Specifying attracting cycles for Newton maps of polynomials

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result due to Plaza and Romero (2011), which shows how to find such a polynomial of degree 2n. Moreover we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n at least 2 and degree d at least 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.Comment: 18 pages, 2 figure

Independence and Alpern Multitowers

Let $T$ be any invertible, ergodic, aperiodic measure-preserving transformation of a Lebesgue probability space (X, \calB, \mu), and \P\, any finite measurable partition of $X$. We show that a (finite) Alpern multitower may always be constructed whose base is independent of \P

Navigating the Past: Brown University and the Voyage of the Slave Ship Sally, 1764-65

James T. Campbell, asssociate professor of American Civilization, Africana Studies, and History, Brown University, asks: “What happens if we see our past whole? How might we take full ownership of our history, not only of the aspects that are gracious and honorable but also of those that are grievous and horrifying? What responsibilities, if any, rest upon us in the present as inheritors of this mixed legacy? Brown’s Steering Committee on Slavery and Justice represents one institution’s attempt to answer this question.” In this essay, originally given as the keynote address for Imagining America’s 2007 conference, James Campbell examines the university’s historical implication in slavery and injustice. Campbell details fully the reliance on the slave trade of both the Brown family, for whom the university is named, and of the entire Providence business community. Slave ships departing from that port required the services of riggers, caulkers, ironwrights, distillers, butchers, bakers, candlestick makers, apothecaries, surgeons, and more. In his description of the preparations for the middle passage, Campbell draws scrupulously on historical documents to narrate the suffering, deaths, and insurrections on board one particular voyage, the Sally, in 1764-65, commissioned by the Brown family. By applying the scholarly tools of the academic trade to an encounter with Brown University’s own history and contemplating our subsequent responsibilities in the present, Campbell, on behalf of the entire Committee, invites all of us to hold our institutions accountable to their pasts

Infection-acquired versus vaccine-acquired immunity in an SIRWS model

Despite high vaccine coverage, pertussis has re-emerged as a public health concern in many countries. One hypothesis posed for re-emergence is the waning of immunity. In some disease systems, the process of waning immunity can be non-linear, involving a complex relationship between the duration of immunity and subsequent boosting of immunity through asymptomatic re-exposure. We present and analyse a model of infectious disease transmission to examine the interplay between infection and immunity. By allowing the duration of infection-acquired immunity to differ from that of vaccine-acquired immunity, we explore the impact of the difference in durations on long-term disease patterns and prevalence of infection. Our model demonstrates that vaccination may induce cyclic behaviour, and its ability to reduce the infection prevalence increases with both the duration of infection-acquired immunity and duration of vaccine-acquired immunity. We find that increasing vaccine coverage, while capable of leading to an increase in overall transmission, always results in a reduction in prevalence of primary infections, with epidemic cycles characterised by a longer interepidemic period and taller peaks. Our results show that the epidemiological patterns of an infectious disease may change considerably when the duration of vaccine-acquired immunity differs from that of infection-acquired immunity. Our study highlights that for any particular disease and associated vaccine, a detailed understanding of the duration of protection and how that duration is influenced by infection prevalence is important as we seek to optimise vaccination strategies.Comment: 21 pages, 5 figure

CD11b+ Migratory Dendritic Cells Mediate CD8 T Cell Cross-Priming and Cutaneous Imprinting after Topical Immunization

Topical antigen application is a focus of current vaccine research. This immunization route mimics natural antigen exposure across a barrier tissue and generates T cells imprinted for skin-selective homing. Soluble antigens introduced through this route require cross-presentation by DC to generate CD8 T cell responses. Here we have explored the relative contribution of various skin-derived DC subsets to cross-priming and skin-selective imprinting. In our model, DC acquire soluble Ag in vivo from immunized murine skin for cross-presentation to naïve CD8 T cells ex vivo. We find CD11b+ migratory DC to be the relevant cross-priming DC in this model. Both Langerin+ and Langerin- CD11b+ migratory DC can cross-present antigen in our system, but only the Langerin+ subset can induce expression of the skin-selective addressin E-selectin ligand. Thus, the CD11b+ Langerin+ migratory DC population, comprised primarily of Langerhans cells, both cross-primes naïve CD8 T cells and imprints them with skin-homing capabilities

Geostationary earth climate sensor: Scientific utility and feasibility, phase A

The possibility of accurate broad band radiation budget measurements from a GEO platform will provide a unique opportunity for viewing radiation processes in the atmosphere-ocean system. The CSU/TRW team has prepared a Phase 1 instrument design study demonstrating that measurements of radiation budget are practical from geosynchronous orbit with proven technology. This instrument concept is the Geostationary Earth Climate Sensor (GECS). A range of resolutions down to 20 km at the top of the atmosphere are possible, depending upon the scientific goals of the experiment. These tradeoffs of resolution and measurement repeat cycles are examined for scientific utility. The design of a flexible instrument is shown to be possible to meet the two goals: long-term, systematic monitoring of the diurnal cycles of radiation budget; and high time and space resolution studies of regional radiation features

Solar sail formation flying for deep-space remote sensing

In this paper we consider how 'near' term solar sails can be used in formation above the ecliptic plane to provide platforms for accurate and continuous remote sensing of the polar regions of the Earth. The dynamics of the solar sail elliptical restricted three-body problem (ERTBP) are exploited for formation flying by identifying a family of periodic orbits above the ecliptic plane. Moreover, we find a family of 1 year periodic orbits where each orbit corresponds to a unique solar sail orientation using a numerical continuation method. It is found through a number of example numerical simulations that this family of orbits can be used for solar sail formation flying. Furthermore, it is illustrated numerically that Solar Sails can provide stable formation keeping platforms that are robust to injection errors. In addition practical trajectories that pass close to the Earth and wind onto these periodic orbits above the ecliptic are identified

Perspectives of Pedagogical Change within a Broadcast STEM Course

As calls for pedagogical transformation of undergraduate science, technology, engineering, and mathematics (STEM) instruction intensify, the pace of change remains slow. The literature shows that research-based instructional strategies transfer only sporadically into STEM instructional practice. Difficulties associated with implementation and sustainment of instructional change may appear daunting— if not insurmountable—to many STEM change agents and teaching faculty. Subsequently, the path towards systematic and lasting pedagogical transformation in post-secondary STEM stands largely uncharted. To understand how challenges faced by STEM educators engaged in pedagogical change may be overcome, this paper uses qualitative inquiry to explore an emergent process of teacher change. The change process took place during implementation of an online innovation within an undergraduate engineering calculus course taught via synchronous broadcast at a mid-size, Western, public university. The instructional innovation required first year calculus students to participate in an asynchronous, online discussion forum for graded credit. Data, consisting of written reflections and transcribed interviews, were gathered from three STEM faculty members who each played a different role in the change process: a mathematics instructor implementing the online forum within his course; an engineering faculty peer-mentor assisting with the implementation of the online forum; and a STEM education faculty member evaluating the implementation and observing the process of change. Situated within the interpretive research paradigm, this study uses exploratory thematic analysis of narrative data to understand the ways in which contextual factors may influence pedagogical change

Quantifying quantum speedups: improved classical simulation from tighter magic monotones

Consumption of magic states promotes the stabilizer model of computation to universal quantum computation. Here, we propose three different classical algorithms for simulating such universal quantum circuits, and characterize them by establishing precise connections with a family of magic monotones. Our first simulator introduces a new class of quasiprobability distributions and connects its runtime to a generalized notion of negativity. We prove that this algorithm has significantly improved exponential scaling compared to all prior quasiprobability simulators for qubits. Our second simulator is a new variant of the stabilizer-rank simulation algorithm, extended to work with mixed states and with significantly improved runtime bounds. Our third simulator trades precision for speed by discarding negative quasiprobabilities. We connect each algorithm's performance to a corresponding magic monotone and, by comprehensively characterizing the monotones, we obtain a precise understanding of the simulation runtime and error bounds. Our analysis reveals a deep connection between all three seemingly unrelated simulation techniques and their associated monotones. For tensor products of single-qubit states, we prove that our monotones are all equal to each other, multiplicative and efficiently computable, allowing us to make clear-cut comparisons of the simulators' performance scaling. Furthermore, our monotones establish several asymptotic and non-asymptotic bounds on state interconversion and distillation rates. Beyond the theory of magic states, our classical simulators can be adapted to other resource theories under certain axioms, which we demonstrate through an explicit application to the theory of quantum coherence.Comment: 24+13 pages, 8 figures; final author copy. Since v1: restructured with additional discussion, proof sketches and examples. Since v3: minor revisions to improve clarity, additional acknowledgment