Groups Acting on Tensor Products

Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and present effective algorithms to construct generators for these groups. We also discuss applications of our methods to algorithmic problems for which all currently known methods require an exponential amount of work. (C) 2013 Elsevier B.V. All rights reserved.

A VEGETATION RECONSTRUCTION OF UNION COUNTY, PENNSYLVANIA DURING THE EARLY HISTORICAL PERIOD (1755-1855)

This thesis assesses relationships between vegetation and topography and the impact of human tree-cutting on the vegetation of Union County during the early historical era (1755-1855). I use early warrant maps and forestry maps from the Pennsylvania historical archives and a warrantee map from the Union County courthouse depicting the distribution of witness trees and non-tree surveyed markers (posts and stones) in early European settlement land surveys to reconstruct the vegetation and compare vegetation by broad scale (mountains and valleys) and local scale (topographic classes with mountains and valleys) topography. I calculated marker density based on 2 km x 2 km grid cells to assess tree-cutting impacts. Valleys were mostly forests dominated by white oak (Quercus alba) with abundant hickory (Carya spp.), pine (Pinus spp.), and black oak (Quercus velutina), while pine dominated what were mostly pine-oak forests in the mountains. Within the valleys, pine was strongly associated with hilltops, eastern hemlock (Tsuga canadensis) was abundant on north slopes, hickory was associated with south slopes, and riparian zones had high frequencies of ash (Fraxinus spp.) and hickory. In the mountains, white oak was infrequent on south slopes, chestnut (Castanea dentata) was more abundant on south slopes and ridgetops than north slopes and mountain coves, and white oak and maple (Acer spp.) were common in riparian zones. Marker density analysis suggests that trees were still common over most of the landscape by 1855. The findings suggest there were large differences in vegetation between valleys and mountains due in part to differences in elevation, and vegetation differed more by topographic classes in the valleys than in the mountains. Possible areas of tree-cutting were evenly distributed by topographic classes, suggesting Europeans settlers were clearing land and harvesting timber in most areas of Union County.