We prove that any {0,1 }-preserving homomorphism of finite distributive lattices can be realized as the restriction of the congruence relations of a finite planar lattice with no nontrivial automorphisms to an ideal of that lattice, where this ideal also has no nontrivial automorphisms. We also prove that any {0,1 }-preserving homomorphism of finite distributive lattices with more than one element and any homomorphism of groups can be realized, simultaneously, as the restriction of the congruence relations and, respectively, the restriction of the automorphisms of a lattice L to those of an ideal of L; if the groups are both finite, then so is the lattice L.

We consider the location of zeros of four related classes of polynomials, one of which is the class of chromatic polynomials of graphs. All of these polynomials are generating functions of combinatorial interest. Extensive calculations indicate that these polynomials often have only real zeros, and we give a variety of theoretical results which begin to explain this phenomenon. In the course of the investigation we prove a number of interesting combinatorial identities and also give some new sufficient conditions for a polynomial to have only real zeros.

We study path properties of kernel generated two-time parameter, not necessarily stationary, Gaussian processes. We establish large deviation results for some increments of these processes and use these results to prove some of their moduli of continuity and other path properties.

The usual super-Brownian motion is a measure-valued process that arises as a high density limit of a system of branching Brownian particles in which the branching mechanism is critical. In this work we consider analogous processes that model the evolution of a system of two such populations in which there is inter-species competition or predation.

We first consider a competition model in which inter-species collisions may result in casualties on both sides. Using a Girsanov approach, we obtain existence and uniqueness of the appropriate martingale problem in one dimension. In two and three dimensions we establish existence only. However, we do show that, in three dimensions, any solution will not be absolutely continuous with respect to the law of two independent super-Brownian motions. Although the supports of two independent super-Brownian motions collide in dimensions four and five, we show that there is no solution to the martingale problem in these cases.

We next study a prédation model in which collisions only affect the "prey" species. Here we can show both existence and uniqueness in one, two and three dimensions. Again, there is no solution in four and five dimensions. As a tool for proving uniqueness, we obtain a representation of martingales for a super-process as stochastic integrals with respect to the related orthogonal martingale measure.

We also obtain existence and uniqueness for a related single population model in one dimension in which particles are killed at a rate proportional to the local density. This model appears as a limit of a rescaled contact process as the range of interaction goes to infinity.

This is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

The Lp-norm convergence of weighted averages μnf is ergodic theory is equivalent to the pointwise convergence of the Fourier transforms . If h(γ) = , then the behavior of h determines when the Lp-norm limit of μnf is ∫f dm. The nature of such limit functions h is the focus of this article.

We characterize Banach spaces that admit Markuševič bases with various properties connected with weak countable determining or weak Lindelöf determining of spaces or with various norming properties.

In this paper the Corwin's conjecture is proved, which says that if d is a function analytic near ∞, then the hypoellipticity of the convolution operator Ad, defined by for every u ∊ S'(ℝn), implies that P(x)/ logx → ∞ as x → ∞, where P(x) is the distance from x ∊ ℝn to the set of complex zeros of d.