28,688 research outputs found
Minimal simple pairs in the cubic grid
International audiencePreserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. This paper constitutes an introduction to the study of non-trivial simple sets in the framework of cubical 3-D complexes. A simple set has the property that the homotopy type of the object in which it lies is not changed when the set is removed. The main contribution of this paper is a characterisation of the non-trivial simple sets composed of exactly two voxels, such sets being called minimal simple pairs
Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions
The parameter space for monic centered cubic polynomial maps
with a marked critical point of period is a smooth affine algebraic curve
whose genus increases rapidly with . Each consists of a
compact connectedness locus together with finitely many escape regions, each of
which is biholomorphic to a punctured disk and is characterized by an
essentially unique Puiseux series. This note will describe the topology of
, and of its smooth compactification, in terms of these escape
regions. It concludes with a discussion of the real sub-locus of
.Comment: 51 pages, 16 figure
Improving convergence in smoothed particle hydrodynamics simulations without pairing instability
The numerical convergence of smoothed particle hydrodynamics (SPH) can be
severely restricted by random force errors induced by particle disorder,
especially in shear flows, which are ubiquitous in astrophysics. The increase
in the number NH of neighbours when switching to more extended smoothing
kernels at fixed resolution (using an appropriate definition for the SPH
resolution scale) is insufficient to combat these errors. Consequently, trading
resolution for better convergence is necessary, but for traditional smoothing
kernels this option is limited by the pairing (or clumping) instability.
Therefore, we investigate the suitability of the Wendland functions as
smoothing kernels and compare them with the traditional B-splines. Linear
stability analysis in three dimensions and test simulations demonstrate that
the Wendland kernels avoid the pairing instability for all NH, despite having
vanishing derivative at the origin (disproving traditional ideas about the
origin of this instability; instead, we uncover a relation with the kernel
Fourier transform and give an explanation in terms of the SPH density
estimator). The Wendland kernels are computationally more convenient than the
higher-order B-splines, allowing large NH and hence better numerical
convergence (note that computational costs rise sub-linear with NH). Our
analysis also shows that at low NH the quartic spline kernel with NH ~= 60
obtains much better convergence then the standard cubic spline.Comment: substantially revised version, accepted for publication in MNRAS, 15
pages, 13 figure
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