DrRocket

What is abundantly clear is that communication with Hal is not possible. I was under the general impression that the purpose of language was to facilitate communication. Is it not ? N'est pas ?

Calculus on Manifolds -- Michael Spivak

Much of the current theory IS well-proved. But the current theory is not, an no one has claimed otherwise, the final word. It is well known that there are shortcomings to the standard model, and there have been attempts, so far unsuccessful to extend it. Physics remains a vibrant research subject. Nevertheless, the standard model has been very successful within its domain of validity and any successor theory will of necessity extend and refine the standard model, not overthrow it. I suspect that you are confusing "causal" with "deterministic". That is not surprising as Wiki is similarly confused. QM predicts a deterministic and causal evolution of the state function. Only the results of specific measurements are stochastic -- in other words QM describes a deterministic and causal evolution of probability measures. This may not satisfy you, but experimental evidence shows that it describes nature. And THAT is scientific truth -- whether you like it or not. Perhaps in the future some new theory will replace the current quantum field theories. But that theory will have to explain the experiments in which the same initial conditions do not always result in the same outcome, although the frequencies of outcomes in many trials do conform to predicted probabilities. No one has been brain washed or is just reciting dogma. Rather the body of evidence has been considered, and that body of evidence fully supports the current theory, within limits. The limits are known, and research is in progress to extend the frontiers of knowledge. Real scientists are pursuing real research with real expertise and real imagination. Crackpots need not apply.

If you are interested in science at any level beyond being a technician assisting a scientist, a college degree is essential. At the research level a PhD is the norm. There are lots of scholarships and other financial aid available to qualified students. 34 is pretty late to be starting out, but not unheard of. A very good friend received a PhD about a year ago. He was just shy of 60 at the time.

Electrical engineering includes, electronic devices, electronic circuit design, fields and waves (including antenna design), communication and information theory, control theory, optical circuits, electrical machinery, power distribution, etc. Mechanical engineering includes, classical dynamics, structural analysis (including finite element methods), fluid mechanics, thermodynamics, design of mechanisms, composite materials design & analysis, heat transfer, etc Systems engineering can be many things and is not well-defined. In some circumstances it involves logistics and requirements tracking. In many aerospace circumstances it is interdisciplinary and involves the definition and allocation to subsystems of requirements for complex products, and management of the technical development of the product. In other circumstances it is a highly mathematical outgrowth of modern control theory, abstract computer science and information theory.

One fact that has not been made clear in this thread is that the mathematical theory of probability starts with probabilities being given -- starts with a probability space which includes events ( a sigma algebra of measurable sets) and probabilities (a probability measure). The actual determination of probabilities is either based on empirical models, statistics, or an outright assumption. Probability theory has nothing to say about the matter. Probabilities can be almost anything subject to only a few basic constraints: the probability that at least one of all possible events occurs must be one; the probability any event must be non-negative and no greater than 1; the probability that at least one of countably many disjoint events occurs is the sum of the individual probabilities. If there are finitely many primitive events, any assignment of non-negative numbers that sum to 1 defines a legitimate probability space -- only outside considerations can determine if it is a physically accurate model.

No one ever said QM is just mathematical. QM is a physical theory. It predicts physical phenomena. But to understand QM at any fdepth you must understand the mathemativcs that is the language. "To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’. To those who do not know mathematics it is difficult to get across a real feeling as the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once." – Richard P. Feynman in The Character of Physical Law "Philosophers say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong." – Richard P. Feynman http://depts.washington.edu/ssnet/Weinberg_SSN_1_14.pdf

I watched as much of that trash as I could stand (not a lot). For a decent presentation of the golden ratio watch this clip from "Donald in Mathemagic Lad" --

t is time. n is an integer. [math]\omega t = 2n \pi [/math] is required in order that [math] cos( \omega t) = 1 [/math] [math]\omega t = \theta[/math] in uniform circular motion. It would help if you indicated your level of education and where you are encountering this problem.

This is true for t=0 and ωt = 2n pi I assume ar is radial acceleration and not a x r. In any case velocity is not constant. Speed is constant. If ar = v2/r and v2 is constant then ar is not zero unless v is also zero. If ar were 0 there would be no such thing as centripetal acceleration. nope I would approach this by noting that for uniform circular motion x = r cos( ωt) and y = r sin(ωt) and then differentiate to get velocity and acceleration. Your equations are not true in general, but x'' + ω2 r = 0 is true for t=0 while y'' + ω2 r = 0 is not.

What is the basis for this assertion. Wishful thinking ? QM is certainly not beyond description in terms of mathematical models. QM is in fact a mathematical model. But classical imagery simply does not apply. Probably not. If you wish to be recognized as a scientist it would be advisable to put your statements in scientific terms and back them up with real data and real mathematics. Vague philosophical statements are of no value. No one is particularly happy with the various interpretations of quantum theories, but the theory makes extraordinarily accurate predictions and it is rather difficult to discount that success criteria.

The model IS quantum mechanics. But it appears that you cannot accept QM as the model, because it does not admit of classical interpretations. In that case either you are doomed to never understand, or you will have to invent and validate some alternative to quantum mechanics. Since an army of professional physicists has found no alternative to quantum theory in roughly a century since its discovery, I would personally bet that you are doomed. "There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe that there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper, a lot of people understood the theory of relativity in some way or other, certainly more than twelve. On the other hand, I can safely say that nobody understands quantum mechanics." – Richard P. Feynman in The Character of Physical Law

You are asking about advanced topics, topics treated in depth only at the graduate level. Such topics are not particularly amenable to on-line treatments. There is a reason for the residency requirement that one finds in most PhD programs. Learning at that level is a full-time job.

The pressure inside the tube must be higher than the outside pressure in order for flow from inside to outside to exist. To a first order, the flow rate will be proportional to the pressure difference. See Poiseuille's Law. http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation

No we are two different crazy mixed-up hyper-mathematicians.

Yes. You understand. I understand. All the non-mathematicians are scratching their heads and pointing at the crazy mathematicians, with some justification.

You need to use the definition of linear dependence and prove your assertion.

I hope this is not part of your strategy for writing popularizations of mathematics.

The "wave" that you are describing would not explain why space itself is expanding and expanding uniformly in all directions. Space is not a fabric. The analogy to which you refer is not aqccurate -- it is meant only to give a very rough idea of the notion of curvature. It probably generates as much confusion as enlightenment.

You shoulde be able to determine for yourself when you have a valid proof for a given proposition. But,, no, this is not a valid proof for the proposition that you stated. Your statements in fact assume the conclusion of the proposition.

In radians the angle is associated with the arc length of an arc at radius 1 subtended by the angle. So arctan is the arc, or angle, that produces some value for the tangent function -- the inverse. Since tan is pi periodic, the inverse is only defined modulo pi. arctan is the inverse of tan restricted to (-pi/2, pi/2).

Look at the volume element. The"r" in the volume element cancels the "1/r" in the integrand. It is not obvious. It takes some work to show the connection. I don't have a ready reference. ?????????? Nothing has infinite probability. What is infinite, in dimensions 1 and 2 is the expected number of returns to a neighborhood of a point in infinitely many steps. Yes. A drunk on the street with continually walk into the lamp pole, but a drunk astronaut will wander off to infinity. You wind up with the divergent integrand [math]\frac {1}{r}[/math] -- look at the volume element. good idea

Sure Consider [math]f(x) = \frac {1}{r}[/math](i.e. [math]\frac {1}{||x||}[/math]) In dimension 3 0r above. Since the volume element in spherical coordinates is [math] r^{n-1}dr \times [/math] (other stuff involving angles cosines and sines) the integral converges on finite balls centered at 0. This is why random walks in dimensions 1 and 2 are recurrent but random walks in dimension 3 and above are non-recurrent. (With reasonable constraints on the probability measure). Or if you want to stick to dimension 1 see below. Sure. Take [math] f(x)=\frac{1}{x^2}[/math] for [math] x \ge 1[/math] and [math]f(x)= \frac{1}{\sqrt x}[/math] for [math]0<x<1[/math]

Theoretical physics is the development of mathematical models that describe the behavior of nature, through a minimal number of basic assumptions. It requires imagination, which is applicable to art. It also requires rigor and logical constraints that may not be applicable. Good theories have, thus far, been beautiful, so in some sense theoretical physics, and more so mathematics, can be considered to be art forms. It takes a good deal of knowledge and aptitude to appreciate the asthetics of mathematics and of physical law.. Theoretical physics is not at all limited to what is known. Rather it is constrained by what is known -- a theory that is contradicted by what is known is not a valid theory. A theory that can only predict what is already known is no better than existing theory. The power of a new theory is measured by its ability topredict new phenomena which are subsequently verified by experiment. A theory that predicts something new is provisional, neither fact nor fiction, until experiment either verifies the new phenomenon or shows that it does not exist. In the former case the validity of the theory is supported. In the latter case the theory is falsified and discarded. "To summarize , I would use the words of Jeans, who said that ‘the Great Architect seems to be a mathematician’. To those who do not know mathematics it is difficult to get across a real feeling as the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures. I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once." – Richard P. Feynman in The Character of Physical Law

Any text on linear algebra. Hoffman and Kunze would do.