ElijahJones

The equation sum(a^1,i=0 to n)=(a^(n+1)-1)/(a-1) is not the same as the euqation I wrote above. Look at it carefully and you will see this to be true. You can substitute your factorization of the gemoetric sequence (which everyone knows by the way) for the sum on the left but the fact remains that the formula I present is factorization formula with far reaching implications and it is related to the cyclotomic polynomials. Look again at the formula. It is a factorization of the geometric series while the standard formula for the sum of a gemeotric series is rather a factorization of the term a^(n+1)-1. In fact the standard formula tells us that a-1 always divides a^(n+1)-1 for a>1 and is a result applicable to the real numbers. I am only considering my formula in relation to integral values of a greater than or equal to 1. Notice that in my formula if a equals one we reduce to the familiar multiplication tables for integers. This suggests that the result is somehow an extension of the familiar multiplicative structure on the integers, which in fact it is. The difficulties in extending it further have to do with the notion of the outer product, polynomial multiplication, and information that is lost in that step. But I am working on some p.d.f's and algorithms that will look into that. The fastest way to factor any number is going to be in base two and at best you are going to get log time, this seems to follow from the theorem when a=2. Consider this for odd integer N and in base 2, N = (2x[1]+1)(2y[1]+1) Suppose x[1] and y[1] are two big to guess simply, then write x[1]=2x[2]+1 or x[1] = 2x[2] y[1]=2y[2]+1 or y[1] = 2y[1] There are four choices of form then for the ordered pair (x[2],y[2]) but for all of them x[2] <= x[1]/2 and y[2] <= y[1]/2 so the algorithm will end in Log[2](N) steps but because we have no way of guessing which of the choices searching the treee of possibilities quickly becomes untenable for large N. The formula I gave here defines those polynomials in a for which the process is simple because polynomial multiplication proceeds in a special way. Hope that helps.

The points are good ones, thanks that settles that it has application in the real world. Astronomy also has application in the real world but not every venture of astronomy is essential. My point is that science should not roll of on mad adventure of useless inquiry when we have so many real world problems that need to be addresed. I know all the arguments about how you never know what might tutrn out to be important and all that, yes. Oh, your argument that because so many universities have astronomy departments implies that astronomy must be practically applicable is not correct. Astronomy is useful though, cosmology is not a necessary study, not unless you can convince me that the Earth's imminent demise is at stake. Anyways I am not going to argue. We all have our little things that are 'SO' important. I'll check back in the Spring and see how you guys are getting on. Good Luck, EJ

Ok, ok its important. So you are saying Swansont that there are say biological systems that because we think we know how they have evolved we can better predict which medications will work to treat illness? I am a little vague on the specific way that a knowledge of evolution is essential to our current goals of desease prevention, the advancement of medicine, understanding the human genome (which will probably not evolve in the time we are going to be putzing with it). In general isn't it kind of like saying that cosmology and GR are essential to a civil engineer. They are not, Newton works just fine. Yes, Mokele, some process gave us our big brain that lead to our social tendencies and development. Is knowing what that process was and exactly how it happened essential to understanding human society? I can see where it might be important if evolution were seen to constrain certain needed social changes. I will write down two questions in order of their importance and then expect a nice tongue lashing, but at least we are having a healthy debate. 1) What exists right now and how does it work? 2) What is the relationship between the past, present and future? So if we cannot give at least one specific reason for spending lots of money and time on broad elegant theories maybe we should stick to number one. Do you have one specific example of how evolutionary studies are currently essential to a real world problem affecting the immediate future of mankind? Remember I do believe in evolution as a process that is acting now as surely as I believe that there are black holes gobbling up galaxies somewhere in the universe. I am just not entirely convinced that it should be anything near to a research priority at this moment in time or that it should become the premise of some type of reverse witch-hunt among people whose knowledge of it is useless to them. They donlt get it, they want their kids raised to think the earth is six thousand years old, good. As long as their kids contribute to society by pulling levers down at the local factory, more power to them. Let ID, GD, SD, MD all be options on the school plate. Those who are evolutionarily suited to grasping the truth will get it. Or can we thwart evolution by teaching our kids they are cheerios and the world is a bowl of milk? I think my point comes out pretty clear. Mess with the natives long enough and you eventually find out that you are outnumbered the natives breed much faster than you do. Cheers, EJ

My thesis is on math models of ecological succession but it looks at this from a few perspectives. There is the old Markov chain models which suffer from not being very realistic in terms of the processes that cause the change. Then there are the Lotka-Volterra type competition models, which can at least contain a term for any important quantity. My real goal is to build spatial models that incorporate all of the important aspects, matter cycling, energy flux, competition, hierarchy. The models will have to include both deterministic and stochastic processes. I am learning now how to do finite element method. Its hard to find an easy to understand textbook though and I have a degree in mathematics. Probably the big X-factor in the whole thing is the microbes. We just seem to know so little about where they are and what they do even how many there are, but screwing that system up can have huge impacts. Like for instance encouraging a certain type of microbe through warming climate can stimulate CO2 emission from the soil which then feeds back as more trapped sunlight into a thousand different positive and negative loops. The system is nudged into a new trajectory just by that forcing function of GHGs and it could be very sensitive to our particular way of polluting. Well so in forest succession if the temperature is rising some species are facing selective pressures and may disappear, but if they were part of an assemblage that evolved together their decline will effect perhaps every other species in the forest to some extent, changing the future population distributions thus impacting even the course of evolution(apparently).

I'm starting to get the vibe that you guys have this all figured out. Mokele you example is a good one but it does not need to be esoterically worded (not saying it is). Basically you collected data on current populations, then you used a theory to assume some traits of fossil population and then you looked at the fitted curve that seemed most reasonable or you used an ANOVA test to determine whether the treatment between two samples was significant or the change over time was moving in the direction your theory suggested. I have never said I don't believe evolution is happening or that it is not happening largely as we believe. However inference from data is almost always less certain when you have less data. So those parts of physics that do not allow mucho data are less certain. The fossil record is vast by some estimates but scant in other important ways (don't ask me to say how I read it a while ago) so evolution is probably less certain (some claims at least) than gravity which is constantly observed. So my point is that inference can be rational and therefore sharp or it can be statistical and therefore a little broad, but rational models are not valid unless they can be significantly linked to real world data. This requires statistics and hypothesis tests and confidence intervals. What I would not mind seeing is a list of the top five reasons why evolution matters to a world with so many troubles. Economists (at least a few of whom I know by the way) can say that supply and demand will affect the amount of money in my wallet. What does evolution have to offer to humanity today?

Yeah I think it is easiest to simply realize that you are evaluating a constraint (sort of like a boundary condition) f(x)=ax^2+6 g(x)=x Such that f'(x)=g'(x), so 2ax = 1, x = 1/(2a) This is the information we gathered from evaluating the constraint. But look we are really free to pick either a or x to suit our fancy. Nowhere does the problem stipulate that f(x) should equal g(x) only that at some point the slopes are equal. So lets say we want a parabola that has slope one at the point x=1 then we get 1=1/2a, a = 1/2 So yes (a,x) = (1/2,1) is one solution, but for every x there is a corresponding a. In fact this is really a rather clever question. You can see that the real solution is a hyperbola by graphing x = 1/(2a). You could also think of this as a map that associates to every value x, a function f(x)=ax^2+6 Oh yeah also look at the parabola ax^2 + 6. The slope at any point is not affected by adding six (we know this from the derivative) and the parabola is still symmetric with respect to the y-axis. So if you want a parabola that has a slope of 1 at x=-1 pick a =-1/2. The reason this is interesting is that it flips the parabola upside down. This type of change with respect to a parameter is called a bifurcation, meaning the object is quantitatively different depending on whether a certain parameter (in this case x) has crossed a certain threshold (in this case zero). In more complicated systems of differential equations important properties of the system can change dramatically as parameters vary. I know that is far beyond the level of this question but maybe it will peak your interest.

Look at it like this. The space spanned by your set of vectors is really the xy plane. your space is an xyzw four space. Okay but lets pretend you space was actually and xyz three space, the projection of (1,2,3,0) onto the xy plane is (1,2,0,0) and has length SQRT(5). Can you visualize this? You can actually make a nice diagram of this. Unless something really spooky happens when we move to four dimensions the projection of (1,2,3,4) onto the xy plane is still (1,2,0,0). You can convince yourself of this more by arguing that the projection of (1,2,0,4) onto the xy plane is also (1,2,0,0). The tricky part is putting it back into the basis you have. So you have to form a linear combination of (1,0,0,0) and (3,4,0,0) that is equal to (1,2,0,0). x*(1,0,0,0)+y*(3,4,0,0)=(1,2,0,0) x+3y = 1 4y=2 y=1/2 x=-1/2 (-1/2,1/2,0,0) in the basis you were given. You can draw this too and verify why the first element is now negative. A bit rusty on linear algebra but I think that is correct. EJ Note: my method here was to assume that (1,2,3,4) was taken over the standard basis. That the point was to find the projection to the xy plane then transform it into the basis {(1,0,0,0),(3,4,0,0)}. The question really depends on what basis you have.

Yeah, this thread seems to have gone from poking fun at our own extreme tendencies to some kind of whacko extremist mele (spelling?). Who cares if you eat vegetables or not? the only time eating an animal bothers me is if I know it personally. That's why I don't eat dog, I know how close a person can be to a dog. I suppose if I had a child and we were raising pigs and the day came to make some pork chops and my kids said "Daddy, that pigs my friend!" I would understand. But it would not stop me from raising pigs and eating pork chops. I like the quote of the pelican in 'Finding Nemo'. He meats Nemo and apologizes if he ever tried to eat him and says 'Sorry if I ever took a snap at you. You know fish gotta swim, birds gotta eat.' Jim Crow? What?

Come one now! By testable he obviously means by experiment and no evolution beyond microbes is not testable, it is inferred and is a strong inferrence given the apparent evidence. Verifiable, I'm not sure what he means, probably whether we can prove evolution is true and in fact no you cannot prove it is absolutely true. But it does appear to be the best theory based upon science and reason that fits the observations (fossils and limbs and...). I am certain that if we could get a precise record of how evolution proceeded we would find many surprises. I am also pretty sure that evolution is the major force that has shaped life on earth within the constraints of the physical universe and the peculiar environments of earth. But it does not rise to the level of gravity or electromagnetism on the scale of scientific certainty, although it is more scientific than say economics for various reasons. Interestingly we have come to a level of development where our actions are overriding or should I say altering the course that evolution would have taken humanity. An example might be: notice if you will that many humans today need glasses. In a world where eyesight was crucial to not being eaten people like me (I'm as blind as a bat) would not survive very well even in groups. The genes for poor eyesight, bad reflexes, obesity, cancer, probably many more would not survive. But they survive today because we have used our minds to overcome some of those frailties. So this leaves us with a very interesting question. How can evolution produce something that supercedes evolution? EJ

No, I am not overestimating mathematics. I do agree with your point though, but I guess what I am saying is that we are engaging in genetic research with the hopes of understanding DNA. If its just going to come out as a new way to try to cure the common cold we could skip the whole genetic thing and start testing cocktails of jungle herbs for much less money and technical strain. no the goal of DNA research has to be to gain as much understanding and ability in genetics we possibly can. Will this come ot the point of custom engineered genomes and dinosaur wanna be's living in real Jurassic Parks. I don't know but I have seen some of the things that are done in mathematics and they are much more powerful and complicated than alot of people realize. I say this with a sense of realizing the issues here, we have a scale issue. Trillions of combinations to try even if we can cleverly narrow the field. Then of course is the complexity of all those trillions of choices interacting, which as you point out Mokele is of a whole other order of magnitude. The problem itself however is probably not a shard to describe as some make it seem. Basically the genes are a blueprint and a timeline. Understanding the timeline and event triggers is just as important as understanding the blueprint for individual proteins. These things are intersting to me because I am a grad student in envinromental science working on math modeling of ecosystem change. The obvious question once one starts to feel good about that is how does evolution interact with ecosystem change, is it of a completely different time scale or can evolution appear as a factor within a model? This of course leads to the issue of genetics etc. How do genes mutate, how fast, how do creatures change in response to changes in genes? Someday I hope to look at the whole issue from a mathematical perspective but the next few years are tied up. Although I would not claim to be an expert in DNA by a long shot as an area of scientific inquiry it cannot be beyond a scientist to say something useful about it. Have you ever taken a look at the subspecies of Cutthroat that live out west, there are like 15 different varieties and each one has a slightly different morphology. It seems like Darwin's finches live right here in the US, only they are fish. The watersheds they live in were cut off from each other after the last Ice Age, and in ten thousand years one and only one narrow band of morphological changes has taken root in each subspecies. It cannot be random can it? Perhaps it is a response to the unique chemistry of each watershed changing the original species (the common ancestor)? How does one account for the fact that fish of an identical species seperated for ten thousand years all developed into distinct subspecies, but only one per watershed? What factors preserve (or alter) morphology in the face of random genetic variation? Cheers, EJ

As much as it pains me to admit it, you are right. If 4 divides 7 then 7 is even If 4 divides 2 then 2 is even Are true because they are examples of If 4 divides x then x is even But notice you cannot form TRUE and FALSE = FALSE Also what happens if we do this If 4 divides x then x is odd Now of course the statement is always false no matter what the status of the prepositional parts. The cases that arise are chracterized by If 4 divides 2 then 2 is odd (False and False = False) If 4 divides 2n+1 then 2n+1 is odd (False and True = False) If 4 divides 4n then 4n is odd (True and False = False) TRUE and TRUE cannot be formed. I must humble myself and ask you to forgive me. But I still think it is arrogant to deride people who request explanations for the things you say. Noone is to be so trusted as to be taken without question. It is at just that point that pride leads us to a fall.

Your arrogance is starting to bother me again. Sorry studly but the proposition cannot be true by fiat, it must be determined true by examining the clauses. Insert x = 2 and you get If 4 divides 2 then 2 is even. Since four does not divide 2 it is impossible to arrive at the conclusion that 2 is even. The statement is bullshit. Insert x = 7 and you get If 4 divides 7 then 7 is even Again the statement is bullshit, and I gurantee that if you develop proofs bye taking this as a true statement you will be producing bullshit. This probably explains why much of modern mathematics is useless to the real world. Since you are so adept at proving things perhaps you can derive us a solution to the oil crisis. If I use your reasoning I could start with, "If elephants are pink then invading Iraq will solve our energy problems." By the way the wall paper on your website has some symbols that appear to meaningless. Its sort of like you said f(x)=f(x).

I was just on my way back to correct that P({})={{}} so the cardinality is one. I'm not sure whether to be insulted about your polya comment or not that you would care. What would be the value of having a different model of set theory?

How is it that photons interact with electric charge?