Toroidal Decomposition Research Articles

Concordance of decompositions given by defining sequences

In this paper, we study the concordance and cobordism of decompositions associated with defining sequences and we relate them to some invariants of toroidal decompositions and to the cobordism of homology manifolds. These decompositions are often wild Cantor sets and they arise as nested intersections of knotted solid tori. We show that there are at least uncountably many concordance classes of such decompositions in the 3-sphere.

A generalized poloidal–toroidal decomposition and an absolute measure of helicity

In fluid mechanics and magneto-hydrodynamics it is often useful to decompose a vector field into poloidal and toroidal components. In a spherical geometry, the poloidal component contains all of the radial part of the field, while the curl of the toroidal component contains all of the radial current. This paper explores how they work in more general geometries, where space is foliated by nested simply connected surfaces. Vector fields can still be divided into poloidal and toroidal components, but in geometries lacking spherical symmetry it makes sense to further divide the poloidal field into a standard part and a ‘shape’ term, which in itself behaves like a toroidal field and arises from variations in curvature.The generalised P–T decomposition leads to a simple definition of helicity which does not rely on subtracting the helicity of a potential reference field. Instead, the helicity measures the net linking of the standard poloidal field with the toroidal field as well as the new shape field. This helicity is consistent with the relative helicity in spherical and planar geometries. Its time derivative due to motion of field lines in a surface has a simple and intuitively pleasing form.

Efficient use of mobile devices for quantification of pressure injury images.

Pressure Injuries are chronic wounds that are formed due to the constriction of the soft tissues against bone prominences. In order to assess these injuries, the medical personnel carry out the evaluation and diagnosis using visual methods and manual measurements, which can be inaccurate and may generate discomfort in the patients. By using segmentation techniques, the Pressure Injuries can be extracted from an image and accurately parameterized, leading to a correct diagnosis. In general, these techniques are based on the solution of differential equations and the involved numerical methods are demanding in terms of computational resources. In previous work, we proposed a technique developed using toroidal parametric equations for image decomposition and segmentation without solving differential equations. In this paper, we present the development of a mobile application useful for the non-contact assessment of Pressure Injuries based on the toroidal decomposition from images. The usage of this technique allows us to achieve an accurate segmentation almost 8 times faster than Active Contours without Edges (ACWE) and Dynamic Contours methods. We describe the techniques and the implementation for Android devices using Python and Kivy. This application allows for the segmentation and parameterization of injuries, obtain relevant information for the diagnosis and tracking the evolution of patient’s injuries.

Simplicial volume of links from link diagrams

AbstractThe hyperbolic volume of a link complement is known to be unchanged when a half-twist is added to a link diagram, and a suitable 3-punctured sphere is present in the complement. We generalise this to the simplicial volume of link complements by analysing the corresponding toroidal decompositions. We then use it to prove a refined upper bound for the volume in terms of twists of various lengths for links.

The toroidal decomposition of the vector potential of a magnetic field and its applications

The toroidal (orthogonal) decomposition of the vector potential of a magnetic field is proposed. Based on this decomposition, equations of electrodynamics of non-varying and varying electromagnetic fields are derived. Applications of the developed theory to natural phenomena and laboratory experiments are discussed.

Can We Determine Electric Fields and Poynting Fluxes from Vector Magnetograms and Doppler Measurements?

The availability of vector-magnetogram sequences with sufficient accuracy and cadence to estimate the temporal derivative of the magnetic field allows us to use Faraday’s law to find an approximate solution for the electric field in the photosphere, using a Poloidal–Toroidal Decomposition (PTD) of the magnetic field and its partial time derivative. Without additional information, however, the electric field found from this technique is under-determined – Faraday’s law provides no information about the electric field that can be derived from the gradient of a scalar potential. Here, we show how additional information in the form of line-of-sight Doppler-flow measurements, and motions transverse to the line-of-sight determined with ad-hoc methods such as local correlation tracking, can be combined with the PTD solutions to provide much more accurate solutions for the solar electric field, and therefore the Poynting flux of electromagnetic energy in the solar photosphere. Reliable, accurate maps of the Poynting flux are essential for quantitative studies of the buildup of magnetic energy before flares and coronal mass ejections.

ALFVÉN SIMPLE WAVES: EULER POTENTIALS AND MAGNETIC HELICITY

The magnetic helicity characteristics of fully nonlinear, multi-dimensional Alfven simple waves are investigated, by using relative helicity formulae and also by using an approach involving poloidal and toroidal decomposition of the magnetic field and magnetic vector potential. Different methods to calculate the magnetic vector potential are used, including the homotopy and Biot-Savart formulae. Two basic Alfven modes are identified: (1) the plane one-dimensional Alfven simple wave given in standard texts, in which the Alfven wave propagates along the z-axis with wave phase {psi} = k{sub 0}(z - {lambda}t), where k{sub 0} is the wave number and {lambda} is the group velocity of the wave and (2) the generalized Barnes simple Alfven wave in which the wave normal n moves in a circle in the xy-plane perpendicular to the mean field, which is directed along the z-axis. The plane Alfven wave (1) is analogous to the slab Alfven mode and the generalized Barnes solution (2) is analogous to the two-dimensional mode in Alfvenic, incompressible turbulence. The helicity characteristics of these two basic Alfven modes are distinct. The helicity characteristics of more general multi-dimensional simple Alfven waves are also investigated. Applications to nonlinear Alfvenic fluctuations and structures observed in the solar windmore » are discussed.« less

An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.

A spectral method for the wave equation of divergence-free vectors and symmetric tensors inside a sphere

The wave equation for vectors and symmetric tensors in spherical coordinates is studied under the divergence-free constraint. We describe a numerical method, based on the spectral decomposition of vector/tensor components onto spherical harmonics, that allows for the evolution of only those scalar fields which correspond to the divergence-free degrees of freedom of the vector/tensor. The full vector/tensor field is recovered at each time-step from these two (in the vector case), or three (symmetric tensor case) scalar fields, through the solution of a first-order system of ordinary differential equations (ODE) for each spherical harmonic. The correspondence with the poloidal–toroidal decomposition is shown for the vector case. Numerical tests are presented using an explicit Chebyshev-tau method for the radial coordinate.

Poloidal–toroidal decomposition in a finite cylinder: II. Discretization, regularization and validation

The Navier–Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous calculations of axisymmetric vortex breakdown and of onset of non-axisymmetric helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective.

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

The Navier–Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number.

The shrinkability of bing-whitehead decompositions

THIS PAPER is a study of a special class of toroidal decompositions of 3-manifolds called Bing-Whitehead decompositions. It is well known that a Bing-Whitehead decomposition of a 3-manifold is shrinkable if all successive stages are Bing nested; but it is not shrinkable if all successive stages are Whitehead nested. (See Figs 1 and 2.) Consider a Bing-Whitehead decomposition of a 3-manifold which is defined by h, successive Bing nested stages. followed by 1 Whitehead nested stage, followed by h, successive Bing nested stages, followed by 1 Whitehead nested stage, . . The principal result of this paper is that this decomposition is shrinkable if and only if

Toroidal decompositions of $S^3$ and a family of 3-dimensional ANR's (AR's)

Toroidal decompositions of $S^3$ and a family of 3-dimensional ANR's (AR's)

Spheroidal Decompositions of E 4

This paper investigates a generalization to E4 of the notion of toroidal decomposition of E3. A certain type of this kind of upper semicontinuous decomposition is shown to be shrinkable and hence yield E4 as its decomposition space.

Spheroidal decompostions of $E\sp{4}$

This paper investigates a generalization to ${E^4}$ of the notion of toroidal decomposition of ${E^3}$. A certain type of this kind of upper semicontinuous decomposition is shown to be shrinkable and hence yield ${E^4}$ as its decomposition space.

Toroidal decompositions of $E^3$ which yield $E^3$

Toroidal decompositions of $E^3$ which yield $E^3$

Toroidal decompositions of Ea

Toroidal decompositions of Ea