The Tutte polynomial $T(G;x,y)$ of a graph evaluates to many interesting combinatorial quantities at various points in the $(x,y)$ plane, including the number of spanning trees, number of forests, number of acyclic orientations, the reliability polynomial, the partition function of the Q-state Potts model of a graph, and the Jones polynomial of an alternating link. The exact computation of $T(G;x,y)$ has been shown by Vertigan and Welsh [8] to be #P-hard at all but a few special points and on two hyperbolae, even in the restricted class of planar bipartite graphs. Attention has therefore been focused on approximation schemes. To date, positive results have been restricted to the upper half plane $y>1$, and most results have relied on a condition of sufficient denseness in the graph. In this paper we present an approach that yields a fully polynomial randomized approximation scheme for $T(G;x,y)$ for $x>1,\ y=1$, and for $T(G;2,0)$, in a class of sparse graphs. This is the first positive result that includes the important point $(2,0)$.

As usual, let us write $G_{n,p}$ for a random graph with vertex set $[n]=\{1, 2, \dots , n\}$, in which the edges are chosen independently, with probability $p$. Similarly, $G_{n,m}$ is a random graph on $[n]$ with $m$ edges. For $p=m/({{n}\atop{2}})$, in many respects, the random graphs $G_{n,m}$ and $G_{n,p}$ are practically indistinguishable. (See [4] for an introduction to random graphs.) When in the late 1950s and early 1960s Erdős and Rényi founded the theory of random graphs, one of the most important problems they left open was the determination of the chromatic number of a random graph $G_{n,m}$. The original question concerned the case $m=O(n)$, but when in the 1970s Erdős popularized the problem, he was asking for good estimates on the chromatic number of $G_{n,m}$ for $m \sim n^2/4$ (equivalently, the chromatic number of $G_{{n,1/2}}$).

Let ${\bf C}$ denote the field of complex numbers and $\Omega_n$ the set of $n$th roots of unity. For $t = 0,\ldots,n-1$, define the ideal $\Im(n,t+1) \subset {\bf C}[x_0,\ldots,x_{t}]$ consisting of those polynomials in $t+1$ variables that vanish on distinct $n$th roots of unity; that is, $f \in \Im(n,t+1)$ if and only if $f(\omega_0,\ldots,\omega_{t}) = 0$ for all $(\omega_0,\ldots,\omega_{t}) \in \Omega_n^{t+1}$ satisfying $\omega_i \neq \omega_j$, for $0 \le i < j \le t$.

In this paper we apply Gröbner basis methods to give a Combinatorial Nullstellensatz characterization of the ideal $\Im(n,t+1)$. In particular, if $f \in {\bf C}[x_0,\ldots,x_{t}]$, then we give a necessary and sufficient condition on the coefficients of $f$ for membership in $\Im(n,t+1)$.

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

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.

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.

Let $A$ be a commutative noetherian local ring, $I$ an ideal of $A$, and $B\,{=}\,A/I$. Assume that the André-Quillen homology functors $H_n (A,B,-) = 0$ for all $n \,{\ge}\, 3$. Then $A$ is Gorenstein if and only if $B$ is.

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 $q\,{>}\,C(n)$.

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.

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 $(X,\mathcal{F},\mu)$ 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.

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 $f=(\,f_n)$.

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.

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.

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 $\Z G_1\,{\cong}\, \Z G_2$.

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.

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.