We present a necessary and sucient condition for a holomorphic mapping to be a generator of a ow on any nite Cartesian product of Hilbert balls. A related null point theorem is also established.

The object of the present paper is to investigate a sucient condition and an angular estimation for -like functions. Our result contains the previous results as special cases.

We consider some solution methods for large sparse linear systems of equations which arise from second-order elliptic finite element problems defined on composite meshes. Historically these methods were called FAC and AFAC methods. Optimal bounds of the condition number for certain AFAC iterative operator are established by proving a strengthened Cauchy-Schwarz inequality using an interpolation theorem for Hilbert scales. This work completes earlier work by Dryja and Widlund. We also apply an extension theorem for finite element functions to get a weaker bound under some more general assumptions. The optimality of the FAC methods, with exact solvers or spectrally equivalent inexact solvers being used, is also proved by using similar techniques and some ideas from multigrid theory..

We study the first kind integral equation $$ \displaystyle\int^{+\infty}_0 k(x-y)\sigma (y)dy = g(x) $$ by the wavelet method. The integral equation is discretized with respect to two different wavelet bases. We then have two different linear systems. One of them is a Toeplitz system and the other one is a system with condition number $\kappa =O(1)$ after a diagonal scaling. By using the preconditioned conjugate gradient (PCG) method with the fast wavelet transform (FWT) and the fast Fourier transform (FFT), we can solve the systems in $O(n \log n)$ operations where $n$ is the size of the systems.

Using the scaling equations, quadratures involving polynomials and scaling (or wavelet) functions can be evaluated by linear algebraic equations (which are theoretically exact) instead of numerical approximations. We study two matrices which are derived from these kinds of quadratures. These particular matrices are also seen in the literature of wavelets for other purposes.

Let $E(X)$ and $F(Y)$ be vector-valued sequence spaces and A be an operator-valued infinite matrix which maps $E(X)$ into $F(Y)$. In this paper, we establish the continuity and boundedness results for matrix A which generalize the scalar results.

Let $R$ be a prime ring with extended centroid $C$ and $f(X_1, \ldots, X_t)$ a polynomial over $C$ which is not central-valued on $RC$. Suppose that $d$ and $\delta$ are two derivations of $R$ such that $$ d(f(x_1, \ldots, x_t))f(x_1, \ldots, x_t) -f(x_1, \ldots, x_t) \delta (f(x_1, \ldots, x_t)) \in C $$ for all $x_1, \ldots, x_t$ in $R$. Then either $d = 0 = \delta$, or $\delta = - d$ and ${f(X_1, \ldots, X_t)}^2$ is central-valued on $RC$, except when $\rm charR =2$ and $\dim_CRC = 4$.

We show that interesting dividing formulas such as, Chinese theorem, Fermat's little theorem, and Euler's theorem can easily be derived from some well-known iterated maps. Other dividing formulas concerning Fibonacci numbers, generalized Fibonacci numbers of degree m, and numbers of other types can also be derived. The results show that iterated maps offer a systematic and unified way for obtaining nontrivial dividing formulas $n|Q(n)$, and we can thus understand the dividing formulas from the point of view of iterated maps.

An extension of Kato's stability condition for nonautonomous Cauchy problems is presented. It is proved that in the commutative case this condition and a mild regularity assumption imply wellposedness. If one supposes the Kato-stability, then the solutions are given by an integral formula. By means of examples we show that in general these stability conditions cannot be omitted in our results. Moreover, it is seen that the Kato-stability is not necessary for wellposedness.

Let $X$ be a complex Banach spcce, and denote by $T$ a strongly continuous semigroup of linear operators defined on $X$ and by $C$ a cosine function of operators with associated sine function $S$ defined on $X$. In this note we characterize in terms of spectral properties of the infinitesimal generator those semigroups $T$ and cosine functions $C$ such that $\{T(t) - I : t \geq 0 \}$, $\{C(t) - I : t \in {\Bbb R}\}\;$ and $\{S(t) : t \in {\Bbb R}\}$ are collectively compact sets of bounded linear operators.