Archive for February, 2006

The Glory of Salads

Sunday, February 5th, 2006

Today I’m going to talk to you about salads. This is a subject I feel quite strongly about, so the post is going to be full of hyperbole and over the top language. But, let’s be honest, when are my posts not?

You probably think salads are boring. Some lettuce, cucumber, maybe tomatoes and carrots if you’re lucky. In order to make them interesting you need to pile them with dressing. Right?

Wrong! Wrong wrong wrong.

Indeed, mere words cannot express how wrong this is. So instead I am going to have to refer you to some higher authorities.

Here is what the bible has to say on the subject:

“Thou shalt not put the cucumber and a measly supermarket tomato on the lettuce and call it a salad, for that is an abomination.”

After a consultation with the Eschaton, it was convinced that the matter was of sufficiently great importance that the following appeared across the galaxy.

“4. Thou shalt not make boring salads within my historic light cone. Or else.”

Finally, if these have not convinced you of the severity of the situation, if you make boring salads then these cute kittens will cry.

So, on reflection, if you make boring salads then you will go to hell, your civilisation will be wiped out by a passing asteroid, and kittens will cry. Moreover, you will have a boring salad.

Now, I must explain how one goes about making an interesting salad.

The first myth to be disposed of is that a lot of dressing will make an interesting salad. If you put dressing on a boring salad then what you have is a boring salad covered in dressing. This might be edible, but it’s not an interesting salad.

The second thing to bear in mind about dressing is that, given a decent salad, it isn’t neccesary. I’m not saying it’s a bad thing – I really like salad dressing. But the more ingredients your salad has, the more the dressing becomes just an accompaniment to the salad – it’s not an integral part of it, and can quite happily be left out.

Implicit in this is that real salads should have a number of different ingredients. If you’re only going to have a few ingredients then they should be interesting ones.

For example, the salad I had for lunch today contained the following ingredients: Romaine lettuce (never ever use iceberg lettuce. It is the devil’s leaf.), sundried tomatoes, half an orange sweet pepper, two hard boiled eggs and a banana (do not doubt the use of banana in salad until you have tried it. It is awesome.).

I considered this to be a fairly basic salad. Ideally I’d have added some avocado, maybe a few capers, some seared red onion, etc. to it, but I didn’t have the first two and couldn’t be bothered to cook the third.

So, here is a list of some worthwhile salad ingredients. It is in no way exhaustive, and I’m definitely not suggesting you use all of them in a single salad.

  • Lettuce of course. My favourites are Romaine and little gems, but there is a wide variety of opinion on this. However the people who think iceberg lettuce is appropriate are simply wrong.
  • Cucumber fulfills a similar role to lettuce – it’s nice, bulks up the salad a bit, and has a simple flavour to it. It isn’t however in itself very interesting.
  • Carrots. Good quality carrots have a wonderful texture to them, and chopped or shredded (this is distinct from grated) carrot in a salad is very nice.
  • Good quality fresh tomatoes. None of those boring tasteless default supermarket tomatoes.
  • Avocado.
  • Sundried tomatoes.
  • Capers
  • Egg. Either scrambled (to the point where it’s dry rather than runny) or hard boiled.
  • Roast squash.
  • Sweet peppers. Either raw or cooked.
  • Good cheeses. Especially feta or mozzarrela.
  • Banana.
  • Raisins or sultanas.
  • Green beans.
  • Chickpeas.
  • Kidney beans.
  • Seared onion. Red is best here. You can also include them raw, but I don’t like it.
  • Tuna fish.
  • Anchovies.
  • Artichoke hearts.
  • Just about anything else that’s edible cold.

One particular combination (which I can’t eat any more) that works really well is that of banana, sundried tomatoes and feta. I know you’re probably looking skeptical at this, but try it anyway and then come back and yell at me if you’re still not convinced.

Having put together these salads, you can then drizzle dressing over them – vinaigrette, honey-mustard, sesame and soy sauce, whatever you feel like as long as it’s interesting.

So, spread the word. Salads can – and should – be interesting, and people who make boring salads will be the first against the wall when the revolution comes.

Silly proofs 2

Sunday, February 5th, 2006

I swear this was supposed to be Silly proofs three, but obviously my memories of having done two silly proofs are misleading.

This proof isn’t actually that silly. It’s a proof of the [tex]L^2[/tex] version of the Fourier inversion theorem.

We start by noting the following important result:

[tex]\int_{-\infty}^{\infty} e^{itx} e^{-\frac{1}{2}x^2} = \sqrt{2 \pi} e^{-\frac{1}{2}t^2}[/tex]

Thus if we let [tex]h_0 = e^{-\frac{1}{2}x^2}[/tex] then we have [tex]\hat{h_0} = h_0 [/tex] (where [tex]\hat{f}[/tex] denotes the fourier transform of [tex]f[/tex])

Let [tex]h_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2}[/tex]

This satisfies:

[tex]h_n – xh_n = -h_{n+1}[/tex]

So taking the Fourier transform we get

[tex]ix \hat{h_n} – i \frac{d}{dx} \hat{h_n} = -\hat{h_{n+1}}[/tex]

So, [tex]h_n[/tex] and [tex](-i)^n h_n^[/tex] satisfy the same recurrence relation. Further [tex]\hat{h_0} = h_0[/tex]

Hence we have that [tex]\hat{h_n} = (-i)^n h_n[/tex].

Now, the functions [tex]h_n[/tex] are orthogonal members of [tex]L^2[/tex], and so form an orthonormal basis for their closed span.

On this span we have the map [tex]h \to \hat{h}[/tex] is an isometric linear map with each [tex]h_n[/tex] an eigenvector. Further [tex]\hat{\hat{h_n}} = (-1)^n h_n[/tex]. Thus the fourier transform is a linear isometry from this space to itself.

Now, [tex]h_n[/tex] is odd iff n is odd and even iff n is even. i.e. [tex]h_n(-x) = (-1)^n h_n[/tex]

Thus [tex]\hat{\hat{ h_n(x)} } = (-1)^n h_n(-x)[/tex].

And hence [tex]\hat{\hat{h}}(x) = (-1)^n h(-x)[/tex] for any [tex]h[/tex] in the span. As both sides are continuous, it will thus suffice to show that the span of the [tex]h_n[/tex] is dense.

Exercise: The span of the [tex]h_n[/tex] is precisely the set of functions of the form [tex]p(x) e^{- \frac{1}{2} x^2 }[/tex], where [tex]p[/tex] is some polynomial.
It will thus suffice to prove the following: Suppose [tex]f[/tex] is in [tex]L^2[/tex] and [tex]\int x^n e^{-\frac{1}{2}x^2 } f(x) dx = 0[/tex] for every x. Then [tex]f = 0[/tex].

But this is just an application of the density of the polynomial functions in [tex]L^2[a, b] [/tex]: pick a big enough interval so that the integral of [tex]|f(x)|^2[/tex] over that interval is within [tex]\epsilon^2[/tex] of [tex]||f||^2[/tex], and this shows that the integral of [tex]|f(x)|^2[/tex] over that interval is [tex]0[/tex]. Thus [tex]||f||_2 < \epsilon[/tex], which was arbitrary, hence [tex]||f||_2 = 0[/tex]. (Note: When editing this for the new blog site I noticed that this proof is wrong. I haven't been able to fix it yet, but will update this when I do).
I’ve dodged numerous details here, like how the [tex]L^2[/tex] Fourier transform is actually defined, but this really can be turned into a fully rigorous proof – nothing in this is wrong, just a little fudged. The problem as I see it is that – while the [tex]L^2[/tex] Fourier theory is very pretty and cool – this doesn’t really convert well to a proof of the [tex]L^1[/tex] case, which is in many ways the more important one.