Some Conjectures Out of The Thesis
By A. H. Haji
Conjecture 1:
Instead of computing the slope in a scaling region near the smallest box sizes, a more precise approach to compute the correlation dimension may be extrapolating for box sizes equal to zero.
However the correlation dimension vs. box size may be not differentiable. Also regarding a multi fractal i.e. a fractal which shows different scaling regions, how would this approach be explained? (A multi fractal can be free from a predictable scheme.)
Conjecture 2:
In order to compute the minimum embedding dimension for reconstruction of an attractor using a time series according to the Takens theorems, we first compute the correlation dimension for several embedding dimensions. Then the embedding dimension after which the correlation dimension remains unchanged is taken as the minimum embedding dimension or saturation dimension.
However if we plot the curves of correlation sum vs. box size (which are used to compute the correlation dimension) for several embedding dimensions all in one coordinate, it can be seen that the scaling regions contract and new scaling regions are created with increasing the embedding dimension.
Why do scaling regions contract and new scaling regions come into existence with increasing the embedding dimension?
In order to make the multi fractal completely uniform does a negative embedding dimension have any meaning?
Fig. 1: typical curves of correlation and embedding dimension computations
A combined Conjecture (1 & 2): for embedding dimension calculation it may be not required to extrapolate the correlation dimension for zero box size. Perhaps we can say that two curves of correlation sum vs. box size which show similar slopes on some scaling regions have the same slope on each scaling region.
Conjecture 3:
Is the correlation dimension unaffected under a diffeomorphism? The mathematical form is as follows:
Suppose
is
a one – to – one correspondence. Also
and
are
continuous. Show that from:
(1)
we have:
(2)
where:
is
the heavy – side function.
The crucial statement to be proven is:
(3)
where:
(4)
Can we say the necessary and
sufficient condition is that the 
should
be bounded or more generally 
?
Does statement (3) mean that the difference of two diffeomorphic fractals has zero dimension?
By the way what does diffeomophism mean for discrete sets (The case which we encounter in the time series analysis.)?
Conjecture 4:
For the time series under reconstruction it is maybe better to infinitesimally move the Poincare section randomly. This is the case when the data are not acquired with a very high accuracy sampling rate i.e. the states recorded for the time series are in some neighboring of our desired Poincare section rather than exactly belonging to it. However this is the natural case!
I have found correlation dimension curves of an experimental 2000 data showing a saturation region around high embedding dimensions like 12. This is unexpectedly because for such a large dimensional attractor to be reconstructed we need so much more data. Also the determinism uncertainty test achieved for this time series shows a low uncertainty.
Conjecture 5:
Phenomena under effect of competing forces lead to fractal patterns. Among these systems are more famous cases such as diffusion processes. City extension pattern is under effect of several competing factors such as closeness to important centers, remoteness from high pollutant areas and etc.
So it is sensible that we have actually fractal cities which perhaps obey the same rules as other diffusion systems. I suggest computing the correlation dimension of some city structural features like street networks and comparison to that of a diffusion system as the first step. Such a fractal feature may reflect some dynamical aspects of an alive system i.e. the City. This quantifier or signature of life may further more be reasonable to search for in other systems from human body to even an extremely far away galaxy.
Conjecture 6:
Life may have a scale less structure propagating from our galaxy or even the whole Universe to the Earth and to our bodies and beyond. So in order to find the life in the space it may be better to search for living galaxies (which are also capable of producing life in scales of the Earth) using for example a life fractal signature.