## Two supply optimization problems

### October 28, 2013

1. Computer program power consumption
A programming language that “minimizes” power consumption through minimal interconnect usage (e.g. memory calls).
2. Food sourcing power consumption
Farmland supply to cities: how to optimize land usage? What part of the produce can be made local e.g. made at the consumer or turned to hydroponic and its culture brought within the city itself?

Both these problems require a grammar of solutions, rather than a single instance, due to the diversity of the operating/boundary conditions that are encountered.
As such, I don’t think that a “proof of correctness” for either can be hoped for, but perhaps a number of heuristic checks might prove the point.
The former is addressed by a single technology, whereas the second requires a diverse array of strategies.

General considerations

• Area and land usage
Arbitrary rearrangement of the resources is not trivial: CPUs are designed with CAD tools that favor periodicity and reuse, and farmland restricts supply due to physiological productivity/rest cycles.
• Time and flow
Time plays a part as well: the edges in these supply nets do not handle a constant flow. In the first case, storage is regulated by registers, queues and stacks, whereas in the second, the flowing entities are subject to seasonal variation, degrade with time etc.

This framework is intentionally generic in order to highlight similarities, and it is of course a work in progress.
Both these problems in fact have broad political implications, which leaves plenty of space for many juicy discussions. Looking forward.

Literature

1. An article from the NYT: A Balance Between the Factory and the Local Farm (Feb. 2010) highlights both the high costs of local (i.e. small-scale) green production, citing The 64\$ Tomato, and the related climatic issues (e.g. cultivation on terrain located in the snow belt).
The article closes with “Localism is difficult to scale up enough to feed a whole country in any season. But on the other extreme are the mammoth food factories in the United States. Here, frequent E. coli and salmonella bacteria outbreaks […] may be a case of a manufacturing system that has grown too fast or too large to be managed well.
Somewhere, there is a happy medium.” — an optimum, if you will.

Side questions

• Why do large-scale economics “work better”? i.e. have a larger monetary efficiency, which drives down the prices for the end user? More effective supply chain, waste minimization, minimization of downtime …

## On plant growth and form

### September 28, 2013

How quickly does the number of leaves on this plant increase, and why?

The photo above is of a Schlumbergera truncata (or “Christmas cactus”), which my mother (from whom I got the parent plant) calls “Woman’s tongue”. The latter might be seen as a rather sexist definition (pointy and forked..) but I guess it’s an old heritage, from when the world was younger..

Anyway, how fast do the leaves increase in number, and what drives the general form of the plant?
Say we have initially N branches, each that can generate one to three leaves (branching factor p), more or less at random.
If the branching factor were constant and identical for all branches, after one generation we would see 2N to 4N leaves; after two, 3N to 16N, or, for the j-th branch: $L_j = \sum\limits_i{l_{i\,j}}$.
The number of leaves $l_i$ at each level i is seen to be $\Theta(l_{i-1})$ where the bound constants are $\min(p)$ and $\max(p)$.

In the real world, a branch has spatial extent, with each leaf occupying a volume, with no intersections or overlaps in $\mathbb{R}^3$. Moreover, the leaves seek and thrive in sunlight (phototropism), which conditions their orientation and size.
Therefore we could represent a plant as a vector field, with “streamlines” (nutrient direction) and “polarization” (orientation to sunlight).
Gravity, through the self-weight loading of the branches and capillary diffusion of nutrients, obviously plays a part.
We can sketch the processes at play in the following, informal manner. First some notation: for leaf k of level i, the leaf size $s_{i\,k}$, orientation $\alpha_{i\,k}$, sunlight exposure (intensity flux) $\phi_{i\,k}$, viability (function of exposure and distance from the root) $v_{i,k}$, repulsion potential (function of size and distance to nearest leaf neighbors) $\rho_{i\,k}$, branch node stiffness $\sigma_{i\,k}$, which increases as the leaf matures.
Moreover we can define the branch vitality $v_b$ as the sum over all the parent leaves (viability upstream).
$s \rightarrow \phi , \rho, \sigma$ (leaf size influences exposure, repulsion and stiffness)
$\phi, v_b, \sigma \rightarrow s, v, \alpha$ (exposure, viability and stiffness influence leaf size, viability and leaf angle)
$\rho \rightarrow \phi$ (repulsion influences exposure through position: model requires collision and ray-tracing, which require considerable computational effort)
$\alpha_{i-1\,k} \rightarrow \alpha_{i\,k}$
$\alpha \rightarrow \phi$, but also
$\phi \rightarrow \alpha$
As each leaf matures, it becomes larger in size, stiffer and possibly its exposure efficiency reaches a plateau.