The heart of a convex body

We investigate some basic properties of the {\it heart} $\heartsuit(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $\heartsuit(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K});$ we also prove an upper estimate for the area of $\heartsuit(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.Comment: 15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-6

MOCVD approach to perovskite based thin films: From high T_csuperconductors to giant dielectric constant materials

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One Pot Grafting of Tetrairon(III) Single Molecule Magnets on Silicon

Herein we report on the Si grafting of two Fe4 derivatives, [Fe4(Li)2(tmhd)6], in which tmhd is 2,2,6,6-tetramethylheptane-3,5-dionate and H3Li = R–C(CH2OH)3 is a tripodal ligand with R = CH2=CH–CH2–O–CH2 (H3L1) and CH2=CH–(CH2)9–O–CH2 (H3L2). These complexes were specifically designed to be directlyanchored on the H-terminated silicon surface via the hydrosilylation reaction. The complexes were grafted by a one pot route based on the photoinduced hydrosilylation followed by a ligand exchange step in the same reaction solution. The resulting decorated surfaces were characterized using X-ray photoelectronspectroscopy (XPS), attenuated total reflection infrared spectroscopy (ATR-IR) and atomic force microscopy (AFM)