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

Coulomb blockade in a Si channel gated by an Al single-electron transistor

We incorporate an Al-AlO_x-Al single-electron transistor as the gate of a narrow (~100 nm) metal-oxide-semiconductor field-effect transistor (MOSFET). Near the MOSFET channel conductance threshold, we observe oscillations in the conductance associated with Coulomb blockade in the channel, revealing the formation of a Si single-electron transistor. Abrupt steps present in sweeps of the Al transistor conductance versus gate voltage are correlated with single-electron charging events in the Si transistor, and vice versa. Analysis of these correlations using a simple electrostatic model demonstrates that the two single-electron transistor islands are closely aligned, with an inter-island capacitance approximately equal to 1/3 of the total capacitance of the Si transistor island, indicating that the Si transistor is strongly coupled to the Al transistor.Comment: 3 pages, 4 figures, 1 table; typos corrected, minor clarifications added; published in AP

Proximity and anomalous field-effect characteristics in double-wall carbon nanotubes

Proximity effect on field-effect characteristic (FEC) in double-wall carbon nanotubes (DWCNTs) is investigated. In a semiconductor-metal (S-M) DWCNT, the penetration of electron wavefunctions in the metallic shell to the semiconducting shell turns the original semiconducting tube into a metal with a non-zero local density of states at the Fermi level. By using a two-band tight-binding model on a ladder of two legs, it is demonstrated that anomalous FEC observed in so-called S-M type DWCNTs can be fully understood by the proximity effect of metallic phases.Comment: 4 pages, 4 figure

Magnetization reversal through synchronization with a microwave

Based on the Landau-Lifshitz-Gilbert equation, it can be shown that a circularly-polarized microwave can reverse the magnetization of a Stoner particle through synchronization. In comparison with magnetization reversal induced by a static magnetic field, it can be shown that when a proper microwave frequency is used the minimal switching field is much smaller than that of precessional magnetization reversal. A microwave needs only to overcome the energy dissipation of a Stoner particle in order to reverse magnetization unlike the conventional method with a static magnetic field where the switching field must be of the order of magnetic anisotropy.Comment: 4 pages, 5 figure

Application of the 'ESCIMO' theory to turbulent diffusion flames

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A Millikelvin Scanned Probe for Measurement of Nanostructures

We demonstrate a scanning force microscope, based upon a quartz tuning fork, that operates below 100 mK and in magnetic fields up to 6 T. The microscope has a conducting tip for electrical probing of nanostructures of interest, and it incorporates a low noise cryogenic amplifier to measure both the vibrations of the tuning fork and the electrical signals from the nanostructures. At millikelvin temperatures the imaging resolution is below 1 um in a 22 um x 22 um range, and a coarse motion provides translations of a few mm. This scanned probe is useful for high bandwidth measurement of many high impedance nanostructures on a single sample. We show data locating an SET within an array and measure its coulomb blockade with a sensitivity of 2.6 x 10^-5 e/Hz^1/2.Comment: 5 pages, 5 figures, submitted to RS