Hastings-Levitov aggregation in the small-particle limit

We establish some scaling limits for a model of planar aggregation. The model is described by the composition of a sequence of independent and identically distributed random conformal maps, each corresponding to the addition of one particle. We study the limit of small particle size and rapid aggregation. The process of growing clusters converges, in the sense of Caratheodory, to an inflating disc. A more refined analysis reveals, within the cluster, a tree structure of branching fingers, whose radial component increases deterministically with time. The arguments of any finite sample of fingers, tracked inwards, perform coalescing Brownian motions. The arguments of any finite sample of gaps between the fingers, tracked outwards, also perform coalescing Brownian motions. These properties are closely related to the evolution of harmonic measure on the boundary of the cluster, which is shown to converge to the Brownian web

Laplacian growth with separately controlled noise and anisotropy

Conformal mapping models are used to study competition of noise and anisotropy in Laplacian growth. For that, a new family of models is introduced with the noise level and directional anisotropy controlled independently. Fractalization is observed in both anisotropic growth and the growth with varying noise. Fractal dimension is determined from cluster size scaling with its area. For isotropic growth we find d = 1.7, both at high and low noise. For anisotropic growth with reduced noise the dimension can be as low as d = 1.5 and apparently is not universal. Also, we study fluctuations of particle areas and observe, in agreement with previous studies, that exceptionally large particles may appear during the growth, leading to pathologically irregular clusters. This difficulty is circumvented by using an acceptance window for particle areas.Comment: 13 pages, 15 figure

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure

Scaling exponent of the maximum growth probability in diffusion-limited aggregation

An early (and influential) scaling relation in the multifractal theory of Diffusion Limited Aggregation(DLA) is the Turkevich-Scher conjecture that relates the exponent \alpha_{min} that characterizes the ``hottest'' region of the harmonic measure and the fractal dimension D of the cluster, i.e. D=1+\alpha_{min}. Due to lack of accurate direct measurements of both D and \alpha_{min} this conjecture could never be put to serious test. Using the method of iterated conformal maps D was recently determined as D=1.713+-0.003. In this Letter we determine \alpha_{min} accurately, with the result \alpha_{min}=0.665+-0.004. We thus conclude that the Turkevich-Scher conjecture is incorrect for DLA.Comment: 4 pages, 5 figure

Can forest management based on natural disturbances maintain ecological resilience?

Given the increasingly global stresses on forests, many ecologists argue that managers must maintain ecological resilience: the capacity of ecosystems to absorb disturbances without undergoing fundamental change. In this review we ask: Can the emerging paradigm of natural-disturbance-based management (NDBM) maintain ecological resilience in managed forests? Applying resilience theory requires careful articulation of the ecosystem state under consideration, the disturbances and stresses that affect the persistence of possible alternative states, and the spatial and temporal scales of management relevance. Implementing NDBM while maintaining resilience means recognizing that (i) biodiversity is important for long-term ecosystem persistence, (ii) natural disturbances play a critical role as a generator of structural and compositional heterogeneity at multiple scales, and (iii) traditional management tends to produce forests more homogeneous than those disturbed naturally and increases the likelihood of unexpected catastrophic change by constraining variation of key environmental processes. NDBM may maintain resilience if silvicultural strategies retain the structures and processes that perpetuate desired states while reducing those that enhance resilience of undesirable states. Such strategies require an understanding of harvesting impacts on slow ecosystem processes, such as seed-bank or nutrient dynamics, which in the long term can lead to ecological surprises by altering the forest's capacity to reorganize after disturbance

Reduced gray matter volume in ventral prefrontal cortex but not amygdala in bipolar disorder:significant effects of gender and trait anxiety

Neuroimaging studies in bipolar disorder report gray matter volume (GMV) abnormalities in neural regions implicated in emotion regulation. This includes a reduction in ventral/orbital medial prefrontal cortex (OMPFC) GMV and, inconsistently, increases in amygdala GMV. We aimed to examine OMPFC and amygdala GMV in bipolar disorder type 1 patients (BPI) versus healthy control participants (HC), and the potential confounding effects of gender, clinical and illness history variables and psychotropic medication upon any group differences that were demonstrated in OMPFC and amygdala GMV. Images were acquired from 27 BPI (17 euthymic, 10 depressed) and 28 age- and gender-matched HC in a 3T Siemens scanner. Data were analyzed with SPM5 using voxel-based morphometry (VBM) to assess main effects of diagnostic group and gender upon whole brain (WB) GMV. Post-hoc analyses were subsequently performed using SPSS to examine the extent to which clinical and illness history variables and psychotropic medication contributed to GMV abnormalities in BPI in a priori and non-a priori regions has demonstrated by the above VBM analyses. BPI showed reduced GMV in bilateral posteromedial rectal gyrus (PMRG), but no abnormalities in amygdala GMV. BPI also showed reduced GMV in two non-a priori regions: left parahippocampal gyrus and left putamen. For left PMRG GMV, there was a significant group by gender by trait anxiety interaction. GMV was significantly reduced in male low-trait anxiety BPI versus male low-trait anxiety HC, and in high- versus low-trait anxiety male BPI. Our results show that in BPI there were significant effects of gender and trait-anxiety, with male BPI and those high in trait-anxiety showing reduced left PMRG GMV. PMRG is part of medial prefrontal network implicated in visceromotor and emotion regulation

Diffusion-limited aggregation in channel geometry

We performed extensive numerical simulation of diffusion-limited aggregation in two-dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D=1.712+/-0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers