Topological Invariants in Hydrodynamics and Hydromagnetics

In this thesis we are studying possible invariants in hydrodynamics and hydromagnetics. The concept of flux preservation and line preservation of vector fields, especially vorticity vector fields, have been studied from the very beginning of the study of fluid mechanics by Helmholtz and others. In ideal magnetohydrodynamic flows the magnetic fields satisfy the same conservation laws as that of vorticity field in ideal hydrodynamic flows. Apart from these there are many other fields also in ideal hydrodynamic and magnetohydrodynamic flows which preserves flux across a surface or whose vector lines are preserved. A general study using this analogy had not been made for a long time. Moreover there are other physical quantities which are also invariant under the flow, such as Ertel invariant. Using the calculus of differential forms Tur and Yanovsky classified the possible invariants in hydrodynamics. This mathematical abstraction of physical quantities to topological objects is needed for an elegant and complete analysis of invariants.Many authors used a four dimensional space-time manifold for analysing fluid flows. We have also used such a space-time manifold in obtaining invariants in the usual three dimensional flows.In chapter one we have discussed the invariants related to vorticity field using vorticity field two form w2 in E4. Corresponding to the invariance of four form w2 ^ w2 we have got the invariance of the quantity E. w. We have shown that in an isentropic flow this quantity is an invariant over an arbitrary volume.In chapter three we have extended this method to any divergence-free frozen-in field. In a four dimensional space-time manifold we have defined a closed differential two form and its potential one from corresponding to such a frozen-in field. Using this potential one form w1 , it is possible to define the forms dw1 , w1 ^ dw1 and dw1 ^ dw1 . Corresponding to the invariance of the four form we have got an additional invariant in the usual hydrodynamic flows, which can not be obtained by considering three dimensional space.In chapter four we have classified the possible integral invariants associated with the physical quantities which can be expressed using one form or two form in a three dimensional flow. After deriving some general results which hold for an arbitrary dimensional manifold we have illustrated them in the context of flows in three dimensional Euclidean space JR3. If the Lie derivative of a differential p-form w is not vanishing,then the surface integral of w over all p-surfaces need not be constant of flow. Even then there exist some special p-surfaces over which the integral is a constant of motion, if the Lie derivative of w satisfies certain conditions. Such surfaces can be utilised for investigating the qualitative properties of a flow in the absence of invariance over all p-surfaces. We have also discussed the conditions for line preservation and surface preservation of vector fields. We see that the surface preservation need not imply the line preservation. We have given some examples which illustrate the above results. The study given in this thesis is a continuation of that started by Vedan et.el. As mentioned earlier, they have used a four dimensional space-time manifold to obtain invariants of flow from variational formulation and application of Noether's theorem. This was from the point of view of hydrodynamic stability studies using Arnold's method. The use of a four dimensional manifold has great significance in the study of knots and links. In the context of hydrodynamics, helicity is a measure of knottedness of vortex lines. We are interested in the use of differential forms in E4 in the study of vortex knots and links. The knowledge of surface invariants given in chapter 4 may also be utilised for the analysis of vortex and magnetic reconnections.

Dielectric Behavior of Natural Rubber Composites in Microwave Fields

In this article, we report the preparation of conducting natural rubber (NR) with polyaniline (Pani). NR was made into a conductive material by the compounding of NR with Pani in powder form. NR latex was made into a conductive material by the in situ polymerization of aniline in the presence of NR latex. Different compositions of Pani- NR semi-interpenetrating networks were prepared, and the dielectric properties of all of the samples were determined in microwave frequencies. The cavity perturbation techpique was used for this study. A HP8510 vector network analyzer with a rectangular cavity resonator was used for this study. S bands 2-4 GHz in frequency were used. Thermal studies were also carried out with thermogravimetric analysis and differential scanning calorimetry.

Dielectric Behavior of Natural Rubber Composites in Microwave Fields

In this article, we report the preparation of conducting natural rubber (NR) with polyaniline (Pani). NR was made into a conductive material by the compounding of NR with Pani in powder form. NR latex was made into a conductive material by the in situ polymerization of aniline in the presence of NR latex. Different compositions of Pani- NR semi-interpenetrating networks were prepared, and the dielectric properties of all of the samples were determined in microwave frequencies. The cavity perturbation techpique was used for this study. A HP8510 vector network analyzer with a rectangular cavity resonator was used for this study. S bands 2-4 GHz in frequency were used. Thermal studies were also carried out with thermogravimetric analysis and differential scanning calorimetry.

Invariant density for a class of initial distributions under quadratic mapping

For the discrete-time quadratic map xt+1=4xt(1-xt) the evolution equation for a class of non-uniform initial densities is obtained. It is shown that in the t to infinity limit all of them approach the invariant density for the map.

Regression models for bivariate survival data

Multivariate lifetime data arise in various forms including recurrent event data when individuals are followed to observe the sequence of occurrences of a certain type of event; correlated lifetime when an individual is followed for the occurrence of two or more types of events, or when distinct individuals have dependent event times. In most studies there are covariates such as treatments, group indicators, individual characteristics, or environmental conditions, whose relationship to lifetime is of interest. This leads to a consideration of regression models.The well known Cox proportional hazards model and its variations, using the marginal hazard functions employed for the analysis of multivariate survival data in literature are not sufficient to explain the complete dependence structure of pair of lifetimes on the covariate vector. Motivated by this, in Chapter 2, we introduced a bivariate proportional hazards model using vector hazard function of Johnson and Kotz (1975), in which the covariates under study have different effect on two components of the vector hazard function. The proposed model is useful in real life situations to study the dependence structure of pair of lifetimes on the covariate vector . The well known partial likelihood approach is used for the estimation of parameter vectors. We then introduced a bivariate proportional hazards model for gap times of recurrent events in Chapter 3. The model incorporates both marginal and joint dependence of the distribution of gap times on the covariate vector . In many fields of application, mean residual life function is considered superior concept than the hazard function. Motivated by this, in Chapter 4, we considered a new semi-parametric model, bivariate proportional mean residual life time model, to assess the relationship between mean residual life and covariates for gap time of recurrent events. The counting process approach is used for the inference procedures of the gap time of recurrent events. In many survival studies, the distribution of lifetime may depend on the distribution of censoring time. In Chapter 5, we introduced a proportional hazards model for duration times and developed inference procedures under dependent (informative) censoring. In Chapter 6, we introduced a bivariate proportional hazards model for competing risks data under right censoring. The asymptotic properties of the estimators of the parameters of different models developed in previous chapters, were studied. The proposed models were applied to various real life situations.

Dynamics of coupled Josephson junctions under the influence of applied fields

We investigate the effect of the phase difference of appliedfields on the dynamics of mutually coupledJosephsonjunctions. A phase difference between the appliedfields desynchronizes the system. It is found that though the amplitudes of the output voltage values are uncorrelated, a phase correlation is found to exist for small values of applied phase difference. The dynamics of the system is found to change from chaotic to periodic for certain values of phase difference. We report that by keeping the value of phase difference as π, the system continues to be in periodic motion for a wide range of values of system parameters. This result may find applications in devices like voltage standards, detectors, SQUIDS, etc., where chaos is least desired.

Studies on Fuzzy Matroitls and Related Topics

The doctoral thesis focuses on the Studies on fuzzy Matroids and related topics.Since the publication of the classical paper on fuzzy sets by L. A. Zadeh in 1965.the theory of fuzzy mathematics has gained more and more recognition from many researchers in a wide range of scientific fields. Among various branches of pure and applied mathematics, convexity was one of the areas where the notion of fuzzy set was applied. Many researchers have been involved in extending the notion of abstract convexity to the broader framework of fuzzy setting. As a result, a number of concepts have been formulated and explored. However. many concepts are yet to be fuzzified. The main objective of this thesis was to extend some basic concepts and results in convexity theory to the fuzzy setting. The concept like matroids, independent structures. classical convex invariants like Helly number, Caratheodoty number, Radon number and Exchange number form an important area of study in crisp convexity theory. In this thesis, we try to generalize some of these concepts to the fuzzy setting. Finally, we have defined different types of fuzzy matroids derived from vector spaces and discussed some of their properties.

Benthic ecology of selected prawn culture fields and ponds near Cochin

The thesis on the"Benthic ecology of selected prawn culture fields and ponds near Cochin” was taken up with a view to provide information on the qualitative and quantitative distribution of benthos and their relationships to prawnproduction of different culture ecosystems and to the physico-chemical parameters influencing their production. A two-year observation was carried out in nine selected prawn culture systems including perennial ponds (stations 1 to 4) seasonal fields (stations 5 to 7) and contiguous canals (stations 8 and 9) during December 1988 to November 1989. All macro- and meiobenthic organisms contributing to the faua were identified and their abundance, distribution, diversity, biomass and trophic relationships between benthos and prawns were studied. The environmental variables studied were temperature pH, salinity, dissolved oxygen, alkalinity, nitrite-nitrogen, nitrate-nitrogen, amonianitrogen, phosphate and silicate of bottom water and organic carbon and texture of the soil The thesis is presented in 4 Chapters. Chapter I presents an’ INTRODUCTION to the topic of study and a review of relevant works to bring an awareness to the present status of research in benthos and benthic ecology. Chapter 11, MATERIALS AND MTHODS, includes the techniques of sampling, preservation of samples and methods of analyses of various physico-chemical factors and area covered under the study is also given in this chapter. Chapter III, HYDROGRAPHY deals with the results of investigation and discussion onthe physico-chemical parameters of water and Chapter IV, SEDIMENT covers the sedimentoloical characteristics of the different culture systems followed by a detailed discussion. Chapter V, BOTTOM FAUNA presents an account on the various aspects of benthos and benthic ecology and the details of prawn production. A discussion on the overall assessment of interrelations between abiotic and biotic factors is given in Chapter VI, DISCUSSION. A critical evaluation of the implication of benthic production on prawn production under culture conditions and trophic relationships are also included in this chapter. An executive SUMMARY of the observations made during this study is presented in the final section of the thesis .

Ecological characteristics of prawn culture fields in the Cochin area

Prawn culture by traditional method forms an important occupation for the people in these areas, especially in the Vypeen island. Though short term studies have been made on various aspects of prawn culture field and its ecology, a study of detailed nature covering perennial, seasonal, fields and canals between coconut plantation is lacking from these areas. This study will also enable to assess the relative productivity of different systems during different seasons and the influence of the environment on the production potentials. Therefore the present study is taken upto throw more light on the ecological characteristics of these fields with special emphasis on its primary, secondary and tertiary production. The present area of investigation includes the prawn culture fields adjacent to Cochin backwater. The Cochin backwater (O9° 58'N 76° 28'E) is a shallow semi-enclosed body of water of tropical estuary. A narrow gut, about 450 M wide forms its main connection with the Arabian sea and this region is subjected to regular tidal influenceertiary production.

Studies in molecular dynamics: new criteria for evaluation of intramolecular force fields

The present thesis deals with some studies in molecular dynamics using spectroscopic data. Two new approximation procedures the variable method and the average bonding energy criterion have been developed for a reliable calculation of molecular force fields and applied to several molecular species belonging to the xy2 type.

Nature and Ecological significance of Nutrient Regeneration in Different Prawn Culture Fields

The present work "Nature and Ecological Significance of Nutrient Regeneration in different Prawn Culture Fields" was undertaken to understand the seasonal variation of nutrients, nutrient cycling and primary productivity of the prawn culture systems. The main emphasis was to find the qualitative and quantitative estimates of distribution of total phosphorus, inorganic phosphorus, organic phosphorus, total nitrogen and nitrogen fractions in the water. The effect of nutrient cycling on primary productivity and concentration of metals also form one part of the study. The entire thesis comprise of only one major chapter with subchapters such as, Introduction (I), Review of Literature (2), Material and Methods (3), Results (14), Discussion (5), Executive Summary (6) and Biblio~ graphy (7)