Hall\u27s Relations in Finite Groups

Let H be a finite group, and let θ be an automorphism of H whose order divides n. Hall proved that the number of elements x of H that satisfy the relation x*xθ*xθ2 ***xθn-1 = 1 is a multiple of gcd(n,|H|). For a prime factor p of gcd(n,|H|), if such a number is not a multiple of gcd(pn,|H|), then a Sylow p-subgroup of H is exceptional

Cerebellar Blood Flow and Gene Expression in Crossed Cerebellar Diaschisis after Transient Middle Cerebral Artery Occlusion in Rats

Crossed cerebellar diaschisis (CCD) is a state of hypoperfusion and hypometabolism in the contralesional cerebellar hemisphere caused by a supratentorial lesion, but its pathophysiology is not fully understood. We evaluated chronological changes in cerebellar blood flow (CbBF) and gene expressions in the cerebellum using a rat model of transient middle cerebral artery occlusion (MCAO). CbBF was analyzed at two and seven days after MCAO using single photon emission computed tomography (SPECT). DNA microarray analysis and western blotting of the cerebellar cortex were performed and apoptotic cells in the cerebellar cortex were stained. CbBF in the contralesional hemisphere was significantly decreased and this lateral imbalance recovered over one week. Gene set enrichment analysis revealed that a gene set for "oxidative phosphorylation" was significantly upregulated while fourteen other gene sets including "apoptosis", "hypoxia" and "reactive oxygen species" showed a tendency toward upregulation in the contralesional cerebellum. MCAO upregulated the expressions of nuclear factor erythroid 2-related factor 2 (Nrf2) and heme oxygenase-1 (HO-1) in the contralesional cerebellar cortex. The number of apoptotic cells increased in the molecular layer of the contralesional cerebellum. Focal cerebral ischemia in our rat MCAO model caused CCD along with enhanced expression of genes related to oxidative stress and apoptosis

A comparison of the prevalence and risk factors of complications in intracranial tumor embolization between the Japanese Registry of NeuroEndovascular Therapy 2 (JR-NET2) and JR-NET3

BACKGROUND: The Japanese Registry of NeuroEndovascular Therapy 2 (JR-NET2) and 3 (JR-NET3) were nationwide surveys that evaluated clinical outcomes after neuroendovascular therapy in Japan. The aim of this study was to compare the prevalence and risk factors of complications of intracranial tumor embolization between JR-NET2 and JR-NET3. METHODS: A total of 1018 and 1545 consecutive patients with intracranial tumors treated with embolization were enrolled in JR-NET2 and JR-NET3, respectively. The prevalence of complications in intracranial tumor embolization and related risk factors were compared between JR-NET2 and JR-NET3. RESULTS: The prevalence of complications in JR-NET3 (3.69%) was significantly higher than that in JR-NET2 (1.48%) (p = 0.002). The multivariate analysis in JR-NET2 showed that embolization for tumors other than meningioma was the only significant risk factor for complication (odds ratio [OR], 3.88; 95% confidence interval [CI], 1.13-12.10; p = 0.032), and that in JR-NET3 revealed that embolization for feeders other than external carotid artery (ECA) (OR, 3.56; 95% CI, 2.03-6.25; p  CONCLUSIONS: Embolization for feeders other than ECA and use of liquid materials could increase the complication rate in intracranial tumor embolization