89,834 research outputs found

    Rank n swapping algebra for the PSL(n,R) Hitchin component

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    Binomial coefficients, Catalan numbers and Lucas quotients

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    Let pp be an odd prime and let a,ma,m be integers with a>0a>0 and m≢0(modp)m \not\equiv0\pmod p. In this paper we determine k=0pa1(2kk+d)/mk\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k mod p2p^2 for d=0,1d=0,1; for example, k=0pa1(2kk)mk(m24mpa)+(m24mpa1)up(m24mp)(modp2),\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 {un}n0\{u_n\}_{n\geqslant0} is the Lucas sequence given by u0=0u_0=0, u1=1u_1=1 and un+1=(m2)unun1u_{n+1}=(m-2)u_n-u_{n-1} for n=1,2,3,n=1,2,3,\ldots. As an application, we determine 0<k<pa,kr(modp1)Ck\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k modulo p2p^2 for any integer rr, where CkC_k denotes the Catalan number (2kk)/(k+1)\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

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    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+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

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    We study the role of periodically driven time-dependent Rashba spin-orbit coupling (RSOC) on a monolayer graphene sample. After recasting the originally 4×44\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