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

Magnetic properties of a long, thin-walled ferromagnetic nanotube

We consider magnetic properties of a long, thin-walled ferromagnetic nanotube. We assume that the tube consists of isotropic homogeneous magnet whose spins interact via the exchange energy, the dipole-dipole interaction energy, and also interact with an external field via Zeeman energy. Possible stable states are the parallel state with the magnetization along the axis of the tube, and the vortex state with the magnetization along azimuthal direction. For a given material, which of them has lower energy depends on the value \gamma=R^2d/(L \lambda_x^2), where R is the radius of the tube, d is its thickness, L is its length and \lambda_x is an intrinsic scale of length characterizing the ration of exchange and dipolar interaction. At \gamma<1 the parallel state wins, otherwise the vortex state is stable. A domain wall in the middle of the tube is always energy unfavorable, but it can exist as a metastable structure. Near the ends of a tube magnetized parallel to the axis a half-domain structure transforming gradually the parallel magnetization to a vortex just at the edge of the tube is energy favorable. We also consider the equilibrium magnetization textures in an external magnetic field either parallel or perpendicular to the tube. Finally, magnetic fields produced by a nanotube and an array of tubes is analyzed

A stochastic space-time model for intermittent precipitation occurrences

Modeling a precipitation field is challenging due to its intermittent and highly scale-dependent nature. Motivated by the features of high-frequency precipitation data from a network of rain gauges, we propose a threshold space-time $t$ random field (tRF) model for 15-minute precipitation occurrences. This model is constructed through a space-time Gaussian random field (GRF) with random scaling varying along time or space and time. It can be viewed as a generalization of the purely spatial tRF, and has a hierarchical representation that allows for Bayesian interpretation. Developing appropriate tools for evaluating precipitation models is a crucial part of the model-building process, and we focus on evaluating whether models can produce the observed conditional dry and rain probabilities given that some set of neighboring sites all have rain or all have no rain. These conditional probabilities show that the proposed space-time model has noticeable improvements in some characteristics of joint rainfall occurrences for the data we have considered.Comment: Published at http://dx.doi.org/10.1214/15-AOAS875 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Onset of unsteady horizontal convection in rectangle tank at Pr=1Pr=1

The horizontal convection within a rectangle tank is numerically simulated. The flow is found to be unsteady at high Rayleigh numbers. There is a Hopf bifurcation of $Ra$ from steady solutions to periodic solutions, and the critical Rayleigh number $Ra_c$ is obtained as $Ra_c=5.5377\times 10^8$ for the middle plume forcing at $Pr=1$, which is much larger than the formerly obtained value. Besides, the unstable perturbations are always generated from the central jet, which implies that the onset of instability is due to velocity shear (shear instability) other than thermally dynamics (thermal instability). Finally, Paparella and Young's [J. Fluid Mech. 466 (2002) 205] first hypotheses about the destabilization of the flow is numerically proved, i.e. the middle plume forcing can lead to a destabilization of the flow.Comment: 4pages, 6 figures, extension of Chin. Phys. Lett. 2008, 25(6), in pres

A Modern Look at Freedman's Box Model

This paper revisits the box model, a metaphor developed by David Freedman to explain sampling distributions and  statistical inference to introductory statistics students. The basic idea is to represent all random phenomena in terms of drawing tickets at random from a box. In this way, random sampling from a population can be described in the same way as everyday phenomena, like coin tossing and card dealing. For Freedman, box models were merely a thought experiment; calculations were still done using normal approximations. In this paper, we propose a more modern view that treats the box model as a practical simulation framework for conducting inference. We show how concepts in introductory statistics and probability classes can be motivated by simulating from a box model. To facilitate this simulation-based approach to teaching box models, we developed an online, open-source "box model simulator"

Phonon anomalies in pure and underdoped R{1-x}K{x}Fe{2}As{2} (R = Ba, Sr) investigated by Raman light scattering

We present a detailed temperature dependent Raman light scattering study of optical phonons in Ba{1-x}K{x}Fe{2}As{2} (x ~ 0.28, superconducting Tc ~ 29 K), Sr{1-x}K{x}Fe{2}As{2} (x ~ 0.15, Tc ~ 29 K) and non-superconducting BaFe{2}As{2} single crystals. In all samples we observe a strong continuous narrowing of the Raman-active Fe and As vibrations upon cooling below the spin-density-wave transition Ts. We attribute this effect to the opening of the spin-density-wave gap. The electron-phonon linewidths inferred from these data greatly exceed the predictions of ab-initio density functional calculations without spin polarization, which may imply that local magnetic moments survive well above Ts. A first-order structural transition accompanying the spin-density-wave transition induces discontinuous jumps in the phonon frequencies. These anomalies are increasingly suppressed for higher potassium concentrations. We also observe subtle phonon anomalies at the superconducting transition temperature Tc, with a behavior qualitatively similar to that in the cuprate superconductors.Comment: 5 pages, 6 figures, accepted versio