Isomorphismes

This has confused me ever since I learned about evolution.

As Mr. Shepherd explained it to me in the second grade, zebras are hard to kill because, when the zeal runs, their collective stripes make it hard for a predator to visually pick out an individual zebra to tackle.

So the benefit of stripes comes from

• high contrast
• highly structured shapes
• in a group

In economic terms, increasing returns to scale — and increasing returns to coördinated group action (“Let’s all be stripey”).

COÖPERATION by Selfish Genes?

So how can that process get started, if the returns to stripes don’t pile up until you get to a really high level of contrast, a very certain structure, and group coördination? Even with a constant scaling factor, how could genes that promote themselves, inside an individual zebra, coördinate to get many zebras to all adopt stripes at once?

It’s not like genes can talk to each other. How could they have coöperated to achieve a better group outcome? Humans do so with language. If there’s a fire in the building and everyone’s crowding the exit, someone can yell: “Stop! We all need to back up and wait in a line. Then we’ll get through this exit faster and all live. None of us will get through if we push each other.” But the genes don’t have a way to communicate for better group outcomes — right?

Maybe stripes could evolve if it only took a few mutations to turn on high-contrast stripes (then the possibility of coöperation arising randomly would be greater … but still small). But I also wonder if there isn’t a population dynamics answer. Or a game theory answer. Is sexual selection involved? Is the sexual transfer of genes involved?

EVOLUTION OF STRINGS

I think about evolution in an abstract way. Even though I know there’s meiosis and specific proteins and RNA’s change the expression and so on, … I just think about genes-as-strings. They mutate and cross-pollinate, with the sexier strings pollinating more. Nature selects (in a Brownian manner) from the pool for the next generation.

Does anybody know a good book or paper about the mathematics of sexual selection, like in a dynamical systems model? Or some other explanation for how the zebra got its stripes?

ANSWERS FROM READERS

Readers answered with a lot of mathematical biology links (great!). As I read through what they’ve sent, I’ll add to this list:

• The pattern of zebra stripes is fully determined by day 21-35 of embryonic development (out of a year-long gestation). Melanoblasts mark out the patterns on the zembryo. Turing Patterns in Animal Coats, via Artemy Kolchinsky

You know what’s surprising?

• Rotations are linear transformations.

I guess lo conocí but no entendí. Like, I could write you the matrix formula for a rotation by θ degrees:

But why is that linear? Lines are straight and circles bend. When you rotate something you are moving it along a circle. So how can that be linear?

I guess 2-D linear mappings ℝ²→ℝ² surprise our natural 1-D way of thinking about “straightness”.

When I was a math teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story.

You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession — Peano map, Peano map, Peano map, Peano map. Sure. But that’s an example of the number 4, not the number 4 itself.

So is it even possible to say what a number is? No, let’s ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.

Willard van Orman Quine had an interesting answer. He said that the number seventeen “is” the equivalence class of all sets of with 17 elements.

Accept that or not, it’s at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and ∞ serve a practical purpose just like the concept of ⅓ (you know, the obvious moral cap on income tax). For instance

• if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)
• ∞ allows for limits and therefore derivatives and calculus. Just one example apiece.

Do 6-dimensional spheres exist? Do matrices exist? Do power series exist? Do vector fields exist? Do eigenfunctions exist? Do 400-dimensional spaces exist? Do dynamical systems exist? Yes and no, in the same way.

Not talking about camping outside. I mean the elements as in the periodic table of the elements.

How did they come to be? Why are there more of some and less of others? Why does 29 protons conduct electricity 10 times better than 31 protons? Etc.

There are six salient facts that a theory of the origin of the elements must explain:

1. H and He are by far the most abundant.
2. Elemental abundances generally drop with increasing atomic number.
3. Even atomic numbers are more common than odd atomic numbers.
4. Li, Be, and B are anomalously rare.
5. Fe is anomalously abundant.
6. Tc, Pm, and elements above Bi are extremely scarce or nonexistent — except for U and Th.

(Source: ocw.mit.edu)

π of course is the distance around a circle. √π is the area under ∫exp (−x²), and exp (−x²) is the key ingredient in the normal distribution.

That’s more or less what √π means—the area under the Bell curve.

But what does it mean mean? I mean, if π is a distance and √ is used to turn areas into distances — is it, like, shrinking the π even one more time? Are we talking about a half-dimension here?

edit: hmm, the end of this post seems to have been deleted by the rare weirdness of tumblr’s mass editor. I’ll see if I can’t remember how it ended. Umm, something about the moment-generating function? (i.e. going around the complex unit circle)

Every time I hear that bond investors are flocking to quality, and that quality = lending to the U.S. government, I think the following:

• what’s so high quality about U.S. government debt?
• is this really just because business schools use Treasury’s as the canonical example of a “risk-free” asset?
• what’s a “good” debt-to-income ratio for a government, anyway?

The U.S. economy is big

The U.S. produces roughly $10 trillion worth of goods and services per year, and its government can tax that — to a degree. But other countries also can tax their citizens. And if you added together some smaller countries you could find another $10 trillion of taxable income. (Governments can tax old assets too, like real estate.)

If you’re not a giant fund looking to plunk all your money in one place, you could lend money to 5000 municipalities or cities or something, and is California really any more interested in defaulting than the U.S.? And California is like the world’s 4th largest economy, if you broke it off from the rest of the States. Not only that, but “diversification” is the buzzword for panacea in finance.

“I’ll believe in whatever they believe in”

I get that, like, people think the US Government won’t default on its debt. Partially that’s due to a feedback loop—so many people believe it to be the case that the government can cheaply borrow to cheaply pay back the current debts with cheap future money. After the “financial meltdown” people and institutions from all over the world were shoveling so much money into the Treasury’s coffers that the government would have been stupid to turn them away. This when Uncle Sam was spending, what, $2 trillion on the bailout?

But a feedback loop the other way — a sudden loss of trust — could cause Uncle Sam’s lenders to jack up the API on his credit card, and each dollar of interest might spike to $1.05, $1.15, $1.25, who knows. The laws of calculus would no longer apply, i.e., small changes might become huge jumps and suddenly the default looks a lot more attractive — thus raising the interest due even more.

_Why?

Really, I don’t get it. I have never invested in a bond in my life. I have never analyzed a municipality’s balance sheet, a county’s, a state’s, or a country’s. I’m not even sure if I should be writing Treasuries, Treasurys, or Treasury’s. But still I wonder, why do people consider an investment in U.S. Treasurys to be risk-free?

This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

Or, in fancy notation:

Or, in C++:

long big = 9999999999;
single frac = .70;
double total = 0;

for ( i = 0, big, i++)
{
  total += frac^i;
  }

cout << total;                 # in this case, prints 1 / .3 = 10/3
cout << total - 1/(1-frac)     # prints 0 for any value of frac


Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

Take that, Zeno!

Here is a really broad question.  What’s the lay of the land regarding maps of continuous surfaces to discrete ones?  I’m thinking here of credit scores. Someone’s credit score is continuous, one-dimensional — and derives from a multitude of measures that are both continuous and discrete.  As a lender, you have to decide at some point, whether or not to lend to this person — a yes/no proposition.  Granted, you can charge different interest rates.  So maybe it’s not a continuous-to-discrete problem?  Nevertheless it has that flavour.

Logistic maps come to mind.  As does the expansion of a point into several branches.  Maybe someone can lay this out better….  Takers?