On local convergence of sequential quadratically-constrained quadratic-programming type methods,with an extension to variational problems

We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098–1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-Moré type condition. Research of the second author is partially supported by CNPq Grants 300734/95-6 and 471780/2003-0, by PRONEX–Optimization, and by FAPERJ.     

On association schemes all elements of which have valency 1 or 2

In the present note, we investigate schemes S in which each element s satisfies n_s2 and n_s^*s≠2. We show that such a scheme is schurian. More precisely, we show that it is isomorphic to G//t, where G is a finite group and t an involution of G weakly closed in C_G(t).

Groups G with an involution t weakly closed in C_G(t) have been described in Glauberman's Z^*-Theorem [G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966) 403–420] with the help of the largest normal subgroup of G having odd order.     

Small deviation probability via chaining

We obtain several extensions of Talagrand’s lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method.     

Director configurations in nematic droplets with tilted surface anchoring

The polymer dispersed nematic liquid crystal (LC) with the tilted surface anchoring has been studied. The droplet orientational structures with two point surface defects – boojums and the surface ring defect – are formed within the films. The director tilt angle α = 40° ± 4° at the droplet interface and LC surface anchoring strength W_s ~ 10^–6 (J m^?2) have been estimated. The bipolar axes within the studied droplets of oblate ellipsoidal form can be randomly oriented are oriented randomly relatively to the ellipsoid axes as opposed to the droplets with homeotropic and tangential anchoring.     

A Geometric Level-Set Formulation of a Plasma-Sheath Interface

In this paper, we present a new geometric level-set formulation of a plasma-sheath interface arising in plasma physics. We formally derive the explicit dynamics of the interface from the Euler-Poisson equations and study the local-time evolution of the interface and sheath in some special cases.     

A step toward direct fullerene synthesis: C60 fullerene precursors with fluorine in key positions

Several fluorine containing polycyclic aromatic hydrocarbons with exact carbon atom topology of the C₆₀ fullerene have been synthesized. Different numbers of fluorine atoms were introduced in the key positions, as needed for an efficient intramolecular condensation to the fullerene molecule. The polycyclic aromatic compounds obtained represent attractive precursors for rational, high-yield fullerene synthesis by flash vacuum pyrolysis.     

Tri‐ and Tetrasubstituted N‐Phthalimidoaziridines in 1,3‐Dipolar Cycloaddition Reactions

The thermal reactions of the 2,2,3‐trisubstituted N‐phthalimidoaziridine 1a with dimethyl acetylenedicarboxylate (DMAD), thioketones 4a – 4d , and dimethyl azodicarboxylate ( 5 ) proceed even at room temperature leading to the five‐membered cycloadducts 2a, 6 – 8 , and 12 , respectively, with retention of the spatial arrangement of the aziridine substituents, in contrast to the expectation based on the conservation of orbital symmetry in concerted reactions. The analogous reactions of the tetrasubstituted phthalimidoaziridine 1b with thioketones at 40° lead to the 1,3‐thiazolidine derivatives 10 and 11 as mixtures of diastereoisomers. These unexpected results may be explained by either the isomerization of the intermediate azomethine ylides or a non‐concerted stepwise cycloaddition reaction of these ylides with the dipolarophiles. The structures of some adducts have been determined by X‐ray crystallography.