A variant of the Mukai pairing via deformation quantization

We give a new method to prove a formula computing a variant of Caldararu's Mukai pairing \cite{Cal1}. Our method is based on some important results in the area of deformation quantization. In particular, part of the work of Kashiwara and Schapira in \cite{KS} as well as an algebraic index theorem of Bressler, Nest and Tsygan in \cite{BNT},\cite{BNT1} and \cite{BNT2} are used. It is hoped that our method is useful for generalization to settings involving certain singular varieties.Comment: 8 pages. Comments and suggestions welcom

Boiling Point of Binary Liquid Mixtures Using Ultrasonic Investigation

AbstractUltrasonic velocity is a useful tool to find thermal, physical and chemical parameters. In this work using ultrasonic velocity, the boiling point has been calculated for the following binary mixtures, Toluene+Heptan-1-ol, toluene +Octan-1-ol, toluene +Decan-1-ol,  1,1,1 Tricholoroethane +1-propanol, 1,1,1 Tricholoroethane+1-Butanol, 1,1,1 Tricholoroethane +1-Pentanol, 1,1,1Tricholoroethane+1-Hexanol,1,1,1 Tricholoroethane +1-Heptanol, Benzonitrile +Butan-1-ol, Benzonitrile+Pentan-1-ol, Benzonitrile+Methylpropan-2-ol and Benzonitrile+ toluene. The results are discussed

Calcibiocavitological investigations

Calcibiocavitation is a major poroblem in the marine environment and a detailed study on these aspects has been undertaken and the salient findings are presented here. Gregarious molluscs such as the sacred chank Xancus pvrum, mussels (both green and brown), rock oysters (Crassostrea spp.), pearl oysters, Thais rudolphi (Lam.) and corals which inhabit the southwest coast of Kerala and the Gulf of Mannar have been investigated. The wide fluctuations noted in the abundance and population structure of the various pests in the molluscan beds during the short period of two years clearly indicated that they were in severe competetion for suitable substrata and the shells of gregarious mollusks which inhabit this area provide ample opportunities for the pests to flourish. The incidence of boring sponges is found to be rather high among raft-cultured pearl oysters both at Tuticorin and Vizhinjam. It is difficult to control the infection of boring animals in the natural beds, but the low rate of incidence recorded year after year under suggests that the nature controls this to lower level.Another important observation was the wide distribution of the boring sponge C.vastifica in the Ashtamudi Lake, Quilonwhich may form a major threat to our future rock oyster farms along the estuaries

Abel-Jacobi maps for hypersurfaces and non commutative Calabi-Yau's

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non Pfaffian hypersurface but the dual Calabi-Yau becomes non commutative.Comment: 34 pages; exposition of Hochschild homology expanded; references added; introduction re-written; some imrecisions, typos and the orbit diagram in the last section correcte

Underwater observations in the lagoons

The paper deals with the results of the underwater observations carried out on the faunistic and topographical features by direct underwater observations at selected localities covering the entire length and breadth of the lagoons of Minicoy, Suheli paar, Kalpeni, Kavaratti Androth, Agatti, Bangaram, Amini, Kadmat, Kiltan, Bitra and Chetlat between January and April 1987. In each island a week long survey was carried out by means of diving The scope of this report is limited and by no means exhaustive It enables us to provide first hand information on the nature of substratum, disposition of coral reefs, fauna and flora of the lagoon

Invasion of Cliona margaritifera Dendy and C.lobata Hancock on the molluscan beds along the Indian coast

In the recent past the invasion of two new sponge pests Cliona margaritifera Dendy and C. lobata Hancock on the molluscan beds of the southwest coast of India is reported. These two new pests made their first appearance on cultured pearl oysters on raft at Vizhinjam in 1980 and thence started spreading to the economically important molluscan beds in and around Vizhinjam. The spreading of these pests along the southwest coast of India was rather fast and from this coast C. margaritifera could migrate to the raft-cultured pear! oysters at Tuticorin and C. lobata to the chank beds off Tbiruchendur (southeast coast) within two years i.e. by 1982

The spawning of the holothurian Actinopyga mauritiana (Quoy and Gaimard) on board FORV Sagar Sampada

Four specimens of Actinopyga mauHtlana collected on 27th September 1988 in Andaman waters south of Burmanalla near Port Blair, were left in the Aquarium on board FORV Sagar Sampada for observations. The surface temperature of the sea water was 25.0° C and the sea water in the Aquarium was 29.2° C. The higher temperature in the Aquarium triggered the male holothurians to spawn first followed by the female. The fertilized eggs developed only upto four-celled stage

Molluscan resources

A critical review of literature on the fauna of Lakshadweep reveals that there is only scanty reports on the occurrence of molluscs from various islands (Smith, 1906; Appukuttan 1973; Nair and Dharmaraj, 1983 and George ef a/., 19S6). A survey on the fisheries potential of Lakshadweep was undertaken by the scieniists of Central Marine Fisheries Research Institute to provide adequate informations on the molluscan fisheries potential of the islands for future planning and development. The present status of fishery of potentially important molluscs, fishing methods, occurrence and abundance are dealt with in the present account. The island surveyed for molluscan resources are Minicoy, Suheh pear, Kalpeni, Kavaratti, Androth, Agathi, Bangaram, Amini, Kadamat, Kiltan, Bitra and Chetlat. To understand the distribution pattern, population density and habitat, transect method of sampling was adopted uniformly in ail the 'slands. The sampling areas were broadly classified intointertidal area, lagoon, reef cres^ in the lagoon side and leward side and reef slopes of both sides. Quadrat method of sampling for few bivalves and gastropods were done for quantitative estimates. By diving and hand-picking, samples were collected from various stations and relative abundance were noted by visual estimates. The maps of islands shows the distribution of commercially importtant molluscs

From Atiyah Classes to Homotopy Leibniz Algebras

A celebrated theorem of Kapranov states that the Atiyah class of the tangent bundle of a complex manifold $X$ makes $T_X[-1]$ into a Lie algebra object in $D^+(X)$, the bounded below derived category of coherent sheaves on $X$. Furthermore Kapranov proved that, for a K\"ahler manifold $X$, the Dolbeault resolution $\Omega^{\bullet-1}(T_X^{1,0})$ of $T_X[-1]$ is an $L_\infty$ algebra. In this paper, we prove that Kapranov's theorem holds in much wider generality for vector bundles over Lie pairs. Given a Lie pair $(L,A)$, i.e. a Lie algebroid $L$ together with a Lie subalgebroid $A$, we define the Atiyah class $\alpha_E$ of an $A$-module $E$ (relative to $L$) as the obstruction to the existence of an $A$-compatible $L$-connection on $E$. We prove that the Atiyah classes $\alpha_{L/A}$ and $\alpha_E$ respectively make $L/A[-1]$ and $E[-1]$ into a Lie algebra and a Lie algebra module in the bounded below derived category $D^+(\mathcal{A})$, where $\mathcal{A}$ is the abelian category of left $\mathcal{U}(A)$-modules and $\mathcal{U}(A)$ is the universal enveloping algebra of $A$. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of $L/A$ and $E$, and inducing the aforesaid Lie structures in $D^+(\mathcal{A})$.Comment: 36 page