An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈$A$,$G$^{α}may not cover$G$_{α}as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about the$G$^{α}/G_{α}is that each is an extension of a trivially ordered subgroup by a subgroup of$R$. It$B$is a plenary subset of$A$, then there exists a$v$-isomorphism μ of$G$into$V(B, G$^{β}/G_{β}), but whether or not μ is an$o$-isomorphism is not known.

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AbstractWe introduce a sub-cell shock capturing method for scalar conservation laws built upon the Jin-Xin relaxation framework. Here, sub-cell shock capturing is achieved thanks to an original defect measure correction technique. The proposed correction exactly restores entropy shock solutions of the exact Riemann problem and moreover, it produces monotone and entropy satisfying approximate self-similar solutions. These solutions are then sampled using Glimm's random choice method to advance in time. The resulting scheme combines the simplicity of the Jin-Xin relaxation method with the resolution of the Glimm's scheme in the sharp capture of discontinuities. The strong benefit of using defect measure corrections over usual sub-cell shock capturing methods is that the scheme can be easily made entropy satisfying with respect to infinitely many entropy pairs. Consequently, under a classical CFL condition, the method is proved to converge to the unique entropy weak solution of the Cauchy problem for general non-linear flux functions.