College Math Teaching

June 15, 2016

Elementary Math in the news: elections

Filed under: calculus, elementary mathematics, news — Tags: — collegemathteaching @ 9:11 pm

Ok, mostly I am trying to avoid writing up the painful details of a proposed mathematics paper.
But I do follow elections relatively closely. In the California Democratic primary, CNN called the election for Hillary Clinton late on June 7; at the time she lead Bernie Sanders 1,940,588-1,502,043, which is a margin of 438,537 votes. Percentage wise, the lead was 55.8-43.2, or 12.6 percentage points.

But due to mail in balloting and provisional ballot counting, there were still many votes to count. As of this morning, the totals were:

2,360,266-1,887,178 for a numerical lead of 473,088 votes. Percentage wise, the lead was 55.1-44.0, or 11.1 percentage points.

So, the lead grew numerically, but shrunk percentage wise.

“Big deal”, you say? Well, from reading social media, it is not obvious (to some) how a lead can grow numerically but shrink as a percentage.

Conceptually, it is pretty easy to explain: suppose one has an election involving 1100 voters who MUST choose between candidates. Say the first 100 votes that are counted happened to come from a strongly pro-Hillary group, and the tally after 100 was 90 Hillary, 10 Bernie. Then suppose the next 1000 was closer, say 550 for Hillary and 450 for Bernie. Then the lead grew by 100 votes (80 to 180) but the percentage lead shrunk from 80 percentage points to a 16.36 percentage point lead (58.18 to 41.82 percent). And it is easy to see that if the rest of the vote was really 55 percent Hillary, her percent of the vote would asymptotically shrink to close to 55 percent as the number of votes counted went up.

So, how might one have students model it? Let H(t), B(t) be increasing functions of t which represent the number of votes for Hillary and Bernie as a function of time. Assume no mistakes, hence H(t), B(t) can be assumed to be increasing functions. So we want a case there D(t) = H(t)-B(t) is an increasing function but P(t) = \frac{H(t)}{H(t)+ B(t)} decreases with time.

Without calculus: rewrite P(t) = \frac{1}{1+\frac{B(t)}{H(t)}} and note that P(t) decreases as \frac{B(t)}{H(t)} increases; that is, as B(t) outgrows H(t) . But H(t) must continue to outgrow B(t) . That is, the new ballots must still include more Hillary Bernie ballots, but the ratio of Bernie ballots to Hillary ballots must be going down.

If we use some calculus, we see that H'(t) must exceed B'(t) but to make P(t) decrease, use the quotient rule plus a tiny bit of algebra to conclude that H'(t)B(t)-B'(t)H(t) must be negative, or that \frac{B'(t)}{B(t)} > \frac{H'(t)}{H(t)} . That is, the Bernie ballots must be growing at a higher percentage rate than the Hillary ballots are.

None of this is surprising, but it might let the students get a feel of what derivatives are and what proportional change means.

June 7, 2016

Pop-math: getting it wrong but being close enough to give the public a feel for it

Space filling curves: for now, we’ll just work on continuous functions f: [0,1] \rightarrow [0,1] \times [0,1] \subset R^2 .

A curve is typically defined as a continuous function f: [0,1] \rightarrow M where M is, say, a manifold (a 2’nd countable metric space which has neighborhoods either locally homeomorphic to R^k or R^{k-1}) . Note: though we often think of smooth or piecewise linear curves, we don’t have to do so. Also, we can allow for self-intersections.

However, if we don’t put restrictions such as these, weird things can happen. It can be shown (and the video suggests a construction, which is correct) that there exists a continuous, ONTO function f: [0,1] \rightarrow [0,1] \times [0,1] ; such a gadget is called a space filling curve.

It follows from elementary topology that such an f cannot be one to one, because if it were, because the domain is compact, f would have to be a homeomorphism. But the respective spaces are not homeomorphic. For example: the closed interval is disconnected by the removal of any non-end point, whereas the closed square has no such separating point.

Therefore, if f is a space filling curve, the inverse image of a points is actually an infinite number of points; the inverse (as a function) cannot be defined.

And THAT is where this article and video goes off of the rails, though, practically speaking, one can approximate the space filling curve as close as one pleases by an embedded curve (one that IS one to one) and therefore snake the curve through any desired number of points (pixels?).

So, enjoy the video which I got from here (and yes, the text of this post has the aforementioned error)

Infinite dimensional vector subspaces: an accessible example that W-perp-perp isn’t always W

Filed under: integrals, linear albegra — Tags: , — collegemathteaching @ 9:02 pm

This is based on a Mathematics Magazine article by Irving Katz: An Inequality of Orthogonal Complements found in Mathematics Magazine, Vol. 65, No. 4, October 1992 (258-259).

In finite dimensional inner product spaces, we often prove that (W^{\perp})^{\perp} = W My favorite way to do this: I introduce Grahm-Schmidt early and find an orthogonal basis for W and then extend it to an orthogonal basis for the whole space; the basis elements that are not basis elements are automatically the basis for W^{\perp} . Then one easily deduces that (W^{\perp})^{\perp} = W (and that any vector can easily be broken into a projection onto W, W^{\perp} , etc.

But this sort of construction runs into difficulty when the space is infinite dimensional; one points out that the vector addition operation is defined only for the addition of a finite number of vectors. No, we don’t deal with Hilbert spaces in our first course. 🙂

So what is our example? I won’t belabor the details as they can make good exercises whose solution can be found in the paper I cited.

So here goes: let V be the vector space of all polynomials. Let W_0 the subspace of even polynomials (all terms have even degree), W_1 the subspace of odd polynomials, and note that V = W_0 \oplus W_1

Let the inner product be \langle p(x), q(x) \rangle = \int^1_{-1}p(x)q(x) dx . Now it isn’t hard to see that (W_0)^{\perp} = W_1 and (W_1)^{\perp} = W_0 .

Now let U denote the subspace of polynomials whose terms all have degree that are multiples of 4 (e. g. 1 + 3x^4 - 2x^8 and note that U^{\perp} \subset W_1 .

To see the reverse inclusion, note that if p(x) \in U^{\perp} , p(x) = p_0 + p_1 where p_0 \in W_0, p_1 \in W_1 and then \int^1_{-1} (p_1(x))x^{4k} dx = 0 for any k \in \{1, 2, ... \} . So we see that it must be the case that \int^1_{-1} (p_0(x))x^{4k} dx = 0 = 2\int^1_0 (p_0(x))x^{4k} dx as well.

Now we can write: p_0(x) = c_0 + c_1 x^2 + ...c_n x^{2n} and therefore \int^1_0 p_0(x) x^{4k} dx = c_0\frac{1}{4k+1} + c_1 \frac{1}{2 + 4k+1}...+c_n \frac{1}{2n + 4k+1} = 0 for k \in \{0, 1, 2, ...2n+1 \}

Now I wish I had a more general proof of this. But these equations (for each k leads a system of equations:

\left( \begin{array}{cccc}  1 & \frac{1}{3} & \frac{1}{5} & ...\frac{1}{2n+1} \\  \frac{1}{5} & \frac{1}{7} & \frac{1}{9}...&\frac{1}{2n+5} \\  ... & ... & ... & ... \\  \frac{1}{4k+1} & \frac{1}{4k+3} & ...& \frac{1}{10n+4}     \end{array} \right)       \left( \begin{array}{c}  c_0 \\  c_1  \\  ...  \\  c_n   \end{array} \right) =     \left( \begin{array}{c}  0 \\  0  \\  ...  \\  0  \end{array} \right)

It turns out that the given square matrix is non-singular (see page 92, no. 3 of Polya and Szego: Problems and Theorems in Analysis, Vol. 2, 1976) and so the c_j = 0. This means p_0 = 0 and so U^{\perp} = W_1

Anyway, the conclusion leaves me cold a bit. It seems as if I should be able to prove: let f be some, say…C^{\infty} function over [0,1] where \int^1_0 x^{2k} f(x) dx = 0 for all k \in \{0, 1, ....\} then f = 0 . I haven’t found a proof as yet…perhaps it is false?

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