Translation invariance and instability of phase coexistence in the two dimensional Ising system

It is shown that any Gibbs state of the two dimensional ferromagnetic Ising system is of the form ₊+(1–)_–, with some [0, 1]. This excludes the possibility of a locally stable phase coexistence and of translation symmetry breaking, which are known to occur in higher dimensions. Use is made in the proof of the stochastic aspects of the geometry of the interface lines.Supported in part by the U.S. National Science Foundation under the grant PHY — 7825390     

Consistent Initialization for Nonlinear Index-2 Differential–Algebraic Equation: Large Sparse Systems in MATLAB

An important component of any initial-value solver for higher-index differential–algebraic equations consists in the computation of consistent initial values. In a recent paper [5], an algorithm is proposed which is applicable to a very general class of index-2 systems. Unfortunately, the computational expense is rather high. We present a modification of this approach, which gives rise to a MATLAB implementation capable of handling systems of moderate dimension (several thousands of unknowns). The algorithm is illustrated by examples.     

Independence Sequences of Well-Covered Graphs: Non-Unimodality and the Roller-Coaster Conjecture

A graph G is well-covered provided each maximal independent set of vertices has the same cardinality. The term s_k of the independence sequence (s₀,s₁,,s) equals the number of independent k-sets of vertices of G. We investigate constraints on the linear orderings of the terms of the independence sequence of well-covered graphs. In particular, we provide a counterexample to the recent unimodality conjecture of Brown, Dilcher, and Nowakowski. We formulate the Roller-Coaster Conjecture to describe the possible linear orderings of terms of the independence sequence. We are grateful to Jason Brown for a helpful discussion in December of 1999 and to Richard Stanley for valuable feedback in August of 2000. The work of the second author was partially supported by the Naval Academy Research Council and ONR grant N0001400WR20041     

Iwasawa μ-invariants of elliptic curves and their symmetric powers

In an earlier paper we considered the effects that finite submodules can have on μ-invariants of Selmer groups. In this paper we examine some of the consequences of that theory to elliptic curves and their symmetric powers. One main result is the construction of an isogeny class of elliptic curves, all of which have positive μ-invariants. A second result is a connection between the behaviors of μ-invariants associated with the symmetric powers of an elliptic curve over , and the behavior of the μ-invariant of that elliptic curve over different extensions of .     

Cores and retracts

Theretracts (idempotent, isotone self-maps) of an ordered set are naturally ordered as functions. In this note we characterize the possible ways that one retract can cover another one. This gives some insight into the structure of the ordered set of retracts and leads to a natural generalization of the core of an ordered set.Supported by NSERC Operating Grant 41702.