η\eta-invariant and flat vector bundles

We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $\eta$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with the analytic continuation procedure appearing in the original article of Atiyah, Patodi and Singer.Comment: 6 page

Superconnection and family Bergman kernels

We establish an asymptotic expansion for families of Bergman kernels. The key idea is to use the superconnection as in the local family index theorem.Comment: C. R. Math. Acad. Sci. Pari

Bergman kernels and symplectic reduction

We generalize several recent results concerning the asymptotic expansions of Bergman kernels to the framework of geometric quantization and establish an asymptotic symplectic identification property. More precisely, we study the asymptotic expansion of the $G$-invariant Bergman kernel of the spin^c Dirac operator associated with high tensor powers of a positive line bundle on a symplectic manifold. We also develop a way to compute the coefficients of the expansion, and compute the first few of them, especially, we obtain the scalar curvature of the reduction space from the $G$-invariant Bergman kernel on the total space. These results generalize the corresponding results in the non-equivariant setting, which has played a crucial role in the recent work of Donaldson on stability of projective manifolds, to the geometric quantization setting. As another kind of application, we generalize some Toeplitz operator type properties in semi-classical analysis to the framework of geometric quantization. The method we use is inspired by Local Index Theory, especially by the analytic localization techniques developed by Bismut and Lebeau.Comment: 132 page

Effect of Ceramic Properties and Depth-of-penetration Test Parameters on the Ballistic Performance of Armour Ceramics

Through an analysis on the relationship among ceramic properties, the depth of penetration (DOP) test parameters and the ballistic performance of armour ceramics based on literatures, the effects of ceramic type, tile thickness and projectile velocity on the ballistic performance of different kinds of ceramics were investigated systematically. The results show that the ballistic performance of different armour ceramics mainly depends on its density, and by using thin ceramic tiles or under high velocity impact, the ceramic composite armour could not provide effective ballistic protection. Furthermore, the differences in the ballistic performance of armour ceramic are found due to the different ballistic performance criteria and DOP test conditions. Additionally, the slope of the depth of penetration (not include tile thickness) (Pa) versus tile thickness has negative correlation with flexural strength of ceramics, indicating the flexural strength can be one of the criteria to evaluate the performance of armour ceramics