Mahi Mahi with Tomatoes and Peas

Right before Christmas, I usually crave fish and vegetables. I guess it’s my body preparing for the onslaught of meat, gravy and pie that is coming. The other day, I improvised this recipe and decided that I had to share it. It’s quick, easy, delicious and healthy (well, minus the bacon).

Ingredients:

• 1 piece mahi mahi
• 1 slice of bacon
• ~15 cherry tomatoes
• 1 small clove of garlic
• A handful of peas
• 1/4 cup white wine
• Salt
• Black pepper
• Oregano

Start by frying the bacon, on medium heat, until it is crispy. Meanwhile, chop the garlic and halve the tomatoes.

Pour the wine into a small pot and start simmering it with the garlic and a little bit of oregano. When the bacon is done, place it on a paper towel to remove excess fat. Pour out excess bacon fat from the pan, and start frying the mahi mahi, skin side down for about 3-4 minutes or until the skin is nice and crispy. Add the tomatoes to the wine and spices. Generously salt and pepper the top of the mahi mahi before flipping it over.

Let the fish finish cooking for another 3-4 minutes. Add the peas to the tomatoes and let it simmer for a few more minutes. You want the tomatoes to be cooked, but not falling apart. Place the fish and vegetable on a plate, and crumble the bacon slice over it. Enjoy!

Interesting property of matrices

A few weeks ago, Gilbert Strang from MIT gave my department’s colloquium on the topic of banded matrices. A diagonal matrix is a 1-banded matrix, one where the semi-diagonals above and below the diagonal are non-zero as well is a 3-banded matrix, and so on. An example of a 3-banded matrix is .

Now, the interesting property of matrices is as follows: take a -matrix, say . Notice that the ratio on the diagonal is  . Now square the matrix: . The ratio on the diagonal is still . If we cube we get . Now the ratio on the diagonal is . It actually turn out that thus ratio along the diagonal is preserved for all powers of a -matrix. There are several ways to prove that it holds: but the most obvious is by induction on the powers of . This proof is fairly mechanical, so I will leave it as an exercise for the reader 🙂 (I’ve always wanted to write that).

The first question we ask as mathematicians is “how general is that property”. For these matrices, it turns out that it doesn’t generalise to general -matrices. However, it does generalise to 3-banded matrices. All the ratios along the diagonal of a 3-banded matrix are preserved under powers of the matrix. This is once again easy to prove by induction on the power of .

I think this is a really neat little fact, and I’ve never seen it before. I don’t know if it shows up in any applications, so if you know of any please leave a comment! I’ll be back after Christmas with some graph theory 🙂 Merry Christmas!

Christmas brownies

It would seem I have lapsed in my blogging. But, this time I’m making up for it with a recipe for Christmas brownies. My first semester of grad school is over (at least until the grades are released) and I decided it called for brownies. Then I had some more realisations: it is almost Christmas, and the best part of Christmas is peppermint mochas. So I decided to modify my tried and true briwnie recipe into peppermint mocha brownies. I also decided that these brownies needed a little extra in the form of spiced rum. As always with brownies, you need to use a good quality chocolate that you enjoy eating on its own, as this is what your brownies will taste like. I used Ghiradelli’s 70% cocoa chocolate chips for mine, and it turned out delicious.

Ingredients for brownie:

• 4oz dark chocolate
• 1/2 cup butter
• 2 tsp instant coffee
• 1/2 tsp peppermint extract
• 2 tbsp dark, spiced rum
• 3/4 cup granulated sugar
• 2 eggs
• 3/4 cup flour

Ingredients for topping:

• 4oz dark chocolate
• 2tbsp heavy cream
• 1 candy cane, crushed

Preheat the oven to 325 degrees Fahrenheit. Melt the chocolate and butter in a pot over low heat. Stir carefully to make sure it doesn’t burn (if you are worried, melt it in a steel bowl over a water bath). Take the pan off the heat and stir in the instant coffee, peppermint extract and rum. Add in the sugar, and then the eggs, mixing carefully after each addition. Finally add in the flour. Pour the batter into a 9×9 inch brownie pan and bake for 20-25 minutes or until a toothpick inserted in the center comes out with a few moist crumbs. Closer to 20 minutes will yield a more fudge-like brownie, and closer to 25 will give you a more cake-like one. Let the brownie cool completely before attempting to decorate it.

When it is cooled, melt the dark chocolate (on the stove or in the microwave) and stir in the heavy cream. Spread this over the brownie, sprinkle the crushed candy cane over it and then chill for about an hour. Cut into 9-16 pieces, depending on how big you like your brownies (the correct answer is as big as possible without getting sick from too much sugar). Serve room temperature or cold, possibly with some vanilla ice cream or whipped cream.

I didn’t realise how much a crushed candy cane sprinkled over the chocolate topping would add to the cake until after I took the pictures: hence the lack of red and white sprinkles there.

These are the perfect thing to enjoy on a rainy December evening with a good book. They are dense, chocolatey and with a certain oomph from the rum, coffee and peppermint. I’m currently obsessed with Ken Follett, and if my grad student budget allowed me to I would have bought all of his books as a Christmas present to myself. As it stands I have to ration them a little. But now I’m off to find any brownies my roommate and boyfriend haven’t eaten and finish Winter of the World. Enjoy your brownies!