Discrete transformations in the Thomson Problem

LaFave, Tim

doi:10.1016/j.elstat.2013.11.007

Cited by 10 publications

(4 citation statements)

References 28 publications

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“…The criminal law generally penalizes actions, not refusals to act. Moreover, a physician cannot be punished for failing to provide a ventilator that does not exist, as will be true if the supply of ventilators is insufficient …”

Section: The Risk Of Criminal and Civil Liabilitymentioning

confidence: 99%

Potential Legal Liability for Withdrawing or Withholding Ventilators During COVID-19

Cohen

Crespo

White

2020

JAMA

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With an anticipated shortage of ventilators for patients with coronavirus disease 2019 (COVID-19), hospitals, physicians, and nurses may have to make an unprecedented decision: should they withdraw or withhold ventilators from some patients and use them for other patients who have a better chance of survival? It is not uncommon for care teams to decide against initiating or continuing mechanical ventilation when such treatment would not achieve a patient's goals or directives. COVID-19 presents a different case: patients who do not receive a ventilator could benefit, perhaps living for many additional years, if they receive short-term mechanical ventilation. Denying patients such treatment, against their wishes, most likely will result in their death, but it will also make this scarce resource available to other patients who are more likely to survive if they receive ventilator support.Recently developed protocols expressly call for the rationing and reallocation of ventilators, in a manner that aims to save the greatest number of lives. 1 These protocols are broadly accepted by medical ethicists. 1,2 But ethics aside, there are potential legal ramifications of either withholding or withdrawing a ventilator from a patient who would ordinarily receive such aid in the absence of a public health emergency.In this Viewpoint, we assess the legal risks that physicians, other health care workers, and hospital systems confront in such scenarios and recommend that states explicitly and immediately adopt legal protections for health care workers, modeled on provisions in place in Maryland.

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Section: The Risk Of Criminal and Civil Liabilitymentioning

confidence: 99%

Potential Legal Liability for Withdrawing or Withholding Ventilators During COVID-19

Cohen

Crespo

White

2020

JAMA

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“…For N ¼ 4, 6, and 12, they correspond to Platonic solids. The global minimum structures for larger N have been tackled computationally [32,33,[40][41][42][43][44][45][46][47][48]. Most of these studies, however, only focus on the global minima.…”

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confidence: 99%

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Mehta

Chen

et al. 2016

Phys. Rev. Lett.

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Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modied, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N ¼ 132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

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“…This is a considerable reduction of the difficulty of the problem given that 3D integrals over a typical regular body generally lead to analytical results. On the other hand, this is also a useful simplification for numerical calculations [19][20][21][22] of the electrostatic selfenergy of charged bodies with irregular shape since a numerical integration with three variables can be done faster and more accurately than its counterpart with six variables.…”

Section: Discussionmentioning

confidence: 99%

Fourier transform method for the electrostatic self-energy of a solid sphere with uniform volume charge density

Ciftja¹

2021

Eur. J. Phys.

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The problem of a solid sphere with uniform volume charge density is encountered in virtually all undergraduate calculus-based physics textbooks dealing with the topic of electromagnetism. This example illustrates well the use of Gauss’s law and from there one can easily derive all the quantities of interest such as electrostatic field, potential, self-energy, and so on. Undergraduate physics majors are also well aware of the theory of Fourier transforms from having taken mathematics courses. Nevertheless, despite its great utility, the Fourier transform method is rarely mentioned as a powerful tool to solve physics problems at this level. To address this shortcoming, in this work we propose a possible scenario which may allow an instructor to introduce this powerful method to a proper undergraduate audience without any major pedagogical drawback. The case study that we choose is that of a solid sphere with uniform volume charge density. Specifically, we show the calculation of its electrostatic self-energy by using Fourier transform techniques. The main idea of this work is to draw reader’s attention to the versatility of the approach that can, in principle, be applied to other more geometrically complicated bodies where Gauss’s law does not lead to simple solutions. Concurrently, this work also provides instructional approaches that intertwine content-specific and pedagogical viewpoints that can be useful to all undergraduate students and teachers who wish to enhance their command of the subject.

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Discrete transformations in the Thomson Problem

Cited by 10 publications

References 28 publications

Potential Legal Liability for Withdrawing or Withholding Ventilators During COVID-19

Potential Legal Liability for Withdrawing or Withholding Ventilators During COVID-19

Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem

Fourier transform method for the electrostatic self-energy of a solid sphere with uniform volume charge density

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