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.

Biochemical & Pharmacological evaluation of liver oils of selected deep sea sharks and chimaeras of the Indian EEZ

With a seacoast of 8,1 18 km, an exclusive economic zone (EEZ) of 2 million square km, and with an area of about 30,000 square km under aquaculture, lndia produces close to six million tonnes of fish, over 4 per cent of the world fish production. While the marine waters upto 50m depth have been fully exploited, those beyond, remain unexplored. There is an ever increasing demand for fishery resources as food. The coastal fishery resources of the country are dwindling at a rapid pace and it becomes highly imperative that we search for alternate fishery resources for food. The option we have is to hunt for marine fishery resources. Studies pertaining to proximate composition, amino acid and fatty acid composition are essential to understand the nutraceutical values of these deep sea fishery resources. The present study was aimed to carry out proximate composition of deep sea fishery resources obtained during cruises onboard the FORV Sarise Sampada, to identify fishery resources which have appreciable lipid content and thereby analyse the bioactive potentials of marine lipids, to study the amino acid profile of these fishery resources, to understand the contents of SPA, MUFA and PUFA and to calculate the n3/n6 fatty acid contents. Though the presence of nutraceuticals was identified in the marine fishery resources their use as potential food resources deserve further investigation. So the study were carried out to calculate the hepatosomatic indices of sharks & chimaeras and conduct biochemical characterisation of liver oils of Apristurus indicus, Cenlrophorus scalprams, Centroselachus crepidater, Neoharriotta raleighana, and Harriotta pinnata obtained during cruises onboard the FORV Sugar Sampada.Therapeutic use of shark liver oil is evident from its use for centuries as a remedy to heal wounds and fight flu (Neil er al. 2006). Japanese seamen called it 'samedava' or "cure all". Shark liver oil is being promoted worldwide as a dietary supplement to boost the immune system, fight infections, to treat cancer and to lessen the side effects of conventional cancer treatment. These days more emphasis is laid on the nutritive benefits of shark liver oils especially on the omega 3 polyunsaturated fatty acids ( PUFAs) (Anandan er al. 2007) and alkylglycerols (AKGs) (Pugliese er al. I998) contained in them due to the high rise of inflammatory disorders such as arthritis, asthma and neurodegenerative diseases like Alzheimer’s, Parkinson’s and Schizophrenia. So the present study also evaluate the pharmacological properties with respect to analgesic, anti-inflammatory, anti pyretic and anti-ulcer effects of four different liver oils of sharks belonging to the Indian EEZ and to identify the components of oil responsible for these activities.The analgesic and anti-inflammatory activities of liver oils from Neoharriotra raleighana (NR), Centrosymnus crepidater (CC), Apristurus indicus (AI), and Centrophorus sculpratus (CS) sharks caught from the Arabian Sea and the Indian Ocean were compared. The main objectives also include determination of the cholesterol lowering effects of liver oils of Neoharriotra raleighana (NR) and Centrophorus sculpratus (CS) on the high fat diet induced dyslipidemia and to compare the impact of four isolipidemic diets, on levels of serum diagnostic marker enzymes, on lipid profile of blood and liver and antioxidant status of heart in male Albino rats. And also to study the efficacy of Centrophorus sculpratus (CS) liver oil against Complete Freund’s Adjuvant-induced arthritis and to compare the anti-inflammatory activity of this oil with a traditionally used anti-inflammatory substance gingerol (oleoresin extracted from ginger.). The results of the present study indicated that both (Centrophorus sculpratus liver oils as well as gingerol extracts proved to be effective natural remedies against CFA-induced arthritis in Albino rats.

Form-invariant bivariate weighted models

In this paper the class of continuous bivariate distributions that has form-invariant weighted distribution with weight function w(x1, x2) ¼ xa1 1 xa2 2 is identified. It is shown that the class includes some well known bivariate models. Bayesian inference on the parameters of the class is considered and it is shown that there exist natural conjugate priors for the parameters

Performance Study of an Improved Legendre Moment Descriptor as Region-Based Shape Descriptor

This paper reports a novel region-based shape descriptor based on orthogonal Legendre moments. The preprocessing steps for invariance improvement of the proposed Improved Legendre Moment Descriptor (ILMD) are discussed. The performance of the ILMD is compared to the MPEG-7 approved region shape descriptor, angular radial transformation descriptor (ARTD), and the widely used Zernike moment descriptor (ZMD). Set B of the MPEG-7 CE-1 contour database and all the datasets of the MPEG-7 CE-2 region database were used for experimental validation. The average normalized modified retrieval rate (ANMRR) and precision- recall pair were employed for benchmarking the performance of the candidate descriptors. The ILMD has lower ANMRR values than ARTD for most of the datasets, and ARTD has a lower value compared to ZMD. This indicates that overall performance of the ILMD is better than that of ARTD and ZMD. This result is confirmed by the precision-recall test where ILMD was found to have better precision rates for most of the datasets tested. Besides retrieval accuracy, ILMD is more compact than ARTD and ZMD. The descriptor proposed is useful as a generic shape descriptor for content-based image retrieval (CBIR) applications