Critical percolation of free product of groups

In this article we study percolation on the Cayley graph of a free product of groups. The critical probability $p_c$ of a free product $G_1*G_2*...*G_n$ of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups $G_1,G_2,...,G_n$. For finite groups these equations are polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that $p_c$ for the Cayley graph of the modular group $\hbox{PSL}_2(\mathbb Z)$ (with the standard generators) is $.5199...$, the unique root of the polynomial $2p^5-6p^4+2p^3+4p^2-1$ in the interval $(0,1)$. In the case when groups $G_i$ can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of $G_1*G_2*...*G_n$ and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite. We show that the critical point, introduced by Schonmann, $p_{\mathrm{exp}}$ of the free product is just the minimum of $p_{\mathrm{exp}}$ for the factors

Microwave properties of DyBa_2Cu_3O_(7-x) monodomains and related compounds in magnetic fields

We present a microwave characterization of a DyBa$_{2}$Cu$_{3}$O$_{7-x}$ single domain, grown by the top-seeded melt-textured technique. We report the (a,b) plane field-induced surface resistance, $\Delta R_s(H)$, at 48.3 GHz, measured by means of a cylindrical metal cavity in the end-wall-replacement configuration. Changes in the cavity quality factor Q against the applied magnetic field yield $\Delta R_s(H)$ at fixed temperatures. The temperature range [70 K ; T_c] was explored. The magnetic field $\mu_0 H <$ 0.8 T was applied along the c axis. The field dependence of $\Delta R_s(H)$ does not exhibit the steep, step-like increase at low fields typical of weak-links. This result indicates the single-domain character of the sample under investigation. $\Delta R_s(H)$ exhibits a nearly square-root dependence on H, as expected for fluxon motion. From the analysis of the data in terms of motion of Abrikosov vortices we estimate the temperature dependences of the London penetration depth $\lambda$ and the vortex viscosity $\eta$, and their zero-temperature values $\lambda(0)=$165 nm and $\eta(0)=$ 3 10$^{-7}$ Nsm$^{-2}$, which are found in excellent agreement with reported data in YBa$_{2}$Cu$_{3}$O$_{7-x}$ single crystals. Comparison of microwave properties with those of related samples indicate the need for reporting data as a function of T/T_c in order to obtain universal laws.Comment: 6 pages, 4 figures, LaTeX, submitted to Journal of Applied Physic

Evaluation of exercises designed to improve children's use of descriptive words in oral language

Thesis (Ed.M.)--Boston Universit

Emergence of long memory in stock volatility from a modified Mike-Farmer model

The Mike-Farmer (MF) model was constructed empirically based on the continuous double auction mechanism in an order-driven market, which can successfully reproduce the cubic law of returns and the diffusive behavior of stock prices at the transaction level. However, the volatility (defined by absolute return) in the MF model does not show sound long memory. We propose a modified version of the MF model by including a new ingredient, that is, long memory in the aggressiveness (quantified by the relative prices) of incoming orders, which is an important stylized fact identified by analyzing the order flows of 23 liquid Chinese stocks. Long memory emerges in the volatility synthesized from the modified MF model with the DFA scaling exponent close to 0.76, and the cubic law of returns and the diffusive behavior of prices are also produced at the same time. We also find that the long memory of order signs has no impact on the long memory property of volatility, and the memory effect of order aggressiveness has little impact on the diffusiveness of stock prices.Comment: 6 pages, 6 figures and 1 tabl

Adaptive constraints for feature tracking

In this paper extensions to an existing tracking algorithm are described. These extensions implement adaptive tracking constraints in the form of regional upper-bound displacements and an adaptive track smoothness constraint. Together, these constraints make the tracking algorithm more flexible than the original algorithm (which used fixed tracking parameters) and provide greater confidence in the tracking results. The result of applying the new algorithm to high-resolution ECMWF reanalysis data is shown as an example of its effectiveness

Fluid/solid transition in a hard-core system

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition

Influence of oxygen ordering kinetics on Raman and optical response in YBa_2Cu_3O_{6.4}

Kinetics of the optical and Raman response in YBa_2Cu_3O_{6.4} were studied during room temperature annealing following heat treatment. The superconducting T_c, dc resistivity, and low-energy optical conductivity recover slowly, implying a long relaxation time for the carrier density. Short relaxation times are observed for the B_{1g} Raman scattering -- magnetic, continuum, and phonon -- and the charge transfer band. Monte Carlo simulations suggest that these two relaxation rates are related to two length scales corresponding to local oxygen ordering (fast) and long chain and twin formation (slow).Comment: REVTeX, 3 pages + 4 PostScript (compressed) figure