Orthogonality

There are some overarching ideas from math that seem to make their way into my everyday life. Orthogonality is one of the newer ones. I've come to appreciate it more recently, seeing it in so many different aspects in so many different areas and applications.

Most engineers and scientists use the term to refer solely to vectors, while the use can be extended to many different types of mathematical objects. The idea however is a very broad one. Parts of code for example can be orthogonal. If changing one part doesn't affect the other parts, this is intuitively like orthogonal vectors. Part of good software design is then a kind of orthogonalization. ( My spell check keeps on warning me "orthogonalization" is technical jargon, but I can't seem to find a suitable expression for it in normal English. )

Analysis is best done by breaking a complicated thing into orthogonal components. This is often an assumed fact. It is clearly intuitive in linear algebra. It is not so clear when it comes to Legendre polynomials for example where it appears to a student that we are adding complexity to make orthogonality. It is not so clear that this makes the computation easier. The best example of the perils of poorly orthogonalized systems might come from statistical modeling in the form of Multicollinearity. A multicollinear model is nothing more than a statistical model whose variables are not well orthogonalized. Multicollinearity is an cause of poor statistical model design in the same way that strongly linked objects are a cause of poor program design.

Christmas shopping.

There's almost too much that could be written about this. I have a feeling that each person's individual personality is reflected in how they go about finding presents for the friends and relatives.  Some people approach the task with glee and others with dread. Some people demand lists from people of acceptable items, and others give them random garbage they probably already have.

To some degree every person ends up giving a gift that somehow reflects what they think of the receiver. My family doesn't seem keen on giving me books and always tries to find an alternative present. I can't really blame them. Looking at my more recent purchases,  I can't seem to give someone a Christmas present that I wouldn't want myself. It's a terrible curse. Sometimes I often think I would enjoy the gift far more than they will. A bit of greed seeps into my thoughts. The possessive part of me thinks I should buy all the gifts I give in double and keep half for myself. Why don't I have these things for myself? I could never really justify their purchase for my own sake, so I hope others will for me.

Last year there was a special on NPR about how the holiday gift giving frenzy is bad for the economy. The idea seemed reasonable. Focusing so much consumer spending in a short time can't be stable. It makes me sad to see my friends suffer working on Black Friday and even worse, when they have to work on Thanksgiving. Additionally, I'm sure its a huge pain to stock the stores like they must for the shopping season. Please just spread your holiday shopping evenly throughout the year, at least for logistic's sake.

Oh I shouldn't forget:

Good-bye, Mandelbrot.

As a Mathematician, if you wish to remembered by the general public, make sure your area of study involves lots of pretty pictures.

Wandering around campus at night, I think I could easily be the ghost of former self haunting my old college haunts.

transitions

What it the latin etymology for the word? well its completely the opposite of what i want. trans means across. the rest is from the verb ir "to go". its wrong to say im in a transition. I want to use a word that expresses the change which has come has come by my own decisions to remain static.

unavoidable.

Death in the Modern World

Death is something much different for me than it was for my parents at my age. It's because the internet makes us manage our friendships differently now.  Its easier to keep in contact with more people, and easier to hear about everything thats happening in their lives. I've heard stories where, in the past, people would learn that their old friends had died - often in things like highschool reunions and chitchat when they ran into people they remembered. I can't even imagine this happening today. These days it seems like with anyone you've known, you learn about their death about as soon as their close loved ones do.

To my parents, it seems as if bad things are always happening to my friends. They're always astounded how many have caught some terrible disease, are in trouble, or have recently died. While they probably attribute it to bad luck, I know it's more the fact my network extends farther than they can imagine. They've never been aware of the social web.

I don't know what this means for death and mourning. Does more exposure to it mean that it will feel like less of a big deal?  Am I less likely to go far out of my way to mourn someone who wasn't a close friend or relative? Or maybe the enormity of it will that much more evident.

Algorithm

I was thinking about an old job interview question the other day:

You have a list of numbers. Each number has a duplicate on the list except for one number. How do you find this number?

This is clearly linear time. A little less obvious is that it can be done in constant space. As far as I know, I was the only one to actually come up with a solution to this one. It's not easy, at least in my appraisal. My solution, however, wasn't exactly the one they were looking for. I'll cover my line of thought, and then cover "the" answer and explain my feelings on them.

The constant space part led me to try adding all the numbers together. Fortunately, the sum of the numbers has an important relation to the exceptional number in the list: if it is even or odd, so is the exceptional number. This is due to the fortunate fact that the sum of two even numbers and two odd numbers is even. I immediately told this to my interviewer and convinced him it was true. He didn't seem to be impressed, so I abstracted my result using group theory and showed that the same process could be essentially used to derive the value of the number bit by bit. (The full thing is a fun bit of math. I'll leave it to the reader until I get around to writing it up.) 

My interviewer wasn't impressed. Here's the official answer:

Bit-wise xor all the numbers together.

It's immediately clear why this works and I might have thought of it had I been more into low level programming. This solution seems so simple compared to the one I promised. So why is it so unappealing to me? It is the illusion of simplicity, nothing more.

There's nothing particularly natural about bitwise Xor. The solution works, but its not clear how you could get to the solution other than to conveniently know a function which does what Xor does. More importantly, the solution doesn't reveal where the power of the solution comes from. Understanding this solution doesn't help with related problems. When I was asked the problem, I approached it abstractly -- which probably hurt my ability to solve it. Thinking more about programming would have helped, but it wouldn't have led to the deeper solution I found. This is why I am more fond of my own solution, and why I believe in abstract mathematical reasoning. Thinking about computers when trying to solve computers tends to lead to concrete thinking, destroying creative thought. Creative thought is rare.