈ D(A1/2)× H, we prove global-in-time convergence with respect to strong topologies.
Moreover, we estimate the convergence rate in the case where (u0, u1)∈ D(A3/2) ∈ D(A1/2), and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for |u′ε(t)| which does not depend on ε. 相似文献
20.
V. V. Yablokov 《Journal of Mathematical Sciences》2006,135(1):2803-2811
The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity
in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε
q (ε is a small positive parameter, q = const > 0) and radius a
ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ2: x
1 = 0} and their number is equal to N
ɛ= O(ɛ−1) as ε → 0. Under the limit condition lim
on the parameters characterizing the geometry of the domain, the weak H
1-limit of the generalized solution of this problem is found.
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005. 相似文献
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