Gradient estimates for a class of elliptic systems

Summary Gradient bounds are proved for solutions to a class of second order elliptic systems in divergence form. The main condition on this class is a generalization of the assumption that the system be the Euler-Lagrange system of equations for a functional depending only on the modulus of the gradient of the solution.     

On critical exponents for some quasilinear parabolic equations

We study the Cauchy problem for the quasilinear parabolic equation where p > 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with ? ? sψ′(s)/ψ(s) ? θ for some θ > 0. If 1 < p < 1 + 2/N, there are no global positive solutions, whereas if p > 1 + 2/N, there are global, positive solutions for small initial data.     

Deposition of DNA rafts on cationic SAMs on silicon [100

We demonstrate a guided self-assembly approach to the fabrication of DNA nanostructures on silicon substrates. DNA oligonucleotides self-assemble into "rafts" 8 x 37 x 2 nm in size. The rafts bind to cationic SAMs on silicon wafers. Electron-beam lithography of a thin poly(methyl methacrylate) (PMMA) resist layer was used to define trenches, and (3-aminopropyl)triethoxysilane (APTES), a cationic SAM precursor, was deposited from aqueous solution onto the exposed silicon dioxide at the trench bottoms. The remaining PMMA can be cleanly stripped off with dichloromethane, leaving APTES layers 0.7-1.2 nm in thickness and 110 nm in width. DNA rafts bind selectively to the resulting APTES stripes. The coverage of DNA rafts on adjacent areas of silicon dioxide is 20 times lower than on the APTES stripes. The topographic features of the rafts, measured by AFM, are identical to those of rafts deposited on wide-area SAMs. Binding to the APTES stripes appears to be very strong as indicated by "jamming" of the rafts at a saturation coverage of 42% and the stability to repeated AFM scanning in air.     

Four-dimensional mapping model for two-frequency electron cyclotron resonance heating

Two-frequency electron cyclotron resonance heating (ECRH) is modelled by a four-dimensional symplectic mapping derived from the nonrelativistic single particle equations of motion. The model includes changes in parallel energy due to the spatially separate resonance zones, not given by previous two-dimensional models. Fixed points are located and their linear stability limits determined. Resonances in action space are calculated along with their widths and used to obtain the adiabatic barrier to heating. Quasilinear diffusion coefficients are derived for the stochastic regime and found to agree well with numerical calculations. The primary diffusion in perpendicular energy can couple to the parallel motion, leading to diffusion in parallel energy. The resulting diffusion coefficient is calculated analytically and compared with numerical results. The much weaker Arnold diffusion along a resonance layer is also treated analytically, yielding diffusion coefficients in reasonable agreement with numerical values.     

Solvability of quasilinear elliptic equations with nonlinear boundary conditions. II

It is shown that solvability of the second order quasilinear elliptic equation Qu = 0 in Ω with first order nonlinear boundary condition Nu = 0 on ?Ω can be inferred from appropriate estimates on solutions of the problem $\text{Qu = ?}$ in Ω, Nu = 0 on ?Ω as ? varies over a suitable function class. This result improves previous work of the author, where estimates were required on solutions of $\text{Qu = ?}$ in Ω, Nu = ? on ?Ω as (?, ?) varies over some function space. The value of this improvement is demonstrated by some examples.     

A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients

In recent years, there has been considerable interest in extending the well-known Calderon-Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the L^p estimates. We shall show that the Morrey space estimates follow from the L^p estimates via an elementary argument which is very similar to that used by Campanato.     

Time scale to ergodicity in the Fermi-Pasta-Ulam system

We study the approach to near-equipartition in the N-dimensional Fermi-Pasta-Ulam Hamiltonian with quartic (hard spring) nonlinearity. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy which is independent of N, one can reach equipartition if one waits for a time proportional to N(2). Below this critical energy the time to equipartition is exponentially long. We develop a theory to determine the time evolution and the excitation of the nonlinear modes based on a resonant normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for equipartition, the time scale to equipartition, and the form of the nonlinear modes below equipartition, in qualitative agreement with the numerical results. (c) 1995 American Institute of Physics.     

An efficient one-pot synthesis of annulated pyridines utilising a directed ortho-metallation/transmetallation approach

The ortho-alkylation of Boc-protected aminopyridines with α,ω-dihaloalkanes followed by in situ cyclisation, resulted in the corresponding annulated pyridine derivatives in good to excellent yields. The effect of the alkylating and chelating agents, the transmetallation additives and the directing group was examined.     

Characterization of the particulate methane monooxygenase metal centers in multiple redox states by X-ray absorption spectroscopy

The integral membrane enzyme particulate methane monooxygenase (pMMO) converts methane, the most inert hydrocarbon, to methanol under ambient conditions. The 2.8-A resolution pMMO crystal structure revealed three metal sites: a mononuclear copper center, a dinuclear copper center, and a nonphysiological mononuclear zinc center. Although not found in the crystal structure, solution samples of pMMO also contain iron. We have used X-ray absorption spectroscopy to analyze the oxidation states and coordination environments of the pMMO metal centers in as-isolated (pMMO(iso)), chemically reduced (pMMO(red)), and chemically oxidized (pMMO(ox)) samples. X-ray absorption near-edge spectra (XANES) indicate that pMMO(iso) contains both Cu(I) and Cu(II) and that the pMMO Cu centers can undergo redox chemistry. Extended X-ray absorption fine structure (EXAFS) analysis reveals a Cu-Cu interaction in all redox forms of the enzyme. The Cu-Cu distance increases from 2.51 to 2.65 A upon reduction, concomitant with an increase in the average Cu-O/N bond lengths. Appropriate Cu2 model complexes were used to refine and validate the EXAFS fitting protocols for pMMO(iso). Analysis of Fe EXAFS data combined with electron paramagnetic resonance (EPR) spectra indicates that Fe, present as Fe(III), is consistent with heme impurities. These findings are complementary to the crystallographic data and provide new insight into the oxidation states and possible electronic structures of the pMMO Cu ions.