Current Development in Theory and Applications of Wavelets
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Abstract: The multi-scale and multi-resolution characteristic was
found to be inherently embeded in a stage and scale-controlled fragmentation
process. Fragmentation is a procedure of decomposition resulting into a concrete
package of differently scaled physical wavelets in forms of crumbs with
different sizes, it can also be alternatively interpreted as a conceptual
package of scale-varied wavelets of energy packs. The inter‑mapping
relation between physical package and energy package facilitates a practical
possibility of quantifying energy constitution through the analysis of
geometrical size-number properties of the sibling fragments. Conceptual
mathematical parities of a stage and scale-controlled fragmentation process with
wavelets analysis include: (1) The simultaneous, dynamical fragmentation action
is, when in view of energy relationships, a superimposing of impacting energy on
an imaginable ‘energy package’ of the parent bulk material, bearing the
similar characters of integration on the processed signal in wavelets analysis;
(2) A stage and scale-controlled fragmentation process is thus treated with two
distinctive properties: scale and resolution, among which the scale is
controlled by sieving with size openings reduced, while the resolution is
the related magnitude of contacting energies of fragments corresponding to its
related scales; (3) A stepped sieving procedure with successively decreased
sieve sizes behaves much the same as a filtering process in signal processing.
Thus, a conceptual mathematical interpretation with wavelets theory on a stage
and scale-controlled process will pave an alternative way for the complexed task
of energy analysis on the general fragmentation process.
Keywords and phrases: fragmentation, stress-wavelets, multi-scale and multi-resolution, fractal, size distribution.
