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    Homotopy invariance through small stabilizations

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    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 KK-theory groups KH_*(I_{S(\fA)}) are homotopy invariant for certain ideals SS including c0c_0 and p\ell^p. Moreover, if either S=c0S=c_0 and \fA is a local CC^*-algebra or S=p,p±S=\ell^p,\ell^{p\pm} 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 S{c0,p,p±}S\in\{c_0,\ell^p,\ell^{p\pm}\} 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

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    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

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    Let MM be a finite module over a commutative noetherian ring RR. For ideals \fa and \fb of RR, the relations between cohomological dimensions of MM with respect to \fa, \fb, \fa\cap\fb and \fa+ \fb are studied. When RR is local, it is shown that MM is generalized Cohen-Macaulay if there exists an ideal \fa such that all local cohomology modules of MM with respect to \fa have finite lengths. Also, when rr is an integer such that 0r<dimR(M)0\leq r< \dim_R(M), any maximal element \fq of the non-empty set of ideals \{\fa : \H_\fa^i(M) is not artinian for some ii, iri\geq r}\} is a prime ideal and that all Bass numbers of \H_\fq^i(M) are finite for all iri\geq r.Comment: 10 pages, to appear in Canadian Mathematical Bulleti
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