11,518 research outputs found
Compactly Supported Tensor Product Complex Tight Framelets with Directionality
Although tensor product real-valued wavelets have been successfully applied
to many high-dimensional problems, they can only capture well edge
singularities along the coordinate axis directions. As an alternative and
improvement of tensor product real-valued wavelets and dual tree complex
wavelet transform, recently tensor product complex tight framelets with
increasing directionality have been introduced in [8] and applied to image
denoising in [13]. Despite several desirable properties, the directional tensor
product complex tight framelets constructed in [8,13] are bandlimited and do
not have compact support in the space/time domain. Since compactly supported
wavelets and framelets are of great interest and importance in both theory and
application, it remains as an unsolved problem whether there exist compactly
supported tensor product complex tight framelets with directionality. In this
paper, we shall satisfactorily answer this question by proving a theoretical
result on directionality of tight framelets and by introducing an algorithm to
construct compactly supported complex tight framelets with directionality. Our
examples show that compactly supported complex tight framelets with
directionality can be easily derived from any given eligible low-pass filters
and refinable functions. Several examples of compactly supported tensor product
complex tight framelets with directionality have been presented
The Growth and Structure of Dark Matter Haloes
In this paper, we analyse in detail the mass-accretion histories and
structural properties of dark haloes in high-resolution N-body simulations.
Modeling the density distribution in individual haloes with the NFW profile, we
find, for all main progenitors of a given halo, there is a tight correlation
between its inner scale radius and the mass within it, , which is
the basic reason why halo structural properties are closely related to their
mass-accretion histories. This correlation can be used to predict accurately
the structural properties of a dark halo at any time from its mass-accretion
history. We also test our model with a large sample of GIF haloes. The build-up
of dark haloes in CDM models generally consists of an early phase of fast
accretion and a late phase of slow accretion [where increases with time
approximately as the expansion rate]. These two phases are separated at a time
when the halo concentration parameter . Haloes in the two accretion
phases show systematically different properties, for example, the circular
velocity increases rapidly with time in the fast accretion phase but
remain almost constant in the slow accretion phase,the inner properties of a
halo, such as and increase rapidly with time in the fast accretion
phase but change only slowly in the slow accretion phase. The potential well
associated with a halo is built up mainly in the fast accretion phase, even
though a large amount of mass (over 10 times) can be accreted in the slow
accretion phase. We discuss our results in connection to the formation of dark
haloes and galaxies in hierarchical models.Comment: 26 pages, including 10 figures. v2: some conceptual changes. Accepted
for publication in MNRA
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