Three-dimensional Schwarzschild spacetimes

In the following we find all of the three-dimensional flat spacetimes which are static and rotationally symmetric.     

Efficient polymer passivation of ligand‐stripped nanocrystal surfaces

Water‐dispersible, polymer‐wrapped nanocrystals are highly sought after for use in biology and chemistry, from nanomedicine to catalysis. The hydrophobicity of their native ligand shell, however, is a significant barrier to their aqueous transfer as single particles. Ligand exchange with hydrophilic small molecules or, alternatively, wrapping over native ligands with amphiphilic polymers is widely employed for aqueous transfer; however, purification can be quite cumbersome. We report here a general two‐step method whereby reactive stripping of native ligands is first carried out using trialkyloxonium salts to reveal a bare nanocrystal surface. This is followed by chemically directed immobilization of a hydrophilic polymer coating. Polyacrylic acids, with side‐chain grafts or functional end groups, were found to be extremely versatile in this regard. The resulting polymer‐wrapped nanocrystal dispersions retained much of the compact size of their bare nanocrystal precursors, highlighting the unique role of monomer side‐chain functionality to serve as effective, conformal ligation motifs. As such, they are well poised for applications where tailored chemical functionality at the nanocrystal's periphery or improved access to their surfaces is desirable. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Shedding light on the mitochondrial matrix through a functional membrane transporter

The first fluorescent probes that are actively channeled into the mitochondrial matrix by a specific mitochondrial membrane transporter in living cells have been developed. The new functional probes (BCT) have a minimalist structural design based on the highly efficient and photostable BODIPY chromophore and carnitine as a biotargeting element. Both units are orthogonally bonded through the common boron atom, thus avoiding the use of complex polyatomic connectors. In contrast to known mitochondria-specific dyes, BCTs selectively label these organelles regardless of their transmembrane potential and in an enantioselective way. The obtained experimental evidence supports carnitine–acylcarnitine translocase (CACT) as the key transporter protein for BCTs, which behave therefore as acylcarnitine biomimetics. This simple structural design can be readily extended to other structurally diverse starting F-BODIPYs to obtain BCTs with varied emission wavelengths along the visible and NIR spectral regions and with multifunctional capabilities. BCTs are the first fluorescent derivatives of carnitine to be used in cell microscopy and stand as promising research tools to explore the role of the carnitine shuttle system in cancer and metabolic diseases. Extension of this approach to other small-molecule mitochondrial transporters is envisaged.

A BODIPY derivative of carnitine enters mitochondria regardless of their membrane potential and in an enantioselective way through a specific mitochondrial membrane transporter in living cells.     

A longitudinal study of MR diffusion changes in normal appearing white matter of patients with early multiple sclerosis

BACKGROUND AND PURPOSE: The stage at which normal appearing white matter (NAWM) abnormalities first appear in multiple sclerosis (MS) is not clear. The aim of our study was to monitor water diffusion changes over time in NAWM of patients with early MS.METHODS: Out of a consecutive series of patients enrolled in a MR study on clinically isolated syndrome (CIS), we selected 19 subjects who had completed a one year follow-up. The MR scans obtained at baseline and at 12 months were reviewed according to the new criteria on the diagnosis of MS. Lesion load on T2 and T1 weighted images and the trace of the apparent diffusion coefficient in NAWM were measured both at baseline and at 12 months in patients and in 12 healthy controls.RESULTS: In three patients the diagnosis of MS was done at baseline based on MR. Thirteen patients developed MS during the study and in three patients the diagnosis remained "possible MS." TADC in NAWM in patients was significantly higher than in controls at the 12 months' follow-up but not at baseline (controls mean tADC +/- sd = 0.745 +/- 0.02 mm(2)/sec x 10(-3); patients mean tADC(12) +/- sd = 0.767 +/- 0.02 mm(2)/sec x 10(-3); p < 0.02). TADC and T2 lesion load at 12 months were significantly correlated (p < 0.01). Patients exhibiting tADC(12) above a confidence interval had a significantly greater EDSS score at the same time period (EDSS(12) +/- sd = 1.9 +/- 0.5 and = 1.1 +/- 0.4 respectively; p < 0.01).CONCLUSIONS: This study suggests that diffusion MR cannot detect alterations in NAWM of patients with a CIS suggestive of MS. After one year, when most patients develop MS, diffusion MR abnormalities in NAWM become apparent. These abnormalities are correlated with T2 lesion load and may contribute to neurological impairment.     

Generalized inverses on normed vector spaces

We extend the concepts, introduced by C.R. Rao for Euclidean norms, of minimum g-inverses and least square g-inverses, using arbitrary norms. We give a characterization of such generalized inverses and an application to the case in which the norm is l_∞. As a result of this application we obtain that when $\text{A?}\text{C}{}^{\mathrm{(n+1)\times n}}$ has rank n, there exists a generalized inverse of A, which serves the same purpose as the Moore-Penrose inverse, when the norm is l_∞.     

Approximations in the l∞ norm and the generalized inverse

We introduce the concept of a strict l_∞-metric projector, based in the definition of strict approximation, to prove that for each matrix A of order m×n with coefficients in the field R of real numbers there exists a set of operators G: R^m → Rⁿ homogeneous and continuous, but not necessarily linear (strict generalized inverse) such that AGA = A and 6AGy?y6 is minimized for all y, when the norm is the l_∞ norm. We investigate the properties of these operators and prove that there are two distinguished operators A^-1_{∞, β} and A^-1_∞ which are extensions of the generalized inverse introduced by Newman and Odell in the case of a strictly convex norm.