Dimension Reduction for the Landau-de Gennes Model on Curved Nematic Thin Films

We use the method of \(\Gamma \)-convergence to study the behavior of the Landau-de Gennes model for a nematic liquid crystalline film attached to a general fixed surface in the limit of vanishing thickness. This paper generalizes the approach in Golovaty et al. (J Nonlinear Sci 25(6):1431–1451, 2015) where we considered a similar problem for a planar surface. Since the anchoring energy dominates when the thickness of the film is small, it is essential to understand its influence on the structure of the minimizers of the limiting energy. In particular, the anchoring energy dictates the class of admissible competitors and the structure of the limiting problem. We assume general weak anchoring conditions on the top and the bottom surfaces of the film and strong Dirichlet boundary conditions on the lateral boundary of the film when the surface is not closed. We establish a general convergence result to an energy defined on the surface that involves a somewhat surprising remnant of the normal component of the tensor gradient. Then we exhibit one effect of curvature through an analysis of the behavior of minimizers to the limiting problem when the substrate is a frustum.     

An axiomatic duality framework for the theta body and related convex corners

Lovász theta function and the related theta body of graphs have been in the center of the intersection of four research areas: combinatorial optimization, graph theory, information theory, and semidefinite optimization. In this paper, utilizing a modern convex optimization viewpoint, we provide a set of minimal conditions (axioms) under which certain key, desired properties are generalized, including the main equivalent characterizations of the theta function, the theta body of graphs, and the corresponding antiblocking duality relations. Our framework describes several semidefinite and polyhedral relaxations of the stable set polytope of a graph as generalized theta bodies. As a by-product of our approach, we introduce the notion of “Schur Lifting” of cones which is dual to PSD Lifting (more commonly used in SDP relaxations of combinatorial optimization problems) in our axiomatic generalization. We also generalize the notion of complements of graphs to diagonally scaling-invariant polyhedral cones. Finally, we provide a weighted generalization of the copositive formulation of the fractional chromatic number by Dukanovic and Rendl from 2010.     

On 2-Connected Spanning Subgraphs with Bounded Degree in K 1,r -Free Graphs

For any integer r > 1, an r-trestle of a graph G is a 2-connected spanning subgraph F with maximum degree Δ(F) ≤ r. A graph G is called K _1,r -free if G has no K _1,r as an induced subgraph. Inspired by the work of Ryjáček and Tkáč, we show that every 2-connected K _1,r -free graph has an r-trestle. The paper concludes with a corollary of this result for the existence of k-walks.     

Novel estradiol derivatives labeled with Ru,W, and Co complexes. Influence on hormone-receptor affinity of several organometallic groups at the 17 alpha position

In order to elucidate the extent to which recognition of the estrogen receptor is influenced by addition of an organometallic substituent at the 17 alpha position, modification of 17 beta-estradiol at this position was carried out by using the organometallic groups -C identical to C(eta 5-C5H4)RuCp, CH2-(eta 5-C5H4)RuCp, -C identical to C-(eta 5-C5H4)-W(CO)3(Me), -(C identical to CCHO)Co2(CO)6, and -(C identical to CCH2OH)Co2(CO)6. The relative binding affinity (RBA) values for estradiol receptor alpha showed that recognition was good (RBA between 20 and 13.5%) when the organometallic moiety was attached at the end of a rigid alkyne spacer. However, the affinity of the modified hormone for the receptor was severely reduced (RBA = 1%) for a substituent such as -CH2-(eta 5-C5H4)RuCP, in which the spacer is reduced to a single flexible sp3 carbon atom, allowing the organometallic moiety greater freedom of movement around the attachment point. The RBA values found were in agreement with results obtained from a molecular-modeling study in which 5, an organometallic hormone with a rigid spacer, or 7, a molecule with a flexible spacer, was inserted into the cavity of the recently characterized Ligand-Binding Domain of estrogen receptor alpha.     

Glycosylation of organic phosphorus thio- and selenoacids—II: The reaction of organic phosphorus thioselenoacids with glycosyl bromides under kinetically and thermodynamically controlled conditions

Ambident anions derived from phosphorus thioselcnoacids were glycosylated with 2,3,4,6 - tetra - O - acetyl - α - d - glucopyranosyl bromide, 2,3,4,6 - tetra - O - acetyl - α - d - galactopyranosyl bromide and 2,3,4 - tri- O - acetyl - α - d - xylopyranosyl bromide. The products were β-Se-glucosyl- and β-S-glucosylthioselenoates. The Se/S ratio of the glycosylated phosphorothioselenoates depends on the reaction conditions. At higher temperatures an equilibrium was observed. As a result of this equilibrium the Se/S ratio of the linkages formed in the glycosylated products was different from that observed under kinetic control. The structures of the glycosylated phosphorothioselenoates were confirmed by spectroscopy, independent synthesis and selective oxidation.     

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lovász and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N₀-, N-, or N₊-rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N₀- or the N-rank. Hence we obtain the ranks for these graphs, and we also present some graph-minor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N₊-ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N₊-rank of a graph.Research of these authors was supported in part by a PREA from Ontario, Canada and research grants from NSERC.Mathematics Subject Classification (2000): 0C10, 90C22, 90C27, 47D20     

CEMS of nitride coatings in agressive environments

The corrosion properties of single layered TiN and CrN films have been compared to bi-layered and multi-layered Ti/TiN films. XPS has showed that in humid SO₂ atmosphere the best corrosion properties have been achieved by a multi-layered Ti/TiN coating. Cyclic voltammetry in acetate buffer has been applied to measure the porousity and corrosion resistance of coatings. The best results have been achieved by multi-layered Ti/TiN and CrN films. Conversion electron Mössbauer spectroscopy has been used to study the changes in the interface Fe/TiN during thermal treatment in UHV. It has been shown that the amount of iron nitrides in the interface increases with increasing temperature.