No abstract uploaded!

In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new %Akizuki-Kodaira-Nakano type vanishing theorems for sheaves of logarithmic differential forms on compact K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

Global geometrical optics method is a new semi-classical approach for the high frequency linear waves proposed by the author in [13]. In this paper, we rederive it in a more concise way. It is shown that the right candidate of solution ansatz for the high frequency wave equations is the extended WKB function, other than the WKB function used in the classical geometrical optics approximation. A new and main contribution of this paper is an interface analysis for the Helmholtz equation when the incident wave is of extended WKB-type. We derive asymptotic expressions for the reflected and/or transmitted propagating waves in the general case. These expressions are valid even when the incident rays include caustic points.

No abstract uploaded!

A simplicial complex Δ is called$flag$if all minimal nonfaces of Δ have at most two elements. The following are proved: First, if Δ is a flag simplicial pseudomanifold of dimension$d$−1, then the graph of Δ (i) is (2$d$−2)-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the$d$-dimensional cross-polytope. Second, the$h$-vector of a flag simplicial homology sphere Δ of dimension$d$−1 is minimized when Δ is the boundary complex of the$d$-dimensional cross-polytope.