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.

Significance of soil microorganisms with special reference to climate change

There are a large number of agronomic-ecological interactions that occur in a world with increasing levels of CO2, higher temperatures and a more variable climate. Climate change and the associated severe problems will alter soil microbial populations and diversity. Soils supply many atmospheric green house gases by performing as sources or sinks. The most important of these gases include CH4, CO2 and N2O. Most of the green house gases production and consumption processes in soil are probably due to microorganisms. There is strong inquisitiveness to store carbon (C) in soils to balance global climate change. Microorganisms are vital to C sequestration by mediating putrefaction and controlling the paneling of plant residue-C between CO2 respiration losses or storage in semi-permanent soil-C pools. Microbial population groups and utility can be manipulated or distorted in the course of disturbance and C inputs to either support or edge the retention of C. Fungi play a significant role in decomposition and appear to produce organic matter that is more recalcitrant and favor long-term C storage and thus are key functional group to focus on in developing C sequestration systems. Plant residue chemistry can influence microbial communities and C loss or flow into soil C pools. Therefore, as research takings to maximize C sequestration for agricultural and forest ecosystems - moreover plant biomass production, similar studies should be conducted on microbial communities that considers the environmental situations

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.