Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.

On Modular Design of Field Robotic Systems

Robots are needed to perform important field tasks such as hazardous material clean-up, nuclear site inspection, and space exploration. Unfortunately their use is not widespread due to their long development times and high costs. To make them practical, a modular design approach is proposed. Prefabricated modules are rapidly assembled to give a low-cost system for a specific task. This paper described the modular design problem for field robots and the application of a hierarchical selection process to solve this problem. Theoretical analysis and an example case study are presented. The theoretical analysis of the modular design problem revealed the large size of the search space. It showed the advantages of approaching the design on various levels. The hierarchical selection process applies physical rules to reduce the search space to a computationally feasible size and a genetic algorithm performs the final search in a greatly reduced space. This process is based on the observation that simple physically based rules can eliminate large sections of the design space to greatly simplify the search. The design process is applied to a duct inspection task. Five candidate robots were developed. Two of these robots are evaluated using detailed physical simulation. It is shown that the more obvious solution is not able to complete the task, while the non-obvious asymmetric design develop by the process is successful.

Omtvedt Innovation Awards Presented to Chris Calkins, Steven Jones, Rodger Johnson, and Sheila Scheideler

It's a great pleasure to welcome you to this very first recognition ceremony for the Omtvedt Innovation Awards. We are present here to honor innovation strengths of the Institute of Agriculture and Natural Resources, and certainly the four faculty members receiving today's awards are greatly deserving of this recognition. Just hearing about their work is gratifying!

Nebraska Builder Award Presented to Dr. Mark Gustafson

Few Nebraskans are as devoted to the University of Nebraska as Mark Gustafson. Driven by his belief that a strong university is key to a strong Nebraska economy, Mark is an advocate for the university in the local, state, and national arenas. He is a Nebraska delegate to the Council for Agricultural, Research, Extension, and Teaching, a national advocacy organization for higher education. Since 1991, he's been a member of Agriculture Builders of Nebraska, Inc., which supports UNL's Institute of Agriculture and Natural Resources, as well as the entire University, and has served three terms as president. He has served on the advisory councils for the UNL chancellor and the NU president and served on UNL's Future Nebraska Taskforce. He holds baccalaureate and master's degrees from UNL and a Ph.D. from the University of California-Berkeley. When he's not volunteering his time, Mark operates the family farm near Mead. He and his wife, Dianne, are the parents of two children - Christopher, a UNL alumnus, and Anneke, a UNL junior.

Omtvedt Innovation Award Presented to Allen G. Blezek

It is such a pleasure to honor innovation and accomplishment in the Institute of Agriculture and Natural Resources today through this 2007 Omtvedt Innovation Award. This award is made possible because of the generosity of Leone and the late Neal Harlan, great friends of the Institute of Agriculture and Natural Resources. The Harlans had the vision and the foresight to realize the importance of recognizing and supporting outstanding and innovative work in the Institute. They honored Irv Omtvedt on his retirement as Vice Chancellor of the Institute with a generous gift of funds to support the Omtvedt Innovation Awards. These awards recognize areas of strength and promise within the Institute, as well as innovative research and programming by our faculty, staff, and students.

Omtvedt Innovation Award Presented to Marilyn Fox

It is such a pleasure to honor innovation and strength in the Institute of Agriculture and Natural Resources today through this 2006 Omtvedt Innovation Award. This award is made possible because of the generosity of Leone and the late Neal Harlan, great friends of the Institute of Agriculture and Natural Resources. The Harlans had the vision and the foresight to realize the importance of recognizing and supporting outstanding and innovative work in the Institute, and honored Irv Omtvedt on his retirement as Vice Chancellor of the Institute with a generous gift of funds to support the Omtvedt Innovation Awards. These awards recognize areas of strength and promise within the Institute, as well as innovative research and programming by our faculty, staff, and students.

Class Notes for Math 918: Cohen Macaulay Modules, Instructor Roger Wiegand

Topics covered are: Cohen Macaulay modules, zero-dimensional rings, one-dimensional rings, hypersurfaces of finite Cohen-Macaulay type, complete and henselian rings, Krull-Remak-Schmidt, Canonical modules and duality, AR sequences and quivers, two-dimensional rings, ascent and descent of finite Cohen Macaulay type, bounded Cohen Macaulay type.