Global bifurcation for the Whitham equation

We prove the existence of a global bifurcation branch of $2\pi$-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of H\"older class $C^{\alpha}$, $\alpha < \frac{1}{2}$. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a `highest', cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation

Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks

The aim of the study was to compare the epidemic spread on static and dynamic small-world networks. The network was constructed as a 2-dimensional Watts-Strogatz model (500x500 square lattice with additional shortcuts), and the dynamics involved rewiring shortcuts in every time step of the epidemic spread. The model of the epidemic is SIR with latency time of 3 time steps. The behaviour of the epidemic was checked over the range of shortcut probability per underlying bond 0-0.5. The quantity of interest was percolation threshold for the epidemic spread, for which numerical results were checked against an approximate analytical model. We find a significant lowering of percolation thresholds for the dynamic network in the parameter range given. The result shows that the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7 +/- 1.4%, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small-world we observe suppression of the average epidemic size dependence on network size in comparison with finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to latency time of the disease.Comment: 13 pages, 6 figure

Optimal Population Codes for Space: Grid Cells Outperform Place Cells

Rodents use two distinct neuronal coordinate systems to estimate their position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial location, grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice. We study how to best construct place and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal’s position from the neuronal population response? A priori, estimating a spatial position from a grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid cells with different spacings; the spatial resolution crucially depends on choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic limit, using Fisher information, and for low spike counts, using maximum likelihood estimation. Achieving high spatial resolution and covering a large range of space in a grid code leads to a trade-off: the best grid code for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate. Optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed. First, short lattice spacings should outnumber long lattice spacings. Second, the grid code should be self-similar across different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial “tuning curves” for each neuron span the same range of firing rates, then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons

Theory and simulation of subwavelength high contrast gratings and their applications in vertical-cavity surface-emitting laser devices

This work intends to fully explore the qualities and applications of subwavelength gratings. Subwavelength gratings are diffraction gratings with physical dimensions less than the wavelength of incident light. It has been found that by tailoring specific dimension parameters, a number of different reflection profiles can be attained by these structures including high reflectivity or low reflectivity with broad and narrow spectral responses. In the course of this thesis the physical basis for this phenomenon will be presented as well as a mathematical derivation. After discussion of the mechanics of the reflection behavior, the methods used in modeling subwavelength gratings and designing them for specific functions will be explored. Following this, the fundamentals of vertical-cavity surface-emitting lasers (VCSELs) will be discussed, and the applications of subwavelength gratings when used with these lasers will follow. Several devices, both theoretical proposals and fabricated examples, will be presented in addition to the available performance measurements. Finally, the fabrication challenges that restrict subwavelength gratings from adoption as standard components in VCSEL design will be considered with regard to ongoing fabrication research

Symbolic Time Series Analysis in Economics

In this paper I describe and apply the methods of Symbolic Time Series Analysis (STSA) to an experimental framework. The idea behind Symbolic Time Series Analysis is simple: the values of a given time series data are transformed into a finite set of symbols obtaining a finite string. Then, we can process the symbolic sequence using tools from information theory and symbolic dynamics. I discuss data symbolization as a tool for identifying temporal patterns in experimental data and use symbol sequence statistics in a model strategy. To explain these applications, I describe methods to select the symbolization of the data (Section 2), I introduce the symbolic sequence histograms and some tools to characterize and compare these histograms (Section 3). I show that the methods of symbolic time series analysis can be a good tool to describe and recognize time patterns in complex dynamical processes and to extract dynamical information about this kind of system. In particular, the method gives us a language in which to express and analyze these time patterns. In section 4 I report some applications of STSA to study the evolution of ifferent economies. In these applications data symbolization is based on economic criteria using the notion of economic regime introduced earlier in this thesis. I use STSA methods to describe the dynamical behavior of these economies and to do comparative analysis of their regime dynamics. In section 5 I use STSA to reconstruct a model of a dynamical system from measured time series data. In particular, I will show how the observed symbolic sequence statistics can be used as a target for measuring the goodness of fit of proposed models.