ISU REU: diverse, research-intense, team-based

This article describes the Iowa State University (ISU) mathematics REU. The emphasis is on how certain choices made have shaped the ISU REU. The ISU REU draws a diverse group of students from a broad spectrum of colleges and universities nationwide. It is research-intense, with no course or workshop component, and results are disseminated through publications and presentations at conferences. Students in the REU work in teams with graduate students and faculty

The copositive completion problem: Unspecified diagonal entries

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207–211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix

Fractional Zero Forcing via Three-color Forcing Games

An $r$-fold analogue of the positive semidefinite zero forcing process that is carried out on the $r$-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional positive semidefinite forcing number of a graph. We develop a fractional parameter based on the standard zero forcing process and it is shown that this parameter is exactly the skew zero forcing number with a three-color approach. This approach and an algorithm are used to characterize graphs whose skew zero forcing number equals zero.Comment: 24 page

Matrix completion problems for pairs of related classes of matrices

For a class X of real matrices, a list of positions in an n×n matrix (a pattern) is said to have X-completion if every partial X-matrix that specifies exactly these positions can be completed to an X-matrix. If X and X0 are classes that satisfy the conditions any partial X-matrix is a partial X0-matrix, for any X0-matrix A and ε\u3e0, A+εI is a X-matrix, and for any partial X-matrix A, there exists δ\u3e0 such that A−δĨ is a partial X-matrix (where Ĩ is the partial identity matrix specifying the same pattern as A) then any pattern that has X0-completion must also have X-completion. However, there are usually patterns that have X-completion that fail to have X0-completion. This result applies to many pairs of subclasses of P- and P0-matrices defined by the same restriction on entries, including the classes P/P0-matrices, (weakly) sign-symmetric P/P0-matrices, and non-negative P/P0-matrices. It also applies to other related pairs of subclasses of P0-matrices, such as the pairs classes of P/P0,1-matrices, (weakly) sign-symmetric P/P0,1-matrices and non-negative P/P0,1-matrices. Furthermore, any pattern that has (weakly sign-symmetric, sign-symmetric, non-negative) P0-completion must also have (weakly sign-symmetric, sign-symmetric, non-negative) P0,1-completion, although these pairs of classes do not satisfy condition (3). Similarly, the class of inverse M-matrices and its topological closure do not satisfy condition (3), but the conclusion remains true, and the matrix completion problem for the topological closure of the class of inverse M-matrices is solved for patterns containing the diagonal

Zero forcing and maximum nullity for hypergraphs

The concept of zero forcing is extended from graphs to uniform hypergraphs in analogy with the way zero forcing was defined as an upper bound for the maximum nullity of the family of symmetric matrices whose nonzero pattern of entries is described by a given graph: A family of symmetric hypermatrices is associated with a uniform hypergraph and zeros are forced in a null vector. The value of the hypergraph zero forcing number and maximum nullity are determined for various families of uniform hypergraphs and the effects of several graph operations on the hypergraph zero forcing number are determined. The hypergraph zero forcing number is compared to the infection number of a hypergraph and the iteration process in hypergraph power domination

Graph theoretic methods for matrix completion problems

A pattern is a list of positions in an n×n real matrix. A matrix completion problem for the class of Π-matrices asks whether every partial Π-matrix whose specified entries are exactly the positions of the pattern can be completed to a Π-matrix. We survey the current state of research on Π-matrix completion problems for many subclasses Π of P0-matrices, including positive definite matrices, M-matrices, inverse M-matrices, P-matrices, and matrices defined by various sign symmetry and positivity conditions on P0- and P-matrices. Graph theoretic techniques used to study completion problems are discussed. Several new results are also presented, including the solution to the M0-matrix completion problem and the sign symmetric P0-matrix completion problem

Relationships between the Completion Problems for Various Classes of Matrices

This is a proceeding from the Eighth SIAM Conference on Applied Linear Algebra (2003). Posted with permission.</p

Applications of analysis to the determination of the minimum number of distinct eigenvalues of a graph

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues of several families of graphs and small graphs

A linear algebraic view of partition regular matrices

Rado showed that a rational matrix is partition regular over N if and only if it satisfies the columns condition. We investigate linear algebraic properties of the columns condition, especially for oriented (vertex-arc) incidence matrices of directed graphs and for sign pattern matrices. It is established that the oriented incidence matrix of a directed graph Γ has the columns condition if and only if Γ is strongly connected, and in this case an algorithm is presented to find a partition of the columns of the oriented incidence matrix with the maximum number of cells. It is shown that a sign pattern matrix allows the columns condition if and only if each row is either all zeros or the row has both a + and −

Your NSF Mathematical Sciences Institutes

This is an article from IMAGE 39 (2007): 17. Posted with permission.</p