Characterization of projective spaces and Pr\mathbb P^r-bundles as ample divisors

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_X$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^n$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^r$-bundle as an ample divisor by the recent work of D.~Litt.Comment: 13 pages. Final version. To appear on Nagoya Mathematical Journa

An alternative implementation of the Lanczos algorithm for wave function propagation

We reformulate the Lanczos algorithm for quantum wave function propagation in terms of variational principle. By including some basis states of previous time steps into the variational subspace, the resultant accuracy increases by several orders. Numerical errors of the alternative method accumulate much slower than that of the original Lanczos method. There is almost no extra numeric cost for the gaining of the accuracy, i.e., the accuracy increase needs no extra operations of the Hamiltonian acting on state vectors, which are the major numeric cost for wave function propagation. A wave packet moving in a 2-dimensional H\'enon-Heiles model serves as an illustration. This method is suitable for small time step propagation of quantum wave functions in large scale time dependent calculations where the operations of the Hamiltonian acting on state vectors are expensive.Comment: To appear in J. Phys.

Graphs with small diameter determined by their DD-spectra

Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},...,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra