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.

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.
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....