[신간의 별자리x] 우리/미술, 그리고 ‘슬픔의 박물관’

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Hopf bifurcations in fast-slow systems of ordinary differential equations can be associated with surprising rapid growth of periodic orbits This process is referred to as canard explosion The key step in locating a canard explosion is to calculate the location of a special trajectory, called a maximal canard, in parameter space A first-order asymptotic expansion of this location was found by Krupa and Szmolyan in the framework of a "canard point"-normal-form for systems with one fast and one slow variable We show how to compute the coefficient in this expansion using the first Lyapunov coefficient at the Hopf bifurcation thereby avoiding use of this normal form Our results connect the theory of canard explosions with existing numerical software, enabling easier calculations of where canard explosions occur

From First Lyapunov Coefficients to Maximal Canards

A large variety of complex systems in ecology, climate science, biomedicine and engineering have been observed to exhibit tipping points, where the internal dynamical state of the system abruptly changes. For example, such critical transitions may result in the sudden change of ecological environments and climate conditions. Data and models suggest that detectable warning signs may precede some of these drastic events. This view is also corroborated by abstract mathematical theory for generic bifurcations in stochastic multi-scale systems. Whether the stochastic scaling laws used as warning signs are also present in social networks that anticipate a-priori {\it unknown} events in society is an exciting open problem, to which at present only highly speculative answers can be given. Here, we instead provide a first step towards tackling this formidable question by focusing on a-priori {\it known} events and analyzing a social network data set with a focus on classical variance and autocorrelation warning signs. Our results thus pertain to one absolutely fundamental question: Can the stochastic warning signs known from other areas also be detected in large-scale social network data? We answer this question affirmatively as we find that several a-priori known events are preceded by variance and autocorrelation growth. Our findings thus clearly establish the necessary starting point to further investigate the relation between abstract mathematical theory and various classes of critical transitions in social networks.

Critical Transitions in Social Network Activity

Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems.

Pathwise mild solutions for quasilinear stochastic partial differential equations

Invasion waves are a fundamental building block of theoretical ecology. In this study we aim to take the first steps to link propagation failure and fast acceleration of traveling waves to critical transitions (or tipping points). The approach is based upon a detailed numerical study of various versions of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation of this work is to contribute to the following question: how much information do statistics, collected by a stationary observer, contain about the speed and bifurcations of traveling waves? We suggest warning signs based upon closeness to carrying capacity, second-order moments and transients of localized initial invasions.

Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts

Invasion waves are a fundamental building block of theoretical ecology. In this study, we aim to take the first steps to link propagation failure and fast acceleration of traveling waves to critical transitions (or tipping points). The approach is based upon a detailed numerical study on various versions of the Fisher–Kolmogorov–Petrovskii–Piscounov equation. The main motivation of this work is to contribute to the following question: how much information do statistics, collected by a stationary observer, contain about the speed and bifurcations of traveling waves? We suggest warning signs based upon closeness to carrying capacity, second-order moments, and transients of localized initial invasions. However, we also show that these warning signs can be difficult to interpret if limited information is available and that the generalization of classical variance-based warning signs is problematic in the context of propagation failure.

Christian Kuehn

Papers

From First Lyapunov Coefficients to Maximal Canards

Critical Transitions in Social Network Activity

Pathwise mild solutions for quasilinear stochastic partial differential equations

Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts

Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts