Some concepts and models usefulninthe analysis of discrete life time data

This thesis entitled Reliability Modelling and Analysis in Discrete time Some Concepts and Models Useful in the Analysis of discrete life time data.The present study consists of five chapters. In Chapter II we take up the derivation of some general results useful in reliability modelling that involves two component mixtures. Expression for the failure rate, mean residual life and second moment of residual life of the mixture distributions in terms of the corresponding quantities in the component distributions are investigated. Some applications of these results are also pointed out. The role of the geometric,Waring and negative hypergeometric distributions as models of life lengths in the discrete time domain has been discussed already. While describing various reliability characteristics, it was found that they can be often considered as a class. The applicability of these models in single populations naturally extends to the case of populations composed of sub-populations making mixtures of these distributions worth investigating. Accordingly the general properties, various reliability characteristics and characterizations of these models are discussed in chapter III. Inference of parameters in mixture distribution is usually a difficult problem because the mass function of the mixture is a linear function of the component masses that makes manipulation of the likelihood equations, leastsquare function etc and the resulting computations.very difficult. We show that one of our characterizations help in inferring the parameters of the geometric mixture without involving computational hazards. As mentioned in the review of results in the previous sections, partial moments were not studied extensively in literature especially in the case of discrete distributions. Chapters IV and V deal with descending and ascending partial factorial moments. Apart from studying their properties, we prove characterizations of distributions by functional forms of partial moments and establish recurrence relations between successive moments for some well known families. It is further demonstrated that partial moments are equally efficient and convenient compared to many of the conventional tools to resolve practical problems in reliability modelling and analysis. The study concludes by indicating some new problems that surfaced during the course of the present investigation which could be the subject for a future work in this area.

Direction Dependent Transmission Characteristics of Dye Mixture Doped Polymer Optical Fibre Preforms

The direction dependant wavelength selective transmission mechanism in poly (methyl methacrylate)(PMMA) rods doped with C 540 dye and C 540:Rh.B dye mixture as a combination has been investigated. When a polished slice of pure C 540 doped polymer rod was used side by side with a C540:Rh B doped rod with acceptor concentration [A] = 7x10-4 m/l , we could notice more than 100% change in the transmitted intensity along opposite directions at the C 540, Rh B emission and the excitation wavelengths . A blue high bright LED emitting at a peak wavelength 465nm was used as the excitation source.

Direction Dependent Transmission Characteristics of Dye Mixture Doped Polymer Optical Fibre Preforms

The direction dependant wavelength selective transmission mechanism in poly (methyl methacrylate)(PMMA) rods doped with C 540 dye and C 540:Rh.B dye mixture as a combination has been investigated. When a polished slice of pure C 540 doped polymer rod was used side by side with a C540:Rh B doped rod with acceptor concentration [A] = 7x10-4 m/l , we could notice more than 100% change in the transmitted intensity along opposite directions at the C 540, Rh B emission and the excitation wavelengths . A blue high bright LED emitting at a peak wavelength 465nm was used as the excitation source.

Direction Dependent Transmission Characteristics of Dye Mixture Doped Polymer Optical Fibre Preforms

The direction dependant wavelength selective transmission mechanism in poly (methyl methacrylate)(PMMA) rods doped with C 540 dye and C 540:Rh.B dye mixture as a combination has been investigated. When a polished slice of pure C 540 doped polymer rod was used side by side with a C540:Rh B doped rod with acceptor concentration [A] = 7x10-4 m/l , we could notice more than 100% change in the transmitted intensity along opposite directions at the C 540, Rh B emission and the excitation wavelengths . A blue high bright LED emitting at a peak wavelength 465nm was used as the excitation source.

A Cryptosystem Using the Concepts of Algebraic Geometric Code

Cryptosystem using linear codes was developed in 1978 by Mc-Eliece. Later in 1985 Niederreiter and others developed a modified version of cryptosystem using concepts of linear codes. But these systems were not used frequently because of its larger key size. In this study we were designing a cryptosystem using the concepts of algebraic geometric codes with smaller key size. Error detection and correction can be done efficiently by simple decoding methods using the cryptosystem developed. Approach: Algebraic geometric codes are codes, generated using curves. The cryptosystem use basic concepts of elliptic curves cryptography and generator matrix. Decrypted information takes the form of a repetition code. Due to this complexity of decoding procedure is reduced. Error detection and correction can be carried out efficiently by solving a simple system of linear equations, there by imposing the concepts of security along with error detection and correction. Results: Implementation of the algorithm is done on MATLAB and comparative analysis is also done on various parameters of the system. Attacks are common to all cryptosystems. But by securely choosing curve, field and representation of elements in field, we can overcome the attacks and a stable system can be generated. Conclusion: The algorithm defined here protects the information from an intruder and also from the error in communication channel by efficient error correction methods.

Excitation wavelength dependence of energy transfer in dye mixture doped polymer optical fibre preforms

Dependence of energy transfer parameters on excitation wavelength has been investigated in poly (methyl methacrylate) (PMMA) optical fibre preforms doped with C 540:Rh B dye mixture by studying the fluorescence intensity and the lifetime variations. A fluorescence spectrophotometer was used to record the excitation spectra of the samples for the emission wavelengths 495 and 580 nm. The fluorescence emission from the polymer rods was studied at four specific excitation wavelengths viz; 445, 465, 488 and 532 nm. The fluorescence lifetime of the donor molecule was experimentally measured in polymer matrix by time correlated single photon counting technique. The energy transfer rate constants and transfer efficiencies were calculated and their dependence on the acceptor concentration was analysed for three excitation wavelengths. It was found that any change in the excitation wavelength leads to significant variations in the quenching characteristics, which in turn affect the calculated energy transfer parameters.

Fabrication and characterization of dye mixture doped polymer optical fiber as a broad wavelength optical amplifier

Rhodamine 6G and Rhodamine B dye mixture doped polymer optical fiber amplifier (POFA), which can operate in a broad wavelength region (60 nm), has been successfully fabricated and tested. Tunable operation of the amplifier over a broad wavelength region is achieved by mixing different ratios of the dyes. The dye doped POFA is pumped axially using 532 nm, 10 ns laser pulses from a frequency doubled Q-switched Nd: YAG laser and the signals are taken from an optical parametric oscillator. A maximum gain of 22.3 dB at 617 nm wavelength has been obtained for a 7 cm long dye mixture doped POFA. The effects of pump energy and length of the fiber on the performance of the fiber amplifier are also studied. There exists an optimum length for which the amplifier gain is at a maximum value.

Photoacoustic study on the photostability of polymethyl methacrylate films doped with Rhodamine 6G–Rhodamine B dye mixture system

The photosensitivity of dye mixture-doped polymethyl methacrylate (PMMA) films are investigated as a function of laser power, concentration of the dyes, modulation frequency and the irradiation wavelength. Energy transfer from a donor molecule to an acceptor molecule affects the emission output of the dye mixture system. Photosensitivity is found to change with changes in donor–acceptor concentrations. PMMA samples doped with the dye mixture are found to be more photosensitive when the dyes are mixed in the same proportion.

On Finite Mixture of Pareto and Beta Distnbutions

Department of Statistics, Cochin University of Science and Technology

Geometric Algebra and Einstein’s Electron

This thesis entitled Geometric algebra and einsteins electron: Deterministic field theories .The work in this thesis clarifies an important part of Koga’s theory.Koga also developed a theory of the electron incorporating its gravitational field, using his substitutes for Einstein’s equation.The third chapter deals with the application of geometric algebra to Koga’s approach of the Dirac equation. In chapter 4 we study some aspects of the work of mendel sachs (35,36,37,).Sachs stated aim is to show how quantum mechanics is a limiting case of a general relativistic unified field theory.Chapter 5 contains a critical study and comparison of the work of Koga and Sachs. In particular, we conclude that the incorporation of Mach’s principle is not necessary in Sachs’s treatment of the Dirac equation.

Some characterization problems associated with the bivariate exponential and geometric distributions

It is highly desirable that any multivariate distribution possessescharacteristic properties that are generalisation in some sense of the corresponding results in the univariate case. Therefore it is of interest to examine whether a multivariate distribution can admit such characterizations. In the exponential context, the question to be answered is, in what meaning— ful way can one extend the unique properties in the univariate case in a bivariate set up? Since the lack of memory property is the best studied and most useful property of the exponential law, our first endeavour in the present thesis, is to suitably extend this property and its equivalent forms so as to characterize the Gumbel's bivariate exponential distribution. Though there are many forms of bivariate exponential distributions, a matching interest has not been shown in developing corresponding discrete versions in the form of bivariate geometric distributions. Accordingly, attempt is also made to introduce the geometric version of the Gumbel distribution and examine several of its characteristic properties. A major area where exponential models are successfully applied being reliability theory, we also look into the role of these bivariate laws in that context. The present thesis is organised into five Chapters

Algebraic Geometric Codes and their relation to Cryptography using Elliptic Curves

Communication is the process of transmitting data across channel. Whenever data is transmitted across a channel, errors are likely to occur. Coding theory is a stream of science that deals with finding efficient ways to encode and decode data, so that any likely errors can be detected and corrected. There are many methods to achieve coding and decoding. One among them is Algebraic Geometric Codes that can be constructed from curves. Cryptography is the science ol‘ security of transmitting messages from a sender to a receiver. The objective is to encrypt message in such a way that an eavesdropper would not be able to read it. A eryptosystem is a set of algorithms for encrypting and decrypting for the purpose of the process of encryption and decryption. Public key eryptosystem such as RSA and DSS are traditionally being prel‘en‘ec| for the purpose of secure communication through the channel. llowever Elliptic Curve eryptosystem have become a viable altemative since they provide greater security and also because of their usage of key of smaller length compared to other existing crypto systems. Elliptic curve cryptography is based on group of points on an elliptic curve over a finite field. This thesis deals with Algebraic Geometric codes and their relation to Cryptography using elliptic curves. Here Goppa codes are used and the curves used are elliptic curve over a finite field. We are relating Algebraic Geometric code to Cryptography by developing a cryptographic algorithm, which includes the process of encryption and decryption of messages. We are making use of fundamental properties of Elliptic curve cryptography for generating the algorithm and is used here to relate both.

Properties of Equilibrium Distributions of Order n

The present work is intended to discuss various properties and reliability aspects of higher order equilibrium distributions in continuous, discrete and multivariate cases, which contribute to the study on equilibrium distributions. At first, we have to study and consolidate the existing literature on equilibrium distributions. For this we need some basic concepts in reliability. These are being discussed in the 2nd chapter, In Chapter 3, some identities connecting the failure rate functions and moments of residual life of the univariate, non-negative continuous equilibrium distributions of higher order and that of the baseline distribution are derived. These identities are then used to characterize the generalized Pareto model, mixture of exponentials and gamma distribution. An approach using the characteristic functions is also discussed with illustrations. Moreover, characterizations of ageing classes using stochastic orders has been discussed. Part of the results of this chapter has been reported in Nair and Preeth (2009). Various properties of equilibrium distributions of non-negative discrete univariate random variables are discussed in Chapter 4. Then some characterizations of the geo- metric, Waring and negative hyper-geometric distributions are presented. Moreover, the ageing properties of the original distribution and nth order equilibrium distribu- tions are compared. Part of the results of this chapter have been reported in Nair, Sankaran and Preeth (2012). Chapter 5 is a continuation of Chapter 4. Here, several conditions, in terms of stochastic orders connecting the baseline and its equilibrium distributions are derived. These conditions can be used to rede_ne certain ageing notions. Then equilibrium distributions of two random variables are compared in terms of various stochastic orders that have implications in reliability applications. In Chapter 6, we make two approaches to de_ne multivariate equilibrium distribu- tions of order n. Then various properties including characterizations of higher order equilibrium distributions are presented. Part of the results of this chapter have been reported in Nair and Preeth (2008). The Thesis is concluded in Chapter 7. A discussion on further studies on equilib- rium distributions is also made in this chapter.