Decomposition of stochastic flows with automorphism of subbundles component

We show that given a $G$-structure $P$ on a differentiable manifold $M$, if the group $G(M)$ of automorphisms of $P$ is big enough, then there exists the quotient of an stochastic flows $phi_t$ by $G(M)$, in the sense that $\phi_t = \xi_t \circ \rho_t$ where $\xi_t \in G(M)$, the remainder $\rho_t$ has derivative which is vertical but transversal to the fibre of $P$. This geometrical context generalizes previous results where $M$ is a Riemannian manifold and $\phi_t$ is decomposed with an isometric component, see Liao \cite{Liao1} and Ruffino \cite{Ruffino}, which in our context corresponds to the particular case of an SO(n)-structure on $M$.Comment: To appear in Stochastics and Dynamics, 201

Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method

The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions

Grid-Oscillator Beam-Steering Array

Recently Liao and York (see IEEE Trans. Microwave Theory Tech., vol.41, no.10, p.1810, 1993) showed that beam-steering can be achieved by detuning the end elements of a coupled-oscillator array. The advantage of this approach is that no phase shifters are required. Liao and York used a single line array of patch antennas. Here we report the results for a pair of 1×4 HEMT line-grid oscillators at 11 GHz. This array can scan from -6.5° to +5° by changing the bia

Comments on two password based protocols

Recently, M. Hölbl et al. and I. E. Liao et al. each proposed an user authentication protocol. Both claimed that their schemes can withstand password guessing attack. However, T. Xiang et al. pointed out I. E. Liao et al.\u27s protocol suffers three kinds of attacks, including password guessing attacks. We present an improvement protocol to get rid of password guessing attacks. In this paper, we first point out the security loopholes of M. Hölbl et al.\u27s protocol and review T. Xiang et al.\u27s cryptanalysis on I. E. Liao et al.\u27s protocol. Then, we present the improvements on M. Hölbl et al.\u27s protocol and I. E. Liao et al.\u27s protocol, respectively