Energy and Vorticity in Fast Rotating Bose-Einstein Condensates

We study a rapidly rotating Bose-Einstein condensate confined to a finite trap in the framework of two-dimensional Gross-Pitaevskii theory in the strong coupling (Thomas-Fermi) limit. Denoting the coupling parameter by 1/\eps^2 and the rotational velocity by $\Omega$, we evaluate exactly the next to leading order contribution to the ground state energy in the parameter regime |\log\eps|\ll \Omega\ll 1/(\eps^2|\log\eps|) with \eps\to 0. While the TF energy includes only the contribution of the centrifugal forces the next order corresponds to a lattice of vortices whose density is proportional to the rotational velocity.Comment: 19 pages, LaTeX; typos corrected, clarifying remarks added, some rearrangements in the tex

Convergence of Ginzburg-Landau functionals in 3-d superconductivity

In this paper we consider the asymptotic behavior of the Ginzburg- Landau model for superconductivity in 3-d, in various energy regimes. We rigorously derive, through an analysis via {\Gamma}-convergence, a reduced model for the vortex density, and we deduce a curvature equation for the vortex lines. In a companion paper, we describe further applications to superconductivity and superfluidity, such as general expressions for the first critical magnetic field H_{c1}, and the critical angular velocity of rotating Bose-Einstein condensates.Comment: 45 page

The Effects of Negative Legacies on the Adjustment of Parentally Bereaved Children and Adolescents

This is a report of a qualitative analysis of a sample of bereaved families in which one parent died and in which children scored in the clinical range on the Child Behavior Check List. The purpose of this analysis was to learn more about the lives of these children. They were considered to be at risk of developing emotional and behavioral problems associated with the death. We discovered that many of these “high risk” children had a continuing bond with the deceased that was primarily negative and troubling for them in contrast to a comparison group of children not at risk from the same study. Five types of legacies, not mutually exclusive, were identified: health related, role related, personal qualities, legacy of blame, and an emotional legacy. Coping behavior on the part of the surviving parent seemed to make a difference in whether or not a legacy was experienced as negative

Vortex density models for superconductivity and superfluidity

We study some functionals that describe the density of vortex lines in superconductors subject to an applied magnetic field, and in Bose-Einstein condensates subject to rotational forcing, in quite general domains in 3 dimensions. These functionals are derived from more basic models via Gamma-convergence, here and in a companion paper. In our main results, we use these functionals to obtain descriptions of the critical applied magnetic field (for superconductors) and forcing (for Bose-Einstein), above which ground states exhibit nontrivial vorticity, as well as a characterization of the vortex density in terms of a non local vector-valued generalization of the classical obstacle problem.Comment: 34 page

Inhomogeneous Vortex Patterns in Rotating Bose-Einstein Condensates

We consider a 2D rotating Bose gas described by the Gross-Pitaevskii (GP) theory and investigate the properties of the ground state of the theory for rotational speeds close to the critical speed for vortex nucleation. While one could expect that the vortex distribution should be homogeneous within the condensate we prove by means of an asymptotic analysis in the strongly interacting (Thomas-Fermi) regime that it is not. More precisely we rigorously derive a formula due to Sheehy and Radzihovsky [Phys. Rev. A 70, 063620(R) (2004)] for the vortex distribution, a consequence of which is that the vortex distribution is strongly inhomogeneous close to the critical speed and gradually homogenizes when the rotation speed is increased. From the mathematical point of view, a novelty of our approach is that we do not use any compactness argument in the proof, but instead provide explicit estimates on the difference between the vorticity measure of the GP ground state and the minimizer of a certain renormalized energy functional.Comment: 41 pages, journal ref: Communications in Mathematical Physics: Volume 321, Issue 3 (2013), Page 817-860, DOI : 10.1007/s00220-013-1697-

The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate

We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as 1 / \eps^2 we consider the asymptotic regime \eps \to 0 with the angular velocity $\Omega$ proportional to (\eps^2|\log\eps|)^{-1} . We prove that if \Omega = \Omega_0 (\eps^2|\log\eps|)^{-1} and $\Omega_0 > 2(3\pi)^{-1}$ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.Comment: 52 pages, PDFLaTex. Minor corrections, sign convention modified. To be published in Commun. Math. Phy

Mappings of least Dirichlet energy and their Hopf differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure

Passing to the Limit in a Wasserstein Gradient Flow: From Diffusion to Reaction

We study a singular-limit problem arising in the modelling of chemical reactions. At finite {\epsilon} > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/{\epsilon}, and in the limit {\epsilon} -> 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier, Savar\'e, and Veneroni, SIAM Journal on Mathematical Analysis, 42(4):1805-1825, 2010, using the linear structure of the equation. In this paper we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the {\epsilon}-problem converge to a solution of the limiting problem.Comment: Added two sections, corrected minor typos, updated reference

Quality of Maternal and Neonatal Care in Albania, Turkmenistan and Kazakhstan: A Systematic, Standard-Based, Participatory Assessment

BACKGROUND: Progress in maternal and neonatal mortality has been slow in many countries despite increasing access to institutional births, suggesting deficiencies in the quality of care. We carried out a systematic assessment of the quality of maternal and newborn care in three CEE/CIS countries, using an innovative approach to identify priority issues and promote action. METHODS: A standard-based tool, covering over 400 items grouped in 13 main areas ranging from support services to case management, was used to assess a sample of ten maternity hospitals in Albania, Kazakhstan and Turkmenistan. Sources of information were visit to services, medical records, observation of cases, and interviews with staff and mothers. A score (range 0 to 3) was attributed to each item and area of care. The assessment was carried out by a multidisciplinary team of international and national professionals. Local managers and staff provided the necessary information and were involved in discussing the findings and the priority actions. RESULTS: Quality of care was found to be substandard in all 13 areas. The lowest scores (between one and two) were obtained by: management of normal labour, delivery, obstetric complications and sick babies; infection prevention; use of guidelines and audits; monitoring and follow-up. Neonatal care as a whole scored better than obstetric care. Interviewed mothers identified lack of information, insufficient support during labour and lack of companionship as main issues. Actions to improve quality of care were identified at facility as well as at central level and framed according to main health system functions. CONCLUSIONS: Quality of care is a key issue to improve maternal and neonatal outcomes, particularly in countries such as CEE/CIS where access to institutional births is nearly universal. Approaches that involve health professionals and managers in comprehensive, action-oriented assessments of quality of care are promising and should be further supported

Augmented Two-Channel Arrhythmia Detection: An Efficient Diagnostic Method for Implantable Devices

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/75624/1/j.1540-8159.1996.tb03164.x.pd