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

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

Polynomial Carleson operators along monomial curves in the plane

We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem.Comment: 27 page

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

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

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