26 research outputs found
Static axisymmetric space-times with prescribed multipole moments
In this article we develop a method of finding the static axisymmetric
space-time corresponding to any given set of multipole moments. In addition to
an implicit algebraic form for the general solution, we also give a power
series expression for all finite sets of multipole moments. As conjectured by
Geroch we prove in the special case of axisymmetry, that there is a static
space-time for any given set of multipole moments subject to a (specified)
convergence criterion. We also use this method to confirm a conjecture of
Hernandez-Pastora and Martin concerning the monopole-quadropole solution.Comment: 14 page
Calculation of, and bounds for, the multipole moments of stationary spacetimes
In this paper the multipole moments of stationary asymptotically flat
spacetimes are considered. We show how the tensorial recursion of Geroch and
Hansen can be replaced by a scalar recursion on R^2. We also give a bound on
the multipole moments. This gives a proof of the "necessary part" of a long
standing conjecture due to Geroch.Comment: 11 page
Static spacetimes with prescribed multipole moments; a proof of a conjecture by Geroch
In this paper we give sufficient conditions on a sequence of multipole
moments for a static spacetime to exist with precisely these moments. The proof
is constructive in the sense that a metric having prescribed multipole moments
up to a given order can be calculated. Since these sufficient conditions agree
with already known necessary conditions, this completes the proof of a long
standing conjecture due to Geroch.Comment: 29 page
Bayesian uncertainty quantification in linear models for diffusion MRI
Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue
microstructure. By fitting a model to the dMRI signal it is possible to derive
various quantitative features. Several of the most popular dMRI signal models
are expansions in an appropriately chosen basis, where the coefficients are
determined using some variation of least-squares. However, such approaches lack
any notion of uncertainty, which could be valuable in e.g. group analyses. In
this work, we use a probabilistic interpretation of linear least-squares
methods to recast popular dMRI models as Bayesian ones. This makes it possible
to quantify the uncertainty of any derived quantity. In particular, for
quantities that are affine functions of the coefficients, the posterior
distribution can be expressed in closed-form. We simulated measurements from
single- and double-tensor models where the correct values of several quantities
are known, to validate that the theoretically derived quantiles agree with
those observed empirically. We included results from residual bootstrap for
comparison and found good agreement. The validation employed several different
models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI)
and Constrained Spherical Deconvolution (CSD). We also used in vivo data to
visualize maps of quantitative features and corresponding uncertainties, and to
show how our approach can be used in a group analysis to downweight subjects
with high uncertainty. In summary, we convert successful linear models for dMRI
signal estimation to probabilistic models, capable of accurate uncertainty
quantification.Comment: Added results from a group analysis and a comparison with residual
bootstra
Orientationally-averaged diffusion-attenuated magnetic resonance signal for locally-anisotropic diffusion
Diffusion-attenuated MR signal for heterogeneous media has been represented
as a sum of signals from anisotropic Gaussian sub-domains. Any effect of
macroscopic (global or ensemble) anisotropy in the signal can be removed by
averaging the signal values obtained by differently oriented experimental
schemes. The resulting average signal is identical to what one would get if the
micro-domains are isotropically (e.g., randomly) distributed, which is the case
for "powdered" specimens. We provide exact expressions for the
orientationally-averaged signal obtained via general gradient waveforms when
the microdomains are characterized by a general diffusion tensor possibly
featuring three distinct eigenvalues. Our results are expected to be useful in
not only multidimensional diffusion MR but also solid-state NMR spectroscopy
due to the mathematical similarities in the two fields.Comment: 13 pages (manuscript) + 12 pages (supplementary material), 4 figure