Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

A Single Thermoresponsive Diblock Copolymer Can Form Spheres,Worms or Vesicles in Aqueous Solution

It is well‐known that the self‐assembly of AB diblock copolymers in solution can produce various morphologies depending on the relative volume fraction of each block. Recently, polymerization‐induced self‐assembly (PISA) has become widely recognized as a powerful platform technology for the rational design and efficient synthesis of a wide range of block copolymer nano‐objects. In this study, PISA is used to prepare a new thermoresponsive poly(N‐(2‐hydroxypropyl) methacrylamide)‐poly(2‐hydroxypropyl methacrylate) [PHPMAC‐PHPMA] diblock copolymer. Remarkably, TEM, rheology and SAXS studies indicate that a single copolymer composition can form well‐defined spheres (4 °C), worms (22 °C) or vesicles (50 °C) in aqueous solution. Given that the two monomer repeat units have almost identical chemical structures, this system is particularly well‐suited to theoretical analysis. Self‐consistent mean field theory suggests this rich self‐assembly behavior is the result of the greater degree of hydration of the PHPMA block at lower temperature, which is in agreement with variable temperature ¹H NMR studies.     

The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability

A brief historical overview is given which discusses the development of classical stability concepts, starting in the seventeenth century and finally leading to the concept of Lyapunov stability at the beginning of the twentieth century. The aim of the paper is to find out how various scientists thought about stability and to which extent their work is related to the stability concepts bearing their names, i.e. Lagrange, Poisson and Lyapunov stability. To this end, excerpts of original texts are discussed in detail. Furthermore, the relationship between the various works is addressed.     

The Dynamics of Power laws: Fitness and Aging in Preferential Attachment Trees

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.     

Numerical Schemes for Linear and Non-Linear Enhancement of DW-MRI

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numerical schemes.