Archive for January, 2006

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Latkes

Thursday, January 12th, 2006

I was bored.

I don’t just mean the sort of boredom that arises from not being very interested in what you are doing, I mean the sort of soul crushing mind numbing boredom that rises up from the deep, saps you of all your energy and causes you to reach the point where anything you might do to prevent the boredom just seems like too much work.

As you can imagine, this sort of boredom is a problem. I get it a lot, and being currently unemployed (which will change soon, hurrah! I just need to decide which offer to accept.) while all my friends are gainfully employed in jobs or PhDs, I’m particularly prone to it at the moment. There are plenty of things I could be doing, but if I get into a slump then they all seem like too much effort.

This time however, a solution arose. Lunchtime.

My stomach grumbled. “David”, it said “I hunger. You should feed me.”

“I don’t know. That sounds like a lot of work.

“Yeah, it is. But if you don’t feed me then I’ll be forced to escape from your body and go on a rampage. Millions will die to sate my hunger, and it will be all your fault.”

“Hmm. Well, that would be bad, yes. But I’m still not sure…”

My brain chimed in. “Yes. You should cook something. I’m bored out of your mind here. Cook something new and write it up for ‘Playing with your food’. That way you’ll have an hour or two of entertainment, and your three readers will get something to amuse them as well.”

“Alright, alright. You’ve convinced me.”

When your body parts conspire against you there’s really nothing to do but to go along with their wishes.

Unfortunately I was still uninspired. This needed to be remedied if I were to effectively produce a new meal. So, Robin, to the blogmobile!

My method of finding new cooking blogs to peruse was very simple. I wandered over to Food, in the main… and clicked on every link on the right hand side of the page.

Eventually I settled on this. I’ve heard about latkes before and I keep thinking “Hmm, I should try to make those.” but never get around to it. Well, the time for procrastination was over. Latkes it is.

Purely by chance, somewhere in between deciding on this and cooking it I thought I’d check my weight (I haven’t in months) and noted that I’d somehow managed to drop down to about my desired ideal weight with no conscious effort on my part. Right after christmas and new years.

How much oil was in that recipe again?

Anyway, time for some cooking.

What I used

About 7 small to medium potatoes, peeled
Three small white onions
1/2 tbsp of salt
Two eggs
1/2 a cup of flour
1/2 a cup of sunflower oil
1/2 tsp baking soda

What I did

As you will probably have noticed if you’ve followed the above link, there’s not really an awful lot of resemblance between my ingredients and Debbie’s ingredients. They’re the same sort of things, but the quantities are only tangentially related. This is in part due to my usual tendency to adapt, in part because I peel potatoes on autopilot and peeled far more potatoes than I actually needed before I knew it, and mostly because the recipe was all the way upstairs and it would be so much work to go up and check it.

I used the grateresque attachment for the food processor and got a large pile of shredded potato. As directed I put it into a metal colander and squeezed as much of the juice out as I could, but it was still quite damp. It was at this point I decided to employ low cunning to complete the task and sprinkled the salt over it, mixed it up thoroughly and went to do the onions. These I decided that rather than shredding I would do with the normal processor blades, turning them into more of an onion puree.

As an idle observation, if you run the grater attachment to the food processor and just drop a potato on top of it then it bounces about in an amusing manner. I can’t help but imagine the little potato pleading for its life as it avoids the spinning blades.

But maybe that’s just me.

Anyway, I’d decided to mix the Latkes in the kenwood. Why? Because I’m lazy, that’s why. And it’s the same amount of washing up, so why not? I transferred the onion to the kenwood bowl and returned my attention to the potato. The salt had done its work quite admirably and I was able to squeeze a lot more liquid out of it. Having done so I transferred it to the kenwood as well.

I then added the eggs, flour and baking soda and mixed it all up. It produced a batter with a texture fairly similar to my normal pancake mix, which was encouraging.

I then poured the oil into a nonstick pan and heated it. At this point I looked at the pan, slightly disbelieving. Buddha on a pogo stick that’s a lot of oil… I know I have recipes which use almost that much oil, but that’s in a large curry. The sole purpose of this oil is for frying things in. This somehow makes it more alarming.

Anyway, I used a heaped spoonfull of batter for each latke. The pan fit about four of them, and there was enough batter to make eight. I fried them for five minutes on each side until they were a darkish brown.

Towards the point where I wanted to flip the first batch I noticed that the best spatula was currently in a pile of washing up leftover from my brother cooking bacon and egg earlier (the irony amused me briefly). I quickly washed it up and moved to flip the latkes.

Point of reference? Putting a wet spatula into very hot oil isn’t a great idea. Ouch.

Anyway, latkes duly flipped, they cooked for another five minutes and then I transferred them to a plate and put the next batch on.

Conclusion

I was already pretty sure these would be good. They smelled wonderful.

Good lord. They tasted even better. These are really really good.

I’d made far more than I was going to be able to eat, so I rushed upstairs to get the Boy. (‘The Boy’ is my affectionate name for my brother).

“Boy”, says I, “Would you like to try something indescribably delicious?”

The Boy is skeptical. I cook strange and unnatural things, with vegetables and hardly any dead animal to speak of. However upon my description of what a latke is (“It’s basically a pancake made out of shredded potato and onion”) he is convinced to give it a go.

The Boy agrees. They’re really very good. He’s rather full from having had the aforementioned bacon and egg, so he only has one, but I make up for this by having three (and another one halfway through writing this post). The rest will keep to be reheated in the oven later.

So, a definite success. They do generate a lot of washing up, and they do stink up the house, but they taste amazing.

The only thing I would do differently in future is make sure that they really were a quite dark brown. Some of them were slightly undercooked and, while still nice, they were a bit to soft inside and lacked the crispness of the really well cooked ones.

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Spicy pumpkin and bean stew

Wednesday, January 11th, 2006

I still haven’t learned my lesson with pumpkins and how much water they give off when you cook them. Eventually I’ll figure it out, but until them I appear to be doomed to have watery pumpkin dishes. In the end it didn’t turn out too badly. It was just a bit more of a soup than I’d intended it to be…

Anyway, on to the recipe. I’m afraid this is another one I didn’t measure very carefully.

What I used

Lots of cinnamon (about 5cm)
Two dried red chillis
About half a tbsp of coarse sea salt
1/4 cup sunflower oil
Three small white onions
Lots of chopped pumpkin (maybe a bit under a kilo?)
Two cans of red kidney beans

What I did

First of all I had a pumpkin to dismember. As I’ve probably mentioned, my mother grows an awful lot of pumpkins. This is great, but when we have a large number of them we don’t manage to sell it means that there’s an awful lot of pumpkin to process.

This is normally a massive chore, but I for christmas I was given some extremely nice knives. All three of which tout themselves as the “World’s sharpest knife”, supplied by the rather dubiously named Twin Towers Trading (I can’t view their site because a) They’re idiots who don’t know how to correctly design a site and b) I don’t have flash installed). Sounds like it’s the lead up to a massive joke and/or disaster, doesn’t it?

It actually didn’t. These turned out to be really good. The serrated carving knife in particular is scarily effective and cuts through the pumpkin almost effortlessly.

Good lord. I sound like an advert. But seriously, it’s true. These things are great.

So, summary version for those of you whose eyes glazed over: One large pumpkin rendered into conveniently sized chunks. A lot of time spent, but relatively little effort. Most of it was bagged for later use or turned into soup, but the aforementioned about a kilo went into making this dinner.

Next, the spices. I dry fried the cinnamon and chillis (having broken them up suitably) and then powdered them in a mortar and pestle with the salt. Yes, this is exactly as much work as you think it is. I really need to get a spice grinder. (I suspect I’ve complained about this before, but blogger is currently down so I can’t actually check. Also on to do list: Make local copies of my blog posts)

Towards the end of this my mother pointed out that the food processor had a mini attachment that would work well for spices. I looked skeptical, but was fed up so decided to give it a try. It worked about as well as I expected. Worse yet, it managed to not screw up in an entertaining fashion which I could relate to my enthralled audience. It just didn’t do anything to them. At this point I declared the spices to be good enough.

Anyway, I now had a food processor that was going to need washing up anyway, and after two long and involved tasks I didn’t really feel like chopping onions, so I just shoved them in the processor.

Now, obvious things ensued. Heat oil, add the onions, fry for about 5 minutes. Yawn. Add the powdered spice mix, fry for another five minutes. Taste the fried onion to make sure I’m not about to poison people (I wasn’t. Mmm… cinnamon and chilli).

More standard obviousness continues. I added the pumpkin and fried for another five minutes. Added the beans and fried for another five minutes.

Hmm. At this point the recipe book says ‘bring to the boil then reduce heat and simmer’. Errr… what? There’s no water in this recipe. I double checked it and confirmed. No water. This presents me with a dilemma: Am I supposed to boil the pumpkin? How do I do that? Wikipedia lists the boiling point of carbon at about 4000k, and my oven barely reaches half that temperature!

Instead I resort to covering it and sticking it in the hot oven to see what would happen. After about 5-10 minutes I was somewhat skeptical about whether or not it was really doing anything, so I added boiling water to half cover the pumpkin and put it back in. This did indeed cook it, but of course once the pumpkin started cooking it began giving off its own water, ending up with what was really closer to a pumpkin soup. After about twenty minutes I uncovered it and put it back for another 15 in an attempt to reduce it a bit. This rather failed, but never mind.

Conclusion

This was really nice. I should have used less water of course (probably about half a cup of boiling water to start it off with steaming is enough), but the taste was great. The cinnamon was my own addition, and I consider it to be a total success – this would have been boring without it.

So, if you have prechopped pumpkin (and I’m going to for ages now) and a sane way of grinding spices then this is a low work recipe which tastes great. Definitely a winner.

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Silly proofs 1

Wednesday, January 11th, 2006

My general style of proof in mathematics is to apply deep and powerful theorems in clever ways. I don’t like going back to basics, and thus will often apply a big theorem in lieu of making explicit estimates, direct calculations, etc. Sometimes this gives you a better proof, other times it just gives you a shorter and more mysterious proof.

This, combined with my general liking of severe overkill, is probably what leads to me enjoying proving things in silly ways. I’m going to use this blog to share some of my favourite silly proofs.

This one came up recently:

Let M_n be a sequence of real numbers with M_n -> inf. Show that there exists a_n > 0 such that sum a_n converges and sum M_n a_n diverges.

Now, if you want to be boring you can construct such a_n explicitly. But my way is much more fun. “Ah ha!” I thought, “If it didn’t then you could define a discontinuous linear functional on l_1, and we know that’s impossible!”. What follows is basically just a sketch of the most elementary version of this argument I could use:

Suppose not. Then for every a_n in l_1 we have sum M_n a_n converges (as it converges absolutely). Define f : l_1 -> R by f(x) = sum M_n x_n. Then f is linear and discontinuous – because ||f|| >= M_n -> inf.

Now, let f_N = sum_{n <= N} M_n x_n. f_N is continuous, and for all x we have f_N(x) -> f(x). So f is a pointwise limit of continuous linear functionals. But the continuous linear functionals are closed under pointwise limits (e.g. because they are Borel measurable, because of the uniform boundedness theorem, because the stars are in the correct alignment, etc), contradicting the fact that f is discontinuous!

I leave the elementary proof as an exercise for the interested reader.

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Chains of null sets

Tuesday, January 10th, 2006

Noam Elkies made a cute observation on his site, which is that there are chains of null subsets of [0, 1] whose union is not null. Proof: Take a maximal chain. The union of this can’t be null or it would contradict the maximality of the chain.

He then went on to ask how ‘large’ their union can be: Under the continuum hypothesis it can be all of [0, 1]. Do you need the continuum hypothesis for this? Without it how large can the union be?

I tinkered around with applying some cardinal invariants to the problem and came up with a nearly complete solution. I’m going to be posting a pdf of it at some point, but here are the highlights:

The solution is for all intents and purposes purely combinatorial. I used essentially no measure theory in proving it except for one lemma which has a comparably easy category analogue.

It basically depends on three cardinal invariants. add, cov and non. The additivity is the smallest cardinal k such that there are k many null sets whose union is not null. cov is the smallest cardinality of a covering of [0, 1] by null sets. non is the smallest cardinality of a non-null set.

We have aleph_1 <= add <= non, cov <= 2^aleph_0. add is regular, the other two don't have to be, and no inequalities between non and cov are provable in ZFC. (I screwed up and claimed in my email to Elkies that in fact non <= cov and they were both regular. This is false, but it only broke a minor part of my conclusions).

It’s relatively easy to see that if non = 2^aleph_0 or add = cov then you can get a chain whose union is [0, 1]. What’s slightly harder is that if such a chain exists then you can find some regular cardinal k (the cofinality of the chain) with cov <= k <= non. So, because you can have non < cov, you can't always find such a chain.

So, in the absence of one with union [0, 1], how big can it be? Can it be measurable? (If it is measurable then of course it has measure > 0). Turns out not, and this is where the tiny bit of measure theory comes in. If A is measurable and m(A) > 0 then mu(A + Q) = 1 (addition is mod 1). So, if we have our chain L_t of null sets whose union is measurable and of positive measure, then replacing each L_t with L_t + Q gives a chain of null sets whose union is of full measure. Then adding in the complement we get it to be all of [0, 1].

The only question I have remaining which I’m not sure about is whether or not the existence of such a regular cardinal is equivalent to the existence of such a chain. I doubt it, but I’m not really sure and probably lack the knowledge of forcing needed to understand a proof either way.

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