Theory of differential inequalities with appliciations to singular perturbiation problems and method of lines

During recent years, the theory of differential inequalities has been extensively used to discuss singular perturbation problems and method of lines to partial differential equations. The present thesis deals with some differential inequality theorems and their applications to singularly perturbed initial value problems, boundary value problems for ordinary differential equations in Banach space and initial boundary value problems for parabolic differential equations. The method of lines to parabolic and elliptic differential equations are also dealt The thesis is organised into nine chapters

Fuzzy Topological Games and Related Topics

The main purpose of study is to extend the concept of the topological game G(K, X) and some other kinds of games into fuzzy topological games and to obtain some results regarding them. Owing to the fact that topological games have plenty of applications in covering properties, it made an attempt to explore some inter relations of games and covering properties in fuzzy topological spaces. Even though the main focus is on fuzzy para-meta compact spaces and closure preserving shading families, some brief sketches regarding fuzzy P-spaces and Shading Dimension is also provided. In a topological game players choose some objects related to the topological structure of a space such as points, closed subsets, open covers etc. More over the condition on a play to be winning for a player may also include topological notions such as closure, convergence, etc. It turns out that topological games are related to the Baire property, Baire spaces, Completeness properties, Convergence properties, Separation properties, Covering and Base properties, Continuous images, Suslin sets, Singular spaces etc.

Studies on the dynamics and rheology of dilute suspensions of Brownian particles in simple shear flow under the action of an external force

We present a novel approach to computing the orientation moments and rheological properties of a dilute suspension of spheroids in a simple shear flow at arbitrary Peclct number based on a generalised Langevin equation method. This method differs from the diffusion equation method which is commonly used to model similar systems in that the actual equations of motion for the orientations of the individual particles are used in the computations, instead of a solution of the diffusion equation of the system. It also differs from the method of 'Brownian dynamics simulations' in that the equations used for the simulations are deterministic differential equations even in the presence of noise, and not stochastic differential equations as in Brownian dynamics simulations. One advantage of the present approach over the Fokker-Planck equation formalism is that it employs a common strategy that can be applied across a wide range of shear and diffusion parameters. Also, since deterministic differential equations are easier to simulate than stochastic differential equations, the Langevin equation method presented in this work is more efficient and less computationally intensive than Brownian dynamics simulations.We derive the Langevin equations governing the orientations of the particles in the suspension and evolve a procedure for obtaining the equation of motion for any orientation moment. A computational technique is described for simulating the orientation moments dynamically from a set of time-averaged Langevin equations, which can be used to obtain the moments when the governing equations are harder to solve analytically. The results obtained using this method are in good agreement with those available in the literature.The above computational method is also used to investigate the effect of rotational Brownian motion on the rheology of the suspension under the action of an external force field. The force field is assumed to be either constant or periodic. In the case of con- I stant external fields earlier results in the literature are reproduced, while for the case of periodic forcing certain parametric regimes corresponding to weak Brownian diffusion are identified where the rheological parameters evolve chaotically and settle onto a low dimensional attractor. The response of the system to variations in the magnitude and orientation of the force field and strength of diffusion is also analyzed through numerical experiments. It is also demonstrated that the aperiodic behaviour exhibited by the system could not have been picked up by the diffusion equation approach as presently used in the literature.The main contributions of this work include the preparation of the basic framework for applying the Langevin method to standard flow problems, quantification of rotary Brownian effects by using the new method, the paired-moment scheme for computing the moments and its use in solving an otherwise intractable problem especially in the limit of small Brownian motion where the problem becomes singular, and a demonstration of how systems governed by a Fokker-Planck equation can be explored for possible chaotic behaviour.

Electrochemical studies on Metal phthalocyanines

The primary aim of these investigations was to probe the elecnuchemical and material science aspects of some selected metal phthalocyanines(MPcs).Metal phthalocyanines are characterised by a unique planar molecular structure. As a single class of compounds they have been the subject of ever increasing number of physicochemical and technological investigations. During the last two decades the literature on these compounds was flooded by an outpour of original publications and patents. Almost every branch of materials science has benefited by their application-swface coating, printing, electrophotography, photoelectrochemistry, electronics and medicine to name a few.The present study was confined to the electrical and electrochemical properties of cobalt, nickel, zinc. iron and copper phthalocyanines. The use of soluble Pes as corrosion inhibitor for aluminium was also investigated.In the introductory section of the thesis, the work done so far on MPcs is reviewed. In this review emphasis is given to their general methods of synthesis and the physicochemical properties.In phthalocyanine chemistry one of the formidable tasks is the isolation of singular species. In the second chapter the methods of synthesis and purification are presented with necessary experimental details.The studies on plasma modified films of CoPe, FePc, ZnPc. NiPc and CuPc are also presented.Modification of electron transfer process by such films for reversible redox systems is taken as the criterion to establish enhanced electrocatalytic activity.Metal phthalocyanines are p- type semiconductors and the conductivity is enhanced by doping with iodine. The effect of doping on the activation energy of the conduction process is evaluated by measuring the temperature dependent variation of conductivity. Effect of thennal treatment on iodine doped CoPc is investigated by DSC,magnetic susceptibility, IR, ESR and electronic spectra. The elecnucatalytic activity of such doped materials was probed by cyclic voltammetry.The electron transfer mediation characteristics of MPc films depend on the film thickness. The influence of reducing the effective thickness of the MPc film by dispersing it into a conductive polymeric matrix was investigated. Tetrasulphonated cobalt phthalocyanine (CoTSP) was electrostatically immobilised into polyaniline and poly(o-toluidine) under varied conditions.The studies on corrosion inhibition of aluminium by CoTSP and CuTSP and By virtue of their anionic character they are soluble in water and are strongly adsorbed on aluminium. Hence they can act as corrosion inhibitors. CoTSP is also known to catalyze the reduction of dioxygen.This reaction can accelerate the anodic dissolution of metal as a complementary reaction. The influence of these conflicting properties of CoTSP on the corrosion of aluminium was studied and compared with those of CuTSP.In the course of these investigations a number of gadgets like cell for measuring the electrical conductivity of solids under non-isothermal conditions, low power rf oscillator and a rotating disc electrode were fabricated.

Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

Enhancement of Calcifications in Mammograms using Volterra Series based Quadratic Filter

The paper summarizes the design and implementation of a quadratic edge detection filter, based on Volterra series, for enhancing calcifications in mammograms. The proposed filter can account for much of the polynomial nonlinearities inherent in the input mammogram image and can replace the conventional edge detectors like Laplacian, gaussian etc. The filter gives rise to improved visualization and early detection of microcalcifications, which if left undetected, can lead to breast cancer. The performance of the filter is analyzed and found superior to conventional spatial edge detectors

Unsharpmaskingusingquadratic filterfortheenhancement of fingerprintsinnoisybackground

The papersummarizesthedesignandimplementationofaquadraticedgedetection filter basedon Volterraseries.The filter isemployedinanunsharpmaskingschemeforenhancing fingerprints inadark and noisybackground.Theproposed filter canaccountformuchofthepolynomialnonlinearities inherent intheinputimageandcanreplacetheconventionaledgedetectorslikeLaplacian,LoG,etc.The application ofthenew filter isinforensicinvestigationwhereenhancementandidentification oflatent fingerprints arekeyissues.Theenhancementofimagesbytheproposedmethodissuperiortothatwith unsharp maskingschemeemployingconventional filters intermsofthevisualquality,thenoise performance and the computational complexity,making it an ideal candidate for latent fingerprint enhancement.

Quadratic Predictor based Differential Encoding and Decoding of Speech Signals

Modeling nonlinear systems using Volterra series is a century old method but practical realizations were hampered by inadequate hardware to handle the increased computational complexity stemming from its use. But interest is renewed recently, in designing and implementing filters which can model much of the polynomial nonlinearities inherent in practical systems. The key advantage in resorting to Volterra power series for this purpose is that nonlinear filters so designed can be made to work in parallel with the existing LTI systems, yielding improved performance. This paper describes the inclusion of a quadratic predictor (with nonlinearity order 2) with a linear predictor in an analog source coding system. Analog coding schemes generally ignore the source generation mechanisms but focuses on high fidelity reconstruction at the receiver. The widely used method of differential pnlse code modulation (DPCM) for speech transmission uses a linear predictor to estimate the next possible value of the input speech signal. But this linear system do not account for the inherent nonlinearities in speech signals arising out of multiple reflections in the vocal tract. So a quadratic predictor is designed and implemented in parallel with the linear predictor to yield improved mean square error performance. The augmented speech coder is tested on speech signals transmitted over an additive white gaussian noise (AWGN) channel.