Electron columns confined magnetically in vacuum evolve in ðr; hÞ as N 10 field-aligned rods of charge, or point vortices. Neglecting discreteness, the column evolves as would vorticity in an inviscid, incompressible fluid, governed by the Euler equations. The macroscopic flow dynamics is readily imaged, including effects such as surface waves and inviscid damping, two vortex merger, and gradie...

To study the dynamics of chemical processes, we often adopt rate equations to observe the change in chemical concentrations. However, when the number of the molecules is small, the fluctuations cannot be neglected. We often study the effects of fluctuations with the help of stochastic differential equations. Chemicals are composed of molecules on a microscopic level. In principle, the number of...

ally perceived as arising from extrinsic disorder that breaks the discrete translational invariance of the perfect crystal lattice. Familiar examples include the localized vibrational phonon modes around impurities or defects (such as atomic vacancies or interstitial atoms) in crystals and Anderson localization of electrons in disordered media.1 The usual perception among solid-state researcher...

Five theories of spoken word production that differ along the discreteness-interactivity dimension are evaluated. Specifically examined is the role that cascading activation, feedback, seriality, and interaction domains play in accounting for a set of fundamental observations derived from patterns of speech errors produced by normal and brain-damaged individuals. After reviewing the evidence fr...

This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sp...

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and t...

Ordinal stochastic volatility (OSV) models were recently developed and fitted by Müller and Czado (2008) to account for the discreteness of financial price changes, while allowing for stochastic volatility (SV). The model allows for exogenous factors both on the mean and volatility level. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is followed to facilitate estimation in these par...

Methods for exact small-sample analyses with categorical data have been increasingly well developed in recent years. A variety of exact methods exist, primarily using the approach that eliminates unknown parameters by conditioning on their sufficient statistics. In addition, a variety of algorithms now exist for implementing the methods. This paper briefly summarizes the exact approaches and de...

The effects of particle discreteness in N -body simulations of Lambda Cold Dark Matter (ΛCDM) are still an intensively debated issue. In this paper we explore such effects, taking into account the scatter caused by the randomness of the initial conditions, and focusing on the statistical properties of the cosmological density field. For this purpose, we run large sets of ΛCDM simulations and an...

ing an object’s divisibility and the other its self-identity. In mathematics it is the concept of whole number, later elaborated into the set concept, that provides an embodiment of the idea of pure discreteness, that is, of the idea of a collection of separate individual objects, all of whose properties—apart from their distinctness—have been refined away. The basic mathematical representation...