Rank decomposition in zero pattern matrix algebras

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H.Bart, A.P.M.Wagelmans (2000). The proof involves elements from integer programming and employs Farkas’ lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred to above can be extended to other zero pattern matrix algebras. It is shown that such a generalization does indeed hold for certain digraphs determining the pattern of zeros. The digraphs in question can be characterized in terms of forests, i.e., disjoint unions of rooted trees.     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

Hilbert Functions, Residual Intersections, and Residually S2 Ideals

Let R be a homogeneous ring over an infinite field, IR a homogeneous ideal, and I an ideal generated by s forms of degrees d ₁,...,d _s so that codim( :I)s. We give broad conditions for when the Hilbert function of R/ or of R/( :I) is determined by I and the degrees d ₁,...,d _s . These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S ₂ ideals. We prove that the residually S ₂ property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.     

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.     

An economic model of short-rotation forestry

This paper presents an economic model of short-rotation forestry and shows that Lemke's linear complementarity algorithm can be used to compute optimal sustainable harvesting strategies. As an example, we apply our method, using existing data, to calculate optimal harvesting strategies for a plantation consisting of two interdependent species of trees.     

Carbon-carbon bond formation between alkylated alkenes and acrylic ester via 2-methoxyalkyl radicals

Methoxymercuration/demercuration reactions of alkenes 10 in $\text{10}$ in the presence of acrylic ester yield products $\text{11}$ in a carbon-carbon bond formation reaction.