Diameter Bounds for Planar Graphs

The inverse degree of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(V-1)-E)/3 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight

Inverse limit spaces satisfying a Poincare inequality

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs (and certain higher dimensional inverse systems of metric measure spaces) which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space, i.e. it satisfies a doubling condition and a Poincare inequality in the sense of Heinonen-Koskela. We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces, and a large class of new examples. Generically our graph examples have the property that they do not bilipschitz embed in any Banach space with Radon-Nikodym property, but they do embed in the Banach space L_1. For Laakso spaces, these facts were discussed in our earlier papers

Recruitment characteristics of nerve fascicles stimulated by a multi-groove electrode

The recruitment characteristics of fascicle-selective nerve stimulation by a multigroove electrode have been investigated both theoretically and in acute experiments. A three-dimensional (3-D) volume conductor model of fascicles in a multigroove device and a model of myelinated nerve fiber stimulation were used to calculate threshold stimuli of nerve fibers in these fascicles. After their exposition, fascicles from rat sciatic nerve were positioned in different grooves of appropriate sizes and stimulated separately. The device appeared to be suitable for fascicle-selective stimulation, because both computer simulations and acute animal experiments showed that crosstalk between neighboring fascicles is not a problem, even when monopolar stimulation was used. The threshold stimulus was lower for a small fascicle than for a large one. When the amount of (conducting) medium between contact and perineurium or its conductivity was reduced, threshold stimuli were lower. Moreover, simulations predict that the slopes of recruitment curves are smaller and inverse recruitment order is less pronounced. Simulations also showed that a small contact is preferable to a large one, because a small contact gives a slightly smaller slope of the recruitment curve. Both experimentally and theoretically a significantly smaller slope of recruitment curves was obtained by stimulation with a cathode and an anode at opposite sides of the fascicle, driven by two current sources giving simultaneous pulses with different, but linearly dependent amplitude