104 research outputs found
Singular functions on metric measure spaces
On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the -Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally -regular, we study quasiconformal invariance property associated with the existence of global singular functions
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
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
Fgf signaling controls the telencephalic distribution of Fgf-expressing progenitors generated in the rostral patterning center
Protein tyrosine phosphatase receptor type O (Ptpro) regulates cerebellar formation during zebrafish development through modulating Fgf signaling
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