Some mathematics for quasi-symmetry

The concept of quasi-symmetry was introduced in (Booozer, 1983) and then distilled into a design principle for stellarators by N¨uhrenberg & Zille (1988). In its strongest sense it means integrability of first-order guiding-centre motion. An excellent survey of the subject was provided by Helander (2014), assuming magnetohydrostatic (MHS) fields, that is, magnetohydrodynamic equilibrium with isotropic pressure and no mean flow. A fundamental step was made by Burby & Qin (2013), who stated necessary and sufficient local conditions for integrability of guiding-centre motion in terms of a continuous symmetry of three differential forms derived from the magnetic field and made clear that quasi-symmetry can be separated from the issue of whether the magnetic field is MHS or not. Perturbative calculations of Garren & Boozer (1991), however, make it look very likely that the only possibility for exact quasi-symmetry for MHS fields with bounded magnetic surfaces is axisymmetry. Our paper gives first steps to deciding whether or not this is true. In this paper we prove many consequences of quasi-symmetry and thereby restrictions on possible quasi-symmetric fields. In the case of a quasi-symmetric MHS field we derive a generalisation of the axisymmetric Grad-Shafranov equation. Burby & Qin (2013) built in an assumption that a quasi-symmetry must be a circleaction. Here we relax this requirement, though prove that under some mild conditions it is actually a circle-action. We write many equations using differential forms. For those unfamiliar with differential forms, (Arnol’d, 1978, chap. 7) is a classic and there is a tutorial (MacKay, 2019) specifically for plasma physicists. Throughout the paper we will assume enough smoothness that the equations we write make sense, at least in a weak sense

Generalized Grad-Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry

Regions without invariant tori of given class for the planar circular restricted three-body problem

A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for planets around a binary star are deduced. It is expected that lessons learnt from this problem will be useful for applications of the method to other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics

Approximate symmetries of guiding-centre motion

In a strong, inhomogeneous magnetic field, charged particle dynamics may be studied in the guiding-centre approximation, which is known to be Hamiltonian. When the magnetic field is quasisymmetric, the first-order guiding-centre Hamiltonian structure admits a continuous symmetry, and therefore a conserved quantity in addition to the energy. Since the first-order guiding-centre system is only an approximation, it is also interesting to consider approximate symmetries of the guiding-centre Hamiltonian structure. We find that any approximate spatial symmetry coincides with quasisymmetry at first order. For approximate phase-space symmetries, we derive weaker conditions than quasisymmetry. The latter include "weak quasisymmetry" as a subcase, recently proposed by Rodriguez et al. Our results, however, show that weak quasisymmetry is necessarily non-spatial at first order. Finally, if the magnetic field is constrained to satisfy magnetohydrostatic force balance then an approximate symmetry must agree with quasisymmetry to first order

Regions without flux surfaces of given class for magnetic fields in toroidal geometry

A converse KAM method for 3D vector fields, establishing regions through which passes no invariant 2-tori transverse to a given direction field, is tested on some helical perturbations of an axisymmetric magnetic field in toroidal geometry. It finds regions corresponding to magnetic islands and chaos for the fieldline flow. The minimization of these regions is proposed as a tool to help in the design of plasma confinement devices of tokamak and stellarator type

Corneal Confocal Microscopy: A novel noninvasive test to diagnose and stratify the severity of human diabetic neuropathy

OBJECTIVE: The accurate quantification of human diabetic neuropathy is important to define at-risk patients, anticipate deterioration, and assess new therapies. ---------- RESEARCH DESIGN AND METHODS: A total of 101 diabetic patients and 17 age-matched control subjects underwent neurological evaluation, neurophysiology tests, quantitative sensory testing, and evaluation of corneal sensation and corneal nerve morphology using corneal confocal microscopy (CCM). ---------- RESULTS: Corneal sensation decreased significantly (P = 0.0001) with increasing neuropathic severity and correlated with the neuropathy disability score (NDS) (r = 0.441, P 3) defined an NFD of 6) defined a NFD cutoff of <20.8/mm2 with a sensitivity of 0.71 (0.42–0.92) and specificity of 0.64 (0.54–0.74). ---------- CONCLUSIONS: CCM is a noninvasive clinical technique that may be used to detect early nerve damage and stratify diabetic patients with increasing neuropathic severity. Established diabetic neuropathy leads to pain and foot ulceration. Detecting neuropathy early may allow intervention with treatments to slow or reverse this condition (1). Recent studies suggested that small unmyelinated C-fibers are damaged early in diabetic neuropathy (2–4) but can only be detected using invasive procedures such as sural nerve biopsy (4,5) or skin-punch biopsy (6–8). Our studies have shown that corneal confocal microscopy (CCM) can identify early small nerve fiber damage and accurately quantify the severity of diabetic neuropathy (9–11). We have also shown that CCM relates to intraepidermal nerve fiber loss (12) and a reduction in corneal sensitivity (13) and detects early nerve fiber regeneration after pancreas transplantation (14). Recently we have also shown that CCM detects nerve fiber damage in patients with Fabry disease (15) and idiopathic small fiber neuropathy (16) when results of electrophysiology tests and quantitative sensory testing (QST) are normal. In this study we assessed corneal sensitivity and corneal nerve morphology using CCM in diabetic patients stratified for the severity of diabetic neuropathy using neurological evaluation, electrophysiology tests, and QST. This enabled us to compare CCM and corneal esthesiometry with established tests of diabetic neuropathy and define their sensitivity and specificity to detect diabetic patients with early neuropathy and those at risk of foot ulceration

Managing digital coordination of design: emerging hybrid practices in an institutionalized project setting

What happens when digital coordination practices are introduced into the institutionalized setting of an engineering project? This question is addressed through an interpretive study that examines how a shared digital model becomes used in the late design stages of a major station refurbishment project. The paper contributes by mobilizing the idea of ‘hybrid practices’ to understand the diverse patterns of activity that emerge to manage digital coordination of design. It articulates how engineering and architecture professions develop different relationships with the shared model; the design team negotiates paper-based practices across organizational boundaries; and diverse practitioners probe the potential and limitations of the digital infrastructure. While different software packages and tools have become linked together into an integrated digital infrastructure, these emerging hybrid practices contrast with the interactions anticipated in practice and policy guidance and presenting new opportunities and challenges for managing project delivery. The study has implications for researchers working in the growing field of empirical work on engineering project organizations as it shows the importance of considering, and suggests new ways to theorise, the introduction of digital coordination practices into these institutionalized settings

The liminality of trajectory shifts in institutional entrepreneurship

In this paper, we develop a process model of trajectory shifts in institutional entrepreneurship. We focus on the liminal periods experienced by institutional entrepreneurs when they, unlike the rest of the organization, recognize limits in the present and seek to shift a familiar past into an unfamiliar and uncertain future. Such periods involve a situation where the new possible future, not yet fully formed, exists side-by-side with established innovation trajectories. Trajectory shifts are moments of truth for institutional entrepreneurs, but little is known about the underlying mechanisms of how entrepreneurs reflectively deal with liminality to conceive and bring forth new innovation trajectories. Our in-depth case study research at CarCorp traces three such mechanisms (reflective dissension, imaginative projection, and eliminatory exploration) and builds the basis for understanding the liminality of trajectory shifts. The paper offers theoretical implications for the institutional entrepreneurship literature

The reuse of digital computer data: Transformation, recombination and generation of data mixes in big data science

This is the author accepted manuscript. the final version is available on open access from Springer via the DOI in this recordThis chapter is concerned with the relationship between the materiality of digital computer data and their reuse in scientific practice. It builds on the case study of a ‘data mash-up’ infrastructure for research with environmental, weather and population health data. I problematise the extent to which scientists reusing digital computer data heavily manipulate the sources through complex and situated calculative operations, as they attempt to re-situate data well beyond the epistemic community in which they originated, and adapt them to different theoretical frameworks, methods and evidential standards. The chapter interrogates the consequent relationship between derivative data and the data sources from which they originate. The deep relationality of scientific computer data is multi-layered and scaffolded, as it depends on relations between various kinds of data, computing technologies, assumptions, theoretical scaffoldings, hypotheses and other features of the situation at hand.European Research Council (ERC)Engineering and Physical Sciences Research Council (EPSRC