Eigenvalue problems associated with Korn's inequalities in the theory of elasticity

Interest in the possible applications of a priori inequalities in linear elasticity theory motivated the present investigation. Korn's inequality under various side conditions is considered, with emphasis on the Korn's constant. In the "second case" of Korn's inequality, a variational approach leads to an eigenvalue problem; it is shown that, for simply-connected two-dimensional regions, the problem of determining the spectrum of this eigenvalue problem is equivalent to finding the values of Poisson's ratio for which the displacement boundary-value problem of linear homogeneous isotropic elastostatics has a non-unique solution.

Previous work on the uniqueness and non-uniqueness issue for the latter problem is examined and the results applied to the spectrum of the Korn eigenvalue problem. In this way, further information on the Korn constant for general regions is obtained.

A generalization of the "main case" of Korn's inequality is introduced and the associated eigenvalue problem is a gain related to the displacement boundary-value problem of linear elastostatics in two dimensions.

Investigations in the theory of electromagnetic scattering from expanding dielectric obstacles

An equation for the reflection which results when an expanding dielectric slab scatters normally incident plane electromagnetic waves is derived using the invariant imbedding concept. The equation is solved approximately and the character of the solution is investigated. Also, an equation for the radiation transmitted through such a slab is similarly obtained. An alternative formulation of the slab problem is presented which is applicable to the analogous problem in spherical geometry. The form of an equation for the modal reflections from a nonrelativistically expanding sphere is obtained and some salient features of the solution are described. In all cases the material is assumed to be a nondispersive, nonmagnetic dielectric whose rest frame properties are slowly varying.

Diffraction properties of volume holographic gratings for an ultrashort pulsed beam with different polarization states readout

The diffraction properties of volume holographic gratings are studied when the gratings are illuminated by an ultrashort pulsed beam with different polarization states. The developed coupled wave theory of Kogelnik is used. Considering the dispersion effect of the grating media, solutions for the diffracted and transmitted intensities, diffraction efficiencies and the bandwidths of the gratings are given in transmission volume holographic gratings and reflection volume holographic gratings. The bandwidths of the gratings are reduced by the dispersion effect of the grating media. They also have different influences on the diffraction of an ultrashort pulsed beam with different polarization states. For different values of the ratio of the spectral bandwidth of the input pulse to that of the grating, the changes of the spectral and temporal distributions of the diffracted intensities, as well as the diffraction efficiencies of the gratings are shown.

Electro-optic modulation of light propagating near the optic axis with any polarization in uniaxial crystals

The electro-optic effect in uniaxial crystals for light propagating near the optic axis with any polarization has been analyzed. The passive and the electrically induced birefringences and the rotation of polarization direction in crystals have been calculated, and the conoscopic interference figures under orthogonal polariscopes for different polarizer directions have been plotted. The extinction areas caused by the rotation of polarization direction in crystals change with the polarizer direction, but the two heads of the induced optical axes do not vary, which are always on the induced principal axis with bigger refractive index. The directions of polariscopes are always extinction, and the +/- 45 degrees directions with polarizer are always complete transmission. The conoscopic interference figures for LiNbO3 crystals have been demonstrated experimentally by rotating polariscopes directions, which accord with the theoretically calculating plots. (c) 2006 Elsevier GmbH. All rights reserved.

Diffraction of an ultrashort pulsed beam with arbitrary polarization state from a volume holographic grating in LiNbO3 crystals

Based on a modified coupled wave theory of Kogelnik, we have studied the diffraction of an ultrashort pulsed beam with an arbitrary polarization state from a volume holographic grating in photorefractive LiNbO3 crystals. The results indicate that the diffracted intensity distributions in the spectral and temporal domains and the diffraction efficiency of the grating are both changed by the polarization state and spectral bandwidth of the input pulsed beam. A method is given of choosing the grating parameters and input conditions to obtain a large variation range of the spectral bandwidth of the diffracted pulsed beam with an appropriate diffraction efficiency. Our study presents a possibility of using a volume holographic grating recorded in anisotropic materials to shape a broadband ultrashort pulsed beam by modulating its polarization state.

Wavelength dependence of light-induced domain nucleation in MgO-doped congruent LiNbO3 crystal

Within the wavelength range from 351 to 799 nm, the different reductions of nucleation field induced by the focused continuous laser irradiation are achieved in the 5 mol % MgO-doped congruent LiNbO3 crystals. The reduction proportion increases exponentially with decreasing irradiation wavelength and decreases exponentially with increasing irradiation wavelength. At one given wavelength, the reduction proportion increases exponentially with increasing irradiation intensity. An assumption is proposed that the reduction of nucleation field is directly related to the defect structure of crystal lattice generated by the complex coaction of incident irradiation field and external electric field. (c) 2007 American Institute of Physics.

Learning to perceive in the sensor motor approach: Piaget's theory of equilibration interpreted dynamically

Learning to perceive is faced with a classical paradox: if understanding is required for perception, how can we learn to perceive something new, something we do not yet understand? According to the sensorimotor approach, perception involves mastery of regular sensorimotor co-variations that depend on the agent and the environment, also known as the "laws" of sensorimotor contingencies (SMCs). In this sense, perception involves enacting relevant sensorimotor skills in each situation. It is important for this proposal that such skills can be learned and refined with experience and yet up to this date, the sensorimotor approach has had no explicit theory of perceptual learning. The situation is made more complex if we acknowledge the open-ended nature of human learning. In this paper we propose Piaget's theory of equilibration as a potential candidate to fulfill this role. This theory highlights the importance of intrinsic sensorimotor norms, in terms of the closure of sensorimotor schemes. It also explains how the equilibration of a sensorimotor organization faced with novelty or breakdowns proceeds by re-shaping pre-existing structures in coupling with dynamical regularities of the world. This way learning to perceive is guided by the equilibration of emerging forms of skillful coping with the world. We demonstrate the compatibility between Piaget's theory and the sensorimotor approach by providing a dynamical formalization of equilibration to give an explicit micro-genetic account of sensorimotor learning and, by extension, of how we learn to perceive. This allows us to draw important lessons in the form of general principles for open-ended sensorimotor learning, including the need for an intrinsic normative evaluation by the agent itself. We also explore implications of our micro-genetic account at the personal level.

Measurements of light scattering from glass substrates by total integrated scattering

A total integrated scattering (TIS) measurement was performed to investigate the surface and volume scattering of K9 glass substrates with low reflectance. Ag layers with thicknesses of 60 nm were deposited on the front and back surfaces of the K9 glass substrates by the magnetron sputtering technique. Surface scattering of the K9 glass substrate was obtained by the TIS measurement of the Ag layers on the assumption that the Ag layers and the K9 substrate had the same surface profile. Volume scattering of the substrates was deduced by subtracting the front and back surface scattering from the total scattering of the substrates. (c) 2005 Optical Society of America.

Hartle's model within the general theory of perturbatiye matchings: the change in mass

Hartle's model provides the most widely used analytic framework to describe isolated compact bodies rotating slowly in equilibrium up to second order in perturbations in the context of General Relativity. Apart from some explicit assumptions, there are some implicit, like the "continuity" of the functions in the perturbed metric across the surface of the body. In this work we sketch the basics for the analysis of the second order problem using the modern theory of perturbed matchings. In particular, the result we present is that when the energy density of the fluid in the static configuration does not vanish at the boundary, one of the functions of the second order perturbation in the setting of the original work by Hartle is not continuous. This discrepancy affects the calculation of the change in mass of the rotating star with respect to the static configuration needed to keep the central energy density unchanged.