Harmonic Shears of Slit and Polygonal Mappings

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations \omega. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.Comment: 17 pages, 6 figures, 2 table

Derivative relationships between volume and surface area of compact regions in R^d

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.Comment: 15 page

DONATIONS FROM ILL-GOTTEN GAIN: A JEWISH LEGAL PERSPECTIVE

© 2008 by Elliot N. Dorff Excerpts of more than 100 words from this publication may not be reproduced without permission. Editorial questions and permission inquiries may be addressed to: Publications Department The Center on Philanthropy at Indiana University 550 W. North St., Suite 301 Indianapolis, IN 46202-3272 Telephone: 317-274-4200 Web site: www.philanthropy.iupui.ed

Vertical Connectivity Influences Secondary Production, Community Diversity, and Resilience in an Ozark Stream

The hyporheic zone, a key component of stream vertical connectivity, supports stream function (e.g., nutrient regeneration) and provides habitat for aquatic biota. The Ozark Highlands ecoregion contains gravel-bed streams that have extensive hyporheic zones and that are subject to recurrent flash floods. I sampled Leuctra tenuis(Pictet) nymphs from the hyporheic habitat (30-45 cm below the streambed) in a gravel-bed reach and an intermittent tributary of an Ozark stream from early instar to adult emergence. Concurrently, I sampled benthic insects in two reaches of the same stream that differed primarily in amount of hyporheic habitat (bedrock vs. gravel) and monitored the hyporheic insect community in the gravel reach for an entire year (October 2017 to September 2018). I calculated secondary production and development of hyporheic L. tenuis. I also monitored the response of benthic and hyporheic insect communities after two floods and documented reductions in richness and densities of benthic insects, especially in the bedrock reach. Pre- and post-flood communities were significantly different between bedrock and gravel benthic habitats. However, differences in community composition were not significant across the entire year, indicating that floods mixed the communities. Insect richness in the hyporheic zone increased post-flood, which may have facilitated quicker recovery in the gravel reach. It is imperative that we understand the role of vertical connectivity to stream secondary production and how the hyporheic zone can contribute to supporting diverse and resilient aquatic communities in order to better protect stream ecosystems