Maximal compactness, min- imal Hausdor ness and reversibility of frames and a study on automorphism group of frames

One can do research in pointfree topology in two ways. The rst is the contravariant way where research is done in the category Frm but the ultimate objective is to obtain results in Loc. The other way is the covariant way to carry out research in the category Loc itself directly. According to Johnstone [23], \frame theory is lattice theory applied to topology whereas locale theory is topology itself". The most part of this thesis is written according to the rst view. In this thesis, we make an attempt to study about 1. the frame counterparts of maximal compactness, minimal Hausdor - ness and reversibility, 2. the automorphism groups of a nite frame and its relation with the subgroups of the permutation group on the generator set of the frame

Social bees and food plant associations in the Nilgiri Biosphere Reserve, India

The diversity of social bees was assessed at 15 sites across five locations of the Nilgiri Biosphere Reserve, Western Ghats, India, from January to December 2007. We also conducted floristic analyses of local vegetation in each site using one-hectare sample plots. All woody species with a dbh (diameter at breast height) : 30 cm were recorded within the plots. A total area of 9.72 ha was assessed for floristic composition. Similarity of floristic composition between sites was determined using the Jaccard's distance measure and a dendrogram constructed based on the hierarchical clustering of floristic dissimilarities between sites. A Bee Importance Index (BII) was developed to give a measure of the bee diversity at each site. This index was a sum of the species richness of bee species in a site and their visitation frequencies to flowers, calculated as mean flower visits hour 1 within 2 focal patches within one hectare plots. The visits of bee species to flowers were also recorded. The Jaccard distance measure indicated that the montane sites were quite dissimilar to the low elevation sites in floristic diversity. The BII was 7-9 for the wet forest sites and ranged from 4-6 for drier forest sites. Seventy three plant species were identified as social bee plants and of them 45% were visited by one species of bee, 37% by two bee species and 18% by more than two bee species, indicating a certain degree of floral specialization among bees.

Variation in the ovarian reserve Is linked to alterations in intrafollicular estradiol production and ovarian biomarkers of follicular differentiation and oocyte quality in cattle

The mechanisms whereby the high variation in numbers of morphologically healthy oocytes and follicles in ovaries (ovarian reserve) may have an impact onovarian function, oocyte quality, and fertility are poorly understood. The objective was to determine whether previously validated biomarkers for follicular differentiation and function, as well as oocyte quality differed between cattle with low versus a high antral follicle count (AFC). Ovaries were removed (n = 5 per group) near the beginning of the nonovulatory follicular wave, before follicles could be identified via ultrasonography as being dominant, from heifers with high versus a low AFC. The F1, F2, and F3 follicles were dissected and diameters determined. Follicular fluid and thecal, granulosal, and cumulus cells and the oocyte were isolated and subjected to biomarker analyses. Although the size and numerous biomarkers of differentiation, such as mRNAs for the gonadotropin receptors, were similar, intrafollicular concentrations of estradiol and the abundance of mRNAs for CYP19A1 in granulosal cells and ESR1, ESR2, and CTSB in cumulus cells were greater, whereas mRNAs for AMH in granulosal cells and TBC1D1 in thecal cells were lower for animals with low versus a high AFC during follicle waves. Hence, variation in the ovarian reserve may have an impact on follicular function and oocyte quality via alterations in intrafollicular estradiol production and expression of key genes involved in follicle-stimulating hormone action (AMH) and estradiol (CYP19A1) production by granulosal cells, function and survival of thecal cells (TBC1D1), responsiveness of cumulus cells to estradiol (ESR1, ESR2), and cumulus cell determinants of oocyte quality (CTSB).

Low genetic diversity in a marine nature reserve: re-evaluating diversity criteria in reserve design

Little consideration has been given to the genetic composition of populations associated with marine reserves, as reserve designation is generally to protect specific species, communities or habitats. Nevertheless, it is important to conserve genetic diversity since it provides the raw material for the maintenance of species diversity over longer, evolutionary time-scales and may also confer the basis for adaptation to environmental change. Many current marine reserves are small in size and isolated to some degree (e.g. sea loughs and offshore islands). While such features enable easier management, they may have important implications for the genetic structure of protected populations, the ability of populations to recover from local catastrophes and the potential for marine reserves to act as sources of propagules for surrounding areas. Here, we present a case study demonstrating genetic differentiation, isolation, inbreeding and reduced genetic diversity in populations of the dogwhelk Nucella lapillus in Lough Hyne Marine Nature Reserve (an isolated sea lough in southern Ireland), compared with populations on the local adjacent open coast and populations in England, Wales and France. Our study demonstrates that this sea lough is isolated from open coast populations, and highlights that there may be long-term genetic consequences of selecting reserves on the basis of isolation and ease of protection.

Limit operators, collective compactness, and the spectral theory of infinite matrices

In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on .

Hankel and Toeplitz transforms on H 1: continuity, compactness and Fredholm properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H 1 and give a new proof of the old result stating that the Hankel operator H a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H a to be compact on H 1. The Fredholm properties of Toeplitz operators on H 1 are studied for symbols in a Banach algebra similar to C + H ∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H 1 and H 2.