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.

On Truncated Versions of Certain Measures of Inequality and Stability

The present study focuses attention on defining certain measures of income inequality for the truncated distributions and characterization of probability distributions using the functional form of these measures, extension of some measures of inequality and stability to higher dimensions, characterization of bivariate models using the above concepts and estimation of some measures of inequality using the Bayesian techniques. The thesis defines certain measures of income inequality for the truncated distributions and studies the effect of truncation upon these measures. An important measure used in Reliability theory, to measure the stability of the component is the residual entropy function. This concept can advantageously used as a measure of inequality of truncated distributions. The geometric mean comes up as handy tool in the measurement of income inequality. The geometric vitality function being the geometric mean of the truncated random variable can be advantageously utilized to measure inequality of the truncated distributions. The study includes problem of estimation of the Lorenz curve, Gini-index and variance of logarithms for the Pareto distribution using Bayesian techniques.

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.

Stability of Random Sums and Extremes

This study is about the stability of random sums and extremes.The difficulty in finding exact sampling distributions resulted in considerable problems of computing probabilities concerning the sums that involve a large number of terms.Functions of sample observations that are natural interest other than the sum,are the extremes,that is , the minimum and the maximum of the observations.Extreme value distributions also arise in problems like the study of size effect on material strengths,the reliability of parallel and series systems made up of large number of components,record values and assessing the levels of air pollution.It may be noticed that the theories of sums and extremes are mutually connected.For instance,in the search for asymptotic normality of sums ,it is assumed that at least the variance of the population is finite.In such cases the contributions of the extremes to the sum of independent and identically distributed(i.i.d) r.vs is negligible.

Effect of the Vulcanization Time and Storage on the Stability and Physical Properties of Sulfur-Prevulcanized Natural Rubber Latex

Prevulcanized natural rubber latex was prepared by the heating of the latex compound at 55°C for different periods of time (2, 4, 6, 8, and 10 h). The changes in the colloidal stability and physical properties were evaluated during the course of prevulcanization. The prevulcanized latex compounds were stored for 300 days, and the properties were monitored at different storage intervals (0, 20, 40, 60, 120, 180, 240, and 300 days). During prevulcanization, the mechanical stability time increased, and the viscosity remained almost constant. The tensile strength increased during storage for a period of 20 days. The degree of crosslinking, modulus, elongation at break, and chloroform number were varied with the time of storage.

On Chaos and Fractals in General Topological Spaces

The present study on chaos and fractals in general topological spaces. Chaos theory originated with the work of Edward Lorenz. The phenomenon which changes order into disorder is known as chaos. Theory of fractals has its origin with the frame work of Benoit Mandelbrot in 1977. Fractals are irregular objects. In this study different properties of topological entropy in chaos spaces are studied, which also include hyper spaces. Topological entropy is a measures to determine the complexity of the space, and compare different chaos spaces. The concept of fractals can’t be extended to general topological space fast it involves Hausdorff dimensions. The relations between hausdorff dimension and packing dimension. Regular sets in Metric spaces using packing measures, regular sets were defined in IR” using Hausdorff measures. In this study some properties of self similar sets and partial self similar sets. We can associate a directed graph to each partial selfsimilar set. Dimension properties of partial self similar sets are studied using this graph. Introduce superself similar sets as a generalization of self similar sets and also prove that chaotic self similar self are dense in hyper space. The study concludes some relationships between different kinds of dimension and fractals. By defining regular sets through packing dimension in the same way as regular sets defined by K. Falconer through Hausdorff dimension, and different properties of regular sets also.

Biochemical Investigations on the Stability of Biological Membranes

In this project, an attempt has been made to study the stability of erythrocyte and lysosomal membranes biochemically. Erythrocytes were chosen for the study because of their ready availability and relative simplicity. Biological membranes forming closed boundaries between compartments of varying composition consist mainly of proteins and lipids. They are asymmetric, fluid structures that are thermodynamically stable and metabolically active. Normal cellular function begins with normal membrane structure and any variation in it may upset the normal functions. The degree of fluidity of a membrane depends on the chain length of its lipids and degree of unsaturation of constituent fatty acids. In response to environmental changes, many cells can regulate composition of their membranes to maintain the overall semi fluid environment necessary for many membrane associated functions. The assembly and Maintenance of membrane structures in cells is a dynamic process. The components are not only synthesized and inserted into a growing membrane but are also continuously degraded at a slower rate. This turnover process varies with each individual molecule.Lysosomes are important in the catabolic processes occurring in the cell. Lysosomes contain hydrolytic enzymes and are stable under normal conditions. In certain pathological conditions, the lysosomal membrane may rupture, releasing the hydrolytic enzymes into the cell and digestion of cell takes place as a whole. This is very dangerous. In normal life processes of multi cellular organisms, lysosomes rupture following the death of a cell and it may have some value as a built in mechanism for selfremoval of dead cells.An attempt has also been made in this project towards developing lysosome membrane stability as an index of fish spoilage during storage. Different membranes within the cell and between cells have different compositions as reflected in the ratio of protein to lipid. The difference is not surprising given the very different functions of membranes

Box Products in Fuzzy Topological Spaces and Related Topics

The topology as the product set with a base chosen as all products of open sets in the individual spaces. This topology is known as box topology. The main objective of this study is to extend the concept of box products to fuzzy box products and to obtain some results regarding them. Owing to the fact that box products have plenty of applications in uniform and covering properties, here made an attempt to explore some inter relations of fuzzy uniform properties and fuzzy covering properties in fuzzy box products. Even though the main focus is on fuzzy box products, some brief sketches regarding hereditarily fuzzy normal spaces and fuzzy nabla product is also provided. The main results obtained include characterization of fuzzy Hausdroffness and fuzzy regularity of box products of fuzzy topological spaces. The investigation of the completeness of fuzzy uniformities in fuzzy box products proved that a fuzzy box product of spaces is fuzzy topologically complete if each co-ordinate space is fuzzy topologically complete. The thesis also prove that the fuzzy box product of a family of fuzzy α-paracompact spaces is fuzzy topologically complete. In Fuzzy box product of hereditarily fuzzy normal spaces, the main result obtained is that if a fuzzy box product of spaces is hereditarily fuzzy normal ,then every countable subset of it is fuzzy closed. It also deals with the notion of fuzzy nabla product of spaces which is a quotient of fuzzy box product. Here the study deals the relation connecting fuzzy box product and fuzzy nabla product

Influence of Processing Variables on Protein Quality and Frozen Storage Stability of Two Commercially Important Species of Squid (Loligo duvaucelii and Doryteuthis sibogae )

Cephalopods are utilized as an important food item in various countries because of its delicacy as raw consumed food. Mainly sepia and loligo are consumed raw by Japanese and Russians. The freshness of the products is very important when the product is consumed raw. The major species that dominate our squid catch are Loligo duvaucelii and Doryteuthis sibogae. There is a noticeable difference in the quality of both the species. The needle squid (Doryteuthis sibogae ) contributes about 35% of the total squid landing. Due to the fast deterioration , a major portion of the needle squid, which is caught during the first few hauls, is thrown back to sea. The catch in the last hauls only are taken to the landing centers. At present the needle squid is processed as blanched rings and the desired quality is not obtained if it is processed as whole, whole cleaned or as tubes. In this study an attempt is made to investigate the biochemical characteristics in both the species of squid in relation to their quality and, the process control measures to be adopted. The effect of various treatments on their quality and the changes in proteolytic and lysosomal enzymes under various processing conditions are also studied in detail.Thus this study can provide the seafood industry with relevant suggestions and solutions for effective utilization of both the species of squid with emphasis on needle squid.

Acid activated montmorillonite: an efficient immobilization support for improving reusability, storage stability and operational stability of enzymes

Three enzymes, α-amylase, glucoamylase and invertase, were immobilized on acid activated montmorillonite K 10 via two independent techniques, adsorption and covalent binding. The immobilized enzymes were characterized by XRD, N2 adsorption measurements and 27Al MAS-NMR spectroscopy. The XRD patterns showed that all enzymes were intercalated into the clay inter-layer space. The entire protein backbone was situated at the periphery of the clay matrix. Intercalation occurred through the side chains of the amino acid residues. A decrease in surface area and pore volume upon immobilization supported this observation. The extent of intercalation was greater for the covalently bound systems. NMR data showed that tetrahedral Al species were involved during enzyme adsorption whereas octahedral Al was involved during covalent binding. The immobilized enzymes demonstrated enhanced storage stability. While the free enzymes lost all activity within a period of 10 days, the immobilized forms retained appreciable activity even after 30 days of storage. Reusability also improved upon immobilization. Here again, covalently bound enzymes exhibited better characteristics than their adsorbed counterparts. The immobilized enzymes could be successfully used continuously in the packed bed reactor for about 96 hours without much loss in activity. Immobilized glucoamylase demonstrated the best results.