Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras

Darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves, defined within a given matrix representation. A Liouville generating function is obtained in completely separated form and shown, through the Liouville-Arnold integration method, to lead to the Abel map linearization of all Hamiltonian flows induced by the spectral invariants. Serre duality is used to define a natural symplectic structure on the space of line bundles of suitable degree over a permissible class of spectral curves, and this is shown to be equivalent to the Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general construction is given for $\frak{g}=\frak{gl}(r)$ or $\frak{sl}(r)$, with reductions to orbits of subalgebras determined as invariant fixed point sets under involutive automorphisms. The case $\frak{g=sl}(2)$ is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics, as well as the quasi-periodic solutions of the cubically nonlinear Schr\"odinger equation. For $\frak{g=sl}(3)$, the method is applied to the computation of quasi-periodic solutions of the two component coupled nonlinear Schr\"odinger equation.Comment: 61 pg

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

Resistance to autosomal dominant Alzheimer's disease in an APOE3 Christchurch homozygote: a case report.

We identified a PSEN1 (presenilin 1) mutation carrier from the world's largest autosomal dominant Alzheimer's disease kindred, who did not develop mild cognitive impairment until her seventies, three decades after the expected age of clinical onset. The individual had two copies of the APOE3 Christchurch (R136S) mutation, unusually high brain amyloid levels and limited tau and neurodegenerative measurements. Our findings have implications for the role of APOE in the pathogenesis, treatment and prevention of Alzheimer's disease

Reduced hippocampal activation during episodic encoding in middle-aged individuals at genetic risk of Alzheimer's Disease: a cross-sectional study

BACKGROUND: The presence of the apolipoprotein E (APOE) ε4 allele is a major risk factor for the development of Alzheimer's disease (AD), and has been associated with metabolic brain changes several years before the onset of typical AD symptoms. Functional MRI (fMRI) is a brain imaging technique that has been used to demonstrate hippocampal activation during measurement of episodic encoding, but the effect of the ε4 allele on hippocampal activation has not been firmly established. METHODS: The present study examined the effects of APOE genotype on brain activation patterns in the medial temporal lobe (MTL) during an episodic encoding task using a well-characterized novel item versus familiar item contrast in cognitively normal, middle-aged (mean = 54 years) individuals who had at least one parent with AD. RESULTS: We found that ε3/4 heterozygotes displayed reduced activation in the hippocampus and MTL compared to ε3/3 homozygotes. There were no significant differences between the groups in age, education or neuropsychological functioning, suggesting that the altered brain activation seen in ε3/4 heterozygotes was not associated with impaired cognitive function. We also found that participants' ability to encode information on a neuropsychological measure of learning was associated with greater activation in the anterior MTL in the ε3/3 homozygotes, but not in the ε3/4 heterozygotes. CONCLUSION: Together with previous studies reporting reduced glucose metabolism and AD-related neuropathology, this study provides convergent validity for the idea that the MTL exhibits functional decline associated with the APOE ε4 allele. Importantly, these changes were detected in the absence of meaningful neuropsychological differences between the groups. A focus of ongoing work in this laboratory is to determine if these findings are predictive of subsequent cognitive decline

Florbetapir F 18 amyloid PET and 36-month cognitive decline:a prospective multicenter study

This study was designed to evaluate whether subjects with amyloid beta (Aβ) pathology, detected using florbetapir positron emission tomorgraphy (PET), demonstrated greater cognitive decline than subjects without Aβ pathology. Sixty-nine cognitively normal (CN) controls, 52 with recently diagnosed mild cognitive impairment (MCI) and 31 with probable Alzheimer's disease (AD) dementia were included in the study. PET images obtained in these subjects were visually rated as positive (Aβ+) or negative (Aβ−), blind to diagnosis. Fourteen percent (10/69) of CN, 37% (19/52) of MCI and 68% (21/31) of AD were Aβ+. The primary outcome was change in ADAS-Cog score in MCI subjects after 36 months; however, additional outcomes included change on measures of cognition, function and diagnostic status. Aβ+ MCI subjects demonstrated greater worsening compared with Aβ− subjects on the ADAS-Cog over 36 months (5.66±1.47 vs −0.71±1.09, P=0.0014) as well as on the mini-mental state exam (MMSE), digit symbol substitution (DSS) test, and a verbal fluency test (P<0.05). Similar to MCI subjects, Aβ+ CN subjects showed greater decline on the ADAS-Cog, digit-symbol-substitution test and verbal fluency (P<0.05), whereas Aβ+ AD patients showed greater declines in verbal fluency and the MMSE (P<0.05). Aβ+ subjects in all diagnostic groups also showed greater decline on the CDR-SB (P<0.04), a global clinical assessment. Aβ+ subjects did not show significantly greater declines on the ADCS-ADL or Wechsler Memory Scale. Overall, these findings suggest that in CN, MCI and AD subjects, florbetapir PET Aβ+ subjects show greater cognitive and global deterioration over a 3-year follow-up than Aβ− subjects do

Early detection of cryptic memory and glucose uptake deficits in pre-pathological APP mice

Earlier diagnosis and treatment of Alzheimer's disease would greatly benefit from the identification of biomarkers at the prodromal stage. Using a prominent animal model of aspects of the disease, we here show using clinically relevant methodologies that very young, pre-pathological PDAPP mice, which overexpress mutant human amyloid precursor protein in the brain, exhibit two cryptic deficits that are normally undetected using standard methods of assessment. Despite learning a spatial memory task normally and displaying normal brain glucose uptake, they display faster forgetting after a long delay following performance to a criterion, together with a strong impairment of brain glucose uptake at the time of attempted memory retrieval. Preliminary observations suggest that these deficits, likely caused by an impairment in systems consolidation, could be rescued by immunotherapy with an anti-β-amyloid antibody. Our data suggest a biomarker strategy for the early detection of β-amyloid-related abnormalities

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Familial Mediterranean fever, Inflammation and Nephrotic Syndrome: Fibrillary Glomerulopathy and the M680I Missense Mutation

BACKGROUND: Familial Mediterranean fever (FMF) is an autosomal recessive disease characterized by inflammatory serositis (fever, peritonitis, synovitis and pleuritis). The gene locus responsible for FMF was identified in 1992 and localized to the short arm of chromosome 16. In 1997, a specific FMF gene locus, MEFV, was discovered to encode for a protein, pyrin that mediates inflammation. To date, more than forty missense mutations are known to exist. The diversity of mutations identified has provided insight into the variability of clinical presentation and disease progression. CASE REPORT: We report an individual heterozygous for the M680I gene mutation with a clinical diagnosis of FMF using the Tel-Hashomer criteria. Subsequently, the patient developed nephrotic syndrome with biopsy-confirmed fibrillary glomerulonephritis (FGN). Further diagnostic studies were unremarkable with clinical workup negative for amyloidosis or other secondary causes of nephrotic syndrome. DISCUSSION: Individuals with FMF are at greater risk for developing nephrotic syndrome. The most serious etiology is amyloidosis (AA variant) with renal involvement, ultimately progressing to end-stage renal disease. Other known renal diseases in the FMF population include IgA nephropathy, IgM nephropathy, Henoch-Schönlein purpura as well as polyarteritis nodosa. CONCLUSION: To our knowledge, this is the first association between FMF and the M680I mutation later complicated by nephrotic syndrome and fibrillary glomerulonephritis