Hi Nad, ok, thanks for the clarifications. There no urgency concerning the forum, and I am not presently trying to do any housekeeping on the wiki. It was only by coincidence that the certificate on the wiki just expired. I am doing things like t…
Nad, there is no scrap yard. There are some older backups which I may be able to recover the article from. If this matters to you, send me a DM with some search query for the title, and I will restore it to another database and do my best to recover…
Fair enough. I had thought that you hadn't been active here for many years, but on closer examination I see that you have been commenting from time to time. (There have been some gaps in my following of the forum.) Going forward I will post no…
@zskoda there have been some problems with spam on the wiki, and presently I don't have the bandwidth to deal with this. So I put it in a read-only mode.
As a work around, you can describe any changes that you want made right in this thread. I w…
Hi Éricles,
Welcome to the forum! Presently there are only a few people active at this time, so, even though it is a good idea, it is possible that you may not get much response to this specific suggestion. Time will tell. In any case, feel fre…
This blog will cover various topics in mathematical science. We begin with some primers on epidemic modeling. No prior background in math or epidemiology is assumed here. These articles are kept short; suitable for morning coffee.
Series:…
Now let's review the above presentation of the model.
Granted, it is a qualitative description. But does it correctly telegraph the essence of the algorithm?
(For the exact details, we do have the paper itself to refer to.)
Once the description…
Computer visualization
Just by varying the parameters \(\alpha\) and \(\beta\), Xia shows that the algorithm produces a variety of natural looking leaves.
In figure 3, when \(\alpha = 0.68\) and \(\beta = 0.38\), a leaf is produced which looks rem…
On the other hand, if \(\alpha\) is not "close to zero," then growth will stop.
The specific result that Xia proves is that if \(\alpha\) exceeds 0.5 then growth will eventually stop.
Suppose the leaf was giant.
And suppose the algorithm would attempt to add a new cell \(x\) at the fringe of this giant leaf.
\(x\) needs the standard rate of fluid flow to support it. Call this flow amount \(y\).
Now \(y\) would have to be adde…
What happens when \(\alpha\) is very small, close to zero?
Then \(w(e)^\alpha\) is close to 1, and this multiplicative factor drops out of the picture.
Which is to say, when \(\alpha\) is small, close to zero, there is such an economy of scale tha…
Note that the flow rate is conceptually proportional to the cross-sectional area of the pipe - since wider pipes are needed to support greater flow rates.
Recall from comment 14 that the cost for transporting fluid through a pipe \(e\) is \(w(e)^\alpha \cdot L(e)\), where \(L\) is its length and \(w(e)\) is the flow rate through the pipe.
And the parameter \(\alpha\) was stipulated to be between 0 an…
Well, as the leaf gets bigger, the points on its fringe get further and further away from from the root.
It therefore gets more and more expensive to add new cells to the edge of the leaf, as the cost of transporting fluid to them gets larger and l…
Clearly, real leaves in nature do stop growing.
So, if this is to be a plausible empirical model, we should hope to be able to prove that the algorithm will in fact terminate at some point.
Optimization substep
This substep begins with the completion of the growth substep - which has given us an extended leaf (by adding new cells at the fringe) along with a transport system for the extended leaf.
Now what we do is to look at the cell…
Note that the added cost of a cell depends upon which parent is chosen for it in the current leaf.
So the check for whether a cell should be added implicitly also includes the determination of what its parent cell will be.