Optimal extension to Sobolev rough paths

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $\alpha >1/p>0$. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.Comment: Typos fixed. To appear in Potential Analysi

Scaling property of the critical hopping parameters for the Bose-Hubbard model

Recently precise results for the boundary between the Mott insulator phase and the superfluid phase of the homogeneous Bose-Hubbard model have become available for arbitrary integer filling factor g and any lattice dimension d > 1. We use these data for demonstrating that the critical hopping parameters obey a scaling relationship which allows one to map results for different g onto each other. Unexpectedly, the mean-field result captures the dependence of the exact critical parameters on the filling factor almost fully. We also present an approximation formula which describes the critical parameters for d > 1 and any g with high accuracy.Comment: 5 pages, 5 figures. to appear in EPJ

Prevalence of Exocrine Pancreatic Insufficiency in Women with Obesity Syndrome: Assessment by Pancreatic Fecal Elastase 1

Background. Previous research on the combined association of 25-hydroxyvitamin D [25(OH)D] and exocrine pancreas insufficiency may have been limited by restricted age variability and a lack of representation of both body weight and body mass index. There are still too few conclusive reports about conspicuous vitamin D metabolism according to pancreatic fecal elastase 1 (FE1) in obese patients. Methods. Between May 2004 and July 2008, we investigated in 125 female patients with obesity syndrome at an average age of approximately 52.9 years as well as in age-matched 80 healthy female controls the prevalence of pancreas insufficiency. Serum levels of PTH, total calcium, and D3 vitamins calcitriol and calcifediol, as well as the concentration of fecal elastase 1 (FE1) were determined in patients and controls. Results. In 75 female nondiabetic patients with obesity syndrome (BMI 35 ≤ 40 kg/m2), calcifediol was markedly decreased (25.0 ± 4.9 ng/mL) compared to controls (50.2 ± 14.7 nmol/L; P < 0.01). FE1 level was significantly decreased in obese subjects compared to controls ( P < 0.01). Calcifediol was significantly lower in patients with morbid obesity (for calcifediol,  P < 0.05). Conclusion. In obese females, pancreatic FE1 in feces confirms the extent of vitamin D supply, and thus shows a vitamin D3 deficiency, depending on the loss of stool content. There seems to be a connection between the loss of exocrine function and the increasing body mass index. Pancreas insufficiency, as detected by low FE1 concentrations, is frequent in obese patients. However, the BMI is an additional factor for lowered fecal excretion of FE1

The Evolutionarily Dynamics of Aposematism: a Numerical Analysis of Co-Evolution in Finite Populations

The majority of species are under predatory risk in their natural habitat and targeted by predators as part of the food web. During the evolution of ecosystems, manifold mechanisms have emerged to avoid predation. So called secondary defences, which are used after a predator has initiated prey-catching behaviour, commonly involve the expression of toxins or deterrent substances which are not observable by the predator. Hence, the possession of such secondary defence in many prey species comes with a specific signal of that defence (aposematism). This paper builds on the ideas of existing models of such signalling behaviour, using a model of co-evolution and generalisation of aversive information and introduces a new methodology of numerical analysis for finite populations. This new methodology significantly improves the accessibility of previous models. In finite populations, investigating the co-evolution of defence and signalling requires an understanding of natural selection as well as an assessment of the effects of drift as an additional force acting on stability. The new methodology is able to reproduce the predicted solutions of preceding models and finds additional solutions involving negative correlation between signal strength and the extent of secondary defence. In addition, genetic drift extends the range of stable aposematic solutions through the introduction of a new pseudo-stability and gives new insights into the diversification of aposematic displays

The application of temporal difference learning in optimal diet models

An experience-based aversive learning model of foraging behaviour in uncertain environments is presented. We use Q-learning as a model-free implementation of Temporal difference learning motivated by growing evidence for neural correlates in natural reinforcement settings. The predator has the choice of including an aposematic prey in its diet or to forage on alternative food sources. We show how the predator's foraging behaviour and energy intake depend on toxicity of the defended prey and the presence of Batesian mimics. We introduce the precondition of exploration of the action space for successful aversion formation and show how it predicts foraging behaviour in the presence of conflicting rewards which is conditionally suboptimal in a fixed environment but allows better adaptation in changing environments

Network growth models and genetic regulatory networks

We study a class of growth algorithms for directed graphs that are candidate models for the evolution of genetic regulatory networks. The algorithms involve partial duplication of nodes and their links, together with innovation of new links, allowing for the possibility that input and output links from a newly created node may have different probabilities of survival. We find some counterintuitive trends as parameters are varied, including the broadening of indegree distribution when the probability for retaining input links is decreased. We also find that both the scaling of transcription factors with genome size and the measured degree distributions for genes in yeast can be reproduced by the growth algorithm if and only if a special seed is used to initiate the process.Comment: 8 pages with 7 eps figures; uses revtex4. Added references, cleaner figure

Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass

We investigate the low-lying compression modes of a unitary Fermi gas with imbalanced spin populations. For low polarization, the strong coupling between the two spin components leads to a hydrodynamic behavior of the cloud. For large population imbalance we observe a decoupling of the oscillations of the two spin components, giving access to the effective mass of the Fermi polaron, a quasi-particle composed of an impurity dressed by particle-hole pair excitations in a surrounding Fermi sea. We find $m^*/m=1.17(10)$, in agreement with the most recent theoretical predictions.Comment: 4 pages, 4 figures, submitted to PR

Stochastic analysis with modelled distributions

Using a Besov topology on spaces of modelled distributions in the framework of Hairer's regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type $2$. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness results for mild solutions of semilinear stochastic partial differential equations in these spaces of modelled distributions and for distribution-valued SDEs. Furthermore, we provide a Fubini type theorem allowing to interchange the order of stochastic integration and reconstruction.Comment: corrected some typo