Null spaces of correlation matrices

AbstractLet R be the real numbers and Rn the vector space of all column vectors of length n. Let Cn be the convex set of all real correlation matrices of size n. If V is a subspace of Rn of dimension k, we consider the face FV of Cn consisting of all A∈Cn such that V⊂N(A), i.e., AV=0. If FV is nonempty, we say that V is realizable. We give complete geometric descriptions of FV in the cases k=1, n=4, and k=2, n=5. For k=2, n=5, we provide a simple algebraic method for describing FV

Multicolored Asian lady beetle (2000)

Throughout much of the country, the multicolored Asian lady beetle (Harmonia axyridis) has become an unwanted houseguest, especially during the spring and fall months. Adults are about one-quarter inch long, oval in shape and about two-thirds as wide as they are long. The color of the wing covers (the domelike shell over most of the body) can range from beige to yellow to yellowish orange to bright reddish orange with anywhere from 0 to 19 black spots (see Figure 1). The area just behind the head and in front of the wing covers is white with a black, M-shaped mark.New 5/00/7M

Determinantal formulae for matrices with sparse inverses, II: asymmetric zero patterns

AbstractIn an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A−1. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A−1. However, just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries, the symmetry of the zero pattern is not essential for these formulae. We describe here how analogous formulae for det A may be obtained in the asymmetric-zero-pattern case by introducing a directed spanning tree. We also examine the converse question of determining all possible zero patterns of A−1 which guarantee that a certain determinantal formula holds

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of distinct eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a