Group-theoretic algorithms for matrix multiplication

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.Comment: 10 page

Maximizing Expected Value of Information in Decision Problems by Querying on a Wish-to-Know Basis.

An agent acting under uncertainty regarding how it should complete the task assigned to it by its human user can learn more about how it should behave by posing queries to its human user. Asking too many queries, however, may risk requiring undue attentional demand of the user, and so the agent should prioritize asking the most valuable queries. For decision-making agents, Expected Value of Information (EVOI) measures the value of a query, and so given a set of queries the agent can ask, the agent should ask the query that is expected to maximally improve its performance by selecting the query with highest EVOI in its set. Unfortunately, to compute the EVOI of a query, the agent must consider how each possible response would influence its future behavior, which makes query selection particularly challenging in settings where planning the agent's behavior would be expensive even without the added complication of considering queries to ask, especially when there are many potential queries the agent should consider. The focus of this dissertation is on developing query selection algorithms that can be feasibly applied to such settings. The main novel approach studied, Wishful Query Projection (WQP), is based on the intuition that the agent should consider which query to ask on the basis of obtaining knowledge that would help it resolve a particular dilemma that it wishes could be resolved, as opposed to blindly searching its entire query set in hopes of finding one that would give it valuable knowledge. In implementing WQP, this dissertation contributes algorithms that are founded upon the following novel result: for myopic settings, when the agent can ask any query as long as the query has no more than some set number of possible responses, the best query takes the form of asking the user to choose from a specified subset of ways for the agent to behave. The work presented shows that WQP selects queries with near-optimal EVOI when the agent's query set is (1) balanced in the range of queries it contains; and (2) rich in terms of the highest contained query EVOI.PhDComputer Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120772/1/rwcohn_1.pd

Hypercholesterolemia is a risk factor for bioprosthetic valve calcification and explantation

AbstractObjectiveThere are pathophysiologic similarities between calcification and atherosclerosis. We wished to determine whether risk factors for atherosclerosis were linked to bioprosthetic valve calcification and dysfunction.MethodsWe performed a retrospective cohort study on 144 patients at a single institution who had bioprosthetic aortic or mitral valves removed, serum cholesterol levels recorded, and valve calcification assessed on the basis of hematoxylin and eosin staining and radiography of the valve. We also performed case-control analysis of a group of 66 patients whose tissue valves were explanted and compared them with an age- and position-matched group of 66 patients with similar duration of implantation. We also compared mean serum cholesterol levels.ResultsIn the retrospective cohort study cholesterol (P = .035), younger age at implantation (P = .014), and coronary artery disease (P = .017) were linked to calcification of the valve by means of univariate analysis. In stepwise multiple regression analysis only the mean serum cholesterol level was linked to calcification (P = .02). Sex, hypertension, smoking, diabetes, and implant position were not linked to calcification. In the case-control analysis the mean serum cholesterol level of the explanted valve group was significantly higher (189 vs 163 mg/dL, P < .0001) than that of the group whose valves did not require explantation. For those whose serum cholesterol levels were greater than 200 mg/dL, the odds ratio was 3.9 (95% confidence interval, 1.7-8.9) for valve explantation.ConclusionsIncreased serum cholesterol level may be a risk factor for bioprosthetic valve calcification requiring explantation

Algebraic Solutions of the Lam\'e Equation, Revisited

A minor error in the necessary conditions for the algebraic form of the Lam\'e equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lam\'e's differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lam\'e equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lam\'e equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.Comment: 20 pages, elsart document class, no figure

Treadle Driven Lathe

The main objective of this senior design project was to design and build a treadle-driven, metal-cutting lathe that has the ability to cut 3/4 diameter material and produce small parts up to 3 long. The main challenge in this project was to successfully generate enough power through the treadle driven system to successfully cut metal at the proper specifications, while simultaneously meeting all of the user-needs. The entire system needed to be operated and pedaled by a single person, which put constraints on the location of the treadle relative to the lathe itself so the user could comfortably operate both at the same time. Furthermore, in order to generate adequate power and RPM to cut metal, implementing an energy-storage element to the treadle system was crucial. A heavy flywheel (salvaged from an old exercise bike) was the main source of energy storage in the system. The flywheel, coupled with a drive-wheel and a proper gear ratio proved to generate enough power to drive the system. To further aid the user, especially when initially pedaling the treadle to get started, springs were added at the pivot point of the pedal to help the treadle make full revolutions while getting up to speed. Wood and metal supports were placed in critical locations in order to reduce vibrations and increase the structural rigidity of the system in order to withstand the continual input forces from the user. Once the treadle-driven lathe was in working condition, the last step was to ensure that the lathe could operate at multiple speeds, which was achieved. We were successfully able to meet all of our key design metrics and user needs