Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

The median function on graphs with bounded profiles

The median of a profile = (u1, . . . , uk ) of vertices of a graph G is the set of vertices x that minimize the sum of distances from x to the vertices of . It is shown that for profiles with diameter the median set can be computed within an isometric subgraph of G that contains a vertex x of and the r -ball around x, where r > 2 − 1 − 2 /| |. The median index of a graph and r -joins of graphs are introduced and it is shown that r -joins preserve the property of having a large median index. Consensus strategies are also briefly discussed on a graph with bounded profiles.

Geophysical Constraints on Structure and Tectonics of the Eastern Arabian Sea and the Adjoining West Coast of India with Special Reference to the Kerala Basin

The main objective of the present study is to model the gravity fields in terms of lithospheric structure below the western continental margin of India (WCMI) identify zones of crustal mass anomalies and attempt to infer the location of Ocean Continent transition in the Arabian Sea. In this study, the area starting from the western shield margin to the region covering the deep oceanic parts of the Arabian Sea which is bounded by Carlsberg and Cerg and Central Indian ridges in the south, eastern part of the Indus Cone in the west and falling between 630E and 800E longitudes, and 50N - 240N latitudes has been considered. The vast amount of seismic reflection and refraction data in the form of crustal velocities, basement configuration and crustal thicknesses available for the west coast as well as the eastern Arabian Sea has been utilized for this purpose

A Study on the Transport and Distribution of Atmospheric Aerosols and its Influence on Regional Meteorological Parameters over the Indian Subcontinent

This doctoral thesis addresses the growing concern about the significant changes in the climatic and weather patterns due to the aerosol loading that have taken place in the Indo Gangetic Plain(IGP)which includes most of the Northern Indian region. The study region comprises of major industrial cities in India (New Delhi, Kanpur, Allahabad, Jamshedpur and Kolkata). Northern and central parts of India are one of the most thickly populated areas in the world and have the most intensely farmed areas. Rapid increase in population and urbanization has resulted in an abrupt increase in aerosol concentrations in recent years. The IGP has a major source of coal; therefore most of the industries including numerous thermal power plants that run on coal are located around this region. They inject copious amount of aerosols into the atmosphere. Moreover, the transport of dust aerosols from arid locations is prevalent during the dry months which increase the aerosol loading in theatmosphere. The topography of the place is also ideal for the congregation of aerosols. It is bounded by the Himalayas in the north, Thar Desert in the west, the Vindhyan range in the south and Brahmaputra ridge in the east. During the non‐monsoon months (October to May) the weather in the location is dry with very little rainfall. Surface winds are weak during most of the time in this dry season. The aerosols that reach the location by means of long distance transport and from regional sources get accumulated under these favourable conditions. The increase in aerosol concentration due to the complex combination of aerosol transport and anthropogenic factors mixed with the contribution from the natural sources alters the optical properties and the life time of clouds in the region. The associated perturbations in radiative balance have a significant impact on the meteorological parameters and this in turn determines the precipitation forming process. Therefore, any change in weather which disturbs the normal hydrological pattern is alarming in the socio‐economic point of view. Hence, the main focus of this work is to determine the variation in transport and distribution of aerosols in the region and to understand the interaction of these aerosols with meteorological parameters and cloud properties.

Role of sea-surface wind and transport on enhanced aerosol optical depth observed over the Arabian Sea

The objective of this study is to understand the reasons for the enhancement in aerosol optical depth (AOD) over the Arabian Sea observed during June, July and August. During these months, high values of AOD are found over the sea beyond 10◦ N and adjacent regions. The Arabian Sea is bounded by the lands of Asia and Africa on its three sides. So the region is influenced by transported aerosols from the surroundings as well as aerosols of local origin (marine aerosols). During the summer monsoon season in India, strong surface winds with velocities around 15 m s−1 are experienced over most parts of the Arabian Sea. These winds are capable of increasing sea spray activity, thereby enhancing the production of marine aerosols. The strong winds increase the contribution of marine aerosols over the region to about 60% of the total aerosol content. The main components of marine aerosols include sea salt and sulphate particles. The remaining part of the aerosol particles comes from the western and northern land masses around the sea, of which the main component is transported dust particles. This transport is observed at higher altitudes starting from 600 m. At low levels, the transport occurs mainly from the Indian Ocean and the Arabian Sea itself, indicating the predominance of marine aerosols at these levels. The major portion of the total aerosol loading was contributed by coarse-mode particles during the period of study. But in the winter season, the concentration of coarse-mode aerosols is found to be less. From the analysis, it is concluded that the increase in marine aerosols and dust particles transported from nearby deserts results in an increase in aerosol content over the Arabian Sea during June, July and August.