共查询到20条相似文献,搜索用时 406 毫秒
1.
For a q × q matrix x = (x i, j ) we let ${J(x)=(x_{i,j}^{-1})}For a q × q matrix x = (x
i, j
) we let J(x)=(xi,j-1){J(x)=(x_{i,j}^{-1})} be the Hadamard inverse, which takes the reciprocal of the elements of x. We let I(x)=(xi,j)-1{I(x)=(x_{i,j})^{-1}} denote the matrix inverse, and we define K=I°J{K=I\circ J} to be the birational map obtained from the composition of these two involutions. We consider the iterates Kn=K°?°K{K^n=K\circ\cdots\circ K} and determine the degree complexity of K, which is the exponential rate of degree growth d(K)=limn?¥( deg(Kn) )1/n{\delta(K)=\lim_{n\to\infty}\left( deg(K^n) \right)^{1/n}} of the degrees of the iterates. Earlier studies of this map were restricted to cyclic matrices, in which case K may be represented by a simpler map. Here we show that for general matrices the value of δ(K) is equal to the value conjectured by Anglès d’Auriac, Maillard and Viallet. 相似文献
2.
If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì Rd, 1 £ d £ 3,{\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,} modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X
c
is the critical state, then it is proved that
ò¥0m(O\Ot0)dt < ¥,\mathbbP-a.s.{\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}} and
limt?¥ òO|X(t)-Xc|dx = l < ¥, \mathbbP-a.s.{\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}} Here, m is the Lebesgue measure and Otc{\mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=Xc(x)}{\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}} and X
c
(ξ) ≤ X(0, ξ) a.e. x ? O{\xi\in\mathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), limt ? ¥ òK|X(t)-Xc|dx = 0{\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0} exponentially fast for all compact K ì O{K\subset\mathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0. 相似文献
3.
Tomá? Dohnal Michael Plum Wolfgang Reichel 《Communications in Mathematical Physics》2011,308(2):511-542
We consider the nonlinear Schr?dinger equation
(-D+V(x))u = G(x) |u|p-1u, x ? \mathbb Rn(-\Delta +V(x))u = \Gamma(x) |u|^{p-1}u, \quad x\in {\mathbb R}^n 相似文献
4.
We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}
5.
This paper considers Hardy–Lieb–Thirring inequalities for higher order differential operators. A result for general fourth-order
operators on the half-line is developed, and the trace inequality
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