On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

Normal form for Mueller Matrices in Polarization Optics

Abstract

The normal (canonical) form for Mueller matrices in polarization optics is derived: it is shown that a non-singular real 4 × 4 matrix M qualifies to be the bona fide Mueller matrix of some physical system if and only if it has the canonical form M = L′ ΛL, where L and L′ are elements of the proper orthochronous Lorentz group L ^↑ ₊, and where Λ = diag (λ₀, λ₁, λ₂, λ₃) with λ₀ ≥ ¦λ_j¦ > 0. It is further shown that λ₁ and λ₂ can be taken to be positive so that the signature of λ₃ is the same as that of det M. Several experimentally measured Mueller matrices are analysed in the light of the normal form. The case of singular Mueller matrices is briefly discussed as a limiting case.     

Eigenvalues of damped structures: Vectoriteration in the original space of DOF

The equations of motion in structural dynamics as well as the corresponding eigenvalue problem are governed by 3 matrices for mass, damping and stiffness of order n which equals the number of degrees of freedom. High-performance eigenvalue-solvers are developed for only pairs A, B of matrices. Nevertheless, to benefit from these solvers, the original eigenvalue problem (λ² M+λD+K)x=0 is transformed into a linear eigenvalue representation with only two hypermatrices of double order 2n. Consequently the total numerical effort depends on this order 2n. This paper presents a vectoriteration process which actually works in the original space of order n and which needs no special actions like simultaneous iteration if complex conjugate eigenvalues λ, λˉ with identical norm have to be calculated. The theoretical foundation of this process still goes back to the pair of hypermatrices.     

Inclusion Complexes of Tolnaftate with β-Cyclodextrin and Hydroxypropyl β-Cyclodextrin

Abstract

Tolnaftate, an antifungal agent, was found to form inclusion complexes with both β-cyclodextrin (β-CD) and hydroxypropyl β-cyclodextrins (HPBCDs) with two different degrees of substitution [HPBCD(A)-8% and HPBCD(B)-3%]. Complex formation in the solution state was studied using phase solubility and spectral shift methods. Solid complexes were prepared by the coprecipitation method. Solubilities and dissolution rates were determined for each solid complex, its corresponding physical mixture, and free drug. The increase in solubility of tolnaftate with added HPBCD was found to be significantly greater than with added β-CD. For both HPBCD(A) and HPBCD(B), over the concentration range 0–0.05 M. 1:1 complexes with stability constants of 1460 ± 139 M^-1 and 1860 ± 165 M^-1 were observed, respectively. Over the β-CD concentration range 0–0.02 M, a 1:1 complex with a stability constant of 1190 ± 105 M^-1 was observed. At higher HPBCD concentrations, the increase in solubility was observed to show a positive deviation from linearity (type A_p phase diagram). Using the spectral method, in a 2 5% v/v methanol in water system, the stability constants were determined to be 1020 ± 150 M^-1 1110 ± 120 M^-1 and 1100 ± 260 M^-1 for HPBCD(A), HPBCD(B) and β-CD, respectively. The solid complexes prepared showed improved dissolution over physical mixtures and free drug.     

Scheduling flexible manufacturing cells using Tabu Search

Scheduling is an important aspect in the overall control of a flexible manufacturing system. The research presented focuses on production scheduling of jobs within a flexible manufacturing cell (FMC)–one type of flexible manufacturing system. Due to the complexity of the FMC scheduling problem, a 0–1 mixed-integer linear programming (MILP) model is formulated for M machines and N jobs with alternative routings. Although small instances of the problem can be solved optimally with MILP models, a two-stage Tabu Search (TS² ) algorithm that minimises the manufacturing makespan (MS) is proposed to solve medium-to-large-scale problems more efficiently. During Stage I (construction phase), two heuristics are utilised to generate an initial feasible sequence and an initial MS solution. In Stage II (improvement phase), the acquired initial solutions from Stage I are combined with a Tabu Search meta-heuristic procedure that provides improved MS solutions. The TS² algorithm provides tremendous savings in computational time for medium/large-sized multi-machine FMC problems.     

Application of 99mTc Labeling for Fast Comparative Screening of Preformed Liposomes in vivo

Abstract

Labeling of preformed liposomes with ^99mTc is a simple, quick, and highly versatile method, and its potential in the comparative screening of preformed liposomes with regard to their early biodistribution properties, such as the BLOOD/RES ratios at 2 and 30 min postadministration, was investigated. Liposomes differing in lipid composition and size were prepared and labeled under optimum conditions with ^99mTc using a slight modijication of a previously described method (1). The amount of SnCl₂ used to reduce pertechnetate affected liposome biodistribution. The labeling method employed was capable of detecting anticipated changes in liposome biodistribution caused by alterations in liposome composition or size. It could also reveal the effect of a relative immiscibility of monosialganglioside GM₁ with di-myristoy1-phosphatidyl-choline:di-myristoyl-phosphatidyl-glycerol (DMPC:DMPG) on the biodistribution of DMPC:DMPG:GM, liposomes. It is proposed that ^99mTc labeling of liposomes provides a tool for fast comparative screening of preformed liposomal preparations according to their early biodistribution.     

Torsion testing – plastic deformation to high strains and strain rates

Abstract

A new method of calculating stress–strain curves from torsion measurements is described. Contrary to the usual calculation of stress and strain at the surface of the specimen – where the material properties are distorted by microcracks, notch effects, etc. – stress and strain are determined at a ‘critical radius’ inside the specimen. For this purpose an initial approximation for the flow curve of the type σ_f ~ ΦⁿΦ^m is improved by calculating a ‘correction function’ from the test results. This calculation is based on a Taylor series expansion which converges more strongly if tubular specimens are used instead of solid ones. Tubular specimens are therefore used if the flow curve appears to deviate strongly from the exponential law. The new method of test evaluation makes use of the fact that contrary to tensile and upsetting tests, in torsion tests the measured curve gives an almost undistorted linear projection of the flow curve. In fact the test evaluation consists only of ‘calibrating’ the measured torque and angle of twist in terms of yield stress or equivalent strain, respectively.

MST/94     

Beam combination of a linear array: Hermite-Gaussian superposition model

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral and using a Hermite-Gaussian superposition model, an analytical propagation equation of the beam emitted from a linear array is derived and illustrated numerically. Furthermore, the beam quality of the resulting beam in terms of the M ²-factor and power in the bucket (PIB) is analysed. Interestingly, the M ²-factor in the × direction is dependent upon the separation distance of beamlets, but the PIB is not, because the latter is determined by the irradiance distribution in the far field and the bucket size chosen.     

Beam propagation factor of truncated laguerre—gauss beams

Abstract

Beam propagation factor (usually referred to as M ² factor) and brightness of high-order symmetrical Laguerre–Gauss (LG) beams that diffract upon a hard-aperture are investigated. We demonstrate that a circular aperture is able to decrease the M ² factor of a high-order symmetrical LG beam and also to enhance its brightness although losses are induced.     

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C ¹-residual (dense G _δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C ¹ zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ? M there exists Y arbitrarily close to X such that Y ^t has an elliptical closed orbit through U.     

Fast Algorithms for Structured Least Squares and Total Least Squares Problems

We consider the problem of solving least squares problems involving a matrix M of small displacement rank with respect to two matrices Z₁ and Z₂. We develop formulas for the generators of the matrix M ^HM in terms of the generators of M and show that the Cholesky factorization of the matrix M ^HM can be computed quickly if Z₁ is close to unitary and Z₂ is triangular and nilpotent. These conditions are satisfied for several classes of matrices, including Toeplitz, block Toeplitz, Hankel, and block Hankel, and for matrices whose blocks have such structure. Fast Cholesky factorization enables fast solution of least squares problems, total least squares problems, and regularized total least squares problems involving these classes of matrices.     

The pressure field of a spherical diffusion flame

Summary It is usually assumed that the pressure in the neighbourhood of a diffusion flame is constant, and the momentum problem is then de-coupled from the temperature and composition fields. The observation that pressure gradient at the discontinuous flame sheet must itself be discontinuous is sufficient to motivate an examination of the true form of the pressure field, which evidently is not constant. A simple analysis of the spherical discontinuous sheet model leads to the conclusion that the structure of the intense reaction region must play an essential role in fixing local pressure variations. The amplitude of these variations is found to be uniformlyO(M ²), whereM is a typical flow Mach number (M 1), but significant changes within this range can take place over distances which areO(M ^2/3) for a simple chemical reaction.     

Overview Overcoming instabilities in aluminium reduction cells: a route to cheaper aluminium

Abstract

The amount of energy currently used to reduce alumina to aluminium in electrolysis cells is staggering, around 10¹¹ kWh/year. Yet much of this energy (almost one-half) is lost in the form of I ²R heating of the highly resistive electrolyte. Strenuous efforts have been made to minimise these losses by reducing the volume of electrolyte in the cells. However, the aluminium industry has come up against a fundamental problem: when the depth of the electrolyte is reduced below a critical threshold (around 4–5 cm), the liquids in the cell start to ‘slosh around’ in an uncontrolled fashion. This is an instability, fuelled by the intense currents which pass through the liquids. At present, cells operate just above the critical electrolyte depth, but if this depth were reduced from, say, 4·5 to 4 cm, then the annual savings would exceed £10⁸. After a number of false starts, we now have a clear understanding of the physical mechanisms which underpin the instability, and it turns out that these are remarkably simple. These mechanisms are described here and it is shown that, although the cell geometry is too complex to produce an accurate mathematical model, the underlying mechanisms are so simple that it becomes clear how to suppress the instability. Thus, after two decades of research, we are finally in a position to design inherently stable cells. For example, it is shown that slow, small movements of the anode assembly can lower the critical electrolyte depth to around 2 cm. Such a control system could be retrofitted to most existing cells.     

Expansive transitive sets for robust and generic diffeomorphisms

Let f be a diffeomorphism on a compact smooth Riemannian manifold M, and let Λ be a closed f-invariant transitive subset of M. In this paper, we show that f|_Λ is C¹-stably expansive if and only if Λ is a hyperbolic basic set. Moreover, C¹-generically, a locally maximal transitive set Λ is expansive if and only if it is a hyperbolic basic set.     

Influence of conditions on the potential of the couple of nonoxygenated U5+/U4+ ions bound in complexes with the P2W17O 6110? and SiW11O 398? anions

The formal potential of the couple of nonoxygenated U⁵⁺/U⁴⁺ ions bound in complexes with unsaturated heteropoly anions (HPAs) P₂W₁₇O₆₁¹⁰⁻ (I) and SiW_{11 O39}⁸⁻ (II) in 0–1 M NaNO₃ and 1 M (NaCl + HCl) in the range of pH 0.7–4.7 was measured. In 1 M NaNO₃ solutions at pH 4.7–3.0 for I and 4.3–3.9 for II, the formal potentials are constant: 0.820 and 0.730 V, respectively. They preserve approximately the same value with a decrease in pH to 0.7 in 1 M (NaCl + HCl). The potential noticeably decreases with a decrease in the NaNO₃ or NaCl concentration from 1 M to 0 (pH 4.1–4.7): to 0.09 and 0.05–0.06 V for I and II, respectively. Approximate constancy of the potential of the U⁵⁺/U⁴⁺ couple with a decrease in pH to 1 and lower distinguishes this couple from the M⁴⁺/M³⁺ couples (M = Ce, Am, Bk) whose potential appreciably grows with increasing acidity. This is due to the fact that the U⁵⁺ and M⁴⁺ ions in acid solutions remain in the form of complexes with the ratio M: HPA = 1: 2, whereas the M³⁺ ions pass into the form of 1: 1 complexes. Thus, variation of the formal potentials of all the M ^{n + 1}/M ⁿ⁺ couples in the presence of H⁺ and Na⁺ ions is associated with variation of the stability constants of the complexes M(HPA)₂, which, in turn, is caused by interaction of single-charged ions with HPA. However, the H⁺ and Na⁺ ions interact with HPA by different mechanisms and therefore affect the potential of the U⁵⁺/U⁴⁺ couple differently. Original Russian Text ? V.P. Shilov, A.B. Yusov, A.M. Fedoseev, Ph. Moisy, 2008, published in Radiokhimiya, 2008, Vol. 50, No. 5, pp. 393–396.     

The Effect of Environmental Moisture and Temperature on the Physical Stability of Effervescent Tablets in Foil Laminate Packages Containing Minute Imperfections

Abstract

The effect of environmental moisture on the physical stability of effervescent tablets in foil laminate packages containing microscopic imperfections (openings) was examined. Packaged tablets were stored at different relative humidity (RH) and temperature conditions and evaluated for physical stability at predetermined time intervals. Physical stability was assessed by noting if the tablet components reacted prematurely to yield soft tablets during storage. A penetrating dye solution test was used to determine if the foil packages contained imperfections which might allow transmission of moisture. The results of the investigation indicated that absolute moisture integrity of the foil package is required for product stability.     

The Lie-Group Shooting Method for Thermal Stress Evaluation Through an Internal Temperature Measurement

In the present work we study numerical computations of inverse thermal stress problems. The unknown boundary conditions of an elastically deformable heat conducting rod are not given a priori and are not allowed to measure directly, because the boundary may be not accessible to measure. However, an internal measurement of temperature is available. We treat this inverse problem by using a semi-discretization technique, of which the time domain is divided into many sub-intervals and the physical quantities are discretized at these node points of discrete times. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the spatial direction by the Lie-group shooting method to find unknown boundary conditions. The key point is based on one-step Lie group elements: G(r) = G(y⁰,y^l). We are able to search missing boundary conditions through a minimum discrepancy from the targets in terms of a weighting factor r ∈ (0,1). Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the measured temperature is disturbed by large noise, the Lie group shooting method is stable to recover the boundary conditions very well.     

Rheological Modeling with Hookean Bead-Spring Cubes (SC, BBC and FCC)

In this study a general bead-spring model is used for predicting some rheological properties of a cubic bead-spring structure of arbitrary size immersed in a Newtonian solvent. The topology of this bead-spring structure is based upon the well-known cubic crystals (SC, BCC or FCC) and it consists of equal Hookean springs and beads with equal friction coefficients, while hydrodynamic interaction is not included. An appropriate combination of the equations of motion, the expression for the stress tensor and the equation of continuity leads to an explicit constitutive equation with three sets of relaxation times belonging to the three types of bead-spring cubes (SC, BCC or FCC). For small-amplitude oscillatory shear flow it is found that the three relaxation spectra, which are significantly different, result in dynamic moduli which differ mainly in one aspect: the characteristic SC, BCC and FCC time scales are different. The BCC and FCC time scales can be obtained by multiplication of the SC time scale by the ratios M ^sc/M ^bcc and M ^sc/M ^fcc respectively, where M ^sc, M ^bcc and M ^fcc denote the number of springs in the three types of cubic bead-spring structures.     

Beam quality characterization of a radial laser array

Abstract

With the Hermite-Gaussian mode used to describe radial laser arrays and the M²-factor and power in the bucket (PIB) chosen as criteria for characterizing beam quality, a detailed study of beam quality of coherent and incoherent radial laser arrays is made. The closed-form expression for the M²-factor of incoherent radial arrays, which depends upon the Hermite-Gaussian order m, inverse radial fill factor r' and aspect ratio w _0x /w _0y , is derived. Numerical examples are given to illustrate and compare beam quality of coherent and incoherent radial arrays.     

Conditions for the Physical Realizability of Polarization Matrices Characterizing Passive Systems

Abstract

We derive conditions for the physical realizability of polarization matrices characterizing passive systems or scattering media. By physically realizable, we mean that 0  g  1 where g ≡ (output intensity/input intensity). Using the singular-value decomposition of an arbitrary 2 × 2 complex-valued matrix, we prove that a Jones matrix T _J is physically realizable if 0  det T _J ⁺ T _J  1. Consequently singular Jones matrices (i.e. det T _J = 0) completely extinguish the output intensity irrespective of the input intensity because g ≡ 0. Corresponding results are obtained for Mueller-Jones matrices (the 4 × 4 real-valued matrices which are the four-dimensional representations of the two-dimensional 2 × 2 complex-valued Jones matrices). We also study the problem for general Mueller matrices; however because of their phenomenological character they do not admit of such criteria as do the Jones and Mueller-Jones matrices. This is because g now depends upon the matrix elements of the Mueller matrix and the input Stokes parameters; whereas for the Jones and Mueller-Jones matrices, g only depends upon the matrix elements. Finally we study the problem of relating the input and output mean randomness.