Let$f$be an entire function of order at least 1/2,$M(r)$=max_{|}$z$|=$r$|$f(z)$|, and$n(r, a)$the number of zeros of$f(z)-a$in |$z$|≤$r$. It is shown that lim sup_{$r$→∞}$n(r, a)$/log$M (r)$≥1/2π for all except possibly one$a$∈C.

Let$M$be a connected, noncompact, complete Riemannian manifold, consider the operator$L=Δ+∇V$for some$V∈C$^{2}(M) with exp[$V$] integrable with respect to the Riemannian volume element. This paper studies the existence of the spectral gap of$L$. As a consequence of the main result, let ϱ be the distance function from a point o, then the spectral gap exists provided lim_{ϱ→∞}sup$L$_{ϱ<0}while the spectral gap does not exist if o is a pole and lim_{ϱ→∞}inf$L$_{ϱ≥0}. Moreover, the elliptic operators on$R$^{$d$}are also studied.

We propose an algorithm for the state feedback pole assignment problem. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newtons method. Eleven illustrative examples, comparing our algorithm with three other existing ones, are given.

The purpose of this research monograph is to survey some recent developments in the analysis of shock reflection-diffraction, to present our original mathematical proofs of von Neumann’s conjectures for potential flow, to collect most of the related results and new techniques in the analysis of partial differential equations achieved in the last decades, and to discuss a set of fundamental open problems relevant to the directions of future research in this and related areas.

In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rigid Calabi-Yau manifolds poses a serious challenge to the conjecture. The proposed mirror partners for them are higher dimensional generalized Calabi-Yau manifolds. For example, the mirror partner for a certain K3 surface is a cubic fourfold and its Fano variety of lines is birational to the Hilbert scheme of two points on the K3. This hyperk¨ahler manifold can be interpreted as the SYZ mirror of the K3 by considering singular special Lagrangian tori. We also compare the geometries between a CY and its associated generalized CY. In particular we present a new construction of Lagrangian submanifolds.