On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Two isoperimetric inequalities for the Sobolev constant

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.Comment: 11 page

The Bosma effect revisited - I. HI and stellar disc scaling models

The observed proportionality between the centripetal contribution of the dynamically insignificant HI gas in the discs of spiral galaxies and the dominant contribution of DM - the "Bosma effect" - has been repeatedly mentioned in the literature but largely ignored. We have re-examined the evidence for the Bosma effect by fitting Bosma effect models for 17 galaxies in the THINGS data set, either by scaling the contribution of the HI gas alone or by using both the observed stellar disc and HI gas as proxies. The results are compared with two models for exotic cold DM: internally consistent cosmological NFW models with constrained compactness parameters, and URC models using fully unconstrained Burkert density profiles. The Bosma models that use the stellar discs as additional proxies are statistically nearly as good as the URC models and clearly better than the NFW ones. We thus confirm the correlation between the centripetal effects of DM and that of the interstellar medium of spiral galaxies. The edificacy of "maximal disc" models is explained as the natural consequence of "classic" Bosma models which include the stellar disc as a proxy in regions of reduced atomic gas. The standard explanation - that the effect reflects a statistical correlation between the visible and exotic DM - seems highly unlikely, given that the geometric forms and hence centripetal signatures of spherical halo and disc components are so different. A literal interpretation of the Bosma effect as being due to the presence of significant amounts of disc DM requires a median visible baryon to disc DM ratio of about 40%.Comment: Accepted by A&A (Paper I

Carbon Dioxide Embolisms During Laparoscopic Surgery

With the advancement of technology over the past few decades, laparoscopic surgery has significantly increased in popularity among many surgical subspecialties including: general surgery, gynecology, and urology. Many procedures that traditionally required open access with large incisions have been replaced with laparoscopic approaches as it offers many advantages such as, faster recovery, less postoperative pain, and shorter hospital stays. Despite its many advantages, it creates numerous anesthesia challenges and considerations when caring for patients undergoing laparoscopic procedures. Among these challenges, are carbon dioxide embolisms. This manuscript involves research and previous case studies that were conducted on carbon dioxide embolisms during laparoscopic surgery. The purpose of this manuscript is to have a solid foundation of research in order to evaluate an event of a suspected carbon dioxide embolism during a laparoscopic, hand-assisted nephrectomy

The Link between the Baryonic Mass Distribution and the Rotation Curve Shape

The observed rotation curves of disc galaxies, ranging from late-type dwarf galaxies to early-type spirals, can be fit remarkably well simply by scaling up the contributions of the stellar and HI discs. This `baryonic scaling model' can explain the full breadth of observed rotation curves with only two free parameters. For a small fraction of galaxies, in particular early-type spiral galaxies, HI scaling appears to fail in the outer parts, possibly due to observational effects or ionization of the HI. The overall success of the baryonic scaling model suggests that the well-known global coupling between the baryonic mass of a galaxy and its rotation velocity (known as the baryonic Tully-Fisher relation), applies at a more local level as well, and it seems to imply a link between the baryonic mass distribution and the distribution of total mass (including dark matter).Comment: 10 pages, accepted for publication in MNRA

Aneesh, A. Neutral Accent: How Language, Labor, and Life Become Global. Durham and London: Duke University Press, 2015.

Book review

Mass-to-light ratio gradients in early-type galaxy haloes

Since the near future should see a rapidly expanding set of probes of the halo masses of individual early-type galaxies, we introduce a convenient parameter for characterising the halo masses from both observational and theoretical results: \dML, the logarithmic radial gradient of the mass-to-light ratio. Using halo density profiles from LCDM simulations, we derive predictions for this gradient for various galaxy luminosities and star formation efficiencies $\epsilon_{SF}$. As a pilot study, we assemble the available \dML\ data from kinematics in early-type galaxies - representing the first unbiassed study of halo masses in a wide range of early-type galaxy luminosities - and find a correlation between luminosity and \dML, such that the brightest galaxies appear the most dark-matter dominated. We find that the gradients in most of the brightest galaxies may fit in well with the LCDM predictions, but that there is also a population of fainter galaxies whose gradients are so low as to imply an unreasonably high star formation efficiency $\epsilon_{SF} > 1$. This difficulty is eased if dark haloes are not assumed to have the standard LCDM profiles, but lower central concentrations.Comment: 17 pages, 13 figures. Accepted for publication on MNRA

The ethics of digital well-being: a multidisciplinary perspective

This chapter serves as an introduction to the edited collection of the same name, which includes chapters that explore digital well-being from a range of disciplinary perspectives, including philosophy, psychology, economics, health care, and education. The purpose of this introductory chapter is to provide a short primer on the different disciplinary approaches to the study of well-being. To supplement this primer, we also invited key experts from several disciplines—philosophy, psychology, public policy, and health care—to share their thoughts on what they believe are the most important open questions and ethical issues for the multi-disciplinary study of digital well-being. We also introduce and discuss several themes that we believe will be fundamental to the ongoing study of digital well-being: digital gratitude, automated interventions, and sustainable co-well-being