Blaschke, Privaloff, Reade and Saks Theorems for Diffusion Equations on Lie Groups

We prove some asymptotic characterizations for the subsolutions to a class of diffusion equations on homogeneous Lie groups. These results are the diffusion counterpart of the classical Blaschke, Privaloff, Reade and Saks Theorems for harmonic functions.     

On some notions of good reduction for endomorphisms of the projective line

Let Φ be an endomorphism of ${\mathbb{P}^1_{\overline{\mathbb{Q}}}}$ , the projective line over the algebraic closure of ${\mathbb{Q}}$ , of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds also for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ _v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ _v is separable, then Φ has simple good reduction at v.     

Poset embeddings of Hilbert functions

For a standard graded algebra $R$ , we consider embeddings of the poset of Hilbert functions of $R$ -ideals into the poset of $R$ -ideals, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements–Lindström is proved. We exhibit a condition on the embedding that ensures that the classification of Hilbert functions is obtained with images of lexicographic segment ideals.     

On the 2D Cahn–Hilliard Equation with Inertial Term

Galenko et al. proposed a modified Cahn–Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solutions. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.     

A Radiation Condition for the 2-D Helmholtz Equation in Stratified Media

We study the 2-D Helmholtz equation in perturbed stratified media, allowing the existence of guided waves. Our assumptions on the perturbing and source terms are not too restrictive. We prove two results. Firstly, we introduce a Sommerfeld–Rellich radiation condition and prove the uniqueness of the solution for the studied equation. Then, by careful asymptotic estimates, we prove the existence of a bounded solution satisfying our radiation condition.     

Analysis of the Cahn–Hilliard Equation with a Chemical Potential Dependent Mobility

The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn–Hilliard equations with a mobility depending on the chemical potential. Such models arise from generalizations of the (classical) Cahn–Hilliard equation due to Gurtin.     

Controlling stereochemical outcomes of asymmetric processes by catalyst remote molecular functionalizations: chiral diamino-oligothiophenes (DATs) as ligands in asymmetric catalysis

The synthesis, characterization, and structure-guided application of a new class of highly versatile chiral C(2)-symmetric diamine-oligothiophene ligands in Pd-catalyzed asymmetric transformations are presented. Experimental investigations of the intimate role of pendant pi-conjugate oligothiophenes in determining the catalytic activity of the corresponding chiral Pd complexes are described. Their unusual behavior opens up new routes toward the logical design of finely tuned organometallic catalysts by remote structural functionalizations.     

One-dimensional thermotropic dilatation area of lipid headgroups within lamellar lipid/DNA complexes

Using simultaneous synchrotron small- and wide-angle X-ray diffraction (SWAXD), we investigated the thermotropic behavior of a cationic lipid mixture of DOTAP-DOPC (1,2-dioleoyl-3-trimethylammonium-propane-dioleoylphosphatidylcholine) liposomes complexed with calf thymus DNA. The DOTAP-DOPC/DNA complex reacts to temperature change by a bilayer compression normal to its surface and an expansion of the DNA in the plane of the rod lattice. By applying two independent recently developed models, we show here for the first time that the thermotropic dilatation area of lipid headgroups within the complexes is not isotropic but occurs parallel to the 1D DNA lattice (i.e., along the direction perpendicular to the DNA axis). Our results shed light on the role of spatial dimensionality in the DNA packing density within lamellar lipoplexes and provide experimental evidence that the interaction between DNA molecules confined between lipid bilayers can be regarded as a 1D problem.     

300 microJ noncollinear optical parametric amplifier in the visible at 1 kHz repetition rate

We demonstrate an order-of-magnitude energy scaling of a white-light seeded noncollinear optical parametric amplifier in the visible. The generated pulses, tunable between 520 and 650 nm with sub-25-fs duration, had energies up to 310 microJ with 20% blue-pump-to-signal energy conversion efficiency at 540 nm. This new ultrafast source will make possible numerous extreme nonlinear optics applications. As a first application, we demonstrate the generation of tunable vacuum ultraviolet pulses.     

Tailoring (n,m) structure of single-walled carbon nanotubes by modifying reaction conditions and the nature of the support of CoMo catalysts

The (n,m) population distribution of single-walled carbon nanotubes obtained on supported CoMo catalysts has been determined by photoluminescence and optical absorption. It has been found that the (n,m) distribution can be controlled by varying the gaseous feed composition, the reaction temperature, and the type of catalyst support used. When using CO as a feed over CoMo/SiO2 catalysts, increasing the synthesis temperature results in an increase in nanotube diameter, without a change in the chiral angle. By contrast, by changing the support from SiO2 to MgO, nanotubes with similar diameter but different chiral angles are obtained. Finally, keeping the same reaction conditions but varying the composition of the gaseous feed results in different (n,m) distribution. The clearly different distributions obtained when varying catalysts support and/or reaction conditions demonstrate that the (n,m) distribution is a result of differences in the growth kinetics, which in turn depends on the nanotube cap-metal cluster interaction.