Paraxial Maxwell beams: transformation by general linear optical systems

Extending the work of earlier papers on the relativistic-front description of paraxial optics and the formulation of Fourier optics for vector waves consistent with the Maxwell equations, we generalize the Jones calculus of axial plane waves to describe the action of the most general linear optical system on paraxial Maxwell fields. Several examples are worked out, and in each case it is shown that the formalism leads to physically correct results. The importance of retaining the small components of the field vectors along the axis of the system for a consistent description is emphasized.

Kinetics and Mechanism of the Transformation of Vaterite to Calcite

The transformation of vaterite to calcite was investigated systematically. The transition temperature and the energetics of the transformation were determined from differential thermal curves. The variations of lattice constants and crystallite size, accompanying the transformation were studied by X-ray diffractometry. The kinetics of transformation were investigated in the temperature range 460–490°C. The kinetic data were analysed with the help of three separate solid-state models.

Design of laterally loaded piles in clays based on cone penetration test data: a reliability-based approach

The behaviour of laterally loaded piles is considerably influenced by the uncertainties in soil properties. Hence probabilistic models for assessment of allowable lateral load are necessary. Cone penetration test (CPT) data are often used to determine soil strength parameters, whereby the allowable lateral load of the pile is computed. In the present study, the maximum lateral displacement and moment of the pile are obtained based on the coefficient of subgrade reaction approach, considering the nonlinear soil behaviour in undrained clay. The coefficient of subgrade reaction is related to the undrained shear strength of soil, which can be obtained from CPT data. The soil medium is modelled as a one-dimensional random field along the depth, and it is described by the standard deviation and scale of fluctuation of the undrained shear strength of soil. Inherent soil variability, measurement uncertainty and transformation uncertainty are taken into consideration. The statistics of maximum lateral deflection and moment are obtained using the first-order, second-moment technique. Hasofer-Lind reliability indices for component and system failure criteria, based on the allowable lateral displacement and moment capacity of the pile section, are evaluated. The geotechnical database from the Konaseema site in India is used as a case example. It is shown that the reliability-based design approach for pile foundations, considering the spatial variability of soil, permits a rational choice of allowable lateral loads.

Formal description, compression and transformation of digital pictures II. Picture algebra and geometry

In an earlier paper (Part I) we described the construction of Hermite code for multiple grey-level pictures using the concepts of vector spaces over Galois Fields. In this paper a new algebra is worked out for Hermite codes to devise algorithms for various transformations such as translation, reflection, rotation, expansion and replication of the original picture. Also other operations such as concatenation, complementation, superposition, Jordan-sum and selective segmentation are considered. It is shown that the Hermite code of a picture is very powerful and serves as a mathematical signature of the picture. The Hermite code will have extensive applications in picture processing, pattern recognition and artificial intelligence.

Formal description, compression and transformation of digital pictures

This paper describes the application of vector spaces over Galois fields, for obtaining a formal description of a picture in the form of a very compact, non-redundant, unique syntactic code. Two different methods of encoding are described. Both these methods consist in identifying the given picture as a matrix (called picture matrix) over a finite field. In the first method, the eigenvalues and eigenvectors of this matrix are obtained. The eigenvector expansion theorem is then used to reconstruct the original matrix. If several of the eigenvalues happen to be zero this scheme results in a considerable compression. In the second method, the picture matrix is reduced to a primitive diagonal form (Hermite canonical form) by elementary row and column transformations. These sequences of elementary transformations constitute a unique and unambiguous syntactic code-called Hermite code—for reconstructing the picture from the primitive diagonal matrix. A good compression of the picture results, if the rank of the matrix is considerably lower than its order. An important aspect of this code is that it preserves the neighbourhood relations in the picture and the primitive remains invariant under translation, rotation, reflection, enlargement and replication. It is also possible to derive the codes for these transformed pictures from the Hermite code of the original picture by simple algebraic manipulation. This code will find extensive applications in picture compression, storage, retrieval, transmission and in designing pattern recognition and artificial intelligence systems.

Kinetics and mechanism of the transformation in antimony trioxide from orthorhombic valentinite to cubic senarmontite

The kinetics of the polymorphic transformation in antimony trioxide from metastable orthorhombic valentinite to cubic senarmontite has been studied in polycrystalline material between 490 and 530°C. Quantitative analysis of the mixtures was done using infrared spectrophotometry. The kinetic data was analyzed and the activation energy for the process was obtained: (i) On the basis of Avrami's equation, which is derived on the basis of a nucleation and growth mechanism; and (ii) from the time required for a constant fraction of the transformation to take place. The values obtained were 50.8 and 46.0 kcal/mole. Observations have also been made on partly transformed single crystals of valentinite using a polarizing microscope. The latter studies and the value of the activation energy suggest that a better understanding of the transformation could be obtained on the basis of a vapor phase mechanism.

Transformation microstructures in a Ti-31A1-13Mo alloy

Some initial results are presented on the formation of the y phase, based on the intermetallic TiA1 (LIo, c/a = 1.02) from the phase, based on the intermetallic TidAl (DOts , c/a = 0.801) in a Ti-31wt. Al-13wt Mo alloy. The study is part of a programme to evaluate microstractures and properties of alloys containing the y phase in the Ti-Mo-AI ternary system.

A critical analysis of the shear difference method

The shear difference method which is commonly used for the separation of normal stresses using photoelastic techniques depends on the step-by-step integration of one of the differential equations of equilibrium. It is assumed that the isoclinic and the isochromatic parameters measured by the conventional methods pertain to the state of stress at the midpoint of the light path. In practice, a slice thin enough for the above assumption to be true and at the same time thick enough to give differences in the shear-stress values over the thickness is necessary. The paper discusses the errors introduced in the isoclinic and isochromatic values by the conventional methods neglecting the variation of stresses along the light path. It is shown that while the error introduced in the measurement of the isochromatic parameter may not be serious, the error caused in the isoclinic measurement may lead to serious errors. Since the shear-difference method involves step-by-step integration the error introduced will be of a cumulative nature.

A differential transformation technique for the analysis of a class of non-linear time varying systems

The scope of the differential transformation technique, developed earlier for the study of non-linear, time invariant systems, has been extended to the domain of time-varying systems by modifications to the differential transformation laws proposed therein. Equivalence of a class of second-order, non-linear, non-autonomous systems with a linear autonomous model of second order is established through these transformation laws. The feasibility of application of this technique in obtaining the response of such non-linear time-varying systems is discussed.

Mixed mode (I and II) crack tip fields in bulk metallic glasses

Stationary crack tip fields in bulk metallic glasses under mixed mode (I and II) loading are studied through detailed finite element simulations assuming plane strain, small scale yielding conditions. The influence of internal friction or pressure sensitivity on the plastic zones. notch deformation, stress and plastic strain fields is examined for different mode mixities. Under mixed mode loading, the notch deforms into a shape such that one part of its surface sharpens while the other part blunts. Increase in mode If component of loading dramatically enhances the normalized plastic zone size, lowers the stresses but significantly elevates the plastic strain levels near the notch tip. Higher internal friction reduces the peak tangential stress but increases the plastic strain and stretching near the blunted part of the notch. The simulated shear bands are straight and extend over a long distance ahead of the notch tip under mode II dominant loading. The possible variations of fracture toughness with mode mixity corresponding to failure by brittle micro-cracking and ductile shear banding are predicted employing two simple fracture criteria. The salient results from finite element simulations are validated by comparison with those from mixed mode (I and II) fracture experiments on a Zr-based bulk metallic glass.

Fabrication and Phase Transformation in Crystalline Nanoparticles of PbZrO3 Derived By Sol-Gel

In this research fabrication of crystalline PbZrO3 (PZ) nanoparticles and their phase transformation behavior is investigated. A novel sol-gel method was used for the synthesis of air-stable and precipitate-free diol-based sol of PZ, which was dried at 150 degrees C and then calcined at 300-700 degrees C for 1 h. The morphology, crystallinity and phase formation of as synthesized nanoparticles were studied by the selected-area electron diffraction (SAED), X-ray diffraction (XRD), thermal gravimetric analysis/differential scanning calorimetry (TGA-DSC), and high resolution transmission electron microscope (HRTEM). The XRD, SAED, and TGA-DSC analyses confirmed the tetragonal lead rich zirconia phase (t-Z phase) and monoclinic zirconia phase (m-Z phase) as the intermediate phases during the calcinations process followed by crystallization of single orthorhombic PZ phase at about 700 degrees C. The average PZ particle size was observed about 20 nm as confirmed by TEM study. Energy-dispersive X-ray spectroscopy (EDX) analysis demonstrated that stoichiometric PbZrO3 was formed.

Direct physical evidence for the back-transformation of stress-induced martensite in the vicinity of cracks in pseudoelastic NiTi shape memory alloys

Crack loading and crack extension in pseudoelastic binary NiTi shape memory alloy (SMA) miniature compact tension (CT) specimens with 50.7 at.% Ni (austenitic, pseudoelastic) was investigated using infrared (IR) thermography during in situ loading and unloading. IR thermographic measurements allow for the observation of heat effects associated with the stress-induced transformation of martensite from B2 to BIT during loading and the reverse transformation during unloading. The results are compared with optical images and discussed in terms of the crack growth mechanisms in pseudoelastic NiTi SMAs. Direct experimental evidence is presented which shows that crack growth occurs into a stress-induced martensitic microstructure, which immediately retransforms to austenite in the wake of the crack.

Co(II) and Cr(III) complexes of formate-formamide mixed ligands: synthesis, structures, single crystal-to-single crystal transformation and magnetic behaviour

Solvothermal treatment of an equimolar mixture of Co(NO3)(2)center dot 6H(2)O, HCONH2 and NaN3 in MeOH at 100 degrees C yielded a three-dimensional NaCl type network Co(HCOO)(2)(HCONH2)(2) center dot HCONH2 (1a) containing formamides in the pores of the structure. Solvated pink 1a undergoes single crystal-to-single crystal (SCSC) transformation at 215 degrees C to form the desolvated dark brown product Co(HCOO)(2)-( HCONH2)(2) (1b) with the retention of the original framework. Reversible single crystal-to-single crystal transformation of 1b (brown) to 1a (pink) in the presence of excess formamide was also established at room temperature. The coordination environment around Co(II) in both 1a and 1b is octahedral with a CoN2O4 coordination composition. A similar reaction replacing Co(II) by Cr(III) produced a heterometallic 3D extended network Na[Cr(HCOO)(4)(HCONH2)(2)]center dot 2H(2)O (2a) at 100 degrees C. An increase in reaction temperature to 150 degrees C produced a simple mononuclear complex Cr(HCOO)(3)(HCONH2)(3) center dot 3H(2)O (2b). Variable temperature magnetic studies revealed the presence of a canting phenomena in both 1a and 1b, and hysteresis loop in the field dependent magnetisation plot at 2 K whereas complex 2a is simply paramagnetic in nature.

On the hardness of shear bands in amorphous alloys

The nanoindentation hardness of individual shear bands in a Zr-based metallic glass was investigated in order to obtain a better understanding of how shear band plasticity is influenced by non-crystalline defects. The results clearly showed that the shear band hardness in both as-cast and structurally relaxed samples is much lower than the respective hardness of undeformed region. Interestingly, inter-band matrix also exhibited lower hardness than undeformed region. The results are discussed in terms of the influence of structural state and the prevailing mechanism of plastic deformation.

Synthesis, Structure, and Transformation Studies in a Family of Inorganic-Organic Hybrid Framework Structures Based on Indium

Eight new open-framework inorganic-organic hybrid compounds based on indium have been synthesized employing hydrothermal methods. All of the compounds have InO6, C2O4, and HPO3/HPO4/SO4 units connected to form structures of different dimensionality Thus, the compounds have zero- (I), two- (II, III, IV, V, VII, and VIII), and three-dimensionally (VI) extended networks. The formation of the first zero-dimensional hybrid compound is noteworthy In addition, concomitant polymorphic structures have been observed in the present study. The molecular compound, I, was found to be reactive, and the transformation studies in the presence of a base (pyridine) give rise to the polymorphic structures of II and III, while the addition of an acid (H3PO3) gives rise to a new indium phosphite with a pillared layer structure (T1). Preliminary density functional theory calculations suggest that the stabilities of the polymorphs are different, with one of the forms (II) being preferred over the other, which is consistent with the observed experimental behavior. The oxalate units perform more than one role in the present structures. Thus, the oxalate units connect two In centers to satisfy the coordination requirements as well as to achieve charge balance in compounds II, IV, and VI. The terminal oxalate units observed in compounds I, IV, and V suggest the possibility of intermediate structures. Both in-plane and out-of-plane connectivity of the oxalate units were observed in compound VI. The 31 compounds have been characterized by powder X-ray diffraction, IR spectroscopy, thermogravimetric analysis, and P-31 NMR studies.