36,742 research outputs found
Large values of the Gowers-Host-Kra seminorms
The \emph{Gowers uniformity norms} of a function f: G \to
\C on a finite additive group , together with the slight variant
defined for functions on a discrete interval , are of importance in the modern theory of counting additive
patterns (such as arithmetic progressions) inside large sets. Closely related
to these norms are the \emph{Gowers-Host-Kra seminorms} of a
measurable function f: X \to \C on a measure-preserving system . Much recent effort has been devoted to the question of
obtaining necessary and sufficient conditions for these Gowers norms to have
non-trivial size (e.g. at least for some small ), leading in
particular to the inverse conjecture for the Gowers norms, and to the Host-Kra
classification of characteristic factors for the Gowers-Host-Kra seminorms.
In this paper we investigate the near-extremal (or "property testing")
version of this question, when the Gowers norm or Gowers-Host-Kra seminorm of a
function is almost as large as it can be subject to an or
bound on its magnitude. Our main results assert, roughly speaking, that this
occurs if and only if behaves like a polynomial phase, possibly localised
to a subgroup of the domain; this can be viewed as a higher-order analogue of
classical results of Russo and Fournier, and are also related to the
polynomiality testing results over finite fields of Blum-Luby-Rubinfeld and
Alon-Kaufman-Krivelevich-Litsyn-Ron. We investigate the situation further for
the norms, which are associated to 2-step nilsequences, and find that
there is a threshold behaviour, in that non-trivial 2-step nilsequences (not
associated with linear or quadratic phases) only emerge once the norm is
at most of the norm.Comment: 52 pages, no figures, to appear, Journal d'Analyse Jerusalem. This is
the final version, incorporating the referee's suggestion
Splitting method for elliptic equations with line sources
In this paper, we study the mathematical structure and numerical
approximation of elliptic problems posed in a (3D) domain when the
right-hand side is a (1D) line source . The analysis and approximation
of such problems is known to be non-standard as the line source causes the
solution to be singular. Our main result is a splitting theorem for the
solution; we show that the solution admits a split into an explicit, low
regularity term capturing the singularity, and a high-regularity correction
term being the solution of a suitable elliptic equation. The splitting
theorem states the mathematical structure of the solution; in particular, we
find that the solution has anisotropic regularity. More precisely, the solution
fails to belong to in the neighbourhood of , but exhibits
piecewise -regularity parallel to . The splitting theorem can
further be used to formulate a numerical method in which the solution is
approximated via its correction function . This approach has several
benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D
right-hand side belonging to , a problem for which the discretizations and
solvers are readily available. Secondly, it makes the numerical approximation
independent of the discretization of ; thirdly, it improves the
approximation properties of the numerical method. We consider here the Galerkin
finite element method, and show that the singularity subtraction then recovers
optimal convergence rates on uniform meshes, i.e., without needing to refine
the mesh around each line segment. The numerical method presented in this paper
is therefore well-suited for applications involving a large number of line
segments. We illustrate this by treating a dataset (consisting of
line segments) describing the vascular system of the brain
On the accuracy of simulations of turbulence
The widely recognized issue of adequate spatial resolution in numerical simulations of turbulence is studied in the context of two-dimensional magnetohydrodynamics. The familiar criterion that the dissipation scale should be resolved enables accurate computation of the spectrum, but fails for precise determination of higher-order statistical quantities. Examination of two straightforward diagnostics, the maximum of the kurtosis and the scale-dependent kurtosis, enables the development of simple tests for assessing adequacy of spatial resolution. The efficacy of the tests is confirmed by examining a sample problem, the distribution of magnetic reconnection rates in turbulence. Oversampling the Kolmogorov dissipation scale by a factor of 3 allows accurate computation of the kurtosis, the scale-dependent kurtosis, and the reconnection rates. These tests may provide useful guidance for resolution requirements in many plasma computations involving turbulence and reconnection
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