Quiver varieties and crystals in symmetrizable type via modulated graphs

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.Comment: 15 pages. v2: minor revision

Modular representations in type A with a two-row nilpotent central character

We study the category of representations of $\mathfrak{sl}_{m+2n}$ in positive characteristic, whose p-character is a nilpotent whose Jordan type is the two-row partition (m+n,n). In a previous paper with Anno, we used Bezrukavnikov-Mirkovic-Rumynin's theory of positive characteristic localization and exotic t-structures to give a geometric parametrization of the simples using annular crossingless matchings. Building on this, here we give combinatorial dimension formulae for the simple objects, and compute the Jordan-Holder multiplicities of the simples inside the baby Vermas (in special case where n=1, i.e. that a subregular nilpotent, these were known from work of Jantzen). We use Cautis-Kamnitzer's geometric categorification of the tangle calculus to study the images of the simple objects under the [BMR] equivalence. The dimension formulae may be viewed as a positive characteristic analogue of the combinatorial character formulae for simple objects in parabolic category O for $\mathfrak{sl}_{m+2n}$, due to Lascoux and Schutzenberger

Categorification via blocks of modular representations II

Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$ using singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $\mathfrak{sl}_k$-action, following Sussan's approach, by considering more singular blocks of modular representations of $\mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $\mathfrak{sl}_n$.Comment: 24 page

User centric web search using ontology

Semantic information retrieval systems query the World Wide Web based on context information, and are intended to provide more pertinent search results. However, most of the existing systems overlook one important aspect ‘the user’. They are more focused on eliminating the obscure results that a conventional or non-semantic search engine would throw up and hence, they are pretty much static. On the other hand, our effort would channel its focus more towards providing a more user-centric service using ontology and involving learning and prediction. By studying the usage statistics of the user, context information can be built and used effectively to produce better search results. Such an approach also entails that the knowledge that is accrued, be organized such that the relationships between the data elements can be elicited easily and unambiguously. The ontology would be described using OWL. Latent semantic indexing algorithm is used for context analysis and retrieva

Biodiesel Production from Waste Cooking Oil

Biodiesel is receiving increased attention as an alternative, non-toxic, biodegradable and renewable diesel fuel and contributes a minimum amount of net greenhouse gases, such as CO2, SO2 and NO emissions to the atmosphere. Exploring new energy resources, such as biofuel is of growing importance in recent years. The possibility of obtaining oil from plant resources has created a great importance in several countries. Vegetable oil after esterification being used as bio diesel, Considering the cost and demand of the edible oil is bearable, so it may be preferred for the preparation of bio diesel in India.  The transesterification of waste cooking oils with methanol as well as the main uses of the fatty acid methyl esters are reviewed. The general aspects of this process and the applicability of different types of catalysts (acids, alkaline metal hydroxides, alkoxides and carbonates, enzymes and non-ionic bases, such as amines, amides, and guanidine and triamino (imino) phosphoranes) are described. Transesterification is carried in a reaction cavity, once the reaction is complete, glycerine and biodiesel are gravity separated.