Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), given by L = partial derivative/partial derivative t + (a(x) + ib(x))partial derivative/partial derivative x, b not equivalent to 0, near the characteristic set Sigma = {0} x S(1). We show that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. (C) 2008 Elsevier Inc. All rights reserved.

Experience With Lacosamide in a Series of Children With Drug-Resistant Focal Epilepsy

We report our pediatric experience with lacosarnide, a new antiepileptic drug, approved by the US Food and Drug Administration as adjunctive therapy in focal epilepsy in patients more than 17 years old. We retrospectively reviewed charts for lacosamide use and seizure frequency outcome in patients with focal epilepsy (Wilcoxon signed rank test). Sixteen patients (7 boys) were identified (median dose 275 mg daily, 4.7 mg/kg daily; mean age 14.9 years, range 8-21 years). Patients were receiving a median of 2 antiepileptic drugs (interquartile range [IQR] 1.7-3) in addition to having undergone previous epilepsy surgery (n = 3), vagus nerve stimulation (n = 9), and ketogenic diet (n = 3). Causes included structural (encephalomalacia and diffuse encephalitis, 1 each; stroke in 2) and genetic abnormalities (Aarskog and Rett syndromes, 1 each) or cause not known (n = 10). Median seizure frequency at baseline was 57 per month (IQR 7-75), and after a median follow-up of 4 months (range 1-13 months) of receiving lacosamide, it was 12.5 per month (IQR 3-75), (P < 0.01). Six patients (37.5%; 3 seizure free) were classified as having disease that responded to therapy (>= 50% reduction seizure frequency) and 10 as having disease that did not respond to therapy (<50% in 3; increase in 1; unchanged in 6). Adverse events (tics, behavioral disturbance, seizure worsening, and depression with suicidal ideation in 1 patient each) prompted lacosamide discontinuation in 4/16 (25%). This retrospective study of 16 children with drug-resistant focal epilepsy demonstrated good response to adjunctive lacosamide therapy (median seizure reduction of 39.6%; 37.5% with >= 50% seizure reduction) without severe adverse events. (C) 2011 Elsevier Inc. All rights reserved.

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

Motivated in part by the study of Fadell-Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n >= 1, the class of m-string braid groups B(m)(S(2)\{x(1), ... , x(n)}) of the n-punctured sphere includes the usual Artin braid groups B(m) (for n = 1), those of the annulus, which are Artin groups of type B (for n = 2), and affine Artin groups of type (C) over tilde (for n = 3). We first consider the case n = 1. Motivated by the study of almost periodic solutions of algebraic equations with almost periodic coefficients, Gorin and Lin calculated the commutator subgroup of the Artin braid groups. We extend their results, and show that the lower central series (respectively, derived series) of B(m) is completely determined for all m is an element of N (respectively, for all m not equal 4). In the exceptional case m = 4, we obtain some higher elements of the derived series and its quotients. When n >= 2, we prove that the lower central series (respectively, derived series) of B(m)(S(2)\{x(1), ... , x(n)}) is constant from the commutator subgroup onwards for all m >= 3 (respectively, m >= 5). The case m = 1 is that of the free group of rank n - 1. The case n = 2 is of particular interest notably when m = 2 also. In this case, the commutator subgroup is a free group of infinite rank. We then go on to show that B(2)(S(2)\{x(1), x(2)}) admits various interpretations, as the Baumslag-Solitar group BS(2, 2), or as a one-relator group with non-trivial centre for example. We conclude from this latter fact that B(2)(S(2)\{x(1), x(2)}) is residually nilpotent, and that from the commutator subgroup onwards, its lower central series coincides with that of the free product Z(2) * Z. Further, its lower central series quotients Gamma(i)/Gamma(i+1) are direct sums of copies of Z(2), the number of summands being determined explicitly. In the case m >= 3 and n = 2, we obtain a presentation of the derived subgroup, from which we deduce its Abelianization. Finally, in the case n = 3, we obtain partial results for the derived series, and we prove that the lower central series quotients Gamma(i)/Gamma(i+1) are 2-elementary finitely-generated groups.