Compactness and convergence rates in the combinatorial integral approximation decomposition

The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the \(\hbox {weak}^*\) topology of \(L^\infty \) if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.

     

Galois extensions as functors of comodules

Let A be a finite Hopf algebra over a commutative ring k. We show a one-to-one correspondence between the A-Galois extensions of k and certain functors from the category of A-comodules to the category of k-modules.     

Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.     

Radical polymerization of n-substituted methacrylamides

The relationship between the structure of monomer and kinetics of the radical polymerization of N-ethylmethacrylamide, N-butylmethacrylamide and N-phenylmethacrylamide in methanol and in dimethylsulphoxide was investigated. The reaction order with respect to initiator is 0·5 in all cases; the order with respect to monomer is independent of the type of substituent but depends on the solvent and on the viscosity of the reaction mixture. The polymerization rate, the value of $\text{κ}{}_{\mathrm{p}}\text{/κ}{}_{\mathrm{t}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, and the initiator efficiency decrease in the series N-phenylmethacrylamide, N-ethylmethacrylamide and N-butylmethacrylamide. The overall activation energy of polymerization for the monomers lies between 16 and 20 kcal/mol.     

Origin of Schottky barriers in gold contacts on GaAs110

Gold contacts on n-type GaAs(110) have been investigated using scanning tunneling microscopy and spectroscopy in cross-sectional configuration. In spatially resolved current voltage spectroscopy the Schottky barrier potential is visible. We find signatures of delocalized gap states at the interface decaying into the semiconductor and observe a defect density at the interface below 3 x 10(13) cm(-2). Both findings support that the Fermi level pinning at the Au/GaAs(110) interface is dominated by metal-induced gap states.     

Growth of Pt thin films on WSe2

The morphology and structure of Pt deposited on a WSe₂(0 0 0 1) van der Waals surface have been investigated by reflection high energy electron diffraction and scanning tunneling microscopy. At room temperature, the initial growth is characterized by the formation of three-dimensional fcc Pt islands with (1 1 1) orientation. In contrast, at higher temperatures of about 450 °C the formation of a novel chemically ordered Pt-Se alloy is observed. Based on the diffraction patterns, a tetragonal DO₂₂-type structure of a Pt₃Se compound is suggested. With increasing Pt thickness, this chemically ordered alloy disappears and an additional superstructure occurs, which is accompanied by the coalescence of the islands. The observed superstructure is attributed to a strong Se diffusion towards the growth surface, forming most likely a PtSe₂ alloy with the CdI₂-type layered structure on the top surface. Due to the lateral lattice mismatch between the Pt(1 1 1) layers and the PtSe₂(1 1 1) top layer, a Moiré pattern with a period of 1.1 nm is created, which might be used as a long-range atomic pattern for further nanostructure growth.     

Evidence for charge orbital and spin stripe order in an overdoped manganite

Overdoped La0.42Sr1.58MnO4 exhibits a complex ordering of charges, orbitals, and spins. Neutron diffraction experiments reveal three incommensurate and one commensurate order parameters to be tightly coupled. The position and the shape of the distinct superstructure scattering as well as higher-order signals are inconsistent with a harmonic charge and spin-density-wave picture but point to a stripe arrangement in which ferromagnetic zigzag chains are disrupted by excess Mn(4+).