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

Pair Interaction Potentials of Colloids by Extrapolation of Confocal Microscopy Measurements of Collective Structure

A method for measuring the pair interaction potential between colloidal particles by extrapolation measurement of collective structure to infinite dilution is presented and explored using simulation and experiment. The method is particularly well suited to systems in which the colloid is fluorescent and refractive index matched with the solvent. The method involves characterizing the potential of mean force between colloidal particles in suspension by measurement of the radial distribution function using 3D direct visualization. The potentials of mean force are extrapolated to infinite dilution to yield an estimate of the pair interaction potential, $U(r)$. We use Monte Carlo (MC) simulation to test and establish our methodology as well as to explore the effects of polydispersity on the accuracy. We use poly-12-hydroxystearic acid-stabilized poly(methyl methacrylate) (PHSA-PMMA) particles dispersed in the solvent dioctyl phthalate (DOP) to test the method and assess its accuracy for three different repulsive systems for which the range has been manipulated by addition of electrolyte.Comment: 35 pages, 14 figure

Kinetics and mechanism of formic acid decomposition on Ru(001)

The steady-state rate of decomposition of formic acid on Ru(001) has been measured as a function of surface temperature, parametric in the pressure of formic acid. The products of the decomposition reaction are C0_2, H_2, CO, and H_2)0, i.e., both dehydrogenation and dehydration occur on Ru (001). A similar product distribution has been observed on Ni(110), Ni(100), Ru(100), Fe(100), and Ni(111) surfaces; whereas only dehydrogenation to C0_2 and H_2 occurs on the Cu(100), Cu(110), and Pt(111) surfaces. Only reversible adsorption and desorption of formic acid is observed on the less reactive Ag(110) surface at low temperatures, whereas the more reactive Mo(100) surface is oxidized by formic acid at low temperatures with the products of this reaction being H_2, CO, and H_(2)O (Ref. 10). We report here the confirmation of earlier observations of the occurrence of both dehydrogenation and dehydration of formic acid on Ru(001), and more importantly, we provide a detailed mechanistic description of the steady-state decomposition reaction on this surface in terms of elementary steps

Strategy for designing broadband epsilon-near-zero metamaterial with loss compensation by gain media

A strategy is proposed to design the broadband gain-doped epsilon-near-zero (GENZ) metamaterial. Based on the Milton representation of effective permittivity, the strategy starts in a dimensionless spectral space, where the effective permittivity of GENZ metamaterial is simply determined by a pole-zero structure corresponding to the operating frequency range. The physical structure of GENZ metamaterial is retrieved from the pole-zero structure via a tractable inverse problem. The strategy is of great advantage in practical applications and also theoretically reveals the cancellation mechanism dominating the broadband near-zero permittivity phenomenon in the spectral space

Modelling and control of the flame temperature distribution using probability density function shaping

This paper presents three control algorithms for the output probability density function (PDF) control of the 2D and 3D flame distribution systems. For the 2D flame distribution systems, control methods for both static and dynamic flame systems are presented, where at first the temperature distribution of the gas jet flames along the cross-section is approximated. Then the flame energy distribution (FED) is obtained as the output to be controlled by using a B-spline expansion technique. The general static output PDF control algorithm is used in the 2D static flame system, where the dynamic system consists of a static temperature model of gas jet flames and a second-order actuator. This leads to a second-order closed-loop system, where a singular state space model is used to describe the dynamics with the weights of the B-spline functions as the state variables. Finally, a predictive control algorithm is designed for such an output PDF system. For the 3D flame distribution systems, all the temperature values of the flames are firstly mapped into one temperature plane, and the shape of the temperature distribution on this plane can then be controlled by the 3D flame control method proposed in this paper. Three cases are studied for the proposed control methods and desired simulation results have been obtained

Quantum decoherence of excitons in a leaky cavity with quasimode

For the excitons in the quantum well placed within a leaky cavity, the quantum decoherence of a mesoscopically superposed states is investigated based on the factorization theory for quantum dissipation. It is found that the coherence of the exciton superposition states will decrease in an oscillating form when the cavity field interacting with the exciton is of the form of quasimode. The effect of the thermal cavity fields on the quantum decoherence of the superposition states of the exciton is studied and it is observed that the higher the temperature of the environment is, the shorter the decoherence characteristic time is.Comment: 1 figure, 7 page

Blood lactate clearance after maximal exercise depends on active recovery intensity

AIM: High-intensity exercise is time-limited by onset of fatigue, marked by accumulation of blood lactate. This is accentuated at maximal, all-out exercise that rapidly accumulates high blood lactate. The optimal active recovery intensity for clearing lactate after such maximal, all-out exercise remains unknown. Thus, we studied the intensity-dependence of lactate clearance during active recovery after maximal exercise.&lt;p&gt;&lt;/p&gt; METHODS: We constructed a standardized maximal, all-out treadmill exercise protocol that predictably lead to voluntary exhaustion and blood lactate concentration &#62;10 mM. Next, subjects ran series of all-out bouts that increased blood lactate concentration to 11.5±0.2 mM, followed by recovery exercises ranging 0% (passive)-100% of the lactate threshold.&lt;p&gt;&lt;/p&gt; RESULTS: Repeated measurements showed faster lactate clearance during active versus passive recovery (P&#60;0.01), and that active recovery at 60-100% of lactate threshold was more efficient for lactate clearance than lower intensity recovery (P&#60;0.05). Active recovery at 80% of lactate threshold had the highest rate of and shortest time constant for lactate clearance (P&#60;0.05), whereas the response during the other intensities was graded (100%=60%&#62;40%&#62;passive recovery, P&#60;0.05).&lt;p&gt;&lt;/p&gt; CONCLUSION: Active recovery after maximal all-out exercise clears accumulated blood lactate faster than passive recovery in an intensity-dependent manner, with maximum clearance occurring at active recovery of 80% of lactate threshold

Thickness dependent magnetotransport in ultra-thin manganite films

To understand the near-interface magnetism in manganites, uniform, ultra-thin films of La_{0.67}Sr_{0.33}MnO_3 were grown epitaxially on single crystal (001) LaAlO_3 and (110) NdGaO_3 substrates. The temperature and magnetic field dependent film resistance is used to probe the film's structural and magnetic properties. A surface and/or interface related dead-layer is inferred from the thickness dependent resistance and magnetoresistance. The total thickness of the dead layer is estimated to be $\sim 30 \AA$ for films on NdGaO_3 and $\sim 50 \AA$ for films on LaAlO_3.Comment: 11 pages, 4 figure

Spatial coherence resonance on diffusive and small-world networks of Hodgkin-Huxley neurons

Spatial coherence resonance in a spatially extended system that is locally modeled by Hodgkin-Huxley (HH) neurons is studied in this paper. We focus on the ability of additive temporally and spatially uncorrelated Gaussian noise to extract a particular spatial frequency of excitatory waves in the medium, whereby examining also the impact of diffusive and small-world network topology determining the interactions amongst coupled HH neurons. We show that there exists an intermediate noise intensity that is able to extract a characteristic spatial frequency of the system in a resonant manner provided the latter is diffusively coupled, thus indicating the existence of spatial coherence resonance. However, as the diffusive topology of the medium is relaxed via the introduction of shortcut links introducing small-world properties amongst coupled HH neurons, the ability of additive Gaussian noise to evoke ordered excitatory waves deteriorates rather spectacularly, leading to the decoherence of the spatial dynamics and with it related absence of spatial coherence resonance. In particular, already a minute fraction of shortcut links suffices to substantially disrupt coherent pattern formation in the examined system.Comment: 8 two-column pages, 6 figures; accepted for publication in Chao