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1.
V. F. Gaposhkin 《Mathematical Notes》1998,64(3):316-321
The asymptotic behavior asn → ∞ of the normed sumsσn =n
−1 Σ
k
=0n−1
Xk for a stationary processX = (X
n
,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup
k≥n
|σ
k
| ≥ε) asn → ∞ are obtained.
Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998. 相似文献
2.
Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type 总被引:17,自引:0,他引:17
W. A. Kirk 《Israel Journal of Mathematics》1974,17(4):339-346
LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup
i→∞{sup
y∈K
‖t
ix−Tiy∼−‖x−y‖}≦0. IfT
N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k
i‖x−y∼,x,y∈K, wherek
i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of
convexity, ɛ0 (X), be less than one.
Research supported by National Science Foundation Grant GP 18045. 相似文献
3.
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
4.
Consider the retarded difference equationx
n
−x
n−1
=F(−f(x
n
)+g(x
n−k
)), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞.
Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan 相似文献
5.
For each k ≥ 2, let ρ
k
∈ (0, 1) be the largest number such that there exist k-uniform hypergraphs on n vertices with independent neighborhoods and (ρ
k
+ o(1))(
k
n
) edges as n → ∞. We prove that ρ
k
= 1 − 2logk/k + Θ(log log k/k) as k → ∞. This disproves a conjecture of Füredi and the last two authors. 相似文献
6.
Walter Bergweiler 《Journal d'Analyse Mathématique》1994,63(1):121-129
Let (zj) be a sequence of complex numbers satisfying |zj|→ ∞ asj → ∞ and denote by n(r) the number of zj satisfying |zj|≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫∞ (ϕ(t)t logt)−1
dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here.
These results answer a question by A. A. Gol’dberg. 相似文献
7.
A. Račkauskas 《Lithuanian Mathematical Journal》1997,37(4):402-415
Let (ξ
k
,F
k
) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS
n
= ∑
k=1
n
∑
j=1
k−1
β
n
k−j
χ
k
χ
j
, where β
n
asn→∞. The main conclusions aboutP}n
−1
S
n
>x
n
}, wherex
n
→∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space.
Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published
in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997. 相似文献
8.
Vilius Stakénas 《Lithuanian Mathematical Journal》1999,39(1):86-101
This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ1, λ2). An asymptotic formula for the sifting function is derived under the assumption that (λ1, λ2)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p
1
p
2...p
s
,p
1≥p
2>-..., ξk)_is defined by ξ(k)=(logp
1/logk, logp
2/logk,..., logp
s
/logk, 0, ...); allp
i
are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical
behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence
to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density
of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel.
Supported by the Lithuanian State Science and Studies Foundation.
Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1,
pp. 108–127, January–March, 1999.
Translated by V. Stakénas 相似文献
9.
We show that for any sequence tn→∞ and any global weak solution (ρ(t), u(t)) of barotropic compressible Navier-Stokes equations in 2 and 3 space dimensions
with potential force there exists a subsequence {sn} such that ϱ(sn) in L
ω
r
(Ω) of spatially periodic functions, where ϱ∞ is an equilibrium density, r>1 suitable. If the equilibrium is unique then the convergence holds for all t→∞. No smallness
of data is assumed.
Entrata in Redazione il 22 febbraio 1999. Ricevuta nuova versione il 24 agosto 1999.
This work was started and completed during the stay of the second author at the University of Toulon in March 1997 and April
1998, which is acknowledged for the financial support, and also partially supported by the Grant Agency of the Czech Republic,
Grant No. 201/96/0313. 相似文献
10.
We show that there is a function α(r) such that for each constantr≧3, almost everyr-regular graph onn vertices has a hole (vertex induced cycle) of size at least α(r)n asn→∞. We also show that there is a function β(c) such that forc>0 large enough,G
n, p
,p=c/n almost surely has a hole of size at least β(c)n asn→∞. 相似文献
11.
We obtain the exact asymptotics (as n ) of the best L
1-approximations of classes
of periodic functions by splines s S
2n, r – 1 and s S
2n, r + k – 1 (S
2n, r
is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives. 相似文献
12.
H. A. Dzyubenko 《Ukrainian Mathematical Journal》2009,61(4):519-540
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y
i
∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y
i
}
i∈ℤ of points y
i
= y
i+2s
+ 2π such that the function f does not decrease on [y
i
, y
i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T
n
of order ≤n that changes its monotonicity at the same points y
i
∈ Y as f and is such that
*20c || f - Tn || £ \fracc( k,s )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega k( f(r),1 \mathord | / |
\vphantom 1 n n ), f ? C(r), r 3 2 ), \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} 相似文献
13.
A. P. Oskolkov 《Journal of Mathematical Sciences》1998,91(2):2840-2859
Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations
of Kelvin-Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes W
∞
r
(ℝ+;W
2
2+k
(Ω)), W
2
r
(ℝ+;W
2
2+k
(Ω)) and S
2
r
(ℝ+;W
2
2+k
(Ω)) (r=1,2; k=0,1,2, …) under the condition that the right-hand sides f(x,t) belong to the classes W
∞
r-1
(ℝ+;W
2
k
(Ω)), W
2
r-1
(ℝ+;W
2
k
(Ω)) and S
2
r-1
(ℝ+;W
2
k
(Ω)), respectively, and for the solutions of the first and second T-periodic boundary-value problems for the same equations
in the smoothness classes W
∞
r−1
(ℝ; W
2
2+k
(Ω)) and W
2
r−1
(0, T; W
2
2+k
(Ω)) (r=1,2, k=0,1,2…) under the condition that f(x,t) are T-periodic and belong to the spaces W
∞
r−1
(ℝ+; W
2
k
(Ω)) and W
2
r−1
(0,T; W
2
k
(Ω)), respectively. It is shown that as ɛ→0, the smooth solutions {vɛ} of the perturbed initial boundary-value and T-periodic boundary-value problems for the penalized equations of Kelvin-Voight
fluids converge to the corresponding smooth solutions (v,p) of the initial boundary-value and T-periodic boundary-value problems
for the equations of Kelvin-Voight fluids. Bibliography: 27 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 214–242.
Translated by T. N. Surkova. 相似文献
14.
Let {X
n
; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set
S
n
= Σ
k=1
n
X
k
, M
n
= max
k≤n
|S
k
|, n ≥ 1. Suppose σ
2 = EX
12 + 2Σ
k=2∞ EX
1
X
k
(0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M
n
−σɛ√n log n}+ and E{|S
n
| − σɛ√n log n}+ as ɛ ↘ 0 and E{σɛ√π
2
π/8logn − M
n
}+ as ɛ ↗ ∞ are obtained. 相似文献
15.
Harish Seshadri 《Proceedings Mathematical Sciences》2009,119(2):197-201
Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim
r → ∞ e2r
s(r) = 0, then (M, g) has to be isometric to ℍ
n
.
The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim
r → ∞
r
2
s(r) = 0, then (M, g) is isometric to ℝ
n
, a result due to Greene and Wu.
Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B
p
(R) in M and K = a on ∂B
p
(R), then K = a on B
p
(R). 相似文献
16.
M. Zippin 《Israel Journal of Mathematics》1981,39(4):359-364
It is proved that there is a positive function Φ(∈) defined for sufficiently small ∈>0 such that lim∈→0 Φ(∈)=0 and for every integerk and everyk-dimensionalP
1+∈ spaceE, d(E, l
∞
k
)<1+Φ(∈).
Author was partially supported by N.S.F. Grant MCS 79-03042.
An erratum to this article is available at . 相似文献
17.
The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ
k
which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ
k
for almost all (in induced measure) points x onM according as the sum Σ
r
= 1/∞
Ψ(r)r
k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also,
the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r
−t
. τ >k − 1. 相似文献
18.
Martin Kružík 《Applications of Mathematics》2007,52(6):529-543
We study convergence properties of {υ(∇u
k
)}k∈ℕ if υ ∈ C(ℝ
m×m
), |υ(s)| ⩽ C(1+|s|
p
), 1 < p < + ∞, has a finite quasiconvex envelope, u
k
→ u weakly in W
1,p
(Ω; ℝ
m
) and for some g ∈ C(Ω) it holds that ∫Ω
g(x)υ(∇u
k
(x))dx → ∫Ω
g(x)Qυ(∇u(x))dx as k → ∞. In particular, we give necessary and sufficient conditions for L
1-weak convergence of {det ∇u
k
}
k∈ℕ to det ∇u if m = n = p.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday
This work was supported by the grants IAA 1075402 (GA AV ČR) and VZ6840770021 (MŠMT ČR). 相似文献
19.
Letf(t) = ∑a
k
e
ikt
be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖n‖ converges to a finite limitl asn → ∞ (l
2 = sec(arga),a = (f′(0))2 -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn
-1/2) for the Fourier coefficientsa
nk
off
n
, which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn
-1 are present!) for ‖f
n
‖; (iii) the fact that ifi
j
f
(j)(0) is real forj = 1,2,..., 2h + 2 then ‖f
n
‖ = l + o(n
-h
),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times. 相似文献
20.
Let f ∈ L
w
1
[−1, 1], let r
n,m(f) be the best rational L
w
1
-approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim
n → ∞ (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r
n,m
(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006. 相似文献
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