A Poincar\'e-Hopf type formula for Chern character numbers

For two complex vector bundles admitting a homomorphism with isolated singularities between them, we establish a Poincar\'e-Hopf type formula for the difference of the Chern character numbers of these two vector bundles. As a consequence, we extend the original Poincar\'e-Hopf index formula to the case of complex vector fields (to appear in Mathematische Zeitschrift)Comment: 10 page

A note on the Lichnerowicz vanishing theorem for proper actions

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.Comment: 3 page

Holomorphic quantization formula in singular reduction

We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2].Comment: Only the abstract and the introduction are changed, in which the incorrect comments regarding Teleman's work are now correcte

Casson-Lin's invariant of a knot and Floer homology

A. Casson defined an intersection number invariant which can be roughly thought of as the number of conjugacy classes of irreducible representations of $\pi_1(Y)$ into $SU(2)$ counted with signs, where $Y$ is an oriented integral homology 3-sphere. X.S. Lin defined an similar invariant (signature of a knot) to a braid representative of a knot in $S^3$. In this paper, we give a natural generalization of the Casson-Lin's invariant to be (instead of using the instanton Floer homology) the symplectic Floer homology for the representation space (one singular point) of $\pi_1(S^3 \setminus K)$ into $SU(2)$ with trace-free along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number of such a symplectic Floer homology is the negative of the Casson-Lin's invariant.Comment: 22 pages, AmsLaTe

Lu Qi-Keng Conjectue and Hua Domain

The first part I talk about the motivation for Lu Qi-Keng conjecture and the results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in ${\bf{C^n}}$. The second part I summarize the main results on Hua domains, such as the explicit Bergman kernel function, comparison theorem for the invariant metrics, explicit complete Einstein-K\"ahler metrics, the equivalence between the Einstein-K\"ahler metric and the Bergman metric etc.Comment: Dedicated to Lu Qi-Keng on the occasion of his 80th birthday, 16 page

Explicit singular minimal surface solutions for gravitational instantons

We construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimesional Euclidean space and gravitational instantons possessing two killing vectors. By Calabi's correspondence, we derive a family of explicit maximal surface solution for spacelike surface with zero mean curvature equation.Comment: 4 page

Circle actions and Z/k-manifolds

We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.Comment: 6 pages. Minor changes for the published versio

Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows

In this paper we consider the equations of the unsteady viscous, incompressible, and heat conducting magnetohydrodynamic flows in a bounded three-dimensional domain with Lipschitz boundary. By an approximation scheme and a weak convergence method, the existence of a weak solution to the three-dimensional density dependent generalized incompressible magnetohydrodynamic equations with large data is obtained.Comment: 32page

Loss Rate Estimators and the Properties for the Tree Topology

A large number of explicit estimators are proposed in this paper for loss rate estimation in a network of the tree topology. All of the estimators are proved to be unbiased and consistent instead of asymptotic unbiased as that obtained in [1] for a specific estimator. In addition, a set of formulae are derived for the variances of various maximum likelihood estimators that unveil the connection between the path of interest and the subtrees connecting the path to observers. Using the formulae, we are able to not only rank the estimators proposed so far, including those proposed in this paper, but also identify the errors made in previous works. More importantly, using the formulae we can easily identify the most efficient explicit estimator from a pool that makes model selection feasible in loss tomographyComment: 17 pages, 1 figure. arXiv admin note: text overlap with arXiv:1205.624

A mod 2 index theorem for pin−^- manifolds

We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin$^-$ manifolds. The analytic index is the reduced $\eta$ invariant of (twisted) Dirac operators and the topological index is defined through $KO$-theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-orientable manifolds.Comment: 21 pages. MSRI Preprint No. 053-9