Supersolutions for a class of semilinear heat equations

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $\phi\in L^{q}(\Omega)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.Comment: Expanded version of the previous submission arXiv:1111.0258v1. 14 page

An improved method of computing geometrical potential force (GPF) employed in the segmentation of 3D and 4D medical images

The geometric potential force (GPF) used in segmentation of medical images is in general a robustmethod. However, calculation of the GPF is often time consuming and slow. In the present work, wepropose several methods for improving the GPF calculation and evaluate their efficiency against theoriginal method. Among different methods investigated, the procedure that combines Riesz transformand integration by part provides the fastest solution. Both static and dynamic images have been employedto demonstrate the efficacy of the proposed methods

Shearlets and Optimally Sparse Approximations

Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and 3D as a suitable model class, and approximation properties were measured by considering the decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date the only representation system, which provide optimally sparse approximations of this model class in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a theory for compactly supported shearlet frames was derived which moreover also satisfy this optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse approximations of cartoon-like images by band-limited and also compactly supported shearlet frames as well as a reference for the state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data", Birkh\"auser-Springe

Radial and angular derivatives of distributions

When expressing a distribution in Euclidean space in spherical coordinates, derivation with respect to the radial and angular co-ordinates is far from trivial. Exploring the possibilities of defining a radial derivative of the delta distribution 8{x) (the angular derivatives of S(x) being zero since the delta distribution is itself radial) led to the introduction of a new kind of distributions, the so-called signumdistributions, as continuous linear functionals on a space of test functions showing a singularity at the origin. In this paper we search for a definition of the radial and angular derivatives of a general standard distribution and again, as expected, we are inevitably led to consider signumdistributions. Although these signumdistributions provide an adequate framework for the actions on distributions aimed at, it turns out that the derivation with respect to the radial distance of a general (signum)distribution is still not yet unambiguous

Analytic and Gevrey Hypoellipticity for Perturbed Sums of Squares Operators

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying H\"ormander's condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity

A phase 1 trial dose-escalation study of tipifarnib on a week-on, week-off schedule in relapsed, refractory or high-risk myeloid leukemia.

Inhibition of farnesyltransferase (FT) activity has been associated with in vitro and in vivo anti-leukemia activity. We report the results of a phase 1 dose-escalation study of tipifarnib, an oral FT inhibitor, in patients with relapsed, refractory or newly diagnosed (if over age 70) acute myelogenous leukemia (AML), on a week-on, week-off schedule. Forty-four patients were enrolled, two patients were newly diagnosed, and the rest were relapsed or refractory to previous treatment, with a median age of 61 (range 33-79). The maximum tolerated dose was determined to be 1200 mg given orally twice daily (b.i.d.) on this schedule. Cycle 1 dose-limiting toxicities were hepatic and renal. There were three complete remissions seen, two at the 1200 mg b.i.d. dose and one at the 1000 mg b.i.d. dose, with minor responses seen at the 1400 mg b.i.d. dose level. Pharmacokinetic studies performed at doses of 1400 mg b.i.d. showed linear behavior with minimal accumulation between days 1-5. Tipifarnib administered on a week-on, week-off schedule shows activity at higher doses, and represents an option for future clinical trials in AML

On small-noise equations with degenerate limiting system arising from volatility models

The one-dimensional SDE with non Lipschitz diffusion coefficient $dX_{t} = b(X_{t})dt + \sigma X_{t}^{\gamma} dB_{t}, \ X_{0}=x, \ \gamma<1$ is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels [J-D. Deuschel, P.~Friz, A.~Jacquier, and S.~Violante. Marginal density expansions for diffusions and stochastic volatility, part II: Applications. 2013, arxiv:1305.6765] suggests to work with the rescaled variable $X^{\varepsilon}:=\varepsilon^{1/(1-\gamma)} X$: while allowing to turn a space asymptotic problem into a small-$\varepsilon$ problem with fixed terminal point, the process $X^{\varepsilon}$ satisfies a SDE in Wentzell--Freidlin form (i.e. with driving noise $\varepsilon dB$). We prove a pathwise large deviation principle for the process $X^{\varepsilon}$ as $\varepsilon \to 0$. As it will become clear, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell--Freidlin theory. As for applications, the $\varepsilon$-scaling allows to derive exact log-asymptotics for path functionals of the process: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving such an SDE as a component.Comment: 21 pages, 1 figur

Weighted Sobolev spaces of radially symmetric functions

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.Comment: 38 page

Lp Fourier multipliers on compact Lie groups

In this paper we prove Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.Comment: 22 pages; minor correction

Comparison of contact patterns relevant for transmission of respiratory pathogens in Thailand and the Netherlands using respondent-driven sampling

Understanding infection dynamics of respiratory diseases requires the identification and quantification of behavioural, social and environmental factors that permit the transmission of these infections between humans. Little empirical information is available about contact patterns within real-world social networks, let alone on differences in these contact networks between populations that differ considerably on a socio-cultural level. Here we compared contact network data that were collected in the Netherlands and Thailand using a similar online respondent-driven method. By asking participants to recruit contact persons we studied network links relevant for the transmission of respiratory infections. We studied correlations between recruiter and recruited contacts to investigate mixing patterns in the observed social network components. In both countries, mixing patterns were assortative by demographic variables and random by total numbers of contacts. However, in Thailand participants reported overall more contacts which resulted in higher effective contact rates. Our findings provide new insights on numbers of contacts and mixing patterns in two different populations. These data could be used to improve parameterisation of mathematical models used to design control strategies. Although the spread of infections through populations depends on more factors, found similarities suggest that spread may be similar in the Netherlands and Thailand