The Octonions

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction

Prognostic indicators in primary biliary cirrhosis: significance of revised IAHG (International Autoimmune Hepatitis Group) score

Background/AimsPrimary biliary cirrhosis (PBC) is a slowly progressing autoimmune disease of the liver that is characterized by portal inflammation and immune-mediated destruction of the intrahepatic bile ducts. Serum total bilirubin is one of the various prognostic factors that have been proposed. A recent study found that PBC with accompanying autoimmune hepatitis (AIH) carries a negative prognosis. This study examined the clinical characteristics of PBC and analyzed the factors that affect its prognosis.MethodsPatients diagnosed with PBC between January 1998 and December 2010 based on clinical and histopathological findings were compiled and analyzed retrospectively.ResultsAmong 27 patients, 24 (1 male and 23 females, ages 50.0±9.3 years) were followed up. The follow-up period was 8.6±0.9 years. Of the 24 patients, 9 patients progressed to liver cirrhosis (LC). Comparison between patients who did and did not progress to LC revealed statistically significant differences in the patients' serum total bilirubin (2.7±1.8 vs. 0.8±0.4, P=0.012), the Mayo risk score (5.1±0.7 vs. 3.9±0.6, P=0.001), the revised IAHG (International Autoimmune Hepatitis Group) score (9.2±2.3 vs. 5.4±3.0, P=0.004) and frequency of AIH overlap (5/9 [55.6%] vs. 0/15 [0%], P=0.001) at the time of diagnosis.ConclusionsWe propose that serum total bilirubin, the Mayo risk score, and the revised IAHG score at the time of diagnosis are helpful for predicting PBC prognosis. In particular, since all of the patients with accompanying AIH progressed to LC, the presence of overlap syndrome at the time of diagnosis is helpful for predicting PBC prognosis and providing an adequate treatment

CSF total tau/α-synuclein ratio improved the diagnostic performance for Alzheimers disease as an indicator of tau phosphorylation

Abstract Background Recently, several studies suggested potential involvements of α-synuclein in Alzheimers disease (AD) pathophysiology. Higher concentrations of α-synuclein were reported in cerebrospinal fluid (CSF) of AD patients with a positive correlation towards CSF tau, indicating its possible role in AD. We analyzed the CSF biomarkers to verify whether α-synuclein could be an additional supported biomarker in AD diagnosis. Methods In this cross-sectional study, CSF samples of 71 early-onset AD, 34 late-onset AD, 11 mild cognitive impairment, 17 subjective cognitive decline, 45 Parkinsons disease, and 32 healthy control (HC) were collected. CSF amyloid-β1-42 (A), total tau (N), and phosphorylated tau181 (T) were measured by commercial ELISA kits, and in-house ELISA kit was developed to quantify α-synuclein. The cognitive assessments and amyloid-PET imaging were also performed. Results CSF α-synuclein manifested a tendency to increase in AD and to decreased in Parkinsons disease compared to HC. The equilibrium states of total tau and α-synuclein concentrations were changed significantly in AD, and the ratio of total tau/α-synuclein (N/αS) was dramatically increased in AD than HC. Remarkably, N/αS revealed a strong positive correlation with tau phosphorylation rate. Also, the combination of N/αS with amyloid-β1-42/phosphorylated tau181ratio had the best diagnosis performance (AUC = 0.956, sensitivity = 96%, specificity = 87%). In concordance analysis, N/αS showed the higher diagnostic agreement with amyloid-β1-42 and amyloid-PET. Analysis of biomarker profiling with N/αS had distinctive characteristics and clustering of each group. Especially, among the group of suspected non-Alzheimers disease pathophysiology, all A−T+N+ patients with N/αS+ were reintegrated into AD. Conclusions The high correlation of α-synuclein with tau and the elevated N/αS in AD supported the involvement of α-synuclein in AD pathophysiology. Importantly, N/αS improved the diagnostic performance, confirming the needs of incorporating α-synuclein as a biomarker for neurodegenerative disorders. The incorporation of a biomarker group [N/αS] could contribute to provide better understanding and diagnosis of neurodegenerative disorders

Gosset polytopes in integral octonions

summary:We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_{4}$, and the Gosset polytope $4_{21}$ with the symmetry $E_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_{8}$ or $D_{4}$ actions on the $4_{21}$ or the $24$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope $2_{31}$ with the symmetry $E_{7}$, and it is a dual polytope to a Gosset polytope $3_{21}$ with the symmetry $E_{7}$ which is the set of the unit integral octonions with $\operatorname {Re}=1/2$

Cognitive functions and stereopsis in patients with Parkinson's disease and Alzheimer's disease using 3-dimensional television: a case controlled trial.

Stereopsis or depth perception is an awareness of the distances of objects from the observer, and binocular disparity is a necessary component of recognizing objects through stereopsis. In the past studies, patients with neurodegenerative disease (Alzheimer dementia, AD; Parkinson's disease IPD) have problems of stereopsis but they did not have actual stimulation of stereopsis. Therefore in this study, we used a 3-dimensional (3D) movie on 3D television (TV) for actual stereopsis stimulation. We propose research through analyzing differences between the three groups (AD, IPD, and Controls), and identified relations between the results from the Titmus Stereo Fly Test, and the 3D TV test. The study also looked into factors that affect the 3D TV test. Before allowing the patients to watch TV, we examined Titmus stereo Fly Test and cognitive test. We used the 3D version of a movie, of 17 minutes 1 second duration, and carried out a questionnaire about stereopsis. The scores of the stereopsis questionnaire were decreased in AD patients, compared with in IPD and controls, although they did not have any difference of Titmus Stereo Fly Test scores. In IPD patients, cognitive function (Montreal cognitive assessment, MoCA) scores were correlated with the scores of the stereopsis questionnaire. We could conclude that Titmus fly test could not distinguish between the three groups and cognitive dysfunction contributes to actual stereopsis perception in IPD patients. Therefore the 3D TV test of AD and IPD patients was more effective than Titmus fly test

Associated results of the Titmus Stereo Fly Test, in idiopathic Parkinson disease patients, Alzheimer’s disease patients, and Controls.

<p>IPD: Idiopathic Parkinson’s disease, AD: Alzheimer’s disease.</p><p>Associated results of the Titmus Stereo Fly Test, in idiopathic Parkinson disease patients, Alzheimer’s disease patients, and Controls.</p

Correlations of stereopsis questionnaire scores, and Age, MoCA and the Titmus stereo Fly Test.

<p>AD: Alzheimer’s disease, IPD: Idiopathic Parkinson’s disease.</p><p>Total: IPD+AD+Controls, Montreal cognitive assessment</p><p>* p<0.05</p><p>**p<0.01</p><p>Correlations of stereopsis questionnaire scores, and Age, MoCA and the Titmus stereo Fly Test.</p

The number of patients, according to stereopsis questionnaire scores, in three groups.

<p>IPD: Idiopathic Parkinson’s disease, AD: Alzheimer’s disease.</p

The number of patients, according to the Titmus stereo Fly Test, in three groups.

<p>IPD: Idiopathic Parkinson’s disease, AD: Alzheimer’s disease.</p