Proved are two results showing connections between the Hankel transplantation and a transplantation for a certain kind of Laguerre and Jacobi expansions. An asymptotic formula of Hilb's type for Laguerre and Jacobi polynomials is used. As an application of this link we obtain an extension of Guy's transplantation theorem for the Hankel transform to the case $\alpha,\gamma>-1$ also with more weights allowed. This is done by transferring a corresponding transplantation result for Jacobi expansions which was proved by Muckenhoupt. In the case when $\alpha,\gamma\ge-1/2$ the same is obtained by using Schindler's explicit kernel formula for the transplantation operator.

We investigate the classification problem of hypersurfaces with affine normal parallel second fundamental (cubic) form. A new method of approaching the solution to this problem is here presented; it consists in showing and using the equivalence of the mentioned problem with the classification of a certain class of solutions to the equation of Monge-Ampère type $\det(\pa_{ij}f)=\pm1$.

Let $k$ be a number field and $\ti k$ a fixed quadratic extension of $k$. In this paper and its companions, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\ti k$ contained in the biquadratic extensions of $k$ containing $\ti k$.

Nevanlinna showed that for two nonconstant meromorphic functions on the complex plane, if they have the same inverse images counting multiplicities for four distinct values, then they coincide up to a Möbius transformation, and if they have the same inverse images for five distinct values, then they coincide. Fujimoto and Smiley extended Nevanlinna's uniqueness theorems to the case of meromorphic mappings of several complex variables into the complex projective space for hyperplanes. Recently, Motivated by Ru Min and Stoll's accomplishment of the second main theorem for moving targets, Li Baoqin and Shirosaki proved some uniqueness theorems of entire functions in several complex variables and meromorphic functions in one complex variable, respectively, for moving targets. Using the techniques of value distribution theory in several complex variables, we prove some uniqueness theorems of meromorphic mappings of several complex variables into the complex projective space for moving targets.

We consider permanence of an SIR epidemic model with distributed time delays. Based on some known techniques on limit sets of differential dynamical systems, we show that, for any time delay, the SIR epidemic model is permanent if and only if an endemic equilibrium exits.

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved using toric Mori theory due to Reid.