## Friday, December 26, 2008

### Limits of applicability

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.

## Saturday, November 29, 2008

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

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).

## Thursday, November 20, 2008

### Geometrical representation of numbers

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.

## Thursday, November 13, 2008

### Math wallpapers and some other images

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:

## Tuesday, November 11, 2008

### Dividing gold in a rational way

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.

## Tuesday, October 28, 2008

### Politics

The only good thing in politics, in my opinion, is that most of the time it doesn't influence you directly. Unfortunately, this is not the case for me now. The rest of the post is my complains, so feel free not to read it.
First of all, there are elections to the mayor office at Jerusalem at the middle of next month. As of now, I don't even intend to vote because I don't have the time nor the desire to check the information available on the candidates. Also, I will probably be too busy studying anyways. What annoys my most is that for some reason I got a call asking me to help with the election process - to take attendance notes. I have no idea why they called me with this..

Secondly, next week I suppose to start a new semester. However the university already sent letters telling all the students that the semester will not open because the government doesn't give them the money they need. They even included a long list of emails in the letter to where we can send complains. I really doubt that anything will happen to the semester, but who knows...

Thirdly, due to government policies and economical crisis the apartments in Jerusalem are rather expensive now, and my family and I are currently looking for a new apartment. Horrible timing... I certainly hope that we will find something soon.

On a somewhat brighter note, here are link to some political jokes I found:

Two political candidates were having a hot debate. Finally one of
them jumped up and yelled at the other: "What about the powerful
interests that control you?"

The other guy screamed back, "You leave my wife out of this."

Also check the Circle vs. Square webcomic, its author did an excellent job making fun of politics without insulting anybody.

## Tuesday, October 21, 2008

### Free Books

I found today a fairly large collection of books in pdf format. The books cover a lot of subject, but there are also a few books about programing and math. If you are interested follow the link.

## Friday, October 17, 2008

### Series Part 4

Unlike the previous posts on this subject, in this post instead of talking about the subject in general I will just fully solve one rather nontrivial problem. The problem is to find the formula for the n-th term of the Fibonacci progression.

The Fibonacci progression is a progression defined by the following relation:

a(1)=a(2)=1, a(n)=a(n-1)+a(n-2)

It is of course possible to find any term in the progression by simple calculations, but buy using this relation we are required to calculate all the terms which come before the one we want to know. It turn out that for high values of n this task is not practical even for a computer.

To find the formula we will need to use linear algebra. Firstly, lets define a group of progressions which we will call generalized Fibonacci progressions. Those are all the progressions that follow the recursive rule, but the first terms are allowed to be any numbers. Now, lets look on the vector space of all infinite progressions. Obviously, the generalized Fibonacci progressions belong to this space. Moreover they form a subspace of dimension two in the vector space.
The last sentence requires a proof - all we need to show is that the group of the generalized Fibonacci progressions is closed under multiplication and addition. We will take two such series, a(n) and b(n) and a scalar r:

a(n)=a(n-1)+a(n-2)
g(n)=ra(n)=ra(n-1)+ra(n-2)=g(n-1)+g(n-2)

Thus, g(n) is also a generalized Fibonacci progression, and we proved that the group is closed to multiplication.

c(n)=a(n)+b(n)=[a(n-1)+a(n-2)]+[b(n-1)+b(n-2)]
c(n)=a(n-1)+b(n-1)+a(n-2)+b(n-2)=c(n-1)+c(n-2)

The only thing we need to prove is that the dimension is two. To do this it is enough to show that there is a basis for this subspace that has two vectors in it. This is simple, the basis is the two progressions a(n)=1,0,... , b(n)=0,1,....
To prove that this is a basis we need to show that the vectors are independent (left as an exercise) and to show that any other vector is a linear combination of these two. Let c(n) be a vector in this space. All the generalized Fibonacci progressions are uniquely defined be their two first terms, so lets suppose that the two first terms are c(1)=d and c(2)=r. Then, f(n)=da(n)+rb(n)=d,r,....
Since we are in a vector space the resutl is another progression, and because of the unique definiton by the first two terms, we get that c(n)=f(n).

Now, is there a genarilized Fibonacci progression which is also a geometrical progression? We get the following conditions:

a(n)=aq^(n-1), a(n)=a(n-1)+a(n-2)
aq^(n-1)=aq^(n-2)+aq^(n-3)
q^(n-1)=q^(n-2)+q^(n-3)
q^2=q+1
q^2-q-1=0

There is obviously a solution to this equation. Moreover, there are obviuosly two different solutions, and therefore two different progressions. Lets suppose that q(1) and q(2) are the solutions. We can also select a=1 because it doesn't influence the solution. We get then two progressions: a(n) and b(n). The only important thing left to do is to show that they are a basis to our vector space. Because I already prooved that the dimension is two, we just need to show that they are lineary independant. It ois enough to show that the solution of the following equations is a=b=0:

a+b=0, aq(1)+bq(2)=0

Since q(1) is not equal to q(2) this is indeed so and therefore we have a basis.

Now we know that any progression in our space can be written as c(n)=da(n)+fb(n) for some d,f. We also know that a(n)=q(1)^(n-1), b(n)=q(2)^(n-1). Deriving the final formula from this is easy, but since I still cannot use latex in blogger, completing the last step is left to the reader.

## Wednesday, October 15, 2008

### Series Part 3

As promised, in this post I will shortly discuss infinite series. I don't want to say a lot about this because this would require developing precise definition, so I will just describe the main ideas and give some classical examples.

The main question we can ask about an infinite series is if it has a limit, or in a case of an infinite sum, the result of the summation.
Firstly, lets define what a limit of a series is. Simply put, a limit of a series is a single number to which the series is close from some point. For example, the limit of the series:

1,1,1,1,1,.....

Is 1 (we assume that the series continues to infinity following the same pattern). The same is true is we change the first number - the limit will remain the same. In a counter example, the series: 0,1,0,1,01.... Doesn't have a limit at all, because it is wrong to say that there is single number to which the series is getting close to.
It is important to notice that the definition I gave is not exact. Moreover, if you will write such a definition on a exam you will get null point for it. It doesn't mean that it is wrong - but it is not useful. It is simple to fix, all that is needed is to define what does it mean "close to" and "from some point". Unfortuantely, the script I used to write latex in posts no longer works, so I cannot write a definition (it requires writing symbols). Therefore completing the definition is left as an exercise for the reader :).

Since we have the "definition" lets try to use it. Lets look on the series:

a(n)=2^(-n)

It is very easy to see to what number this series is getting close to. All you need to do is to imagine that n is very big (or use a calculator). Either way you will see immediately that the result is very close to zero. However, according to the definition we need to show that there is only one number that the series is getting close to. To do this lets suppose that r is another number that the series is close to. Since the series is positive, if r is negative than the series is always closer to zero than to r, so r is not the limit. Otherwise, r is positive. So lets find n such that 2^(-n)<0.25r. Because the series is monotonically decreasing, we get again that from some point the series is closer to zero than to r.

Lets look on the series: a(n)=4+n. According to our definition it doesn't have a limit because the limit must be a number and it is clear that this series grows "to infinity". (It is of course possible to define the limit differently). From this two examples we get a simple result - all arithmetic series don't have a limit and all the geometric series for which |q|<1 have a limit. Also, for all such geometric series the limit is non other that zero, because we get: a(n)=aq^n. If q is less that one, this number approaches zero.

Lets now look on infinite sums. We will say that a sum converges if the sum is a real number. (Again, the definition is not precise.) This brings a question - how is it possible at all? After all it doesn't matter how small the numbers we sum, if we sum an infinite number of them the result "should" also be infinite. At least that is what our logic tries to say. But, this is wrong. Lets see an example:
1, 05, 0.25, ....

This is a simple geometric series, in which q=0.5 and a=1. From the previous post the sum is:

S(n)=(q^n-1)/(q-1)

We can think about this as a new series - a series of partial sums of the original infinite series. Does it has a limit? Of course - it is obvious that the limit is simply 1/(1-q). Now, if we will define the infinite sum as the limit of the series of the partial sums, we will get the immediate result - the sum of the infinite series. In the case of p=0.5, the sum is 2.

In the next (and final) post I want to discuss an example of a series that is not an arithmetical or a geometrical progression.

## Tuesday, October 14, 2008

### What is persistence?

The following story is from Clientopia. I do not know if it is real or not, but after reading it you will surely understand what does it means to completely concentrate on your goal leaving behind any doubts...
---------------------------------------------------------------

I have had my company mobile for nearly 2 years, out of the 50 odd calls I've had, only 3 were legitimate, the rest were wrong numbers. This is a transcript of the one man who has called me more than 35 times.

Me: Hello, XXX IT, XXXX speaking, how can I help?

Random: Is that Brian?

Me: No, this is XXXX

Random: Are you sure?

Me: I should think I am. I've never been called Brian, and there isn't a Brian involved with this mobile

Random: Oh, can you put me through to him?

Me: No, as there isn't a Brian here, or in XXX IT

Random: Now I know you're lying, this is the number he gave me, now put him on the phone!

Me: I'm sorry, you've got the wrong number

Random: NO I HAVE NOT! BRIAN GAVE ME THIS NUMBER

Me: Then Brian got the wrong number

Random: I THINK HE KNOWS HIS OWN MOBILE NUMBER

Me: On the current evidence, I would say he doesn't, I cannot help, goodbye. *click*

----

He rings back, I don't answer.

----

12 (!) calls later he gives up. Then calls back the next day:

Me: I'm sorry you've got the wrong number, you rang yesterday several times.

Random: THIS NUMBER WAS GIVEN TO ME SO PUT ME THROUGH

Me: *click*

He has since called back several more times.

## Saturday, October 11, 2008

### Ubuntu Intrepid

I am installing the latest version of Ubuntu now. It is still a beta version, the stable release is due on the 30 of this month. Hopefully I will not break my system.... If everything will go fine, I will update this post with a short review of this beta release.

Update: Well, the installation itself went just fine. However I am unable to connect to the internet. To be precise, I a,m able to connect to my ISP and to ping sites, but I am unable to view any web pages. I am also unable to download files using wget or to install new packages. However amule works fine. From what I found on the net this is a rather common problem but there is still no solution.
Right now I am using a liveCD to connect to the internet. I will probably reinstall Ubuntu Hardy tomorrow (unless I will find a solution).

As promised - a short review of the new release. I noticed three main changes. The first one is tabbed browsing in Nautilus. It works well, but for some reason if you start a second session it is not opened automatically as a tab in the first one. It is possible that this behavior can be configured somewhere, but in my opinion it is a required addition.
The second change is that the shutdown button is now changed into a log-out button. I cannot say that a like or dislike this change, but it seems a good idea.
The third one is a change is the network manager. For me it was a disaster - I am no longer able to connect to the internet because of this. It also looks much more complicated than the previous one.
There is also new artwork is this release. In this area there is a clear improvement.

Update2: I did a clean install of Ubuntu Hardy today and everything is working fine again. This is the first time I was forced to reinstall Ubuntu...

## Friday, October 10, 2008

### Series Part 2

Continuing from the previous post on this subject, lets find the formulas for the sum of the two series I mentioned in the previous post, the arithmetic and the geometric progressions:

Arithmetic progression:
Firstly, lets look on a very simple case - the progression 1+2+3+...+n=S. If we just look on it, it is not very clear what the sum is. However, if we will write it in a slightly different manner the answer will be obvious:

1,2,3,4,5,6,7,8,9,...n
n...,9,8,7,6,5,4,3,2,1

Obviously the sum of all the numbers is 2S. We also know that there are n columns and the sum of any column is n+1. Therefore 2S=n(n+1). From this we get the formula by division.

Now lets look on the general case. The general arithmetic series is:

a+(a+d)+(a+2d)+...+(a+(n-1)d)=S
na+d+2d+3d+...+(n-1)d=S
na+d(1+2+...+(n-1))=S
na+0.5d(n-1)n=S
0.5[2a+(n-1)d]n=S

By the way, this formula requires to know the first number in the progression. However, if you know the last one instead of the first you can also use this formula - after changing d to (-d).

Geometric progression:
Firstly, lets again consider a specific case. Lets look on the sum of the following progression:

1+q+q^2+q^3+..+q^(n-1)=S
q+q^2+q^3+..+q^(n-1)+q^n=Sq
q^n-1=Sq-S
S=(q^n-1)/(q-1)

Getting from this to the general case is extremely easy. The only thing we need to do is to remember that a general geometric progression is just a multiplication of the case we took care of by a constant number a. Therefore, the formula for the sum is:

a+aq+aq^2+....+aq^(n-1)=S
S=a(q^n-1)/(q-1)

In the next post I will talk about infinite series and their sums.

## Tuesday, October 7, 2008

### University vs the Government

I received an email from the university today, as expected it was an official statement that unless the government will provide the necessary funds the upcoming semester will be canceled. The previous year the situation was the same, and in the last moment the government decided to give the necessary funds. I wonder what will happen this year...
The letter also contained a call for students to join the fight with the government. While I am not going to participate in such activities, I am sure there will be enough people who will.
I just hope that the semester will open in time and in an orderly fashion..

## Thursday, October 2, 2008

### Series - Part 1

In math a series is a function from the natural number to some set B. The set B can be the real numbers but it can also be a vector space or some other set.
The most basic question we can ask about a series is finding the generating formula. Such formula can be extremely difficult to find, and since a series of random number is by all means a series such a formula may not even exist. Lets start with a few simple series:

Arithmetic progression:
This is probably the simplest example. An arithmetic progression is obtained by adding a fixed number to the previous number in the progression. For example, if we choose to start with the number 5 and to add seven, we will get:

5, 13, 20, 27, 34.....

In the general case we get:

a1=a, a2=a+d, a3=a+2d, a4=a+3d,...

It is very simple to find the generating formula for this progression. To do this we need to look on the difference:

a2-a1=d
a3-a2=d
.
.
a(n)-a(n-1)=d

If we add all of the equations we will get the generating formula:

a(n)-a1=(n-1)d

This method of looking on the difference can be used for many series. For it to work, we need to know how to sum the right side of the equation. In the example it was very simple to do, but usually it is much more difficult. Also, since the differences are in fact a series of its own, we can try to find the formula for the difference, and from it get the formula to the original series.

Geometric progression:
A geometric progression is a progression in which the ratio of the successive terms is fixed, that is a(n+1)/a(n)=const for all n. For example:

1,2,4,8,16....

And the general progression:

a,aq,aq^2,aq^3,aq^4...

In this case, finding the generating formula is trivial. Just from looking on the progression we get that:

a(n)=aq^(n-1)

However, if we would try to find this formula by looking on the difference we would just get the original progression back. Therefore to get the formula this way we would need to know the sum of the geometrical progression.
In the next post I will develop the formulas for the sum of both arithmetical and geometrical progressions.

## Wednesday, October 1, 2008

### I am back

I didn't post anything for a lot of time, but I am still here. Sorry for suddenly stopping posting, I was busy with exams so I had to stop posting for some time. I already got all the exam results, I did good in all except two (but this was expected from the start, so this is not much of a problem).
Originally I planned to stop posting only for a few days, maximum a week, but for some reason not posting anything was so relaxing that I just couldn't force myself to write anything. I even didn't read my rss :).

The next semester will start only next month, so I still have a lot of time to prepare for it. I already have some books for next year, and I intend to buy more. Hopefully I will be able to prepare well - learning seven courses in one semester is not a nice thing...

## Thursday, September 4, 2008

### Sudoku solver and some other problems

I got a comment today on my post Algorithm for solving Sudoku puzzles with a link to a program for solving sudoku online. So if you have problems with a particular hard puzzle, give it a try. The program is called Sudoku solver.

I also stumbled on an excellent collection of simple physic problems. You can find the collection here. Unfortunately there are no solution to the problems there, but they seems all to be of the type that requires understanding of physics and not knowledge of math and formulas. Since I am trying to write a math blog, it would be pointless for me to write solutions to these problems. However, I want to solve the first problem:

You are given two identical steel balls of radius 5 cm. One ball is resting on a table, the other ball is hanging from a thin string. Both balls are heated (e.g., with a blow torch) until their radii have increased to the same value of 5.01 cm. Which ball absorbed more heat and why?

The answer - the ball that lays on the table will absorb more heat. The reason to this is that the heat can escape better from the ball when the ball touches another object, in this case the table. When the ball is suspended in the air, the heat also escapes but the rate of the escape is lower. This is due to the fact that air is much less dense that a table.
Since more heat escapes, more hear will be required to heat the ball on the table.

## Tuesday, September 2, 2008

### Water balls

Apparently there is a chemical way to cause water to turn into solid balls at room temperature. For instruction see the video below:

I am not sure if this one is not fake, however. The thing that makes me suspicious is that water turns into balls when it is out of the tank. In my opinion, there are simply glass balls of the same color inside of the water. While in the water they are invisible.

## Sunday, August 31, 2008

### How old!?

I didn't post anything humorous for a lot of time, so here goes. As far as I know this one is a real story.
---------------------------

Customer: “I’d like two tickets for [movie], please.”

Coworker: “That movie is rated R. Can I see your ID?”

Customer: *shows an ID that states she is 18*

Coworker: “You need to be 21 in order to purchase an R-rated ticket for someone else.”

Customer: “But it’s for my son!”

Coworker: “How old is your son?”

Customer: “16…”

Coworker: “So you’re 18… and you have a 16 year old son?”

Customer: “That’s right!”

Coworker: “Let me get my manager…”

Manager: “Ma’am, you need to be 21 to purchase a ticket for a minor.”

Customer: “But he’s my son!”

Manager: “You’re telling me you gave birth when you were two years old?”

Customer: “YES! It happens, I promise you!”

--------------------
I found this story here.

## Wednesday, August 27, 2008

As you probably noticed, I didn't post for a few days. The reason for this is that I have to prepare to exams. Next week, the posting will continue as usual. Meanwhile, you are welcomed to use the skribit suggestion widget to suggest topics for me to write about.

## Wednesday, August 20, 2008

### Running in the rain

I was sent a letter yesterday which asked me to look and comment on the last claim on this page. Since that page might not remain forever, I am copying the paragraph in question:

You get less wet by running in the rain. Actual mathematical equations devoted to this popular question have suggested it is true, though not for the simple reasons you might think. Complexities include factoring in the number of rain drops hitting the walker’s head versus smacking the runner’s chest.

Well, lets do the calculations. While this sound like a complex problem it is actually a simple case of Galilean relativity, so it is easy to calculate.
The first step in dealing with a problem like this is to formulate it correctly. As I understand this problem, we need to prove or disprove the statement that if you stand still under the rain for some time T, you will be more wet than if you were running for the same time T.
The second step is to simplify the problem. In this case the simplification is to assume that the rain is uniform. That is, the amount of rain drops per square cm is the same in all the are we are dealing with, and there are no sudden changes in the wind.
Now, what we want to calculate is how wet will a person become. Clearly this depends on many factors - the size of the drops, the body area of the person, the speed of the rain drops etc. Because of the uniformity assumption we made, we can say that all the factors except for the speed of the rain are fixed - they don't change with time, and they don't change if the person is moving. We can now define the "wetness" (B) as a simple multiplication between the matrix A and a vector v. "A" will be the constant which we get from the problem condition. Since we are in 3d space A is a diagonal matrix with A1,A2,A3 (the corresponding constant for each one of the three directions) on the diagonal. "v" is the speed of the rain ( it is a vector v=(v1,v2,v3)). The multiplication result, B, is also a vector. The size of B multiplied by the time T is then the answer.

So if the person stands still under the rain we will get:

B=Av

T|B|=T$\sqrt{(A_{1}v_{1})^2+(A_{2}v_{2})^2+(A_{3}v_{3})^2}$

Now lets look what happens if a person starts to run. Since we can choose the coordinate system in any way we want we can assume the the person runs on the x axis with some speed w (for simplicity, we will assume that the speed is constant). Now it is time to use the Galilean transformation. From the coordinate system of the person the speed of the rain is no longer v=(v1,v2,v3). Instead it now becomes v'=(v1-w,v2,v3). Since nothing else changed, we will get:

B'=Av'

T|B'|=T$\sqrt{(A_{1}v_{1}-A_{1}w)^2+(A_{2}v_{2})^2+(A_{3}v_{3})^2$

It is perfectly possible that this is less than the previous result, but it can also be large. It all depends on the direction of w. If for example w=v1, you will get less wet by running. But if w=-v1 you will get more wet if you will run.

This solution was done under the assumption that the rain is uniform. However, while this assumption is not realistic it is very close to reality and on a short time interval it should be extremely close to reality. It is possible to solve the problem without this assumption, but it will only introduce extra steps without changing the final result.

## Monday, August 18, 2008

### Fire and ice

I just stumbled on this image: ( click on the image to see the full size)

The reason I am posting it is that the image is in fact a fractal from the Mandelbrot set. It was edited pretty heavily, but the main part of the image is the classic Mandelbrot fractal.

By the way, if you are using Linux, there is an excellent program for generating fractals you can install. It also allows to add some effects to the resulting images generating impressive artwork. In Ubuntu you can install it by typing:
`sudo aptitude install xaos`

## Sunday, August 17, 2008

### The Department Quilt

Today I decided to do something I do very rarely - to look on the recommended feeds in Google Reader. I usually don't look on them, because I have almost 100 subscriptions already, and I don't want to add any extra feeds. However, today I decided to look on those feeds. In one of the feeds I suddenly saw a very familiar image:

These two images are the front and back sides of a quilt that can be found on the second floor of the mathematics department of Hebrew University Jerusalem. At first I thought that the blog author whose rss feed I was reading visited HUJI, and took those photos, but then I noticed that the post is called Our Department’s Quilt.

This is how because of a Quilt a found the blog of Gil Kalai - Combinatorics and more. He is a professor of mathematics in Huji. His main areas of interest are Combinatorics and convexity, so if you are interested in anyone of them, it is probably a good idea to visit his blog.

## Saturday, August 16, 2008

### Incredible military skills

Our present society is technology centered. It wasn't always like this, but it is so for a long time. However, better technology is not always the answer. As you can see in the movie below, it doesn't always help to point a gun to somebody head...

## Friday, August 15, 2008

### Wireless electricity

I was sent an interesting article yesterday about Atlantis. Most of this article is, as I call it, "conspiracy theories" - the most hilarious part is where the author claims that a mountain in Canada was shaped to resemble a head of an Indian. There is indeed such mountain, it was discovered by the users of Google earth. However, I really fail to see any reason to believe that this was done by humans and not by wind. I am so skeptical because if we assume that it was indeed build be someone, we get an even harder question: Why was it built? Constructions of this size are not built without a purpose, and I don't see any purpose to this one. It is not even art, since it can be viewed only from the air. The article in question also doesn't answer this.

Some of the claims in the article are clearly false (at least according to what I know), while some are true. In my opinion the only purpose of this article is to promote the theory it author has, and not to talk about facts. However it still makes an interesting read. Now to the main part of this post. One of the claims made in this article is that Atlantis used wireless electricity. This electricity was distributed from 13 towers (pillars) which were built in each of the 13 colonies. The author also refereed to Tesla, claiming that he started working on similar construction so this is perfectly possible. Moreover, Tesla was claiming that his device will provide everybody with free electricity from the earth ionosphere. But is this true? And if true, why don't we use this, relaying on wires instead?

Wireless power
Is it possible to power a device without wires? Of course it is. While working on his project Tesla build a room in which lamps (florescent) started to work even when not connected to the power outlets. This is based on the Faraday law - if we have a changing magnetic field we will get voltage. Tesla created a strong, constantly changing electromagnetic field in his room which powered his florescent lamp. There is no fiction in this, this principal is well known and is used on a daily basis - this is exactly how radio works. In a radio station, we have a transmitter that sends a changing electromagnetic wave in space, and this wave creates voltage on the radio antenna. in reality, this is slightly more complicated, but I don't want to get into the detailt of how a radio works.
Another way to transfer power without wires is to use displacement currents. This method was also known to Tesla, but it requires one wire so it is not exactly wireless.

Is it dangerous
From the above paragraph it is clear that if we want wireless power what we need is to create a strong electromagnetic field. Will such field have negative influence on people/other objects?
The answer to this is no and yes. People are not harmed by magnetic fields. The reason to this is that such fields influence only charges in motion or paramagnetic materials. Our bodies don't have any charge, and organic tissue is not paramagnetic. But, a computer in such a field would be fried immediately. The reason to this is that such field would not only supply power, it would effect all the circuits in the computer, and this would simply destroy them, probably with a little fire. It is possible to create a defense against such fields, there are materials which can block an electromagnetic field. But such a defense will have to be large and heavy.

Can it be done
So, we know that it is possible to transmit power without wires and we know how to protect computers and other gentle electronics from it. But can we use it? No. The reason for this is very simple. It is not a problem to create a strong magnetic field, but such fields become much less stronger with the distance. To be precise, the electromagnetic field drops as the square of the distance from the source. To bring some perspective, lets suppose that we have a field of 1000 Tesla. After what distance it will be only as strong as the field of the magnet you put on your refrigerator? The equation we get is:

$\frac{1000}{x^2}=0.1$

And the solution is 100. This means that after 100 meters from the power source the field will be very small. But this is only a 1000 Tesla, right? So if we will take more it will be fine. Unfortunately, no. Tesla is a very large unit. It is simply impossible with current technologies to create magnetic fields so strong. The strongest steady magnet created until now is under 45 Tesla. Also, even if we will be capable of creating electromagnetic fields strong enough, such fields will also cause little pieces of iron to fly to the magnet like a bullet. Besides, think about all the power loses.
Because of the above mentioned problems, it is practically impossible to operate a city on such wireless power source. However, the method I discussed is only a simplified version of what Tesla wanted to build.

Was Tesla wrong?
Surprisingly, no. I am sure that all the points I mentioned above were known to him. However, he found a way to solve this problem. I am not going to describe his idea in detail, but basically the plan was to use the resonance frequency of the earth and the ionosphere as a conduction layer. Tesla even claimed that he was able to transmit power across 25 miles with only 5% power loss. So, theoretically it is indeed possible that a civilization that is described in the article existed and used such power source.
Now, after saying this it looks like an ideal moment to say what is the main problem I see with this article. This article is focused on the idea "how great was Atlantis". It even goes so far as to say that it was utopia to which our race will eventually return, by technological development. But even from the article itself it is clearly not true. And the reason for this is Faraday law. If you indeed use wireless electricity you don't "magically" transport it from one point to another. It still has to pass all the mid points. If there will be metal on its path, this metal according to Faraday law will act like a small generator - it will get small voltages an small currents ("small" is the best scenario). If you want to power a city in this way you need firstly to remove the metal from it. Which means to remove all technology, because as I explained above the electromagnetic field will not only power a circuit, but it will effect all of the parts of the circuit. For complex circuits this means that they will be simply fried (and this cannot be solved by technology advances). This sounds a bit paradoxical - if we cannot have technology in the city, then what good is wireless power? The answer is that such power would still allow two basic devices to work - lamps and motors. I don't know what you think about such life, but I clearly don't like it.
But if you will read the article, this is exactly how the cities are described. Cities build from stone - because they couldn't put metal inside the walls. Cities without writing, civilization without knowledge.

Bottom line
I do not believe that Atlantis existed. There are evidence that support the theory that at some point there was an advanced civilization on the earth. According to these evidence, there are reasons to believe that even space flight was within their reach. But this is not enough to conclude anything about what really happen. For example, from the information we now have it is perfectly possible to get to the conclusion that at some point the earth was visited by aliens, who "helped" humanity, and then after a war (either among themselves or with humans) left the earth. But all such theories go against the basic principal of science - the Occam razor.
Also, Atlantis has zero importance for us. If at some point in the future we will indeed meet aliens that will admit that they visited the earth and influenced our civilization, it would be an interesting historical fact but no more. This is why there is just no point or reason to search for it, whatever the search will bring no longer matters because we have a civilization of our own. There is no reason to try and think about new "conspiracy theories" about the past.

### The tallest buildings in the world

This video shows the current tallest buildings, and also the buildings which are planned to be build.

I doubt that some of them will be ever built, but as a concept they are surely impressive. Especially the pyramid shaped ones...

## Wednesday, August 13, 2008

Some time ago I wrote a post about a Firefox extension called stylish. This extension is basically a script manager that allows you to run scripts that modify web pages. In some aspects it is superior to greasemonkey, but this is not what I want to talk about in this post. One of the scripts I was running using this extension is Gmail Redesigned. This script makes Gmail look significantly better, at least in my opinion. However, this script is no longer available. Instead of it we now have a standalone extension which can be downloaded and installed here. It is now called Google Redesigned, but it includes scripts for Gmail and Gcal only. Before installing it make sure to disable any other scripts that have similar functionality.

### What a good exam

Today I did my first exam of this semester, and I am very pleased of how it went. There is a tradition on this course to give on exam questions which are not very solvable. Those who write the exams must of course make sure that the material that will appear in the exam is what was learned during the semester, but they always try to give tricky questions. Instead of making questions that just test your knowledge of the material, they give questions that require you to think about some trick.
However, this semester one of the lecturers on this course is new and he decided to break this tradition. I know about this because he told the class himself. Near the end of the semester he came to class and said that he will make sure that the questions in the final exam will be solvable, unlike what was in the previous years. He told that he looked on the exams from the previous years and concluded that even a person who knew all the course material might not be able to solve the questions. He later told us that the other lecturer and other people who have influence on the questions that will go into the exam, say that this is the course tradition to give such tricky questions.

The exam was today, and as it looks he won the fight (and my guess that it was indeed a fight). Some of the questions were more difficult than others, but overall the exam was very straightforward and even simple.

## Saturday, August 9, 2008

### Amazingly funny

I just read a post on Astroengine about a recent development concerning LHC. I rare write about post written on other blogs, but this time I felt is was just unfair not to share this. The main idea is that some person named Tia Aumiller decieded to open a group called: "People for the Ethical Treatment of Hadrons" (PETH).

It is not 1 July today, but I really hope that this is not a prank... Anyway, this organization has already protested in front of CERN. Their claims are:

“You’ve got these subatomic particles accelerated at great speeds for the sole purpose of being destroyed. No one thinks of the ethical implications of this. There’s a limited supply of hadrons in the universe. Do we just want to go around destroying them? What if we run out? What if the hadrons can feel pain? Will we look back at this hundreds of years from now and regret it? Kinda like we do with the killing of bacteria with antibiotics now.”

It is just unbelievable and extremely hilarious. I really have no idea how crazy somebody has to be to really believe in this.

Update: After checking this a bit, this story turned out to be fake. However it is still very funny so I am not removing this post from my blog. I guess I should learn a lesson from this - no posting of things that look fake, even if they are funny...

## Friday, August 8, 2008

### Why you shouldn't blog

I have been blogging on Math pages for almost a year now, and I also have a blog on StumbleUpon which is older, larger and has more subscribes. While I don't claim to be expert blogger, for this visit problogger, I do have some experience I think might be useful to those who are thinking about starting there own blog. Some of the points in this post are specifically for math blogs, but most of them are true for any blog.

Time:
It doesn't matter how many post you write per month and how long are they, it still takes time to blog. Do you really have this time? Also, the more popular your blog becomes the more comments and emails you will get. Do you have time for answering?

Goal:
The first thing to think about before opening a blog is what are your goals. A lot of blogs ate opened with dollar signs in the eyes. People know that it is possible to get an income blogging and they try to become professional bloggers. Such blogs usually close after three months. The reasons for this are different, but for a math blog to be closed in such a way there is only one reason - there is no way enough people will visit it to make it profitable. Math is just not poplar enough.
For me the blog is a way to relax a bit and it also helps me to organize my thoughts. I wouldn't mind to earn something from it, but I don't believe that it will happen.

Content:
Every blog should have a topic. The topic doesn't have to be very specific, but it must be there. your readers must know what kind of content you usually post. For a math blog there are seem to be three options:
1. Math - you are writing posts about mathematics, you proof theorems, explain formulas etc. Such posts are full of math and a very hard to write, because there is no easy way to write math formulas in posts. Also, only someone who studies math will read such a blog.
2. Ideas - instead of writing formulas you can write explanation of different mathematical ideas, or write posts about math history. Such posts require very little math and a written to be readable by someone who doesn't know much about math. Depending on how good you write, you can get a lot of readers. The problem of this approach is that if will likely end with you answering endless email and comments from people who don't know much about the subject but are eager to contribute some "groundbreaking" thoughts.
3. Personal - Instead of writing about topics you like, you can use your blog as a diary. This is a very popular way to blog, but it requires the ability to write interestingly. Be warned however that this will also put your life before other people and make it possible to gather a lot of information about you.

People you rather not meet:
This sounds not very nice, but unfortunately this is how things work on the net. From time to time you will meet someone whom you really would prefer not to meet. It seems to me that such people are only around one percent of all the people I know online, but it happens. Such people usually fall two categories:
1. Spammers/SEO - a lot of people try to make money online, but some of them choose ways which the rest of us don't like. I am contacted from time to time by people who ask me to review/link there site/blog, and I am getting spam comments ocasionally. The spam is easy to deal with, and for the reveiws Ia hve a simple policy - if I like the site/blog I am asked to review/link I write a short post about this. Otherwise, I don't do anything.
2. Over productive people - some people have too much free time and think that so do you. So they will send you so much emails/comments than you will feel buried by it. This is rare but happens. I am not speaking about those who email you daily - I am speaking about those who see nothing wrong in 10+ email each day.
3. You offline friends and other people you know offline - I am not saying that you don't want to meet them, but depending on the theme of your blog and your personal charasteristics you might prefer nobody you know in your offline life to know that you have a blog. Especially if your blog is a personal one. I am sure that you don't want your boss to read your personal blog....

Lack of feedback:
This one probably sounds strange, but unless you have a lot of visitors nobody will comment on what your write. If this continues for long enough you will start thinking that blogging is meaningless and purposeless, because you will see no result of your work. While I cannot say that this is a serious emotional challenge, it certainly exists. It can be felt especially well when you get an email or a comment thanking you for a post you wrote, you discover that something was missing.
For me, the most annoying aspect of this is that my posts about Linux get more comments and page views than any of my math posts.

The bottom line:
Despite writing a post of why you shouldn't blog, I must admit that I enjoy blogging a lot. It is fun, and it allows me to meet new interesting people. I am sure that the more time I will continue blogging the more new people I will meet. Also, I know that some of my posts were helpful for others. So if you are thinking about starting your blog, consider the points above and give it a try. It also might be a good idea to blog on a private blog for a month or two in the beginning to get some practice, but this is your choice.

## Thursday, August 7, 2008

### The Riemann condition

Continuing a topic I started a few posts ago, Definite Integrals, I want to discuss some properties of the definition of the integral according to Darbo, and to prove a theorem called Riemann condition.
This post builds heavily on the previous post on this subject, so it will be a good idea to read it before starting to read this post.

In this post f(x) is a bounded function on the interval [a,b] to the real numbers, and P is a division of the interval [a,b].

Firstly, lets show that the infimum of the Upper Darbo Sums (U(P,f)) is alway large or equal to the supremum of the Lower Darbo Sums (L(P,f)). To proof this it is enough to show that for any two divisions P1 and P2 the upper sum according to P2 is greater or equal to the lower sum according to P1. To do this we will need a simple lemma: "If all the points in P are also in P' than

$U(P,f)>U(P',f), L(P,f))<(P',f)$"

The inequalities are all weak. I will proof one of them and the second one can be proofed exactly the same. Lets suppose that there is only one extra point in P'. We can suppose that this new point is between the first two points in P. Than:

$L(P',f)-L(P,f)=\sum m'_{i}(x'_{i}-x'_{i-1})-\sum m_{i}(x_{i}-x_{i-1})=$
=$m'_{1}(x_{t}-x_{0})+m'_{2}(x_{1}-x_{t})-m_{1}(x_{1}-x_{0})$

Now, since m is the infimum of f on the segment x(i)-x(i-1) we get immediately that the result is greater than zero (or equal), as needed. Now by induction we can show that this is true for any number of extra points in P'. And with this the lemma is proven.

Now we can use this lemma. Lets look on P3 - the union of P1 and P2. According to the lemma:

$L(P1,f)

And this is exactly what I wanted to show.

Riemann condition
The next step is to prove the Riemann condition theorem. This theorem says that: "The function f(x) has an integral only and only if:
$\forall \epsilon>o \exist P$ $U(P,f)-L(P,f)<\epsilon$

In the first direction, because of the theorem I proofed above we get immediately that if the above condition is true than:

$infU(P,f)-supL(P,f)<\epsilon$

And since it is true for any epsilon, they must be equal. But if they are equal we get that f has an integral according to the definition. In the second direction, if we will suppose that the function has an integral than we will get that according tot he definition of infimum and supremum, for any epsilon large than zero, there is a P1 and P2 such that:

$L(P1,f)>supL(P,f)-\frac{\epsilon}{2}$ $U(P2,f)

Now all we need to do is to look on P3 - the union of P1 and P2. Than according to the theorem I proofed in the beginning of this post:

$L(P3,f)>supL(P,f)-\frac{\epsilon}{2}$ $U(P3,f)

Since the integral exists we can write than that:

$U(P3,f)-L(P3,f)<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$

And we are done prooving the Riemann condition.

In the next post on this subject I will show (and proof) another condition, which is very similar to Riemann condition but is more easy to work with.

## Wednesday, August 6, 2008

### Lets do math!

I just found the best cat photo for a long time:

Original unknown

You are not going to say no to him, right?
Unfortunately, this is not a question that is likely to be solved soon. It already stood for a long time, and only a few weeks ago there was another unsuccessful attempt to prove it. Well, to be precise there are such "attempt" every week, but most of them are just attempt to get attention (and to win the one million dollar prize) by people who simply don't know math well enough to prove this theorem. The attempt I mentioned was a paper submitted by a professor who already had some previous work done about the RH. While this proof was met without much optimism, it looked serious. But even in his paper an error was found in 24 hours.

By the way, while there is a prize for proving RH, there is no prize for showing that it is wrong. In my opinion the reason for this is that showing it wrong doesn't necessary require mathematical advance. All you need to show that it is wrong is to find a zero of the Riemann zeta function which is not on the critical line. This can be done using a computer, without bringing any new ideas to math.

### Online symbolic calculator

I was told about an interesting website today, and decided to share it with you. The website in question is an online symbolic calculator. While it cannot replace a program like Mathematica or Matlab, it is free and has a nice set of features:

This free online symbolic calculator enables you to define variables and functions as well as to evaluate expressions containing numbers in any number system from 2 (binary) over 8 (octal), 10 (decimal) and 16 (hexadecimal) to 36, roman numerals, complex numbers, intervals, variables, matrices, function calls, Boolean values (true and false) and operators (and, or, not ...), relations (e.g. greater than) and the if-then-else control structure. Comments are C-style /* */ or //.

It also looks like it is being well maintained by its creator, so it is possible that new features will be added in the future.

## Tuesday, August 5, 2008

### Nonconstructive proofs

What is the simplest method to prove a statement? Well, this depends on the statement you want to proof. If, for example the statement you want to prove is: "There exists a positive number large than 2 on the real line", you can just choose a number, for example 3, and show that according to the order axioms it is large than 2. Thus, you will have an example that you constructed.

However, it is not that always that simple. Even when you are asked to show that something exists, sometimes it turns out that to proof this existence you will not need to actually construct anything. An example of this is the theorem: "Not all numbers real numbers are rational". While it is possible to proof this by simple construction, there is another way to show this which doesn't require any construction, but only three theorems from Set Theory:
1. R is uncountable
2. Q is countable
3. For any two sets A-countable, B-uncountable, B\A- is uncountable.

The statement than follows immediately, and even in a stronger version - instead of showing that there are irrational numbers we showed that "most" of the real numbers are irrational. However, we don't get a concrete number from this proof. The fact that such proves are possible was a cause to a rather major disagreement between mathematicians in the past. Now it is a well excepted.

In the example above, I showed a statement which can be proved by construction and without construction. But are there statements that cannot be proved by construction, but can be proved without construction? It turns out that there are such statements. Lets prove that: "There exists an irrational number which when raised to an irrational power will be rational".
It sound hard, but it is surprisingly easy. Lets look on the number:

$\sqrt{2}^{\sqrt{2}}$

If the number is rational, the statement is proven, because the square root of two is irrational. If it is not rational, lets look on:

$(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}$=$\sqrt{2}^{\sqrt{2}*\sqrt{2}}$=$\sqrt{2}^2$=2

Since 2 is rational, we are done.

The problem is that we have don't know if $\sqrt{2}^{\sqrt{2}}$ is rational or not, and therefore we don't have a construction. But we know for sure that the statement is true.

## Monday, August 4, 2008

### The semester has finally ended

It was a long year... While officially there were only two semesters, we actually did three because the professors where on strike during the first one. I still have exams to do, but the last lecture was today.
Well, it was a fun year. I hope the next one will be even better.
I am sorry for not posting for a few days, I was extremely tired and busy. Hopefully I will be able to continue posting as usual during the exam season. I am sure that while I would be revising I would get a lot of ideas for things to write about on Math pages.

Also, it is a sad day to the math blogger community today. Craig Laughton, the author of Gooseania, wrote his last post today. He finished his math Phd, but decieded that he doesn't want to be a mathematician. His blog was created to document the process of doing a Phd, and it contains a lot of information. It is clearly visible how his enthusiasm for math slowly gets way to depression. He decided not to continue posting on the blog in the future. Hope he will find his place in this world....

## Thursday, July 31, 2008

### Math kitty

Photo by Dave Hogg

Apparently cats love studying math, only they are rarely caught doing so...

### Kde4 with Compiz

According to the video, the plugin used to achieve this effect is called the fish eye plugin. I never heard about this one before, but it looks very impressive. The things that computers can do...

### Using kde4

The first part of this post is a general rant about Kde4. The second part is a simple how-to for making kde4 look better and be more usable.

Every month or so I have an urge to try new things... Usually this means installing new (preferably beta) software on my computer. The program that got installed this time is kde4. I am using Linux, so if you are using windows this post will probably be of little interest to to you.

To make the story short, using kde is a horrible experience. The only reason I am doing this is stubbornness. The first thing I noticed after logging to kde4 is that the desktop looks very beautiful. Whoever designed it did an excellent job. However after doing this he let some other people turn the rest of it into a designer nightmare.

Firstly, the Firefox looks horrible. You can use another browser, konquer, but I wouldn't recommend doing so. A lot of sites (including Google and Yahoo) claim that this browser is unknown or very old.
The same is true for many other applications. However, this is possible to fix partially. All you need to do is to install the package gtk-qt-engine-kde4:

`sudo apt-get install gtk-qt-engine-kde4`

You’ll need to log out of KDE for the new options to appear. Once you’re logged back in, open System Settings and select and then GTK Styles and Fonts. Select the Use my KDE style in GTK applications option. Click Apply, and log out to make the changes take effect. I found this tip here. This is not a perfect solution, but it is significantly better than what was before.

After this rant, I guess I should say something positive. The file manager, Dolphin, while being a designer nightmare, works well and faster than Nautilus. The plasma widgets are also nice, but I never used widgets and I doubt I ever will, so this is of minor importance for me. The start menu also looks very good, but some of the icons are missing. Also, the desktop manager (at least I think this is what it is called) crashes sometimes. Well it happens I lose the panel, the widgets and the desktop background but all the rest works fine. Since I am using kiba-dock and Gnome-do, I don't really care about this, but I feel somewhat uncomfortable without the panel. I am writing this post from Kde4, and the desktop manager crashed a few minutes ago. I didn't yet find a way to get it back to work without restarting the X server.

In the next days I hope I will manage to solve at least some of the problems I am having with Kde4. If it will happen, I will update this post with some extra tips.

Splash-Screen:
If you installed kde4 using the Kubuntu package, you will also get your Upsplash changed to say "Kubuntu" instead of "Ubuntu". If you care (I do), the simplest way to get it back is to use the start up manager: System->Administartion->StartUp-Manager. Go to the Appearence tab and change the upsplash theme to th one you want.

Remove games:
I don't play games, so I decided to remove all of them except for sudoku. This turned out to be extremely difficult. Firstly I attempted to use synaptic to remove them. By doing so I discovered that some programmer thought it would be nice to make the package kde4 depend on the package kdegames, which depends on every single game. The result was that an attempt to uninstal any game using aptitude ended with Kde4 being removed, and this set all the other packages connected to kde4 to auto removable. And since aptitude automatically removes such packages, by removing one game I removed all of kde4. To solve this I used adept package manager to uninstall all the games I could, and then I used synaptic to mark all the packages that were set to auto removable as manually installed. After this I finally got rid of the rest of the games.

Make Firefox look better:
The first thing to do after installing kde4 is to make gnome applications look normal in it. See above for instructions. However, this fix doesn't completely solve the problem with Firefox. It still doesn't render well. One possible solution is to install a theme for Firefox which was designed for Kde4. It is still experimental, so you will need to register an account on the Mozilla addons site.

## Wednesday, July 30, 2008

### Warp Drive Engine

I didn't intend to write any more about faster than light travel, but I was sent a link which is too good not to write about. I am not going to comment on the science involved, I am not qualified to talk about anything that concerns 11 dimensions. However, this article raises an interesting point which I didn't mention in my posts about faster than light travel. If you read those post, you know that I started this topic with a simple proof of the impossibility of faster than light travel. I also showed that there are exceptions to this rule because quantum mechanics and relativity doesn't play well together. But what about large objects? Is there a way for them to somehow escape relativity?

In the article I linked, you can read about a simple (but nearly impossible to do) idea - expanding the space behind the object and shrinking it in front of the object. Thus you will create a bubble that slides in space. Inside the bubble the object will move slower than light, but the bubble itself can move faster, because it is not even matter but space itself. This is called a wrap drive.
This sounds possible, but it is easy to see that the method I used to show to show the impossibility of time travel, still works for this example.

However, in this particular case my prove doesn't apply. The reason for this is very simple - while it is bot obvious the proof is build on a simplistic assumption that the universe is the same on large scales. Unfortunately, this is not true. This is only a simplification used for ease of calculations, although it is very close to being correct. If you allow universe to be extended and contracted (and this is exactly what ruins this assumption), there is nothing that forbids faster than light travel.

To finish this post - a little joke:
An experimental physicist finished running a very complex experiment. After plotting the data on a graph, he got to the conclusion that he doesn't understand why the graph looks the way it does. So he went to a theoretical physicist, showed him graph and asked to explain the very high peak in one of the points. The theoretical physicist looked on the graph for a second and said "Oh, there is a perfectly good reason for this peak". And started talking. During the explanation the experimental physicist suddenly looked on the graph and said" Wait a a second it is with the wrong side up". The theoretical physicist looked on it, and said "Oh, there is a perfectly good reason why there is this very law value in this point" And started talking.....

## Tuesday, July 29, 2008

This post is a little follow up for the series of posts I wrote about faster than light travel. In one of the posts on this subject I brought a simple example of how in Quantum mechanics faster than light travel is possible, despite it being impossible according to relativity. The example I talked about in that post was very simple, and it was easy to explain why this is indeed what happens. But this example talked only about faster than light travel on a very small scale. In this post I want to talk about another example, which is far more complex but it shows that faster than light travel is possible also on large, even cosmic distances.

Lets consider the following situation. Suppose you have two balls, one is pink and the other is green. However, the color property of the balls is quantum - both of the colors are in superposition, so both of the balls are pink and green in the same time. But if you will measure one of them, the superposition will collapse to one of the options. What is interesting is that if you will measure one, you will cause the other one also to collapse, because you now know its color as well so it is no longer in superposition.
There are no such balls in the real world, however it is possible to create particles with all the required properties. I don't want to talk about a specific example in this post so we will agree that the balls stand for some object that have a quantum property which we will call color.

Now, lets suppose that you create two such ball in the laboratory and give one of these balls to your friend. Lets label this ball A. You friend happens to be an astronaut and he flies to the moon with this ball. When he gets there, you measure your ball (B) and discovery that it is pink. This measurement causes your ball to collapse - it is no longer in superposition of ping and green, but it also causes the second ball to collapse, in the exact same instance.But even light travel to the moon in over a second. So, something changed in the second ball, A, without a reason to this being in its "Cone of light". This again means that the information of the measurement traveled faster than the speed of light.

This is known as the Einstein paradox. He originally presented it in an attempt to prove that quantum mechanics is incomplete. He claimed that it is not correct that quantum processes are probabilistic - "God doesn't play dice". In this mind experiment there is nothing impossible from the position of quantum mechanics, but allowing faster than light travel we allow time travel, and give place to a lot of other paradoxes. He offered a solution, to add a unknown property lambda which we cannot yet measure but that decides the outcome of the measurement. This works because quantum mechanics says that from all the properties of the object we now about we cannot deduct its state (in our example the ball color) so until we measure it, the object is in superposition. But if we allow for such lambda to exist we get that there is no superposition the balls are always the same color - there is no longer probability involved, all is determent from the beginning.

Interestingly, Einstein was wrong. It took some time but eventually a test that checks if such lambda exist was performed. The test was an experiment that returned a value, we will cal it S. If S is less or equal to two, then there is lambda. If S is bigger there might be lambda, but faster than light travel (in the case of the Einshtein paradox) is possible. To be even more specific, if quantum mechanics is correct $S=2\sqrt{2}$. However, in physics to show that something is equal exactly is nearly impossible, so the main point here is to check if S is less than two or not. This experiment was performed a lot of times. In the beginning the equipment wasn't sensitive enough, but after a few decades the result was that $S=2.5\pm0.35$. Since this is large than two, the case was closed, faster than light travel is possible. It is still unknown if there is lambda. It turned out that we can design a theory with lambda and without it, and they both will work always. They both manage to explain all the results of all the experiments conducted until now.

## Sunday, July 27, 2008

### The weirdest "Hello" ever

Today in the morning I was just entering the Einstein Institute Building in the Hebrew University when another student was going outside. I talked with him a few time before, he is also studying mathematics, and is a really nice person. We both were in a hurry, because it was almost the time for our lectures to begin, so we just said "Hello" and went on. Well, almost. I just said "hello". He said" Oh! Hello professor, how are you doing?". I am not a professor, and nobody was standing behind me. Since it was totally unexpected, and I was in a hurry I even didn't react in any way to this. That was a weird morning... While I want to do an advanced degree in math, I am sure I never told him about it, so I really fail to see why he would joke like this.. Well, it might be prophetic. :)

On a more serious note, the semester ends on 4/9. Oh should I say, the exam month begins on 4/9. Anyway you put it, it is good news actually. I feel that I really want to start learning new courses, that is I want to start the next semester. I need to study a lot for two of the exams, I don't feel that I know these courses material well enough. But I am sure I will manage.

## Saturday, July 26, 2008

### Mersenne prime search

This post is about the Mersenne primes. Mersenne primes are prime numbers of the form $2^n-1$. A bit of trivia: This formula for generating prime numbers is known for a few centuries, however it continues to be interesting even today. It is easy to see that this formula generates primes, for example for n=2 we get 2, n=3 we get 7. However, for n=4 we get 16-1=15 and this number is composite. Generally, the result might be prime only if n is prime.

I am not going to write much about the properties of the Mersenne primes, it is easy to find this information on the net. What I want is to write why those primes are interesting to me and perhaps I will manage to interest you as well.

Firstly, it was proven by Euclid that all the numbers of the form $2^{n-1}(2^n-1)$ are perfect if $2^n-1$ is prime. Later is was proven that all the even perfect numbers are of this form. It is still unknown if there exists an odd perfect number. Despite all the attempts to solve this problem it stands for over 2000 years.... The Mersenne formula therefore can be seen as an connection between two fascinating problems in modern mathematics - the Riemann Hypothesis and the odd perfect number problem.

Secondly, there is a project on the internet whose goal is to find large Mersenne primes. This project is a typical example of distributed computer network - you download their program, and it uses your computer idle cycles to try and factor numbers produced by the formula. They even have a prize offered, but it is pointless to participate only to try and win the prize in my opinion. The last prime they found was found in 2006... However if you want to put your computer idle cycles to some use, this is the place to go to. Since my computer is now staying idle most of the time so I am starting to consider this use.

Thirdly, there is an interesting theorem which happens to concern them. This theorem (proven by Gauss) says that it is possible to divide a circle into p equal segments using square roots (that is, only square roots are required to find the solution - points in a plane), only and only if p is a Merssene prime. To be precise, Gauss grooved only one part of this theorem - he showed that if p is Merssene prime than the construction is possible. The second part was proofed latter. From this we see that there is also a connection between these primes and geometry.

I am sure it is possible to think about more interesting properties of Mersenne primes, but those are the properties I know about and find interesting. What do you find interesting in them?

## Thursday, July 24, 2008

### Formula for Primes

I stumbled on two interesting formulas for primes today. The first formulas allows to check if a certain number is prime or not, and the second formulas gives you the Nth prime. They both use factorials, so neither of them is efficient for large numbers. And since n! becomes large rather rapidly, it means that without a computer it would be a problem to use this formula even for n=20 for example.

Both of the formulas were invented by C.P Willans. The | | stands for the floor function.

If x is prime, the result will be 1 else it will be zero.

As you can see both of them are higly ineficcient - the numbers becomes laege extremely fast.

## Wednesday, July 23, 2008

### How people react when they discover you are strudying math?

The main reason I am writing this post is because of an interesting post I recently read. Beans at Me Or My Maths recently wrote about the different reactions he got from people who heard that he studies math. To sum it up, the reactions he gets are of the type "you are crazy". He is not the only blogger from the UK who says he gets this type of reaction, so apparently it is a rather usual thing in that part of the world....

As you probably understood from the last part of the previous paragraph, such reaction is not common in Israel. I remember very clearly how when I had to study math for 4 hours straight in school (with one half hour brake) some other pupils said that they cannot believe that I do this, but there reaction was surprised but not negative. Perhaps they were just glad that they don't have to do this. Also, when asked about this I answered were simply "I enjoy studying math". I guess after this they preferred not to talk to me... Not much of a loss. I am not cynical, it is simply that my interests were very different from theirs.

Unlike Beans I don't do my homework in trains, and I don't speak with random people about my math, but from time to time I share my enthusiasm for math with someone who has no clue of what I am talking about. By sharing enthusiasm I mean that I start talking and I don't really care if I am understood. However, they usually don't faint or run away. They either remain polite or just ignore what I say. Writing this blog helps to control such bursts of enthusiasm, but sometimes I feel the desire to speak with someone....

The above doesn't mean that Israelis love math. However, it might mean that they really don't care. Most of the people here study as little math as possible in school, and then even if they go to college they are likely to never hear about it again. Also, just today I talked with someone who said that he can hardly wait until the end of the semester - unless he will fail in something, he will not have to study any math next year. But even he didn't say anything negative about math, or about those who study it.

Now, to the more mathematical part of the post. I have been thinking about a certain integral in the past few days, but I am still unable to show that it converges (or not). I am probably missing something simple here, this shouldn't be a hard problem:

$\int (sin(x^2))^3$$dx$

The limits are zero and plus infinity. If you have the solution you are welcomed to post it, if no solutions will be posted before I will solve this thing, I will write the solution here (unless I will forget about this). Good luck...