A comparison theorem for the isoperimetric profile under curve shortening flow

We prove a comparison theorem for the isoperimetric profiles of simple closed curves evolving by the normalized curve shortening flow: If the isoperimetric profile of the region enclosed by the initial curve is greater than that of some `model' convex region with exactly four vertices and with reflection symmetry in both axes, then the inequality remains true for the isoperimetric profiles of the evolved regions. We apply this using the Angenent solution as the model region to deduce sharp time-dependent upper bounds on curvature for arbitrary embedded closed curves evolving by the normalized curve shortening flow. A slightly different comparison also gives lower bounds on curvature, and the result is a simple and direct proof of Grayson's theorem without use of any blowup or compactness arguments, Harnack estimates, or classification of self-similar solutions.Comment: 32 pages, 6 figure

Eschatology of the epistle to the Hebrews

The approach adopted for this study was to select certain temporal references in the epistle and to examine them in the light of the events or periods of time to which they refer. The purpose of the approach was to gain an understanding, in so far as it i3 possible, of the concept of time which underlies the writing of Hebrews. Though some of the references seem casual or unclear, a rather distinct picture of the author’s chronological framework emerges when the texts are examined more closely and related to one another

Paul\u27s Source and Use of Tradition as Reflected in his Writings to the Corinthians

The present study seeks to explore the question whether an adapted form of the Jewish concept of tradition constituted a significant part of Paul\u27s work as a Christian apostle in the congregation at Corinth

Mechanisms Of Decreased Brainstem Arousal In Limbic Seizures: In Vivo Whole-Cell Recordings

Seizures are commonly regarded as disorders of neuronal hyperactivity, but evidence shows that temporal lobe seizures also cause reduced activity in subcortical arousal nuclei, including cholinergic neurons in the pedunculopontine tegmental nucleus (PPT), a key node of the reticular activating system. The synaptic mechanisms underlying the reduced ictal activity of cholinergic neurons in the PPT are unknown. Whole-cell in vivo recordings were made from PPT neurons in head-fixed rats to distinguish active inhibition from withdrawal of excitatory input as the driver of reduced ictal neuronal firing. A subset of PPT neurons exhibited reduced firing and hyperpolarization during seizures and stained positive for choline acetyltransferase. These PPT cholinergic neurons also showed increased input resistance, reduced membrane potential variance and fewer EPSP-like events during seizures. These data weigh against active ictal inhibition and support withdrawal of excitatory input as the dominant mechanism of decreased activity of cholinergic neurons in the PPT