Bernoulli, Ramanujan, Toeplitz and the triangular matrices

By using one of the definitions of the Bernoulli numbers, we prove that they solve particular odd and even lower triangular Toeplitz (l.t.T.) systems of equations. In a paper Ramanujan writes down a sparse lower triangular system solved by Bernoulli numbers; we observe that such system is equivalent to a sparse l.t.T. system. The attempt to obtain the sparse l.t.T. Ramanujan system from the l.t.T. odd and even systems, has led us to study efficient methods for solving generic l.t.T. systems. Such methods are here explained in detail in case n, the number of equations, is a power of b, b = 2, 3 and b generic

The syndrome of polymicrogyria, thalamic hypoplasia, and epilepsy with CSWS

Objective: We explored the long-term follow-up of continuous spike-and-wave complexes during sleep (CSWS) in polymicrogyria and the anatomic volumetric variables that influence the risk of developing this age-related epileptic encephalopathy. Methods: We performed prospective follow-up of 27 patients with polymicrogyria/CSWS (mean follow-up 14.3 years; range 2-31 years) and comparative volumetric analysis of the polymicrogyric hemispheres and ipsilateral thalami vs 3 subgroups featuring polymicrogyria without CSWS, benign rolandic epilepsy (BRE), and headache. Receiver operator characteristic analysis of the power of volumetric values was determined to predict CSWS. Results: CSWS peaked between 5 and 7 years (mean age at onset 4.7 years). Remission occurred within 2 years from onset in 21%, within 4 years in 50%, and by age 13 years in 100%. We found smaller thalamic and hemispheric volumes in polymicrogyria/CSWS with respect to polymicrogyria without CSWS (p 0.0021 for hemispheres; p 0.0003 for thalami), BRE, and controls with headache (p < 0.0001). Volumes of the malformed hemispheres and ipsilateral thalami reliably identified the risk of incurring CSWS, with a 68-fold increased risk for values lower than optimal diagnostic cutoffs (436,150 mm 3 for malformed hemispheres or 4,616 mm 3 for ipsilateral thalami; sensitivity 92.54%; specificity 84.62%). The risk increased by 2% for every 1,000 mm 3 reduction of the polymicrogyric hemispheres and by 15% for every 100 mm 3 reduction of ipsilateral thalami. Conclusions: The polymicrogyria/CSWS syndrome is likely caused by a cortico-thalamic malformation complex and is characterized by remission of epilepsy within early adolescence. Early assessment of hemispheric and thalamic volumes in children with polymicrogyria and epilepsy can reliably predict CSWS

T cells in patients with narcolepsy target self-antigens of hypocretin neurons.

Narcolepsy is a chronic sleep disorder caused by the loss of neurons that produce hypocretin. The close association with HLA-DQB1*06:02, evidence for immune dysregulation and increased incidence upon influenza vaccination together suggest that this disorder has an autoimmune origin. However, there is little evidence of autoreactive lymphocytes in patients with narcolepsy. Here we used sensitive cellular screens and detected hypocretin-specific CD4 &lt;sup&gt;+&lt;/sup&gt; T cells in all 19 patients that we tested; T cells specific for tribbles homologue 2-another self-antigen of hypocretin neurons-were found in 8 out of 13 patients. Autoreactive CD4 &lt;sup&gt;+&lt;/sup&gt; T cells were polyclonal, targeted multiple epitopes, were restricted primarily by HLA-DR and did not cross-react with influenza antigens. Hypocretin-specific CD8 &lt;sup&gt;+&lt;/sup&gt; T cells were also detected in the blood and cerebrospinal fluid of several patients with narcolepsy. Autoreactive clonotypes were serially detected in the blood of the same-and even of different-patients, but not in healthy control individuals. These findings solidify the autoimmune aetiology of narcolepsy and provide a basis for rapid diagnosis and treatment of this disease