Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

For a general subcritical second-order elliptic operator P   in a domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ (or noncompact manifold), we construct Hardy-weight W which is optimal   in the following sense. The operator P−λW$P-\lambda W$ is subcritical in Ω   for all λ<1$\lambda <1$, null-critical in Ω   for λ=1$\lambda =1$, and supercritical near any neighborhood of infinity in Ω   for any λ>1$\lambda >1$. Moreover, if P   is symmetric and W>0$W>0$, then the spectrum and the essential spectrum of W⁻¹P$W^{-1}P$ are equal to [1,∞)$[1,\infty )$, and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu=0$Pu=0$, the existence of which depends on the subcriticality of P in Ω.     

Quantitative uniqueness for Schrödinger operator with regular potentials

We give a sharp upper bound on the vanishing order of solutions to the Schrödinger equation with electric and magnetic potentials on a compact smooth manifold. Our main result is that the vanishing order of nontrivial solutions to Δu + V · ? u + Wu = 0 is everywhere less than . Our method is based on quantitative Carleman type inequalities, and it allows us to show the following uniform doubling inequality which implies the desired result. Copyright © 2013 John Wiley & Sons, Ltd.     

Tunneling in hydrogen-transfer isomerization of n-alkyl radicals

The role of quantum tunneling in hydrogen shift in linear heptyl radicals is explored using multidimensional, small-curvature tunneling method for the transmission coefficients and a potential energy surface computed at the CBS-QB3 level of theory. Several one-dimensional approximations (Wigner, Skodje and Truhlar, and Eckart methods) were compared to the multidimensional results. The Eckart method was found to be sufficiently accurate in comparison to the small-curvature tunneling results for a wide range of temperature, but this agreement is in fact fortuitous and caused by error cancellations. High-pressure limit rate constants were calculated using the transition state theory with treatment of hindered rotations and Eckart transmission coefficients for all hydrogen-transfer isomerizations in n-pentyl to n-octyl radicals. Rate constants are found in good agreement with experimental kinetic data available for n-pentyl and n-hexyl radicals. In the case of n-heptyl and n-octyl, our calculated rates agree well with limited experimentally derived data. Several conclusions made in the experimental studies of Tsang et al. (Tsang, W.; McGivern, W. S.; Manion, J. A. Proc. Combust. Inst. 2009, 32, 131-138) are confirmed theoretically: older low-temperature experimental data, characterized by small pre-exponential factors and activation energies, can be reconciled with high-temperature data by taking into account tunneling; at low temperatures, transmission coefficients are substantially larger for H-atom transfers through a five-membered ring transition state than those with six-membered rings; channels with transition ring structures involving greater than 8 atoms can be neglected because of entropic effects that inhibit such transitions. The set of computational kinetic rates were used to derive a general rate rule that explicitly accounts for tunneling. The rate rule is shown to reproduce closely the theoretical rate constants.     

A new approach to the determination of concentration profiles in atom probe tomography

Atom probe tomography (APT) provides three-dimensional analytical imaging of materials with near-atomic resolution using pulsed field evaporation. The processes of field evaporation can cause atoms to be placed at positions in the APT reconstruction that can deviate slightly from their original site in the material. Here, we describe and model one such process--that of preferential retention of solute atoms in multicomponent systems. Based on relative field evaporation probabilities, we calculate the point spread function for the solute atom distribution in the "z," or in-depth direction, and use this to extract more accurate solute concentration profiles.     

Silaproline Helical Mimetics Selectively Form an All‐trans PPII Helix

The polyproline II helix (PPII) is increasingly recognized as an important element in peptide and protein structures. The discovery of pertinent PPII peptidomimetics is of great interest to tune physical properties of the targeted structure. A series of silaproline oligomers from dimer to pentamer were synthesized. CD studies, NMR spectroscopy and molecular modeling revealed that the ribbon preferentially populates the polyproline type II secondary structure in both [D]chloroform and [D₄]MeOH. The characteristics of this new lipophilic PPII‐like helix were determined.     

Modelling Photosynthesis with ZnII‐Protoporphyrin All‐DNA G‐Quadruplex/Aptamer Scaffolds

All‐DNA scaffolds act as templates for the organization of photosystem I model systems. A series of DNA templates composed of Zn^II‐protoporphyrin IX (Zn^IIPPIX)‐functionalized G‐quadruplex conjugated to the 3′‐ or 5′‐end of the tyrosinamide (TA) aptamer and Zn^IIPPIX/G‐quadruplex linked to the 3′‐ and 5′‐ends of the TA aptamer through a four‐thymidine bridge. Effective photoinduced electron transfer (ET) from Zn^IIPPIX/G‐quadruplex to bipyridinium‐functionalized tyrosinamide, TA‐MV²⁺, bound to the TA aptamer units is demonstrated. The effectiveness of the primary ET quenching of Zn^IIPPIX/G‐quadruplex by TA‐MV²⁺ controls the efficiency of the generation of TA‐MV^+.. The photosystem‐controlled formation of TA‐MV^+. by the different photosystems dictates the secondary activation of the ET cascade corresponding to the ferredoxin‐NADP⁺ reductase (FNR)‐catalysed reduction of NADP⁺ to NADPH by TA‐MV^+., and the sequestered alcohol dehydrogenase catalysed reduction of acetophenone to 1‐phenylethanol by NADPH.     

Pressure-Induced Conversion of a Paramagnetic FeCo Complex into a Molecular Magnetic Switch with Tuneable Hysteresis

A key challenge in the design of magnetic molecular switches is to obtain bistability at room temperature. Here, we show that application of moderate pressure makes it possible to convert a paramagnetic Fe^III₂Co^II₂ square complex into a molecular switch exhibiting a full dia- to paramagnetic transition: Fe^IICo^III ⇔ Fe^IIICo^II. Moreover, the complex follows a rare behavior: the higher the pressure, the broader the magnetic hysteresis. Thus, the application of an adequate pressure allows inducing a magnetic bistability at room temperature with predictable hysteresis width. The structural studies at different pressures suggest that the pressure-enhanced bistability is due to the strengthening of intermolecular interactions upon pressure increase. An original microscopic Ising-like model including pressure effects is developed to simulate this unprecedented behavior. Overall, this study shows that FeCo complexes could be very sensitive piezo switches with potential use as sensors.     

Stabilization of polymer colloid dispersions with pH-sensitive poly-acrylic acid brushes

Polyelectrolyte brushes are widely used for surface modification of nano-and colloidal particles because of their ability to dramatically change their conformation, hydrophobicity, polarity, charge, etc., as a response to smooth variations in environmental conditions. In this work, we have studied experimentally the stability behavior of polymer colloids with grafted poly-acrylic acid (PAA) surface brushes. We have measured the Fuchs stability ratio (W) as a function of electrolyte concentrations at different pH. It is observed that at pH?<?3, the W values with 1 % and 2 % PAA brushes do not differ significantly from those without PAA, indicating that in their protonated state, the carboxylic groups do not contribute significantly to the colloidal stability. At the intermediate pH?~?5, the PAA brushes are partially deprotonated, and their contribution to the colloidal stability is substantial and increases as the length of the PAA brushes increases. Under alkaline conditions (pH?>?8), since most of the carboxylic groups are ionized, the colloidal stability is much higher than that at pH?~?5. However, the W values are basically the same with 1 % and 2 % PAA, implying that the contribution of the ionized AA in the two cases is practically the same. This experimental evidence indicates that only the ionized AA groups in the outer region of long brushes contribute to colloidal stability, thus supporting the hypothesis of local electroneutrality in the inner region of long brushes (LEA).