Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications

We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush-Kuhn-Tucker conditions do not provide necessary optimality criteria. In order to obtain reasonable first order conditions under very weak assumptions, we introduce several MPVC-tailored constraint qualifications, discuss their relation, and prove an optimality condition which may be viewed as the counterpart of what is called M-stationarity in the MPEC-field.     

An oxo‐bridged centrosymmetric tetranuclear titanium compound

The title compound, octa‐tert‐butoxybis­[μ₃‐2,2′‐(N‐methyl­imino)­diethanolato]­di‐μ‐oxo‐tetratitanium(IV), [Ti₂O{(OCH₂CH₂)₂(NCH₃)}{(CH₃)₃CO}₄]₂ or [Ti₄(C₅H₁₁NO₂)₂(C₄H₉O)₈O₂], lies about an inversion centre, and displays the less usual zigzag configuration. One O atom of the N‐methyl­diethoxo­amine ligand bridges the symmetry‐related Ti atoms, while the other bridges the two independent Ti atoms, with the N atom binding to give a facial configuration. Four ^tBuO⁻ ligands and a bridging oxide complete the respective five‐ and sixfold coordination of the two Ti atoms. The Ti—O bond lengths range in a self‐consistent fashion from 1.7624 (17) to 2.0878 (18) Å, while the Ti—N bond length is 2.374 (2) Å.     

Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

We show that under some natural ergodicity assumptions, extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^∞-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.     

Correspondences between the classical electrostatic Thomson problem and atomic electronic structure

Correspondences between the Thomson problem and atomic electron shell-filling patterns are observed as systematic non-uniformities in the distribution of potential energy necessary to change configurations of N ≤ 100 electrons into discrete geometries of neighboring N ? 1 systems. These non-uniformities yield electron energy pairs, intra-subshell pattern similarities with empirical ionization energy, and a salient pattern that coincides with size-normalized empirical ionization energies. Spatial symmetry limitations on discrete charges constrained to a spherical volume are conjectured as underlying physical mechanisms responsible for shell-filling patterns in atomic electronic structure and the Periodic Law.     

An algebraic multigrid method for high order time-discretizations of the div-grad and the curl-curl equations

We present an algebraic multigrid algorithm for fully coupled implicit Runge–Kutta and Boundary Value Method time-discretizations of the div-grad and curl-curl equations. The algorithm uses a blocksmoother and a multigrid hierarchy derived from the hierarchy built by any algebraic multigrid algorithm for the stationary version of the problem. By a theoretical analysis and numerical experiments, we show that the convergence is similar to or better than the convergence of the scalar algebraic multigrid algorithm on which it is based. The algorithm benefits from several possibilities for implementation optimization. This results in a computational complexity which, for a modest number of stages, scales almost linearly as a function of the number of variables.     

The influence of emulsion structure and stability on lipid digestion

The digestion and metabolism of lipids continues to generate considerable scientific interest, with food emulsions increasingly being seen as a mechanism by which lipid uptake may be controlled. Scientific advancement in this field is partly being driven by the ongoing need to address the obesity crisis, for which the enhancement of satiety and/or reduction of energy intake is seen as a positive solution in achieving more effective weight management. Yet the ability to regulate lipid uptake is also seen as beneficial in other areas, such as improved nutrition for the young and/or elderly and in cardiovascular protection.Because of the complexity of food digestion, the majority of research in this area has been applied to model or highly controlled systems. Through this approach it is becoming increasingly apparent that food emulsion structure and stability does have a contributing role on lipid digestion and metabolism. There is now a clear indication of how emulsion stability within the stomach affects emptying rates. There have been considerable developments in understanding the relationship between interfacial composition and lipolysis in both the gastric and intestinal regions, and how this relates to lipid uptake/metabolism. There is also an emerging understanding of the contribution of gastrointestinal biophysics to emulsion structure and stability, and how intestinal motility is in turn impacted by structural aspects, such as relative changes in particle size.Understanding of lipid digestion has been progressed through recent advancements in the sophistication of in vitro models. These are now seen as providing a more realistic representation of physiological conditions, both in terms of biochemical environment, and the biophysics of the gastrointestinal tract. Improvements in the validity of such models against in vivo and clinical behaviours is allowing aspects of emulsion digestion to be observed without the immediate need of costly human trials. Accordingly, emulsion systems with increasing structural complexity are now able to be characterised in terms of digestion behaviours. The ability to design food emulsions with specific lipid digestion profiles may allow the development of mainstream foods with particular physiological properties, such as enhanced satiation, or targeted delivery.     

Prediction of yield and long‐term failure of oriented polypropylene: Kinetics and anisotropy

The time‐dependent yield and failure behavior of off‐axis loaded uniaxially oriented polypropylene tape is investigated. The yield and failure behavior is described with an anisotropic viscoplastic model. A viscoplastic flow rule is used with an equivalent stress, based on Hill's anisotropic yield criterion, and the Eyring flow theory combined with a critical equivalent strain definition. This model is based on factorization of the rate and draw ratio dependence and is capable of quantitatively predicting the rate, angle and draw ratio dependence of the yield stress as well as time‐to‐failure in various off‐axis tensile loading conditions characterized solely from the transverse direction. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2026–2035, 2009     

The synthesis and self‐assembly of ABA amphiphilic block copolymers containing styrene and oligo(ethylene glycol) methyl ether methacrylate in dilute aqueous solutions: Elevated cloud point temperatures for thermoresponsive micelles

A series of ABA amphiphilic triblock copolymers possessing polystyrene (PS) central hydrophobic blocks, one group with “short” PS blocks (DP = 54–86) and one with “long” PS blocks (DP = 183–204) were synthesized by atom transfer radical polymerization. The outer hydrophilic blocks were various lengths of poly(oligoethylene glycol methyl ether) methacrylate, a comb‐like polymer. The critical aggregation concentrations were recorded for certain block copolymer samples and were found to be in the range circa 10⁻⁹ mol L⁻¹ for short PS blocks and circa 10⁻¹² mol L⁻¹ for long PS blocks. Dilute aqueous solutions were analyzed by transmission electron microscopy (TEM) and demonstrated that the short PS block copolymers formed spherical micelles and the long PS block copolymers formed predominantly spherical micelles with smaller proportions of cylindrical and Y‐branched cylindrical micelles. Dynamic light scattering analysis results agreed with the TEM observations demonstrating variations in micelle size with PS and POEGMA chain length: the hydrodynamic diameters (D_H) of the shorter PS block copolymer micelles increased with increasing POEGMA block lengths while maintaining similar PS micellar core diameters (D_C); in contrast the values of D_H and D_C for the longer PS block copolymer micelles decreased. Surface‐pressure isotherms were recorded for two of the samples and these indicated close packing of a short PS block copolymer at the air–water interface. The aggregate solutions were demonstrated to be stable over a 38‐day period with no change in aggregate size or noticeable precipitation. The cloud point temperatures of certain block copolymer aggregate solutions were measured and found to be in the range 76–93 °C; significantly these were ∼11 °C higher in temperature than those of POEGMA homopolymer samples with similar chain lengths. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 7739–7756, 2008