Curves on K3 surfaces and modular forms

We study the virtual geometry of the moduli spaces of curves and sheaves on K3 surfaces in primitive classes. Equivalences relating the reduced Gromov-Witten invariants of K3 surfaces to characteristic numbers of stable pairs moduli spaces are proven. As a consequence, we prove the Katz-Klemm-Vafa conjecture evaluating $\lambda_g$ integrals (in all genera) in terms of explicit modular forms. Indeed, all K3 invariants in primitive classes are shown to be governed by modular forms. The method of proof is by degeneration to elliptically fibered rational surfaces. New formulas relating reduced virtual classes on K3 surfaces to standard virtual classes after degeneration are needed for both maps and sheaves. We also prove a Gromov-Witten/Pairs correspondence for toric 3-folds. Our approach uses a result of Kiem and Li to produce reduced classes. In Appendix A, we answer a number of questions about the relationship between the Kiem-Li approach, traditional virtual cycles, and symmetric obstruction theories. The interplay between the boundary geometry of the moduli spaces of curves, K3 surfaces, and modular forms is explored in Appendix B by A. Pixton.Comment: An incorrect example in Appendix A, pointed out to us by Dominic Joyce, has been replaced by a reference to a new paper arXiv:1204.3958 containing a corrected exampl

Studies on Beach Placers of Kerala Coast

In the present paper, studies carried out on beach placer samples of Kerala coast in general and contribut-ions of NML and its current activities in particular on beach sands of Kerala have been discussed. While earlier beneficiation study at NML was on purification of zircon, present study, under network project, consist of charac-terization of samples from different areas to identify prospective area that will be taken up for development of process technology. Characterisation studies involve size and chemical analyses, sink and float studies, optical microscopy and scanning electron microscopy. Samples from four different areas (Valarpattanam Azhikod (VA), Chava-kkad Ponnani (CP), Neendakara Kayamkulam (NK) and Warkala coast) were studied. It has been found that while heavy mineral concentration at NK and Warkala coast is quite high, the same at CP is rather low in comparison and at VA it's not significant

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page

Beneficiation studies on Bolani iron ore

The iron ore washing plants generate slime on an average of 10 - 20% by weight of the ore and is wasted. Apart from loss of resource, these slimes pose problem of disposal and consequent damage to environment. To reduce problems of slime, it should be treated, as a part of beneficiation, to minimise waste generation. Out of five number of different varieties of Bolani iron ore bulk samples, two bulk composite samples were prepared in the proportions as decided by the sponsor. The first sample was found to be of very good quality assaying 66.69% Fe, 1.27% SiO, and 1.8% Al,O and only crushing and sizing (dry) were done for physical characterisation. The second sample (assaying 63.68% Fe, 2.05% SiO, and 2.6% AI,O,) was beneficiated to produce washed lump weighing 49.9% and with assay of 65.0% Fe. 1.8% SiO, and 2.1% Al,Or Combined fines weighed 42.7% assaying 63.95% Fe, 1.81% Si02 and 2.54% A1,03 leaving slime reject 7.4% by weight and assaying 51.27 % Fe, 5.6 % SiO, and 8.82% Al,O3 which not only reduced waste generation but improved recovery also

Reduction of Environment Pollution in Processing of Barite

One of the largest indigenous manufacturing plants of barium chemicals is located at Cuddapah in Andhra Pradesh. During production of barium carbonate, two types of rejects (flue ash and sludge) have been generated and accumulate for last few decades at the Cuddapah plant. Earlier, briquetting of charge was suggested to restrict physical losses of material as flue ash. Afterwards, to generate product having > 90% barium sulphate which could be reused, studies have been carried out with the rejects currently produced. Detailed investigations suggested seperate treatment of sludge because the flue ash sample was found to be suitable as feed to the reduction circuit. Both gravity and flotation processes were found to generate product meeting specification. It was also found that more than 55% of the combined rejects currently generated at Cuddapah plant could be used again for the production of barium carbonate. In the process, recovey of barite value could be more than 64

Minerological aspects of lead sintering

A brief overview on lead sinter microstructure is presented. Characteristic micro-structural features of a good and bad sinter are highlighted and these are used in a case study involving use of a low grade and complex concentrate of lead (-40% Pb) in the sintering operation. The plant sinter produced exhibited low strength and its nticrostructural examination revealed non-uniform distribution of porosity along with unsintered galena and low melting lead silicate phase. Part replacement of limestone by lime helped in producing sinter with good physical properties and desirable microstructure. The sinter with modified feed chemistry had more uniform distribution of porosity and presence of primarily a Pb-Fe silicate phase characterised by a (Pb+Fe):Si mole ratio of 3:1. Ca-Pb-Zn-Fe-Al-silicate phase identified as hardysonite and a spine! phase of the type (Fe,Zn)O.(Fe,Al),OJ. Lead nietal/oxide/sulphide occurred in the sinter only rarely. The likely implications of lime addition to the sinter charge mix are discussed Key Words: Lead. Complex and low grade concentrate. Sintering. Process Mineralog

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page

Topological wave functions and heat equations

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian symmetric tube domain $G/K$, we show that the general solution of the anomaly equations is a matrix element \IP{\Psi | g | \Omega} of the Schr\"odinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification adde

Instanton counting, Macdonald function and the moduli space of D-branes

We argue the connection of Nekrasov's partition function in the \Omega background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N=2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters \epsilon_1, \epsilon_2 of toric action on C^2 factorizes correctly as the character of SU(2)_L \times SU(2)_R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F_0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T^2 action allows us to obtain the generating functions of equivariant \chi_y and elliptic genera of the Hilbert scheme of n points on C^2 by the method of topological vertex.Comment: 33 pages, 2 figures, (v2) minor changes, references added, (v3) Comments and more references adde

Heterozygous disruption of Flk-1 receptor leads to myocardial ischaemia reperfusion injury in mice: application of affymetrix gene chip analysis

This study addresses an important clinical issue by identifying potential candidates of vascular endothelial growth factor (VEGF) signalling through the Flk-1 receptor that trigger cardioprotective signals under ischaemic stress. Isolated working mouse hearts of both wild-type (WT) and Flk-1+/− were subjected to global ischaemia (I) for 30 min. followed by 2 hrs of reperfusion (R). Flk-1+/− myocardium displayed almost 50% reduction in Flk-1 mRNA as examined by quantitative real-time RT-PCR at the baseline level. Flk-1+/− mouse hearts displayed reduction in left ventricular functional recovery throughout reperfusion (dp/dt 605 versus 884), after 2 hrs (P < 0.05). Coronary (1.9 versus 2.4 ml) and aortic flow (AF) (0.16 versus 1.2 ml) were reduced in Flk-1+/− after 2 hrs of reperfusion. In addition, increased infarct size (38.4%versus 28.41%, P < 0.05) and apoptotic cardiomyocytes (495 versus 213) were observed in Flk-1+/− knockout (KO) mice. We also examined whether ischaemic preconditioning (PC), a novel method to induce cardioprotection against ischaemia reperfusion injury, through stimulating the VEGF signalling pathway might function in Flk-1+/− mice. We found that knocking down Flk-1 resulted in significant reduction in the cardioprotective effect by PC compared to WT. Affymetrix gene chip analysis demonstrated down-regulation of important genes after IR and preconditioning followed by ischaemia reperfusion in Flk-1+/− mice compared to WT. To get insight into the underlying molecular pathways involved in ischaemic PC, we determined the distinct and overlapping biological processes using Ingenuity pathway analysis tool. Independent evidence at the mRNA level supporting the Affymetrix results were validated using real-time RT-PCR for selected down-regulated genes, which are thought to play important roles in cardioprotection after ischaemic insult. In summary, our data indicated for the first time that ischaemic PC modifies genomic responses in heterozygous VEGFR-2/Flk-1 KO mice and abolishes its cardioprotective effect on ischaemic myocardium