Math Pages Blog

My library

noreply@blogger.com (Anatoly) — Tue, 22 Sep 2009 07:19:00 +0000

Over the years I made a little online collection of math and other books. In order to organize and manage this collection better, as well as to share it with other people, I decided to post links to all these books here, on my blog. I hope that you will find this collection of links useful. As of now it is rather small, but I plan to add more books. There is a surprising number of such books available online for free - but it takes time to find them. Some If you have a problem with your book linked from here, please let me know and I will remove the link immediately.
Since most of these books are available for free on the author page, I just linked those pages. In some case, the book is on a site that has no connection with the author. In such a case it is possible that it is an illegal copy - use it on your own risk. I provide the links for educational use only.

If you want to recommend a link for addition, you can leave a comment. In order to keep this post as tidy as possible I will not publish the comment, but I will consider adding the book(s). In order for a book to "qualify" it must be about math, physics or programing and it must be large enough (that is, not an article but an actual book).

Math:
1. Elements of Abstract and Linear Algebra - E. H.Connell (author page)
2. Foundations of Combinatorics with Applications - Edward A. Bender, S. Gill Williamson (author page)
3. Graph Theory 3rd Edition - Springer-Verlag Heilderberg (order / author page)
4. Algebraic Topology - Allen Hatcher (author page)
5. A Problem Course in Mathematical Logic - Stefan Balaniuk (author page)
6. Multivariable Calculus - George Cain and James Herod (author page)
7. Calculus - Gilbert Strang (author page )
8. Linear Methods of Applied Mathematics - Evans M. Harrell II and James V. Herod (author page - this book has a rather nasty license)
9. Complex Analysis - George Cain (author page)
10. Linear Algebra, Infinite Dimensional Spaces, and MAPLE - James Herod (author page)
11. Linear Algebra - Jim Hefferon (author page ) This book has complete solutions of exercises.
12. The Geometry and Topology of Three-Manifolds - William P. Thurston (author page)
13. Introduction to Probability - Charles M.Grinstead (author page)
14. Elementary Linear Algebra - Keith Matthews (author page) This book has complete solutions of exercises.
15. Understanding Calculus (author page - this is an online book, you can download it for 5$)
16. Elementary Calculus: An Infinitesimal Approach - H. Jerome Keisler (author page)
17. Combinatorics - Russell Merris (link)

Programing:
1. Dive into Greasemonkey (author page)

General:
1. Flatland: A Romance of Many Dimensions - Edwin A. Abbott (link - there are lots of other books on this site)

Storing files on the Internet

noreply@blogger.com (Anatoly) — Sun, 20 Sep 2009 15:28:00 +0000

This post is a response to a comment that was left on my post Google OS is already here. I don't usually post responses to comments in such a way, but the answer to this comment ended up being rather long (I though initially that I can split it into two comments, but in the end it is almost as large as two regular posts) . Also, while the initial comment was about Glide, in order to properly answer the points raises I needed to give a little summary of the ways I use to share and store files on the web. This post can be thought of as a list of services that provide the functionality that is provided by Glide, but are not "packed" in an online OS. (This obviously brings us to the following question - Why use many services if there is one that provides all that functionality? The answer to this will be given in the end of the post).

Firstly, concerning the main point of the comment, I agree that Glide has features that a remote desktop doesn't have. I didn't say it clear enough in the previous post so I will say it in this one - the part that Glide cloned from Windows is the visual aspect. The functionality that it has is close to what one would expect from an online OS, but the problem is the "frame". As I see it, Glide attempts to easy the transition from other operating system to itself, but by doing so it is bound to compromise on the visual aspect and thus on the user experience. With this in mind lets look on the features Glide provides that were mentioned in the comment and what alternatives to them are available (if any).

The first point to consider is compatibility. I must say that I have been using Linux for about 3 years and I never had a problem opening any file I got my hands on. I did have some problem with a .gbi file a few weeks ago, but in the end I found a way to open it is well. In the latest Linux distributions there is more then enough support for main file formats, and unless you happen to live in a country with draconian copyright laws, there is no problem to install support for many other file types. In Ubuntu, there is an official package that installs all the needed programs. Besides, there are sites (like Google Docs and DivShare) that automatically convert files to a format that most devices don't have a problem with. If your files are on the web anyway, it doesn't mater what site they are on exactly. All you need to do is to send a link.

The second point is synchronization. I never had real need for synchronizing files between many computers. However, I am aware of some programs that allow easy synchronization as long as the operating system is the same. For Ubuntu you have a nice program called Ubuntu One. It offers a free 2GB online storage that is automatically synchronized across all of your computers (right now it is in beta, but I doubt it will stay like this for a long time). I don't think that it is a good idea to use it for large files, but as long as the files are small enough (lets say under 5mb) it is a perfect solution. For Windows, Diino offers similar functionality. They provide much more space (100GB and more), but it is a paid service.
It is also worth mentioning that Picasa allows to sync albums. The free account is rather small, only 1GB, but as you are probably aware it is possible to buy more storage space. I personally prefer not to sync files across computers, but to store them on the web and if I need them to access them on the web. In this way, my files are scattered around my computers, but they are all available on the web. Although, even there they may be on different sites. For example right now my files are distributed in the following way:
1. Google Docs - for small documents. Most of my documents are well within the size limit.
2. DivShare - for large Documents and Video/Audio. This site provides 5GB of free storage (you can buy more). It works well for storing large (but not huge) files.
3. Picasa and flickr- for photos. They both don't reduce the photo quality and allow an easy enough privacy management.
4. Photobucket - for photos that I use on my SU blog. Photobucket is excellent for monitoring bandwidth use, so since the photos I post on SU are usually small, it is a perfect solution for me.

The third point is accessibility. As you all know it is possible to use Glide from any device that has an Internet connection and a browser. Thus, all your files are always accessible from any computer with an Internet connection. But using an online OS is not the only way to get this. This is another thing that I apparently didn't say clearly enough in my previous post - my example with a phone. It is well known that it is possible to store files on a mobile phone. But you can do more than this. It is possible to install an operating system on it. A modern mobile phone can act as an USB drive. And it is possible to boot a computer from such a disc and thus it is possible to install a full operating system on the phone. Then, for example, if you installed Ubuntu like this, you can add it to your Ubuntu one account and all your files will be synchronized. In addition to this, you don't need an Internet connection to use the OS. This means that all the programs you have on your computer are always with you. The only downside to this is that you need to connect the phone to another computer to use the OS, while Glide is possible to use from the phone itself.

I was asked in the comment: "How would you share a 1GB video with someone in a distant location when you were traveling with your mobile phone away from all of your computers?" - The answer to this depends on the location of the file. If it is on my phone, I would probably use Filemail. It allows sending files up to 2GB for free. If I have the file somewhere on the net, I would just send a download link. Obviously, if the file is only on my computer I would have no way of sending it, but I try to have an online copy of all the important files I have on my computer. The fact that my files a scattered around the web makes it a little disorganized, but I don't think that this is a bad thing. Moreover, did you hear about a site named eggdisk? They offered a really nice online storage service and delivered it for sometime. But then one day (without any warning), they stop providing the service, turned the site into ads and didn't even allowed the people to get their files. Because of this case I prefer to keep my files scattered like this around the web.

The forth and last point is integration of services. About this I agree completely. Right now the level of integration between Google services is rather lacking. However, it is probably worth mentioning that too much of integration will effectively force the user to use only the services provided by Google. But this is just a remark, I most certainly hope that the level of integration will increase. This point is, I think, an important one for those who want a system "that just works". Because of this, Glide is better for those who need to use an online OS for work.

To wrap it up, for different people and situations different solutions are needed. For me Glide fails to be anything else but a remote desktop. Since I don't want my staff to be in only one location on the net, I cannot use Glide in a way other people do effectively. Also, because of this I prefer to use all the services I mentioned in this post, and I am always looking for new sites that provide functionality useful to me. However, since this leads to a loss of time, there are people who prefer their files to be centralized and they want to have one simple way to access them all. For such people an online OS like Glide is clearly the perfect solution. Even the fact that it clones Windows appearance is only a bonus to them.

Indescribable numbers

noreply@blogger.com (Anatoly) — Fri, 18 Sep 2009 12:11:00 +0000

This post is an attempt to explain what the term indescribable number means. Unfortunately, while this is a relatively well know term I frequently see it being misused. To understand it, we must firstly look on the proof that such numbers exist. It is a rather basic proof from set theory. What we need to do is to define two sets:

A={all the mathematical symbols and all the letters}
B={finite words in A}

Now, it is obvious that A is finite. B is not finite but it is only countably infinite (this is the smallest infinity). Therefore, B is smaller than the set of all numbers - R. Since all the possible descriptions are in B we conclude that there are numbers in R that cannot be described at all. Moreover, if you take away all the numbers that can be described the size of R will not change (this is a basic theorem in set theory). From this we can conclude that in fact most numbers are indescribable. This is a perfectly valid example of nonconstructive proof.

Unfortunately, this simple and short proof (I didn't proof all I said, but it is all just basic theorems of set theory) does little to explain what an indescribable number is. Lets consider some examples. For the first example, look on the following set:

C={words in A shorter than one hundred letters}

It is obvious that C is finite. Therefore there is a maximum number described by C. The next integer number (lets call it Y) is thus "the first integer number that cannot be described in 100 letters". But this description is less then 70 letters long. And this means that this number should be in C. Obviously this is a paradox. We got a number that is both in C and not in C.

Here is another example of a similar problem. It is possible to proof that if you randomly choose a real number the probability of it being an indescribable number is 1. So lets randomly choose a number (since we are choosing only one number we don't need the choice axiom). Naturally we get an indescribable number. Well, lets call this number "indescribable number 1". In the case you didn't notice I just gave a description to an indescribable number.

What really is going on is just an indexing problem. It is important to understand that both B and C are just sets of index numbers. If we have such a set we can use it to index another (in our case the set R). Mathematically, description as it was used in those examples is just a function from B to R (or from C to R). But the way we apply the indexing is arbitrary - we can choose any function we want. Lets first look on the 100 letters case. When we defined the set C what we really defined is the pair (C, f). In this pair f is a description function that for any x in C returns a specific number in R (I suppose that all the words in C describe some number, but you can do without it). Since we cannot define Y without firstly defining (C,f) the word ""the first integer number that cannot be described in 100 letters" is assigned by f to some random number. Then when we got a description for Y, we basically created a new pair (C, g). In this pair g is a new function that agrees with f on all C except for one word - ""the first integer number that cannot be described in 100 letters". To this word it assigns Y. For us this may seem illogical, because we think about meaning of words. But in this proof meaning is not important - the words are just a way to index.

With this in mind, lets consider the second example. In this case the number we choose belongs to a set R\f(B). When we gave it a description all we did was to change f in such a way that now this number belongs to f(B). It is obviously not a problem to do such a change (if we wanted to change the function for an infinite amount of values it might have been a problem, but for one index it is always easy to do).

So, what is an indescribable number? After all, we just saw that it is possible to describe any random number. The answer to this is actually simple. Mathematically it is a number that was not indexed by the function f. In normal language it means that it is a number that wasn't described. Form a normal person point of view this is a weird definition, but mathematically it actually makes a lot of sense. The basic idea is that while we have the option to choose any function f, we can only choose one and we cannot change our choice to another function latter on (this is because we need our language to be consistent). Under this conditions it becomes obvious that both examples are just a misunderstanding of what an indescribable number is. You cannot take a number that is not described and give it a description, because the description is already in use and you can have only one number for one description .
But then, what is the point in saying that such numbers exit? It is after all obvious that there are numbers that are not described as of now. Unfortunately this is not what the theorem is about. The point of this theorem is not to say that there are numbers that we didn't describe. This theorem says that no matter what function f we use there are always numbers that are not in f(B). In other words, there are always will be numbers that we didn't describe even if we used all of B for the purpose of such description.

Google OS is already here

noreply@blogger.com (Anatoly) — Wed, 16 Sep 2009 14:32:00 +0000

As you all probably know Google recently announced that they are going to build a natural extension to Chrome. Obviously, a natural extension to a web browser is an OS (and no this is not a joke). Well, I wonder when I will get to use this.... I just hope I will manage to restrain myself from installing the first beta version available to the public.... Sometimes I feel that I love installing software a bit too much. But anyway, lately I feel that the final release of Google OS will be more of a formal step than anything else. The reason for this is that the services provided be Google are already close to being a real on-line OS. Before explaining what that means, lets look on a typical online OS. For example, Glide. If you look on it (here is a little video), you will see that it is basically an attempt to clone a regular OS on the web. The reason I am saying this is simple - just look on the way it looks. It is just like Windows. Obviously some details are different, but the overall idea didn't change. And this is a problem. There is no reason for an online OS to look like a regular OS - the whole point in the transition to online OS is to find a new concept, a new look for the OS. Moreover, the system offered by Glide is nothing more than an remote desktop. The only advantage it has over a normal system is the fact that it is available from any computer with Internet. But this is not enough for a system to be an online OS. Actually, if you want you files to always be available, you can install an operating system on you mobile phone, and then you will be able to use it on any computer with an USB port- even an Internet connection is not needed. Obviously this is better than what such an "online" OS offers.

Google on the other hand is close to making a real online OS. Even now their web services cover most of the things we expect from an OS. To better understand this point lets look on what we expect to get with an install of Windows and attempt to find the corresponding functionality in what Google currently offers. Naturally we will be only looking from the end user point of view.

The most obvious thing we get when installing Windows is "My Documents" folder. In the two latests releases there are attempts to further divide this folder into pictures, documents music and videos. Google provides us with Google Docs, YouTube, Google Video and Picasa Web Albums. Each one of them is meant for only one type of content and they are much better for managing your files that what you get with a default installation of Windows. (I don't know any good place to store music online, but I am sure that there are ways to do this as well, although it is not provided be Google). It is important to remember that while right now Google doesn't do mass file storage, there are sites on the Internet that do just that, and there are rumors flying around the Internet for years about Google storing 100% of our files. Besides, any really large files (like video longer that what you can put on YouTube) you probably don't want to store on the net because getting it from there takes a bit too much time with current Internet connection speeds.

The second thing we get is Office (and notepad, for those who use it). It is not provided be default but it is a basic enough product. Google gives us Google Docs which are for most users a good enough replacement for Microsoft office. Also if you don't like Google Docs you can use other products like Zoho for example. On this note, it is also possible to do light photo editing on the web. While I don't know about sites good enough for professional graphic design, I do know about a nice site for simple graphic editing - FotoFlexer.

The third and final step is communication. This is not something provided be Windows, but the reason we buy computers is to be able to communicate with other people. Since we are talking about an online OS, things like communication become something that is only natural for an OS to provide. And Google does just that. We get Gmail, Blogger, YouTube, Google Video, Google Docs and etc.. All of these products are either build for communication or have the option for collaboration like Google Docs. Right now we view them as services but for an online OS, those are major components.

The only thing Google is missing right now is a "frame". In order for all those services to become an OS we need something that will put them together and offer them to the public as an OS. The building of this frame is being done in three stages. The first stage is the links to other Google products that appear on the Google main page. Those links make all those services connected and easy to reach from each other. The second stage is the browser. The browser has a lot of importance. The way it works and looks is extremely important for an online OS (the reasons should be obvious). For this we now have Google Chrome. Did you notice that by default it has only two lines of menus compared to Firefox five lines? It is obvious that the developers attempt to make it even look like a frame, like the status bar in Windows.
The third and final stage is the Kernel. It is rather pointless to have two OS on one computer in the same time. If we are too use Google OS from Windows (or a Linux distro) we are basically using one OS to connect to another. This is something that needs fixing and Google is now doing just that - they are making a desktop component for the online OS, a component that will remove the need to use two OS in the same time. The only thing that this component needs to do is to boot up, start Google chrome and connect to the Internet. This simplicity is also the reason why they say that the OS is a natural extension of the browser. All it has to do is to make the browser start without anything else on the computer. Obviously, this also means that all the configurations that are needed to be done for the computer to work will be done from inside the browser (at least I think so). Doing this will require a lot of work on Google Chrome, but with Google resources and the help from the Linux community this project will almost surely receive, doing this all in a reasonable time is perfectly possible.

Update:
There was an interesting comment on this post that caused me to write another post about Glide and the functions it provides. The post is titled Storing files on the Internet.

New software wave

noreply@blogger.com (Anatoly) — Mon, 14 Sep 2009 16:49:00 +0000

This post is basically a collection of reviews of different programs (not related to math) and my opinion about their uses. It also includes my opinion (and hopes) for some upcoming products.

Windows 7
Yesterday I finally got my hands on Windows 7. Since I am using Linux I obviously tend to think that Windows is inferior to Linux, but I believe that it is important to at least keep an eye on other operating systems. Also, from what I found about it on the net this particular release of Windows is worth looking into. In this short review I am going to only talk about installation and the overall feel of the system. The rest I'd rather leave to more competent people.
I used qemu to install it on a virtual machine, so I didn't test all the features (especially not all the graphic features) but I did get a general feel of the system. Somewhat surprisingly I will that I neither like nor dislike it - in another words it failed to make any impression. The install process went without any trouble and I think that it was faster than that of Win Xp (it is hard to tell when using qemu). It also looked much better in terms of graphic (although compared to linux it is ages behind, live CD is much better for installing an OS).
The desktop itself looks rather impressive. It is also good that you have a sticky note program as part of the OS already installed. I personally prefer to use a Google to-do list gadget, but for others notes might work better. The start menu is also rather nice, although I think I would like it to consist only of its left side - there is no need for the right side of the start menu to appear all the time, most users don't even use the options you have there (except for the shut down button). I also think that the way it slides instead of opening new menus is definitely an improvement. A similar menu is available for Ubuntu, you can get the deb file here.
Virtual folders are another wonderful idea. I would really like to test them using a few thousands photos. Hope they will appear in linux soon enough.
If you want a more in-depth review of Windows 7, I suggest these two posts: Windows 7 End user experience and Windows 7 Performance.

However, despite all those improvements Windows 7 is still well behind Linux. It requires a lot of space to install, it is less safe and it is still behind in usability. The graphics is clearly improving but there are less useful effects than I have in Linux (frankly there is no way that I will use an OS that doesn't support multiply desktops). I hope that in the future Microsoft will adapt more fearutres from linux, especially the desktop cube. There are other areas in which it is lacking but they didn't changed from XP so I m not going to repeat them here.

Overall, this is a very nice version of Windows. I am definitely going to recommend it to those who still insist on using Windows. The only thing to be careful about is RAM. You need at least 2GB of it so if you are upgrading from XP you must make sure that you have enough RAM. In my simulation I was running it on a 3GH single core virtual processor, so you don't need to think much about processors. Any computer bought in the last 3-4 years should be capable to run it just fine.

Google Chrome
Since I don't like downloading software that is not available for Ubuntu from a repository, I installed Chrome only recently. I was quite impressed with it. It is a very well build browser. It is fast and has a nice collection of features. Its main shortcoming is the lack of extensions. While some extensions are available, they are just too few. Actually I considered switching to it from Firefox, but decided not to. If there were more extensions I would switch in a second.
Since it is going to be an important part of the future Google OS it is quite encouraging to see it doing so well even now. It is also interesting to note that it looks like a frame. The developers clearly put a lot of effort in thinking ways to maximize the part of the browser that actually displays the web page. Frankly it looks like they are already thinking about how Chrome will be part of their OS.

Google Wave
I obviously didn't try this one myself - it will be available to the public only at the end of the month. But there is already enough information about it to make an opinion. Overall it looks like a very nice example of software evolution. It is definitely a great concept, especially for those who communicate with lots of friends using the Internet. Since I don't communicate with many people, I don't know how much I will be effected by this product but some of the ways it can be used are clearly interesting to me as well. For example there is an idea to use it for managing comments on blogs. In this way commenting will be done in real time. Also, it will probably be possible to comment on a blog post "inline". Such an option would allow one to talk with the author about the post in a wave. A typical case can be asking for futher information or explanation from the author or other readers (a bit like Wikipedia edit page) and then the author will be able to automatically update the actual post with the explanation. Obviously a feature like this is rather interesting for those who have blogs.
The only thing that seems to be missing (for now) is the ability to make a video call from inside a wave. However, this is a simple enough matter of integrating Google talk into it, so it will be probably done soon enough (if not by Google than by someone else). Despite the fact that all the information I have about it comes from a few videos, it appears that Google wave is a perfect candidate for a communication center of the upcoming Google OS.

In the next post I plan to write a bit about Google Chrome OS and what is an online OS.

Math Pages is two years old today

noreply@blogger.com (Anatoly) — Sat, 12 Sep 2009 15:32:00 +0000

I wonder if I should write that time went by slowly or should I instead say that it went by without me noticing? I suppose both will be correct. Anyway, I am rather happy to celebrate the second anniversary of Math Pages blog. I once read that most blogs only survive for about 6 month, after this period the person who made the blog leaves it for one reason or another. On the other hand, the blogs that manage to survive after six month usually stay around for a long time. Well, I certainly intend to continue posting, at least untill I am no longer a student.
Unfortunately, I must admit that my posting is really irregular. Originally I was aiming for one post per two days, but it didn't work out on the long term. Occasionally I do manage to post this many posts (and even more on some weeks) but there are periods when I don't post anything for more than a week. Right now the ratio of posts to days is one post per 4 days. As you can see that is far from what I want it to be... Well, since this is an anniversary post I suppose it would be logical to resolve to be more committed to posting, but I seriously doubt that that will really help.

It also worth to mention that while originally I intended for this blog to be about math with mentions of physics, that didn't work out. Instead I found out that I don't like to be restricted or even directed in my writing. I prefer to simply wright about what seems to be interesting to me at this particular moment. This is probably one of the main reasons why I couldn't put to use the skribit suggestion widget... But anyway, this blog ended up being mainly about math but I also occasionally write about computers (most linux). In the beginning I wrote a bit about physics, but with time it seems that my interests shifted away from it.

This post is probably a good opportunity to announce a little project that I am trying to start. Over the years I managed to collect a small library of math books in pdf format. Right now it is rather small, but it is easy enough to expand. There are after all lots of free math books available (legally) on the net. What I want to do is to create a post with links to all these books I collected and plan to collect (currently stored on my Google docs account). Having such an index will clearly make it easier for me to use my library, and will also make it easier for you to find those books if you need them. I already started to work on this idea, it will probably appear here in about two days (maybe more if I run in some kind of problem). Meanwhile if you have math (programing and physics will do as well) books in pdf format that you don't mind sharing with others, or you know about a site where such books can be found please contact me. While I am not going to promise to use what you sent, I will definitely consider it.

Fractals

noreply@blogger.com (Anatoly) — Wed, 09 Sep 2009 16:34:00 +0000

A few days ago I stumbled on an excellent collection of fractals. According to the site the collection is no longer updated, but nevertheless the images there are among the best fractal images I ever saw. There is also more fractals by the same author available here. I am somewhat worried about the site disappearing so I am going to download all the images on it to my computer and then upload them to a private Picasa album. This way I am not breaching the copyright on the images and if the page indeed closed down, please leave a comment on this post and I will link my Picasa album to this post.

In the case that you want even more fractal images, here is another site with hundreds of excellent fractals - Fractal world gallery.

Also, I noticed that a lot of visitors to my blog are actually after mathematical wallpapers. Form my own attempts to find such wallpapers I know that they are somewhat of a rarity (at least good ones). I currently have 14 math wallpapers in my Picasa albums, but this is obviously too little (and unfortunately not all of them are good enough to put on a screen). Since there seems to be a demand for such wallpapers I am going to try and get more maybe I will even attempt to draw some myself.

Correct math wrong results

noreply@blogger.com (Anatoly) — Mon, 07 Sep 2009 13:22:00 +0000

In the previous post I mentioned that in some situations even the math we used is correct, the result we arrive at might not be correct or it might not apply in the real world. In this post I intend to discuss three such examples.

The first example is known as the Tompson lamp problem. Imagine the following situation: You switch a lamp on, than after one minute you switch it off. After 30 seconds you switch it on again, and then after 15 seconds you turn it off. We continue like this for two minutes. Now, is the lamp on or off? There is no real mathematical solution to this problem. One proposed solution is to say that the state of the lamp after two minutes is independent of its state before. So for all we know, after two minutes the lamp could have mutated into a pumpkin. Seriously.
Another solution originates from noticing that the behavior of the lamp can be though of as the infinite sum: 1-1+1-1+1-1+.... So if we find the sum we will get the solution. Consider the following:

S=1-1+1-1+1-1+...
1-S=1-(1-1+1-1+1-1+...)=1-1+1-1+1-1+...=S
1-S=S
1=2S
S=0.5

And thus we found the sum. The result is usually interrupted as the lamp being half on. I don't know about you but I never saw a lamp being in that state.
What would happen if we are to do this switching in reality? Thats simple - the switch will break.
As you can see in this case modeling the situation mathematically fails completely.

The second example is an implementation of a theorem to a situation it cannot be applied in. The theorem I am talking about is the Brouwer's fixed point theorem. It is one of many fixed point theorems, which state that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. Sometime ago I read a statement that according to this theorem, if you take a glass of water and then mix it in anyway, there always will be a small part of the water that didn't change its position. If you are not careful this may seem reasonable. After all, mixing the water is a continues process. Unfortunately, this is simply wrong. The theorem itself is of course correct, but it cannot be used to discribe water in a glass. For this theorem to be used the body it is used on must be continues, but the water is discrete - it is composed from atoms. Because of this the theorem cannot be applied to such a situation, and the result we get by applying it forcefully is wrong.

The third example is known as the Banach–Tarski paradox. It is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. Obviously this is not something that is possible to do in reality.
Again, the theorem is perfectly fine. The problem is that it cannot be applied to actual balls. During the proof of the theorem (at least the proof I am familiar with) we get a countable infinity of finite degree polynomials of the form p(sinx)=q. We need to choose x in such a way that for all the polynomials q is not 0. Since any polynomial have a finite number of roots, there is at most a countable infinity of values for x that doesn't give us what we want. Therefore we can always choose a value that will work.
Unfortunately x is the angel of rotation of the ball. If we want to choose a specific x we need firstly to make sure that we can rotate the ball by such an angel. Surprisingly this is not always possible. The reason for this is physical and not mathematical, so I am not going to explain it in detail, but the main idea is that we cannot make "moves too small" in the real world.

Mathematics is often said to be describing the real world. Personally I don't think so. Those and other similar examples have one common trait - correct math that cannot be applied in our world (at least not the way we want it to). But it can be used to describe a world of math.

Painting the roads

noreply@blogger.com (Anatoly) — Sat, 05 Sep 2009 05:10:00 +0000

Consider the following problem - given a road system is it possible to have a set of instruction which if followed will cause you to end in one specific point regardless of were you started? Obviously this depends on the road system. It is very easy to give examples of road systems for both cases. It turns out however that there are really simple conditions that guarantee the existence of such an instruction set.
In order to discuss this mathematically we will need to define a few things. Firstly we are not going to talk about roads but about finite directional graphs. Secondly for every vortex on the graph we will allow exactly d (d is the same for all vortexes) edges to come out from it and an unspecified number of edges to go to it. In such situation talking about "turning right" is meaningless so we will instead use colors. Thus every "color" is a function that for a given vortex tells us were to go. Naturally there is d colors. In this blog post I will denote colors with letters - for example (a), (b). Only one color means doing only one "step", so for longer steps we need "words" instead of letters. For example (abdc) means to execute (c) then (d) then (b) and finally (a). In this notation we are looking for a word that when executed will result in us landing on a vortex V regardless of our starting point. We will call such a word a sync word.

Required conditions
Firstly, lets see what obvious conditions are required. Basically we want a graph that is possible to color in such a way that there will be a sync word. So lets demand the following conditions:
1. The graph is strongly connected - there is a path between any two different vortexes.
2. The graph is not cyclic. To be more precise what we want is:
a. The greatest common divisor of the lengths of all the closed paths in the graph is 1.
b. It is not possible to divide the graph to a finite number of sets S1,....Sk such that for all i the edges go from Si to Si+1.

Theorem
The conditions listed above are enough.

Prove history
I am not going to prove this theorem here. It is possible, but it requires a bit too much explanations and the explanations are hard to understand without drawings. Because of this I am not going to prove the theorem, but I will shortly discuss the history behind the prove and some important points behind the proof. If you want to read the actual prove it should be available online (somewhere).

There were three main stages in the attempt to prove the theorem. The first stage was done by Friedman. He defined a weight function of the graph, and showed that if the weigh of the graph is a prime number than there is a way to get a sync word unless the maximum weight (the maximum weight is the weight of the largest (by weight) subset of the graph that can be synced) is 1. As you can see this result is very far from proving the theorem, but it turned out to be an important step - the weight function he defined turned out to be useful in future attempts to prove the theorem.
The second stage was done by Kari. He proved that if the graph also has a fixed number of edges entering each vortex and this number is also d, then there is a way to get the sync word. While this is not what is needed this is a much better result than the previous one. It is also interesting to note that he managed to use induction in his prove, by finding a way to reduce the problem of finding a sync word for a large graph to finding such a word in a smaller graph.
The third and last stage was done by Abraham Tracman. This was done only in 2007. As I already said his prove requires complex explanations so I am not going to say much about it. The only thing that is important to say is that in the prove we look closely on the results of continuously using the same function on the graph. By analyzing the result we find out that in all the cases we can get two vortexes that satisfy a condition needed for Kari proof to work. Basically he showed that it is not needed to ask for a fixed number of edges entering the vortex, and by this completed the proof. In a way this is an interesting example of how people build on other people work.

Conclusion
I hope that by now it is clear that the problem has absolutely nothing to do with roads. Well, except for the name. Originally this problem originated from symbolic dynamics. However the name road coloring sound much better and it is possible to use the result in road building, although it is rather pointless to do so. All we managed to do is to show that there is a way to color the graph, so if we color the roads in the appropriate way we will be able to get the needed instruction set but it will require a more complex road system that what we have now.

In this particular case mathematical analysis of a problem gave a solution that is accurate and possible to use in the real world - but is this always correct? In the next post I will discuss some rather famous cases where correct math leads to solutions that are impossible in the real world.

Windows XP on Ubuntu with qemu

noreply@blogger.com (Anatoly) — Thu, 03 Sep 2009 08:25:00 +0000

If you were reading this blog for a long time you already know that I ditched Windows and moved to linux years ago. I have been using Linux for about 3 years now and I never regretted moving to it. However, annoyingly enough, there is one tiny thing that I cannot do on linux but I can do on Windows. In the Hebrew University, in order to get the exercises you need to download them from the university site. Most of the site works just fine in Firefox, but for some reason it is impossible to download the exercise with it. It is only possible to do so in IE.

The students have been asking the university to fix the site for the last two years, but without much success. It is of course possible to go to an university computer and to download everything there, but it is inconvenient. I tried installing IE using wine, but it didn't work. So in the end the only option I could think about was to install Windows using qemu. This post is a short how-to that shows how to install and then configure windows under qemu for it to work without doing problems.

The first step is obviously to obtain a windows cd or an iso file. The next is to install all the required packages:

sudo aptitude install qemu kqemu-common kqemu-source samba smbfs

This will install qemu and also samba for sharing files between Windows and Ubuntu. Next is to configure kqemu - it is an accelerator used by qemu:

sudo module-assistant prepare
sudo module-assistant auto-install kqemu
sudo addgroup --system kqemu
sudo adduser $USER kqemu
sudo modprobe kqemu

Now you need to log out for the changes to take effect. After logging in, the next step is to create a disk image. The minimum size is 4GB (because of the SP3 and other updates), but I used 10GB because I want to be able to install programs latter on without worrying about free space. The image will change it size dynamically, so don't worry about throwing away too much space. To create the image type:

qemu-img create -f qcow windows.img 10G

Now we can start the install process. If you have a CD insert it and type:

qemu -localtime -cdrom /dev/cdrom -m 512 -boot d windows.img

I set memory to 512MB but you can enter something else, preferably at least 384. If you want to install from an iso, put the iso in your home directory and type:

qemu -localtime -cdrom cdimagefile.iso -m 512 -boot d windows.img

After doing this, the install process will start. It worth to note that it will go on for longer than a regular install, so you will have to be patient. Once the install is done, you can start using Windows. To run windows you type:

qemu windows.img -localtime -m 512

Optionally you can create a launcher on the panel or the desktop. For this you will need an appropriate icon. I used this one:

As we all know, windows is much less safe than Linux. Luckily qemu has some options that allow us to protect the installation. The first such option is to create an overlay. This will allow you to use windows while saving all the changes in the overlay file and not in the original img file. If you do this and at some point at time Windows becomes corrupt you can just delete the overlay and create a new one without installing Windows from the beginning. To create an overlay type:

qemu-img create -b windows.img -f qcow windows.ovl

To use the overlay you will need to type:

qemu windows.ovl -localtime -m 512

For increased safety you can also use the snapshot mode. In this mode all the changes you make are written to a temporally file which is removed when you close qemu. To use this just add "-snapshot" to the end of the command.
The final step is to make it possible to share files between Windows and Ubuntu. To do this we can use samba. The first step is to create a shared folder. In my case I created a folder named AnatolySharedFiles in my home directory. Now we can setup samba to share this folder:

sudo addgroup samba
sudo adduser $USER samba
sudo aptitude install system-config-samba

Now you can go to System->Administration->Samba, and use it to add the folder you want to share and to add a user password for yourself. Now all that remains is to configure Windows. The first step is to go to My Computer->Properties->Computer name and enter the correct data. Then go to Network places to create the network. The next step is to mount the shared folder as a network drive (this seems to be the best option to me,buy you can access it from network places as well), this is done by clicking on My Computer->Map network drive.

After doing all this you hopefully have a working Windows and I have an easy way to get my math homework...

Math and Firefox wallpapers

noreply@blogger.com (Anatoly) — Wed, 10 Jun 2009 12:36:00 +0000

I added new wallpapers to my collections of Firefox and math wallpapers, follow the links to download.

From Firefox Wallpapers

Ordinals

noreply@blogger.com (Anatoly) — Sun, 07 Jun 2009 10:25:00 +0000

I once read about a theory that said that numbers can be described as a common property of two groups that have nothing in common excluding their size. For example the number three is a common property of the following groups - three deers, three stones and three trees.
In modern mathematics we have a sort of an extension to this idea - ordinals. An ordinal is a well ordered set such that if A is an ordinal and x is in A and y is in x then y is in A. The first ordinals are phi (=empty set), {phi}, {phi,{phi}}, {phi,{phi}, {phi,{phi}}}. Those ordinals correspond to 0,1,2,3.
As you probably noticed there is a very simple rule that produces the next ordinal - if A is an ordinal than A(union){A} is the next ordinal. From this we can conclude that: The set of all ordinals is a well ordered set and the union of any number of ordinals is an ordinal.

What makes the ordinals truly interesting for me is the fact that in for them "infinity plus one" is not equal to infinity. This is very simple to see, infinity is the so called least infinite ordinal - w. It can be defined as the union of all finite ordinals. The next ordinal is w+1=w(union){w}. It is rather obvious that the two sets are not equal and therefore w+1 is not equal to w.
Ordinals are not the only example of infinity not being equal to infinity and one, but in my opinion they are extremely intuitive in this regard. After all, all we basically do with ordinals is to constantly "add one". This is the same thing we did with natural numbers long ago, but it appears that the natural numbers don't follow our basic intuition that says that "it is always possible to add one"

In the beginning of the post I told that numbers can be described as a common natural property. This however brings an interesting philosophical question - if our intuition is a product of our world than why do natural numbers that come from it don't follow our intuition after a specific point? A possible answer is that "infinity is not natural" and therefore there is no reason for it to follow our intuition in any way. However, infinity appeared as a concept a lot of time ago. At first it appeared as "many" which basically told that there was no known number large enough.When a new number (or even a number system) where invented the "many" was replaced by an appropriate number. And this brings us to the following thought: Is it possible that we are in the same condition again? That is, should we use ordinals instead of natural numbers? After all, they are pretty much an extension of the natural numbers.

Unique captchas

noreply@blogger.com (Anatoly) — Mon, 01 Jun 2009 05:00:00 +0000

Apparently some sites use captchas to assure the intelligence level of their users:

Project Euler

noreply@blogger.com (Anatoly) — Sat, 11 Apr 2009 17:08:00 +0000

I recently found an interesting site called Project Euler. This site attempts to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context. This is being done by publishing different mathematical problems of varying difficulty.
I liked the concept so I thought about joining the site, but after taking a closer look on the problems I lost my motivation - most of the problems are meant to be solved using a computer. I just can't feel motivated to write a program in order to sum all the primes less than 2 million (this is problem number 10). However, if you like this type of problems this is clearly an excellent site. They have lots of problems of varying difficulty and new problems are constantly added.

For those who like my don't like using computer to solve problems there are problems that don't require a computer to solve - for example the first problem: "Add all the natural numbers below one thousand that are multiples of 3 or 5." This one is a very simple problem, so I guess it will be fine to post a hint to a solution. All you need to do is to sum the arithmetic progressions 3,6,9,..... and 5,10,15..... If you add the sums you will get the result, but the numbers that are multiples of both 3 and 5 will be counted twice.
I also really liked problem number 205. It is pretty easy to solve, but it requires some thinking and there is no need for a computer.

Infinitesimals

noreply@blogger.com (Anatoly) — Wed, 18 Mar 2009 17:29:00 +0000

In the previous post, I wrote about division by zero. In this post I want to talk about one particular case when such division, and its definition are important. As you probably guessed from the title, this post is about calculus. (In this post I am talking only about one variable calculus).

One of the most basic questions in calculus is finding slopes of functions. The simplest example of such a problem is to find the slope of a linear function, f(x)=mx+b.
In this case we get a straight line, so the slope is uniform. To find is we need to calculate the difference in y divided by the difference in x:

(f(x+h)-f(x))/h=(m(x+h)+b-mx-b)/h=m(h)/h=m

For a less friendlier functions, we cannot talk about globe slope, but only about slope of a certain part of the function, or even only in one single point. And this is were the problem appeared. We want to know the slope in every point, but how can we calculate it? Firstly we need to define what such a slope is. For example lets look on the function f(x)=x^2 and on the point x=3. Lets look on the points f(3.5) and f(3). If we connect those two points by a straight line, we can calculate the slope of that line. If we draw this on a paper, you will see that the function and the line are really close to each other on a small area around 3. Therefore we can thing about the slope of this line as an approximation to the slope of the function. But, obviously if instead of 3.5 we will take 3.1, we will get a better approximation. In the end we can think about the slope as:

slope=(f(3+h)-f(3))/h , h=0

And here we have it - devision by zero. It is obvious that there is no way around it. If h is not zero we get only an approximation and one which can be improved easily enough. The solution to this problem was found by Newton and Leibniz. Their idea was to define a non negative "number" that is smaller that any positive number. Such a number is called infinitesimal. The only real infinitesimal is zero, but if we agree to imagine that there is another such number, they we get many such numbers. This is because if dx is an infinitisimal that 0.5dx is also an infinitisimal. Since dx is not zero, we can divide by it. And becasue it is smaller than any positive number we can disregard it as if it was zero. It is simple to find the slope of x^2 usinig dx:

slope=(f(3+dx)-f(3))/dx=(9+6dx+dx^2-9)/dx=(6dx+dx^2)/dx=6+dx=6

Although the result is correct, we no longer use infinitesimals, but use limits instead. The reason for this is that infinitesimals are problematic. The problem lies in the very definition - it is not clear what do we mean by a number that is smaller than any positive number, but not negative or zero. We also treat it as both zero and as not zero. However, they are still in use in physics. The reasons for this is that while they are not rigorous enough for mathematicians to use, they give good intuition and they appear rather naturally in physical problems.

Division by Zero

noreply@blogger.com (Anatoly) — Sun, 08 Mar 2009 13:37:00 +0000

We all know that division by zero is undefined. But what does it mean? Firstly, it is not "completely" undefined. For example, it is possible to say that:

1/0=lim1/x, x->0

This particular definition is not very good, because the limit doesn't exist. But we can always suppose that we look only on positive x and the the limit is defined to be infinity. Alternatively, it is possible to use geometric series. The formula for the sum of the geometric series says that:

1+q+q^2=q^3+...=1/1-q,

If we will take q to be 1 we will get: 1/0=1+1+1+1+....=infinity

As you can see I just wrote two different definitions to division be zero. However, this doesn't solve anything. The problem is that infinity is not a number. We could just as well say that 1/0=watermelon. Mathematically it is basically the same. Therefore when we divide by zero we no longer deal with numbers, and this what makes such division undefined.
Now, what about other definitions? A very long time ago, when "zero" appeared in mathematics there was an attempt to define division be zero by simply stating that division by zero gives a fraction whose denominator is zero. But this is again not a number.
We also cannot define 1/0=a where a is a number. The reason for this is that in this case we get:

1=0*1/0=0*a=0

And if this happens we get that the only number that we have is 0, because all the "other" numbers are equal to it. Obviously this is not an interesting situation. Because of this it is necessary that the devision by zero is not defined.

Lets look on some other example of undefined identities. The first one is 0^0. The reason this one is undefined is very simple:

0^0=0^(1-1)=0^1*0^-1=0/0=0*1/0

And we are back to division by zero. It is important to note that it is sometimes assumed that 0^0=1, but this is mostly done as an alternative to saying that in a polynomial x^n in which n can be zero x is not zero.
A more complex example is (-3)^x where is a real number (that is, x is not rational). Obviously, there is nothing special in number (-3), this expression is undefined for all negative numbers. For positive numbers the definition uses limits. For example:

3^x=lim(3^q), limq=x, q- rational

This definition works well for positive numbers, but for negarive numbers we have the problem that the square root is not real. Because of this if we don't allow complex numbers we get many undefined points in the series, and if we allow such numbers the series doesn't converge.

As a bonus, here is a little proof that 2=1. Can you see where is the error?

a=b.
a^2=ab ,
a^2+a^2=a^2+ab,
2a^2=a^2+ab,
2a^2-2ab=a^2-ab,
2(a^2-ab)=1(a^2-ab)
2(a^2-ab)/(a^2-ab)=1(a^2-ab)/(a^2-ab)
2=1

The reliability of wikipedia

noreply@blogger.com (Anatoly) — Wed, 04 Mar 2009 08:01:00 +0000

As you all know there are many articles on wikipedia, and their quality varies greatly. However the main problem is that in many cases it is impossible to tell if a particular article is accurate or not. For example, I just found the following article on wikipedia:

Beard-second
The beard-second is a unit of length inspired by the light year, but used for extremely short distances such as those in nuclear physics. The beard-second is defined as the length an average physicist's beard grows in a second, or about 5 nanometers[1].

One beard-second equals 50 Ångströms (10^-10 m). 20000 Beard seconds equal 1 RCH. 2000 Beard seconds = 1 RBC.

Google search supports the beard-second for unit conversions.[2]

Unsurprisingly, this article is being considered for deletion in accordance with Wikipedia's deletion policy. Suprisingly, the part about Google supporting beard second in unit conversion is actually true. After a quick search on the internet, it seems that the beard second was invented as a teaching exercise a few years ago.

In this case, the article seems to be a joke, but in reality it is about a funny teaching exercise....

Changes

noreply@blogger.com (Anatoly) — Fri, 27 Feb 2009 10:46:00 +0000

As you all probably noticed I didn't post anything for a long time. The main reason for this is that I was overloaded with homework. Also, the things I wanted to do started to pile up. The unread count in google reader went to about 800, sent pages on SU are now at over 100 etc. I also have unanswered comments that piled up..
In the following days I hope to clean this all up, and continue to post more regularly. Also in order to prevent this pile up from happening in the future, I decided to take some measures. Those that are relevant to this blog are written below:

1. Removing the skribit widget - while I got some votes and even one suggestion from it, it is not active enough to be really useful. Update: I will of course firstly write about all the topics that are currently in the skribit widget.
2. Closing the comments for antonymous users - I don't get too much comments, but anonymous comments are somewhat annoying when you have two or more such comments on one post.
3. Closing the comments on my kiba dock posts - there are lots of comments there wich ask for help, and I don't really have the time to help people with computer problems over the internet.

Now, if you have an opinion about this changes you are welcomed to voice it. As of now none of the changes are made, and if enough people will comment saying that they want things to remain as they are now, I will reconsider the changes.

The future

noreply@blogger.com (Anatoly) — Thu, 26 Feb 2009 11:07:00 +0000

Found on the web - Pizza 2015:

Operator: "Thank you for calling Pizza Hut. May I have your..."

Customer: "Hi, I'd like to order."

Operator: "May I have your NIDN first, sir?"

Customer: "My National ID Number, yeah, hold on, eh, it's 6102049998-45-54610."

Operator: "Thank you, Mr. Sheehan. I see you live at 1752 Meadowland Drive, and the phone number's 494-2399. Your office number over at Lincoln Insurance is 745-2302 and your cell number's 266-2566. Which number are you calling from, sir?"

Customer: "Huh? I'm at home. Where d'ya get all this information?"

Operator: "We're wired into the system, sir."

Customer: (Sighs) "Oh, well, I'd like to order a couple of your All-Meat Special pizzas..."

Operator: "I don't think that's a good idea, sir."

Customer: "Whaddya mean?"

Operator: "Sir, your medical records indicate that you've got very high blood pressure and extremely high cholesterol. Your National Health Care provider won't allow such an unhealthy choice."

Customer: "Dang. What do you recommend, then?"

Operator: "You might try our low-fat Soybean Yoghurt Pizza. I'm sure you'll like it."

Customer: "What makes you think I'd like something like that?"

Operator: "Well, you checked out 'Gourmet Soybean Recipes' from your local library last week, sir. That's why I made the suggestion."

Customer: "All right, all right. Give me two family-sized ones, then. What's the damage?"

Operator: "That should be plenty for you, your wife and your four kids, sir. The 'damage,' as you put it, heh, heh, comes to $49.99."

Customer: "Lemme give you my credit card number."

Operator: "I'm sorry sir, but I'm afraid you'll have to pay in cash. Your credit card balance is over its limit."

Customer: "I'll run over to the ATM and get some cash before your driver gets here."

Operator: "That won't work either, sir. Your checking account's overdrawn."

Customer: "Never mind. Just send the pizzas. I'll have the cash ready. How long will it take?

Operator: "We're running a little behind, sir. It'll be about 45 minutes, sir. If you're in a hurry you might want to pick 'em up while you're out getting the cash, but carrying pizzas on a motorcycle can be a little awkward."

Customer: "How the heck do you know I'm riding a bike?"

Operator: "It says here you're in arrears on your car payments, so your car got repo'ed. But your Harley's paid up, so I just assumed that you'd be using it."

Customer: "@#%/$@&?#!"

Operator: "I'd advise watching your language, sir. You've already got a July 2006 conviction for cussing out a cop."

Customer: (Speechless)

Operator: "Will there be anything else, sir?"

Customer: "No, nothing. Oh, yeah, don't forget the two free liters of Coke your ad says I get with the pizzas."

Operator: "I'm sorry sir, but our ad's exclusionary clause prevents us from offering free soda to diabetics."

Tautology and theories

noreply@blogger.com (Anatoly) — Mon, 23 Feb 2009 05:21:00 +0000

What is a tautology? Simply put it is a statement that is always true. More specifically, it is a statement that is true because of its structure. Usually such a statement is not informative. The easiest way to explain this is by example, so lets look on the following statements:

1. All apples are round.
2. All apples are either round or not round.

I have never seen an apple that wasn't round, so the first statement is a correct one. However it is not a tautology. It is perfectly possible for a square shaped apple to exist, you just need to make it grow inside a box. Therefore this statement is not always true.
The second statement is always true, and therefore a tautology, but it doesn't say anything. That is, if all we know about apples is that an apple is either round or not round we don't know anything about apples.
It doesn't mean that tautologies are useless, they have both use and importance in certain cases.

Lets look on tautologies in a more formalistic way. In the example above I assumed we have an object "apple" and a property "round". However, this is not necessary. All we need to write this example is two symbols: P, Q. If we rewrite the example using this symbols we get:
1. P->Q
2. P->(Q(or)(notQ))

I don't want to explain the notation, if you don't know it you are welcome to use wikipedia.
This notation is far better that the previous one. With this notation we are free to chose the meaning of the symbols. For example we can suppose that the meaning of P is x=7, and the meaning of Q is x+4=0. In this case 1 is usually false (but not always) but 2 is true.
From this we see that the second statement is completely independent from reality. It doesn't matter if we use it to tell something about apples or about mathematics, it will always remain true. This is the true meaning of being a tautology.

Sometime ago I read a post that claimed that all theories are tautologies. From the above it should be obvious that this is wrong. But there is a certain moment that causes this confusion. A theory is basically a collection of statements (theorems) that can be logically concluded from a certain set of prepositions. Those prepositions are divided in two groups - tautologies and axioms. (We don't really need to include tautologies, but because they are always true they get included automatically). Axioms are not tautologies. They are just "rules" that we choose by ourselves. In physics, those rules are based on reality and experiment, but this doesn't have to be the case. For example - the parallel postulate of Euclid. It seems natural to us to think that it is correct. But it is possible to built a geometry without it. Naturally in such a geometry all the statement that were derived from the parallel postulate are no longer correct (they might be correct, but not necessarily).

To sum up this discussion, a theory is correct as long as the axioms from which it was built are correct. However, if we try to apply it to a "world" in which the axioms are not true, the theory will not work. By the way, this is a major problem for physics and economics. They can build wonderful theory only to find out that the world we live in is different from their initial assumptions. In mathematics this is not a problem because there is no desire to describe our world, but to built a mathematical structure.
This also makes easy to explain the statement that a theorem once proved is true forever - a theorem is dependent only on its own conditions, so as long as they are satisfied the theorem will always be true.

Limits of applicability

noreply@blogger.com (Anatoly) — Fri, 26 Dec 2008 11:26:00 +0000

A question: In physics, the world is considered to be continues. We can see this for example in the fact that we use integrals instead of finite sums in mechanics. However, we know that the objects in our world are made of atoms, which are in turn made of more elementary particles so they cannot be considered continues in a mathematical meaning. So can we apply mathematical result that are based on the assumption that the world is continues, to solve physical problems?

Since as I already said we use integrals in mechanics, it should be obvious that we can do this. But this is not true in all cases. For example lets consider the following theorem:

For any compact and convex set K in R^n and any continues function F from K to K, there is a point x in K such that F(x)=x.

This theorem is known as the general version of the Brouwer Fixed Point Theorem.

It should be obvious that a glass of water can be viewed as a compact and convex set in R^3. Also mixing the water using a spoon can be considered a continues function. Therefore, according to the theorem there is a point that didn't move. But this is obviously wrong. This is because the theorem gives us a point, but that point doesn't have to correspond to a particle and therefore saying that it didn't move is meaningless.

Why can we consider the world to be continues in one case and not the other? The reason for this is size. When we want to solve a problem in our scale (that is, we are dealing with objects that we can see and with results that can be observed directly), we can safely assume that the world is continues and use the corresponding tools. But this is only because the atoms are so small that for us there is no practical difference if we assume that the objects are continues.
But when like in the example with the water, the result we want to calculate (in that case a point) is not something we can see with our own eyes (we cannot see if an atom moved or not) assuming that the world is continues is wrong.

The above classification is very basic and inaccurate but it gives a general understating of the situation. In the next post, I plan to continue with this topic by discussing applicability of theories to reality.

I wonder if I am pushing myself too hard...

noreply@blogger.com (Anatoly) — Sat, 29 Nov 2008 18:09:00 +0000

Despite my best efforts I barely manage to finish all of my homework in time. Partially it is due to the fact that I was ill for a couple of days which caused me to slightly fall behind, but it seems that getting ahead is more hard that I thought. I also spent half a day yesterday fixing things in the new apartment. As it looks now I will finish all the exercises (of this week) on time, but this will require studying all day long (for the next three days). I don't mind doing so, but after a certain number of hours my head just stop working (and I get a headache). I also think that I am not getting enough sleep...

I was told once that the workload for math students raises with every year (unlike for biology students, who on fourth year can work on two part time jobs). It certainly seems true.
I am also planning on tutoring a bit, it seems line I have an hour or two once a week that can be used for such a purpose. It probably would be better to study during this time, but I am rather sure that I wouldn't manage to do this anyway - my homework requires much more energy to do that I have at that time of week, tutoring somebody will surely be simpler so I hope to manage.

As of now, my main concern is to manage to get ahead in doing homework, so that I will be able to read my notes and to look the material ahead. I really hope to mange to d0 it this week (I hoped the same the last week,so wish me luck.. ).

Now to brighten the mood a bit - a little math fun fact ( the link goes to wikipedia, and the explanation there is not very good, it is too difficult to read. I will try to find something better or to write it myself).

Geometrical representation of numbers

noreply@blogger.com (Anatoly) — Thu, 20 Nov 2008 15:36:00 +0000

This post s a response to a comment I got today. I was asked if there are numbers that cannot be represented at all using geometry.
The answer to this question depends greatly on what you consider to be a number, and what you consider to be a geometrical representation.

What is a number?
One possible approach is to define that x is a number if and only if it belongs to R (the set of all real numbers). There are different ways to define this set, but it can be proven that all those definitions give the same set, so this definition of a number doesn't have any problems.
However, there are objects that don't belong to R but are considered numbers (at least by some people). The most obvious example is the complex numbers. It can be easily shown that the "number" sqrt(-1)=i doesn't belong to R, so if we want to consider it as a number we need to extend our definition to include all the complex numbers - in other words x is a number if and only if it belongs to C (the set of all complex numbers).
It turns out however that even this definition can be extended. In addition to i we have other imaginary "numbers" - the infinitesimals (a non negative number smaller than any positive number) and infinity. Neither of them belong to R or to C.
Surprisingly, even this is not the end. There are also cardinal numbers - a cardinal number is basically a size of a set, so for finite sets this is just a natural number. For infinite sets a cardinal is a "number" (and there are infinitely many different sizes of infinite sets, so there is an infinite os such cardinals) that doesn't belong to any of the sets mentioned above (except for the cardinal of the set of the natural numbers).
It is possible to bring more examples of objects that can be called numbers but, in my opinion the best definition is the first one - x is a number if and only if it belongs to R. But you are free to chose the one you like.
Geometrical representation
It is important to notice that there are two different definition of what a geometrical representation is. The definitions are:
1. A number has a geometrical representation if there is a point on the real line that corresponds to this number.
2. A number has a geometrical representation if a line segment of a corresponding length can be constructed geometrically (using compass and straightedge alone).

Since R can be viewed as the real line, it follows immediately that all the numbers in R has a geometrical representation according to the first definition. It turn out that if you extend this definition of the numbers to include all C, there is also a geometrical representation, because every complex number can represented by a point on a plane. Infinitesimals, infinity and cardinals don't have such a representation.

The second definition is much more strict. The ancient Creeks never asked if there are numbers that cannot be represented in such a way, but it turned out (at about the 16 century I think) that there are such numbers. To better understand this, lets first see some examples. Lets look on the following numbers - 2, 9, sqrt(2) , pi.
It is obvious that we can construct the first two. All we need to do is to decide what we call a line segment of length one, and we are done. We can now draw 2 such segment to get 2 and 9 to get 9. For sqrt(2) it is a bit more complex, we need to make a right triangle with sides 1 and then we will get that the third side is sqrt(2). Generally it has been proven that a number that is a root of a polynomial with rational coefficients can be somehow constructed under the restrictions we put on ourselves. It was also proven that a number that is not a root of any such polynomial cannot be constructed in such a way. Not so long ago (in the last century) it was shown by Cantor that most numbers (numbers according to the first definition) are not constructible. Such numbers are called transcendental.
It is important to note that numbers that belong to C also can be constructed (not all of them, but some of them) this happens because C and R are sets of the same cardinality so you can assign a number in R for any number in C.

Math wallpapers and some other images

noreply@blogger.com (Anatoly) — Thu, 13 Nov 2008 18:19:00 +0000

When I checked Google webmaster tools today to see on what searches my blog showed up lately, I was surprised to see that it some people got here while searching for "math wallpapers" and "photo of Protagoras". I don't remember posting anything like this (I did post a link to a collection of Firefox wallpapers, but it wasn't related to math...), so I decided to post a collection of links to some math wallpapers I found on the net. If you are enthusiastic about math, and don't know how to show it, you can start but putting one of those wallpapers on your desktop. :)

From Deviant art:
Fractal Pi - probably the best wallpaper I ever seen with Pi in it.
Integrating - for all those who love calculus..
Integration fun - somewhat less simplistic than the above.
Pi - A classic of its kind the letter pi on the background of the first "insert large number here" digits of pi.
Infinity - I feel like there is too much objects in this image, but your opinion might be different.
Einstein - Somewhat small, but looks good.
Lambda - Rather simplistic.
Life is math - The title is great, but it is hard to figure the connection to math by looking on the image. However, if you add a caption to it in photoshop it will surely look great...
Three number lost -for those who love weird graphs...

From other sources:
Mathematica -photos of some famous historical figures, on a background with different number systems.

Well, this is all I found. I would really like to add to this short list, but unfortunately I don't have time to hunt for photos now... :(. In the case the links above don't work, you can download the wallpapers from my picasa album - Math wallpapers.
Last but not least - a photo of the Statue of Pythagoras in Pythagorion:

Dividing gold in a rational way

noreply@blogger.com (Anatoly) — Tue, 11 Nov 2008 06:13:00 +0000

I finally got internet connection in my new apartment, I am really lucky it didn't take more time. Also, as I expected the semester in the university started on time despite the threats to not open it unless the government pays. I am doing 8 courses this semester, so I am really busy most of the week. Yesterday I was in the university from 8:00 to 20:00... Gladly today the lectures start at 16:00 so I can stay at home in the morning.

Now, about the title. One of the courses I am doing this semester is game theory. When I got the exercise for this course, one of the questions was about dividing gold between pirates (surprisingly, the other questions were rather difficult questions in analysis). What is interesting in this question is that the solution seems unbelievable, so I decided to post both the question and the solution on this blog:

The situation is as follows - five pirates got 50 coins of gold. They must find a way to divide those. The pirates all have different rank from 1 to five. They decide on how to divide the gold in a very simple manner: the pirate with the highest rank offers a way to divide the gold, and then the rest vote for or against his proposal. If the absolute majority is against he is killed and the process starts over with 4 pirates. There are three further assumptions. Firstly, the pirate who offers how to divide the gold wants to get as much gold as possible. Secondly, all the pirates are rational. Thirdly, if a pirate have no reason to vote for or against the proposal, he will vote against it.
A tip: go from the end to the beginning.

The solution is that the first pirate (who has the highest rank) will get 48 coins and the pirates 3 and 5 will each get 1 coin. I don't want to post the full solution, if is obtained by repeating a few simple steps on the situation, so I will just write the beginning of the solution:
Suppose there is only one pirate, he then takes all the gold. If there are two pirates (number 4 and 5), no matter what way to divide the gold the forth will offer the fifth will be always against, because then the forth will die and he will get all the gold. If there are three (3, 4 and 5) then the fifth will be always against and the forth doesn't have a reason to vote for or against (we assume that his life is not important to him), so he will vote against unless he gets something - at least one coin. In this situation, if the third will give him one coin, the forth will vote for him, so the third can keep 49 coins.