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

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 $\operatorname {Re}=1/2$

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 $\operatorname {Re}=1/2$

Clinical Factors Associated with Obstructive Coronary Artery Disease in Patients with Out-of-Hospital Cardiac Arrest: Data from the Korean Cardiac Arrest Research Consortium (KoCARC) Registry

Background: Although coronary artery disease (CAD) is a major cause of out-of-hospital cardiac arrest (OHCA), there has been no convinced data on the necessity of routine invasive coronary angiography (ICA) in OHCA. We investigated clinical factors associated with obstructive CAD in OHCA. Methods: Data from 516 OHCA patients (mean age 58 years, 83% men) who underwent ICA after resuscitation was obtained from a nation-wide OHCA registry. Obstructive CAD was defined as the lesions with diameter stenosis >= 50% on ICA. Independent clinical predictors for obstructive CAD were evaluated using multiple logistic regression analysis, and their prediction performance was compared using area under the receiver operating characteristic curve with 10,000 repeated random permutations. Results: Among study patients, 254 (49%) had obstructive CAD. Those with obstructive CAD were older (61 vs. 55 years, P < 0.001) and had higher prevalence of hypertension (54% vs. 36%, P < 0.001), diabetes mellitus (29% vs. 21%, P = 0.032), positive cardiac enzyme (84% vs. 74%, P = 0.010) and initial shockable rhythm (70% vs. 61%, P = 0.033). In multiple logistic regression analysis, old age (>= 60 years) (odds ratio [On 2.01; 95% confidence interval [CI], 1.36-3.00; P = 0.001), hypertension (OR, 1.74; 95% CI, 1.18-2.57; P = 0.005), positive cardiac enzyme (OR, 1.72; 95% CI, 1.09-2.70; P = 0.019), and initial shockable rhythm (OR, 1.71; 95% CI, 1.16-2.54; P = 0.007) were associated with obstructive CAD. Prediction ability for obstructive CAD increased proportionally when these 4 factors were sequentially combined (P < 0.001). Conclusion: In patients with OHCA, those with old age, hypertension, positive cardiac enzyme and initial shockable rhythm were associated with obstructive CAD. Early ICA should be considered in these patients.Y

Eculizumab in refractory generalized myasthenia gravis previously treated with rituximab: subgroup analysis of REGAIN and its extension study

Introduction/Aims Individuals with refractory generalized myasthenia gravis (gMG) who have a history of rituximab use and experience persistent symptoms represent a population with unmet treatment needs. The aim of this analysis was to evaluate the efficacy and safety of eculizumab in patients with refractory anti‐acetylcholine receptor antibody‐positive (AChR+) gMG previously treated with rituximab. Methods This post hoc subgroup analysis of the phase 3 REGAIN study (NCT01997229) and its open‐label extension (OLE; NCT02301624) compared baseline characteristics, safety, and response to eculizumab in participants who had previously received rituximab with those who had not. Rituximab use was not permitted within the 6 months before screening or during REGAIN/OLE. Results Of 125 REGAIN participants, 14 had received rituximab previously (7 received placebo and 7 received eculizumab). In the previous‐rituximab group, 57% had used at least four other immunosuppressants compared with 16% in the no‐previous‐rituximab group. Myasthenia Gravis Activities of Daily Living total scores from eculizumab baseline to week 130 of eculizumab treatment improved in both the previous‐rituximab and no‐previous‐rituximab groups (least‐squares mean −4.4, standard error of the mean [SEM] 1.0 [n = 9] and least‐squares mean −4.6, SEM 0.3 [n = 67], respectively; difference = 0.2, 95% confidence interval −1.88 to 2.22). In addition, in both groups, most patients who were treated with eculizumab for 130 weeks achieved a Myasthenia Gravis Foundation of America post‐intervention status of minimal manifestations (66.7% and 65.0%, respectively). The eculizumab safety profile was similar between groups and consistent with its established profile. Discussion Eculizumab is an effective therapy for patients with refractory AChR+ gMG, irrespective of whether they had received rituximab treatment previously