On the well-posedness of the stochastic Allen-Cahn equation in two dimensions

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.Comment: 21 pages, 4 figures; accepted by Journal of Computational Physics (Dec 2011

Decreasing intensity of open-ocean convection in the Greenland and Iceland seas

The air–sea transfer of heat and fresh water plays a critical role in the global climate system. This is particularly true for the Greenland and Iceland seas, where these fluxes drive ocean convection that contributes to Denmark Strait overflow water, the densest component of the lower limb of the Atlantic Meridional Overturning Circulation (AMOC). Here we show that the wintertime retreat of sea ice in the region, combined with different rates of warming for the atmosphere and sea surface of the Greenland and Iceland seas, has resulted in statistically significant reductions of approximately 20% in the magnitude of the winter air–sea heat fluxes since 1979. We also show that modes of climate variability other than the North Atlantic Oscillation (NAO) are required to fully characterize the regional air–sea interaction. Mixed-layer model simulations imply that further decreases in atmospheric forcing will exceed a threshold for the Greenland Sea whereby convection will become depth limited, reducing the ventilation of mid-depth waters in the Nordic seas. In the Iceland Sea, further reductions have the potential to decrease the supply of the densest overflow waters to the AMOC

Rayleigh loops in the random-field Ising model on the Bethe lattice

We analyze the demagnetization properties of the random-field Ising model on the Bethe lattice focusing on the beahvior near the disorder induced phase transition. We derive an exact recursion relation for the magnetization and integrate it numerically. Our analysis shows that demagnetization is possible only in the continous high disorder phase, where at low field the loops are described by the Rayleigh law. In the low disorder phase, the saturation loop displays a discontinuity which is reflected by a non vanishing magnetization m_\infty after a series of nested loops. In this case, at low fields the loops are not symmetric and the Rayleigh law does not hold.Comment: 8pages, 6 figure

Discovering study-specific gene regulatory networks

This article has been made available through the Brunel Open Access Publishing Fund.Microarrays are commonly used in biology because of their ability to simultaneously measure thousands of genes under different conditions. Due to their structure, typically containing a high amount of variables but far fewer samples, scalable network analysis techniques are often employed. In particular, consensus approaches have been recently used that combine multiple microarray studies in order to find networks that are more robust. The purpose of this paper, however, is to combine multiple microarray studies to automatically identify subnetworks that are distinctive to specific experimental conditions rather than common to them all. To better understand key regulatory mechanisms and how they change under different conditions, we derive unique networks from multiple independent networks built using glasso which goes beyond standard correlations. This involves calculating cluster prediction accuracies to detect the most predictive genes for a specific set of conditions. We differentiate between accuracies calculated using cross-validation within a selected cluster of studies (the intra prediction accuracy) and those calculated on a set of independent studies belonging to different study clusters (inter prediction accuracy). Finally, we compare our method's results to related state-of-the art techniques. We explore how the proposed pipeline performs on both synthetic data and real data (wheat and Fusarium). Our results show that subnetworks can be identified reliably that are specific to subsets of studies and that these networks reflect key mechanisms that are fundamental to the experimental conditions in each of those subsets

Pre-registration of CT pulmonary volumetric image data

Bakalárska práca sa zaoberá predregistráciou pľúcnych objemových CT obrazových dát. Predregistrácia je riešená metódou fázovej korelácie pri rozklade 3D obrazu na 2D rezy usporiadané za sebou. Práca ďalej popisuje geometrické transformácie, interpolácie, výpočet podobnostných kritérií, optimalizáciu registrácie obrazu a proces samotnej registrácie obrazu. Predregistračný softvér je navrhnutý v programovom prostredí MATLAB^®, kde prebieha predregistrácia 3D reálnych CT obrazových dát s dôrazom na rýchlosť procesu.This bachelor thesis is dealing with pre-registration of CT pulmonary volumetric image data. Pre-registration is solved by phase correlation method, which decomposes 3D images into 2D slices arranged in a row. It further describes the geometric transformations, interpolation, calculations of similarity criteria, optimization of registration of images and the image registration process itself. The pre-registration software runs in MATLAB^®, which works with 3D images of real CT image data with an emphasis on process speed.

Tricritical Points in the Sherrington-Kirkpatrick Model in the Presence of Discrete Random Fields

The infinite-range-interaction Ising spin glass is considered in the presence of an external random magnetic field following a trimodal (three-peak) distribution. The model is studied through the replica method and phase diagrams are obtained within the replica-symmetry approximation. It is shown that the border of the ferromagnetic phase may present first-order phase transitions, as well as tricritical points at finite temperatures. Analogous to what happens for the Ising ferromagnet under a trimodal random field, it is verified that the first-order phase transitions are directly related to the dilution in the fields (represented by $p_{0}$). The ferromagnetic boundary at zero temperature also exhibits an interesting behavior: for $0<p_{0}<p_{0}^{*} \approx 0.30856$, a single tricritical point occurs, whereas if $p_{0}>p_{0}^{*}$ the critical frontier is completely continuous; however, for $p_{0}=p_{0}^{*}$, a fourth-order critical point appears. The stability analysis of the replica-symmetric solution is performed and the regions of validity of such a solution are identified; in particular, the Almeida-Thouless line in the plane field versus temperature is shown to depend on the weight $p_{0}$.Comment: 23pages, 7 ps figure

Generating synthetic fjord bathymetry for coastal Greenland

Bed topography is a critical boundary for the numerical modelling of ice sheets and ice-ocean interactions. A persistent issue with existing topography products for the bed of the Greenland Ice Sheet and surrounding sea floor is the poor representation of coastal bathymetry, especially in regions of floating ice and near the grounding line. Sparse data coverage, and the resultant coarse resolution at the ice-ocean boundary, poses issues in our ability to model ice flow advance and retreat from the present position. In addition, as fjord bathymetry is known to exert strong control on ocean circulation and ice-ocean forcing, the lack of bed data leads to an inability to model these processes adequately. Since the release of the last complete Greenland bed topography-bathymetry product, new observational bathymetry data have become available. These data can be used to constrain bathymetry, but many fjords remain completely unsampled and therefore poorly resolved. Here, as part of the development of the next generation of Greenland bed topography products, we present a new method for constraining the bathymetry of fjord systems in regions where data coverage is sparse. For these cases, we generate synthetic fjord geometries using a method conditioned by surveys of terrestrial glacial valleys as well as existing sinuous feature interpolation schemes. Our approach enables the capture of the general bathymetry profile of a fjord in north-west Greenland close to Cape York, when compared to observational data. We validate our synthetic approach by demonstrating reduced overestimation of depths compared to past attempts to constrain fjord bathymetry. We also present an analysis of the spectral characteristics of fjord centrelines using recently acquired bathymetric observations, demonstrating how a stochastic model of fjord bathymetry could be parameterised and used to create different realisations.This study was supported by UK NERC grant NE/M000869/1

Neighbour identity hardly affects litter-mixture effects on decomposition rates of New Zealand forest species.

The mass loss of litter mixtures is often different than expected based on the mass loss of the component species. We investigated if the identity of neighbour species affects these litter-mixing effects. To achieve this, we compared decomposition rates in monoculture and in all possible two-species combinations of eight tree species, widely differing in litter chemistry, set out in two contrasting New Zealand forest types. Litter from the mixed-species litter bags was separated into its component species, which allowed us to quantify the importance of litter-mixing effects and neighbour identity, relative to the effects of species identity, litter chemistry and litter incubation environment. Controlling factors on litter decomposition rate decreased in importance in the order: species identity (litter quality) >> forest type >> neighbour species. Species identity had the strongest influence on decomposition rate. Interspecific differences in initial litter lignin concentration explained a large proportion of the interspecific differences in litter decomposition rate. Litter mass loss was higher and litter-mixture effects were stronger on the younger, more fertile alluvial soils than on the older, less-fertile marine terrace soils. Litter-mixture effects only shifted percentage mass loss within the range of 1.5%. There was no evidence that certain litter mixtures consistently showed interactive effects. Contrary to common theory, adding a relatively fast-decomposing species generally slowed down the decomposition of the slower decomposing species in the mixture. This study shows that: (1) species identity, litter chemistry and forest type are quantitatively the most important drivers of litter decomposition in a New Zealand rain forest; (2) litter-mixture effects—although statistically significant—are far less important and hardly depend on the identity and the chemical characteristics of the neighbour species; (3) additive effects predominate in this ecosystem, so that mass dynamics of the mixtures can be predicted from the monocultures

A proposal for a coordinated effort for the determination of brainwide neuroanatomical connectivity in model organisms at a mesoscopic scale

In this era of complete genomes, our knowledge of neuroanatomical circuitry remains surprisingly sparse. Such knowledge is however critical both for basic and clinical research into brain function. Here we advocate for a concerted effort to fill this gap, through systematic, experimental mapping of neural circuits at a mesoscopic scale of resolution suitable for comprehensive, brain-wide coverage, using injections of tracers or viral vectors. We detail the scientific and medical rationale and briefly review existing knowledge and experimental techniques. We define a set of desiderata, including brain-wide coverage; validated and extensible experimental techniques suitable for standardization and automation; centralized, open access data repository; compatibility with existing resources, and tractability with current informatics technology. We discuss a hypothetical but tractable plan for mouse, additional efforts for the macaque, and technique development for human. We estimate that the mouse connectivity project could be completed within five years with a comparatively modest budget.Comment: 41 page