Isospectral Flow and Liouville-Arnold Integration in Loop Algebras

A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and the sine-Gordon equation. Each system has an associated invariant spectral curve and may be integrated via the Liouville-Arnold technique. The linearizing map is the Abel map to the associated Jacobi variety, which is deduced through separation of variables in hyperellipsoidal coordinates. More generally, a family of moment maps is derived, identifying certain finite dimensional symplectic manifolds with rational coadjoint orbits of loop algebras. Integrable Hamiltonians are obtained by restriction of elements of the ring of spectral invariants to the image of these moment maps. The isospectral property follows from the Adler-Kostant-Symes theorem, and gives rise to invariant spectral curves. {\it Spectral Darboux coordinates} are introduced on rational coadjoint orbits, generalizing the hyperellipsoidal coordinates to higher rank cases. Applying the Liouville-Arnold integration technique, the Liouville generating function is expressed in completely separated form as an abelian integral, implying the Abel map linearization in the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199

Strong "quantum" chaos in the global ballooning mode spectrum of three-dimensional plasmas

The spectrum of ideal magnetohydrodynamic (MHD) pressure-driven (ballooning) modes in strongly nonaxisymmetric toroidal systems is difficult to analyze numerically owing to the singular nature of ideal MHD caused by lack of an inherent scale length. In this paper, ideal MHD is regularized by using a $k$-space cutoff, making the ray tracing for the WKB ballooning formalism a chaotic Hamiltonian billiard problem. The minimum width of the toroidal Fourier spectrum needed for resolving toroidally localized ballooning modes with a global eigenvalue code is estimated from the Weyl formula. This phase-space-volume estimation method is applied to two stellarator cases.Comment: 4 pages typeset, including 2 figures. Paper accepted for publication in Phys. Rev. Letter

Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size

We construct affinization of the algebra $gl_{\lambda}$ of ``complex size'' matrices, that contains the algebras $\hat{gl_n}$ for integral values of the parameter. The Drinfeld--Sokolov Hamiltonian reduction of the algebra $\hat{gl_{\lambda}}$ results in the quadratic Gelfand--Dickey structure on the Poisson--Lie group of all pseudodifferential operators of fractional order. This construction is extended to the simultaneous deformation of orthogonal and simplectic algebras that produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Comment: 29 pages, no figure

Activation of brain regions vulnerable to Alzheimer\u27s disease: The effect of mild cognitive impairment

This study examined the functionality of the medial temporal lobe (MTL) and posterior cingulate (PC) in mild cognitive impairment amnestic type (MCI), a syndrome that puts patients at greater risk for developing Alzheimer disease (AD). Functional MRI (fMRI) was used to identify regions normally active during encoding of novel items and recognition of previously learned items in a reference group of 77 healthy young and middle-aged adults. The pattern of activation in this group guided further comparisons between 14 MCI subjects and 14 age-matched controls. The MCI patients exhibited less activity in the PC during recognition of previously learned items, and in the right hippocampus during encoding of novel items, despite comparable task performance to the controls. Reduced fMRI signal change in the MTL supports prior studies implicating the hippocampus for encoding new information. Reduced signal change in the PC converges with recent research on its role in recognition in normal adults as well as metabolic decline in people with genetic or cognitive risk for AD. Our results suggest that a change in function in the PC may account, in part, for memory recollection failure in AD. © 2005 Elsevier Inc. All rights reserved

Global alterations to the choroid plexus blood-CSF barrier in amyotrophic lateral sclerosis

© 2020 The Author(s). The choroid plexus (CP) is a highly vascularized structure located in the ventricles that forms the blood-CSF barrier (BCSFB) and separates the blood from the cerebrospinal fluid (CSF). In addition to its role as a physical barrier, the CP functions in CSF secretion, transport of nutrients into the central nervous system (CNS) and a gated point of entry of circulating immune cells into the CNS. Aging and neurodegeneration have been reported to affect CP morphology and function and increase protein leakage from blood to the CSF. Amyotrophic lateral sclerosis (ALS) is a neurodegenerative disease associated with both upper and lower motor neuron loss, as well as altered proteomic and metabolomic signatures in the CSF. The role of the BCSFB and the CP in ALS is unknown. Here we describe a transcriptomic and ultrastructural analysis of BCSFB and CP alterations in human postmortem tissues from ALS and non-neurologic disease controls. ALS-CP exhibited widespread disruptions in tight junctional components of the CP epithelial layer and vascular integrity. In addition, we detected loss of pericytes around ALS blood vessels, accompanied by activation of platelet aggregation markers vWF and Fibrinogen, reminiscent of vascular injury. To investigate the immune component of ALS-CP, we conducted a comprehensive analysis of cytokines and chemokine panels in CP lysates and found a significant down-regulation of M-CSF and V-CAM1 in ALS, as well as up-regulation of VEGF-A protein. This phenotype was accompanied by an infiltration of MERTK positive macrophages into the parenchyma of the ALS-CP when compared to controls. Taken together, we demonstrate widespread structural and functional disruptions of the BCSFB in human ALS increasing our understanding of the disease pathology and identifying potential new targets for ALS therapeutic development

IC‐P2‐074: Differentiating amnestic MCI converting to probable AD from stable amnestic MCI using FDG‐PET and an AD‐related hypometabolism overlap index

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/152594/1/alzjjalz200805067.pd