Speaker: Henning Ulfarsson (University of Reykjavik)
Title: Pattern avoiding permutations and non-crossing subgraphs of polygons
Abstract: We will explain a surprising connection between independent sets arising from the study of permutations avoiding the pattern 132 and the classical problem of enumerating non-crossing subgraphs in a complete graph drawn on a regular polygon. This leads to a generalization of n-gons, where some internal edges have been doubled. These can be used to count subclasses of permutations avoiding the pattern 1324, the only principal permutation class of length 4 that remains unenumerated. No prior knowledge on permutation patterns is assumed. Joint work with Christian Bean and Murray Tannock (also at Reykjavik University)
Speaker: David Ross (UH)
Title: A field-extension proof that R is uncountable
Abstract: For some (but not all!) non-Archimedean ordered field extensions
F of the rationals, you can construct the reals from F by identifying
elements which differ by an infinitesimal. I’ll describe necessary and
sufficient conditions on F for the construction to actually produce the
reals, and then give a new proof for uncountability of R that uses this
condition in an essential way.
Title: Asteroid Rendezvous Missions using Indirect Methods of Optimal Control
Abstract:
The main objective of this dissertation is to assess the feasibility of space missions to a new population of near Earth asteroids which temporarily orbit Earth, called minimoons, by designing optimal missions to a large sample of simulated minimoons. The Pontryagin maximum principle and indirect methods of optimal control theory are used, and continuation-based techniques are developed to address the well-known difficulty of initializing such algorithms. Time-minimal and time-constrained fuel-minimal rendezvous missions are computed to a catalog of over sixteen-thousand simulated minimoons. Analysis of the results provides insight into the characterization of minimoons which are most suitable for rendezvous, as well as the characterization of locations along a given minimoon trajectory which are most suitable for rendezvous.
As a first approach, the time-minimization rendezvous problem is investigated. The Circular Restricted Three-Body Problem is used to model the gravitational effects of the Earth and Moon on the spacecraft. Continuation-based techniques which rely on the knowledge of an existing solution are used to initialize the algorithms. For a spacecraft with 1 Newton maximum thrust, our methods successfully compute time-minimal transfers to over 96% of the 16,923 simulated minimoons, with transfer times on the order of one month. For a sample of 250 minimoons, continuation techniques further reduce the maximum thrust as low as 0.1 Newtons with transfer times less than four months.
The time-minimal results give some understanding of a lower bound for the transfer times, but have high fuel requirements. To improve the results and investigate fuel constraints, the fuel-minimization problem is investigated. The spacecraft is assumed to start on a Halo orbit around the Earth-Moon L2 Lagrangian point. The Circular Restricted Four-Body Problem is used to model the gravitational effects of the Earth, Moon, and Sun, and the mass variation of the spacecraft is modeled. The structure of the control is fixed to three boosts, and the transfer times are constrained to be less than six months. Again indirect methods are employed to identify fuel-minimal transfers, and a continuation-based “cloud” technique is developed to overcome the initialization difficulty. For a spacecraft with 22 Newton maximum thrust and 230 second specific impulse, our methods produce rendezvous missions with delta-v values under 500 meters per second for over 50% of the simulated asteroids, and for some transfers delta-v values less than 100 meters per second.
Most importantly, the work presented in this dissertation strongly suggests that minimoons are accessible via spacecraft for low-cost and should continue to be investigated.
Title:
Mathematical Modeling of the Evolution and Development of Myelin
Abstract:
Nerve impulses exhibit an increased conduction velocity in axons that are wrapped with myelin. Although unmyelinated and myelinated axons are well studied, the process of myelination that allow the formation of myelin on previously unmyelinated axons is not well understood. In this paper we develop a mathematical model of myelination. By varying a parameter that represents the tightness of myelin, this model was made to describe the unmyelinated axon, the fully myelinated axon, and the transitional states in between. As myelin tightens, a slowdown in conduction velocity occurs in the transitional states before a speedup. Varying other geometric parameters such as the thickness and length of myelin shows that some sequences of geometric changes are more optimal than others.