Converged high-speed schlieren for shock wave boundary layer interaction study

Scanning-PIV using the drum scanner is employed to study the impingement of a vortex ring onto a solid surface. Both the wall-normal and wall-parallel planes are measured. The former reveals the expansion of the vortex ring and the generation of the secondary and even the tertiary rings in the impingement. The latter successfully visualised the development of the azimuthal instability in the primary and secondary rings. The well-understood long-wavelength instability is found to occur when the secondary ring moves into the primary one. Apart from the azimuthal instability of long-wavelength, another azimuthal instability phenomenon of short-wavelength is visualised. This shortwavelength instability along the primary ring happens when it approaches the surface and being stretched

Floquet spin states in graphene under ac driven spin-orbit interaction

We study the role of periodically driven time-dependent Rashba spin-orbit coupling (RSOC) on a monolayer graphene sample. After recasting the originally $4\times 4$ system of dynamical equations as two time-reversal related two-level problems, the quasi-energy spectrum and the related dynamics are investigated via various techniques and approximations. In the static case the system is a gapped at the Dirac point. The rotating wave approximation (RWA) applied to the driven system unphysically preserves this feature, while the Magnus-Floquet approach as well as a numerically exact evaluation of the Floquet equation show that this gap is dynamically closed. In addition, a sizable oscillating pattern of the out-of-plane spin polarization is found in the driven case for states which completely unpolarized in the static limit. Evaluation of the autocorrelation function shows that the original uniform interference pattern corresponding to time-independent RSOC gets distorted. The resulting structure can be qualitatively explained as a consequence of the transitions induced by the ac driving among the static eigenstates, i.e., these transitions modulate the relative phases that add up to give the quantum revivals of the autocorrelation function. Contrary to the static case, in the driven scenario, quantum revivals (suppresions) are correlated to spin up (down) phases.Comment: 10 pages, 8 figures. Typos corrected. Accepted for publication in PR

Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

Vision-based hand gesture interaction using particle filter, principle component analysis and transition network

Vision-based human-computer interaction is becoming important nowadays. It offers natural interaction with computers and frees users from mechanical interaction devices, which is favourable especially for wearable computers. This paper presents a human-computer interaction system based on a conventional webcam and hand gesture recognition. This interaction system works in real time and enables users to control a computer cursor with hand motions and gestures instead of a mouse. Five hand gestures are designed on behalf of five mouse operations: moving, left click, left-double click, right click and no-action. An algorithm based on Particle Filter is used for tracking the hand position. PCA-based feature selection is used for recognizing the hand gestures. A transition network is also employed for improving the accuracy and reliability of the interaction system. This interaction system shows good performance in the recognition and interaction test