A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour.     

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

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properties of truncated Toeplitz operators;

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Carleson-type embedding theorems for the coinvariant subspaces;

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factorizations of pseudocontinuable functions from H¹.

These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.     

Poly(vinyl alcohol) physical hydrogels: new vista on a long serving biomaterial

Poly(vinyl alcohol), PVA, and physical hydrogels derived thereof have an excellent safety profile and a successful history of biomedical applications. However, these materials are hardly in the focus of biomedical research, largely due to poor opportunities in nano- and micro-scale design associated with PVA hydrogels in their current form. In this review we aim to demonstrate that with PVA, a (sub)molecular control over polymer chemistry translates into fine-tuned supramolecular association of chains and this, in turn, defines macroscopic properties of the material. This nano- to micro- to macro- translation of control is unique for PVA and can now be accomplished using modern tools of macromolecular design. We believe that this strategy affords functionalized PVA physical hydrogels which meet the demands of modern nanobiotechnology and have a potential to become an indispensable tool in the design of biomaterials.     

NCN-chelated organoantimony(III) and organobismuth(III) phosphates: synthesis and solid-state and solution structures

.Organoantimony(III) and organobismuth(III) phosphates (LM)(3)(PO(4))(2) [M = Sb (3) and Bi (4)], containing the NCN-chelating ligand L [L = 2,6-(CH(2)NMe(2))(2)C(6)H(3)], were prepared by the simple treatment of parent oxides 1 and 2 with H(3)PO(4). Both compounds were characterized by elemental analysis, electrospray ionization mass spectrometry, and IR and NMR spectroscopy and in the case of 3 by X-ray diffraction techniques. Compound 3 has an interesting behavior in solution, i.e., the formation of two possible conformational isomers, which was studied by (1)H, (13)C, and (31)P NMR spectroscopy.     

From random sphere packings to regular pillar arrays: effect of the macroscopic confinement on hydrodynamic dispersion

Flow and mass transport in bulk and confined chromatographic supports comprising random packings of solid, spherical particles and hexagonal arrays of solid cylinders (regular pillar arrays) are studied over a wide flow velocity range by a numerical analysis scheme, which includes packing generation by a modified Jodrey-Tory algorithm, three-dimensional flow field calculations by the lattice-Boltzmann method, and modeling of advective-diffusive mass transport by a random-walk particle-tracking technique. We demonstrate the impact of the confinement and its cross-sectional geometry (circular, quadratic, semicircular) on transient and asymptotic transverse and longitudinal dispersion in random sphere packings, and also address the influence of protocol-dependent packing disorder and the particle-aspect ratio. Plate height curves are analyzed with the Giddings equation to quantify the transcolumn contribution to eddy dispersion. Confined packings are compared with confined arrays under the condition of identical bed porosity, conduit cross-sectional area, and laterally fully equilibrated geometrical wall and corner effects on dispersion. Fluid dispersion in a regular pillar array is stronger affected by the macroscopic confinement and does not resemble eddy dispersion in random sphere packings, because the regular microstructure cannot function as a mechanical mixer like the random morphology. Giddings' coupling theory fails to preserve the nature of transverse dispersion behind the arrays' plate height curves, which approach a linear velocity-dependence as transverse dispersion becomes velocity-independent. Upon confinement this pseudo-diffusive behavior can outweigh the performance advantage of the regular over the random morphology.     

Intercalation and dynamics of hydrated Fe<Superscript>2+</Superscript> in the vermiculites from Santa Olalla and Ojén

Although the intercalation of Fe³⁺ into layered phyllosicilicates—especially into smectites—attracted much attention in the past two decades, the information about Fe²⁺ loaded phyllosilicates is sparse. Here we present an investigation of the Fe²⁺ exchanged vermiculites from Santa Olalla and Ojén (Andalusia, Spain) by means of Mössbauer spectroscopy. The room temperature Mössbauer spectra are very similar to those of the starting compounds (Na forms) except for a decrease of the contribution of structural Fe³⁺ and a concomitant increase of the contribution of Fe²⁺ sites, indicating an internal redox process. The extent of this redox reaction is different for the two vermiculites. Thus, the intercalated Fe²⁺ acts as an electron mediator from the external medium to the structural Fe³⁺ ions. A new component attributable to intercalated Fe²⁺ is practically invisible in the room temperature Mössbauer spectra, but increases strongly and continuously during cooling to 4.2 K, where it is the dominant feature of the Mössbauer patterns. At 4.2 K, its quadruple splitting amounts to 3.31 mm/s, which is in excellent agreement with the quadrupole slitting of Fe²⁺ coordinated to six water molecules in a highly symmetric octahedral arrangement. The strong decrease of the Mössbauer–Lamb factor of this component with increasing temperature indicates a weak bonding of the Fe²⁺ in the interlayer space.