Convolutions of Certain Analytic Functions

Abstract. Ruscheweyh and Sheil-Small proved the Polya-Schoenberg conjecture that the classes of convex functions, starlike functions and close-to-convex functions are closed under convolution with convex functions. By making use of this result, the radii of starlikeness of order α, parabolic starlikeness, and strong starlikeness of order γ of the convolution between two starlike functions are determined. Similar convolution results for two classes of analytic functions are also obtained

Supporting parents to read to their infants on a level 2 neonatal unit: launch of a neonatal library.

Growing evidence shows that preterm infants can be at increased risk of communication andliteracy difficulties as they acquire skills during their early years. The impact of these difficultiescan reduce inclusion in education and social situations for children born preterm. This articlesummarises the steps taken on a level 2 unit to introduce parents to reading books to theirinfants through use of a unit library on the neonatal unit

Local behavior of p-harmonic Green's functions in metric spaces

We describe the behavior of p-harmonic Green's functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been propose

The prognostic value of multivoxel magnetic resonance spectroscopy determined metabolite levels in white and grey matter brain tissue for adverse outcome in term newborns following perinatal asphyxia

Magnetic resonance spectroscopy can identify brain metabolic changes in perinatal asphyxia by providing ratios of metabolites, such as choline (Cho), creatine (Cr), N-acetyl aspartate (NAA) and lactate (Lact) [Cho/Cr, Lact/NAA, etc.]. The purpose of this study was to quantify the separate white and grey matter metabolites in a slab cranial to the ventricles and relate these to the outcome. A standard 2D-chemical shift imaging protocol was used for measuring a transverse volume of interest located cranial to the ventricles allowing for direct comparison of the metabolites in white and grey matter brain tissue in 24 term asphyxiated newborns aged 3 to 16 days. Cho, NAA and Lact showed significant differences between four subgroups of asphyxiated infants with more and less favourable outcomes. High levels of Cho and Lact in the grey matter differentiated non-survivors from survivors (P = 0.003 and P = 0.017, respectively). In perinatal asphyxia the levels of Cho, NAA and Lact in both white and grey matter brain tissue are affected. The levels of Cho and Lact measured in the grey matter are the most indicative of survival. It is therefore advised to include grey matter brain tissue in the region of interest examined by multivoxel MR spectroscopy. aEuro cent Magnetic resonance spectroscopy can identify brain metabolic changes in perinatal asphyxia. aEuro cent Choline and lactate levels in grey matter seem the best indicators of survival. aEuro cent Both grey and white matter should be examined during spectroscopy for perinatal asphyxia