Built on Facts

Sunday Function

And now, two quick notes before we get to business:

1. God help me, but I’ve joined the Twitter bandwagon. Here I am, @BuiltOnFacts. Though it goes under this blog’s name, it is more of a “personal” account. So you’ll be reading some incomprehensible personal minutiae, random observations, wild assertions, and somewhat more politics that I typically introduce here. But you may enjoy it nonetheless, and if you ever have physics questions that can be answered in 140 characters feel free to fire away.

2. Speaking of politics, this paragraph is political and I dislike it when people shoehorn politics into non-political posts. As such I will not even be slightly offended in you skip this paragraph. Onward: I’m a little horrified at some of the the horror that’s followed Citizens United v. FEC. Some of it’s due to misreporting (corporate campaign contribution limits were upheld, for instance). Some of it’s rabbit-chasing with the “corporate personhood” debate, which actually has pretty much nothing to do with the case, which should have come out the same even if corporations weren’t “persons” (the 1A doesn’t confer rights on people (except for assembly), it prohibits the government from making laws that abridge certain things). Finally, pretty much all speech that rises above personal pamphleteering is corporate speech in some sense – including the speech you’re reading now. The decision is certified ACLU-kosher, and I highly recommend constitutional law professor Ilya Somin’s excellent look at the issue. You can certainly be a proud liberal and still cheer the decision.

Ok, how about an actual function? I’m picking out a function that contains oceans of depth, but we’re just going to dip our toe into the water to see what the water feels like. You may be familiar with the Legendre polynomials, which we’ve talked about on a few occasions. They’re just a set of ordinary everyday polynomials that happen to have certain useful properties. They’re numbered by the order of the polynomial. The zeroth order polynomial is 1, the first order is x, the second is .5(3x^2 – 1), well the coefficients can get a little complicated but they are just regular polynomials. Let me plot the first four on the same graph:

i-7b7142f1efa09b47c3b8242f89bcb9f4-graph.png

You can see that some of them are even (they’re symmetric about the y-axis) and some are odd (inverted with respect to the other side of the y-axis). We’ve explored this property in the past, too. More on this in a second.

Our function contains all of the Legendre polynomials in one fell swoop. It’s the generating function of the Legendre polynomials:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

The Legendre polynomials are defined as the functions Pn(x) that make that equation true. Every single one of the properties of the Legendre polynomials from the recursion relationships to the differential equation to their orthogonality can be derived directly from the generating function. It can get somewhat complicated, so we’ll only derive one of the easiest properties today. Let’s derive the parity (even/odd) characteristics of the Legendre polynomials.

First, x and t are variables so we can let them be whatever we want. Let’s replace t with -t and x with -x:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

Negative times negative is positive, so actually the middle expression is the same both here and in the original expression. Equating equals with equals, this means:

i-d18063683dcc0d42b9be45451a84d1e3-3.png

We can pull the -1 out from the parentheses:

i-8d7070ce28f0af28047c7a5fb8246d53-4.png

Now here’s the key thing. This is a sum so we can’t just cancel the t willy-nilly. But we can recognize that each t^n is attached to a unique Pn and effectively serves just as a label. In other words, what’s true for the sum is also true term-by-term. If you’re skeptical, just spend a few moments thinking about it, or write down a few terms explicitly to see how it works. This means:

i-bb5c2c6b0452df43a61e3974bd9b473f-5.png

Now we can cancel the t^n:

i-9ef27a331268eac0961c3fd4d1a55446-7.png

Which is precisely the statement of alternating even/odd parity that we expected from the graph. Ok, I admit this one was a little esoteric. But I think it’s still pretty cool!

Comments

  1. #1 Andy from the UK
    January 24, 2010

    Hey Matthew

    Sorry – I’m not on Twitter but I noticed someone asked for something like this: http://www.funnyordie.co.uk/videos/0e4a1fa827/hitler-finds-out-about-another-downfall-parody

    All the best from Englandshire!

  2. #2 Uncle Al
    January 24, 2010

    Legendre polynomials are orthogonal – and easier to spell than Chebsh, Chebys, Kebsev… Gegenbauer.

  3. #3 Lyle
    January 24, 2010

    Even given the decision on corporate speech, the situation is better than it was in the past when lobbyists passed out money in the form of bribes to law makers. (Recall for a while the NY Leg was said to be the best money could buy, the question was what did it keep a law maker to stay bought?) It is interesting that the reaction from a large group of businessmen of leave me alone I don’t want to contribute to you blanky blank campaign did not get much coverage.

  4. #4 James Brennan
    January 24, 2010

    “…and if you ever have physics questions that can be answered in 140 characters feel free to fire away”

    So, what’s the deal with string theory?

  5. #5 Alex
    January 24, 2010

    You missed the more obvious defence of the decision, which was that most attacking it were attacking it because it came to a conclusion they didn’t like, rather than attacking it based on whether or not they felt it was legally correct according to the Constitution.

  6. #6 Seth Finkelstein
    January 25, 2010

    @Alex – If four justices indeed felt it was NOT legally correct according to the Constitution – and it had not been legally correct according to the Constitution for many years before – then frankly, the Constitution can’t be all that clear on it.

  7. #7 Alex
    January 25, 2010

    If four justices indeed felt it was NOT legally correct according to the Constitution – and it had not been legally correct according to the Constitution for many years before – then frankly, the Constitution can’t be all that clear on it.

    Your conclusion doesn’t follow – it could easily be the case that the 4 justices are just plain wrong.

    But anyway you missed my point which was that most people arguing against this decision were doing so because it will produce (in their opinion) bad results, and not whether or not it is constitutional to ban the political speech of corporations and unions etc.

  8. #8 Alex
    January 25, 2010

    Oh yeah, and Seth:

    Is segregation constitutional or not? Do you oppose Brown because it overturned Plessy? Is the constitution not clear on segregation, or perhaps were the 7 justices in Plessy just plain wrong?

    And may I draw you to the dissenting opinion in that case:

    The white race deems itself to be the dominant race in this country. And so it is in prestige, in achievements, in education, in wealth and in power. So, I doubt not, it will continue to be for all time if it remains true to its great heritage and holds fast to the principles of constitutional liberty. But in view of the constitution, in the eye of the law, there is in this country no superior, dominant, ruling class of citizens. There is no caste here. Our constitution is color-blind, and neither knows nor tolerates classes among citizens. In respect of civil rights, all citizens are equal before the law. The humblest is the peer of the most powerful. The law regards man as man, and takes no account of his surroundings or of his color when his civil rights as guaranteed by the supreme law of the land are involved.

    Justice Harlan saw that, while he saw blacks as inferior to whites, that was irrelevant to him as a justice. All that is relevant is what the law says.

  9. #9 Paul Murray
    January 26, 2010

    Absolutely do not understand this post. What’s g(t, x)? Why are there three things, all equal to each other? Does the first equal sign actially mean “is defined as being” rather than “is numerically equal to”? Why are there two variables x and t – is “t” simply a replacement for “y” in y=.5(3x^2 – 1) ? Or is t some sort of constant which you choose, and which then gives you a sreies of polynomials for that t?

  10. #10 annuity quotes
    January 27, 2010

    There used to at least be the “illusion” of interpreting the constitution for these rulings. But it is clear that all of the major rulings fall with a 5-4 decision along party lines.

    Even Gore v Bush was a 5-4 split. So we go through all the motions and intellectual debates only to have conservatives take the conservative position, liberals take the liberal position, and moderates bounce back and forth.

    How is this any different than the other two branches of government…. except for the black robes.

  11. #11 MPL
    February 4, 2010

    As a liberal, I’ll actually mostly agree with you about #2. Most of the complaints are “corporations aren’t people!”, which, as you point out, is relevant to neither side of the decision.

    Honestly, I would be happier if corporations were directly, and transparently spending money on candidates, than the loophole-ridden situation where the spending happened, just funneled in strange ways. The money’s going to be spent, better not to have it hidden.

    I would greatly appreciate the honesty of an ad that went something like “Vote for [name here], paid for by [disreputable company]“. It would make it so much easier to judge things for naked self-interest or real concern.

    In the end, my reaction is mostly pragmatic: campaign finance reform had almost no discernible impact on the amount/source of money in politics, as far as I could tell. Getting rid of it probably won’t matter too much either.

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