We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive interaction supported by a smooth surface in R 3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrdinger type operator with an effective potential expressed in terms of the interaction support curvatures.

We simulate blood flow in patient-specific cerebral arteries. The complicated geometry in the human brain makes the problem challenging. We use a fully unstructured three dimensional mesh to cover the artery, and Galerkin/least-squares finite element method to discretize the incompressible Navier-Stokes equations, that are employed to model the blood flow, and the resulting large sparse nonlinear system of equations is solved by a Newton-Krylov-Schwarz algorithm. From the computed flow fields, we are able to understand certain behavior of the blood flow of this particular patient before and after a stenosis is surgically removed. We also report the robustness and parallel performance of the domain decomposition based algorithm.

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We extend the idea of crossvalidation to choose the smoothing parameters of the double-kernel local linear regression for estimating a conditional density. Our selection rule optimises the estimated conditional density function by minimising the integrated squared error. We also discuss three other bandwidth selection rules, an ad hoc method used by Fan et al. (1996), a bootstrap method of Hall et al. (1999) for bandwidth selection in the estimation of conditional distribution functions, modified by Bashtannyk & Hyndman (2001) to cover conditional density functions, and finally a simple approach proposed by Hyndman & Yao (2002). The performance of the new approach is compared with these three methods by simulation studies, and our method performs outstandingly well. The method is illustrated by an application to estimating the transition density and the Value-at-Risk of treasury-bill data.

Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N=(2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -- recently computed via localization by Benini et al. and Doroud et al. -- yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in {\alpha}'. We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P^7, recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.