T. D. Morley

Eigenvalues of the Laplacian of a graph

Let G be a finite undirected graph with no loops or multiple edges. We define the Laplacian matrix of G,Δ(G)by Δij= degree of vertex i and Δij−1 if there is an edge between vertex i and vertex j. In this paper we relate the structure of the graph G to the eigenvalues of A(G): in particular we prove that all the eigenvalues of Δ(G) are non-negative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

Capillary morphogenesis protein-2 is the major receptor mediating lethality of anthrax toxin in vivo

Anthrax toxin, a major virulence factor of Bacillus anthracis, gains entry into target cells by binding to either of 2 von Willebrand factor A domain-containing proteins, tumor endothelium marker-8 (TEM8) and capillary morphogenesis protein-2 (CMG2). The wide tissue expression of TEM8 and CMG2 suggest that both receptors could play a role in anthrax pathogenesis. To explore the roles of TEM8 and CMG2 in normal physiology, as well as in anthrax pathogenesis, we generated TEM8- and CMG2-null mice and TEM8/CMG2 double-null mice by deleting TEM8 and CMG2 transmembrane domains. TEM8 and CMG2 were found to be dispensable for mouse development and life, but both are essential in female reproduction in mice. We found that the lethality of anthrax toxin for mice is mostly mediated by CMG2 and that TEM8 plays only a minor role. This is likely because anthrax toxin has approximately 11-fold higher affinity for CMG2 than for TEM8. Finally, the CMG2-null mice are also shown to be highly resistant to B. anthracis spore infection, attesting to the importance of both anthrax toxin and CMG2 in anthrax infections.

Positive solutions to X = A−BX-1 B∗

Abstract We study the positive (semidefinite) solutions to the matrix equation X = A− BX -1 B ∗ under the assumption that A⩾0. We show that a positive solution exists if and only if a certain block tridiagonal operator is positive, in which case the solution is given by the shorted operator (or generalized Schur complement) of that operator.

The challenge of collaborative learning in engineering and math

The CoWeb is a collaborative learning technology used in many classes (over 100) at Georgia Institute of Technology (Georgia Tech), USA. The authors present evidence of the success of the tool in supporting learning at a low cost. They also provide anecdotes about the active resistance they have received to use of the CoWeb in engineering, mathematics and some computer science classes. Evidence from interviews and questionnaires points to some of the sources for this resistance.

Ladder networks, fixpoints, and the geometric mean

Using fixed-point arguments, existence and uniqueness results are obtained for the joint resistance of infinite positive operator networks with noncommuting operators in the branches. Explicit representations for the joint resistance are given using the geometric mean of positive operators.

Using a CSCL-Driven shift in agency to undertake educational reform

Computer supported collaborative learning has the potential for driving educational reform efforts, but there are many challenges to face before achieving that vision. Two of the most critical are achieving teacher buy-in and motivating students to participate in a knowledge-building community. We have developed a new CSCL tool whose flexibility is inspiring good teachers to create exciting new collaborative learning activities, and whose persistence is an enticement to students. We are now attempting to use this tool to leverage a particular kind of educational reform goal: Achieving integration across curricular boundaries.

Symmetric function means of positive operators

Abstract Many authors have studied means involving ratios of the elementary symmetric functions. This work involves generalizing these means to an operator theoretic setting, and extending many of the classical scalar inequalities to operators. It is shown that these operator means are related to electrical networks.

A note on the shorted operator

The Schur complement of a partitioned operator \[ A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \] is defined by the formula

Almost Definite Operators and Electro-mechanical Systems

S ( A ) = A_{11} - A_{12} A_{22}^{ - 1} A_{21}

Computer supported collaborative learning for curriculum integration

. In finite dimensions