The theory and applications of writhing

Writhe measures the extent to which a curve is kinked and coiled about itself in space. It has generally been expressed as a double integral. This measure can be interpreted as the average number of signed crossings seen by each viewer, over all possible viewpoints of the curve. This simple geometrical interpretation is used to describe the established properties of the writhe, as applied to closed spacecurves. These descriptions differ from previous work as they do not require the construction of an artificial ribbon structure. A major feature of this thesis concerns the evaluation of the writhe along a preferred direction. A directional measure termed the polar writhe will be developed which can be applied to generic curves (open or closed) . This single integral expression is shown to be equivalent to the double integral writhe measure for closed curves. However for open curves the two measures are shown to differ. Further, it is shown that the polar writhe has distinct advantages when analysing curves with a strong directional bias. The thesis then discusses in detail the properties of both the writhe and the polar writhe measures for open curves. The use of artificial closures for both measures is examined. In the case of the writhe a new closure is defined that allows the evaluation of the writhe using single integral expression via the theorems of Fuller. This closure is unique in that it can be applied to open curves whose end points are in general position. A simple expression for calculating the non-local polar writhe is derived which generalises a closed curve expression defined in (Berger Prior J. Phys. A: Math. Gen. 39, 8321-8348, (2006)). A quantitative study on the effect of the choice of evaluation direction of the polar writhe is conducted. The polar writhe formulation is applied to a simple linear force-free magnetic field model where the field lines form loops above a boundary plane. Loops with a sufficient amount of kinking are generally seen to form S or inverse S (Z) shaped structures. Such field lines structures are commonly observed in the Sun’s corona. A popular measure of the field line morphology is the magnetic helicity. We use the polar writhe, the correct form for the writhe helicity in the coronal region, to challenge some popular assumptions of the field. Firstly, the writhe of field lines of significant aspect ratio (the apex height divided by the foot point width) can often have the opposite sign to that assumed in a recent review paper by Green et al (Solar Phys., 365-391, (2007)). Secondly, we demonstrate the possibility of field lines forming apparent Z shaped structures which are in fact constructed from a pair of S shapes and have a writhe sign expected of an S shaped structure. Such field lines could be misinterpreted without full knowledge of the line’s three dimensional structure. Thirdly, we show that much of the interesting morphological behaviour occurs for field lines located next to separatrices

The evaluation of directionally writhing polymers

We discuss the appropriate techniques for modelling the geometry of open ended elastic polymer molecules. The molecule is assumed to have fixed endpoints on a boundary surface. In particular we discuss the concept of the winding number, a directional measure of the linking of two curves, which can be shown to be invariant to the set of continuous deformations vanishing at the polymer's end-point and which forbid it from passing through itself. This measure is shown to be the appropriate constraint required to evaluate the geometrical properties of a constrained DNA molecule. Using the net winding measure we define a model of an open ended constrained DNA molecule which combines the necessary constraint of self-avoidance with being analytically tractable. This model builds upon the local models of Bouchiat and Mezard (2000). In particular, we present a new derivation of the polar writhe expression, which detects both the local winding of the curve and non local winding between different sections of the curve. We then show that this expression correctly tracks the net twisting of a DNA molecule subject to rotation at the endpoints, unlike other definitions used in the literature

The emergence of braided magnetic fields

We study the emergence of braided magnetic fields from the top of the solar interior through to the corona. It is widely believed that emerging regions smaller than active regions are formed in the upper convection zone near the photosphere. Here, bundles of braided, rather than twisted, magnetic field can be formed, which then rise upward to emerge into the atmosphere. To test this theory, we investigate the behaviour of braided magnetic fields as they emerge into the solar atmosphere. We compare and contrast our models to previous studies of twisted flux tube emergence and discuss results that can be tested observationally. Although this is just an initial study, our results suggest that the underlying magnetic field structure of small emerging regions need not be twisted and that braided field, formed in the convection zone, could suffice

Above- and below-ground biomass accumulation, production, and distribution of sweetgum and loblolly pine grown with irrigation and fertilization.

Abstract: Increased forest productivity has been obtained by improving resource availability through water and nutrient amendments. However, more stress-tolerant species that have robust site requirements do not respond consistently to irrigation. An important factor contributing to robust site requirements may be the distribution of biomass belowground, yet available information is limited. We examined the accumulation and distribution of above- and below-ground biomass in sweetgum (Liqrridambar sfyrac$lua L.) and loblolly pine (Pinus taeda L.) stands receiving irrigation and fertilization. Mean annual aboveground production after 4 years ranged from 2.4 to 5.1 ~g.ha-'.year' for sweetgum and from 5.0 to 6.9 ~g.ha-l.year-l for pine. Sweetgum responded positively to irrigation and fertilization with an additive response to irrigation + fertilization. Pine only responded to fertilization. Sweetgum root mass fraction (RME)in creased with fertilization at 2 years and decreased with fertilization at 4 years. There were no detectable treatment differences in loblolly pine RMF. Development explained from 67% to 98% of variation in shoot versus root allometry for ephemeral and perennial tissues, fertilization explained no more than 5% of the variation in for either species, and irrigation did not explain any. We conclude that shifts in allocation from roots to shoots do not explain nutrient-induced growth stimulations

2022 EULAR points to consider for remote care in rheumatic and musculoskeletal diseases

Background: Remote care and telehealth have the potential to expand healthcare access, and the COVID-19 pandemic has called for alternative solutions to conventional face-to-face follow-up and monitoring. However, guidance is needed on the integration of telehealth into clinical care of people with rheumatic and musculoskeletal diseases (RMD).  Objective: To develop EULAR points to consider (PtC) for the development, prioritisation and implementation of telehealth for people with RMD.  Methods: A multidisciplinary EULAR task force (TF) of 30 members from 14 European countries was established, and the EULAR standardised operating procedures for development of PtC were followed. A systematic literature review was conducted to support the TF in formulating the PtC. The level of agreement among the TF was established by anonymous online voting.  Results: Four overarching principles and nine PtC were formulated. The use of telehealth should be tailored to patient's needs and preferences. The healthcare team should have adequate equipment and training and have telecommunication skills. Telehealth can be used in screening for RMD as preassessment in the referral process, for disease monitoring and regulation of medication dosages and in some non-pharmacological interventions. People with RMD should be offered training in using telehealth, and barriers should be resolved whenever possible. The level of agreement to each statement ranged from 8.5 to 9.8/10.  Conclusion :The PtC have identified areas where telehealth could improve quality of care and increase healthcare access. Knowing about drivers and barriers of telehealth is a prerequisite to successfully establish remote care approaches in rheumatologic clinical practice

Measurement of the νe and total 8B solar neutrino fluxes with the Sudbury Neutrino Observatory phase-III data set

This paper details the solar neutrino analysis of the 385.17-day phase-III data set acquired by the Sudbury Neutrino Observatory (SNO). An array of 3He proportional counters was installed in the heavy-water target to measure precisely the rate of neutrino-deuteron neutral-current interactions. This technique to determine the total active 8B solar neutrino flux was largely independent of the methods employed in previous phases. The total flux of active neutrinos was measured to be 5.54-0.31+0.33(stat.)-0.34+0.36(syst.)×106 cm-2 s-1, consistent with previous measurements and standard solar models. A global analysis of solar and reactor neutrino mixing parameters yielded the best-fit values of Δm2=7.59-0.21+0.19×10 -5eV2 and θ=34.4-1.2+1.3degrees

On the helicity of open magnetic fields

We reconsider the topological interpretation of magnetic helicity for magnetic fields in open domains, and relate this to the relative helicity. Specifically, our domains stretch between two parallel planes, and each of these ends may be magnetically open. It is demonstrated that, while the magnetic helicity is gauge-dependent, its value in any gauge may be physically interpreted as the average winding number among all pairs of field lines with respect to some orthonormal frame field. In fact, the choice of gauge is equivalent to the choice of reference field in the relative helicity, meaning that the magnetic helicity is no less physically meaningful. We prove that a particular gauge always measures the winding with respect to a fixed frame, and propose that this is normally the best choice. For periodic fields, this choice is equivalent to measuring relative helicity with respect to a potential reference field. However, for aperiodic fields, we show that the potential field can be twisted. We prove by construction that there always exists a possible untwisted reference field

The extended polar writhe: a tool for open curves mechanics

A measure of the writhing of a curve is introduced and is used to extend the C?lug?reanu? decomposition for closed curves, as well as the polar decomposition for curves bound between planes. The new writhe measure is also shown to be able to assess changes in linking due to belt-trick and knotting type deformations, and further its utility is illustrated on examples taken from elastic rod parameter-continuation studies. Finally C++ and mathematica codes are made available and shown to be faster than existing algorithms for the numerical computation of the writhe

Spherical winding and helicity

In ideal magnetohydrodynamics, magnetic helicity is a conserved dynamical quantity and a topological invariant closely related to Gauss linking numbers. However, for open magnetic fields with non-zero boundary components, the latter geometrical interpretation is complicated by the fact that helicity varies with non-unique choices of a field's vector potential or gauge. Evaluated in a particular gauge called the winding gauge, open-field helicity in Cartesian slab domains has been shown to be the average flux-weighted pairwise winding numbers of field lines, a measure constructed solely from field configurations that manifest its topological origin. In this paper, we derive the spherical analogue of the winding gauge and the corresponding winding interpretation of helicity, in which we formally define the concept of spherical winding of curves. Using a series of examples, we demonstrate novel properties of spherical winding and the validity of spherical winding helicity. We further argue for the canonical status of the winding gauge choice among all vector potentials for magnetic helicity by exhibiting equivalences between local coordinate changes and gauge transformations