Logarithmic lower bounds for Néel walls

Most mathematical models for interfaces and transition layers in materials science exhibit sharply localized and rapidly decaying transition profiles. We show that this behavior can largely change when non-local interactions dominate and internal length scales fail to be determined by dimensional analysis: we consider a reduced model for Néel walls, micromagnetic transition layers which are observed in a suitable thin-film regime. The typical phenomenon associated with this wall type is the very long logarithmic tail of transition profiles. Recently, we derived logarithmic upper bounds. Here, we prove that the latter result is indeed optimal. In particular, we show that Néel wall profiles are supported by explicitly known comparison profiles that minimize relaxed variational principles and exhibit logarithmic decay behavior. This lower bound is established by a comparison argument based on a global maximum principle for the non-local field operator and the qualitative decay behavior of comparison profiles.Received: 17 June 2003, Accepted: 18 November 2003, Published online: 25 February 2004Mathematics Subject Classification (2000): 78A30, 49S05, 45G15, 35B25     

Tracking regression and progression of atherosclerosis in human carotid arteries using high-resolution magnetic resonance imaging

BACKGROUND AND PURPOSE: Magnetic resonance imaging (MRI) can accurately and reproducibly measure the volume of atherosclerotic plaque in human carotid arteries. Atherosclerotic plaques may either progress or regress over time, depending on individual risk factors and treatment regimens. This study was designed to determine if regression or progression of human carotid atherosclerosis in patients receiving statin therapy over 24 months can be detected by high-resolution MRI. METHODS: In 11 subjects who had undergone unilateral carotid endarterectomy and were on statin therapy, volumes for total carotid artery, concentric wall (normal wall), eccentric wall (plaque), and lumen were quantified at 0, 16 and 24 months using a 1.5-T human imager equipped with 6-cm phased array coils. RESULTS: The interobserver mean coefficient of variation (CV) was lowest for the lumen volume (3.1%) and highest for the plaque volume (9.8%). The interscan mean CV was lowest for the total artery volume (3.2%) and highest for the plaque volume (9.9%). As much as 26% regression and 35% progression were observed in individual subject's carotid artery eccentric wall (plaque) volumes over time. Mean eccentric wall volume increased 5% by 16 months and 8% by 24 months. Mean total wall volume increased slightly at both 16 and 24 months (+1.2% and +1.8%). CONCLUSIONS: High-resolution MRI provides a noninvasive reproducible method of tracking changes in carotid atherosclerosis. This pilot study detected changes in individual subjects at both 16 and 24 months. MRI tracking of changes in atherosclerotic plaques should prove useful in assessing vascular disease risk and monitoring the efficacy of interventions designed to induce regression or retard progression.     

On the Gibbsian Nature of the Random Field Kac Model Under Block-Averaging

We consider the Kac–Ising model in an arbitrary configuration of local magnetic fields = , in any dimension d, at any inverse temperature. We investigate the Gibbs properties of the 'renormalized' infinite volume measures obtained by block averaging any of the Gibbs-measures corresponding to fixed , with block-length small enough compared to the range of the Kac-interaction. We show that these measures are Gibbs measures for the same renormalized interaction potential. This potential depends locally on the field configuration and decays exponentially, uniformly in , for which we give explicit bounds. The construction of the potential is based on a high temperature-type cluster expansion.     

Two Connections Between Random Systems and Non-Gibbsian Measures

In this contribution we discuss the role disordered (or random) systems have played in the study of non-Gibbsian measures. This role has two main aspects, the distinction between which has not always been fully clear: 1) From disordered systems: Disordered systems can be used as a tool; analogies with, as well as results and methods from the study of random systems can be employed to investigate non-Gibbsian properties of a variety of measures of physical and mathematical interest. 2) Of disordered systems: Non-Gibbsianness is a property of various (joint) measures describing quenched disordered systems. We discuss and review this distinction and a number of results related to these issues. Moreover, we discuss the mean-field version of the non-Gibbsian property, and present some ideas how a Kac limit approach might connect the finite-range and the mean-field non-Gibbsian properties.