We prove$BMO$and$L$^{$p$}norm inequalities in$R$^{$n$}for lacunary Walsh and generalized trigonometric series.

We consider subordination chains of simply connected domains with smooth boundaries in the complex plane. Such chains admit Hamiltonian and Lagrangian interpretations through the Löwner–Kufarev evolution equations. The action functional is constructed and its time variation is obtained. It represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in$R$^{$n$}× (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form$u$($x$,$t$) =$b$($t$), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].

Our purpose is to compare how much the F5 algorithm can gain in efficiency compared to the F4 algorithm. This can be achieve as the F5 algorithm uses the concept of signatures to foresee potential useless computation which the F4 algorithm might make represented by zero rows in the reduction of a large matrix. We experimentally show that this is a modest increase in efficiency for the parameters we tested.

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