共查询到20条相似文献,搜索用时 843 毫秒
1.
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
2.
Kwang C. Shin 《Potential Analysis》2011,35(2):145-174
For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u
″(z) + [( − 1)ℓ(iz)
m
− P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays
argz=-\fracp2±\frac(l+1)pm+2\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ
n
. Then we show that if the eigenvalue problem is PT\mathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when
gcd(m,l)=1\gcd(m,\ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PT\mathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results. 相似文献
3.
R. Zarouf 《Journal of Mathematical Sciences》2012,182(5):639-645
We prove a Bernstein type inequality involving the Bergman and Hardy norms for rational functions in the unit disk
\mathbb D {\mathbb D} that have at most n poles all of which are outside the disk
\frac1r \mathbb D \frac{1}{r} {\mathbb D} , 0 < r < 1. The asymptotic sharpness of this inequality is shown as n → ∞ and r → 1—. We apply our Bernstein type inequality to
an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space constrained by the H2-nom. Bibliography: 14 titles. 相似文献
4.
We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the
form ∂
t
μ = L
*
μ for bounded Borel measures on ℝ
d
× (0, T), where L is a second order elliptic operator, for example, Lu = Dxu + ( b,?xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity
ò( ?tu + Lu )dm = 0 \int \left( {{\partial_t}u + Lu} \right)d\mu = 0 相似文献
5.
A. A. Amosov 《Journal of Mathematical Sciences》2011,176(3):361-408
We consider semidiscrete and asymptotic approximations to a solution to the nonstationary nonlinear initial-boundary-value
problem governing the radiative–conductive heat transfer in a periodic system consisting of n grey parallel plate heat shields
of width ε = 1/n, separated by vacuum interlayers. We study properties of special semidiscrete and homogenized problems whose solutions approximate
the solution to the problem under consideration. We establish the unique solvability of the problem and deduce a priori estimates
for the solutions. We obtain error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε) for semidiscrete approximations and error estimates of order O( ?{e} ) O\left( {\sqrt {\varepsilon } } \right) and O(ε
3/4) for asymptotic approximations. Bibliography: 9 titles. 相似文献
6.
We introduce a space of sequences lpl \ell_p^\lambda of nonabsolute type, which is a p-normed space and a BK-space in the cases of 0 < p < 1 and 1 ≤ p < ∞; respectively. Further, we deduce some imbedding relations and construct a basis for the space lpl \ell_p^\lambda , where 1 ≤ p < ∞. 相似文献
7.
We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P
1, …, P
k
∈ Z[m] in one unknown m with P
1(0) = … = P
k
(0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with
1 \leqslant m \leqslant xe1 \leqslant m \leqslant x^\varepsilon, such that x + P
1(m), …, x + P
k
(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P
j
= (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse
and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number
of points over finite fields in certain algebraic varieties. 相似文献
8.
A discrete Fourier analysis on the fundamental domain Ω
d
of the d-dimensional lattice of type A
d
is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis
in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange
interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation
is shown to be in the order of (log n)
d
. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already
appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature
formulas, which provides only the second known example of such formulas. 相似文献
9.
On the trigonometric interpolation and the entire interpolation 总被引:17,自引:0,他引:17
Liu Yongping 《分析论及其应用》1990,6(4):85-106
In this paper, we study a kind of interpolation problems on a given nodal set by trigonometric polynomials of order n and
entire functions of exponential type according as the nodal set is
respectively. We established some equivalent conditions and found the explicit forms of some interpolation functions on the
interpolation problems. As a special case, the explicit forms of fundamential functions of (0,m)-interpolat on by trigonometric
case or entire functions case (in B2
σ) respectively, if they exist, may follow from our results. Besides, we also considered the convergence of the interpolation
functions at above stated.
Suported by the Natural Youth Science Foundation of Beijing Normal University. 相似文献
10.
A. Arkhipova 《Journal of Mathematical Sciences》2011,176(6):732-758
We prove the existence of a global heat flow u : Ω ×
\mathbbR+ ? \mathbbRN {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×
\mathbbR+ {\mathbb{R}^{+}}) ⊂
\mathbbRn {\mathbb{R}^{n}}),
n \geqslant 2 n \geqslant 2 , and
\mathbbRN {\mathbb{R}^{N}}) with boundary ∂
[`(W)] \bar{\Omega } such that φ(∂Ω) ⊂
\mathbbRN {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles. 相似文献
11.
We use a piecewise-linear, discontinuous Galerkin method for the time discretization of a fractional diffusion equation involving
a parameter in the range − 1 < α < 0. Our analysis shows that, for a time interval (0,T) and a spatial domain Ω, the error in L¥((0,T);L2(W))L_\infty\bigr((0,T);L_2(\Omega)\bigr) is of order k
2 + α
, where k denotes the maximum time step. Since derivatives of the solution may be singular at t = 0, our result requires the use of non-uniform time steps. In the limiting case α = 0 we recover the known O(k
2) convergence for the classical diffusion (heat) equation. We also consider a fully-discrete scheme that employs standard
(continuous) piecewise-linear finite elements in space, and show that the additional error is of order h
2log(1/k). Numerical experiments indicate that our O(k
2 + α
) error bound is pessimistic. In practice, we observe O(k
2) convergence even for α close to − 1. 相似文献
12.
For fixed generalized reflection matrix P, i.e. P
T
= P, P
2 = I, then matrix X is said to be generalized bisymmetric, if X = X
T
= PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A
1
X
1
B
1 + A
2
X
2
B
2 + ⋯ + A
l
X
l
B
l
= C where [X
1,X
2, ⋯ ,X
l
] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When
the matrix equation is consistent, for any initial generalized bisymmetric matrix group , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors,
and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized
bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized
bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation
, where . Given numerical examples show that the algorithm is efficient.
Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science
Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017). 相似文献
13.
Manel Sanchón 《Potential Analysis》2007,27(3):217-224
We consider the equation on a smooth bounded domain of with zero Dirichlet boundary conditions where p ≥ 2, λ > 0 and f satisfies typical assumptions in the subject of extremal solutions. We prove that, for such general nonlinearities f, the extremal solution u
* belongs to L
∞ (Ω) if N < p + p/(p − 1) and if N < p(1 + p/(p − 1)).
This work was partially supported by MCyT BMF 2002-04613-CO3-02. 相似文献
14.
We consider equations of the form L*μ = 0 for bounded measures on
_boxclose^d {\mathbb{R}^{d}} , where L is a second order elliptic operator, for example, Lu = Δu + (b,∇u), and the equation is understood as the identity
|