DrRocket

These sentences are contradictory. If a sequence has two subsequences that converge to different limits that in and of itself is enough to show that the original sequence does not converge. So, not only is it a valid approach, you have inadvertently made the essential observation but failed to understand the significance,. Not only is non-convergence not "very difficult to prove", it is trivial with this observation. Perhaps you should gain an understanding of the suubject before offering bogus advice.

I am pretty confident that the range quoted is sufficient to cover most practical designs. Note that the OP did not specify the type of craft, so rockets are fair game. Likewise low speed specialized endurance aircraft like the Voyager, or more ambitious future craft of that genre, could be in the mix.

A sphere minimizes the potential energy of a body under gravitational forces. It is the shape of minimum surface area for a given volume.

An MSDS is required for any chemical composition. Seasrch the internet for Tide MSDS. There should also be an MSDS for the sharpie fluid.

I, on the other hand, am pretty sure.

For a ten foolt fuselage the wing span will be precisely [math] 50 \pm 50 [/math] feet long (give or take a bit), depending on anticipated flight speed, altitude, engine thrust and weight. http://en.wikipedia.org/wiki/Rutan_Voyager http://en.wikipedia.org/wiki/North_American_X-15 http://en.wikipedia.org/wiki/Lockheed_Martin_X-33 http://en.wikipedia.org/wiki/Lockheed_U-2 http://en.wikipedia.org/wiki/Delta_rocket

Why would you take pre-calculus and calculus simultaneously ? If you need the pre-calculus class then you have no business taking calculus. If you don't need it, pre-calculus will be a boring waste of time and likely will not count towards graduation in any sort of science major.

The alcohol in Listerine is ethanol, unsurprisingly. 21.6% per the MSDS http://www.discountofficeitems.com/pub/manu_folders/PFI/PFI42703MSDS1354.pdf I recommend gargling and spitting out Listerine. If one intends to swallow ethanol there are more palatable forms. I recommend a nice sour mash Bourbon, a good Scotch or a nice Merlot. Gargliong isopropanol is not recommended. Neither is drinking it. Ditto for methanol (which is why good moonshiners discard the top cut).

Which has the distinction of having been shown to exist and which is described by a predictive theory, quantum electrodynamics.

Energy is, exactly, the conserved current related to time invariance of the Lagrangian. Google "Noether's theorem".

Please tell me that the PI was misquoted.

Dark energy is the explanation for the observed acceleration of the expansion of the universe. No one knows what the cause of the expansion itself really is. Dark energy is not really an explanation so much as a name to given to something that is not understood. To be an explanation we would need to know that dark energy truly exists and know what it is. There is no "math that supports the cosmological constant." There is observational evidence that the universe is expanding and that the rate of expansion is increasing. The increasing expansion rate requires a positive cosmological constant in cosmological models that are based on general relativity. A positive cosmological constant and dark energy are essentially the same thing. There is no solid explanation for the positive cosmological constant. There is some thought that the positive cosmological constant is the result of the energy of the vacuum in quantum electrodynamics. The vacuum energy results in a negative pressure term, equivalent to a positive cosmological constant, in the stress-energy tensor. However, calculations of the value of the negative pressure over-predicts the observed cosmological constant by a factor of [math]10^{120}[/math], which is the largest over-prediction in the history of physics. In short, nobody understands what is going on. This is one of the major mysteries of current physics. None of this has much to do with the question of the topology of the universe. That too is an open question, and one that is likely to stay open for a long time. The (spatial) universe could be a 3-sphere, it could be flat Euclidean space, it could be hyperbolic, it could be a flat 3-torus or it could be something else. Nobody knows.

No, it has not solved the problem. The extra dimensions are necessary for the consistency of string theory. But string theory has still yet to be clearly defined or to provide any new testable prediction. It remains very preliminary. The many worlds interpretation of Hugh Everett has nothing to do with any extra dimensions. There is no "interpretation problem" just another interpretation of quantum mechanics. What in the world are you talking about ? There is no theory in which "extra dimensions are created every time one of these loops is created" and in fact in general relativity loops are not created, they either exist or they do not. The spacetime manifold contains all of time and all of space and there is nothing that is either "created" or "destroyed". Nor does general relativity either require or admit any extra dimensions, though it might be possible to formulate a theory on a higher dimensional manifold. You have been reading too much nonsense. The consideration of extra dimensions goes back to Kaluza-Klein and the attempt to incorporate electrodynamics into general relativity. That did not work but it did show that additional spatial dimensions offer some potential, from a mathematical perspective, in attempting to formulate theories that may be testable. It may or may not be an approach that will eventually bear fruit, but don't throw the baby out with the bath water. The issue is not whether or not extra dimensions per se are testable, though they may be, but whether or not a mathematically consistent theory which requires extra dimensions makes new physical predictions that are testable. So far that has not happened, but there is no reason to state a priori that such is impossible. We have no direct physical test to show that atoms exist, but the atomic hypothesis has resulted in any number of physical theories that have shown to be incredibly accurate -- not to mention the entire discipline of chemistry. Still no one has any photograph of an atom, crystal "pictures" showing blobs where we think atoms ought to be notwithstanding.

In the atomic world resonances are realized as eigenfunctions of the Hamiltonian.

Nothing in that quote makes any sense. If you want to study physics start with a good physics book. As the subject appears to have (indeterminate) connection to special relativity, Wolfgang Rindler's Introduction to Special Relativity might be a good place to start.

Prove it.

And since those disciplines only pose questions, and never determine final answers, one will not find resolution in that direction either. Science may not answer the question "What is time ?" beyond that it is "what clocks measure", but somehow it manages to work the concept of time into powerful predictive models and muddles right along. Given the subtleties of general relativity, time is rather mysterious, and time in scientific models is actually not a single consistent concept. In general relativity what clocks measure is proper time, and nothing else. But the time of special relativity, hence of quantum field theories, is only consistent with proper time in the complete absence of gravity. Neglecting gravity is often a very good approximation, but it is never exact. Hence from a purely philosophical perspective science adopts a rather pragmatic stance with regard to time that is dependent on the specific issues at hand -- perhaps unsatisfying philosophically, but very effective scientifically. If anyone has a definition of time that will fit all situations precisely, and can demonstrate how that fits into predictive models, then I suggest immediate publication -- and preparation for a trip to Stockholm.

Given that this is a physics forum, rather than offering estimates of distance based on questionable assumptions paltry data, perhaps one ought to identify the physics at work. First, neglecting aerodynamic effects (which may be very important with an irregular body like a potato) the distance traveled over a flat surface is easily calculated if the initial velocity and launch angle are known. Without the launch angle, essentially nothing can be determined. Incidentally in this idealization it can be shown that maximum distance occurs with a launch angle of 45 degrees. Second the muzzle velocity will be dependent on the volume of propelling gas available, and the length of the barrel. If you assume an ideal gas, an infinite volume of available gas (approximated by a reservoir that is large with respect to the volume in the barrel at the time of exit of the potato), and reversible adiabatic expansion then the pressure will be constant and the work done on the potato during travel in the barrel is just pressure x barrel cross-sectional area x barrel length which determines the velocity if the mass of the potato is known. From that and launch angle you can calculate distance. If the available gas volume cannot be approximated as infinite, then the situation is more complicated and you need to know what the gas is, particularly the ratio of specific heats. If the physical situation precludes significant heat transfer during the shot, you can then do an isentropic expansion calculation and again determine the initial velocity. If there is significant heat transfer going on then the situation is much more complicated and you have to deal with coupled equations -- computer simulation time.

The usual scisor illustration is this: The tips of the scissors cannot move faster than c, but the point of intersetion of the cutting blades (the cutting point of the scissors) can move arbitrarily quickly. But that point of intersection is not a physical point, and does not carry information so that there is no contradiction of relativity. A similar analogy is made with the point of intersection of a guillotine blade with the cutting plane. Now, you may object that the scissors could cut a wire and thereby send a signal. The answer to that is that we are not talking about a real pair of scissors made from real material, and in the case of such a physical pair of scissors, the rigid body idealization breaks down, and the scissors cannot really perform per the idealization. In reality the stress wave that propagates down the blades to cause them to close travels at about the speed of sound in the material.

If you had n linearly independent eigenvectors then the matrix would be diagonalizable and you would not need Jordan normal form.

Yep. The formulator of the question should flunk. Special relativity (and really all science) requires precision in the use of language, else confusion arises and one gets an answer to the question as interpreted by the reader (or just a justified huh ?) rather than the question that the interogator intended to pose. This one calls for a "huh ?", which should receive full credit.

Maybe you ought to read this. http://en.wikipedia.org/wiki/Atomic_orbital Then think a bit more deeply about what the term "orbital" means (and what it does not mean) and how the term applies to atoms with a very large number of protons and electrons. quantum mechanics can get messy.

What ajb told you is correct. But your question is rather awkwardly worded and confusing. It occurs to me that you might, by using the expression "The relative speed between the two" be trying to describe the closing speed of two photons approaching one another as determined in some inertial reference frame. In that case the answer is 2c. This is the upper limit for closing speeds in special relativity. This does not in any way contradict special relativity, as there is nothing that is actually moving at 2c. It is possible to have "speeds" that exceed c in special relativity. Closing speeds are one example. Another is the "speed" of motion of the spot of light generated by sweeping a searchlight across a distance screen, which can be arbitrarily large with sufficient distance between the screen and the searchlight. These do not contradict special relativity since nothing is actually moving at greater than c and no information-carrying signal is being transmitted superluminally.

You obviously know nothing of either Fourier analysis or sinusoids. Read what I said again, the whole paragraph, and don't make inane comments about fragments.

Start and end with that which interests you. The subect may change diuring your studies. A PhD takes as long as it takes you to do original research and make a significant contribution to the field of study. I know of people who have done very high quality PhD work withing 3 years of matriculating as a freshman and others who barely squeaked out after a decade in graduate school.