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### 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

### 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

### Discussion on `Characterization of 1-3 piezoelectric polymer composites - a numerical and analytical evaluation procedure for thickness mode vibrations' by C.V. Madhusudhana Rao, G. Prasad, Condens. Matter Phys., 2010, Vol.13, No.1, 13703

In the paper entitled "Characterization of 1-3 piezoelectric polymer
composites - a numerical and analytical evaluation procedure for thickness mode
vibrations", the dependence of the thickness electromechanical coupling
coefficient on the aspect ratio of piezoceramic fibers is studied by finite
element simulation for various volume fractions of piezoceramic fibers in a 1-3
composite. The accuracy of the results is questionable because the boundary
condition claiming that `predefined displacements are applied perpendicularly
on $C^+$ plane on all nodes' is not suitable for the analysis of 1-3 composite
with comparatively large aspect ratio from 0.2 to 1. A discussion regarding
this problem and the suggested corrections are presented in this paper.Comment: 4 pages, 3 figure

### 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

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