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.
The eigenstates of a diagonalizable PT-symmetric Hamiltonian satisfy unconventional completeness and orthonormality relations. These relations reflect the properties of a pair of bi-orthonormal bases associated with non-hermitean diagonalizable operators. In a similar vein, such a dual pair of bases is shown to possess, in the presence of PT symmetry, a Gram matrix of a particular structure: its inverse is obtained by simply swapping the signs of some its matrix elements. 相似文献
New chromoionophores have been developed, focused on NIR applications so that optode membranes may be used in monolithically integrated optical sensors. The wavelength of maximum absorbance has been estimated for a new model compound by the Pariser-Parr-Pople (PPP) method. Several cyanine type dyes have been tested as membrane chromoionophores. Membrane composition has been altered to overcome solubility problems. In this way, simple pH-sensitive optode membranes have been produced. 相似文献
The controlled squeezing of electromagnetic energy into nanometric volumes via surface plasmon-polariton excitations in plasmonic nanoresonators is analyzed using the concept of an effective electromagnetic mode volume Veff, while taking careful account of the plasmon-polariton dispersion and the electromagnetic energy stored in the metal. Together with the quality factor Q of the cavity resonance, this enables a comparison with dielectric optical cavities, where Veff is limited by diffraction. For a Fabry–Perot type planar metallic cavity, a one-dimensional analytic model as well as a three-dimensional finite-difference time-domain simulation reveal that Veff is not bounded by diffraction, and that Q/Veff increases for decreasing cavity size. In this picture, matter–plasmon interactions can be quantified in terms of Q and Veff, and a resonant cavity model for the enhancement of spontaneous Raman scattering is presented. 相似文献
The hexa-4-dodecyloxybenzoyl derivative 1 of azacrown [18]-N6 was originally reported to have a 'tubular' mesophase on the basis of its large central ring and 6-fold symmetry. Starting in the mesophase, annealing of 1 under a cover slip results in formation of a new crystalline phase that melts directly to an isotropic liquid at the temperature previously observed for the mesophase to isotropic transition. Thus the phase behaviour of 1 is kinetically controlled. The analogous hexa-3,4-bisdodecyloxybenzoyl derivative 2 of azacrown [18]-N6 has no kinetic limitations to its phase changes and has an enantiotropic columnar liquid crystalline phase. We have synthesized side chain copolysiloxanes with a (CH2)11 spacer and 75-84 per cent by weight of the same 4-dodecyloxybenzoyl-[18]-N6 mesogen. The polysiloxanes also display a liquid crystalline phase. 相似文献