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Centered numerical fluxes can be constructed for compressible Euler equations which preserve kinetic energy in the semi-discrete finite volume scheme. The essential feature is that the momentum flux should be of the form f m j+ 1 2 = p̃j+ 1 2 +uj+ 1 2 f ρ j+ 2 where uj+ 2 = (uj+uj+1)/2 and p̃j+ 1 2 , f ρ j+ 1 2 are any consistent approximations to the pressure and the mass flux. This scheme thus ...

We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, t...

Global entropy solutions in BV for a scalar nonlocal conservation law with fading memory are constructed as limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in BV are established, which also yield the existence of entropy solutions in L∞ while the initial data is only in L∞. Moreover, if the memory kernel depends on a relaxation parameter ε ...

Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equatio...

We present deterministic sub-linear space algorithms for problems over update data streams, including, estimating frequencies of items and ranges, finding approximate frequent items and approximate φ-quantiles, estimating inner-products, constructing near-optimal B-bucket histograms and estimating entropy. We also present improved lower bound results for several problems over update data streams.

We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shif...