共查询到20条相似文献,搜索用时 46 毫秒
1.
Nakao HAYASHI Pavel I. NAUMKIN 《数学学报(英文版)》2006,22(5):1441-1456
We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut+uux-uxx+uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s (R)∩L^1 (R), where s 〉 -1/2, then there exists a unique solution u (t, x) ∈C^∞ ((0,∞);H^∞ (R)) to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u(t)=t^-1/2fM((·)t^-1/2)+0(t^-1/2) as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 (x) ∈ L^1 (R), then the asymptotics are true u(t)=t^-1/2fM((·)t^-1/2)+O(t^-1/2-γ) where γ ∈ (0, 1/2). 相似文献
2.
Given a semi-convex functionu: ω⊂R
n→R and an integerk≡[0,1,n], we show that the set ∑k defined by
相似文献
3.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
4.
V. A. Kofanov 《Ukrainian Mathematical Journal》2008,60(10):1557-1573
We obtain a new sharp inequality for the local norms of functions x ∈ L
∞, ∞
r
(R), namely,
5.
We are going to discuss special cases of a conditional functional inequality
6.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
7.
R. Nair 《Israel Journal of Mathematics》2009,171(1):197-219
We consider a system of “generalised linear forms” defined at a point x = (x
(i, j)) in a subset of R
d
by
8.
We obtain a strengthened version of the Kolmogorov comparison theorem. In particular, this enables us to obtain a strengthened Kolmogorov inequality for functions x L
x
(r), namely,
9.
O. V. Matveev 《Mathematical Notes》1997,62(3):339-349
Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x|
m
form odd,K(x)=|x|
m
In |x| form even (x∈ℝ
n
),P is the set of real polynomials inn variables of total degree ≤m/2, andx
1,...,x
N
∈ℝ
n
. We construct a function of the form
10.
Let f(x, y) be a periodic function defined on the region D
11.
Abstract
Let Λ = {λ
k
} be an infinite increasing sequence of positive integers with λ
k
→∞. Let X = {X(t), t ∈? R
N
} be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R
d
. Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K
1 and K
2 such that, with unit probability,
12.
LetI be a finite interval andr ∈ ℕ. Denote by △
+
s
L
q
the subset of all functionsy ∈L
q
such that thes-difference △
T
s
y(·) is nonnegative onI, ∀τ>0. Further, denote by △
+
s
W
p
r
the class of functionsx onI with the seminorm ‖x
(r)
‖L
p
≤1, such that △
T
s
x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths
, whereM
n
is the set of all linear manifoldsM
n
inL
q
, dimM
n
≤n, such thatM
n
⋂△
+
s
L
q
≠ 0.
Part of this work was done while the first author visited Tel Aviv University in 2001 and part of it while the second author
was a member of the Industrial Mathematics Institute (IMI), University of South Carolina. 相似文献
13.
Let X
1, X
2, ... be i.i.d. random variables. The sample range is R
n
= max {X
i
, 1 ≤ i ≤ n} − min {X
i
, 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α
k
), (β
k
) then we have
14.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space WN,2([0, ∞); e−x) and the Sobolev-Legendre space WN,2([−1, 1]) with respect to the Sobolev-Laguerre inner product
15.
O. M. Fomenko 《Journal of Mathematical Sciences》2006,133(6):1733-1748
Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let
16.
Let r ∈ N, α, t ∈ R, x ∈ R 2, f: R 2 → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R 2, R + 2 }, and ψ n ∈ L 1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R + such that A ? E, f ∈ L p (G), α ∈ [0, 2π] when G =R 2 and α ∈ [0, π/2] when G = R + 2 Denote Δ n, k = ∫ A t k ψ n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ m, r > 0, Δ m, r + 1 < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K
17.
Here, we solve non-convex, variational problems given in the form
18.
We consider two-phase metrics of the form ϕ(x, ξ) ≔
, where α,β are fixed positive constants and B
α, B
β are disjoint Borel sets whose union is ℝN, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying
19.
The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in
L
pspaces for 1 < p < ∞ and proves that if f(x)∈L
p
[-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈R
n
l such that
20.
Let dα be a measure on R and let σ = (m
1, m
2,...,m
n
), where m
k
≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ω
n
(x) := (x − x
1)(x − x
2)...(x − x
n
) with x
1 ≤ x
2 ≤ ... ≤ x
n
is said to be the nth σ-orthogonal polynomial with respect to dα if the vector of zeros (x
1, x
2, ..., x
n)T is a solution of the extremal problem
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