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

Achieving Effective Innovation Based On TRIZ Technological Evolution

Organised by: Cranfield UniversityThis paper outlines the conception of effective innovation and discusses the method to achieve it. Effective Innovation is constrained on the path of technological evolution so that the corresponding path must be detected before conceptual design of the product. The process of products technological evolution is a technical developing process that the products approach to Ideal Final Result (IFR). During the process, the sustaining innovation and disruptive innovation carry on alternately. By researching and forecasting potential techniques using TRIZ technological evolution theory, the effective innovation can be achieved finally.Mori Seiki – The Machine Tool Compan

Current-Induced Effective Magnetic Fields in Co/Cu/Co Nanopillars

We present a method to measure the effective field contribution to spin-transfer-induced interactions between the magnetic layers in a trilayer nanostructure, which enables spin-current effects to be distinguished from the usual charge-current-induced magnetic fields. This technique is demonstrated on submicron Co/Cu/Co nanopillars. The hysteresis loop of one of the magnetic layers in the trilayer is measured as a function of current while the direction of magnetization of the other layer is kept fixed, first in one direction and then in the opposite direction. These measurements show a current-dependent shift of the hysteresis loop which, based on the symmetry of the magnetic response, we associate with spin-transfer. The observed loop-shift with applied current at room temperature is reduced in measurements at 4.2 K. We interprete these results both in terms of a spin-current dependent effective activation barrier for magnetization reversal and a spin-current dependent effective magnetic field. From data at 4.2 K we estimate the magnitude of the spin-transfer induced effective field to be $\sim 1.5 \times 10^{-7}$ Oe cm$^2$/A, about a factor of 5 less than the spin-transfer torque.Comment: 6 pages, 4 figure

Current Induced Excitations in Cu/Co/Cu Single Ferromagnetic Layer Nanopillars

Current-induced magnetic excitations in Cu/Co/Cu single layer nanopillars (~50 nm in diameter) have been studied experimentally as a function of Co layer thickness at low temperatures for large applied fields perpendicular to the layers. For asymmetric junctions current induced excitations are observed at high current densities for only one polarity of the current and are absent at the same current densities in symmetric junctions. These observations confirm recent predictions of spin-transfer torque induced spin wave excitations in single layer junctions with a strong asymmetry in the spin accumulation in the leads.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let