Serre's Modularity Conjecture

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture

Normal zeta functions of the Heisenberg groups over number rings II -- the non-split case

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.Comment: 19 pages; to appear in Israel J. Mat

Bolza quaternion order and asymptotics of systoles along congruence subgroups

We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of "Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister-Schreier algorithm as implemented in magma to generate these surfaces.Comment: 35 pages, to appear in Experimental Mathematic

Action bias among elite soccer goalkeepers: The case of penalty kicks

In soccer penalty kicks, goalkeepers choose their action before they can clearly observe the kick direction. An analysis of 286 penalty kicks in top leagues and championships worldwide shows that given the probability distribution of kick direction, the optimal strategy for goalkeepers is to stay in the goal's center. Goalkeepers, however, almost always jump right or left. We propose the following explanation for this behavior: because the norm is to jump, norm theory (Kahneman and Miller, 1986) implies that a goal scored yields worse feelings for the goalkeeper following inaction (staying in the center) than following action (jumping), leading to a bias for action. The omission bias, a bias in favor of inaction, is reversed here because the norm here is reversed - to act rather than to choose inaction. The claim that jumping is the norm is supported by a second study, a survey conducted with 32 top professional goalkeepers. The seemingly biased decision making is particularly striking since the goalkeepers have huge incentives to make correct decisions, and it is a decision they encounter frequently. Finally, we discuss several implications of the action/omission bias for economics and management.Decision Making; Uncertainty; Choice Behavior; Sport Psychology; Behavioral Economics; Action Bias; Omission Bias; Commission Bias; Action Effect; Inaction Effect; Actor Effect; Economic Psychology; Heuristics and Biases; Soccer; Goalkeepers; Penalty Kicks; Risk; Norms

Pro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes $\mathfrak{p}$ of $K$ and depending only on the structure of $\Gamma$, the degree $[K : \mathbb{Q}]$, and the cardinality of the residue field $\mathcal{O}_K / \mathfrak{p}$. We show that the factors satisfy a certain uniform rationality and study their dependence on $[K : \mathbb{Q}]$. Explicit computations are given for several families of unipotent groups. These include an apparently novel identity involving permutation statistics on the hyperoctahedral group.Comment: 50 pages. A uniform rationality result, Theorem 1.3, has been added; it is proved in Section 4. Comments are welcome

Evaluation of a Telerehabilitation Consultation Model for Remote Wheelchair Prescription

The purpose of this project was to determine the effectiveness of a telerehabilitation (TR) consultation model to prescribe and procure an appropriate wheeled mobility and seating (WMS) device at a remotely located site. The availability of practitioners with specific expertise in this area was limited particularly in Westerns Pennsylvania. A telerehabilitation service delivery model was developed for a series of studies based on a current model implemented at the Center for Assistive Technology at the University of Pittsburgh Medical Center (CAT-UPMC). In a multi-center non-randomized clinical trial, 96 participants were evaluated: 50 In-Person (IP) at the CAT-UPMC and 46 TR participants at remote sites. The performance-based Functioning Everyday with a Wheelchair-Capacity (FEW-C) tool demonstrated excellent inter-rater reliability coefficients (ICC 2,k = 0.91) and good internal consistency measured by Cronbach's alphas with correlations ranging between 0.82 to 0.91 among the 46 TR participants. Results indicated that using a TR consultation model, a significant improvement in mean differences was observed for the each of the self-report Functioning Everyday with a Wheelchair (FEW) items and for the average FEW scores at the remotely sites. Effect size calculations indicated that nine of the ten items on the FEW as well as the total FEW had very large effect sizes using Cohen's d, indicating the effectiveness of not only the new WMS device but the TR assessment as well. A significant relationship was found between the self-report FEW and performance-based FEW-C tools at baseline measured by Spearman rho's correlations. A significant difference was found for participants previous WMS device evaluation and prescription process compared to their current TR WMS device evaluation and prescription scores as well as patient satisfaction regarding the impact of the technology. The findings based on confidence intervals of post FEW scores indicated that TR was non-inferior to the standard IP care at CAT-UPMC. Telerehabilitation services resulted in decreased travel for participants, improved access to specialized services, education benefits for generalist practitioners, and service stabilization at the remote sites. A TR consultation model offers new alternative and effective opportunities to provide rehabilitation services in clinical settings, especially in rural or underserved locations

Telerehabilitation Clinical and Vocational Applications for Assistive Technology: Research, Opportunities, and Challenges

Rehabilitation service providers in rural or underserved areas are often challenged in meeting the needs of their complex patients due to limited resources in their geographical area. Recruitment and retention of the rural clinical workforce are beset by the ongoing problems associated with limited continuing education opportunities, professional isolation, and the challenges inherent in coordinating rural community healthcare. People with disabilities who live in rural communities also face challenges accessing healthcare. Traveling long distances to a specialty clinic for necessary expertise may be troublesome due to inadequate or unavailable transportation, disability specific limitations, and financial limitations. Distance and lack of access are just two threats to quality of care that now being addressed by the use of videoconferencing, information exchange, and other telecommunication technologies that facilitate telerehabilitation. This white paper illustrates and summarizes clinical and vocational applications of telerehabilitation. We provide definitions related to the fields of telemedicine, telehealth, and telerehabilitation, and consider the impetus for telerehabilitation. We review the telerehabilitation literature for assistive technology applications; pressure ulcer prevention; virtual reality applications; speech-language pathology applications; seating and wheeled mobility applications; vocational rehabilitation applications; and cost effectiveness. We then discuss external telerehabilitation influencers, such as the positions of professional organizations. Finally, we summarize clinical and policy issues in a limited context appropriate to the scope of this paper. Keywords: Telerehabilitation, Telehealth, Telemedicine, Telepractic

Vacuum arc plasma thrusters with inductive energy storage driver

An apparatus for producing a vacuum arc plasma source device using a low mass, compact inductive energy storage circuit powered by a low voltage DC supply acts as a vacuum arc plasma thruster. An inductor is charged through a switch, subsequently the switch is opened and a voltage spike of Ldi/dt is produced initiating plasma across a resistive path separating anode and cathode. The plasma is subsequently maintained by energy stored in the inductor. Plasma is produced from cathode material, which allows for any electrically conductive material to be used. A planar structure, a tubular structure, and a coaxial structure allow for consumption of cathode material feed and thereby long lifetime of the thruster for long durations of time