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Regularity of weak solutions to rate-independent systems in one-dimension
Get PDFWe show that under some appropriate assumptions, every weak solution (e.g.
energetic solution) to a given rate-independent system is of class SBV, or has fi�nite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required
Regularity of weak solutions to rate-independent systems in one-dimension
Get PDFWe show that under some appropriate assumptions, every weak solution (e.g.
energetic solution) to a given rate-independent system is of class SBV, or has fi�nite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required
Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics
No full textWe propose a simple yet efficient method for generating abruptly autofocusing optical beams with arbitrary caustics. In addition we introduce a family of abruptly autodefocusing beams whose maximum intensity suddenly decreases by orders of magnitude right after the target. The method relies on appropriately modulating the phase of a circularly symmetric optical wavefront, such as that of a Gaussian, and subsequently on Fourier-transforming it by means of a lens. If two such beams are superimposed in a Bessel-like standing wave pattern, then a complete mirror-symmetric, with respect to the focal plane, caustic surface of revolution is formed that can be used as an optical bottle. We also show how the same method can be used to produce accelerating 1D or 2D optical beams with arbitrary convex caustics
A Posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation
Get PDFNew Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain
Get PDFWe construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simula- tions suggest that they are probably unstable
The maximum number of faces of the Minkowski sum of three convex polytopes
Get PDFWe derive tight expressions for the maximum
number of -faces, , of the
Minkowski sum, , of three -dimensional convex polytopes , and in ,
as a function of the number of vertices of the polytopes, for any .
Expressing the Minkowski sum as a section of the Cayley polytope of its summands, counting the -faces of reduces to counting the -faces of which meet the vertex sets of the three polytopes.
In two dimensions our expressions reduce to known results,
while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of -polytopes in , where .
For , the maximum values are attained when
, and are -polytopes, whose vertex sets are chosen appropriately from three distinct -dimensional moment-like curves