103,444 research outputs found
Tensor Lines in Tensor Fields of Arbitrary Order: Tracking Lines in Higher Order Tensor Fields
This paper presents a method to reduce time complexity of the computation of higher–order tensor lines. The method can be applied to higher–order tensors and the spherical harmonics representation, both
widely used in medical imaging. It is based on a gradient descend technique and integrates well into fiber tracking algorithms. Furthermore, the method improves the angular resolution in contrast to discrete sampling methods which is especially important to tractography, since there, small errors accumulate fast and make the result unusable. Our implementation does not interpolate derived directions but works directly on the interpolated tensor information. The specific contribution of this paper is a fast algorithm for tracking lines tensor fields of arbitrary order that increases angular resolution compared to previous approaches
Topological visualization of tensor fields using a generalized Helmholtz decomposition
Analysis and visualization of fluid flow datasets has become increasing important with the development of computer graphics. Even though many direct visualization methods have been applied in the tensor fields, those methods may result in much visual clutter. The Helmholtz decomposition has been widely used to analyze and visualize the vector fields, and it is also a useful application in the topological analysis of vector fields. However, there has been no previous work employing the Helmholtz decomposition of tensor fields. We present a method for computing the Helmholtz decomposition of tensor fields of arbitrary order and demonstrate its application. The Helmholtz decomposition can split a tensor field into divergence-free and curl-free parts. The curl-free part is irrotational, and it is useful to isolate the local maxima and minima of divergence (foci of sources and sinks) in the tensor field without interference from curl-based features. And divergence-free part is solenoidal, and it is useful to isolate centers of vortices in the tensor field. Topological visualization using this decomposition can classify critical points of two-dimensional tensor fields and critical lines of 3D tensor fields. Compared with several other methods, this approach is not dependent on computing eigenvectors, tensor invariants, or hyperstreamlines, but it can be computed by solving a sparse linear system of equations based on finite difference approximation operators. Our approach is an indirect visualization method, unlike the direct visualization which may result in the visual clutter. The topological analysis approach also generates a single separating contour to roughly partition the tensor field into irrotational and solenoidal regions. Our approach will make use of the 2nd order and the 4th order tensor fields. This approach can provide a concise representation of the global structure of the field, and provide intuitive and useful information about the structure of tensor fields. However, this method does not extract the exact locations of critical points and lines
Path Integral for Inflationary Perturbations
The quantum theory of cosmological perturbations in single field inflation is
formulated in terms of a path integral. Starting from a canonical formulation,
we show how the free propagators can be obtained from the well known
gauge-invariant quadratic action for scalar and tensor perturbations, and
determine the interactions to arbitrary order. This approach does not require
the explicit solution of the energy and momentum constraints, a novel feature
which simplifies the determination of the interaction vertices. The constraints
and the necessary imposition of gauge conditions is reflected in the appearance
of various commuting and anti-commuting auxiliary fields in the action. These
auxiliary fields are not propagating physical degrees of freedom but need to be
included in internal lines and loops in a diagrammatic expansion. To illustrate
the formalism we discuss the tree-level 3-point and 4-point functions of the
inflaton perturbations, reproducing the results already obtained by the methods
used in the current literature. Loop calculations are left for future work.Comment: (v1) 28 pages, no figures; (v2) 29 pages, minor changes, matches
published versio
Stress Tensors for Instantaneous Vacua in 1+1 Dimensions
The regularized expectation value of the stress-energy tensor for a massless
bosonic or fermionic field in 1+1 dimensions is calculated explicitly for the
instantaneous vacuum relative to any Cauchy surface (here a spacelike curve) in
terms of the length L of the curve (if closed), the local extrinsic curvature K
of the curve, its derivative K' with respect to proper distance x along the
curve, and the scalar curvature R of the spacetime: T_{00} = - epsilon
pi/(6L^2) - K^2/(24 pi), T_{01} = - K'/(12 pi), T_{11} = - epsilon pi/(6L^2) -
K^2/(24 pi) + R/(24 pi), in an orthonormal frame with the spatial vector
parallel to the curve. Here epsilon = 1 for an untwisted (i.e., periodic in x)
one-component massless bosonic field or for a twisted (i.e., antiperiodic in x)
two-component massless fermionic field, epsilon = -1/2 for a twisted
one-component massless bosonic field, and epsilon = - 2 for an untwisted
two-component massless fermionic field. The calculation uses merely the
energy-momentum conservation law and the trace anomaly (for which a very simple
derivation is also given herein, as well as the expression for the Casimir
energy of bosonic and fermionic fields twisted by an arbitrary amount in
R^{D-1} x S^1). The two coordinate and conformal invariants of a quantum state
that are (nonlocally) determined by the stress-energy tensor are given.
Applications to topologically modified deSitter spacetimes, to a flat cylinder,
and to Minkowski spacetime are discussed.Comment: LaTeX, 28 pages, last term of Eq. (79) correcte
Vacuum polarization in the spacetime of charged nonlinear black hole
Building on general formulas obtained from the approximate renormalized
effective action, the approximate stress-energy tensor of the quantized massive
scalar field with arbitrary curvature coupling in the spacetime of charged
black hole being a solution of coupled equations of nonlinear electrodynamics
and general relativity is constructed and analysed. It is shown that in a few
limiting cases, the analytical expressions relating obtained tensor to the
general renormalized stress-energy tensor evaluated in the geometry of the
Reissner-Nordstr\"{o}m black hole could be derived. A detailed numerical
analysis with special emphasis put on the minimal coupling is presented and the
results are compared with those obtained earlier for the conformally coupled
field. Some novel features of the renormalized stress-energy tensor are
discussed
Braginskii magnetohydrodynamics for arbitrary magnetic topologies: coronal applications
We investigate single-fluid magnetohydrodynamics (MHD) with anisotropic viscosity,
often referred to as Braginskii MHD, with a particular eye to solar coronal applications.
First, we examine the full Braginskii viscous tensor in the single-fluid limit. We pay
particular attention to how the Braginskii tensor behaves as the magnetic field strength
vanishes. The solar corona contains a magnetic field with a complex and evolving
topology, so the viscosity must revert to its isotropic form when the field strength is zero,
e.g. at null points. We highlight that the standard form in which the Braginskii tensor
is written is not suitable for inclusion in simulations as singularities in the individual
terms can develop. Instead, an altered form, where the parallel and perpendicular tensors
are combined, provides the required asymptotic behaviour in the weak-field limit. We
implement this combined form of the tensor into the Lare3D code, which is widely used
for coronal simulations. Since our main focus is the viscous heating of the solar corona,
we drop the drift terms of the Braginskii tensor. In a stressed null point simulation,
we discover that small-scale structures, which develop very close to the null, lead to
anisotropic viscous heating at the null itself (that is, heating due to the anisotropic
terms in the viscosity tensor). The null point simulation we present has a much higher
resolution than many other simulations containing null points so this excess heating is
a practical concern in coronal simulations. To remedy this unwanted heating at the null
point, we develop a model for the viscosity tensor that captures the most important
physics of viscosity in the corona: parallel viscosity for strong field and isotropic viscosity
at null points. We derive a continuum model of viscosity where momentum transport,
described by this viscosity model, has the magnetic field as its preferred orientation.
When the field strength is zero, there is no preferred direction for momentum transport
and viscosity reverts to the standard isotropic form. The most general viscous stress
tensor of a (single-fluid) plasma satisfying these conditions is found. It is shown that
the Braginskii model, without the drift terms, is a specialization of the general model.
Performing the stressed null point simulation with this simplified model of viscosity
reveals very similar heating profiles compared to the full Braginskii model. The new
model, however, does not produce anisotropic heating at the null point, as required.
Since the vast majority of coronal simulations use only isotropic viscosity, we perform the
stressed null point simulation with isotropic viscosity and compare the heating profiles
to those of the anisotropic models. It is shown than the fully isotropic viscosity can
over-estimate the viscous heating by an order of magnitude
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