pp-adic quotient sets

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.Comment: 24 page

Solving for a Bellman Function

The Bellman method is a tool that can be used to solve a wide variety of problems from harmonic analysis. However, the problem of finding the Bellman function for a specific problem was originally only possible with a guess-and-check method, although that problem has since been mitigated. This paper applies the method pioneered by Vasily Vasyunin of finding the Bellman function directly to a problem presented in JanineWittwer’s survey article [5], and also explores how Bellman functions can be used to prove H˝older’s inequality

Programming parameters of subthalamic deep brain stimulators in Parkinson's disease from a controlled trial

Background: Programming algorithms have never been tested for outcome. The EARLYSTIM study showed superior outcomes of deep brain stimulation of the subthalamic nucleus (STN-DBS) over best medical treatment in early Parkinson's disease (PD). Patients were programmed according to common guidelines but customized for each patient. Methods: Stimulation parameters were systematically documented at 1, 5, 12, and 24 month in the cohort of 114 patients who had bilateral STN-DBS at 24 month. We investigated the influence of atypical programming, changes of stimulated electrode contacts and stimulation energy delivered. Outcomes were the Unified Parkinson's Disease Rating Scale (UPDRS) motor and ADL-subscores, health-related quality of life (PDQ-39) summary index and mobility- and ADL-subscores. Results: At 1/5/12/24 months follow up, mean amplitude (1.8/2.5/2.6/2.8 V), impedance (1107/1286/1229/1189 Omega) and TEED (33.7/69.0/84.4/93.0 V2*mu s*Hz/Omega) mainly increased in the first 5 months, while mean pulse width (60.0/62.5/65.1/65.8 mu s), frequency (130/137.7/139.1/142.7 Hz) remained relatively stable. Typical programming (single monopolar electrode contact) was used in 80.7% of electrodes. Double monopolar (11/114) and bipolar (2/114) stimulation was only rarely required. There was no significant difference in clinical outcomes between the patient groups requiring contact changes (n = 32/28.1%) nor between (n = 83/72.8%) versus non-typical programming. Energy used for STN-DBS was higher for the dominant side of PD. Conclusion: In the first 5 months an increase in amplitude is required to compensate for various factors. Monopolar stimulation is sufficient in 80% of patients at 24 months. Homogeneous stimulation strategies can account for the favorable outcomes reported in the Earlystim study