An exact description of nonlinear wave interaction processes ruled by 2 × 2 hyperbolic systems

By means of a reduction approach to 2 × 2 quasilinear hyperbolic homogeneous systems of first-order PDEs, a full and exhaustive analysis of nonlinear wave interactions is achieved. The alteration in the profile as well as in the wave time distortion of the emerging pulses caused by the interaction process is illustrated in detail through exact solutions of initial value problems. Canonical forms of 2 × 2 systems which allow for special (soliton-like) hyperbolic wave interactions and of interest in applications are also determined.     

Selective hydroxylation of methane by dioxiranes under mild conditions

The direct conversion of methane to methanol at low temperatures was achieved selectively using dioxiranes 1a,b either in the isolated form or generated in situ from aqueous potassium caroate and the parent ketone at a pH close to neutrality. Results suggest that the more powerful dioxirane TFDO (1b) should be the oxidant of choice.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

Ostwald ripening in two dimensions—the rigorous derivation of the equations from the Mullins-Sekerka dynamics

We consider a dilute mixture of a finite number of particles and we are interested in the coarsening of the spatial distribution in two space dimensions under Mullins-Sekerka dynamics. Under the appropriate scaling hypotheses we associate radii and centers to each particle and derive equations for the whole evolution.     

New polyurethanes and polyesters for second‐order nonlinear optical applications

New polymers for second‐order nonlinear optical (NLO) applications were synthesized and characterized. They were distinguished by the presence of chromophore groups, with various molecular hyperpolarizability values, used as pendants on substantially rigid backbones. The polymers were prepared through the reaction of tolylene‐2,4‐diisocyanate, or a suitable alkyloxyphthaloyl dichloride, with the N,N‐diethanol‐4‐(phenyl) group azo‐linked to a nitrofluorenone, nitrostilbene, nitrooxadiazole, or nitrothiadiazole moiety. The polymers exhibited good thermal stability, high glass‐transition temperatures, and an absence of crystallinity. The second‐order NLO properties of thin, transparent poled films, prepared by spin coating and thermal corona poling, were investigated for some of the polymers. The second harmonic coefficients, ranging between 18 and 25 pm/V, depended more on the alignment of the chromophore groups along the direction of the poling field than on their molecular hyperpolarizability. The temporal stability of the NLO properties of the polymers was also investigated. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 3013–3022, 2004     

A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫_Ω g(detDu)dx with respect to sequences of functions fromW ^1,n (Ω;ℝ ⁿ ) which converge inL ¹ (Ωℝ ⁿ ) and are uniformly bounded inW ^1,n−1 (Ω;ℝ ⁿ ), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results. Supported by MURST, Gruppo Nazionale 40% Partially supported by Australian Research Council     

Stereoselective dioxirane hydroxylations and the synthesis of tripod boronic acid esters

Methyl(trifluoromethyl)dioxirane (TFDO, 1b), a powerful yet selective oxidant, was employed to achieve in high yield the direct stereoselective hydroxylation at tert-CH of cis,cis-1,3,5-trimethylcyclohexane (4), yielding triol 7 bearing all-axial disposition of the three OH groups. Similarly, TFDO oxidation of 1,3- and of 1,4-dimethylcyclohexane gave the corresponding Z-diols 5 and 6, respectively. Triol 7 was a convenient starting material to synthesize a novel borate—that is, 1-bora-2,8,9-trioxa-3,5,7-trimethyladamantane (8)—having a peculiar cage-shaped ‘tripod’ structure. From triol 7, novel tripod arylboronic Brönsted-assisted Lewis acids (BLA) could be obtained, as exemplified by 10a and 10b.