Measure Preserving Diffeomorphisms of the Torus are Unclassifiable

The isomorphism problem in ergodic theory was formulated by von Neumann in 1932 in his pioneering paper Zur Operatorenmethode in der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). The problem has been solved for some classes of transformations that have special properties, such as the collection of transformations with discrete spectrum or Bernoulli shifts. This paper shows that a general classification is impossible (even in concrete settings) by showing that the collection $E$ of pairs of ergodic, Lebesgue measure preserving diffeomorphisms $(S,T)$ of the 2-torus that are isomorphic is a complete analytic set in the $C^\infty$- topology (and hence not Borel).Comment: 97 pages, 4 figure

Level repulsion in hybrid photonic-plasmonic microresonators for enhanced biodetection

We theoretically analyse photonic-plasmonic coupling between a high Q whispering gallery mode (WGM) resonator and a core-shell nanoparticle. Blue and red shifts of WGM resonances are shown to arise from crossing of the photonic and plasmonic modes. Level repulsion in the hybrid system is further seen to enable sensitivity enhancements in WGM sensors: maximal when the two resonators are detuned by half the plasmon linewidth. Approximate bounds are given to quantify possible enhancements. Criteria for reactive vs. resistive coupling are also established

VRShape: A Virtual Reality Tool for Shaping Movement Compensation

The majority of persons living with chronic stroke experience some form of upper extremity motor impairment that affects their functional movement, performance of meaningful activities, and participation in the flow of daily life. Stroke survivors often compensate for these impairments by adapting their movement patterns to incorporate additional degrees of freedom at new joints and body segments. One of the most common compensatory movements is the recruitment of excessive trunk flexion when reaching with the affected upper extremity. Long-term use of these compensations may lead to suboptimal motor recovery and chronic pain or injury due to overuse. Rehabilitation focuses on repetitive practice with the impaired limb to stimulate motor learning and neuroplasticity; however, few interventions achieve the required repetition dose or address the possible negative effects of compensatory movements. Virtual reality (VR) is an emerging tool in rehabilitation science that may be capable of (1) objectively measuring compensation during upper extremity movement, (2) motivating persons to perform large doses of repetitive practice through the integration of virtual environments and computer games, and (3) providing the basis for a motor intervention aimed at improving motor performance and incrementally reducing, or shaping, compensation. The purpose of this project was to develop and test a VR tool with these capabilities for shaping movement compensation for persons with chronic stroke, and to achieve this we performed three separate investigations (Chapters 2-4).First, we investigated the validity and reliability of two generations of an off-the-shelf motion sensor, namely the Microsoft Kinect, for measuring trunk compensations during reaching (Chapter 2). A small group of healthy participants performed various reaching movements on two separate days while simultaneously being recorded by the two sensors and a third considered to be the gold standard. We found that the second generation Kinect sensor was more accurate and showed greater validity for measuring trunk flexion relative to the gold standard, especially during extended movements, and therefore recommended that sensor for future VR development. Research with a more heterogeneous and representative population, such as persons with stroke, will further improve the evaluation of these sensors in future work.Second, we tested a newly-designed VR tool, VRShape, for use during a single session of upper extremity movement practice (Chapter 3). VRShape integrates the Microsoft Kinect and custom software to convert upper extremity movements into the control of various virtual environments and computer games while providing real-time feedback about compensation. A small group of participants with stroke used VRShape to repetitively perform reaching movements while simultaneously receiving feedback concerning their trunk flexion relative to a calibrated threshold. Our tool was able to elicit a large number of successful reaches and limit the amount of trunk flexion used during a single practice session while remaining usable, motivating, and safe. However, areas of improvement were identified relative to the efficiency of the software and the variety of virtual environments available. Third, we implemented VRShape over the course of a motor intervention for persons with stroke and evaluated its feasibility and effect on compensation during reaching tasks (Chapter 4). A small group of participants took part in 18 interventions session using VRShape for repetitive reaching practice with incrementally shaped trunk compensation. Trunk flexion decreased significantly and reaching kinematics improved significantly as a result of the intervention. Even with extended use, participants were able to complete intense practice and thousands of repetitions while continually rating the system as usable, motivating, engaging, and safe. Our VR tool demonstrated feasibility and preliminary efficacy within a small study, but future work is needed to identify its ideal applications and address its limitations. In summary, this project shows that use of a VR tool incorporating an accurate sensor (Chapter 2) and feedback from initial testing (Chapter 3) is capable of changing the amount of trunk flexion used during reaching movements for persons with stroke (Chapter 4). More research is needed to establish its efficacy and effectiveness, but improvements in motor recovery and associated decreases in compensation associated with the use of VRShape are important rehabilitation goals that may lead to improved participation and quality of life for persons living with long-term impairments due to chronic stroke

The Tree Property

AbstractWe construct a model in which there are no ℵn-Aronszajn trees for any finiten⩾2, starting from a model with infinitely many supercompact cardinals. We also construct a model in which there is noκ++-Aronszajn tree forκa strong limit cardinal of cofinalityω, starting from a model with a supercompact cardinal and a weakly compact cardinal above it