43,999 research outputs found
Minimization Solutions to Conservation Laws with Non-smooth and Non-strictly Convex Flux
Conservation laws are usually studied in the context of sufficient regularity
conditions imposed on the flux function, usually and uniform convexity.
Some results are proven with the aid of variational methods and a unique
minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these
classical results can be extended to a flux function that is not necessarily
smooth or uniformly or strictly convex. Although uniqueness a.e. of the
minimizer will generally no longer hold, by considering the greatest (or
supremum, where applicable) of all possible minimizers, we can successfully
extend the results. One specific nonlinear case is that of a piecewise linear
flux function, for which we prove existence and uniqueness results. We also
approximate it by a smoothed, superlinearized version parameterized by
and consider the characterization of the minimizers for the
smooth version and limiting behavior as to that of the
sharp, polygonal problem. In proving a key result for the solution in terms of
the value of the initial condition, we provide a stepping stone to analyzing
the system under stochastic processes, which will be explored further in a
future paper.Comment: 27 pages, 5 figure
Affine Buildings and Tropical Convexity
The notion of convexity in tropical geometry is closely related to notions of
convexity in the theory of affine buildings. We explore this relationship from
a combinatorial and computational perspective. Our results include a convex
hull algorithm for the Bruhat--Tits building of SL and techniques for
computing with apartments and membranes. While the original inspiration was the
work of Dress and Terhalle in phylogenetics, and of Faltings, Kapranov, Keel
and Tevelev in algebraic geometry, our tropical algorithms will also be
applicable to problems in other fields of mathematics.Comment: 22 pages, 4 figure
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