We consider virtual pullbacks in K-theory, and show that they are bivariant classes and satisfy certain functoriality. As applications to K-theoretic counting invariants, we include proofs of a virtual localization formula for schemes and a degeneration formula in Donaldson-Thomas theory.

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By using in situ neutron reflectivity, we measured the swelling behavior of two types of polymer brushes, deuterated polystyrene with a trichlorosilane end group and deuterated polystyrene-block-poly(4-vinylpyridine) block copolymer, in supercritical carbon dioxide (scCO 2 ). The measurements were conducted in the pressure range of 0.1-20 MPa at 36°C. The pressure dependence of the brush height clearly showed an anomalous peak at the density fluctuation ridge (pressure ϭ 8.2 MPa) that defined the maximum long-range density fluctuation amplitude in the pressure-temperature phase diagram of carbon dioxide (CO 2 ). The density profile of the brush, which could be approximated by a simple step function, and the magnitude of the brush height both indicated that the solvent quality of scCO 2 for the deuterated polystyrene brushes was still poor even at the density fluctuation ridge. In addition, atomic force microscopy images for the frozen polystyrene brush prepared by the rapid drying of CO 2 showed a phase-separated structure, as predicted from the numerical calculations of Grest and Murat, as a function of the variable N, where N is the polymerization index and is the grafting density.

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The purpose of this short article is to prove a product formula relating the log Gromov-Witten invariants of V × W with those of V and W in the case the log structure on V is trivial. Introduction.Product formulas in the literature of Gromov-Witten (GW) theory started with [15] for the genus zero Gromov-Witten invariants. It was soon generalized in [8] to (absolute) GW invariants in any genus. There is also an orbifold version in [7]. In this paper, we extend the product formula to the setting of relative GW theory or more generally log GW theory.

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In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth bases. An essential ingredient in the proof is a quantum splitting principle which reduces a statement in Gromov-Witten theory on non-split bundles to the case of split bundles.