By Laurent Schwartz

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**Example text**

E. that there is an increasing sequence {s j } ⊂ R, possibly empty and without finite accumulation points in case it is infinite, such that the triad is well defined in each interval (s j , s j+1 ); recall that this is the case when ¨ vanishes nowhere in the interval. Tubes in R3 can have a variety of cross sections. We suppose that M ⊂ R2 is an open precompact set which contains zero, and put a := supx∈M |x|. 16) where r, θ are polar coordinates in R2 . In this way we associate with and M a tube := f (R× M).

Then η(M) > 0. Proof Since the Sobolev space H01 (M) is compactly embedded into L 2 (M), the 2 operator − M D − ∂ϕ associated with the quadratic form M (|∇t v|2 + |∂ϕ v|2 ) dxt , v ∈ H01 (M) , has a purely discrete spectrum.

3: the set of closed classical trajectories has zero measure, which is here even more obvious. e. the tip of the barrier, which is an event of probability zero). More about eigenvalues and eigenfunctions can be learned from the numerical solution which is found by the mode-matching method. Since is mirror-symmetric with respect to the line x = 0, the operator − D decomposes into a direct sum of two parts with definite parities, and one can therefore consider the halfstrip problems with Neumann and Dirichlet conditions, respectively, at the cut.