Grace Notes

Skip this post if you are not a math geek.

There’s a problem in Calculus by Spivak which asks the reader to prove that if f and g are convex, and f increasing then g○f, the composition (i.e. g○f(x) = g(f(x) ) is also convex.

It’s false.
Let g(x) = 1/2 * (x^2) – 5*x
Let f(x) = exp(x*2*ln(2))
Both are convex, f increasing.
But graphing g(f(x)) shows it’s definitely not convex on R.

Addendum: It’s problem 5 in the appendix to chapter 11.

Counter example showing the graph of the composite function

I’m sorry to report that I’ve had to disable comments. The amount of spam makes allowing comments not worth the time necessary to filter out the junk.

While not exclusive to one side or the other, there seems to be a much greater tendency on the left to engage in ad hominem attacks, while mainstream commenters on the right tend to refrain from such. Two examples illustrate: the smears, and now attempts to have her removed, of Miss California, and the calls for prosecution, or at least disbarment, of the lawyers who provided guidance on the enhanced interrogation methods. It seems that there are two sets of rules here, and the conservative side has agreed to be more constrained.

That doesn’t seem right. There are important principles at stake here. Trying to destroy or publicly humiliate someone with whom the you have a policy disagreement is to change the game. The would be destroyers ought to at least risk consequences proportionate to what they are attempting to do. Seek to disbar an attorney as a way to shut down debate is maybe to risk a significant financial or other judgment. Attempt to publicly disgrace someone; risk your own public disgrace. The issues matter and need to be debated on the merits, and not have things decided by who is the nastier name caller.

By playing by a set of Marquess of Queensberry rules we are unnecessarily handicapping ourselves. Those who engage in personal destruction and public humiliation cannot be accorded the status of those whom we simply disagree with and with whom we can reason together. They’ve already demonstrated that reasoning together is just a tactic to gain advantage at some later time. It’s time to realize that they are not opponents, but rather enemies.

[Update]

As I think about this, I do not believe that direct turnabout is right. For example, pusillanimous bullies who descend to school yard smears, that bear no relation to the truth, or who try to expose potentially embarrassing details, should not then likewise be villified and investigated with an eye to unearthing their embarrassing histories. Rather, I believe that such behavior should be confronted directly.

Also, there should be a way to mitigate harm. One example is that scurrilous claims are sometimes made to intimidate or tie someone up in legal proceeding and costs (e.g. Sarah Palin). When baseless charges are made, the target should be made whole. His legal fees should be reimbursed, and he should be compensated for time and trouble.

Or diminishing returns. Assistant Village Idiot has a great post about capitalism. One of his points really caught my attention, and seems obvious in retrospect. In a relatively free market, government intervention can focus on items which offer a relatively high return for a relatively low cost. Comparing a state of nature (nasty, brutish, and short) to a civilized society, we can see that the initial actions of the state have a very high payoff indeed.

However, diminishing returns applies here as well. Increasing interventions come at, usually, increasing costs and have, usually, decreasing benefits. So just as we rightly limit (well, up until recently anyway, but that’s another story) the amount of leverage a bank is allowed to have, it would make sense to limit the amount of intervention the government was allowed.

The idea would be to set down some rules to say that even if some program or policy might have some benefit that certain things were just not the state’s business. This is not a new idea. James Madison thought of it over 200 years ago. Unfortunately, this and other reasons for supporting this idea of a government of limited and enumerated powers have been lost or forgotten by a great many. We may see a time when government action crosses over the line where the costs exceed the benefits.

One of the main principles is conservation. Conservation of mass,  of energy, of charge, of lots of things. In games like bridge, cards in one place can’t be somewhere else. Generally, start with some stuff, process it, total up the result and the waste, you have what you began with, or maybe just a little less.

Only very rarely do we see genuine creation. Start with some stuff, process it, and end up with more than you began with. Sometimes a lot more. For example, paint and canvas are relatively cheap, but with the right painter, the value of the painting far exceeds the value of the input materials and the painters time valued by ordinary standards. New businesses come along and create value.

Yet unless we pay close attention, it is easier to think of value as being conserved. One man’s gain must be offset by loses to others. It’s not true, at least not always, because wealth is not fixed. Some things increase it, others decrease it. It’s the same with policy. Some policies serve to increase wealth and well being, others to reduce both.

With toxins, dose matters. Within a certain dosage range, to add 10% to the dose is to increase the effect by about 10%. Of course, above a limit, adding dosage doesn’t have much effect. After all, dead is dead. This could be summarized as more dose, more bad, less dose, less bad. So even a tiny dose causes a tiny bit of bad.

It turns out that this is not quite true. For many items, a very small dose actually strengthens the organism. The term J-curve comes from plotting the relation of harm against dosage.

None of this is new, but it occurs to me that it is important, as it becomes very hard to predict the behavior of nonlinear systems.

There is an old principle of mathematics that states if you put things into boxes and you’ve more things than boxes then some box will get more than one thing. An alternative form says that if there are fewer things than boxes, some box will go empty. This isn’t deep, more like common sense.

Now there is only so much stuff in our economy. As the government directs more of it, less can be used by private parties.

There are only a few things that can be at the top of the news. So one way to prevent criticism of an action is to distract attention, say by declaring a health emergency and pulling a stunt like a low flying plane near Manhattan. My guess is that cramming national health care down our throats via the budget reconciliation process might be it.

What does it mean to say what the probability of something is. For example when the weatherman says that there is a 30% chance of rain, what does that really mean?

Start with the simplest thing, a coin toss. A fair coin toss. The odds of a head coming up are 50%. But when we toss the coin it’s either heads or tails. One possible meaning is that if we toss the coin often enough, we’ll get approximately 50% heads. Mathematicians make this precise, but in terms of probability. So we’ve replaced the probability of the outcome of simple coin toss with a complicated mathematical gadget, again defined in terms of probability.  It is a mathematical advance, but not a philosophical one.

Another way to think about this is to consider wagering on the outcome. Here the rational view would look with equal favor on taking either heads or tails, with each player wagering a like amount. But this does not entirely get us out of the difficulty. There are risk averse people who consider the loss of a dollar worse than the gain of a dollar good, and they would have to be offered compensation before they would be willing to accept the wager. Say heads wind $1.25 while tails loses $1.oo. That certainly doesn’t match the 50-50 probability we associate with a fair coin toss. Or someone who, perhaps for superstitious reason, only wants to bet one way.

The fact that the toss is to occur doesn’t get us out of the difficulty either. Toss the coin into a box where the outcome is unknown. Wagering is still reasonable on whether the outcome will be revealed as a head or a tail. Yet it is already determined.

This leads me to ask: is probability of an event occuring intrinsically bound up with the subject, and not just the object?

Yes, Microsoft knows about this. It’s a carry over from lotus 1-2-3 which did the same thing.

Microsoft points out that correcting the error would cause problems as date numbers in existing spreadsheets would now be off by one, and other issues would arise as well. It is also noted that since dates around 1900 are rare in ordinary applications this error shows up only rarely.

I wonder if carrying the error forward to ease compatibility with 1-2-3 saved work in the long run?

It seems that while people are becoming more entitled to a whole host of things such as education, health care, old age pensions, and so on, they are becoming less entitled to keep the fruits of their labors. This seems backwards, and counterproductive.

The word “people” may be misleading here, as the individuals involved may be different.

This creates some counterproductive  incentives, which have been noted by others more eloquent and perceptive.