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

MHD-RLC discharge model and the efficiency characteristics of plasma synthetic jet actuator

Major factors affecting efficiency of plasma synthetic jet actuator (PSJA) are analyzed based on a new discharge model in the present paper. The model couples the magnetohydrodynamics (MHD) equations with the resistor–inductor–capacitor (RLC) equations, and is able to resolve the time-dependent voltage fall on the sheath region and arc region, which is critical in analyzing energy loss in the heating process. This model is integrated into the commercial CFD software by a two-equation method. Results show that in a typical capacitive discharge at microsecond scale, the maximum energy loss is the sheath energy loss, which accounts for nearly half of the discharge energy, while the radiation loss is less than 5%. The discharge time is an important parameter for the PSJA efficiency. A short discharge time less than 1 μs will effectively reduce the sheath energy loss, while a longer discharge time will decrease the thermodynamic efficiency

On the nature of the lightest scalar resonances

We briefly review the recent progresses in the new unitarization approach being developed by us. Especially we discuss the large $N_c$ $\pi\pi$ scatterings by making use of the partial wave $S$ matrix parametrization form. We find that the $\sigma$ pole may move to the negative real axis on the second sheet of the complex $s$ plane, therefore it raises the interesting question that this `$\sigma$' pole may be related to the $\sigma$ in the linear $\sigma$ model.Comment: Talk presented by Zheng at ``Quark Confinement and Hadron Spectroscopy VI'', 21--25 Sept. 2004, Cagliari, Italy. 3 pages with 2 figure

Modeling and optimization of the multichannel spark discharge

This paper reports a novel analytic model of this multichannel spark discharge, considering the delay time in the breakdown process, the electric transforming of the discharge channel from a capacitor to a resistor induced by the air breakdown, and the varying plasma resistance in the discharge process. The good agreement between the experimental and the simulated results validated the accuracy of this model. Based on this model, the influence of the circuit parameters on the maximum discharge channel number (MDCN) is investigated. Both the input voltage amplitude and the breakdown voltage threshold of each discharge channel play a critical role. With the increase of the input voltage and the decrease of the breakdown voltage, the MCDN increases almost linearly. With the increase of the discharge capacitance, the MDCN first rises and then remains almost constant. With the increase of the circuit inductance, the MDCN increases slowly but decreases quickly when the inductance increases over a certain value. There is an optimal value of the capacitor connected to the discharge channel corresponding to the MDCN. Finally, based on these results, to shorten the discharge time, a modified multichannel discharge circuit is developed and validated by the experiment. With only 6-kV input voltage, 31-channels discharge is achieved. The breakdown voltage of each electrode gap is larger than 3 kV. The modified discharge circuit is certain to be widely used in the PSJA flow control field

The origins of electromechanical indentation size effect in ferroelectrics

Metals exhibit a size-dependent hardening when subject to indentation. Mechanisms for this phenomenon have been intensely researched in recent times. Does such a size-effect also exist in the electromechanical behavior of ferroelectrics?--if yes, what are the operative mechanisms? Our experiments on BaTiO3 indeed suggest an electromechanical size-effect. We argue, through theoretical calculations and differential experiments on another non-ferroelectric piezoelectric (Quartz), that the phenomenon of flexoelectricity(as opposed to dislocation activity) is responsible for our observations. Flexoelectricity is the coupling of strain gradients to polarization and exists in both ordinary and piezoelectric dielectrics. In particular, ferroelectrics exhibit an unusually large flexoelectric response.Comment: in revie