3,049,529 research outputs found
Homotopy invariance through small stabilizations
We associate an algebra \Gami(\fA) to each bornological algebra \fA. The
algebra \Gami(\fA) contains a two-sided ideal I_{S(\fA)} for each symmetric
ideal S\triqui\elli of bounded sequences of complex numbers. In the case of
\Gami=\Gami(\C), these are all the two-sided ideals, and I_S\mapsto J_S=\cB
I_S\cB gives a bijection between the two-sided ideals of \Gami and those of
\cB=\cB(\ell^2). We prove that Weibel's -theory groups KH_*(I_{S(\fA)})
are homotopy invariant for certain ideals including and .
Moreover, if either and \fA is a local -algebra or
and \fA is a local Banach algebra, then
KH_*(I_{S(\fA)}) contains K_*^{\top}(\fA) as a direct summand. Furthermore,
we prove that for the map
K_*(\Gamma^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)}) fits into a long exact
sequence with the relative cyclic homology groups
HC_*(\Gamma^\infty(\fA):I_{S(\fA)}). Thus the latter groups measure the
failure of the former map to be an isomorphism.Comment: 32 pages. The original paper has been split into two parts, of which
this is the first part. The second part is now arXiv:1304.350
Axon diameters and myelin content modulate microscopic fractional anisotropy at short diffusion times in fixed rat spinal cord
Mapping tissue microstructure accurately and noninvasively is one of the
frontiers of biomedical imaging. Diffusion Magnetic Resonance Imaging (MRI) is
at the forefront of such efforts, as it is capable of reporting on microscopic
structures orders of magnitude smaller than the voxel size by probing
restricted diffusion. Double Diffusion Encoding (DDE) and Double Oscillating
Diffusion Encoding (DODE) in particular, are highly promising for their ability
to report on microscopic fractional anisotropy ({\mu}FA), a measure of the pore
anisotropy in its own eigenframe, irrespective of orientation distribution.
However, the underlying correlates of {\mu}FA have insofar not been studied.
Here, we extract {\mu}FA from DDE and DODE measurements at ultrahigh magnetic
field of 16.4T in the aim to probe fixed rat spinal cord microstructure. We
further endeavor to correlate {\mu}FA with Myelin Water Fraction (MWF) derived
from multiexponential T2 relaxometry, as well as with literature-based
spatially varying axonal diameters. In addition, a simple new method is
presented for extracting unbiased {\mu}FA from three measurements at different
b-values. Our findings reveal strong anticorrelations between {\mu}FA (derived
from DODE) and axon diameter in the distinct spinal cord tracts; a moderate
correlation was also observed between {\mu}FA derived from DODE and MWF. These
findings suggest that axonal membranes strongly modulate {\mu}FA, which - owing
to its robustness towards orientation dispersion effects - reflects axon
diameter much better than its typical FA counterpart. The {\mu}FA exhibited
modulations when measured via oscillating or blocked gradients, suggesting
selective probing of different parallel path lengths and providing insight into
how those modulate {\mu}FA metrics. Our findings thus shed light into the
underlying microstructural correlates of {\mu}FA and are (...
Artinian and non-artinian local cohomology modules
Let be a finite module over a commutative noetherian ring . For ideals
\fa and \fb of , the relations between cohomological dimensions of
with respect to \fa, \fb, \fa\cap\fb and \fa+ \fb are studied. When
is local, it is shown that is generalized Cohen-Macaulay if there exists an
ideal \fa such that all local cohomology modules of with respect to \fa
have finite lengths. Also, when is an integer such that , any maximal element \fq of the non-empty set of ideals \{\fa :
\H_\fa^i(M) is not artinian for some , is a prime ideal and
that all Bass numbers of \H_\fq^i(M) are finite for all .Comment: 10 pages, to appear in Canadian Mathematical Bulleti
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