Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

A method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system ${r,\theta,\phi}$. The time harmonic displacement field $\mathbf{u}(r,\theta ,\phi)$ is expanded in a separation of variables form with dependence on $\theta,\phi$ described by vector spherical harmonics with $r$-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as $\mathbf{u}(r,\theta)$, admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.Comment: 15 page

The "Artificial Mathematician" Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Reuben Hersh confided to us that, about forty years ago, the late Paul Cohen predicted to him that at some unspecified point in the future, mathematicians would be replaced by computers. Rather than focus on computers replacing mathematicians, however, our aim is to consider the (im)possibility of human mathematicians being joined by “artificial mathematicians” in the proving practice—not just as a method of inquiry but as a fellow inquirer

Development of a core outcome set for evaluative research into paediatric cerebral visual impairment (CVI), in the UK and Eire

This is the final version. Available on open access from BMJ Publishing Group via the DOI in this recordData availability statement: Data are available on reasonable request.Objectives Cerebral visual impairment (CVI) comprises a heterogeneous group of brain-related vision problems. A core outcome set (COS) represents the most important condition-specific outcomes according to patients, carers, professionals and researchers. We aimed to produce a COS for studies evaluating interventions for children with CVI, to increase the relevance of research for families and professionals and thereby to improve outcomes for affected children.Design We used methods recommended by the Core Outcome Measures in Effectiveness Trials Initiative. These included a proportionate literature review of outcomes used in previous studies; qualitative interviews with children and families; a two-round Delphi survey involving parents, children and professionals and a consensus meeting to ratify the most important outcomes.Setting Telephone interviews and online Delphi surveys of participants who all lived in UK or Eire.Participants Eighteen parents and six young people were interviewed. Delphi participants (n=80 did both rounds) included professionals working with children who have CVI (teachers, orthoptists, ophthalmologists, optometrists, qualified teachers for visually impaired, family members (parents and siblings) and affected children.Results The literature review included 13 studies yielding 37 outcomes. Qualitative interviews provided 22 outcomes. After combining and refining similar items, the first round contained 23 outcomes and the second 46. At the consensus meeting, 5 attendees recommended 27 outcomes for inclusion in the CVI COS, of which 15 were ratified as most important, including 4 related to vision; 1 to family well-being; 1 to adults around the child being informed about CVI and the rest to the child’s abilities to engage with people and surroundings.Conclusions Good engagement from participants led to the development of a COS. Future research will be useful to identify the best ways to measure COS items and potentially to update this COS as more interventions for CVI are developed.Trial registration number ISRCTN13762177.Data are available on reasonable request.National Institute for Health Research (NIHR

Ordering in voter models on networks: Exact reduction to a single-coordinate diffusion

We study the voter model and related random-copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random-copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales, and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are hard to access by existing theoretical approaches.Comment: 37 pages, 10 figures. Revised version with additional discussion and simulation results to appear in J Phys

Byzantine Gathering in Networks

This paper investigates an open problem introduced in [14]. Two or more mobile agents start from different nodes of a network and have to accomplish the task of gathering which consists in getting all together at the same node at the same time. An adversary chooses the initial nodes of the agents and assigns a different positive integer (called label) to each of them. Initially, each agent knows its label but does not know the labels of the other agents or their positions relative to its own. Agents move in synchronous rounds and can communicate with each other only when located at the same node. Up to f of the agents are Byzantine. A Byzantine agent can choose an arbitrary port when it moves, can convey arbitrary information to other agents and can change its label in every round, in particular by forging the label of another agent or by creating a completely new one. What is the minimum number M of good agents that guarantees deterministic gathering of all of them, with termination? We provide exact answers to this open problem by considering the case when the agents initially know the size of the network and the case when they do not. In the former case, we prove M=f+1 while in the latter, we prove M=f+2. More precisely, for networks of known size, we design a deterministic algorithm gathering all good agents in any network provided that the number of good agents is at least f+1. For networks of unknown size, we also design a deterministic algorithm ensuring the gathering of all good agents in any network but provided that the number of good agents is at least f+2. Both of our algorithms are optimal in terms of required number of good agents, as each of them perfectly matches the respective lower bound on M shown in [14], which is of f+1 when the size of the network is known and of f+2 when it is unknown

Phase separation transition in liquids and polymers induced by electric field gradients

Spatially uniform electric fields have been used to induce instabilities in liquids and polymers, and to orient and deform ordered phases of block-copolymers. Here we discuss the demixing phase transition occurring in liquid mixtures when they are subject to spatially nonuniform fields. Above the critical value of potential, a phase-separation transition occurs, and two coexisting phases appear separated by a sharp interface. Analytical and numerical composition profiles are given, and the interface location as a function of charge or voltage is found. The possible influence of demixing on the stability of suspensions and on inter-colloid interaction is discussed.Comment: 7 pages, 3 figures. Special issue of the J. Phys. Soc. Ja

Massively parallel computing on an organic molecular layer

Current computers operate at enormous speeds of ~10^13 bits/s, but their principle of sequential logic operation has remained unchanged since the 1950s. Though our brain is much slower on a per-neuron base (~10^3 firings/s), it is capable of remarkable decision-making based on the collective operations of millions of neurons at a time in ever-evolving neural circuitry. Here we use molecular switches to build an assembly where each molecule communicates-like neurons-with many neighbors simultaneously. The assembly's ability to reconfigure itself spontaneously for a new problem allows us to realize conventional computing constructs like logic gates and Voronoi decompositions, as well as to reproduce two natural phenomena: heat diffusion and the mutation of normal cells to cancer cells. This is a shift from the current static computing paradigm of serial bit-processing to a regime in which a large number of bits are processed in parallel in dynamically changing hardware.Comment: 25 pages, 6 figure

The effect of pose variability and repeated reliability of segmental centres of mass acquisition when using 3D photonic scanning

Three-dimensional (3D) photonic scanning is an emerging technique to acquire accurate body segment parameter data. This study established the repeated reliability of segmental centres of mass when using 3D photonic scanning (3DPS). Seventeen male participants were scanned twice by a 3D whole-body laser scanner. The same operators conducted the reconstruction and segmentation processes to obtain segmental meshes for calculating the segmental centres of mass. The segmental centres of mass obtained from repeated 3DPS were compared by relative technical error of measurement (TEM). Hypothesis tests were conducted to determine the size of change required for each segment to be determined a true variation. The relative TEMs for all segments were less than 5%. The relative changes in centres of mass at ±1.5% for most segments can be detected (p < 0.05). The arm segments which are difficult to keep in the same scanning pose generated more error than other segments

First Light Measurements of Capella with the Low Energy Transmission Grating Spectrometer aboard the Chandra X-ray Observatory

We present the first X-ray spectrum obtained by the Low Energy Transmission Grating Spectrometer (LETGS) aboard the Chandra X-ray Observatory. The spectrum is of Capella and covers a wavelength range of 5-175 A (2.5-0.07 keV). The measured wavelength resolution, which is in good agreement with ground calibration, is $\Delta \lambda \simeq$ 0.06 A (FWHM). Although in-flight calibration of the LETGS is in progress, the high spectral resolution and unique wavelength coverage of the LETGS are well demonstrated by the results from Capella, a coronal source rich in spectral emission lines. While the primary purpose of this letter is to demonstrate the spectroscopic potential of the LETGS, we also briefly present some preliminary astrophysical results. We discuss plasma parameters derived from line ratios in narrow spectral bands, such as the electron density diagnostics of the He-like triplets of carbon, nitrogen, and oxygen, as well as resonance scattering of the strong Fe XVII line at 15.014 A.Comment: 4 pages (ApJ letter LaTeX), 2 PostScript figures, accepted for publication in ApJ Letters, 200