Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

Let $(X,\mathcal{F},\mu)$ be a complete probability space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. We consider the extreme points of the closed unit ball $\mathbb{B}(\mathcal{A})$ of the normed space $\mathcal{A}$ whose points are the elements of $L^\infty(X,\mathcal{F},\mu)$ with the norm$\Vert\, f \Vert = \Vert\Phi(\vert\, f \vert)\Vert_\infty$, where $\Phi$ is the probabilistic conditional expectation operator determined by $\mathcal{B}$. No $\mathcal{B}$- measurable function is an extreme point of the closed unit ball of $\mathcal{A}$, and in certain cases there are no extreme points of $\mathbb{B}(\mathcal{A})$.

For an interesting family of examples, depending on a parameter $n$, we characterize the extreme points of the unit ball and show that every element of the open unit ball is a convex combination of extreme points. Although in these examples every element of the open ball of radius $\frac{1}{n}$ can be shown to be a convex combination of at most $2n$ extreme points by elementary arguments, our proof for the open unit ball requires use of the $\lambda$-function of Aron and Lohman. In the case of the open unit ball, we only obtain estimates for the number of extreme points required in very special cases, e.g. the $\mathcal{B}$-measurable functions, where $2n$ extreme points suffice.

In this paper, we discuss the structure of the global attractor of a positively bounded system. In particular, we are concerned with the existence of connecting orbits and the relation between maximal elliptic sectors and connecting orbits. For the systems with two singular points a necessary and sufficient condition for the existence of connecting orbits is given.

With the help of some $p$-adic formal series over $p$-adic number fields and the estimates of character sums over Galois rings, we prove that there is a constant $C(n)$ such that there exists a primitive polynomial $f(x)\,{=}\,x^{n}-a_{1}x^{n-1}+\cdots +(-1)^{n}a_{n}$ of degree $n$ over $F_{q}$ with the first $m=\lfloor\frac{n-1}{2}\rfloor$ coefficients $a_{1},\ldots ,a_{m}$ prescribed in advance if $q\,{>}\,C(n)$.

A free boundary problem for a parabolic system arising from the mathematical theory of combustion will be considered in the one dimensional case. The existence and uniqueness of the classical solution locally in time will be obtained by the use of a fixed point theorem. Also the existence of the classical solution globally in time and a convergence result with respect to a parameter $\lambda$ will be proved under some reasonable assumptions.

We obtain estimates for the degree of bilipschitz determinacy of quasihomogeneous function-germs.

Let $G$ be a locally soluble-by-finite group in which every non-subnormal subgroup has finite rank. It is proved that either $G$ has finite rank or $G$ is soluble and locally nilpotent (and even a Baer group). On the other hand, a group $G$ is constructed that has infinite rank and satisfies the given hypothesis, but does not have every subgroup subnormal.

Let $A$ (resp. $B$) be a bounded linear operator on a complex Hilbert space $ {\mathcal H}$ (resp. $ {\mathcal K}$). We show that the tensor product $ A \otimes B $ is log-hyponormal if and only if $A$ and $B$ are log-hyponormal, and that a similar result holds for class $A(s,t)$ operators.

We prove that every ergodic amenable action of an algebraic group over a local field of characteristic zero is induced from an ergodic action of an amenable subgroup.

A class of operators on a tensor product of separable Hilbert spaces is considered. It contains various traditional operators. Invertibility, positive invertibility conditions and estimates for the norm of the resolvents are established. In addition, bounds for the spectrum are suggested. Applications to partial integral and integro-differential operators are discussed.

Let $A$ be a uniform algebra on a compact Hausdorff space $X$ and $m$ a probability measure on $X$. Let $H^p(m)$ be the norm closure of $A$ in $L^p(m)$ with $1 \le p < \infty$ and $H^\infty(m)$ the weak $\ast$ closure of $A$ in $L^\infty(m)$. In this paper, we describe a closed ideal of $A$ and exhibit a closed invariant subspace of $H^p(m)$ for $A$ that is of finite codimension.

We investigate the constancy of the Milnor number of one parameter deformations of holomorphic germs of functions $f:(\C^n,0) \to (\C,0)$ with isolated singularity, in terms of some Newton polyhedra associated to such germs.

When the Jacobian ideals $J(\hspace*{1.8pt}f_t) = \left\langle {\partial f_t}/{\partial x_{1}} \ldots ,{\partial f_t}/{\partial x_{n}}\right\rangle $ of a deformation $f_t(x) = f(x)+ \sum_{s=1}^{\ell}\delta_s(t)g_s(x)$ are non-degenerate on some fixed Newton polyhedron $\Gamma_+$, we show that this family has constant Milnor number for small values of $t$, if and only if all germs $g_s$ have non-decreasing $\Gamma$-order with respect to $f$. As a consequence of these results we give a positive answer to Zariski's question for Milnor constant families satisfying a non-degeneracy condition on the Jacobian ideals.

The equi-attraction properties concerning the global attractors $\a_\lam$ of dynamical systems $S_\lam(t)$ with parameter $\lam\in\Lam$, where $\Lam$ is a compact metric space, are investigated. In particular, under appropriate conditions, it is shown that the equi-attraction of the family $\{\a_\lam\}$ is equivalent to the continuity of $\a_\lam$ in $\lam$ with respect to the Hausdorff distance.

Let $(\Omega,\, \Sigma,\,\Prob)$ be a nonatomic probability space and let $\F=(\F_n)_{n{\in}\Z}$ be a filtration. If $f=(\,f_n)_{n{\in}\Z}$ is a uniformly integrable $\F$-martingale, let $\A_{\F}f=(\A_{\F}f_n)_{n{\in}\Z}$ denote the martingale defined by $\A_{\F}f_n =\E[|\,f_{\infty}||\F_n]\; (n \,{\in}\, \Z)$, where $f_{\infty}=\lim_n f_n$ a.s. Let $X$ be a Banach function space over $\Omega$. We give a necessary and sufficient condition for $X$ to have the property that $S(\,\hspace*{.2pt}f\hspace*{.3pt}) \,{\in}\, X$ if and only if $S(\A_{\F}f) \,{\in}\, X$, where $S(\,\hspace*{.2pt}f\hspace*{.3pt})$ stands for the square function of $f=(\,f_n)$.

We prove, under quite general assumptions, the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet boundary conditions. The same results for weakly coupled reaction-diffusion systems are also given.

We consider the isomorphism problem for partial group rings $R_{\hbox{\scriptsize\it par}}G$ and show that, in the modular case, if $\textit{char}(R)\,{=}\,p$ and $R_{\hbox{\scriptsize\it par}}G_1\,{\cong}\, R_{\hbox{\scriptsize\it par}}G_2$ then the corresponding group rings of the Sylow $p$-subgroups are isomorphic. We use this to prove that finite abelian groups having isomorphic modular partial group algebras are isomorphic. Moreover, in the integral case, we show that the isomorphism of partial group rings of finite groups $G_1$ and $G_2$ implies $\Z G_1\,{\cong}\, \Z G_2$.

In this paper we show that the normal parts of quasisimilar log-hyponormal operators are unitarily equivalent. A Fuglede-Putnam type theorem for log-hyponormal operators is proved. Also, it is shown that a log-hyponormal operator that is quasisimilar to an isometry is unitary and that a log-hyponormal spectral operator is normal.

In this paper we study a seemingly unnoticed property of bases in a Hilbert space that falls in the general area of constructing new bases from old, yet is quite atypical of others in this regard. Namely, if $\{x_n\}$ is any normalized basis for a Hilbert space $ H $ and $ \{\,f_n\}$ the associated basis of coefficient functionals, then the sequence $\{x_n+f_n\} $ is again a basis for $ H $. The unusual aspect of this observation is that the basis $ \{x_n+f_n\} $ obtained in this way from $\{x_n\} $ and $ \{\,f_n\} $ need not be equivalent to either, in contrast to the standard techniques of constructing new bases from given ones by means of an isomorphism on $ H $. In this paper we study bases of this form and their relation to the component bases $ \{x_n\} $ and $ \{\,f_n\}$.

A ring $R$ satisfies the dual of the isomorphism theorem if $R/Ra\cong \mathtt{l}(a)$ for every element $a\in R.$ We call these rings left morphic, and say that $R$ is left P-morphic if, in addition, every left ideal is principal. In this paper we characterize the left and right P-morphic rings and show that they form a Morita invariant class. We also characterize the semiperfect left P-morphic rings as the finite direct products of matrix rings over left uniserial rings of finite composition length. J. Clark has an example of a commutative, uniserial ring with exactly one non-principal ideal. We show that Clark's example is (left) morphic and obtain a non-commutative analogue.

An ${\cal L}$-isomorphism between inverse semigroups $S$ and $T$ is an isomorphism between their lattices ${\cal L}(S)$ and ${\cal L}(T)$ of inverse subsemigroups. The author and others have shown that if $S$ is aperiodic – has no nontrivial subgroups – then any such isomorphism $\Phi$ induces a bijection $\phi$ between $S$ and $T$. We first characterize the bijections that arise in this way and go on to prove that under relatively weak ‘archimedean’ hypotheses, if $\phi$ restricts to an isomorphism on the semilattice of idempotents of $S$, then it must be an isomorphism on $S$ itself, thus generating a result of Goberstein. The hypothesis on the restriction to idempotents is satisfied in many applications. We go on to prove theorems similar to the above for the class of completely semisimple inverse semigroups.