A common misunderstanding

A few days ago, one of the lectures in the university told us about a funny (but real) news report he once heard on TV. It was shortly after the discovery that 2^(42,643,801)-1 is a prime. (For more information about this go to Mersenne prime search website). On this TV program a reporter was interviewing a math professor. The conversation went like this: (R= reporter, P=professor)

R: So what do you have to say about the discovery of the largest prime number 2^(42,643,801)?
P: The number 2^(42,643,801) is not prime since it is an even number. You must have ment to say 2^(42,643,801)-1.
R: Well, they are close enough. The important thing is that this is the largest prime number.
P: It is not the largest. Euclid proved that there is an infinite number of prime numbers so there is no such thing as the largest prime.
R: Is it still correct today that there are infinity many prime numbers?

I really find it hilarious how some people think that a mathematical proof is something that is subject to changes. Sure, sometimes we have errors or we find better proof, but the there is no change in the fact itself. I suppose it is somewhat understandable why people act like this - they are too used to seeing things change. But it is still hilarious to watch, as long as you not part of the discussion.

Social Accounts

We all have our own way of using the net and the services available on it. In this post I want to present my own current method of organizing my online activity. This doesn't mean that this post is about promoting yourself on the net. I do not have any problem with those who try to promote their content by networking with similar people and influencing the social networks, but the result that these methods give are the opposite of what I want. I suppose it is a bit surprising for you that I am writing about this practice from a negative point of view. After all, I have my own blog, accounts on some social sites as well as accounts on Digg and StumbleUpon. However, I decided long ago that I have no reason to try and promote my blog or any account I have. The reason for this is rather simple. I just don't have the time to do this or to manage a popular blog. Being in the spotlight is great and all, but it also requires time and concentration. I feel that I rather have a not really popular blog, but I will write about what I find interesting and I will spend only as much time working on my blog as I want to. Because of this I don't ask people to promote my posts, even if they offer it. With this cleared, lets move on to the actual post.

As you all know, there are lots of different sites and services available on the net that require opening an account to use (or at least have extra features for those who opened an account). As a result one can find himself with lots of accounts. Some of them are not even used anymore - but they are still there. Some accounts are even forgotten by those that open them. A few days ago I happened upon someones Google profile. It had links to his other accounts. The problem is that there were so many accounts that I didn't even bother to try and take a look on them. The question is, how many of those accounts are actually being used? Or an even better question - How many accounts can one person make use of?

Having accounts that are not used or used really rare is in no way a problem. I myself have accounts that I no longer use. I think I have about 10-20 such accounts. While I try to keep tabs on them, they tend to multiply almost on their own :). Actually, a few years ago I was rather surprised to find out that I somehow got an Yahoo email account when I am completely sure that I never opened one. It turned out that I got it automatically because of my flickr account.
However, if something is not being used it practically doesn't exist and as such is not a problem. But, unfortunately, some people believe that it is fine to use many low quality accounts in order to get the result that they want. In other words, they open lots of accounts on different social networks and then try to keep all of them active and use them mostly to promote their own content. This obviously end up with duplicated content and overall low quality account on all of the social networks. But, if such a user is active enough he may manage to attract people. Especially others like him. Since I own a blog, I am approached from time to time by such people. They usually ask me to review their posts or to digg them, and offer to do the same in return. As long as the amount of requests is small enough, I do not see this as a problem. While I do not ask people to promote my blog, I do not mind helping them - as long as I have nothing against their content. For example, there is absolutely no way that I will help promote anything about music or poetry. I hate music (I cannot really say why, this is just how it is) and lately I am starting to feel the same about poetry (don't ask why).

What follows is a story of how I came to use the current accounts that I have and what I am using them for. It is in a somewhat chronological order, but I naturally don't remember when I started to use what account. The accounts listed are not all of my accounts, but all the main accounts I have on social networks are mentioned.

StumbleUpon
Perhaps somewhat surprisingly it all started with StumbleUpon. I installed the toolbar when I was looking for interesting extensions to add to Firefox. It proved to be interesting enough, so I started to use if a bit - at first I only used it to stumble on sites. After some time (about a year maybe) I decided to start adding some simple content to my SU blog. As a result people started subscribing to it. When I saw that what I did was interesting to other people I started to increase the amount of staff I posted. I also started to write reviews of other people and at some point even wrote some bits of advice about using StumbleUpon. I cannot say that I made any friends on SU, but it was and is a rather nice experience. Currently I use SU as a photoblog. I post photos that I find on the Internet with a short comment. At first I tried to post other staff as well, but as time went by I decided that what I want is to post photos only. After some time I decided that trying to post 4 good photos per day works best for me.

Photobucket
I got the account after reading posts about how great photobucket is. I wanted to take a look on it, so I got an account. Months later, I noticed that some photos on my SU blog were gone and others were loading slowly. So I decided to start uploading all the photos I post to photobucket and then link them from there. The free account is probably too small for some people, but I usually post only 4 photos per day and they are rather small. Who knows, maybe Photobucket will upgrade the amount of space it gives before I ran out. If not, I will just start storing photos on another site, probably Flickr.

Picasa
Originally I started using it because of its ability to find all the photos you have on your computer and then present them and organize them rather well. Later I started to use Web Albums to keep an online backup of my photos. The only problem I have is the storage limit. Picasa cannot be used to store all of your photos online, but it gives me the ability to keep backups of photos I found on the net. While some of these photos I also have on my SU blog and therefore on Photobucket, Picasa allows me to keep an organized collection of photos I found on my computer so I use web albums to backup these collections. Eventually, when I will run out of space to use, I will start uploading photos to Flickr, or maybe some other site. It obviously depends on my activity on the net, but I think that I will run out of space next year.


Facebook
If I remember correctly I got the account because I was bored. Even now I admit that I do not see what is so great about it. The only really good thing is the way it is connected to other sites. For me it means that I for example can post an YouTube video on Facebook with just one click on YouTube. Frankly this is also the only real use I have for this account. I have been trying to think ways to use Facebook to do something that I do not already do using some other site, but the only thing I came up with is to use it to post some random interesting staff I found on the net. However, it appears that Google reader shared staff is doing it better. While it makes some sense to have similar content in Google reader shared items and my Facebook account, I am not really sure if this is what I want. I hope that I will find some permanent use for this account eventually. Maybe I will use it to store photos, they seem to have some fair support for this.

Digg
I don't really remember why I opened this account. I think it was because I hoped that having it will help me to stay on top of the current news. If so I probably forgot that since I am not really interested in news I will not use this account unless I have some sort of motivation to go to it. As a result this account was left alone for some time. Eventually I found the correct motivation - all those nice people who ask me to help them promote their sites on Digg. By doing so I visit the main site and as a result I see other popular posts.

Google Reader and FriendFeed
These two sites as well as Facebook can be considered an activity feed. Their main purpose is to show my activity collected from different sites. In other words, it is a mix of updates. These sites work with different services and differ in quality. Somewhat surprisingly, the best one by far is Google Reader. But all three of them have their uses. FriendFeed is totally overloaded because of my StumbleUpon activity and I noticed that it misses some entires I have made on other sites. However it provides the most information out of all three services. Facebook provides an option to put FriendFeed into a tab in your account. This means that all of your activity can be seen from your Facebook. In addition to this it also allows to post links to different pages on the web, somewhat similar to SU. However, the format in which it is done is more fitting for a status update than a blog.
Google Reader is much more like a blog any of the other two. While it allows to post status updates, the format it uses is more suited for longer posts. It also deals well with showing videos that I added to favorites on YouTube. Basically it is an activity feed that doubles as a blog. As I already said it is the best one of the three. However, it lacks visual polish and unlike the other two there is no obvious community around the service.
I really don't know what will happen with activity feeds in the future. It is obvious that they are here to stay, but it seems to me that it is rather possible that I will find it reasonable to have more than one activity feed for a long time.

YouTube
This account is currently in the making. I have some plans for it, but it is probable that they will change as time goes by. Right now I am considering making it into a collection of math\science related videos. I am not really planning to upload anything at this time (although it is possible), but from what I have seen there is more than enough content on YouTube already - it just waits to be found.

Wikipedia
This account I opened so that my really rare edits to Wikipedia will be attributed to me. I doubt that I will ever use it for anything more than this, but time will tell. Right now it is just an about me page with some links to my other profiles.

Google profile
This particular profile is a good example of the recent attempts to provide centralized account and identity on the net. Google is not the only one who is trying to do so, but it will come as no surprise if they will be the ones to actually make it happen. While the profile is rather simple, this is actually a good thing, because it doesn't encourage users to overload it with information. Instead it allows one to focus on main function of this profile - provide a centralized gateway to all your other profiles.
I am not really sure that it would be a good idea to have one profile for all sites, but as long as we can provide different information on any site in addition to what is written in our main profile, the idea itself is something I have been waiting for for a long time. Right now my Google profile is just a description of who I am and what I do with links to my profiles around the net.


Conclusion
As you can see from this list I have a rather small number of accounts. My activity is divided between two blogs - SU and Math Pages. In addition to this I also keep two activity feeds updated - Facebook and Google Reader shared items. All the rest of my accounts are either updated as part of my activity on the mentioned sites or are rarely used, if at all. However, as you probably noticed, sometimes I don't update my blogs and activity feeds for relatively long periods of time. Maybe if I had less of them it would be easier, but I doubt it. Right now my activity is divided rather well between my interests, so if I fail to update it just means that I am to busy or have some other reason that prevents me from updating.

Analysis and Combinatorics

We often hear about how all mathematics is interconnected, but rarely see clear and simple examples of such connections. In this post I want to show one example in which theorems developed in Calculus are used to solve a range of combinatoric problems.

To begin lets consider the following problem. For any natural number K what is the number of ways it can be written as a sum of powers of two, if we allow each of them to be used only ones? Lets suppose that the number of ways is a(k). It is obvious that a(1)=1, a(2)=1. Lets define a function:

f(x)=a(0)+a(1)x+a(2)x^2+....
a(0)=1

It is easy to see that if a(k) is defined this way then it is also true that:

f(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)......

This is true because if we open the brackets we will get that x^k appears exactly a(k) times. Now lets multiply f(x) by (1-x). It is easy to show that:

(1-x)f(x)=1

You show this by looking on a finite multiplication and then taking limit. However, now we got that:

f(x)=1/(1-x)=1+x+x^2+x^3+x^4+.....

This is true because this is the formula for the sum of the geometric series for any 0<1. style="text-align: center;">F(n)=F(n-1)+F(n-2) , F(0)=0, F(1)=1
f(x)=F(0)+F(1)x+F(2)x^2+....

Lets look on the following multiplications xf(x), x^2f(x). Because of the recursive formula we get:

f(x)(1-x-x^2)=F(0)+(F(1)-F(0))=1

f(x)=1/(1-x-x^2)

The only thing left to do is to calculate the series and we are done. This part is left as an exercise for the bored reader. The important thing is that the same idea works for any different series - this is not something that is true only in a specific case. As such this is indeed an example of how mathematics is interconnected and how calculus that is the study of infinite can be used to solve finite combinatoric problems.

Gabriel's Horn

I didn't write anything about paradoxes for a long time, so here is a little something. Lets look on the volume we get by rotating the graph of the function y=1/x, for x>1. The object we get is called Gabriel's Horn. It is easy enough to show that its volume is finite and in is equal to pi. If we cut the horn in any finite point a we will get that the volume is exactly:If we now take the limit when a goes to infinity we will get that the volume is indeed pi. However the surface area is infinite. For any finite a we will get that it is exactly:


But the limit of this expression is infinity. Now that we know this , we can go on to the description of the paradox. Suppose that you want to paint the Horn with finite amount of paint. Obviously, it is not possible because the surface area is infinite. But you can fill it with a finite amount of paint. Lets now suppose that Horn is made from a transparent plastic. In this case, filling it with paint is the same thing as painting it.

As a result we get that it is both impossible and possible to paint the Horn with finite amount of paint. So which one is true? The solution is in fact rather simple. Firstly lets look on the graph of y=1/2x. Obviously this is again Gabriel's horn, but in a scaled down version. Lets put it inside the original while it is still filled with paint. In this way we painted it from the outside. How did we do it? The answer to this is in the distribution of paint. The thickness of paint is given by g=1/x-1/2x=1/2x. And this is the whole trick. We can paint even an infinite surface, the only thing we need to worry about is allowing the thickness of paint to approach zero in a way similar to this example (we need the integral of the paint distribution to be finite).

Collecting and storing books

For a long time buying books was considered at the very least practical. As long as it was economically sound, having a nice small library at home was without doubt a useful thing. However is this still true now? Naturally, I am not talking about buying fiction (this is after all a math blog).

To better illustrate the question, lets consider the following example. About a year ago some friends of my grandfather gave me a good multi volume encyclopedia. Obviously it is not something that is expected to be used everyday, but I didn't open it even once in all this time. The reason for this is simple - if I want information about some specific subject, it is easier for me to search in the Internet. It is almost certain that there will be an article on this subject on Wikipedia, or some other place.
To a certain degree the same is true even for my math textbooks\notes. Frequently enough I prefer to search on the Internet for a specific definition or proof of a certain theorem. Unfortunately, this is often less successful than searching for staff one can find in an encyclopedia.

As a result we get the following situation - while we have lots of available books it doesn't seem practical to invest money in buying them. This is especially true about buying new editions of books we already have, or books that cover the same topic but use different approaches. This is especially true considering how overpriced some books are.
A possible solution to this is downloading books (for free). While not all books can be downloaded for free from the net, it is possible to find good books on any given topic. For example there is currently a collection of over 600 math books available on bittorent. There is also a nice collection of calculus books on the same site. The only problem with those collections is that they will not remain available forever. In other words, it is a good idea to download it even if you don't need it right now.

This however brings us to a second problem. While it is possible to download lots of books from the net, we also need some way to organize them so that it will be possible to use them. Another problem is keeping an up to date backup (you wouldn't want to lose 10GB of books suddenly would you?).
I must admit that I don't feel that I managed to make any serious progress in solving either of these two problems. For backup, I long ago decided that burning my files to CD or DVD is not a good idea. It becomes difficult to keep track of the backups, and also the discs tend to be damaged so it is not very safe. Another option, is to keep a copy of your files on the web. I personally use Google Docs. It can only be used for pdf files up to 10mb, so some books I cannot upload, but it is really reliable and the way it is build makes organizing books relatively easy. Some times ago I tried to use Scribd for storing some of the large books I have. Unfortunately, it didn't work. They check the files that you upload, and if they notice that you have books that are copyrighted they will delete them. I also tried to use DivShare, but it is rather unreliable and overall not something I would recommend.

In the end the decision whether to have a digital book library or not is a personal one. In my case I decided to do it out of pure love for books. I just cannot say no to an opportunity to have a library. I do hope however that I will manage to make use of all those books I collected...