Control of Multi-level Voltage States in a Hysteretic SQUID Ring-Resonator System

In this paper we study numerical solutions to the quasi-classical equations of motion for a SQUID ring-radio frequency (rf) resonator system in the regime where the ring is highly hysteretic. In line with experiment, we show that for a suitable choice of of ring circuit parameters the solutions to these equations of motion comprise sets of levels in the rf voltage-current dynamics of the coupled system. We further demonstrate that transitions, both up and down, between these levels can be controlled by voltage pulses applied to the system, thus opening up the possibility of high order (e.g. 10 state), multi-level logic and memory.Comment: 8 pages, 9 figure

Quenching and Propagation of Combustion Without Ignition Temperature Cutoff

We study a reaction-diffusion equation in the cylinder $\Omega = \mathbb{R}\times\mathbb{T}^m$, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of $T$ larger than three as $T\to 0$ and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger for the no-flow case. We also consider shear flows with large amplitude and show that if the reaction power-law decay is larger than three and the flow has only small plateaux (connected domains where it is constant), then any compactly supported initial datum is quenched when the flow amplitude is large enough (which is not true if the power is smaller than three or in the presence of a large plateau). This extends results of Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our work carries over to the case $\Omega = \mathbb{R}^n\times\mathbb{T}^m$, when the critical power is $1 + 2/n$, as well as to certain non-periodic flows

Scaling, renormalization and statistical conservation laws in the Kraichnan model of turbulent advection

We present a systematic way to compute the scaling exponents of the structure functions of the Kraichnan model of turbulent advection in a series of powers of $\xi$, adimensional coupling constant measuring the degree of roughness of the advecting velocity field. We also investigate the relation between standard and renormalization group improved perturbation theory. The aim is to shed light on the relation between renormalization group methods and the statistical conservation laws of the Kraichnan model, also known as zero modes.Comment: Latex (11pt) 43 pages, 22 figures (Feynman diagrams). The reader interested in the technical details of the calculations presented in the paper may want to visit: http://www.math.helsinki.fi/mathphys/paolo_files/passive_scalar/passcal.htm

A mean-field version of the Nicodemi-Prisco SSB model for X-chromosome inactivation

Nicodemi and Prisco recently proposed a model for X-chromosome inactivation in mammals, explaining this phenomenon in terms of a spontaneous symmetry-breaking mechanism [{\it Phys. Rev. Lett.} 99 (2007), 108104]. Here we provide a mean-field version of their model

Explicit methods for stiff stochastic differential equations

Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the Euler-Maruyama method, face severe stepsize restriction when applied to stiff problems. Fully implicit methods are usually not appropriate for stochastic problems and semi-implicit methods (implicit in the deterministic part) involve the solution of possibly large linear systems at each time-step. In this paper, we present a recent generalization of explicit stabilized methods, known as Chebyshev methods, to stochastic problems. These methods have much better (mean-square) stability properties than standard explicit methods. We discuss the construction of this new class of methods and illustrate their performance on various problems involving stochastic ordinary and partial differential equations

Optimal Hedging in Incomplete Markets

We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of a Lévy-Ito market, where assets are driven jointly by an n-dimensional Brownian motion and an independent Poisson random measure on an n-dimensional state space. Given a position in need of hedging and the instruments available as hedges, we demonstrate the existence of an optimal hedge portfolio, where optimality is defined by use of an expected least squared-error criterion over a specified time frame, and where the numeraire with respect to which the hedge is optimized is taken to be the benchmark process associated with the designated pricing kernel

The effect of climate variations on the dynamics of pasture–livestock interactions under cooperative and noncooperative management

It is well known from Hardin's “Tragedy of the Commons” [Hardin G (1968) Science 162:1243–1248] that, if open access is allowed, overgrazing typically results. Hardin, and most authors of the subsequent literature, adopted a static view of the underlying ecosystem. Here we extend this tragedy of the commons to consider the dynamics of the involved ecosystem as well. We consider a general model that allows for a variable carrying capacity of the pastures (due to variation in precipitation) and a stimulating effect on plant growth due to grazing. Our analysis further emphasizes the tragedy; in addition to overgrazing, the ecosystem may approach limit cycles. Thus, unless the pastoralists are able to coordinate themselves, the human capability of long-term planning will generally not stabilize the system. Although numerical optimization shows that a cooperative optimum would yield a high and stable harvest, the open-access system may produce limit cycles, in which even the peak harvest may be below the stable cooperative optimal harvest. Such fluctuations cause both losses in biomass production and utility losses. Our dynamic analysis also demonstrates that, in the absence of cooperation between herders, too much rain in an otherwise dry area might (temporally) destabilize the ecological grazing system through overstocking, subsequently leading to further overgrazing (which will be observed in, but not caused by, the typically dry conditions of landscapes where pastoralism is practiced). In short, through this study we have brought time (and temporal dynamics) into the Hardin's tragedy of the commons and show that the tragedy might be profoundly worsened