共查询到20条相似文献,搜索用时 31 毫秒
1.
J. K. Langley 《Archiv der Mathematik》2002,78(4):291-296
We prove that if
n \geqq 3 n \geqq 3 and A0, ?, An-2 A_0, \ldots, A_{n-2} are entire functions of small growth, not all polynomials, then the linear differential equation¶¶ w(n) + ?j=0n-2 Aj w(j) = 0 w^{(n)} + \sum\limits_{j=0}^{n-2} A_j w^{(j)} = 0 ¶¶ cannot have a fundamental set of solutions each with few zeros. 相似文献
2.
Jaume Llibre Clàudia Valls 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(5):657-679
In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and
their isochronicity for the polynomial differential systems in
\mathbbR2{\mathbb{R}^2} of degree d that in complex notation z = x + i
y can be written as
[(z)\dot] = (l+i) z + (z[`(z)])\fracd-52 (A z4+j[`(z)]1-j + B z3[`(z)]2 + C z2-j[`(z)]3+j+D[`(z)]5), \dot z = (\lambda+i) z + (z \overline{z})^{\frac{d-5}{2}} \left(A z^{4+j} \overline{z}^{1-j} + B z^3 \overline{z}^2 + C z^{2-j} \overline{z}^{3+j}+D \overline{z}^5\right), 相似文献
3.
Lawrence A. Fialkow 《Integral Equations and Operator Theory》2003,45(4):405-435
We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz 2 ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g(2n):g00,g01,g10,?,g0,2n,g1,2n-1,?,g2n-1,1,g2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that gij=ò[`(z)]izj dm (0 £ 1+j £ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ 2, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number gn,n+1 \gamma_{n,n+1} satisfying gn+1,n o [`(g)]n,n+1=Agn,n-1+Bgn,n+Cgn+1,n-1+Dgn,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k. 相似文献
4.
For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò02p s( \fracjn,eiq) e-i(j-k)q dq, j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s](n+1)E[s] \text as n?¥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear. 相似文献
5.
Given g { l\fracn2 g( lj x - kb ) }jezjezn ,where lj \left\{ {\lambda ^{\frac{n}{2}} g\left( {\lambda _j x - kb} \right)} \right\}_{j\varepsilon zj\varepsilon z^n } ,where\;\lambda _j > 0 and b > 0. Sufficient conditions for the wavelet system to constitute a frame for L
2(R
n
) are given. For a class of functions g{ ezrib( j,x ) g( x - lk ) }jezn ,kez\left\{ {e^{zrib\left( {j,x} \right)} g\left( {x - \lambda _k } \right)} \right\}_{j\varepsilon z^n ,k\varepsilon z} to be a frame. 相似文献
6.
7.
Yu Liu 《Monatshefte für Mathematik》2012,127(2):41-56
Let
Lf(x)=-\frac1w?i,j ?i(ai,j(·)?jf)(x)+V(x)f(x){\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)} with the non-negative potential V belonging to reverse H?lder class with respect to the measure ω(x)dx, where ω(x) satisfies the A
2 condition of Muckenhoupt and a
i,j
(x) is a real symmetric matrix satisfying l-1w(x)|x|2 £ ?ni,j=1ai,j(x)xixj £ lw(x)|x|2.{\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. } We obtain some estimates for VaL-a{V^{\alpha}\mathcal{L}^{-\alpha}} on the weighted L
p
spaces and we study the weighted L
p
boundedness of the commutator [b, Va L-a]{[b, V^{\alpha} \mathcal{L}^{-\alpha}]} when b ? BMOw{b\in BMO_\omega} and 0 < α ≤ 1. 相似文献
8.
H. Crauel 《Archiv der Mathematik》2000,75(6):472-480
Let x1,..., xn be points in the d-dimensional Euclidean space Ed with || xi-xj|| £ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?i, j=1n|| xi-xj|| 2\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?i, j=1n|| xi-xj|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x1,...,xn are chosen under the same constraints as above. 相似文献
9.
S. Wada 《Archiv der Mathematik》2001,77(5):415-422
A generalized Hlawka's inequality says that for any n (\geqq 2) (\geqq 2) complex numbers¶ x1, x2, ..., xn,¶¶ ?i=1n|xi - ?j=1nxj| \leqq ?i=1n|xi| + (n - 2)|?j=1nxj|. \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. ¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶ ||A - Tr A ||1 \leqq ||A||1 + (n - 2)| Tr A|. ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. ¶¶ We consider also the related inequalities for p-norms (1 \leqq p \leqq ¥) (1 \leqq p \leqq \infty) on matrices. 相似文献
10.
Liviu I. Ignat Julio D. Rossi 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,170(1):918-925
In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely,
u¢n(t) = ?j ? \mathbbZd Jn-juj(t)-un(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and
n ? \mathbbZdn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies
?n ? \mathbbZdJn = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform. 相似文献
11.
Let
W í \Bbb C\Omega \subseteq {\Bbb C}
be a simply connected domain in
\Bbb C{\Bbb C}
, such that
{¥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}]
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
[`(W)]\bar{\Omega}
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
z ? [`(W)]\zeta \in \bar{\Omega}
we denote
SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
12.
J. Hu 《Transformation Groups》2010,15(2):333-370
Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} | / | \vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) are both independent of K, and the natural homomorphism from \mathfrakBn( - 2m ) \mathord | / | \vphantom ( - 2m ) \mathfrakBn(f) \mathfrakBn(f) {\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{(f)}}}} \right.} {\mathfrak{B}_n^{(f)}}} to \textEn\textdK\textSp(V)( V ?n \mathord | / | \vphantom V ?n V ?n V ?n\mathfrakBn(f) ) {\text{En}}{{\text{d}}_{K{\text{Sp}}(V)}}\left( {{{{V^{ \otimes n}}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}} {{V^{ \otimes n}}}}} \right.} {{V^{ \otimes n}}}}\mathfrak{B}_n^{(f)}} \right) is always surjective. We show that HTf ?n \mathcal{H}\mathcal{T}_f^{ \otimes n} has a Weyl filtration and is isomorphic to the dual of V ?n\mathfrakBn(f) \mathord | / | \vphantom V ?n\mathfrakBn(f) V V ?n\mathfrakBn( f + 1 ) {{{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} \mathord{\left/{\vphantom {{{V^{ \otimes n}}\mathfrak{B}_n^{(f)}} V}} \right.} V}^{ \otimes n}}\mathfrak{B}_n^{\left( {f + 1} \right)} as an \textSp(V) - ( \mathfrakBn( - 2m ) \mathord | / |
\vphantom ( - 2m ) \mathfrakBn( f + 1 ) \mathfrakBn( f + 1 ) ) {\text{Sp}}(V) - \left( {{\mathfrak{B}_n}{{\left( { - 2m} \right)} \mathord{\left/{\vphantom {{\left( { - 2m} \right)} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right.} {\mathfrak{B}_n^{\left( {f + 1} \right)}}}} \right) -bimodule. We obtain an
\textSp(V) - \mathfrakBn {\text{Sp}}(V) - {\mathfrak{B}_n} -bimodules filtration of V
⊗n
such that each successive quotient is isomorphic to some
?( l) ?zg,l\mathfrakBn \nabla \left( \lambda \right) \otimes {z_{g,\lambda }}{\mathfrak{B}_n} with λ ⊢ n 2g, ℓ(λ)≤m and 0 ≤ g ≤ [n/2], where ∇(λ) is the co-Weyl module associated to λ and z
g,λ is an explicitly constructed maximal vector of weight λ. As a byproduct, we show that each right
\mathfrakBn {\mathfrak{B}_n} -module
zg,l\mathfrakBn {z_{g,\lambda }}{\mathfrak{B}_n} is integrally defined and stable under base change. 相似文献
13.
Bodo Dittmar 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,95(1):565-568
For a simply connected and normalized domain D in the plane it was proven by Pólya and Schiffer in 1954 for the fixed membrane eigenvalues
|