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We study a subRiemannian geometry induced by 2 specific vector fields in R 3 , and obtain the canonical curve whose tangents provide the missing direction.
The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Grušin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study… (More)
The Grusin operator ∆G = 1 2 (∂ 2 x + x 2 ∂ 2 y), x, y ∈ R, is studied by Hamilton-Jacobi theory. In particular, we find all the geodesics of ∆G of the induced nonholonomic geometry, construct a modified complex action f which allows us to obtain the heat kernel Pt of ∆G. The small time asymptotics of Pt at all critical points of f are computed. Finally we… (More)
We prove geodesic completeness and global conectivity for a step 2k + 2 subRie-mannian manifold. Using complex Hamiltonian mechanics we also calculate some subRiemannian distances. 1. Introduction. SubRiemannian geometry starts with the Carathéodory's for-malization of thermodynamics  where the quasi-static adiabatic processes are related to the integral… (More)
2 i Preface This volume grew from lectures that the authors gave at City University of Hong Kong. Its aim is to cover a variety of interconnected topics that we have found particularly interesting, with some historical background, some motivation, and enough detail (however sketchy) to convey the main ideas. We are most grateful to Professors Wong and… (More)
BACKGROUND MSdialog, a web- and mobile-based software application, captures data on self-administration of subcutaneous interferon β-1a, clinical outcomes, and patient-reported outcomes in patients with multiple sclerosis outside the clinic. METHODS Patient and healthcare professional reactions to MSdialog were surveyed; participants rated benefits of… (More)
BACKGROUND AND STUDY AIMS Gastrointestinal bleeding in children and adolescents accounts for up to 20 % of referrals to gastroenterologists. Detailed management guidelines exist for gastrointestinal bleeding in adults, but they do not encompass children and adolescents. The aim of this study was to assess gastrointestinal bleeding in pediatric patients and… (More)
I shall discuss some of the reasons for the study of the ∂-Neumann problem and speculate about its nonelliptic analogue the so called ∂ b-Neumann problem, plus the possibility of a nonelliptic index. The talk will point to the necessity of understanding CR-geometry more precisely than is understood at present.