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.

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

Development of Shape Descriptors Based on Legendre Polynomials and Content-Based Retrieval of Scoliosis Images

The wealth of information available freely on the web and medical image databases poses a major problem for the end users: how to find the information needed? Content –Based Image Retrieval is the obvious solution.A standard called MPEG-7 was evolved to address the interoperability issues of content-based search.The work presented in this thesis mainly concentrates on developing new shape descriptors and a framework for content – based retrieval of scoliosis images.New region-based and contour based shape descriptor is developed based on orthogonal Legendre polymomials.A novel system for indexing and retrieval of digital spine radiographs with scoliosis is presented.

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.

Edge excitations and topological order in rotating Bose gas

The edge excitations and related topological orders of correlated states of a fast rotating Bose gas are studied. Using exact diagonalization of small systems, we compute the energies and number of edge excitations, as well as the boson occupancy near the edge for various states. The chiral Luttinger-liquid theory of Wen is found to be a good description of the edges of the bosonic Laughlin and other states identified as members of the principal Jain sequence for bosons. However, we find that in a harmonic trap the edge of the state identified as the Moore-Read (Pfaffian) state shows a number of anomalies. An experimental way of detecting these correlated states is also discussed.

Topological magnetic irreversibility in superconducting Pb samples of various shapes

The magnetic-field dependence of the magnetization of cylinders, disks, and spheres of pure type-I superconducting lead was investigated by means of isothermal measurements of first magnetization curves and hysteresis cycles. Depending on the geometry of the sample and the direction and intensity of the applied magnetic field, the intermediate state exhibits different irreversible features that become particularly highlighted in minor hysteresis cycles. The irreversibility is noticeably observed in cylinders and disks only when the magnetic field is parallel to the axis of revolution and is very subtle in spheres. When the magnetic field decreases from the normal state, the irreversibility appears at a temperature-dependent value whose distance to the thermodynamic critical field depends on the sample geometry. The irreversible features in the disks are altered when they are submitted to an annealing process. These results agree well with very recent high-resolution magneto-optical experiments in similar materials that were interpreted in terms of transitions between different topological structures for the flux configuration in the intermediate state. A discussion of the relative role of geometrical barriers for flux entry and exit and pinning effects as responsible for the magnetic irreversibility is given.

Automatic Image Annotation Using SURF Descriptors

In recent years there is an apparent shift in research from content based image retrieval (CBIR) to automatic image annotation in order to bridge the gap between low level features and high level semantics of images. Automatic Image Annotation (AIA) techniques facilitate extraction of high level semantic concepts from images by machine learning techniques. Many AIA techniques use feature analysis as the first step to identify the objects in the image. However, the high dimensional image features make the performance of the system worse. This paper describes and evaluates an automatic image annotation framework which uses SURF descriptors to select right number of features and right features for annotation. The proposed framework uses a hybrid approach in which k-means clustering is used in the training phase and fuzzy K-NN classification in the annotation phase. The performance of the system is evaluated using standard metrics.

The Effect of SIFT Features as Content Descriptors in the Context of Automatic Writer Identification in Malayalam Language

The span of writer identification extends to broad domes like digital rights administration, forensic expert decisionmaking systems, and document analysis systems and so on. As the success rate of a writer identification scheme is highly dependent on the features extracted from the documents, the phase of feature extraction and therefore selection is highly significant for writer identification schemes. In this paper, the writer identification in Malayalam language is sought for by utilizing feature extraction technique such as Scale Invariant Features Transform (SIFT).The schemes are tested on a test bed of 280 writers and performance evaluated

State-of-the-Art: Transformation Invariant Descriptors

As the popularity of digital videos increases, a large number illegal videos are being generated and getting published. Video copies are generated by performing various sorts of transformations on the original video data. For effectively identifying such illegal videos, the image features that are invariant to various transformations must be extracted for performing similarity matching. An image feature can be its local feature or global feature. Among them, local features are powerful and have been applied in a wide variety of computer vision aplications .This paper focuses on various recently proposed local detectors and descriptors that are invariant to a number of image transformations.

Boundaries and Topological Algorithms

This thesis develops a model for the topological structure of situations. In this model, the topological structure of space is altered by the presence or absence of boundaries, such as those at the edges of objects. This allows the intuitive meaning of topological concepts such as region connectivity, function continuity, and preservation of topological structure to be modeled using the standard mathematical definitions. The thesis shows that these concepts are important in a wide range of artificial intelligence problems, including low-level vision, high-level vision, natural language semantics, and high-level reasoning.