Stationary Multiple Spots for Reaction-Diffusion Systems

In this paper, we review analytical methods for a rigorous study of the existence and stability of stationary, multiple spots for reaction-diffusion systems. We will consider two classes of reaction-diffusion systems: activator-inhibitor systems (such as the Gierer-Meinhardt system) and activator-substrate systems (such as the Gray-Scott system or the Schnakenberg model). The main ideas are presented in the context of the Schnakenberg model, and these results are new to the literature. We will consider the systems in a two-dimensional, bounded and smooth domain for small diffusion constant of the activator. Existence of multi-spots is proved using tools from nonlinear functional analysis such as Liapunov-Schmidt reduction and fixed-point theorems. The amplitudes and positions of spots follow from this analysis. Stability is shown in two parts, for eigenvalues of order one and eigenvalues converging to zero, respectively. Eigenvalues of order one are studied by deriving their leading-order asymptotic behavior and reducing the eigenvalue problem to a nonlocal eigenvalue problem (NLEP). A study of the NLEP reveals a condition for the maximal number of stable spots. Eigenvalues converging to zero are investigated using a projection similar to Liapunov-Schmidt reduction and conditions on the positions for stable spots are derived. The Green's function of the Laplacian plays a central role in the analysis. The results are interpreted in the biological, chemical and ecological contexts. They are confirmed by numerical simulations

On the Distribution of MIMO Mutual Information: An In-Depth Painlev\'{e} Based Characterization

This paper builds upon our recent work which computed the moment generating function of the MIMO mutual information exactly in terms of a Painlev\'{e} V differential equation. By exploiting this key analytical tool, we provide an in-depth characterization of the mutual information distribution for sufficiently large (but finite) antenna numbers. In particular, we derive systematic closed-form expansions for the high order cumulants. These results yield considerable new insight, such as providing a technical explanation as to why the well known Gaussian approximation is quite robust to large SNR for the case of unequal antenna arrays, whilst it deviates strongly for equal antenna arrays. In addition, by drawing upon our high order cumulant expansions, we employ the Edgeworth expansion technique to propose a refined Gaussian approximation which is shown to give a very accurate closed-form characterization of the mutual information distribution, both around the mean and for moderate deviations into the tails (where the Gaussian approximation fails remarkably). For stronger deviations where the Edgeworth expansion becomes unwieldy, we employ the saddle point method and asymptotic integration tools to establish new analytical characterizations which are shown to be very simple and accurate. Based on these results we also recover key well established properties of the tail distribution, including the diversity-multiplexing-tradeoff.Comment: Submitted to IEEE Transaction on Information Theory (under revision

An Abstract Machine for Unification Grammars

This work describes the design and implementation of an abstract machine, Amalia, for the linguistic formalism ALE, which is based on typed feature structures. This formalism is one of the most widely accepted in computational linguistics and has been used for designing grammars in various linguistic theories, most notably HPSG. Amalia is composed of data structures and a set of instructions, augmented by a compiler from the grammatical formalism to the abstract instructions, and a (portable) interpreter of the abstract instructions. The effect of each instruction is defined using a low-level language that can be executed on ordinary hardware. The advantages of the abstract machine approach are twofold. From a theoretical point of view, the abstract machine gives a well-defined operational semantics to the grammatical formalism. This ensures that grammars specified using our system are endowed with well defined meaning. It enables, for example, to formally verify the correctness of a compiler for HPSG, given an independent definition. From a practical point of view, Amalia is the first system that employs a direct compilation scheme for unification grammars that are based on typed feature structures. The use of amalia results in a much improved performance over existing systems. In order to test the machine on a realistic application, we have developed a small-scale, HPSG-based grammar for a fragment of the Hebrew language, using Amalia as the development platform. This is the first application of HPSG to a Semitic language.Comment: Doctoral Thesis, 96 pages, many postscript figures, uses pstricks, pst-node, psfig, fullname and a macros fil

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Geometric auxetics

We formulate a mathematical theory of auxetic behavior based on one-parameter deformations of periodic frameworks. Our approach is purely geometric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behavior to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction