upsc update

Since I made a post earlier about the undergraduate problem solving contest I figured I might as well do an update post now. As it happens the first problem is the only one that I didn't turn in a solution for (the second one was about efficient encodings and the last one was a relatively easy question about squares) As it happens that actually puts me in third place. As the prize for winning the upsc (thats Undergraduate Problem Solving Competition, I like to pronounce the shortened version as oopsie=upsc) is an expenses paid trip to mathfest 2009 I certainly hope that I continue to pull up in the ranks (an all expense paid nerd vacation awesome). Hopefully at least one or two of the questions posted next semester will be really hard. Otherwise I doubt it will be possible for me to pull into the lead.

More urgently though the Putnam exam is going to be December 6^th. It will have been 1 year and 5 days since I took the 2007 exam. It is hard to believe that it has only been one year since then. I have learned a great deal in that time. Looking at the putnam problems now I feel at least somewhat prepared for the test. Last time I took the Putnam I got a score of 1 point out of a possible 120. Which I am eager to say is above the average score. I have a book with some of the old putnam exams in it and perhaps if I have a spare 6 hours tomorrow I should administer one of them to myself and see how I do. Hearteningly if I get 40 points on the test then I would be in the top 100 of test takers (a few thousand mathematics undergrads and assorted others take the test every year, what can I say doing well on the exam looks good). Realistically I am shooting for 20 points. The 2007 test was a little harder than usual and encouragingly I now could solve 3 of the 12 problems from that exam. before I finish this post off and run away I will give you some example putnam exam problems.

(A1-1976) P is an interior point of the angle whose sides are the rays OA and OB. Locate X on OA and Y on OB so that the line sexment XY contains p and so that the product of distances (px)(py) is a minimum.

A1 and B1 are traditionally the easiest problems and A6 and B6 traditionally the hardest.

(A6-1976) Suppose f(x) is a twice continuously differentiable real valued function defined for all real numbers x and satisfying |f(x)| <= 1 for all x and (f(0))² + (f '(0))² = 4. Prove there exists a point x₀ such that f(x₀) + f ' ' (x₀) = 0.

Numbers are Cool

This post is just a place for you to put comments about specific numbers that are cool or why numbers in general are cool.

Obama Won!!!!!

Now that the election is over I can't help but feel elated. This really is an extra ordinary moment in American history. I thought this merited a bit of space here on the blog. All I really want to do is just say it over and over again Obama won, Obama won, Obama won! Because I have been scared that it wouldn't really happen despite the eventually commanding lead he had. I always thought something terrible would happen before the election and then we would have McCain in office which wouldn't have been the end of the world... until he had a heart attack and Palin took over for him. But I am still rattled by how close the election was in terms of the popular vote. How can it be possible for a McCain Palin ticket to get even 20% of the popular vote let alone 46%? But now that the election is good and truly over I can take a sigh of relief and just wish that January were a little bit closer. I hope this is the beginning of a political movement of change like Obama has said he will try for. As he said after the election "change has come" lets hope so.

Sad but True

The poll closed, and it is now official by a vote of 3 to 2 the universe is going to expand forever. This is actually rather close to the current scientific consensus. Detailed measurements of the rate of expansion of the universe done using supernova actually tell us that the rate of expansion of the universe is getting faster! If you are interested the exact method of using these super nova to give the rate of expansion works something like this. Step 1 you look for a type 1A supernova. I'm not exactly sure how you tell if it is a type 1A nova but I'll trust that it is possible. Steps 2 and 3 record the red shift and brightness of the nova as measured from earth. step next, rinse and repeat. So I am not sure exactly how many type 1A's we have on record but from what we do have we can use to make a measurement of the rate of expansion of the universe and how it has changed over time. The brightness of the supernova as measured from earth gives us info on exactly how far away the nova was because type 1A super novas all have exactly the same brightness when they occur. If memory serves it is because the star that collapses into a black hole accumulates mass from a nearby binary star until in reaches a definite critical mass at which point KAB00M! Since all these events happen when the star reaches the same critical mass the brightness of the explosion is just about the same. So as the apparent brightness from earth drops off we can measure how far away the damn thing was to begin with. Now the light has a particular characteristic spectrum which we would see if the thing were not moving relative to us thanks to good old doppler though the spectrum gets shifted. We can use the spectrum shift to measure how fast these things are moving away from us. There you have it we now have data that gives us how fast things distance x away from us are moving which gives you the rate of expansion of the universe. Course it isn't quite that simple, all the complications come in when you take into account that we don't see the light from the nova's instantaneously but it takes some time for the light to reach us. The farther away stuff is the longer it takes the light to get to us but also the more space has expanded during its flight. When you take this into account you not only get a measurement of the rate of expansion of the universe but a measurement of how that rate of expansion has evolved in time. With each nova being a window to a different epoch of the universe. Turns out the expansion of the universe started to slow down for a while and then started speeding up again! So now what it looks like is all those big crunch theories where the universe eventually ends in one huge gravitational collapse were not meant to be. The universe gets to end as a sea of cold lonely atoms and radiation. The 3 to 2 vote confirms it, sad but true.

Nano again

So November is NAtional NOvel WRIting MOnth or nanowrimo for short. The challenge is to write 50,000 words of fiction during the days of november. I have yet to fulfill this goal even once though I have been participating to various degrees in nano for 3 years (this will be my 4th year) This year I'm really going to do it though! Of course I have slightly less than no idea what I am going do to for the novel this year. I am really open, at the moment it could be anything from a book about a group of jumpers orbiting a post implosion Jupiter, which is now a singularity used for warp research to a book about how the only thing known to kill vampires are cigarettes.

Death to Tyrants

For the few rare souls who actually have visited this blog more than once. Why on earth did you come back? The content of this blog is perhaps what you might aptly call eclectic, yet another apt title might be completely disordered badly written ruminations of a crazed and unstable mind. Because of this I would be willing to bet that the thing that gives cogency to the group that is my readers is none other than personal acquaintance which lends interest to the blog even though. The content is terrible. Is this the case? Please tell me in the comments.

P.S. The title of this post is intended to imply that I am acting as a sort of content tyrant. I cannot stop being the content dictator but if the content is bad then that makes me the content tyrant since my rule is unjust. So give me some feedback and hopefully I can become the benevolent dictator instead of the unjust tyrant.

Do You Know Me Personally?

The poll with that as the question has finally come to a close after a year of polling.

The poll got a total of 22 votes which I hope is less than the number of people that actually visited the blog in a year. Still 22 people isn't bad and I am particularly pleased that only 7 (thats 7/22 or approximately 100/pi % thats 31% ish) of them claimed to know me while the majority 15 (68% ish) of them claimed to not know me at all. This is heartening, since it means my blog is not entirely invisible on the web. Furthermore it means that the blog has more pull than merely the pull afforded by personal acquaintance with this nut.

Fractional Dimensional Spaces and Fourier's Trick

One doesn't usually think of the existence of the standard basis for euclidean vector spaces as related to fourier series. It is easy to show that there is a rather direct correspondence. If we take a slightly non standard method of providing a coordinate system to the points in Rⁿ we let every point p be labeled with a the function defined using the usual dot product as f_p(x) = p * x where * is the vector dot product and x is any other point in Rⁿ Now to obtain the dot product of two points we express each point as a linear combination of the standard basis vectors and then take the sum of the products of the corresponding coefficients in the standard basis expansion. More to the point if you take the dot product of one of the basis vectors and any point you obtain the coordinate of that point with respect to that basis vector. Thus the dot product of two points is the sum over the standard basis of the products of the dot products of the two points under consideration and the i'th vector in the standard basis.

Now consider what happens when we take our above functions mapped to points. We can obtain the dot product of the two points as the integral of the product of the functions associated to the points over the unit sphere of vectors. If the space under consideration is integer dimensional space then relatively simple exercises in linear algebra show that any point can be expressed as the linear combination of d linearly independent vectors (d being the dimension of the space). So (as is kind of obvious) if a (finite) standard basis exists then the dimension of the space is integer.

This is all based on the fact that the space has a vector space structure set up on it. I posit that vector space structure does not in itself imply the dimension of the space to be of integer dimension. If this is true then it means that any non-integer dimensional space would have to have an infinite basis

Problem Solving

At least once in a while I should post something with some personal interest. It is strange how reading something which is happening in someones life that you don't even know can be entertaining. I suppose what is on my mind at the moment is problem solving.

There is an undergraduate problem solving contest here at the U where a new problem is posed once a month during the school year. People submit solutions at the end of the month and are assigned points based on their solutions to the problems 3 points for a correct one and +e points if you win. I have found that I can solve a significant fraction of the problems but that also sometimes I spend dozens of hours working on a problem only to be unable to solve it. This last problem really made me angry when I found out how it admits a really simple solution and even one that I sort of thought about before. you have a 7x7 checkerboard with a black square in the corner. What single tiles can you remove in order to make the board tileable by 2x1 dominoes? The answer is you can remove any black square and not any red squares. I worked on the problem for hours and made it more and more complicated... but I couldn't figure out how to prove that removing a red tile made the board untileable. The amazingly simple thing to realize is that there is one more black square to begin with on the board and every dominoe covers a red and black square thus removing a red square leaves the board untilable since it leaves the number of black and red squares unequal. An even more elegant solution deals with finding even paths on the board which gives you both the black and the red tile cases in one go. The really infuriating thing is that I considered both the fact that every dominoe covers a red and a black tile and the fact that the existence of eulerian paths on the dual graph of the board is related to the tiling.... and yet I didn't solve the damn thing. I guess I am really just not capable of doing personal happenings rants. I just get caught up in the details of whatever little abstract thing I mention. I'll do better next time.

The Fiend of Finance

As it so happens I find myself in a relatively uncomfortable financial situation. I have made it almost the entire length of my undergraduate career without resorting to a job. I have worked of sorts since I have done some research for small grants but I have not entered into an employee employer relationship while in school. I did however have a job before I started and retained that job for some time while I was in school during the first year. I now find that I must come up with some new way of acquiring monies. The prospect of getting some sort of gainful employment however is not something I relish. I suppose I should give myself some sort of period during which I can pursue different paths which are less likely to allow me to have stable financial self support but which are all together cooler than the usual resort of a 9 to 5.

List of ways to get monies

1. Sell things on e-bay: obviously in order for this to work for an extended period so that it can provide some sort of illusion of financial stability I would have to find a way to supply the items which are to be thereby sold... thus making this option not terribly useful without a supporting money making concept, see following items.

2. Sell handmade wooden puzzles: actually I rather like this option as it does not require a great deal of technical prowess, I already have (or more precisely already have access to) a great number of woodworking tools, last but not least I can put as much (or as little) originality and ingenuity into the creation of said puzzles as I like.

3. Create and maintain an original software package: This idea is so varied as to require an entire sublist of the possible sorts of software enterprise that could be undertaken. I could try my hand at engineering a game, a map maker, a fake language generator... etc.

4. Make grandfather clocks: Unlike the wooden puzzles thought these objects would sell for quite significant sums of money and would still be a great deal of fun to make. Apparently I can purchase the actual clockwork parts of these devices online for rather less than I could sell the finished product... however of course the thing that makes the toys attractive is that they can be made and sold easily and shipped as easily. Selling these things would be I imagine rather more difficult. $500 clock anyone? (and that is dirt cheap for a grandfather clock...)

5. Sell my body: Specifically I mean "donate" blood plasma which is relatively easy to do... but it would weaken my immune system probably leave me with scars after a while and is not that pleasant in general. On the bright side it would force me to eat regularly (which is something that I do not currently do). However the monetary gain has severely limited potential since I can only give so much blood plasma a month (shame on them for stopping you before you hurt yourself). I have already done this a little and I must say in most ways I prefer being scared of going hungry to selling blood plasma.

6. Shamelessly beg for money on the internets: this is related to #3 since I would probably have to put my software out for free and then ask for donations, alternatively of course I could put a portion of the software out for free and then have other portions which would have to be purchased but there is something deeply satisfying about knowing that all of the money you get is given not because it was required but because in some way the person giving it felt it was deserved.