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

ESOL teachers’ perceptions and experiences framing pedagogical approaches in the multicultural classroom

The purpose of this study was to collect the lived experiences of English Speakers of Other Languages (ESOL) teachers to amplify the voices of the teachers regarding their needs in teaching multicultural students. This study sought to increase awareness regarding pedagogical approaches in teaching multicultural students, as well as to provide a description of teacher perceptions and experiences about teaching immigrant students. The problem identified in this study, as well as the gap in the existing research literature, is that teachers are not adequately prepared to teach students who immigrate to the United States from other countries. The research methodology was grounded in a qualitative methodology and a narrative research design. Through lived experiences and storytelling, this study explored teacher perceptions and experiences of educating immigrant students. The participants of the study included six teachers that were interviewed, participated in a focus group, and completed journal prompts. The location of the research site was in the state of Georgia. Critical Race Theory was explored in terms of its application to current educational contexts. Three of the tenets of Critical Race Theory were applicable and relevant to this study: understanding the centrality of racism, interest convergence, and centrality of experience. Additionally, five central themes were outlined as a result of the data collected: Instructional Curriculum & Resources, Instructional Time, Instructional Challenges, Teaching Experience & Preparation, and Life Experience. Teachers highlighted issues with the provided instructional curriculum, access to resources, and having inadequate instructional time to teach and re-teach lessons to students. Additional instructional challenges were presented based on levels of preparation to instruct multilingual learners. All of these experiences, including both the personal and professional, framed pedagogical approaches in the multicultural classroom

Thermal DeNOx Optimization: A Case Study in Fluid Catalytic Cracking

This thesis will first provide background information on fluid catalytic cracking (FCC), a highly important unit operation to the process of petroleum refining, and a description of Thermal DeNOx, an environmental treatment system common to FCC units and other process units where high temperature furnaces are used. Next, this thesis will detail a project which I had the chance to lead as a process engineering intern at ExxonMobil’s Baton Rouge Refinery in the fall of 2016. The objective of the project was to optimize Thermal DeNOx systems installed on two FCC units at the refinery. Experiments were conducted on these units at different chemical injection rates and temperatures, in order to determine the operating parameters at which the Thermal DeNOx systems were as efficient and economical as possible. A control scheme was implemented on the unit to maintain operation at these experimentally determined parameters, with a projected economic benefit of $250,000 per year for the refinery. This thesis also includes a discussion of my personal learnings and takeaways from this experience, including the challenges one can face when leading an engineering project in an industrial manufacturing setting