Below are the 5 most recent journal entries recorded in morbidcoriolis' LiveJournal:

Sunday, June 3rd, 2007
11:33 pm	odd perfects ok we shall start at the very beginnning (a very good place to start) when you read you begin with abc, when you deal with number theory you begin with the primes. (as opposed to do rae me) A prime number is a number that is such that it has only two integer numbers which evenly divide it. These two divisors of course being 1 and the number itself. The fundamental theorem of arithmetic is that every counting number has an unique prime factorization, or in other words a unique set of primes which multiply to it. The second beastie we shall meet on the way to odd perfects is the divisor sigma function. The divisor sigma function is merely the sum of all the divisors of a number. The divisor sigma is usually denoted with a lower case sigma but since this is an e-mail I will just dennote it S(n) Looking at a few small numbers we can easily see that S(2) = 2+ 1 = 3 S(3) = 3 + 1= 4 S(4) = 1+ 2+ 4 = 7 S( 5) = 5+ 1 = 6 S(6)= 1+2+ 3 + 6 = 12 The most obvious way to obtain S(n) is to simply check if each number up to n divides it and take the sum of the ones that do. But this gets rather intensive rather quickly and there is a much easier and more useful way of doing things. We take a moment to note that S(a * b) = S(a) * S(b) if and only if a and b share no common factors. (a common factor by the way would be a number other than 1 that divides both a and b if two numbers share no common factors they are said to be co-prime, for instance 15 and 28 are co-prime because even though neither 15 nor 28 are prime they share no primes in their factorizations 3 * 5 and 2^2 * 7 which implies they have no common factor) ok now for the perfects part, a perfect number is a number which satisfies 2n = S(n) such a number is 6 since 12 = 2*6 = S(6) ta da. a number whose S(n) is greater thant 2n is called abundant while a number whos S(n) is between n+1 and 2n is called deficient. Of course it is a little more natural to just ignore n when finding the sum of the divisors of n in which case these classifications make a little more sense since then a number is deficient if its sum is less than itself and abundant if it is greater than itself and of course perfect if the sum is equal to itself. Obviously a number of perfect numbers exist, in fact at least 40+ perfect numbers exist one for each mersenne prime. It is not known whether there is an infinite number of perfects but that is likely the case. Anyway it is not known at all however whether even a single perfect number is odd. obviously an odd perfect number cant have a 2 in its factorization or it wouldn't be odd! Since all the numbers in its factorization are odd we have some product of odd primes n = A1^a1 A2^a2 A3^a4..... Aj^aj and so on consider for a moment that each prime is coprime with the others so we can split the determination of S(n) into S(A1^a1) * S(A2^a2) * .... * S(Aj^aj). All of these sigmas are of the form of the sum of a number which is a power of a prime number. The only proper divisors the power of a prime number P^a are all the powers of the prime P^b with b<=a if the number werent a power of the prime then it wouldn't divide P^a and so wouldn't contribute to S(P^a). Meaning that S(P^a) consists of a sum of a+1 odd numbers (since of course p^0 or 1 is in S(P^a) too!) and so if a is even then S(P^a) will be a sum of an odd number of odds which will itself be odd. But if a is odd then S(P^a) will be even. Going back and taking this piece of information to look at the odd perfect again we see that if more than one of the Aj's had an odd power then it would mean that S(n) would be divisible by 4. But that would mean that S(n)/2 would be even. Which is of course a contradiction since S(n) / 2 = n and we started with the stipulation that n be odd. Therefore n has a single "special" prime in its factorization whose exponent is odd. The rest of the exponents are even giving us the rather interesting insight that n = P^(2m -1) * K^2 (1 Comment |Comment on this)
2:42 pm	5 days left I really have no reason to be writing this other than I haven't updated my lj account for like a full year so I figure it is time that I put something here. Right now the only thing I feel like writing about is that jenna is going to be back from japan in 5 days and I don't know what to do for her. I was going to make that piece of pine board she gave me into a go board and give it to her but I wouldn't be able to make very good pieces to go along with the board and that sort of sapps my enthusiasm for the whole process. I still think it would be a good idea to make her a little gift from out of that board but I am not sure what. She doesn't have any necklaces at all and I contemplated making her a little necklace using that bismuth I have. But I don't know how well it would go over or how much durability such a necklace would have. I guess this entry is meaty enough to deserve to be posted now. (1 Comment |Comment on this)
Thursday, June 22nd, 2006
9:22 am	prime eight you know it is interesting how easily people will let puns slip by them without taking notice. there is so much in the world that people find hilarious when pointed out to them as funny but which they let slide as totally normal if you let them. I suppose this is one of the values of human interaction after all a joke isn't nearly as funny if you are enjoying it by yourself as if you are sharing it with a friend or even inflicting it on strangers. I suppose that humor is essentially a social construct even though people must develop an individual sense of it. In this way it paralells ethics since the ethos of a culture can only be constructed through consensus but it can only have instances that are specific. Though the sense of humor might be somewhat less important than a sense of ethics it makes me wonder all the same if that deviation of the sense of humor might have equally dire consequences in its deviation. After all there must be some sort of steadying influcence that brings the individual senses of humor into line with each other in a society or else they would become so disparate that sitcoms such as friends might never have a chance to become popular. I suppose that simple human interaction would be enough to cause the propagation of the central tennets of a societies sense of humor and it is not necessary that everyone share an identical sense in order to find many things funny in common. However I think we find many things funny not because of our particular sense of humor but by agreement that some things are meant to be funny and therefore are percieved as such. For instance while watching sitcoms we can laugh our heads off at the predicaments of the fictional characters on the screen even though if things like that were happening in real life we would almost certainly find it innapropriate to laugh. This however certainly cannot account for the whole mechanism of the humor of such shows though since that would not account for the general appeal of shows like friends and their differential popularity. Similarly we must remind ourselves that often times while looking back at extremely unfortunate or uncomfortable events in real life we can recount them with mirth and laughter. Perhaps then laughter is also a sort of survival technique that allows us to relax the effects of a stressor once the danger has passed or even while it is still present. People have been known to break into hysterical laughter when confronted by extreme stress. The laugh response might be a way to resolve tension since it certainly serves that function often in human societies when people laugh nervously at each other. Often times a girl will laugh at something she personally finds merely uncomfortable to think about and not funny in the least simply to relieve the tension caused by the expression of that idea. Even if laughter evolved as a tension reduction device however does this completely account for its functioning? Within this framework it seems that we can explain a great deal of laughing behavior since everything from laughing at sitcoms afterwork to laughing at a friends jokes is a relaxing thing to do. However this seems a bit like saying that a thing is caused by its effects. After all if we evolved humor as a way to relax then it is not surprising that it causes us to relax! The question then of whether it has come to be more than that is still an open one and for that matter even if that is why we have laughter in the first place. (Comment on this)
Wednesday, May 24th, 2006
3:49 pm	The nature of originality You know it is one of the primary driving forces in my life to be original. Perhaps it stems from some sort of psychological imbalance and if I were really stable I would be able to sit back and be comfortable as who I am without feeling the need to be different from other people. But I think that even if it signifies something unstable or even unhealthy in my psychology it is still desirable. That drive to be different will eventually lead me or drive me into a realm of the mind that no one has ventured into before that will bear fruit. The drive to be or at least feel different from other people is definitely not something that I alone have and I see other people express it faily commonly. Sometimes a drive to be different can just be a drive to fit in, in disguise. You might express a philosophy that differs radically from the mainstream but if that philosophy is based more on what subgroup in society you belong to than to an active analysis of conditions then that philosophy is probably not really your own. Rather you are just trying to fit into a subgroup by accepting their ideology. Now on the other hand people also have ideas which are purely reactionary and this is striking somewhat closer to the differentness drive as I am talking about it. When people avoid anything that is mainstream just because it is mainstream and dislike the status quo because it is then that is a reactionary sort of difference. But even here your ideology is in some way based on what is mainstream since it is formed around a reaction against that. Therefore you are not really independent from the mainstream and therefore not different in an independent sense. This is sort of like what I think about when I get up in the morning and am about to put on my clothes. I think about what is the easiest to get and what is moderately clean. Every so often I will think about what image my clothes are projecting to other people and this bothers me. I am dissillusioned by the fact that I actually care how I look on some level because this means that I am not independent of societies expectations on my appearance. But now I question my ideas that in order for something to be really original or unique it needs to be totally independent of the things that came before it. Of course varying degrees of originality can be achieved within the framework of the things that came before but then is that true originality in its purest form? I think that originality can be given a more general sort of view than merely independence from other ideas because there seems to be a sort of force in human thought processes that remove people from traveling the paths of the state space of their lives that are really possible. Most people are so similar to each other that I find it somewhat disturbing really. I looked at a message board recently which every student for a class I was in was supposed to make a post on. That was the assignment just make a post any post at all. 93% of the 500 posts that I read were of the form "check me off please" only 7% of them or about 35 out of 500 represented what I deemed to be "original" responses. Why then did these people feel compelled to make their messages occupy such a small portion of the state space possible for their messages? I mean the state space for the messages that they could have posted of that same length is about 10^18. yet only 7% of the messages differed from this format appreciably. meaning that the state space of the messages were all clustered very closely. Even if we account for the fact that human language causes clustering in and of itself within the state space we are left with a residual clustering effect. Using a simple first order markov process based on word frequencies distributed in a zipf's law fasion I can estimate the size of the state space for a message of 4 words in length to be about 6.2 * 10^13 (Comment on this)
Sunday, May 21st, 2006
2:14 am	human potential I was reading a book that mentions the idea that the reason that the athenian culture was so well acchieved is because their culture made the average person believe they were capable of excellence and then demanded that excellence from them. This has made me start to wonder about the nature of human potential not just on an individual scale but the idea of potential for a group. Is the potential of an individual limited by the group in which they exist? So for example is it possible for a person who is a true genius in mathematics to grow up entirely in the company of people who cannot do simple sums? If this sort of independence of group is possible then that bodes well for the prodigies which have been raised up out of the depths of poverty and so forth but the question can be asked would those prodigies have been even more incredible if they had grown up in essentially the same circumstances but in a society that was more adept at the thing in which they were prodigies. For that matter is the society of a prodigy something that produces them or would they be prodigies regardless? I have seen a small amount of psychological research on the subject and I suspect that a prodigy needs to have at least certain rudiments of society before they can become great but that they do not need to have excellence in their relations with other people. But then the potential of the prodigy is sort of only a side issue after all since what is really important is the ability of the masses. Different cultures produce different aptitudes of populations (though one might argue this is caused by the the educational systems alone education is merely a subset of cultural interaction). The culture of america is one that is very strong in the sense of being overbearing at least and american culture is something that america sort of tends to export without thinking about it too much. However inside america that culture is beginning to collapse a little bit. Education is becoming increasingly a problem and other countries consistently out perform american students. One might ask if this is simply a product of the way in which our culture stresses qualities which our system of advancement both in education and business don't find appealing. A quick look at modern successful movies will readily demonstrate the american tendency to side with criminals and theives take oceans eleven or the Italian job or whatever. The trend is not new movies that laud thievery or violence or for that matter any kind of behavior that will quickly lead to a dead end in application in real life are very common. One must ask then if culture emphasizes and lauds one kind of behavior in the media but then attempts to get people to attempt a different kind of behavior in their every day lives then we are robbing ourselves of the ability to have culturally inspired excellence on a very large scale. Current Mood: contemplative Current Music: where eagles dare (Comment on this)