Now With TeX-Support

25.06.10

Since a lot of content on this site revolves around mathematics, I now proudly present TeX-support for writing mathematical expressions. Please note that since this feature is brand new (on this site at least), expect some bugs and quirks to begin with. I still haven't figured out the best font settings so your experiences with the rendering may vary.



It can be used anywhere by everyone (that is, in my posts, in your comments and so on). Just embed your code in two dollar signs.

For instance, $ $ \TeX{} $ $ (without the spaces between the dollar signs) will generate
$$\TeX{}$$
Since TeX is awesome I present some other examples. You can click on the expressions to see the TeX-code behind it.

Euler's identity
$$\mathrm{e}^{i \pi} + 1 = 0$$
The Schrödinger equation
$$iu_t(x,t) + u_{xx}(x,t) = \lambda |u(x,t)|^2 u(x,t)$$
Stokes theorem
$$\int_\omega \mathrm{d}\omega = \oint_{\partial \omega} \omega$$
The Residue theorem
$$\oint_\gamma f(z) \, \mathrm{d}z = 2\pi i \sum_{k=1}^n \mathrm{I}(\gamma,a_k) \mathrm{Res}(f,a_k)$$

Futurama is Back!

24.06.10

YES! FINALLY!

Read this and grin. I've just finished watching the first two episodes of season 6 and it didn't disappoint. It picks up the story (sort of) following the third movie and it immediately sucks you back into the Futurama universe. IT WAS AWESOME!



This is indeed 'good news, everyone'. Can't wait for the next episode. :D

The Strange Behavior of Time

22.06.10

Time is a difficult concept. For what is time? Is it constant? Has it always existed and will it continue to exist forever? Where did time come from? All these questions challenge our minds and philosophers and scientists have devoted a great amount of work studying the idea of time. And all that work gave fruits during the nineteenth century when Albert Einstein released his work on the theory of relativity. The theory of relativity takes many aspects of our universe into account (time, space, matter etc.), but I will only discuss it's ramification on the concept of time.

For starters, it helps to imagine time as another dimension. You have length, width and depth which make up our three space dimensions, but each event happening takes place at a specific point in the time-dimension. Imagine if we were to meet for lunch. We must first agree on where to meet (i.e. where in space) but also where in time to meet. Also, if I am standing still for an hour, I haven’t traveled in space at all, but I have traveled in time. I’ve become one hour older.

Time is however not constant. It is relative. This might sound strange at first, but I will try to explain it using a famous thought experiment involving a moving train. But before I do that, I have to explain what a "frame of reference" is. A frame of reference is, quite simply, a set of coordinates which is attached to a system.

For instance, if I am standing still looking at a car running at constant speed, the car is moving in my frame of reference. I myself on the other hand, am standing quite still. If we now move the frame of reference to the car, the car is, in its moving frame of reference, standing still and I am now moving away from it. You can have other frames of reference, such as attaching coordinates to a rotating system and other interesting stuff, but we will manage with the car-person-frame for this purpose.

The next part is a bit stranger. The theory of relativity tells us that the speed of light must be constant in all reference frames. This is very counterintuitive if you are used to Newtonian physics. You are perhaps used to think in the following way:
Say that you are standing on a train moving at 10 km/h. You now throw a ball in the same direction as the train is going at 2 km/t. In your moving frame of reference, the train is standing still and the ball is moving at the 2 km/h you gave it when you threw it. Now, if I am standing next to the train observing all this, I will see that the ball is moving at the 2 km/h you threw it at, plus the 10 km/h the train is moving, ergo 12 km/h. We have now established that the ball has a different velocity in the two different frames of reference. Adding velocities like this is ok for speeds that are way below the speed of light.

Light however does not behave this way. It always moves at the constant speed of light (which is about 300.000 kilometers per second) in all frames of reference. So if you and I are in two different frames of reference and we both send out light for each other to measure the speed of, we will both reach the same speed. This is unlike the ball-train-example mentioned earlier.

If you find this strange you can at least take comfort in knowing that you are not alone. The very nature of light is something which continues to puzzle scientists even today. We don't really know what light is, as it inhabits the behavior of both waves and particles.

But what does all this have to do with time and the question whether time is constant or not?

Consider now that you have built a clock to measure time with. It works in the following way. You are standing in a hallway with mirrors placed on the floor and in the ceiling. There is also a beam of light traveling up and down between these mirrors. We imagine that the mirrors are "perfect" so that the entire beam is reflected each time it hits the mirror. We can now calculate how much time the beam used to get from the floor to the roof by dividing the distance between the mirrors by the speed of light (which is constant). Let us say that the distance is so that the light used one second. (In other words it is a waaaay tall hallway.)

Imagine now that the hallway is moving, say, placed on a train moving at constant speed. When you are standing inside the train looking at the beam, it makes no difference to the previous example. In your frame of reference the mirrors are still standing perfectly still and the light is moving up and down giving you the same time of 1 second when you do the measuring.

But what if you are watching this from outside the train? You will then see the light leaving the mirror on the floor and as the beam makes its way towards the ceiling, the train moves as well making the path the light has to take slightly longer. I've tried to illustrate the two cases in figure 1 and 2.



This means that if I am standing outside looking at the train, I will measure a time which is slightly longer than the time you’re measuring from inside the train. In other words, time moves faster for you than for me. This effect will only become more and more dramatic as the train starts approaching the speed of light. Oh dear.

Now if we actually conducted this experiment with a real train and thus are limited to speeds way below the speed of light, the effect would not be noticeable. So for almost all purposes on earth, we can say that time is constant, and this is exactly why the idea of time not being constant is so counterintuitive at first. There are however cases when this effect needs to be taken into account, for instance when working with satellites and GPS. The small difference in time must be corrected for or else the satellite sends you off path when you try to navigate.

But such small effects are boring when you think of the bigger picture. This effect opens the very exiting door which has “time travel” written all over it. Oh yes, if you are able to travel at near the speed of light you can indeed travel in time which is incredibly awesome and I want to try it now fast. Unfortunately it has to remain in the idea bin for now – you would need incredible amounts of energy to reach such velocities; in fact you would need infinite energy to reach the same speed as light itself.

Phew, this became a rather long post but I hope I was able to explain this rather strange concept in a sort of understandable way. If you have any questions, please post them below in the comment box and I will try to answer them as best I can.

Now I need to go fantasize and draw another time-machine-sketch.

TO INFINITY-

The Riemann Hypotheses

15.06.10

Prime numbers are strange. Well, not the concept "prime number" by itself, meaning a number which can only be divided by 1 or itself without leaving any reminder. No, it is the the distribution - where they appear on the number line - that really puzzles mathematicians. As far as we know today, the distribution follows no provable pattern. Prime numbers play an important role in algebra, and therefore also in code theory. When you connect to your bank via the Internet, you use secure channels which, among other things, rely on huge prime numbers. To keep it simple, the public code (which includes the secret code) which is sent out on the net can be a result of two really large prime numbers multiplied together. To get a hold of the secret message, you must know the factors of this number (which means that you have to know the two large primes). And since there is no effective way of determining where those prime numbers are, you are stuck with brute force attempts which can take years. So, in essence, we should be happy that large primes are hard to come by.

The Riemann hypotheses


If you were to either prove or disprove this hypotheses, The Clay Mathematics Institute of Cambridge is more than willing to pay to $1.000.000 for your effort. That, and you will go down in history as a truly remarkable mathematician. The hypothesis, proposed by Bernhard Riemann, states that all the non-trivial zeros of the Riemann Zeta function has real part equal to 1/2. That is, the zeros are on the form

z = 1/2 + it

where t is a real number. So far scientists have confirmed this for 10 000 000 000 000 (!!) zeros, but a general proof is yet to be found. Should the hypotheses be proven correct, it will be a huge step towards discovering the nature of prime number distribution which could lead to the downfall of some of the encryption theory we rely on today. Not that I would hold my breath in fear - the hypotheses has shown to be incredibly hard to prove, rendering other money scams (such as this) more lucrative for e-bandits.

Grand Finale

11.06.10

Finally.

Today I had my last exam for this semester, concluding my third year as a student at NTNU. So now it's time to lay back, crack open a few cold ones and let the good times roll. Feels good.



So now, tell me, how are you going to spend your summer? All work? Travel?


PS: I've started writing a small article about the Riemann hypotheses. I will probably post it next week, so stay tuned. :)

Summertime

07.06.10

It is almost time.

Time to stop venturing down to the university every day of the week to read on and on about various mathematical phenomenons in which you will be thoroughly tested. I had my final exam in numerical mathematics today and, I swear, every time I walk from the exam locale after completing an exam, life seems a bit sweeter. The sun seems to shine just a bit brighter and people on the bus seems just a bit friendlier. It’s as if they know. I’m waiting for them to say “hey, great job on the exam, guy! You’ve earned this day!”, but it has yet to happen.



But it is this Friday that my summer spirit can kick in at full force; I’m having my final exam in TMA4165 dynamical systems. An exam which brings this semester to a complete and also marks the end of my third year studying mathematics at NTNU. It is strange, I’ve spent as much time here as I did in secondary school but time has indeed flown by much faster. At least it feels that way.

As for the summer, I am staying put here in Trondheim. In a new flat, actually. Well, not that new, I’ve really just moved all my stuff two floors up to an almost identical flat, but the view is slightly different and I get to open a new mailbox. Oh, and there are tiles on the kitchen wall which is a nice touch.

As for summer plans, I have none. Other than working, that is. Summer job income is a critical pillar in my delicate financial plan throughout the year known as the “leaky money bag”. If I manage to fill the bag just enough during the summer it will slowly dry out during the two next semesters, hopefully leaving me with enough money for bread June 2011. This year the plan has worked exquisitely.

I will try to update this blog with interesting mathematical topics, such as the last two entries about Fermats theorem and the Goldbach conjecture, during the summer. I am thinking posting maybe once or twice a week depending on my ability to dig up interesting stories. Having said that, I am very open to suggestions, so feel free to post them in the comment section and I will get back to you.

I’ve also tried to divide the content on the blog into two subgroups, as you can see on the blog-list-margin-thing to the right. If you click on “science” it lists all the science-related entries (and analogously with the “personal” button). The entry-list now also only expands the current month (the last system was getting out of hand), but you can expand the month of your desire by clicking on it. Give it a try and tell me what you think.

Well, that about sums it up, I am off to get a new cup of coffee. Till next time, take care.


PS: If you have Spotify, check out this summer music

- Mungo Jerry - In The Summertime
- Beach Boys - I Get Around

PPS: Please post some summer song suggestions! :D

The Goldbach Conjecture

06.06.10

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers and is one of the oldest unsolved problems in number theory. The conjecture can easily be verified for small even numbers such as:

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 7 + 3
12 = 5 + 7
14 = 3 + 11

It was stated by the Prussian mathematician Christian Goldbach in a letter to Leonhard Euler on 7 June 1742, as such

Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units.

In the same letter, another conjecture can be found in the margin (Fermat anyone?)

Every integer greater than 2 can be written as the sum of three primes.

Euler obviously had faith in this conjecture; the following can be found in a letter from him dated 30 June, 1742

... every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.

Goldbach also came up with another equivalent conjecture, known today as the "weak" Goldbach conjecture. It states that

All odd numbers greater than 7 are the sum of three odd primes.

A proof of the strong Goldbach conjecture will imply that the weak conjecture is true as well.

Even though no rigorous proof exist of this conjecture, it has been verified by computer science up to 1.609 * 10^18 (!!). In 1930 it was proven that every even number greater than or equal to 4 can be written as the sum of at most 20 primes. The number of primes was reduced to 6 in 1995.

All in all, this conjecture still continues to puzzle mathematicians around the world and, who knows, maybe we will see a proof of it within our life span.