Mathematics Weblog

Dr Who’s Happy Primes

Saturday 19 May 2007 at 8:34 pm | In Articles | 1 Comment

It’s so nice to see mathematics playing an important part in popular television. In tonight’s Dr Who the spaceship crew had to find the next number in the sequence 313, 331, 367, …. Dr Who recognises this as a sequence of happy primesÂ with the next one being 379. It’s all explained at that Wikipedia articleÂ and a longer version of the sequence can be found at the wonderful The On-Line Encyclopedia of Integer Sequences.

It’s interesting that the Dr Who reference was put in the Wikipedia article half-an-hour before the programme was aired so probably an inside job. I was delighted thatÂ in the programme Dr WhoÂ asked if mathematics was so dumbed down that recreational maths wasn’t studied any more. As Russell T.Â Davies, the head writer,Â said in tonight’sÂ Dr Who Confidential, the programme reflects current concerns, so this problem has clearly reached a wider audience than I imagined was the case.

Eigenvalues without determinants

Friday 18 May 2007 at 3:14 pm | In Articles | 9 Comments

Most (all?) undergraduate courses use determinants to introduce eigenvalues and eigenvectors. So the eigenvalues of a matrix (or linear transformation ) are the solutions of $\det(A-\lambda I)=0$ . However, Sheldon Axler published a paper in 1994 called Down with Determinants! where he maintains that determinants should not be used so early in linear algebra courses. He gives a very nice proof of the existence of eigenvalues in finite-dimensional vector spaces (over $\mathbb{C}$ ) which I would like to reproduce here.

Every linear transformation of a finite-dimensional complex vector space has an eigenvalue.

Here is his proof:

Let be a non-trivial finite-dimensional complex vector space and tÂ a linear transformation $t\colon V \to V$ . Let be a fixed non-zero vector in and suppose that $\dim V =n$ . Then the n+1 vectors $v,t(v),t^2 (v),\dots,t^n (v)$ are linearly dependent. Hence there exists complex numbers $\alpha_0,\alpha_1,\dots,\alpha_n$ not all 0 such that

$\alpha_0 v + \alpha_1 t(v) + \dots + \alpha_n t^n (v)=0$ .

and hence

$(\alpha_01+ \alpha_1 t + \dots + \alpha_n t^n)(v)=0$ .

Now, since $\mathbb{C}$ is algebraically closed, Â the polynomial $\alpha_0 + \alpha_1 z+ \dots + \alpha_n z^n$ will factorise so we get

$\alpha_0 + \alpha_1 z+ \dots + \alpha_n z^n=c(z-\lambda_1)(z-\lambda_2)\dots(z-\lambda_m)$ .

where $c, \lambda_1,\dots,\lambda_m$ are complex numbers with $c \neq 0$ . It follows that

$c(t-\lambda_1 1)(t-\lambda_2 1)\dots(t-\lambda _m 1)(v)=0$

which means that, since this is composition of functions, and $v \ne 0$ , then
either
$(t-\lambda _m 1)(v)=0$ soÂ Â $(t-\lambda_m 1)(v)=v$ Â so that $t(v)=\lambda_m v$ and is an eigenvector
or
$(t-\lambda _{m-1} 1)(t-\lambda _m 1)(v)=0$ so $(t-\lambda _m 1)(v)$ Â is an eigenvector
or
…
or
$(t-\lambda_2 1)\dots(t-\lambda _m 1)(v)$ is an eigenvector,
and hence has an eigenvalue. Â Â $\blacksquare$

Discussion on this approach of not using determinants can be found at NeverEndingBooksÂ and The n-category CafÃ©.

Problems, Problems

Saturday 5 May 2007 at 2:22 pm | In Articles | 3 Comments

The Art of Problem Solving (AopS) site encourages, teaches and promotes mathematics competitions from small local ones right up to the International Mathematical Olympiad (IMO). As it says on its front page:

Is math class too easy for you? Looking for a greater challenge?
You’ve come to the right place.

It has an excellent introduction to $\LaTeX$ siteÂ and a forum to discuss problems. The forum has an RSS feed where students pose new problems every day. So if you’re “looking for a greater challenge” subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.

2. Prove that $1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots<2.8$ . 3. Let a,b,c be nonzero real numbers. Find all ordered pairs (a,b,c) such that $\displaystyle\frac{2(a-b-c)}{a^{2}}=\frac{4b-a-2c}{b^{2}}=\frac{4c-a-2b}{c^{2}}$ .

4. is a continuous complex-valued function satisfying:
i) |f(z)| = |z|
ii) |f(z)-z| = |z|
Find f(f(f(2007)))

5. If and are relatively coprime, find all possible values of $\gcd(a+b,a^{2}+b^{2})$ .

6. Let A,B,C be three angles ofÂ $\triangle ABC$ . Prove that $(1-\cos A)(1-\cos B)(1-\cos C)\ge\cos A\cos B\cos C$ .

7. For each function which is defined for all real numbers and satisfies f(xy)=xf(y)+yf(x) and $f(x+y)=f(x^{1993})+f(y^{1993})$ determine the value of $f(\sqrt{5753})$ .