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.     

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.     

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     

On the regularity over positively graded algebras

We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring. In this case both are characterizations of the so-called Castelnuovo–Mumford regularity. Moreover, we can characterize a standard graded polynomial ring as a K-algebra with extremal properties with respect to the Tor- and the local-regularity. For modules of finite projective dimension we get a nice formula relating the two regularity notions. Interesting examples are given to help to understand the relationship between the Tor- and the local-regularity in general.     

Higher Hopf formulae for homology via Galois Theory

We use Janelidze's Categorical Galois Theory to extend Brown and Ellis's higher Hopf formulae for homology of groups to arbitrary semi-abelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the Barr-Beck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A is the category Gp of all groups and B is the category Ab of all abelian groups, this yields a new proof for Brown and Ellis's formulae. We also give explicit formulae in the cases of groups vs. k-nilpotent groups, groups vs. k-solvable groups and precrossed modules vs. crossed modules.     

Synthesis of 5-trichloromethyl-Δ-1,2,4-oxadiazolines and their rearrangement into formamidine derivatives

A series of 5-trichloro-Δ⁴-1,2,4-oxadiazolines have been synthesised by 1,3-dipolar cycloaddition of nitrones to trichloroacetonitrile. These oxadiazolines rearrange into formamidine derivatives, via ring opening and a 1,2-aryl shift from carbon to the adjacent amino nitrogen. Both cycloaddition and rearrangement are facilitated when electron deficient nitriles and electron rich nitrones are used.