Dynamical Networks in Function Dynamics

As a first step toward realizing a dynamical system that evolves while spontaneously determining its own rule for time evolution, function dynamics (FD) is analyzed. FD consists of a functional equation with a self-referential term, given as a dynamical system of a 1-dimensional map. Through the time evolution of this system, a dynamical graph (a network) emerges. This graph has three interesting properties: i) vertices appear as stable elements, ii) the terminals of directed edges change in time, and iii) some vertices determine the dynamics of edges, and edges determine the stability of the vertices, complementarily. Two aspects of FD are studied, the generation of a graph (network) structure and the dynamics of this graph (network) in the system.Comment: 29 pages, 10 figure

Geometric Suppression of Single-Particle Energy Spacings in Quantum Antidots

Quantum Antidot (AD) structures have remarkable properties in the integer quantum Hall regime, exhibiting Coulomb-blockade charging and the Kondo effect despite their open geometry. In some regimes a simple single-particle (SP) model suffices to describe experimental observations while in others interaction effects are clearly important, although exactly how and why interactions emerge is unclear. We present a combination of experimental data and the results of new calculations concerning SP orbital states which show how the observed suppression of the energy spacing between states can be explained through a full consideration of the AD potential, without requiring any effects due to electron interactions such as the formation of compressible regions composed of multiple states, which may occur at higher magnetic fields. A full understanding of the regimes in which these effects occur is important for the design of devices to coherently manipulate electrons in edge states using AD resonances.Comment: 4 pages, 2 figure

Transient dynamics and structure of optimal excitations in thermocapillary spreading: Precursor film model

Linearized modal stability theory has shown that the thermocapillary spreading of a liquid film on a homogeneous, completely wetting surface can produce a rivulet instability at the advancing front due to formation of a capillary ridge. Mechanisms that drain fluid from the ridge can stabilize the flow against rivulet formation. Numerical predictions from this analysis for the film speed, shape, and most unstable wavelength agree remarkably well with experimental measurements even though the linearized disturbance operator is non-normal, which allows transient growth of perturbations. Our previous studies using a more generalized nonmodal stability analysis for contact lines models describing partially wetting liquids (i.e., either boundary slip or van der Waals interactions) have shown that the transient amplification is not sufficient to affect the predictions of eigenvalue analysis. In this work we complete examination of the various contact line models by studying the influence of an infinite and flat precursor film, which is the most commonly employed contact line model for completely wetting films. The maximum amplification of arbitrary disturbances and the optimal initial excitations that elicit the maximum growth over a specified time, which quantify the sensitivity of the film to perturbations of different structure, are presented. While the modal results for the three different contact line models are essentially indistinguishable, the transient dynamics and maximum possible amplification differ, which suggests different transient dynamics for completely and partially wetting films. These differences are explained by the structure of the computed optimal excitations, which provides further basis for understanding the agreement between experiment and predictions of conventional modal analysis

Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept analysis* to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the *limit superior* and *limit inferior* operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram