Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary

We consider two calculi of pseudodifferential operators on manifolds with fibered boundary: Mazzeo's edge calculus, which has as local model the operators associated to products of closed manifolds with asymptotically hyperbolic spaces, and the phi calculus of Mazzeo and the second author, which is similarly modeled on products of closed manifolds with asymptotically Euclidean spaces. We construct an adiabatic calculus of operators interpolating between them, and use this to compute the `smooth' K-theory groups of the edge calculus, determine the existence of Fredholm quantizations of elliptic symbols, and establish a families index theorem in K-theory

The CTA Sensitivity to Lorentz-Violating Effects on the Gamma-Ray Horizon

The arrival of TeV-energy photons from distant galaxies is expected to be affected by their QED interaction with intergalactic radiation fields through electron-positron pair production. In theories where high-energy photons violate Lorentz symmetry, the kinematics of the process $\gamma + \gamma\rightarrow e^+ + e^-$ is altered and the cross-section suppressed. Consequently, one would expect more of the highest-energy photons to arrive if QED is modified by Lorentz violation than if it is not. We estimate the sensitivity of Cherenkov Telescope Array (CTA) to changes in the $\gamma$-ray horizon of the Universe due to Lorentz violation, and find that it should be competitive with other leading constraints.Comment: 13 pages, 4 figures, typos corrected + references added, results unchanged. Matches version accepted by JCA

Resolution of smooth group actions

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a `resolution structure' consisting of equivariant iterated fibrations of the boundary faces. This structure projects to give a similar resolution structure for the quotient. In particular these results apply to give a canonical resolution of the radial compactification, to a ball, of any finite dimensional representation of a compact Lie group; such resolutions of the normal action of the isotropy groups appear in the boundary fibers in the general case.Comment: First half of http://arxiv.org/abs/0907.3211v2, extensively rewritte

Na+ and K+ ion imbalances in Alzheimer's disease

AbstractAlzheimer's disease (AD) is associated with impaired glutamate clearance and depressed Na+/K+ ATPase levels in AD brain that might lead to a cellular ion imbalance. To test this hypothesis, [Na+] and [K+] were analyzed in postmortem brain samples of 12 normal and 16 AD individuals, and in cerebrospinal fluid (CSF) from AD patients and matched controls. Statistically significant increases in [Na+] in frontal (25%) and parietal cortex (20%) and in cerebellar [K+] (15%) were observed in AD samples compared to controls. CSF from AD patients and matched controls exhibited no differences, suggesting that tissue ion imbalances reflected changes in the intracellular compartment. Differences in cation concentrations between normal and AD brain samples were modeled by a 2-fold increase in intracellular [Na+] and an 8–15% increase in intracellular [K+]. Since amyloid beta peptide (Aβ) is an important contributor to AD brain pathology, we assessed how Aβ affects ion homeostasis in primary murine astrocytes, the most abundant cells in brain tissue. We demonstrate that treatment of astrocytes with the Aβ 25–35 peptide increases intracellular levels of Na+ (~2–3-fold) and K+ (~1.5-fold), which were associated with reduced levels of Na+/K+ ATPase and the Na+-dependent glutamate transporters, GLAST and GLT-1. Similar increases in astrocytic Na+ and K+ levels were also caused by Aβ 1–40, but not by Aβ 1–42 treatment. Our study suggests a previously unrecognized impairment in AD brain cell ion homeostasis that might be triggered by Aβ and could significantly affect electrophysiological activity of brain cells, contributing to the pathophysiology of AD