DrRocket

There has been speculation along those lines. But the plain fact is that while the theory is pretty good back to about 10^-33 sec, no one has a clue what happened at the actual moment of the big bang, t=0, and we have no theory remotely capable of addressing that question.

Better yet analyze the specific problem and understand it without an electronic crutch.

Spacetime is not naturally flat. In fact spacetime is flat only in the total absence of gravity, which is nowhere. Euclidean space exists in the mathematical sense, and is necessary for the very definition of a manifold, and spacetime is a manifold. But it is very unlikely that there is any region of our spacetiime that is actually Euclidean. Note that being flat and being Euclidean are not the same thing. There are in fact compact flat manifolds, and at least one is potential model for the topology of "space" (not spacetime) in cosmological models. That manifold is the flat 3-torus (called Pac Man space by Bryan Green in his latest book). IF this manifold represents space then spacetime is just the Cartesian product of a line segment with the 3-torus. No one knows whether spacetime, on the very largest scales, is flat or not. So far there is no evidence of large scale curvature. But the topology of the universe is very sensitive to any curvature, no matter how small, even with the assumption of homogeneity and isotropy. It is quite clear that when smaller scales are considered the spacetiime is most definitely not flat. If it were there would be no gravity whatever. There is no such thing as a "straight line" in the general setting of a manifold. The closest thing to a straight line is a geodesic. But apart from a geodesic the notion of a "straight line" is meaningless in the general setting that one has in general relativity. A geodesic, not a straight line, is the shortest local distance between two points on a Riemannian manifold. Straight lines are geodesics in Euclidean space. However in spacetime light follows geodeesics, and geodesics in spacetime are the LONGEST distance between two points. This is because spacetime is a Lorentzian manifold, not a Riemannian manifold. Not only that but the distance along a timelike curve, as measured with the Lorentzian metric, is the time that is experienced by a body with that world line and that is how one explains the so-called "twin paradox" using general relativity (the stay-at-home twin in free fall follows a geodesic in spacetime and therefore experiences a greater proper time interval than does the accelerating, traveling twin who does not have a geodesic world line).

Any good book oon quantum mechanics -- the books by Griffiths or,Messiah are pretty standard., I am neither blind nor stupid. What you think is your problem. I don't give a damn what Fowler thinks or what he follows. Why would I ? I provided you with the proper statement of the Heisenberg Uncertainty Principle. I am no longer an academic. But that is not important. What is important is the proper statement of the principle and understanding of that statement.

Things are a hell of a lot better understood today than in 1959. Go get that better source. While you are at it understand your own references better. I don't see any significant conflict Born's statements with what I said earlier. But your last reference give the common, incorrect, over-simplfied statement of the uncertainty principle that one finds in popularizations and too-simple undergraduate texts. What I told you in terms of standard deviations is correct. The fact that the Heisenberg uncertainty principle is reflected in the results of experiments, which obviously require measurements, does not imply that the fundamental issue is measurement or that the uncertainty principle is the source of the stochastic nature of quantum mechanics.

That definition is what is used in mathematics as well (good news for physics). It works quite well in a Banach algebra. [math] e^{i \pi}\ = \ -1 [/math]

That itself does not make sense. "measures time to be the same" as what "within his own local reference frame" ? Within a given reference frame there is one and only one notion of time. It is always the same as itself.

Get a better source than a Wiki explanation in "layman's terms". What I told you is correct. The Heisenberg uncertainty principle is a statement regarding the random variables that are observables in quantum mechanics. It is not tied to measurement, but its effects are, of course, seen in the sample paths of random variables that are the result of measurement. The important point is that quantum mechanics is inherently stochastic and the uncertainty principle is a statement about random variables. The uncertainty principle is not the source of the stochastic nature of quantum mechanics but is simply an important statement about what are inherently random variables.

To address this question what is needed is a validated theory that can combine gravity (general relativity) with quantum mechanics (quantum field theories as in the Standard Model of particle physics). Unfortunately no such theory currently exists.

No. If you assume partial subscriptions the 2000 cookie subscription, with the lowest per cookie cost provides the solution. This problem as stated is not amenable to any application of calculus of several variables. It is very non-linear and totally discrete problem. Such problems are both very difficult to solve and completely uninteresting. The reason for the 0 cookies for $300 possibilitb is that it lets you also take 80 subscriptions of 2000m cookies at $1800 each and then fill in the remaining 370 subscriptions to get to the required 450. As I said,this is a totally impractical and uninteresting problem.

No. The problem posed is very discrete in nature. Only a finite number of potential solutions exist and calculus is useless. This problem is a combinatorial nightmare, and totally impractical. You are not looking for a minimum cost to obtain 1,600,000 cookies (that solution is obvious) but a minimum cost with 450 subscriptions, which is quite a different thing. You need to somehow consider all possible solutions. There may be a strategy to reduce the number of cases that must be evaluated in detail, but I don't see it at first blush.

Informed, rational counter-point to emotiona,l knee-jerk reactions. kudos

Bribery of a politician in a tanning salon.

Hard to believe that anyone in China might not have high regard for intellectual property rights isn't it ?

The sine and cosine are not really "represented" through the unit circle so much as defined in terms of the unit circle. So, the basic answer to his question is that, no, you cannot replace the unit circle with something else in any natural way to produce the sine or cosine. You can, of course, utilize computational power and graphics to kluge up some unnatural representation, but to do that would be to miss the point. The graphics in your link do pretty much that. I am quite sure that whoever put together that animation understands the sine and cosine quite well, and the exercise of putting it together would have been an good learning experience. But anyone who does not already understand the sine and cosine will only be baffled by the animation which generates sine and cosing traveling waves. Just because you can concoct some animation based on other geometric shapes does not mean that you should do so. There is no natural way to do it. If one is going to force the issue using some ad hoc approach and lots of computing power, then I suggest that the geometric object be Salma Hayek -- there would be essentially no loss in mathematical content, but the geometric object would be at least be interesting.

In general relativity it is spacetime, not "space" that shows curvature. In fact in general relativity there is no such thing, on a global basis, as either time or space. Time and space are purely local notions, and are dependent on the arbitrary selection of a local coordinate system. Mass and more precisely mass/energy not only warps spacetime, but determines the curvature of spacetime via the stress-energy tensor and the Einstein field equations. The statement that "Gravity affects Motion & Duration" is meaningless babble. It is not even wronog, just nonsense words. If you want to understand what general relativity really says, ralther than make a ridiculous criticism of a theory that you manifestly don't begin to understand, then you need to read a real book on the subject. Gravitation by Misner, Thorne and Wheeler would be a good start.

Actually, he standard deviation is associated with the concept of a random variable and it has nothing to do with measuurements. There is a notion of a "sample standard deviation", which is slightly different from the actual standard deviation (aka "population standard deviation"). You find the sample deviation discussed in the context of a measurement from a subset of the total population and it is the optimal non-biased estimate of the "population standard deviation" from the sample taken. But the standard deviation itself is associated with a random variable. The reason that randomness is inherent in the statement of the Heisenberg uncertainty principle is that it is in fact a statement about random variables. The statement makes no sense in the context of a deterministic system. General relativity is a completely deterministic theory. It has absolutely nothing to do with the uncertainty principle. In fact the most fundamental conflict between general relativity and quantum theory is that the former is completely deterministic while tthe latter is stochastic.

Time does not stop ever, including at absolute zero. Time has nothing to do with the issue at hand. Classically one might expect all motion to cease at absolute zero, but the atomic world is quantum mechanical, not classical. The fact that absolute zero is not achievable in a finite number of thermodynamic steps has nothing to do with the fact that there is still motion at absolute zero. The inability to reach absolute zero in a finitte number of steps is the Third Law of Thermodynamics. Absolute zero is simply a ground state in quantum mechanic,s and the Pauli exclusion prohibits all fermionic particles from occupying the same state, even at absolute zero.

A sine is not a circle. The sine is a well-defined function. There is no such thing as "other types of sine-waves". You can represent functions in terms of functions other than sine waves. This is something that you can study when you study Hilbert spaces or "orthogonal functions". One set of functions that are sometimes used are Walsh functions which are trains of square waves. The secant can be graphed as the secant. It too is a well-defined function. It has the shape of a secant curve, just as the siine has the shape of a sine curve. What in the world does it mean to add circles ? If you want a bigger circle just open up your compass and draw one.

This is just flat wrong. What it demonstrates is fundamental lack of understanding of general relativity. Curvature of spacetime (not "space") is what we call "gravity". Both "space" and "time" are local, coordinate-dependent notions. Lack of spacetime curvture is evidence of total absence of gravity. Since gravity permeates the universe there is no lack of curvature. Now go read Gravitation by Misner, Thorne and Wheeler.

You have succumbed to inaccurate descriptions of the uncertainty principle, common in popularizations. What the uncertainty principle actually says is that for two complementary observables (position and momentum are complimentary) that if one takes particles prepared in identical quantum states and then does repeated measurements of position x followed by momentum p (or momentum p followed by position x) that [math] \sigma_x \sigma_p \ge \frac{\hbar}{2} [/math] where [math]\sigma_x[/math] and [math]sigma_p[/math] are the standard deviations associated with the random variables [math]x[/math] and [math]p[/math] respectively. Inherent in this statement are that position and momentum are not deterministic. They are ranndom variables associated with a stochastic system. It is NOT a statement that there will be any definite error in the measurement of either position or momentum. It is in fact a statement that there is no such thing as a definite measurable value of either position or momentum associated with a quantum particle.

Wrong. Once you make the obvious observation that the sequence has more than one cluster point, you have proved -- using the definition -- that the sequence fails to converge. The fact that any two distinct points are contained in disjoint intervals (which you can formalize using epsilons if you like bhy takikng epsilon to be one third of the distance separating the points) is all that is needed. The fact that you fail to recognize that fact speaks volumes.

No, he asked to solve the problem using the definition. If you understood mathematics, which you clearly do not, then it would be quite obvious how to use "epsilons" to prove that no convergent sequence can have more than one cluster point. This result remains true in any metric space and indeed in any Hausdorff topological space. Using epsilons to show that the real numbers constitute a Hausdorff space is the heart of the matter, though that terminilogy is not usually introduced in an elementary calculus class. But the gist of the idea is that any two distimct points lie in disjooint small open intervals and no sequence can be eventually in both of them, though it can be in both frequently. I, in fact, did provide him guidance toward a very valid approach to solve the problem -- using the definition. You very clearly failed to recognize that fact and also failed to recognize the nature of the problem and the natural (and very easy) solution. Apparently the language and mathematics difficulties lie entirely with you. Please refrain from usiing all capital letters to emphasize stupidit comments. They are quite easy to recognize without the added emphasis.

One tends to fall downhill rather than uphill.

If frogs had wings they wouldn't bump their ass.