Emergence of patterns in random processes. III. Clustering in higher dimensions.

Newman et al. [Phys. Rev.

Emergence of patterns in random processes. II. Stochastic structure in random events.

Random events can present what appears to be a pattern in the length of peak-to-peak sequences in time series and other point processes. Previously, we showed that this was the case in both individual and independently distributed processes as well as for Brownian walks. In addition, we introduced the use of the discrete form of the Langevin equation of statistical mechanics as a device for connecting the two limiting sets of behaviors, which we then compared with a variety of observations from the physical and social sciences.

Role of the Kelvin-Helmholtz instability in the evolution of magnetized relativistic sheared plasma flows.

We explore, via analytical and numerical methods, the Kelvin-Helmholtz (KH) instability in relativistic magnetized plasmas, with applications to astrophysical jets. We solve the single-fluid relativistic magnetohydrodynamic (RMHD) equations in conservative form using a scheme which is fourth order in space and time. To recover the primitive RMHD variables, we use a highly accurate, rapidly convergent algorithm which improves upon such schemes as the Newton-Raphson method.

Emergence of patterns in random processes.

Sixty years ago, it was observed that any independent and identically distributed (i.i.d.

Statistical properties of record-breaking temperatures.

A record-breaking temperature is the highest or lowest temperature at a station since the period of time considered began. The temperatures at a station constitute a time series. After the removal of daily and annual periodicities, the primary considerations are trends (i.

Time-dependent fiber bundles with local load sharing. II. General Weibull fibers.

Fiber bundle models (FBMs) are useful tools in understanding failure processes in a variety of material systems. While the fibers and load sharing assumptions are easily described, FBM analysis is typically difficult. Monte Carlo methods are also hampered by the severe computational demands of large bundle sizes, which overwhelm just as behavior relevant to real materials starts to emerge.

Inverse cascade via Burgers equation.

Burgers equation is employed as a pedagogical device for analytically demonstrating the emergence of a form of inverse cascade to the lowest wavenumber in a flow. The transition from highly nonlinear mode-mode coupling to an ordered preference for large scale structure is shown, both analytically (revealing the presence of a global attractor) and via a numerical example. (c) 2000 American Institute of Physics.

The method of variation of constants and multiple time scales in orbital mechanics.

The method of variation of constants is an important tool used to solve systems of ordinary differential equations, and was invented by Euler and Lagrange to solve a problem in orbital mechanics. This methodology assumes that certain "constants" associated with a homogeneous problem will vary in time in response to an external force. It also introduces one or more constraint equations.

A mathematical model for self-limiting brain tumors.

It is puzzling that certain brain tumors exhibit arrested exponential growth. We have observed in pediatric low-grade astrocytomas (LGA) at a certain volume approximately 100-150 cm(3) that the tumor ceases to grow. This observation led us to develop a macroscopic mathematical model for LGA growth kinetics that assumes the flow through the surface of the astrocytoma of a triggering agent or "promoter" that is uniformly distributed throughout the tumor, thereby providing relatively homogeneous growth.

Identifying coherent structures in nonlinear wave propagation.

Nonlinear wave phenomena are often characterized by the appearance of "solitary wave coherent structures" traveling at speeds determined by their amplitudes and morphologies. Assuming that time intervals exist in which these structures are essentially noninteracting, a method for identifying the number of independent features and their respective speeds is proposed and developed. The method is illustrated with a variety of increasingly realistic specific applications, beginning with a simple nonlinear but analytically tractable Gaussian model, continuing with (numerically generated) data describing multisoliton solutions to the Korteweg-de Vries equation, and concluding with (numerical) data from a realistic simulation of nonlinear wave interactions in plasma turbulence.