Bignose

Halls, I didn't misunderstand anything. It is a driect response to To which the answer is definately no. I don't think that you can decompose any given vector into two crossed vectors. Certainly not uniquely, look at the cross product in Cartesion coordinates: Let w = u x v, then the components of w are going to be given by: [math]w_x = u_y v_z - u_z v_y[/math] [math]w_y = u_z v_x - u_x v_x[/math] [math]w_z = u_x v_y - u_y v_x[/math] given that the components of w are known, you still have 6 unknowns and only 3 equations. OK, you say, so, let's fix one of u or v, well, now the fixed one -- let's just pick v -- is a regular vector. That is, it will change sign under coordinate inversion. But, the other one is going to have to be a pseudovector so we get the correct w. And, the now the properties of pseudovectors apply, namely: [vector] x [pseudovector] = [vector], so we're back to where we started, with a regular vector. So, again, I say (without confusion), no, not all vectors are pseudovectors.

No, every vector is not a pseudovector. A position vector is a classic example example of a non-pseudovector also known as polar vectors, because it will change it's sign upon inversion of the coordinate axes. A pseudovector won't, like your vector p there. Usually, pseudovectors arise from descriptions of some sort of rotation, like the vorticity in a fluid, or angular momentum, or torque.

Sure, but you have to be more careful with the parentheses: [math]\log{a^k} \neq (\log{a})^k[/math] Like was said above [math](\log{a})^2 = (\log{a})*(\log(a))[/math] not [math]2*\log{a}[/math]

Mathworld ( http://mathworld.wolfram.com/KleinBottle.html ) lists a parametric representation of a Klein bottle where the parameters range from 0 to 2*pi. Does that count?

Fred, may I suggest you start your own blog? You clearly are unsatisfied with this forum's responses, and if you started your own blog you could post all the pondering open-ended questions you wanted without having the moderators here closing your threads. Maybe you could start your own forum, too, as another option.

superman, here is the basic question: Once some energy has been used to move water from a higher level to a lower level -- that energy being used to generate electricity -- how is the water going to get back to it's higher level? That is going to cost energy, too, and it will be at least as much energy as you got from the water going from a higher elevation to a lower one. And, if you try to convert that energy to electricity, there are going to be unavoidable losses. So, you end up with less energy than where you started. So, the critical question is, how do you think you can overcome this? What loophole do you think you have found that hasn't been found yet? Where in this picture is the "free" energy? Like I said, I'm not holding my breath.

This is the key sentence right here. You should be able to build a model and show that it works -- you will throw physics completely on it's head, however, since everything we know today tells you and us that it is impossible. That's why you won't receive any funding until you can prove it works. A bathtub scale model shouldn't be too expensive to build and try and prove to yourself that you can't get energy out of nothing. Or, you can draw a diagram and show us, and we can probably show you why you won't get free energy out of the system. But, most people claiming miraculous results won't post diagrams of their ideas because they think someone will steal them, so I don't actually expect you to, either. That's why you're going to have to build a model and going to have to submit it to a great deal of thorough, objective testing. Best of luck, I hope that you'll forgive me if I don't hold my breath.

I would think that any textbook for computer engineering and/or electrical engineering would be chock full of information like you are looking for. I am sure any well-stocked university library would have several to choose from.

Norman in on the right track here. It is a knuckleball effect. To first understand it, you probably need to look up the Magnus effect, the lift force that affects a rotating object. Then, think about what happens when an object, that is not perfectly symmetrical and smooth, rotates: effectively, the non-smoothness gets "blurred" out, because the rotation ensures that all the imperfections in the ball gets exposed to the different parts of the flow field. I.e. the rotation brings a seam to the leading edge, then a non-seam part, and then another seam, etc. Now, think about what happens when a non-smooth ball does NOT rotate. The leading edge will be a seam until a little gust of air pushes that aside, then the leading edge is a non-seam part. The ball seems to tumble, or as you described it, float. In baseball, a good knuckleball doesn't have zero spin. A good knuckleball actually will complete about one and half rotations before it crosses the plate -- the idea again being to expose different leading edges to the front of the flow to cause more tumbling and knuckling action. But, at the heart of the matter is drag, Magnus lift, and the fact that no air flow is ever perfectly uniform -- there will always be little vortices to cause the balls to tumble.

I think you need to be using the Leibniz rule for differentiating an integral: http://mathworld.wolfram.com/LeibnizIntegralRule.html I'll try to take a look at the problem a little later today... Edited to add: Actually, a pretty neat trick would be to put in the series approximation of e^(-x^2) into the integral first, then do the integral, then do the differentiation.

If you are just looking for the series, you can check out MathWorld's page: http://mathworld.wolfram.com/MaclaurinSeries.html As to the specifics of how to calculate it, what steps did you do? Post some details and we may be able to diagnose where you made your mistakes.

The closest packing circles of equal radius can have is [math]\frac{\pi}{\sqrt{12}}[/math]. The closest packing spheres can have is [math]\frac{\pi}{\sqrt{18}}[/math]. You might really like the article "Cannonballs and Honeycombs" by Hales in Notices of the AMS, vol 47, 2000, because it has a neat discussion of these issues and other shapes being close packed.

Right, so you see where difficulties come in? You can multiply and divide by 2t to get the du term, and then try some integration by parts.

hobz, do it and try. The best way to learn is to try to do it. I'd suggest trying to implement a change in variable, something like x=t^2 and go from there. Feel free to ask any further questions along the way.

It has an antiderivative. d erf(x) / dx = e^-(x^2) (there is a 2 over the square root of pi in there, too). It just may not be in a nice looking form, but erf is a function just like all the other ones. Is it unsatisfactory that d sin (x) / dx = cos(x) ? Similarly to the error function, the gamma function ( http://mathworld.wolfram.com/GammaFunction.html ) is the antiderivative of terms that look like t^(z-1)e^(-t) It isn't going to come out in terms of e's and polynomials -- that's why the error function and the gamma function were invented. But, they are both functions just like sine and cosine. They are well defined, and obey lots of rules and are exceptionally well-studied. There are a whole host of "other" functions out there. Confluent hypergeometric functions, Bessel functions, elliptic integrals, Mathieu functions, etc. In general, whenever a problem wasn't amenable by a straightforward approach, someone invented a function to solve the problem. That's where erf came from.

ajb answered your question. You have to use the error function. The error function is just another function that sits in your toolbox ready to come out when needed. It is like sine and cosine and sinh and cosh, but it is the error function. It is the integral of e^-(t^2) terms. If you study diffusion problems, it will come up all the time (not coincidently, because the normal distribution and diffusion are closely related).

I don't think I've ever seen relational operators used on vectors, and don't exactly know what would be meant by v > w. Maybe, somewhere out there, someone has defined the relationships, but there is no such thing in common use today. Like you said, you can always compare components of each vector or their magnitudes, but nothing general has been defined that I am aware of.

I'd still like to know how writing out a 5th order, highly non-linear ODE is better than having just the original polynomial. Both will still have to be solved by computer, if a solution exists. So, I don't see how one is better than the other...

Even if it did, what the heck is the point in making a 5th order, nonlinear ODE to solve something like this? It will have to be solved via computational method. That mess on your website is not going to have an analytic solution. Unless you know of one? It just looks awfully unlikely. So, then, what is the point? You can use a computer to solve for a quintic polynomial pretty fast. Again, like I said in my other post, if you are just posting these things so we click on your links, this seems awfully spam-ish to me. Do you have a larger point, square? Do you have something you want to discuss on the forum?

OK, so, I can't be to only one wondering this. But, square173205, what is the point of these posts? This is probably futile, but, do you want discuss any of the things you post? Are you looking for critiques? Are there questions you want answered? Because, if you are just looking for us to click on your links, I would consider this spam. In short, what is the deal here?

There are actually two different ways a fluid can interact with an object such as a ball to cause lift. There is a spin-induced lift and a shear-induced lift. The spin induced lift is what has been talked about above, this is what happens when the object itself is spinning. The fluid flowing around the object is made to turn, thusly fluid that was approaching the object head-on is made to leave the object in a direction different than 180 degrees from where the fluid approached. Because the fluid doesn't leave in the exact opposite of the way it approached, there is net force. This is why a curve ball breaks in baseball, and a golf ball hangs in the air far longer than if it were dropped. You can read more about this by looking up Magnus force, named after one of the first researchers to describe it. Wikipedia is a decent starting place: http://en.wikipedia.org/wiki/Magnus_effect The other lift is caused by velocity gradients in the fluid motion. That is, if the fluid near the top of an object was moving faster than the fluid at the bottom of an object, then the top of the object would experience more force by the fluid than the bottom, and hence the object is caused to move in a non-square direction -- again called lift. A particle sitting on the a flat plate that has fluid flowing over it experiences this kind of lift -- the particle is lifted off the plate, leaves the boundary layer fluid flow near the plate and enters the bulk flow away from the plate. However, in general, the spin induced lift is far more important than the shear induced lift. Both definitely exist, though, for any object moving through any fluid. So, to answer the OP's question, yes, the rotation of the ball definitely influences it's trajectory. Compare a slider and a screwball thrown at the same velocity. Both start in the same direction, but because they have opposite spins, they break in opposite directions.

I tried to view your website, as insane_alien said, I couldn't get it to work. Nevertheless, my main point is that without a formal proof, it doesn't matter how many examples you create, that isn't "proof". There is a formal proof of the central limit theorem, any good book that goes beyond introductory statistics and probability will have it. Or you can look at http://mathworld.wolfram.com/CentralLimitTheorem.html for a proof. If you can provide something similar to this same level of rigor, then you've got something. Just a bunch of examples has a little meaning, but nothing conclusive. Log on to Amazon.com, and search for "counterexamples in probability" and there are no less than three different books that a full of examples where "intuition" is dead wrong. Intuition can be valuable, but it can also be exceptionally misleading. That's why rigorous proofs are needed, not just examples, even if there are a lot of examples.

I couldn't get the java to work either, but just from the text of your post, I'd like to make one comment. But, just using "experimental verification" is insufficient for proof. As uncool stated, you need to show it in mathematical terms. Intuitively, there is probably some sort of central limit theorem for modes, but intuition can often be wrong, that's why formal proof is needed and not just "experimental verification".

The gamma function is the extension of the factorial to complex and real numbers. [math] \Gamma(n) = (n-1)![/math] and the Gamma function is defined as: [math]\Gamma(n) = \int_0^{\infty} t^{n-1}e^{-t} dt [/math] You can see a lot more at Mathworld: http://mathworld.wolfram.com/GammaFunction.html

That's the general idea. If you are a being whose meals may very well be few and far between, you would want that rare meal to be as beneficial as possible. So, you would prefer a meal that has as many calories as possible. Fats and sugars are prime candidates here. And, we can't quite undo all those years of evolution that told us that calorie-rich foods were awesome, even though now meals (for the average middle-class American) aren't few and far between.