Deforestation of Peano continua and minimal deformation retracts.

Every Peano continuum has a strong deformation retract to a deforested continuum, that is, one with no strongly contractible subsets attached at a single point. In a deforested continuum, each point with a one-dimensional neighborhood is either fixed by every self-homotopy of the space, or has a neighborhood which is a locally finite graph. A minimal deformation retract of a continuum (if it exists) is called its core. Every one-dimensional Peano continuum has a unique core, which can be obtained by deforestation. We give examples of planar Peano continua that contain no core but are deforested.

Arc-reduced forms for Peano continua.

Define a point in a topological space to be homotopically fixed if it is fixed by every self-homotopy of the space, i.e. every self-map of the space which is homotopic to the identity, and define a point to be one-dimensional if it has a neighborhood whose covering dimension is one. In this paper, we show that every Peano continuum is homotopy equivalent to a reduced form in which the one-dimensional points which are not homotopically fixed form a disjoint union of open arcs. In the case of one-dimensional Peano continua, this presents the space as a compactification of a null sequence of open arcs by the homotopically fixed subspace.

Real-time recoverable and irrecoverable energy in dispersive-dissipative dielectrics.

We discuss the recoverable and irrecoverable energy densities associated with a pulse at a point in the propagation medium and derive easily computed expressions to calculate these quantities. Specific types of fields are required to retrieve the recoverable portion of the energy density from the point in the medium, and we discuss the properties that these fields must have. Several examples are given to illustrate these concepts.