Are the Collatz and abc conjectures related?

The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset

Qualitative modelling and simulation of physical systems for a diagnostic purpose

This is a Milton Keynes De Montfort University thesisThe goal of a fault-diagnosis system is to obtain an accurate diagnosis at a low cost. In order to reach this goal, many techniques have been used, e.g. qualitative methods and multiple-models. This research investigates a novel strategy for improving the balance accuracy versus cost of consistency-based fault-diagnosis systems. This new strategy is organised around the notion of entities. These are physical entities. such as water pressure or temperature. The functioning of a physical system can involve numerous entities. Because these entities influence each other's behaviour, multiple-fault situations can occur, where several entities are affected by a fault. These situations are called complex multiple-fault situations. The existing fault-diagnosis systems do not perform satisfactorily on complex multiple-fault situations. This is because the set of entities they investigate is fixed from the start of the diagnostic process. As a consequence, depending on the entities included in this set, existing systems either perform an inaccurate diagnosis, or reach an accurate diagnosis at an unnecessarily high cost. This thesis presents a fault-diagnosis strategy called MVDS (standing for Multiple Variable Diagnosis Strategy) designed specifically for performing the efficient diagnosis of complex multiple-fault situations. The underlying principle of MVDS is that it is not possible to know from the start of the diagnostic process which entities are affected. Thus, a diagnostic process with MVDS is undertaken with the investigation of an initial set of entities, and this set of investigated entities is continuously updated along the process, as intermediate results are obtained. In order to illustrate clearly the functioning of MVDS, a fault-scenario using a small example from the air-conditioning domain is diagnosed and the process studied. The investigation of the performance of MVDS on more complex physical systems is undertaken on a larger case-study using a hot-water and heating system. In MVDS, it is possible to disable the adaptability of the set of investigated entities, so that it can be run with a fixed set. By doing so, the performance of the strategy in MVDS can be compared to the performance of traditional approaches which use a fixed set of investigated entities. The study-case shows that MVDS reaches more accurate results than traditional approaches, and that this accuracy is obtained at a low cost, since unnecessary measurements of entities are avoided. Furthermore, the strategy produces a complete trace of the process that is close to common-sense reasoning. It is also a co-operative strategy where the operator can intervene. Summary of the main research contributions: - The issue of diagnosing complex multiple-fault situations is specifically addressed for the first time. The problem caused by this diagnosis task is defined, and a strategy is constructed in order to diagnose efficiently the complex multiple-fault situations. The strategy is implemented in MVDS and tested on an example and a case-study. - Risk characteristics have been described. They allow to evaluate how prone to complex muItiple-fault situations is a physical system. - Hot-water and heating systems are offered as a new domain of research for consistency-based fault-diagnosis systems. - The inclusion of co-operation into the fault-diagnosis process is a novel approach. Its potential advantages have been identified

A Mathematical Model of the Combined β1- and β2-Adrenergic Signaling System in the Mouse Ventricular Myocyte

The β1- and β2-adrenergic signaling systems play different roles in the functioning of cardiac cells. Experimental data shows that the activation of the β1-adrenergic signaling system produces significant inotropic, lusitropic, and chronotropic effects in the heart, while the effects of the β2-adrenergic signaling system is less apparent. In this dissertation, a comprehensive experimentally-based mathematical model of the combined β1- and β2-adrenergic signaling systems in mouse ventricular myocytes is developed to simulate the experimental findings and make testable predictions of the behavior of the cardiac cells under different physiological conditions. Simulations describe the dynamics of major signaling molecules in different subcellular compartments; kinetics and magnitudes of phosphorylation of ion channels, transporters, and Ca2+ handling proteins; modifications of action potential shape and duration; and [Ca2+]i and [Na+]i dynamics upon stimulation of β1- and β2-adrenergic receptors (β1- and β2-ARs). The model reveals physiological conditions when β2-ARs do not produce significant physiological effects and when their effects can be measured experimentally. Simulations demonstrated that stimulation of β2-ARs with isoproterenol caused a marked increase in the magnitude of the L-type Ca2+ current, [Ca2+]i transient, and phosphorylation of phospholamban only upon additional application of pertussis toxin (PTX) or inhibition of phosphodiesterases of type 3 and 4. The model also made testable predictions of the changes in magnitudes of [Ca2+]i and [Na+]i fluxes, the rate of decay of [Na+]i concentration upon both combined and separate stimulation of β1- and β2-ARs, and the contribution of phosphorylation of PKA targets to the changes in the action potential and [Ca2+]i transient. A comprehensive mathematical model of the mouse ventricular myocyte overexpressing β2-adrenergic receptors was also developed. It was found that most of the β2-adrenergic receptors are active in control conditions in TG mice. Simulations describe the increased basal adenylyl cyclase activity; modifications of action potential; the effects on the L-type Ca2+ current and [Ca2+]i transients upon stimulation of β2-adrenergic receptors in control, after the application of PTX, upon stimulation with zinterol, and upon stimulation with zinterol in the presence of PTX. The model also describes the effects of inverse agonist ICI-118,551 on adenylyl cyclase activity, action potential, and [Ca2+]i transients

Ethical Concerns in an Era of Population Health: A Challenge of Identity for Tomorrow's Hospitals

Background: After focusing on episodic, clinical encounters, U.S. health care is beginning to acknowledge the need for population-level, preventive health strategies. Largely driven by new regulatory measures, new payment programs, and greater appreciation for the social determinants of health, this shift aims to reduce per capita cost and improve population health. In addition to the financial, clinical, and operational challenges, population health strategies raise new ethical questions. Methods: This study used an exploratory sequential mixed methods design. Qualitative data were based on interviews (n=38) with supervisors of Community Benefit or population health management at nonprofit hospitals across the country, recruited through chain referral and stratified sampling methods. The interviews solicited information on organizational structures, community engagement, and ethical concerns in population health work. Ten focus groups in five cities with hospital employees and community partners (n=43) provided examples of ethical concerns in hospital-community collaboration and helped define the characteristics of community health projects for a conjoint analysis. Interviews and focus groups were analyzed with open coding text analysis. Quantitative data were generated from an online survey completed by a convenience sample of hospital employees (n=225) and leaders of nonprofit community organizations (n=136). The survey included a discrete choice experiment, which asked respondents to choose between funding one of two community health initiatives based on the characteristics of those projects. The characteristics of projects were: priority from community need assessment; time until measureable impact; kind of community partnership; type of intervention; evidence of intervention effectiveness; and target population. Discrete choice results were estimated with an effects coded, conditional logit model. The survey included questions on how often respondents have faced certain ethical dilemmas. Sensitivity analyses and latent class analysis were conducted to determine differences between respondent subgroups. Results: Interviews revealed a complex process for Community Benefit and also provided evidence that those in the field do not readily frame population health work in ethical terms. After being prompted, most interviewees gave examples of ethical challenges, and survey respondents indicated they regularly experience ethical dilemmas. The interviews also provided information to map the many challenges nonprofit hospitals face as they allocate limited resources for community health initiatives, including how to best involve community members in the process. The discrete choice experiment showed strong agreement between hospital employees and community members as to what kind of community health initiatives are most important to fund. Projects were more likely to be selected if they included a top priority from the community health needs assessment, were supported by a coalition of community partners, and there was some evidence of intervention effectiveness. Latent class analysis revealed four respondent subgroups who differed on project priorities but who did not differ by demographic characteristics. Discussion: As we enter an era of population health, hospitals must grapple with their changing identity in the community. They are no longer just experts in clinical care; they must also contribute to population-level, preventive strategies. The study identifies several implications for both policy and practice, such as the need to include investment in social determinants of health as part of the Community Benefit portfolio and the need to build educational infrastructure for population health ethics. Rather than using existing ethical principles, this study proposes virtue ethics and a new concept called structures of virtue as resources for evaluating ethical challenges in population health.PHDHealth Services Organization & PolicyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144171/1/mrozier_1.pd