Interface problems have many applications. Mathematically, interface problems usually lead to differential equations whose input data and solutions are non-smooth or discontinuous across some interfaces. The immersed interface method (IIM) has been developed in recent years particularly designed for interface problems. The IIM is a sharp interface method based on Cartesian grids. The IIM makes use of the jump conditions across the interface so that the finite difference/element discretization can be accurate. In this survey paper, we will introduce the immersed interface method for various problems, discuss its recent advances and related software packages, and some of its applications. We also review some other related methods and references in this survey paper.

The purpose of this paper is to give a p-adic version of value distribution theory for hypersurfaces.

We prove that if $T$ is a power-bounded operator of class $C_{*\cdot}$ on a Hilbert space which commutes with a nonzero quasinilpotent operator, then $T$ has a nontrivial invariant subspace. Connections with the questions of convergence of $T^n$ to $0$ in the strong operator topology and of cyclicity of power-bounded operators of class $C_{1\cdot}$ are discussed.

Let a closed, minimal, symplectic 4-manifold $X$ contain a symplectic surface $F$ such that the genus $g$ of $F$ is greater than or equal to one and the value $c_1(TX)[F]\gt g$. Then we show that the space $X$ is rational or ruled.

Let $\mu$ be a probability measure on the real line with finite moments of all orders. Apply the Gram-Schmidt orthogonalization process to the system $\{1, x, x^2, \ldots, x^n, \ldots \}$ to get orthogonal polynomials $P_n(x), \,n\geq 0,$ which have leading coefficient 1 and satisfy $(x-\alpha_n)P_n(x) = P_{n+1}(x) + \omega_n P_{n-1}(x)$. In general it is almost impossible to use this process to compute the explicit form of these polynomials. In this paper we use the multiplicative renormalization to develop a new method for deriving generating functions for a large class of probability measures. From a generating function for $\mu$ we can compute the orthogonal polynomials $P_n(x),\,n\geq 0$. Our method can be applied to derive many classical polynomials such as Hermite, Charlier, Laguerre, Legendre, Chebyshev (first and second kinds), and Gegenbauer polynomials. It can also be applied to measures such as geometric distribution to produce new orthogonal polynomials.

A description of the generalized $(\alpha,\beta)$-derivations $f$, $g$ and $h$ of a prime ring $R$ which satisfying $h=af+gb$ is given.

The relation between directional derivatives of generalized distance functions and the existence of generalized nearest points in Banach spaces is investigated. We show that if the generalized function generated by a closed set has a one-sided directional derivative equal to 1 or -1, then the existence of generalized nearest points follows. We also give a partial answer to an open problem proposed by S. Fitzpatrick.

In 1884, Koenigs showed that when $\varphi$ is an analytic self-map of the unit disk fixing the origin, with $0\lt |\varphi'(0)|\lt 1$, then Schroeder's functional equation, $f\circ\varphi=\varphi'(0)f$, can be solved for a unique analytic function $f$ in the disk with $f'(0)=1$. Here we consider a natural analogue of Schroeder's equation in the unit ball of ${\Bbb C}^N$ for $N\gt 1$, namely, $f\circ\varphi=\varphi'(0)f$ where $\varphi$ is an analytic self-map of the unit ball fixing the origin and $f$ is to be a ${\Bbb C}^N$-valued analytic map on the ball. Under some natural hypotheses on $\varphi$, we give necessary and sufficient conditions for the existence of a solution $f$ satisfying $f'(0)=I$ and then describe all analytic solutions in the ball. We also discuss various phenomena which may occur in the several variable setting that do not occur when $N=1$.

In this paper we prove the existence of mild and strong solutions of an integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using uniformly continuous semigroups and the Schauder fixed point theorem.

For two planar undirected graphs $G$ and $H$, the {\it subgraphs recognition problem} (SRP) is to find and list all subgraphs of $G$ which are isomorphic to $H$. In this paper, we introduce the idea of layer-decomposition with degree and present an algorithm based on this idea for SRP. Since subgraphs isomorphic to each other contain their spanning trees with the same decomposition, the SRP can be decomposed into two subproblems: First, find subtrees of $G$ which have the same layer-decomposition as that of $H$. Then, test whether the induced subgraphs generated by these subtrees are isomorphic to $H$. By this scheme, we greatly decrease the complexity to $O(n(\Delta-1)^{k-1}k^2)$, where $\Delta$ is the degree of $G$ and $n, k$ are the orders of $G, H$ respectively.