ODE Models of Wealth Concentration and Taxation

We refer to an individual holding a non-negligible fraction of the country’s total wealth as an oligarch. We explain how a model due to Boghosian et al. can be used to explore the effects of taxation on the emergence of oligarchs. The model suggests that oligarchs will emerge when wealth taxation is below a certain threshold, not when it is above the threshold. The underlying mechanism is a transcritical bifurcation. The model also suggests that taxation of income and capital gains alone cannot prevent the emergence of oligarchs. We suggest several opportunities for students to explore modifications of the model

ODEs and Mandatory Voting

This paper presents mathematics relevant to the question whether voting should be mandatory. Assuming a static distribution of voters’ political beliefs, we model how politicians might adjust their positions to raise their share of the vote. Various scenarios can be explored using our app at https: //centrism.streamlit.app/. Abstentions are found to have great impact on the dynamics of candidates, and in particular to introduce the possibility of discontinuous jumps in optimal candidate positions. This is an unusual application of ODEs. We hope that it might help engage some students who may find it harder to connect with the more customary applications from the natural sciences

The asymptotic diffusion limit of a linear discontinuous discretization of a two-dimensional linear transport equation

Consider a linear transport problem, and let the mean free path and the absorption cross section be of size [epsilon]. It is well known that one obtains a diffusion problem as [epsilon] tends to zero. We discretize the transport problem on a fixed mesh, independent of [epsilon], consider again the limit [epsilon] --> 0 and ask whether one obtains an accurate discretization of the continuous diffusion problem. The answer is known to be affirmative for the linear discontinuous Galerkin finite element discretization in one space dimension. In this paper, we ask whether the same result holds in two space dimensions. We consider a linear discontinuous discretization based on rectangular meshes. Our main result is that the asymptotic limit of this discrete problem is not a discretization of the asymptotic limit of the continuous problem and thus that the discretization will be inaccurate in the asymptotic regime under consideration. We also propose a modified scheme which has the correct asymptotic behavior for spatially periodic problems, although not always for problems with boundaries. We present numerical results confirming our formal asymptotic analysis.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30231/1/0000625.pd

Diagnosis of bipolar disorders and body mass index predict clustering based on similarities in cortical thickness-ENIGMA study in 2436 individuals

AIMS: Rates of obesity have reached epidemic proportions, especially among people with psychiatric disorders. While the effects of obesity on the brain are of major interest in medicine, they remain markedly under-researched in psychiatry. METHODS: We obtained body mass index (BMI) and magnetic resonance imaging-derived regional cortical thickness, surface area from 836 bipolar disorders (BD) and 1600 control individuals from 14 sites within the ENIGMA-BD Working Group. We identified regionally specific profiles of cortical thickness using K-means clustering and studied clinical characteristics associated with individual cortical profiles. RESULTS: We detected two clusters based on similarities among participants in cortical thickness. The lower thickness cluster (46.8% of the sample) showed thinner cortex, especially in the frontal and temporal lobes and was associated with diagnosis of BD, higher BMI, and older age. BD individuals in the low thickness cluster were more likely to have the diagnosis of bipolar disorder I and less likely to be treated with lithium. In contrast, clustering based on similarities in the cortical surface area was unrelated to BD or BMI and only tracked age and sex. CONCLUSIONS: We provide evidence that both BD and obesity are associated with similar alterations in cortical thickness, but not surface area. The fact that obesity increased the chance of having low cortical thickness could explain differences in cortical measures among people with BD. The thinner cortex in individuals with higher BMI, which was additive and similar to the BD-associated alterations, may suggest that treating obesity could lower the extent of cortical thinning in BD

Mega-analysis of association between obesity and cortical morphology in bipolar disorders:ENIGMA study in 2832 participants

Background: Obesity is highly prevalent and disabling, especially in individuals with severe mental illness including bipolar disorders (BD). The brain is a target organ for both obesity and BD. Yet, we do not understand how cortical brain alterations in BD and obesity interact. Methods: We obtained body mass index (BMI) and MRI-derived regional cortical thickness, surface area from 1231 BD and 1601 control individuals from 13 countries within the ENIGMA-BD Working Group. We jointly modeled the statistical effects of BD and BMI on brain structure using mixed effects and tested for interaction and mediation. We also investigated the impact of medications on the BMI-related associations. Results: BMI and BD additively impacted the structure of many of the same brain regions. Both BMI and BD were negatively associated with cortical thickness, but not surface area. In most regions the number of jointly used psychiatric medication classes remained associated with lower cortical thickness when controlling for BMI. In a single region, fusiform gyrus, about a third of the negative association between number of jointly used psychiatric medications and cortical thickness was mediated by association between the number of medications and higher BMI. Conclusions: We confirmed consistent associations between higher BMI and lower cortical thickness, but not surface area, across the cerebral mantle, in regions which were also associated with BD. Higher BMI in people with BD indicated more pronounced brain alterations. BMI is important for understanding the neuroanatomical changes in BD and the effects of psychiatric medications on the brain.</p

Short title: Complexity of dose calculation methods

Abstract. Grid-based deterministic dose calculation methods for radiotherapy planning require the use of six-dimensional phase space grids. Because of the large number of phase space dimensions, a growing number of Medical Physicists appear to believe that grid-based deterministic dose calculation methods are not competitive withMonte Carlo methods. We argue that this conclusion may bepremature. Our results do suggest, however, that higher than rst order accurate discretizations will probably be needed if grid-based deterministic dose calculation methods are to compete well with Monte Carlo methods. 2 1

The Radiation Therapy Planning Problem

this paper is to describe mathematical aspects of radiation therapy planning to readers with a background in applied mathematics. The use of X-rays for cancer therapy began a few days after their discovery. Wilhelm Rontgen announced the discovery of X-rays on December 28, 1895, and Emil Grubbe used them for cancer therapy on January 12, 1896 [40]. X-rays are still the most common form of radiation used for cancer therapy, but beams of electrons, protons, neutrons, and other particles are used as well. The planning of the radiation treatment of a tumor begins with the creation of a three-dimensional image of the tumor and surrounding healthy tissue, using techniques such as computed tomography or MRI. The treatment planning discussed in this article occurs after the imaging is completed. It involves substantial use of computational algorithms. Radiation therapy planning requires the study of radiation penetrating a background (a portion of a patient&apos;s body and the surrounding air, for instance). Both the radiation and the background are, of course, made up of particles. We shall distinguish between the two by referring to radiation particles and background particles. Background particles can be set in rapid motion as a result of interactions with radiation particles, thereby becoming radiation particles themselves. The transport of the radiation particles through the background is described by a system of coupled Boltzmann transport equations; see for instance Ref. [15], and also Sec. 2 of this article. A solution of this system is a vector of phase space number densities, that is, numbers of radiation particles per unit volume in phase space, i.e. positiondirection -energy space. Different components of this vector correspond to different particle types. Even if the bea..

Complexity of Monte Carlo and deterministic dose calculation methods

. Grid-based deterministic dose calculation methods for radiotherapy planning require the use of six-dimensional phase space grids. Because of the large number of phase space dimensions, a growing number of Medical Physicists appear to believe that grid-based deterministic dose calculation methods are not competitive with Monte Carlo methods. We argue that this conclusion may be premature. Our results do suggest, however, that higher than first order accurate discretizations will probably be needed if grid-based deterministic dose calculation methods are to compete well with Monte Carlo methods. 1. Introduction In current clinical practice, dose calculations for radiotherapy planning are typically performed using methods based on a combination of analysis and laboratory measurements; see Jette (1995) for a survey. However, other types of dose calculation methods are currently under development. Among these, the two main families are the Monte Carlo methods, and grid-based determinis..