Linear thermodynamic systems with memory. II. Necessary and sufficient conditions for applicability of the second law of thermodynamics

The paper is the second work in a series devoted to nonequilibrium thermodynamics of linear systems with memory. A theorem is proved that contains necessary and sufficient conditions which should be satisfied by constitutive equations for such systems in order to meet the second law of thermodynamics.Academic Scientific Complex A. V. Luikov Institute of Heat and Mass Transfer of the Academy of Sciences of Belarus, Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal Vol. 68, No. 5, September–October 1995, pp. 724–738.     

Spatial propagation of coherency matrix in polarization optics

In this work, the question of the coherency matrix propagation of a light beam is addressed by means of the analysis of interpolation processes between two physical situations. These physical situations are defined according to the second order statistical properties of the underlying process. Two states of a light beam or the path in a medium to go from a physical situation at distance z(1) to another one at distance z(2) is related to the correlation between both these physical situations. Equivalence classes are derived from the definition of a group action on the set of coherency matrices. The geodesic curves on each equivalence class define the process of interpolation. The general solution is derived as a symbolic equation, and the solution is explicitly developed for the situation of uncorrelated statistical processes. Interpolating coherency matrix in this particular case describes the propagation of a light beam into a uniform nondepolarizing medium characterized by a differential Jones matrix determined by the far points of the interpolation curve up to a unitary matrix.     

Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absence of the Slater condition. Consequently, we present characterizations of global optimality and the Lagrangian duality for quadratic optimization problems with a single quadratic constraint. Our method of proof makes use of Brickman’s theorem and conjugate analysis, exploiting the hidden link between the convexity and the S-lemma.     

Necessary and sufficient conditions for exponential stability of a family of generalized Liénard differential equations

A class of generalized Liénard differential equations determined by prescribed bilateral bounds for the nonlinear terms is considered. Necessary and sufficient conditions for exponential stability of the system expressed in the indicated bounds are found. The results are compared with known sufficient stability conditions obtained via the Lyapunov function method.     

On necessary and sufficient conditions for differential flatness

This paper is devoted to the characterization of differentially flat nonlinear systems in implicit representation, after elimination of the input variables, in the differential geometric framework of manifolds of jets of infinite order. We extend the notion of Lie-B?cklund equivalence, introduced in Fliess et al. (IEEE Trans Automat Contr 44(5):922–937, 1999), to this implicit context and focus attention on Lie-B?cklund isomorphisms associated to flat systems, called trivializations. They can be locally characterized in terms of polynomial matrices of the indeterminate \fracddt{\frac{d}{dt}} , whose range is equal to the kernel of the polynomial matrix associated to the implicit variational system. Such polynomial matrices are useful to compute the ideal of differential forms generated by the differentials of all possible trivializations. We introduce the notion of a strongly closed ideal of differential forms, and prove that flatness is equivalent to the strong closedness of the latter ideal, which, in turn, is equivalent to the existence of solutions of the so-called generalized moving frame structure equations. Two sequential procedures to effectively compute flat outputs are deduced and various examples and consequences are presented.     

Measurement of the Mueller matrix for ocean water

The normalized light scattering polarization matrix has been measured for ocean water using an electrooptic light scattering polarimeter. Measurements were done on samples from the Atlantic and Pacific Oceans and the Gulf of Mexico. The polarization effects in the matrices were found to have, in general, a form which is similar to polarization effects in the Rayleigh scattering approximation; for example, all off-diagonal matrix elements except S12 and S21 were zero. Mueller matrix elements were calculated using a Mie computer code and compared to the measured matrices for ocean water. A simple one-component distribution was found to produce a reasonably good fit.     

Mueller calculus of polarization change in the cube-corner retroreflector

The optical ray properties of the cube-corner retroreflector (CCR) are first recalled. The change of polarization of the radiation due to CCR reflection is then derived by use of the Mueller matrix calculus. It is found that, in general, when the faces are not ideal reflectors, the useful cross section of the CCR consists of six zones, each of which produces a different change of polarization, i.e., it gives a different Mueller matrix. All the Mueller matrices depend on wavelength. The results are quite general and can be used directly also for partially polarized radiation.     

A characterization and sufficient conditions for the total time on test transform order

The total time on test transform has been used in reliability to compare two random lifetimes. Despite that it has been used widely in this context, it is not very clear in which sense the random variables are being compared. In this paper we provide a characterization which reveals the true nature of this comparison. Given that this comparison is of particular interest when the stochastic order does not hold, we also provide sufficient conditions for the comparisons of the total time on test transforms, when the stochastic order does not hold. Applications to the comparison of several parametric families of distributions are provided.     

Formulation of a Mueller matrix for modeling of depolarization and scattering of nitrobenzene in a Kerr cell

We have studied the depolarization of light from nitrobenzene in a Kerr cell. We observed that absorption in nitrobenzene is electric-field dependent. For modeling a nitrobenzene device we formulated a Mueller matrix for the Kerr-cell assembly, and by operating it on a Stokes vector of the input light we obtained a corresponding Stokes vector for the output light. The first parameter of the output Stokes vector corresponds to the intensity transmittance. It was simulated and compared with the measured intensity transmittance for several orientations of the polarizer-analyzer pair with respect to the applied voltages. The measurement of all unknown coefficients in a Mueller matrix consisting of the superposition of nondepolarizing and depolarizing components predicts the depolarization, scattering, and absorption in the nitrobenzene electro-optic device. The output intensities of the orthogonally polarized and cross-coupled depolarizing coefficients are in good agreement for a semi-isotropic medium. The formulated Mueller matrix agrees with the experimentally measured transmittance.     

Necessary conditions for the adaptation of cement stone in concrete

We establish and formulate necessary conditions for the adaptation of cement stone in concrete to the action of the environment. We prove that the system formed by the environment and cement stone belongs to the class of systems with variable structure. Kharkov State Technical University of Civil Engineering and Architecture, Khar'kov. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 2, pp. 71–75, March–April, 1999.     

Polar decomposition of the Mueller matrix: a polarimetric rule of thumb for square-profile surface structure recognition

In this research, the polar decomposition (PD) method is applied to experimental Mueller matrices (MMs) measured on two-dimensional microstructured surfaces. Polarization information is expressed through a set of parameters of easier physical interpretation. It is shown that evaluating the first derivative of the retardation parameter, δ, a clear indication of the presence of defects either built on or dug in the scattering flat surface (a silicon wafer in our case) can be obtained. Although the rule of thumb thus obtained is established through PD, it can be easily implemented on conventional surface polarimetry. These results constitute an example of the capabilities of the PD approach to MM analysis, and show a direct application in surface characterization.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Necessary convergence conditions for upwind schemes in the two-dimensional case

This paper considers nine-point difference schemes for a two-dimensional boundary value singular perturbation problem without turning points and parabolic boundary layers. Necessary conditions are given for the uniform convergence (in the sense of the maximum norm) of a scheme. Using these conditions, several widely used schemes are analysed. It is shown that some common schemes are not uniformly convergent in ?. and that in some cases we are able to compute uniquely free parameters in the scheme. Some remarks on the treatment of a problem with a parabolic boundary layer are given.     

On experimental characterization of polarization fluctuation dynamics in random optical beams

Polarization correlation functions that characterize the rate of change of the instantaneous polarization state of an arbitrary fluctuating electromagnetic beam were recently introduced. In this work, we describe techniques that enable the measurement of these functions leading to the determination of the so-called polarization time of a random field.     

Analysis of systematic errors in Mueller matrix ellipsometry as a function of the retardance of the dual rotating compensators

A dual rotating compensator ellipsometer based on the optical PC1SC2A configuration described by Collins [1, chap. 7.3] has been developed. The systematic errors for this configuration if the compensators are quarter-wave plates have been already studied [2, 3, 4]. Smith [5] has demonstrated that the optimum retardance of a dual-rotating-retarder (DRR) instrument must be equal to 127° compared to the quarter-wave (90°) retarders generally used. In this condition random errors are optimized. The aim of this work is to used such retarders and verify if the systematic errors due to uncertainties of the optical elements (i.e. analyzer, polarizer, first and second compensators) are improved too. For each optical element in different configurations like single or 4-zone average measurements, the systematic errors are given and compared according to the compensators. It is demonstrated that using a 127° instead of quarter-wave retarders coupled with 4-zone averaging measurement is the best configuration for this instrument. These results were confirmed by a statistical study.     

Preparation and characterization of germanium oxysulfide glassy films for optics

Homogeneous amorphous films in the GeS₂-GeO₂ system have been deposited by a rf sputtering technique. Optical characterizations have shown that the cut-off wavelength and the linear indices increase with an increase in the S/O ratio. Raman spectroscopy indicates the presence of new modes that can be assigned to intermediate germanium oxysulfide structural units. Photo-sensitivity of the oxysulfide films has been demonstrated for irradiation near the band-gap. Diffraction gratings inscribed using 488 nm exposure displayed a limited diffraction efficiency (≤3%) that weakens with a corresponding decrease in the glass S/O ratio.     

Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea

We consider a system of anisotropic layers as an optical model of the eye cornea. Effective refraction indices for normally incident light are calculated with the assumption that each layer consists of closely packed uniform cylinders (fibrils). Jones matrix formalism is used to describe light propagation through the cornea. We calculate the Jones matrices from the experimentally measured Mueller matrices. Two algorithms are used for this purpose. The experiments have shown that ~20% of cornea area studied had the structure well described by the helical model proposed.     

Evaluation of bone matrix and demineralized bone matrix incorporated PLGA matrices for bone repair

The aim of this study was to evaluate the composite matrices prepared using Poly(lactic-co-glycolic acid)- PLGA (85:15) by incorporating human bone matrix (BM) powder or demineralized bone matrix (DBM) powder with the weight ratio of polymer: BM or DBM (75:25) to apply for bone repair. Murine Bone Marrow Stromal Cell (BMSC) attachment was studied with different time points at 30 min, 1 h, 2 h, 4 h, and 6 h for BM/PLGA, DBM/PLGA and PLGA control matrices. All types of matrices were linearly increased the BMSC attachment with the increase of time. Significantly higher number of BMSCs was attached to the both BM/PLGA and DBM/PLGA matrices after 2 h compared to the controls. If BM or DBM is incorporated into biodegradable PLGA matrices and cultured with BMSCs, those composite matrices could be potentially used for bone tissue engineering applications. In addition, particle migration and handling difficulties in DBM powder in clinical applications eliminate using a PLGA matrix. Furthermore, we have observed that DBM/PLGA matrices were structurally stronger compared to the BM/PLGA or control PLGA matrices when they exposed to physiological environment for 72 days.