共查询到20条相似文献,搜索用时 46 毫秒
1.
J. S. Hwang 《数学学报(英文版)》1998,14(1):57-66
Letf(X) be an additive form defined by
wherea
i
≠0 is integer,i=1,2…,s. In 1979, Schmidt proved that if ∈>0 then there is a large constantC(k,∈) such that fors>C(k,∈) the equationf(X)=0 has a nontrivial, integer solution in σ1, σ2, …, σ3,x
1,x
2, …,x
3 satisfying
Schmidt did not estimate this constantC(k,∈) since it would be extremely large. In this paper, we prove the following result 相似文献
2.
J. C. Gupta 《Proceedings Mathematical Sciences》2000,110(4):415-430
Let G
n,k
be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc
n(β1,β2,…, β
k
), β1,β2,…, βk
= 0,1,2,…, β1+β2 + … +β
k
≤n,c
n(0,0,…, 0) = 1 and
whenever β0 ≤n - (β1 + β2 + … + β
k
) where Δc
n(β1,β2,…, β
k
) =c
n(β1 + 1, β2,…, β
k
)+c
n(β1,β2+1,…, β
k
)+…+c
n (β1,β2,…, β
k
+1) -c
n(β1,β2,…, β
k
). Further, let Π
n,k
be the set of all symmetric probabilities on {0,1,2,…,k}
n
. We establish a one-to-one correspondence between the sets G
n,k
and Π
n,k
and use it to formulate and answer interesting questions about both. Assigning to G
n,k
the uniform probability measure, we show that, asn→∞, any fixed section {it{cn}(β1,β2,…, β
k
), 1 ≤ Σβ
i
≤m}, properly centered and normalized, is asymptotically multivariate normal. That is,
converges weakly to MVN[0, Σ
m
]; the centering constantsc
0(β1, β2,…, β
k
) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex
inR
k. 相似文献
3.
Let X, X
1, X
2,… be i.i.d.
\mathbbRd {\mathbb{R}^d} -valued real random vectors. Assume that E
X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms
\mathbbQ[ SN ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S
N
= N
−1/2 (X
1 + ⋯ + X
N
) and show that, without any additional conditions,
DN(a) = supx | \textP{ \mathbbQ[ SN - a ] \leqslant x } - \textP{ \mathbbQ[ G - a ] \leqslant x } - Ea(x) | = O( N - 1 ) \Delta_N^{(a)} = \mathop {{\sup }}\limits_x \left| {{\text{P}}\left\{ {\mathbb{Q}\left[ {{S_N} - a} \right] \leqslant x} \right\} - {\text{P}}\left\{ {\mathbb{Q}\left[ {G - a} \right] \leqslant x} \right\} - {E_a}(x)} \right| = \mathcal{O}\left( {{N^{ - 1}}} \right) 相似文献
4.
Mats Andersson 《Journal d'Analyse Mathématique》1996,68(1):39-58
LetG
1,…,Gm be bounded holomorphic functions in a strictly pseudoconvex domainD such that
. We prove that for each
(0,q)-form ϕ inL
p(∂D), 1<p<∞, there are
formsu
1, …,u
m inL
p(∂D) such that ΣG
juj=ϕ. This generalizes previous results forq=0. The proof consists in delicate estimates of integral representation formulas of solutions and relies on a certainT1 theorem due to Christ and Journé. For (0,n−1)-forms there is a simpler proof that also gives the result forp=∞. Restricted to one variable this is precisely the corona theorem.
The author was partially supported by the Swedish Natural Research Council. 相似文献
5.
The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg
reaction–diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε→0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot
as a logarithmic singularity of unknown strength S
j
for j=1,…,K at unknown spot locations x
j
∈Ω for j=1,…,K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates S
j
and x
j
for j=1,…,K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian.
By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source
strengths S
j
, for j=1,…,K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated
for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from
the Schnakenburg model.
相似文献
6.
Viresh Patel 《Order》2008,25(2):131-152
Given a poset P = (X, ≺ ), a partition X
1, ..., X
k
of X is called an ordered partition of P if, whenever x ∈ X
i
and y ∈ X
j
with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X, ≺ ) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m − 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P. We show that this bound is best possible for k = 2, but we give an improved bound, , for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X, ≺ ) and an integer 2 ≤ k ≤ |X|, we can find an ordered partition of P into k parts that minimises the total number of comparable pairs within parts in time polynomial in the size of P. We prove more general, weighted versions of these results.
Supported by an EPSRC doctoral training grant. 相似文献
7.
S. Reshetov 《Journal of Mathematical Sciences》2010,167(4):537-542
We consider the problem of estimating a vector θ = (θ1, θ2,…) ∈ Θ ⊂ l
2 from observations y
i
= θ
i
+ σ
i
x
i
, i = 1, 2,…, where the random values x
i
are N(0, 1), independent, and identically distributed, the parametric set Θ is compact, orthosymmetric, convex, and quadratically
convex. We show that in that case, the minimax risk is not very different from sup?L( P) \sup {\Re_L}\left( \Pi \right) , where ?L( P) {\Re_L}\left( \Pi \right) is the minimax linear risk in the same problem with parametric set Π, and sup is taken over all the hyperrectangles Π ⊂ Θ.
Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal σ
i
, i = 1, 2,…. Bibliography: 4 titles. 相似文献
8.
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
9.
Let ξ,ξ
1,ξ
2,… be positive i.i.d. random variables, S=∑
j=1∞
a(j)ξ
j
, where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:
10.
A. V. Ustinov 《Journal of Mathematical Sciences》2006,137(2):4722-4738
Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a2 + b2 ≤ R2, are studied. The main result is an asymptotic formula with two meaning terms for the quantity
11.
Let X
1
, X
2
, . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X
i
’s. In this paper, we study the asymptotics for the tail probability of the random sum SN = ?k = 1N Xk {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X
1
> x)), and the distribution function (d.f.) of X
1 is dominatedly varying; (ii) P(X
1
> x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X
1 and N are asymptotically comparable and dominatedly varying. 相似文献
12.
S. E. Pastukhova 《Journal of Mathematical Sciences》2012,181(5):668-700
We consider the operator exponential e
−tA
, t > 0, where A is a selfadjoint positive definite operator corresponding to the diffusion equation in
\mathbbRn {\mathbb{R}^n} with measurable 1-periodic coefficients, and approximate it in the operator norm
|| · ||L2( \mathbbRn ) ? L2( \mathbbRn ) {\left\| {\; \cdot \;} \right\|_{{{L^2}\left( {{\mathbb{R}^n}} \right) \to {L^2}\left( {{\mathbb{R}^n}} \right)}}} with order
O( t - \fracm2 ) O\left( {{t^{{ - \frac{m}{2}}}}} \right) as t → ∞, where m is an arbitrary natural number. To construct approximations we use the homogenized parabolic equation with constant
coefficients, the order of which depends on m and is greater than 2 if m > 2. We also use a collection of 1-periodic functions N
α
(x),
x ? \mathbbRn x \in {\mathbb{R}^n} , with multi-indices α of length
| a| \leqslant m \left| \alpha \right| \leqslant m , that are solutions to certain elliptic problems on the periodicity cell. These results are used to homogenize the diffusion
equation with ε-periodic coefficients, where ε is a small parameter. In particular, under minimal regularity conditions, we construct approximations of order O(ε
m
) in the L
2-norm as ε → 0. Bibliography: 14 titles. 相似文献
13.
T. V. Malovichko 《Ukrainian Mathematical Journal》2008,60(11):1789-1802
We consider the solution x
ε of the equation
14.
Let I≥1 be an integer, ω
0=0<ω
1<⋯<ω
I
≤π, and for j=0,…,I, a
j
∈ℂ, a-j=[`(aj)]a_{-j}={\overline{{a_{j}}}}, ω
−j
=−ω
j
, and aj 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form
|