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Let (R,) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function HI*(n) = ℓ(R/In)) and the corresponding tight Hilbert polynomial PI*(n), where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of PI*(n). We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes. [ABSTRACT FROM AUTHOR]

Cloud computing is a new paradigm for hosting and delivering services using the standard protocols of the Internet. It refers to a service-oriented architecture that offers ubiquitous and convenient computing, greater flexibility, on-demand services, reduced total cost of ownership, reduced information technology overhead for the end-user, and many other things. It allows access to a large amount of computing power in a fully virtualized mode by aggregating a configurable pool of resources that are rapidly provisioned and released dynamically. It also provides the capability to utilize scalable, distributed computing environments within the confines of the Internet with minimal management effort while keeping a single coherent system view. Cloud computing is constantly gaining popularity in the information technology industry, and so more and more issues and challenges are emerging that need to be handled for the success of this paradigm. This paper presents an introduction of cloud computing, and a comparative study of cloud computing and grids. Furthermore, we put forward pinpoint research issues in the cloud computing paradigm and a mathematical understanding of several issues among them. [ABSTRACT FROM AUTHOR]

Dyeing behaviour of four different natural dyes extracted from the leaves collected in a sustainable manner from respective dye containing plants has been studied on cotton fabric. A novel and ecofriendly mordanting technique has been devised for creating aluminium-tannate complex on cotton matrix. The new mordanting technique not only involves application of a modified sequence of mordanting chemicals but also employs addition of specific salts in water used in the dyeing process. The dyeing of the mordanted fabric has been carried out under acidic and alkaline conditions at different temperatures. The modified dyeing technique shows an overall improvement of 140-300% in colour strength along with the excellent fastness properties, such as washing, rubbing, perspiration and light. The overall results indicate that all the four natural dyes can be effectively and economically used for dyeing on cotton fabric. The entire chain of collection, extraction and dyeing methodology is environmentally and ecologically compatible. [ABSTRACT FROM AUTHOR]

We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e1. As an application, we derive bounds on the higher Hilbert coefficient ei in terms of e0. [ABSTRACT FROM AUTHOR]

Let (R, m) be a Cohen–Macaulay local ring and let I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicity which are analogues of ideals studied by J. Sally [Compositio Math.40 (1980), 167–175]. The main theorem describes the Hilbert series of fiber cones of these ideals. [ABSTRACT FROM AUTHOR]

Eighteen genotypes of Brassica napus were crossed to a cytoplasmic male sterile (CMS) line of B. napus BO15 carrying B. tournefortu cytoplasm ('tour' cytoplasm). Fourteen genotypes were found to be stable maintainers of the 'tour' CMS. Of the remaining four genotypes, GSL-1 and 'Asahi-natane' were found to be heterozygous and 'Mangun' and 'Yudal' were homozygous for the restorer gene. Analysis of the F[SUB1] and F[SUB2] progenies of (CMS) BO15 × 'Mangun' and (CMS) BO15 × 'Yudal' showed that fertility restoration is controlled by a single dominant gene.The availability of a number of stable maintainer lines and the simple inheritance pattern of fertility restorer gene makes 'tour' CMS a useful system for hybrid seed production in rapeseed. [ABSTRACT FROM AUTHOR]