[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

Particle moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. We define, study, and compare different natural choices for balance functions and conclude that an efficient compromise is to employ normal infinitesimal Lyapunov exponents. We apply our results to two different model flows: a regularized solid-body rotational flow and the asymmetric Kuhlmann–Muldoon model developed in the context of liquid bridges. To test the balance function approach, we also compute the motion of a finite size particle in an incompressible liquid near a shear-stress interface (invariant wall), using fully resolved numerical simulation. In conclusion, our theoretically developed framework seems to be applicable to models as well as data to understand particle motion near invariant manifolds.

Tracking Particles in Flows near Invariant Manifolds via Balance Functions

Motivated by the normal form of a fast–slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of th...

Discretized Fast-Slow Systems near Pitchfork Singularities

In this paper we investigate the bifurcation structure of the triangular SKT model in the weak competition regime and of the corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be adapted to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the Turing instability analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Epileptic seizures are one of the most well-known dysfunctions of the nervous system During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control In this paper, we aim to contribute towards a better understanding of the dynamical systems phenomena that cause seizures Based on data analysis and modelling, seizure dynamics can be identified to possess multiple spatial scales and on each spatial scale also multiple time scales At each scale, we reach several novel insights On the smallest spatial scale we consider single model neurons and investigate early-warning signs of spiking This introduces the theory of critical transitions to excitable systems For clusters of neurons (or neuronal regions) we use patient data and find oscillatory behavior and new scaling laws near the seizure onset These scalings lead to substantiate the conjecture obtained from mean-field models that a Hopf bifurcation could be involved near seizure onset On the largest spatial scale we introduce a measure based on phase-locking intervals and wavelets into seizure modelling It is used to resolve synchronization between different regions in the brain and identifies time-shifted scaling laws at different wavelet scales We also compare our wavelet-based multiscale approach with maximum linear cross-correlation and mean-phase coherence measures

Christian Kuehn

Papers

Tracking Particles in Flows near Invariant Manifolds via Balance Functions

Discretized Fast-Slow Systems near Pitchfork Singularities

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures