On the approximation of high-dimensional differential equations in the hierarchical Tucker format

The hierarchical Tucker format is a way to decompose a high-dimensional tensor recursively into sums of products of lower-dimensional tensors. The number of degrees of freedom in such a representation is typically many orders of magnitude lower than the number of entries of the original tensor. This makes the hierarchical Tucker format a promising approach to solve ordinary differential equations for high-dimensional tensors. In order to propagate the approximation in time, differential equations for the parameters of the hierarchical Tucker format are derived from the Dirac-Frenkel variational principle. We prove an error bound for the dynamical approximation in the hierarchical Tucker format by extending previous results of Koch and Lubich for the non-hierarchical Tucker format.     

On the Distribution and Scaling of Convective Wavespeeds in the Shear Layers of Heated Supersonic Jets

The noise generated by supersonic plumes is of growing concern given the enormous peak noise intensity radiated by tactical aircraft engines. A key component of this noise is the enhanced radiation of mixing noise caused by large scale eddies convecting supersonically relative to the surrounding quiescent medium. As very little data exist for eddy convection in high Reynolds number, supersonic plumes, our current ability to develop concepts that alter compressible eddy convection is limited. Herein we present new experimental data of eddy convective wavespeeds in the developing shear layer of supersonic heated jets. A new scaling of the wavespeed in radial similarity coordinates is proposed which takes into account the influence of the ratio of static densities between the jet and ambient streams. In particular, we observe a structural change in wavespeed spectra at the end of the potential core—in addition to high turbulence levels, the potential core breakdown region can have enhanced eddy wavespeeds, increasing noise radiation efficiency. The results provide a first examination of the interplay of density ratio effects and the dynamic breakdown process of the potential core in supersonic jets—physics integral to the noise generation process.     

On the structure of finite elation Laguerre planes

Quasi-orderings are introduced in arbitrary affine planes and ternary rings as a common generalization of half-orderings and semiorderings. Quasi-ordered affine planes are described algebraically by quasi-ordered ternary rings. A lot of proper quasi-orderings are constructed in ternary rings. These algebraic examples show that all Lenz-Barlotti-classes, as far as they are known to be non-empty, are containing affine planes with proper quasi-orderings.Dedicated to Professor Dr. H. Lenz on the occasion of his 75th birthday