On Waring's problem: two squares and three biquadrates

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott-Halberstam Conjecture, we show that all large natural numbers n with 8 not dividing n, n not congruent to 2 modulo 3, and n not congruent to 14 modulo 16, are the sum of 2 squares and 3 biquadrates.Comment: to appear in Mathematik

Black Hole Spin Properties of 130 AGN

Supermassive black holes may be described by their mass and spin. When supermassive black holes are active, the activity provides a probe of the state of the black hole system. The spin of a hole can be estimated when the black hole mass and beam power of the source are known for sources with powerful outflows. Seventy-five sources for which both the black hole mass and beam power could be obtained are identified and used to obtain estimates of black hole spins. The 75 supermassive black holes studied include 52 FRII radio galaxies and 23 FRII radio loud quasars with redshifts ranging from about zero to two. The new values are combined with those obtained previously for 19 FRII radio galaxies, 7 FRII radio loud quasars, and 29 radio sources associated with CD galaxies to form samples of 71 FRII radio galaxies, 30 FRII quasars, and a total sample of 130 spin values; all of the sources are associated with massive elliptical galaxies. The new values obtained are similar to those obtained earlier at similar redshift, and range from about 0.1 to 1 for FRII sources. The overall results are consistent with those obtained previously: the spins tend to decrease with decreasing redshift for the FRII sources studied. There is a hint that the range of values of black hole spin at a given redshift is larger for FRII quasars than for FRII radio galaxies. There is no indication of a strong correlation between supermassive black hole mass and spin for the supermassive black holes studied here. The relation between beam power and black hole mass is obtained and used as a diagnostic of the outflows and the dependence of the magnetic field strength on black hole mass.Comment: 12 pages, 12 figures, 5 table

Evaluating the use of particle-spring systems in the conceptual design of grid shell structures

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 63-64).This thesis evaluates particle-spring systems as conceptual design tools in an effort to create efficient grid shell structures. Currently many simulation tools are available to create representations of intricate geometries and forms. However, these forms can become highly complex and challenging upon their realization. A lack of understanding of these forms leads to structures that cannot support their corresponding loads due to their shape, boundary conditions or edge conditions. To create successful grid shells, designers must understand the design principles behind these forms. The goals of this research were achieved through a parametric study that involved manipulating the topology and topography of three global grid shell geometries. It was determined that the ability of particle-spring form finding methods to create good structures is highly dependent on both the mesh type used and the structure's global geometry. A list of implications has been developed and is presented in this work.by Trevor B. Bertin.M.Eng

Anomalous subdiffusion with multispecies linear reaction dynamics

We have introduced a set of coupled fractional reaction-diffusion equations to model a multispecies system undergoing anomalous subdiffusion with linear reaction dynamics. The model equations are derived from a mesoscopic continuous time random walk formulation of anomalously diffusing species with linear mean field reaction kinetics. The effect of reactions is manifest in reaction modified spatiotemporal diffusion operators as well as in additive mean field reaction terms. One consequence of the nonseparability of reaction and subdiffusion terms is that the governing evolution equation for the concentration of one particular species may include both reactive and diffusive contributions from other species. The general solution is derived for the multispecies system and some particular special cases involving both irreversible and reversible reaction dynamics are analyzed in detail. We have carried out Monte Carlo simulations corresponding to these special cases and we find excellent agreement with theory

Knot Graphs

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes

Student understanding of the Boltzmann factor

We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable, nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations on student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions

Natural Regulation of Energy Flow in a Green Quantum Photocell

Manipulating the flow of energy in nanoscale and molecular photonic devices is of both fundamental interest and central importance for applications in light harvesting optoelectronics. Under erratic solar irradiance conditions, unregulated power fluctuations in a light harvesting photocell lead to inefficient energy storage in conventional solar cells and potentially fatal oxidative damage in photosynthesis. Here, we show that regulation against these fluctuations arises naturally within a two-channel quantum heat engine photocell, thus enabling the efficient conversion of varying incident solar spectrum at Earth's surface. Remarkably, absorption in the green portion of the spectrum is avoided, as it provides no inherent regulatory benefit. Our findings illuminate a quantum structural origin of regulation, provide a novel optoelectronic design strategy, and may elucidate the link between photoprotection in photosynthesis and the predominance of green plants on Earth.Comment: 17 pages, 4 figure

Decline and fall:a biological, developmental, and psycholinguistic account of deliberative language processes and ageing

Background: This paper reviews the role of deliberative processes in language: those language processes that require central resources, in contrast to the automatic processes of lexicalisation, word retrieval, and parsing. 10 Aims: We describe types of deliberative processing, and show how these processes underpin high-level processes that feature strongly in language. We focus on metalin- guistic processing, strategic processing, inhibition, and planning. We relate them to frontal-lobe function and the development of the fronto-striate loop. We then focus on the role of deliberative processes in normal and pathological development and ageing, 15 and show how these processes are particularly susceptible to deterioration with age. In particular, many of the commonly observed language impairments encountered in ageing result from a decline in deliberative processing skills rather than in automatic language processes. Main Contribution: We argue that central processing plays a larger and more important 20 role in language processing and acquisition than is often credited. Conclusions: Deliberative language processes permeate language use across the lifespan. They are particularly prone to age-related loss. We conclude by discussing implications for therapy