Its from an old movie

This song has been stuck in my head for the better part of 2 days now

Prego prego
Anywhere you may go
Make each day be a day full of fun
If there's a game or a girl to be won
Do it with a Bing Bang Bong
A Bing Bang Bong

Presto presto
Do your very besto
Don't hang back like a shy little kid
You'll be so glad that you did what you did
If you do it with a Bing Bang Bong
A Bing Bang Bong

Be like Cristobol Columbo
Take a chance
Take a chance
Don't be a dopey or a dumbo
Gotta run in a trance

One step two step
Stepping through a new step
Live your life with a zip and a zing
You'll have the world on the end of a string
If you do it with a Bing Bang Bong
A Bing Bang Bong

A BING BANG BONG


if you recognize it as having come from the movie houseboat my hat is off to you.

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.

Black hole NMR and non-integer dimensions

Lately I have been thinking a great deal about the nature of real dimensional spaces. That is spaces whose dimension can be any real number not just an integral value. A method by a Yurosevich Koloskov is to use a kind of stochastic metrization of the space which is something I don't entirely understand but I think is a very interesting thing to try and look into. I think however that very probably in order to be able to build up the geometry of a probability space that would give rise to the kind of appropriate structure to describe Scroedinger wave type behavior we will need to be able to handle geometries of spaces with non-integer numbers of dimensions. This is backed up in some small degree by the practice of dimensional regularization which is a renormalization technique which assumes a non-integer number of dimensions in order to obtain renormalizations that otherwise could not be achieved. If the geometry that could be used to describe the quantum world is ultimately (even in the limit of the multiverse) necessarily stochastic (a kind of quantum gravitational bell's theorem?) then integer dimensional spaces should suffice to describe its dynamics I am not certain that the stochastic coordinate system mentioned above would be sufficient to encode these stochastic spaces as well but then part of the beauty of the idea of having parallel realities is that they make the geometry static instead of stochastic. The stochastic perception then comes from the constraint of path selection in a probability space which very likely is governed by a kind of action minimization multiplicity peak where the multiplicity of paths with close to minimum action occupying a kind of thermodynamical peak in much the way of the configuration of greatest entropy. At this point I think we finally have enough of a framework to begin to try and make headway on the predictions of this sort of world view.

I recently learned that the gyromagnetic ratio of a nucleus is dependent on the nature of the self interactions of the nucleus with the surrounding vacuum fluctuations. Since a singularity would constitute a kind of point charge with spin then it stands to reason that the gyromagnetic ratio of a singularity would be dependent on its charge its mass and its g-factor which perhaps could be calculated. using QED in much the same way that the calculations are done for the nuclei and point charges like the electron. If there were a reliable way to make a measurement of the charge and the rotation of some black hole as well as the magnetic field strength at a known distance from the singularity then we could test if the gyromagnetic ratio matches the predicted value. That could be a good place to try and get quantum gravitational observational measurements. black hole NMR


Also the g-factor of the singularity which we would be measuring might very possibly be dependent on the number of dimensions in which we are carrying out the measurements. which could be a good way of trying to figure out something about the geometry of spacetime on quantum gravitational territory.

Neural Modeling & immortality

The human brain has about 10¹¹ neurons and for each neuron there is an average of a few thousand connections. If we forget about the details of both the connections and the neurons function etc. To have enough labels to uniquely determine each neuron we would need at least 5 bytes or so and conveniently enough we are moving into an era in which we are doing 64 bit computing. That is we will soon be operating on platforms that are big and bad enough to handle huge structures such as things with 10¹¹ parts. Each neuron since it would have thousands of connections it is reasonable to say we could encode a neuron and all its connections with about 6.4*10⁵ bits (around the amount of space the memory addresses of its connected neurons would take plus a little). Thus just storing the connections without weight or any of the ancilliary information such as connection strength time of signal propagation etc it would take about 6.4*10¹⁶ bits to store the connection information. That is about 10⁵ TB of data. Now massive information storage of that magnitude is not something that you come across in your every day desktop computer, however on the basis of Moores law we could expect desktops to have that (conservatively) within about 32 years. Right now we can (and do) support massive data structures of that size in servers. The amount of computing power necessary to do effective computations using structures of such massive size probably will develop in parallel. The amount of data necessary to store the connections gives you order of magnitude expectations of what is required to store meaningful data on the structure of the brain. If we add extra data into the mix so that we store connection strength information about neuron type and sensitivity and stuff like that you are changing the required data only by some small multiplicitive factor. Although connection structure is definitely not the only important thing about the human brain even if there are hundreds or even thousands of similarly important characteristics you change the size of the required data by two or three orders of magnitude which means you need to wait 12 or 18 years for computers to get better or you need to be willing to make a super computer now that is capable of handling that sort of stuff. At any rate it would definitely appear that within the next 50 years or so computers will be powerful enough to encode and run processes similar to the human brain. Somehow I doubt that by that time we will have scanning technology or knowledge of the human brain sufficient to be able to scan a brain and put that information into a computer. The point is however that some time between 2040 and 2070 I am betting that personal computers will be powerful enough to in principle represent and support a structure equivalent in essence to the functioning of a human brain.

There are possibly some very large holes in this line of reasoning, for instance if the specific spatial layout of the neurons was very important to functioning or if previously unknown structures in the brain actually constitute a large part of its computational power or if chemical diffusion and the specifics of chemical processes in the brain are very important etc. I can't really know how much of the processing power of the brain is tied up in such things as the properties of diffusive chemical kinetics or neural knotting. If it turns out that it takes many orders of magnitude more processing power and space to adequately describe these processes instead of (as I was assuming) a similar amount of of processing power and information to encode and process the connection structure then it could take very considerably longer to really be able to represent the mind. At the same time though this is really kind of the brute force approach, try to make a computer be able to as precisely as possible model exactly what is happening in a human brain. While this approach has some nice allure to it since it is clear to see how its success is on some level relatively certain (assuming, as many people do not, that the physical functioning of the brain is what gives rise to the mind).

The possibility stands however that the particular form of neural computing that the human body employs is really very inefficient when it comes to producing intelligence. Perhaps we will stumble upon a better way of doing things somewhere along the way but I find that unlikely. Very probably we will not understand how we can do intelligence with finesse until we are capable of doing it with sheer brute force.

Collapsing jupiter with sound waves

because the mass of Jupiter is not sufficient to cause collapse into a black hole by itself the only way that we could possibly make this event occur is by means of a sort of burst compression activity. The condition upon which we will have made a successful collapse is that we keep a mass inside some spherical volume with the constraints that we keep the mass there long enough for light to traverse a diameter of the volume (so that the gravitational field definitely has enough time to set up) and also that the amount of mass inside the volume exceeds some threshold. The Schwartzchild radius is r = 2*G*m/c² if you translate that into a necessary density based on the available mass you get m/V = 3 * c⁶ / (32 * Pi * G³ * m ² ) You will notice that the necessary density goes down with increasing mass. Unfortunately c⁶ * G^-3 is a really freaking big number about 10⁸¹ fortunately mass is in there as an inverse square so when you get to really high masses the required density is actually pretty low. If you happen to have a mass of about 10³⁸ Kg involved then the required density is about the density of water.

If you read previous posts on my plan to collapse Jupiter into a black hole then you will know that I was thinking of a strategy that used several million megatons of nuclear bombs to drive a shockwave into the core of Jupiter in the hopes that I could drive the density of the region high enough to cause complete gravitational collapse. Recently I made a visit to INL (Idaho National Labs) and talked to someone there about the possibility of bubble fusion. The concept of bubble fusion is that you release bubbles of deuterium and tritium into some medium and have freaking loud sound waves resonating in the liquid. When the sound waves hit the bubbles of gas they violently compress it causing huge temperatures and pressures which theoretically can be enough to cause fusion inside the bubbles. I don't know if you could really get a positive energy yield out of doing fusion reactions this way. The guy I talked to seemed to think that a positive energy yield was completely out of the question. I genuinely think you could probably get a positive energy yield from a bubble fusion sort of device maybe nucleate the bubbles in a blanket of molten lithium you could get good shielding and breed tritium that way.

Bubble fusion is not the point though, the point is that resonance is cool and that maybe a single burst is not the most efficient way of going about doing it. Although producing many millions of megaton nuclear bombs is something that is theoretically within the technical grasp of humanity I would say that is still somewhat of a brute force strategy. What would otherwise have had to be done all at once in a huge burst of energy could instead be done over a period of years of carefully tuned resonances. Ideally it would be nice to be able to get a radial resonance going inside of Jupiter so that the pressure waves go directly down into the core and then bounce off each other creating amazingly high pressures at Jupiter's core while leaving the devices actually causing the sound waves in relatively safe and low pressure environments near the surface of the planet. How exactly these sound waves are to be produced is a matter for a future post. Also I would be willing to bet that the simplest radial standing wave pattern resonance is not stable and one would have to design something with a little more robustness.

I can has comment?

yes I understand the vile thing I am doing in introducing lolcat syntax into my blog but I wants me some comments. Tell me how horrible I am for making such an attrocious mistake. Slander pickles and grapefuits or people who read new scientist. Just please leave me some comments. Of course if they happen to be vaguely topical that would be lovely and if it is a question (preferably one I know the answer to) awesome! I know you people are reading my blog... well I know that at least one or two people visit it a month... so this is a cry for some feedback. Please leave me some comments! Plus if I can trick you people into investing yourself in the blog enough to leave a comment that has to lead to increase future reading (and also increased posting), which is probably bad for you but good for me.

Ordering

I define an ordering on a set S as a relation < defined on S such that for any a and b that are members of the set if a =\= b then either a < b xor b < a (xor is meant to make it explicit that this is an exclusive or) and such that if a < b and b < c then a < c. You could basically make any ordering you choose to care about for a particular set. We tend to take 3 < 4 but the definition of an ordering would still work just as well if 4 < 3 with all the other appropriate changes made (for instance if 4 < 3 then that would be a fine ordering relation but it would make it damn near impossible to make ordering work with a nice definition of addition) I will define a listing of a set as a injective mapping from the set of objects to the positive integers. If you think about any list you have ever seen you know that you can always number the items on the list 1, 2, 3.... and so on. While many lists don't have this numbering actually done that is not important, here I am simply looking for the quality of what makes something listable. Now for the paradox, the set of real numbers is not listable no matter how you try to list them any list you make will always leave out some real number. You can see this from the cantor diagonalization argument So there exist sets of things which can be ordered but which cannot be listed.
Intuitively it seems that ordering something and listing something are more or less the same. Taking a pile of rocks and putting them in order seems much the same process as taking those rocks and writing them down in order on a piece of paper. In fact it would seem that ordering is an even stronger requirement than listing since if you put the rocks in order along a line then the job of listing the rocks is done for you already. But from the example of the reals we know that this is not in fact the case apparently you can have orderability without having listability. Clearly any finite or even countably infinite set of objects is both orderable and listable, since by definition any countable set can be mapped onto the integers and so we can use the ordering relation on the integers to define one on any countable set. But orderability and listability become different when we move into the realm of sets with the cardinality of the reals. Let us go back and define listable and orderable in a little more detail. Listable means that given a set of objects the set is listable iff there exists an algorithm for reading through the set one element at a time so that you will eventually encounter every element in the set. The definition of orderability also changes slightly. A set is orderable if there exists an algorithm that has basically the same properties as an ordering. If the algorithm is given two numbers the algorithm must be able to tell if one number is less than the other in a finite number of steps.

Here is the fun part an ordering algorithm can be allowed to run for an infinite number of steps if it is given two numbers that are the same. The requirement is only that if the two numbers are not the same number that the algorithm will recognize that in a finite amount of time. Whereas the listing algorithm must conclude in a finite time for all elements in the set no matter what. So if you know something about the chomsky hierarchy that means that in some sense at least things that are orderable are recognizable languages whereas things that are listable are decidable languages. That is the difference between a list and an ordering.

HUH?

My blog is worth $1,129.08.
How much is your blog worth?

I figured it was worth about nil... odd...

semi-pure intelligences (humans)

A "pure intelligence" is something an intelligence that is only intelligence a mind without extra bits. What exactly constitutes "extra bits" stuff that is not absolutely necessary to the existence of a mind is certainly not something anyone can offer a definitive answer on. Clearly no human could ever be a pure intelligence because we have a bunch of extra stuff which we cannot go without our bodies for instance. Whether or not emotions are necessarily part of a pure intelligence I don't know. They cannot be easily be dismissed since emotions in large part are the motivational force behind human actions and one might argue that computation without motivation cannot be intelligence.

Perhaps such a thing as a "pure intelligence" cannot, even in principle, exist. After all every intelligence must have some physical basis in order to have a place in our universe so even a computer intelligence must have a "body" in some sense and therefore have some aspects of its existence which are not essential to what really makes a mind. When I say that humans cannot be pure intelligences because we have bodies I do not mean that having a physical shell is what makes our intelligence less than "pure". What I mean is that our bodies motivate us in ways that are outside of what is essential to our intelligence. One might argue that self preservation and reproductive urges could be essential consequences of real intelligence (though personally I find that hard to believe) but clearly many things that are important to human beings are simply visceral in their nature and nothing more. While sex might have the benefit of reproduction and one could eve say that there can exist a purely intellectual interest in it, people do not engage in sex primarily as an intellectual activity. This is generally true of a great many pass times snowboarding for instance is in part an intellectual activity (plotting your course and avoiding trees etc) but this is only a secondary reason why the activity is fun, the actual physical sensation is tremendously important to the activity.

Saying that this means that a pure intelligence can't have outside stimuli is perhaps going a bit too far. One could imagine a very powerful intelligence simulating an external world and a lesser intelligence interacting with it. One could of course say that the simulated intelligence is then not pure and the greater one because it is self contained is but the distinction is unnecessary in my opinion. Furthermore I am not willing to rob a pure intelligence of the ability to dream or imagine.

Human beings were not made for thinking but for hunting and surviving. Hunting in particular requires pattern recognition, planning and flexibility. Survival requires adaptability either in the form of adapting to ones environment or adapting ones environment to suit ones self. Adapting requires creativity you need to be able to understand your environment and imagine a way to change either yourself or the environment. The combination of pattern recognition and imagination with a sufficiently interesting database is probably sufficient for "intelligence".

Whatever the qualities necessary for intelligence human beings are not terribly good at them. We recognize patterns most readily that would have helped in tracking down food and other patterns that proved useful to our survival. Obviously not every pattern that humans are good at recognizing falls into this narrow view but I imagine a disturbingly large portion of them do. Because all of this overhead of pattern recognition and imagination comes at a cost of energy and development it wouldn't make sense for evolution to make us too good at it. Our brains already take up a large amount of the energy our bodies produce and when it comes to tracking prey once you can tell how a rabit is going to run there is no point to further increasing the power of the ability further and further.

Ultimately I find it rather annoying the degree to which things that are essential parts of who I am are not essential to what constitutes my "intelligence". I admit to being a visceral being. I like sunsets and not because they are interesting or elegant but just because I like to look at them. I like being warm and the way that I seek physical comfort from others and what kinds of ice cream I like are parts of me that are not important ultimately to what really makes my mind but they can be important to what makes me who I am. Sometimes I wish I could divorce myself from the physical aspects of myself and exist as a pure intelligence. More often I think that I would like to deepen my physical experience and experience more. I want to know what it is like to be human in the most general sense. I want to grow old and I want to be depressed and happy and I wish I could be female as well as male at some point (somehow I doubt I am alone in that). I want to know what it is like to be bald and to have long hair I want to break some bones sometime just to know what it is like. This deeply visceral aspect of my nature puzzles me but I suspect that it is an important part of what it means to be the semi-pure intelligence that is human.

MAA conference

I just attended my first real mathematical conference this past weekend and gave my first real presentation of my work (read, in a forum not catering entirely to undergraduate research). There are a lot of new things to think about that I picked up at the conference. This post is going to be basically just a long list of stuff that I thought about while at the conference rather stuff that was sparked by the conference. I want to put some of these thoughts down somewhere so that I won't just forget about all of them. Maybe in a year or two I will look back at this post and pick something I had forgotten about up again, Who knows.

I met Thomas Garrity from Williams College and had what I would like to call a conversation with him. This is really rather exciting for me because I do mean conversation, I do not mean that I asked him a question at the end of his lecture and got brief expository answer from him. I mean that there was actually dialogue. That is not to say that I think I was an equal partner in these discussions I fully realize that the flow of information was primarily from one to the other. But what characterizes a conversation (at least one part of the characterization) is that both parties shape the course of the discussion not just one or the other. In a lecture the one person chooses the subject and the other listens and even perhaps asks questions but they do not really have any real control over what is discussed or how. Especially in the atmosphere of conferences and lectures and the like it can be difficult for a lowly audience member to have a real conversation with a high up. When I went to Lisa Randall's lecture on campus during the science and literature symposium I found it essentially impossible to have any more than completely superficial interaction with her. She certainly didn't seem interested with interacting with one of the swarm. Admittedly the mathematics conference was a very different sort of atmosphere since most of the people attending were not physicists.



Over the course of the two days of the conference I talked to Garrity a number of times and there are four things in particular that I will have to give some greater thought.

The most obvious one is what he gave his lecture on, at the end of the lecture he posed an open problem which is does there exist a way to express real numbers as a sequence of integers so that the sequence is periodic if and only if the number being represented is cubic. Cubic here means that somehow it contains a cube root. Continued fractions do it for square roots and decimal expansions are periodic for rational numbers which is basically the impetus for thinking about the question.


More interesting though were the things that we discussed outside of a lecture structure. First off he made me question whether my recent push to find a method of imposing a coordinate structure onto a fractal space is possible at all. I had always assumed that it was of course possible and even had a vague set of arguments why it should be possible. Something I hadn't considered is that many (maybe all?) fractal spaces are not decidable. By that I mean that probably in order to parameterize a space you probably need to be able to decide what points are in the space and what points are not in the space in a finite number of steps. There is no finite time algorithm for computing the mandelbrot set by which I mean that given any general point in the complex plane you cant always in a finite number of steps decide if the point is in the set or not (assuming that the operations on the reals such as addition and multiplication have a finite computational value which you just make 1 step for convenience). This sort of points out to me that I really don't know exactly what parameterization really is. On the superficial level that I was thinking about it a parameterization was simply some function which mapped the set of points of some space onto an ordered set of real numbers. Clearly such a procedure must exist because the cardinality of the sets are the same and so therefore a one to one mapping exists. At the very lowest level that means that you can construct such a map from one to the other in a direct manner just by looking at all the points and making the map one set of points at a time doesn't it? Actually the answer is no, it doesn't mean that you can make such a map because you cant make a list of the points and so if you cant find a way to characterize all of the points in your set then you clearly can't make the map. So in at least one well defined sense of what you mean by "generalized coordinate system" there is a clear reason that there cannot exist a generalized coordinate system for the space of the mandelbrot and probably most fractal spaces. I don't really have any reason to think that this result would generalize to any fractional dimensional space but I do. There might be very nice fractal dimensional spaces that are decidable but I doubt it. It just so happens that this piece of the puzzle fits with what I have been thinking for a while now, that fractal dimensional spaces cannot be handled within the framework of first order logic


This brings us to the next thing we talked about which I will just mention briefly. That is the suggestion by a philosopher of the name Hintikka that the usual first order logic needs to be extended by game theoretic ideas. I don't know if I agree yet or not but it certainly sounds like something that I should look at.


Lastly, he talked about the applications of the partition function to abstract objects in sets instead of to thermal physics. The partition function being something that I am only just really becoming familiar with in my thermal physics course this semester I was of course intrigued. Apparently the properties of the partition function can be very useful in the analysis of completely abstract collections of things.




I also attended a talk on the ordering dimension of partially ordered sets. I personally thought that the talk was not of particularly high quality and the idea was not very exciting since it was merely a slight further generalization of a set of objects called generalized crowns. However the talk did introduce me to the concept of order dimension and made me think a little bit more about what a dimension really is. While the talk didn't actually go into any real detail about the nature of partially ordered sets or posets since they basically assumed that everyone in the audience was familiar with them (bad assumption). I still don't know in a formal way what a partially ordered set is but basically a partially ordered set is one in which there is a partial ordering relation. Lets say you have 5 objects in the set and let @ denote an ordering of some kind of two objects like for instance a @ b shows that in some sense a < b. A partial ordering on the five objects a, b, c, d, e might say something like a @ b, c @ d, e @ d. in which case we would know nothing about the "ordering" of a and d so the set is only partially ordered. Thinking about this made me think of possible connections to second order logics and also more to the point non-integer dimensional spaces. I don't know if this really sticks but it would make sort of intuitive sense that one requirement for a set to have a countable cardinality is for there to exist a complete ordering on the set. This condition is related to being able to write the set as a list since a list constitutes an ordering on the set (the things that come before something on the list is "less" than it and things that come after are "greater" than it). But it is intriguing to notice that having an ordering relation is not the same as the capacity to write a set as a list. For instance the reals have a complete ordering but they cannot be written as a list. I wonder if perhaps things with the cardinality of the power set of the reals cannot have complete orderings. At any rate any set with at least two elements can have a partial ordering on it (as I understand it) since you can just relate two elements and leave all the others unrelated. Seeing how this is related to non integer dimensions is perhaps a bit of a stretch and this post is primarily meant to write down ideas not expound on them in detail (and it is running long as it is) so I will just leave that one alone for now.

Back in Black

Ok so I haven't been raving about black holes enough lately so lets get a nice good rave about turning Jupiter into one theoretically that is what this site is all about after all. Since in my nanowrimo novel I am using a society which has become an interstellar one by use of a Jovian singularity it wouldn't hurt to think about its implications a bit more.

One interesting thought is that the existence of a Jovian singularity implies that humanity became very certain at some point that it needed one. For some reason humanity can do certain very important things with a singularity that would be impossible without one. This means that before the black hole was created we knew pretty well what we could do with it. If we weren't really very certain then we would almost certainly not have been willing to sacrifice an entire planet just to satisfy curiosity.

Of course that also might say something about the government that would be willing to sacrifice a planet for whatever reason. The chances of a government being willing to get rid of Jupiter though are helped out by the fact that really ultimately Jupiter is valuable only as a research tool and as a gravity well that keeps its moons in orbit. As a research tool Jupiter is an interesting study of fluid mechanics and weather patterns and I'm sure of other things as well. As a black hole though it would provide an unbelievable basic physics laboratory. Plus as an added bonus the gravity well would be unchanged and so all the moons could still hang out in orbit. Of course probably collapse into a black hole of Jupiter would send the moons flying. I'm not sure how much energy would be released from the remains of the planet getting sucked in. If the collapse was quick enough it might just be a brief radiation burst and that's the end of it. But my bet is that it would take quite a while. If the moons of Jupiter house any sort of life (or come to by way of human habitation) then most likely we would not be willing to nuke them with the radiation even if the moons would still stay in orbit. But as far as a sci-fi like setting goes having a government that is willing to blow up Jupiter probably means either some sort of really crazy totalitarianism or plutocracy etc. I doubt a real democracy would ever get close to doing something so destructive but then... there are lots of ways people might be motivated to it. For instance if the earth has been thoroughly trashed and an 18 billion population earth is looking for a new place to trash.

Or maybe we want to go all doctor who and say that our future society is extremely power hungry and they think that a black hole would be a nifty way to get lots of energy at will. Actually this is probably the surest perk to having a local black hole. Since a Jovian singularity would be rather small actually it would put out quite a large amount of energy in the way of hawking radiation. But sucking away the rotational energy would probably be more efficient and if we wanted to we could even feed the black hole some gas and get the energy from the radiation which is probably the best way to go about things.

Terminal Length

It has been a thought in the past that there must be some limiting hair length beyond which it would be infeasible for a person to be able to grow their hair. Because your hair has random breakages and can get pulled out by its roots and of course cut. Lets see if we can make our model predictive of some difficult to measure thing. A reasonable model of the hair length of someone who never cuts their hair (me for instance) is probably a good place to start. For what follows let L represent the average hair length and l represent the length a strand coming from a particular follicle. To begin with though we should put down some assumptions to simplify the model first the hair has a constant growth rate meaning d²l/dt² = 0, which I think is a pretty reasonable assumption. Secondly lets assume that d/dr * d/dt * l = 0 meaning that the rate of growth is the same over the entire head, admittedly this is simply not true but it is still not a horrible assumption. The solution to our system without any breakage or shortening mechanism is simply linear growth both for l and L. I proposed three modes of shortening, breakage, pulling, and cutting. Cutting is an uninteresting case as it is merely setting the initial length distribution of the hair and so the models I develop for the other two cases can easily be applied to cutting to obtain solutions. Breakage is easier to model than cutting so I will begin there. A breakage model of hair will look something like (dl/dt)*t - E*t*(dl/dt) = l. I've broken it into two terms to make it clear what I am doing the first term is the growth term and represents the total length in the absence of breakage the second term is the breakage term where E is a randomly determined factor between 0 and 1. The equation in a more compact form is t*(1-E)(dl/dt) = l. Obviously E must be a monotonically increasing function to make sense. Otherwise breakage would sometimes make the hair grow longer. Though there are many ways one might structure E to have the right kinds of properties I am going to make it a randomly increasing recursive function. in a unit time E has a certain probability M of increasing. if the current value is E₀ the value of E when it increases will be E₁ = E₀ + (1-M)(1-E) admittedly this isn't a terribly accurate model of breakage since the amount of breakage each time is fixed but it does have some nice properties for instance the more probable we take the breakage to be the smaller the breakage and the actual length of hair likely to be lost increases with increasing length (though not directly as a virtue of our function of E) I haven't worked out the function L as a consequence of the model yet and it doesn't look pretty more to come in future.

The stories of my return were greatly exaggerated.

So... apparently I am not back to blogging... in fact I suppose I have never really been that much of a blogger, I suppose I don't have the time at the moment to be a really good blogger. I have been applying for internships and doing homework and on occasion I admit I have just plain been goofing off. I haven't made a single new calculation on the prospect of collapsing Jupiter into a black hole... Sad but true. I have been making some other interesting sorts of computations... by that I mean I have been doing homework problems and calculating things like change in entropy the probability of spin states the distribution of electric fields that sort of thing. I hate the fact that I can do the homework even though I feel totally incompetent when it comes to tackling real problems. I am double majoring in mathematics and yet any physics problem that requires me to do a Fourier transform makes me cringe. This summer when I go to my internship (sounds like going to my rest a little doesn't it?) I think I will require myself to do a great deal of drill. That has always been my weak point I have always been able to understand the concept after doing like 2 problems and then I just go and take the test and if I do well then I don't need to do all the homework problems. So I end a math class having done a very small number of the problems and because the homework problems are easier than the real problems and because the test problems are even easier I can get by that way. Isn't that disgusting?

Towards an applet

I think I am going to try and write and then post an applet on my blog see I figure people like things that are interactive and or things that they can just watch. I could make the applet something that is actually interactive... but then that might be too much to ask for my first one to actually put up. I think instead it will just be something like an applet that does something like show a simulation of some aspect of the plan to collapse Jupiter. Later I could make it dynamic so that you can set parameters and watch it do cool stuff but... maybe that is asking a bit much. does anyone know if you can actually even put up applets on a blogger controlled site? I know how to embed applet tags into the html but I don't know how to upload content... uhh... any content at all actually... ok maybe I should think about putting up a picture before I try to put up something as dynamic as a program.

The Conditional Pascal Wager

For those of you not in the know pascals wager is an argument that runs something like this. Let us analyze belief in god as a gambling wager. The expected return of a wager is simply the sum of the possible returns multiplied by their probabilities. We have two wagers that we can place we can either believe in god or not believe in god. Say there is some probability P(G) that god exists. Lets say we place the wager of believing in god then if god does exist we go to heaven and enjoy eternal bliss, an infinite payoff, if we believe and god doesn't exist then we get nothing. Calculating our payoff we have our average payoff is P(G)*infinity + (1 - P(G)) * 0 = infinity regardless of what the probability of god existing is. On the other hand if we don't believe in god and god exists we go to hell and are eternally tortured which is a sort of negative infinity and if we don't believe in god and he doesn't exist then we again get squat. So according to the pascal argument the expected return of believing in god is infinite while the expected return of not believing in god is negative infinity. That was strong enough reason for pascal to decide to become devout and ditch mathematics. Here is a post on Booms blog expanding on pascal's wager. I disagree with the argument in general but it is the impetus for this post so I figured I should at least link to it.

The biggest problem in the reasoning of Pascal's wager is that it is assumed that the probability that god exists P(G) is independent of the event of believing in god. I have used some standard notation up to this point without declaring it but after this point I am going to need a lot more notation so I had better introduce it formally. An event is the set of all outcomes which share some property. For instance if we were to talk about the event that the third flip in a series of coin flips comes up heads then the event that corresponded to this would be all possible coin flips in which the third flip is heads. The function P(X) is the probability of the event X. The notation P(X | Y) denotes the probability of the event X given that Y occurs. This conditional probability is equivalent to the probability of the event X intersect Y divided by P(Y) So the conditional probability only makes sense when Y has a non zero probability. Let the event of believing in god be denoted B and let the event that god exists be denoted by G also let the complement of any event X be denoted by X*. The complement of an event is the set of all possible outcomes in which X does not occur. Finally let the reward function be R(X | Y) which is the return we get upon the event X occurring when we have made the wager Y.
If two events have no bearing on each other then the probability of the combination of their outcomes is equal to the product of their probabilities. If all this doesn't make perfect sense to you here is a wiki article that might help.

In pascal's wager we take the expected return of believing in god to be P(G)*R(G | B). This calculation implicitly assumes however that the probability of gods existence is independent of our choice to believe or not believe. In other words P(G) = P(G | B) otherwise the calculated return should be P(G | B) * R(G | B). Of course the argument would run something like that whether god exists or not is an aspect of physical reality and therefore it is not a probabilistic event and so our belief or non belief could not possibly alter the existence of god one way or another. Either god exists or he does not and believing one way or another cannot change that so the assumption of this independence of belief and actuality seems consistent. However Pascal's wager is a deterministic partial information game not a complete information probabilistic game but pascal's wager is a probabilistic analysis. Chess is a good example of a perfect information game meaning that there is nothing about the game that is hidden from the players. A partial information game is where each player has only part of the picture such as in a game of poker. Poker is not probabilistic because once the cards have been dealt there is no shuffling or random influence in the game you don't know what cards your opponent has or what is on top of the deck but it is not randomly determined during the game. A game of craps is an example of a complete information probabilistic game because at each stage of the game you know everything there is to know about the state of the game but the outcome is randomly determined.

Even though poker is not a random game but rather a partial information game we treat it as a random game because the laws of probability apply because one can use probability to see how likely any particular initial set up is. Thus because pascal is applying the rules of probability to this partial information game if we are to be consistent with our analysis we must treat the existence of god to be something that is truly random. Just like in poker we let anything that is not yet known to be something that is randomly determined during the course of the game not something that has already been determined otherwise our probabilistic analysis would be inconsistent or at least incomplete.

Now let us consider the probability P(G | B). If there is indeed no interaction between god and the universe meaning that the universe would be exactly the same if god existed as it would be if he did not exist. If the existence of god makes no difference in the universe then our belief must certainly be independent of his existence and therefore the original assumption of the wager holds that P(G | B) = P(G) So in essence the pascal wager considers the condition that gods existence has no effect on the world whatsoever. While this might work for some deists I think most people would like to think god has at least some effect on the universe you know a few miracles here and there and whatnot at the very least resurrect a Jesus or two.

If god does have an effect on the universe then things get a little bit more complicated for the argument. Lets say that our belief in god is a function of a number of independent factors at least one of which is the actual existence of god. Since the case where the existence of god and our belief are in no way correlated under the original pascal wager we will assume that the correlation between our belief and gods existence is greater than zero. Since not everyone believes in god we also know that the correlation is strictly less than 1. Since the existence of god does not change from time to time or from person to person that means that the probability of a person believing in god is affected in a constant way by the existence of god. In other words because the existence of god does not vary any particular persons belief in god can be treated entirely as a function of variables other than the existence of god.

So here we have an experiment, We keep the existence of god constant and vary everything else, race, class, gender, sexuality, intelligence, etc. When we factor in all of these conditions and then look at the belief of these people as a cross section then we can use a statistical technique whose name at the moment escapes me in order to see what the effective dimension of the distribution is or in other words how many things the belief is dependent on. This isn't at all where I was thinking I would end up with this but I like that so I am going to stop there for a bit and I will give you the rest of the tour of the original thought I had in a later blog post.

Lets analyze for a moment shall we the idea of physics as geometry. Now most successful physical theories have at their heart been geometries. For instance Newtonian mechanics was based deeply on euclidean geometry and the deep success of Newtonian mechanics is at least related to how well it sits with Euclidean geometry. Electromagnetics is an area where the connection is less strong but because the laws are so well described by fields the theory has become very much a geometric one though in its inception it was not necessarily so. General relativity is entirely dependent on geometry not only for its success but also for its explanation. Rather the reverse phenomenon occurred in the others where the geometry was a side effect of the explanation but with general relativity the geometry was the explanation. Quantum mechanics however is less than strictly geometrical because although we use geometry to describe the wave function of a quantum mechanical object the theory itself is not a geometrical object we merely use geometry as a useful means of calculating transitions. The wave function is only a probabilistic guide to reality any actual event requires a collapse of the wave function which makes the idea of viewing the wave function as the geometry of the real physical situation at the very least quite difficult. It may seem an odd sort of thing to require of the world but I like to think of the world as being a place that can be described by a static geometry. The uncertainty and indeterminism of quantum mechanics are such that this sort of description makes the description of our world by a static geometry impossible if you keep to the usual tenets of such representations. I think that it is possible that one could describe the world using a static geometric reperesentation if one is willing to accept the parallel universes postulate which is I think a very good argument for it.