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Archive for the ‘Mathematics’ Category

Golden ratio

Steven Strogatz has a nice article about the golden mean in the New York Times. It’s worth a read. I learned a few more factoids about the golden mean.My favorite representation of the golden mean is in term of its continuous fraction representation.

 

\phi = 1+\frac 1{1+\frac 1{1+\frac 1{ 1+\dots}}}

One can use this representation to show that \phi is the most irrational number. I’ll leave that proof for some other day.

 

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Art from math

Occasionally my computer produces plots that can be fun to just post in the absence of context. Then they become artistic.

 
Here is a sample from one that I generated today while trying to understand something related to my current research.

This graph will never be published. At least not in one of my research papers in physics. Hence I publish it in my blog.

 

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A fun identity

So I’ve been working on one of my papers where we need to compute some numbers. They end up being determined by a cubic equation.

However, one often finds surprising identities when Mathematica spits out a bunch of numbers expressed in algebraic form. Here is one of them:

\frac{1}{42} \sqrt[3]{\frac{1}{2} \left(90720 \sqrt{7446}-2859138\right)}-\frac{7893}{7\ 2^{2/3} \sqrt[3]{2859138+90720 \sqrt{7446}}}=0

PS: I use italics in the word surprising above only because if one does not know the origin of these identities they might seem surprising.

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Well, sometimes I get five minutes off to goof around and I find silly facts, which as far as I can tell serve little to no purpose. Hence the name factoid is sometimes applied to them. The random novelty of the day is that the number

1111111111111111111

is prime. Moreover, it is a factor of 1010101010101010101010101010101010101. The only other prime factor is almost as pretty.

909090909090909091

Beware  of 909090909090909090909090909091 which surprise, surprise, it  also happens to be prime. And so seems to be

11111111111111111111111

If someone knows anything more about these primes, please let it be known. I found them by accident, but they seem to have been discovered before my time, so I can not name them after myself, nor get a patent for them.

 

 

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I have written many papers in the last 20 years. Recently, I was taking a trip down memory lane and was reading some of my older stuff. I happened on a pretty identity on one of my papers. Something I had forgotten. That particular identity probably has someone elses name attached to it. I don’t believe that I was the first person who discovered this algebraic identity, but I wouldn’t know where to start looking for the correct precedence.  If the identity has a name attached to it, I wouldn’t be surprised if the culprit for finding it first is pushing daisies.

Mathematics keeps on getting rediscovered after all.

 

The identity is as follows.

Consider a set of of s numbers (they can be real, rational, algebraic, or even more messy commutative algebra objects so long as for the most part their multiplicative inverses are well defined). Let us call this set

S= \{ \alpha_1, \dots , \alpha_s\}

And consider the set of permutations of the first s integers, where \sigma is one such permutation

\sigma\in Perm\{1, \dots, s\}

We are then instructed to take the sum

{\sum_{\sigma\in Perm\{1, \dots, s\}} } {\frac 1{(\alpha_{\sigma(1)}+\alpha_{\sigma(2)}+\dots +\alpha_{\sigma(s)}) }}{\frac 1{(\alpha_{\sigma(2)}+\dots+\alpha_{\sigma(s)})}}\dots {\frac 1{\alpha_{\sigma(s)}}}

The stipulation is that the sum has no infinities (the numbers are generic).

This sum is equal to

{\frac 1{\alpha_1 \cdot \alpha_2 \dots \alpha_s}}

As everyone can see, it’s an  obvious identity so the proof is left to the reader ;-)

For some reason, looking at it I can imagine it appearing miraculously in the twistor formulation of  Yang Mills scattering amplitudes… well, this is just a random speculation.

 

 

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I have been enjoying the hospitality of the Simons Center for Physics and Geometry these past two weeks. I have been attending the workshop on Mathematics and Physics that has been going on yearly for the past few years.

It is a fun filled event where there is about one talk per day (on most days) and where I get to have a lot of conversations with very many people that I don’t see all that often.

The most interesting day of the week is when we have a seminar at the beach. We get together and drive away for about an hour to reach Smith Point Park. We then get to hear a whiteboard talk from some participant under a tent while the rest of the passer-byes gawk at  us: the idea of going all the way to the beach to hear someone speak about physics just floors them.

There are some pictures about those events that you can find in the webpage of the workshop, so you can go there to have a look (if you really must).

Perhaps because of my sense of humor, I kind of imagine us as a flock of kids who suddenly find themselves without their toys: the laptops are missing and we have to have `fun’ without doing calculations. Such activities involve tanning, getting in the sea and riding the waves, walking on the beach and perhaps even play volleyball.

All in all I definitely recommend the experience and welcome the chance to get some rest from my laptop.

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I don’t know how many of the readers here pay attention to what’s happening in Wall Street. Yesterdays trade was quite spectacular, although from many points of view it is terrifying and it is a day that will probably live in infamy.  From the academic point of view I’m sure it will be studied to exhaustion.

All in a day's work.

(more…)

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Tax day

Tax day is here. This is a yearly ritual in the U.S.

I think it is one of the few days where most people get intimate with what they earn, and the ensuing effort that it takes to produce a tax return.

Apart from the fact that doing the work is not rewarding and that it is easy to feel that one is paying too much, I think that the whole exercise is beneficial: people should know how much they are paying the government. I have been in other places where everything is so automated that one just forgets the whole thing and it is as if taxes never happened and when they change people don’t really find out. They also don’t feel the yearly sense of shock about the experience.

In the end, taxes are paying for services that a lot of people here take for granted. For example roads, a working post office, law enforcement and education for their children. They also pay for Social Security and Medicare: some of the most costly entitlement programs that no politician can risk to touch without a backlash from their constituency.

It is also one of the few days where a lot of people get intimate with arithmetic (for a change). It always surprises me how many people seem to have trouble doing this activity, just because they don’t understand addition and substraction (and percentages). Oh well! I’m not going to worry about it.

in the end, like a lot of other people, I filed my taxes at the last minute. I wonder what the statistic of us, last minute filers, really is; as well as the reasons why.

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Today’s puzzle is really simple. It is a single number, and don’t worry about the formating: it is not essential.

Your job, if you decide to take it, is to figure out how this number was chosen.


3983166922118810678205990336564718434224120605664143701183608

7681419052507877828777197836786790614849623650815370930359235

0129704516432578314074470098769495753238915840085811644293357

6455898753759925348456032608650278150327110180834816760596303

09728652360528042842943610644525298135991037793081877485511245

2474332249873412259136132368731917415053983296042291396691439150

43327960030646869030670768248956738775708735485960504256334453

736488021205358726905569574806264667486581193984051009097136483

506525271749484080035733863870881763567000457650973784539927240

2043442156827044195038802613229625783142492845706197046207004041

55134209534719408986385965928333597344225594042950352896

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Sometimes you work hard on a project, and you have some theoretical framework that explains how things should behave in certain limits and how there should be a natural expansion to be able to do a fit for the limit one is studying. Then sometimes you take some data, let us say you do a simulation, and the data just does not conform to these expectations.

You then get stuck with data that you don’t really know how to analyze. And it is terribly frustrating.

You can try to understand what direction the data is pointing to, but it can be more of a ‘nebulous oracle’ than a clear straight arrow pointing to the path you should take. This is normal: it happens all the time.

Think about that.

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