DrRocket

http://en.wikipedia.org/wiki/Edward_Page_Mitchell

No mystery. The OP is The french tourist.

Which is what I said in the first sentence of my post. No disagreement here. I, and I think most mathematicians, distinguish between a function, like the exponential function, which, colloquially, is a machine that takes in an argument and spits out a value, and a formula, which is a simple expression like the formula that defines the Euler characteristic which classifies two-manifolds. Side note: Walter Rudin was an analyst. His feelings toward what is most important in mathematics reflect the natural bias of an analyst. I too am an analyst and share many of those same biases. However, mathematicians with other specialties (his wife Mary Ellen for instance) might find other functions to be more important to them.

The exponential function itself is, in the view of many, the most important function in mathematics. The equation [math] e^{i\pi}=-1[/math] is rather remarkable, and is in fact the basis for a purely analytic definition of [math]\pi[/math]. I find that the simple fact that it includes the numbers [math] e, \ \pi , \ 0, \ 1[/math] to be not particularly important -- rather like a physicists version of numerology -- particularly given that is not known if either [math] e + \pi[/math] or [math]e \pi[/math] are rational (thought they cannot both be rational). However the relationship itself is profound. The equation [math]e^{ix}=cosx +i \ sinx[/math] is in fact the modern definition of the sine and cosine funtions. What is done in a rigorous treatment is to start with the general formula [math] e ^z = \sum_{n=0}^\infty \frac {z^n}{n!}[/math] where [math]z[/math] is a complex variable and show that it is periodic. That period is defined to be [math] 2 i \pi[/math] and the sine and cosine functions are defined as above. This serves as a starting point for the study of analysis of one complex variable. (You can see a rather slick exposition of this approach in the first few pages of Rudin's Real and Complex Analysis.) I would hope that the football team does not consist of a bunch of quantum physicists -- with the HUP they would never be quite sure where the ball was. Another formula due to Euler seems to me to be equally remarkable. That is V-E +F = 2 - 2g which applies to trianulations of surfaces, where V is the number of vertices, E is the number of edges, F is the number of faces and g is the "genus" or number of holes in the surface. The expression on the left-hand side is called the Euler characteristic, and it completely classifies 2-dimensional manifolds up to homeomorphism (topological equivalence). Note that for an ordinary tetrahedrom (a toplogical sphere) that V-E+F = 4 - 6 +4 = 2 so that g=0 (no holes). Any 2-manifold with no holes (simply connected) is topologically equivalent to the 2-sphere. This is in fact one of the motivating thoughts behind the Poincare conjecture in dimension 3, the proof of which remained open for nearly a century until finally solved by Gregori Perelman using extremely powerful methods from analysis and geometry.

Edward Page Mitchell ?

calculus

It's tough to make predictions, especially about the future -- Yogi Berra

I could not possibly have been more sincere. I devooutly hope your promise was equally sincere.

The mere fact that you are asking these questions shows that you are in WAY over your head. No one has any particularly effective trading models of the stock market, or if they do they are not talking. Modern high-frequency traders use models that consider the stock price a millisecond ago, and they do not alway make money -- sometimes they lose quite a bit. There are some models used in the bond market that are effective for large trading houses. They have groups typically staffed by PhD ex-particle physicists and/or algebraic geometers working full-time on sophisticated models. But even these guys don't try to predict the stock market over any significant length of tiime. If you are going to pursue this sort of thing seriously you need to really learn the theory of stochastic proceses and the associated calculus. When you understand both the Ito integral and the Stratonovic integral you will be almost prepared. Until then you might as well skip the middleman and just send your money to Goldman Sachs. Or go to Vegas and have some fun while you lose your money.

First, your parents are wrong. Dead wrong. University education is intended to foster learning, lifelong learning, about whatever it is that interests you. It is not some glorified trade school designed to prepare you for one specific career. Education is not the same thing as training. You should study something that interests you and that will enable you to continue to learn long after your formal course work has ended. Physics could do that. There are perfectly good careers doing academic and government research that is performed by physicists. It is true that there are relatively few such jobs and that competition is stiff. But they do exist and they are good jobs. Moreover, while industry is much more comfortable with engineering degrees, physicists can do well in industry. Jobs over $100K are fairly common. So, bottom line, while there is something to be said for an education that can lead to a decent income and career, it is also important that you study something that interests you. Physics clearly interests you, and physics can lead to that good career. It is your life. Follow your interests, not necessarily those of your parents. Note that there are scholarships available to qualified students that can greatly defray the cost of an education. There are also on-campus jobs, sometimes helping perform research that might interest you. Do not let someone else dictate what you will study. That said, were you to take an undergraduate degree in chemistry you could specialize in physical chemistry and thereby slant your course work towards physics. That would leave you an option of pursuing physics or chemical physics in graduate school -- and presumably with funding available that would put you outside of the immediate control of your parents. Now, as far as mathematics books go, the first step would be to go over calculus again and understand from a conceptual and theoretical basis rather than the symbol pushing calculations typical of high-school calculus. A good book for that is Michael Spivak's Calculus. After that you will need to study calculus of several variables, which requires some linear algebra. For those subjects I recommend Finite Dimensional Vector Spaces by Paul Halmos and Calculus on Manifolds by Michael Spivak and either Elements of Real Analysis by Robert G. Bartle or Principles of Mathematical Analysis by Walter Rudin. Note that these are texts that get you up tothe junior/senior undergraduate level and that it is perfectly reasonable for you to take courses that use these or similar texts, even if you major in chemistry. You did not ask about physics books, but rather than simply studying the relevant mathematics I also suggest that you do some reading in physics. One of the best physics ever written is The Feynman Lecctures on Physics by Feynman, Leighton and Sands. It is not particularly demanding of one's mathematical background, but the physical insight on exhibit from Feynman is pure magic. It might also help to motivate your study of relevant mathematics.

Good to have that in writing.

The geometry of spacetime in general relativity is not dependent on the "frame" of the observer. In fact in general relativity physics is formulted independent of any notion of a "frame" -- this is what Einstein called "general covariance". Curvature is an invariant. It does not depend on any notion of motion or any frame. The curvature tensor is the same for all "observers", though it might look a bit different in different local coordinates. Example: The Einstein curvature tensor is defined by the stress-energy tensor, and the stress-energy tensor accounts for all forms of energy, except for gravitational potential energy itself. So, the individual components of the stress-energy tensor are dependent on the local observer, and observers in relative motion will "see" different components. Nevertheless the tensor itself is not dependent on the observer, and the curvature of space time is invariant -- You cannot cause the sun to create a black hole simply by moving at some high speed relative to it and therefore seeing its relativistic mass as arbitrarily large. At the level of special relativity, while time and length are dependent on the observer, the spacetime interval which includes both space and time is invariant.

You might want to look at a couple of books on special relativity. Introduction to Special Relativity by Wolfgang presents the usual physics approach in which the Lorentz transformations are derived from thought experiments and the two postulates of special relativity. The Geometry of Minkowski Spacetime, an Introduction to the Mathematics of the Theory of Special Relativity by Gregory Nabers derives special relativity by considering transformations that preserve the Minkowski metric. Both are useful perspectives, dependiing on the specific issue at hand.

In thiis case you are talking about an initial point or Cauchy surface in spacetime. There is no good word for it and absolutely nothing is known about it, including whether or not it exists. This entire thread is inane. Maybe that is why it is now into its 34th page.

The twin prime conjecture and the Riemann Hypothesis are two long-standing open problems in mathematics. It should not surprise you that you cannot find proofs for either -- neither can anyone else. The Riemann Hypothesis is widely regarded as the most important and difficult problem in all of mathematics. If you want to pursue these topics further, the first baby step is get a PhD in mathematics -- they really are that difficult.

And if frogs had wings they wouldn't bump their ass.

See the Euler product formula for the zeta function: http://en.wikipedia.org/wiki/Riemann_zeta_function

There are numbers for which it is not know whether they are rational or irrational. [math]e + \pi and e \pi[/math] are such numbers. That does not mean that no proof exists, only that no one has found a proof either way thus far. A proof that no proof exists would be a proof of undecidability of the question. No proof of undecidability has been found either. Your suggestion as to a proof won't work. It won't work because if a number is presented as a ratio orf integers, there is nothing left to do, and if it is not so presented you have no test to determine if any given ratio is the number in question -- try your method on [math] e+ \pi[/math] for instance and you will find that you have no way to make a comparison. Note that while we do not know whether [math] e + \pi[/math] or [math] e \pi[/math] are rational, it is known that at most one of them can be rational. The proof is in fact quite simple.

Everything in this paragraph is either a) wrong or b) nonsensical. "Anything that can happen, will happen" is a very rough translation of the Law of Large Numbers which implies "that any event of positive probability will occur infinitely often in infinitely many independent trials, with probability one". However it most certainly does not imply that there is no event of 0 probability. It does not even imply that an event of probability 0 cannot occur, but only that the occurrence has probability 0. And no, "probability 0 does NOT mean that an occurrence is impossible." In fact, loosely speaking the probability of an event [math]A[/math] is [math]\displaystyle \lim_{n \to \infty} \frac {number \ of \ occurrences \ of \ A}{n}[/math] where [math]n[/math] is the number of trials. If, for instance, there are only a finite number of occurrences of [math]A[/math] in infinitely many trials then the probability of[math] A[/math] is zero, even though it did occur. Note also that in order to have a non-zero probability that [math]A [/math] must occur infinitely many times in infinitely many trials, as indicated above. This is made much more precise in a proper treatment of probability theory based, following Kolmogorov, on measure theory. The statement that "there exist facts that we know they're true, but we can't prove them" is a very bad paraphrase of Godel's first incompleteness theorem, which in fact states that, given any system of axioms that admits ordinary arithmetic there will be true theorems that are not provable by first-order logic. That does not necessarily mean that "we know that they're true" since knowledge in mathematics comes from proof and if we know it is true that means that we can prove it. It also does not mean that statements not provable via first-order logic are not provable via higher-order methods. Moreover, there are assertions, such as the Axiom of Choice and the Continuum Hypothesis that are known to be independent of the basic Zermelo Fraenkel axioms. There is no such thing as a "normalized probability space", though there is such a thing as the "normal probability density" also called the "Gaussian probability density" which figures rather prominently in the theory -- Google Central Limit Theorem. The notion that "everything happens at once" makes no sense in this context. Theere is in fact no need to normalize a probabilty space since, by definition, the total measure of a probability space is always 1.

In fact the parameter required in this instance is just arc length, with respect to the natural metric of spacetime. So what is involved is just the usual trick of parameterizing a curve by arc length. So, one can say that for a timelike curve the length of the curve is the propertime associated with that world line. This in fact applies to any time-like curve in spacetime, whether or not it is a geodesic. If one also notes that with a metric of signature +,-,-,- that geodesics actually maximize arc length (quite different from the Riemannian case) one has a quick resolution of the "twin paradox" (the "non-traveling twin" has a world line that is a geodesic while the traveling twin has a non-geodesic world line, so the non-traveling twin's world line has the greater proper time). I guess you might stretch this to say that the proper time associated with any segment of a null geodesic is zero, but I don't know that you get anything useful out of that stretch.

No. The "trivial zeros" of the Riemann zeta function are the negative real even integers -2, -4,-6, .... They are "trivial" only by comparison with other zeros of the zeta function, and the proof that these points are zeros relies on some knowledge of the zeta function, and in particular a functional equation that it satisgies. Any zero of the zeta function that is not one of trivial zeros is called "non-trivial". The Riemann hypothesis is the conjecture that the real part of any non-trivial zero is 1/2. There are suggestive results indicating that the conjecture may be true, but no one knows. So the hypothesis is that to input a value s to the zeta function and receive 0 as an output, either s must be a negative even integer or else s must have real part 1/2. The Riemann hypothesis has been a major, in some minds the major, open problem in mathematics since David Hilbert's address of 1900 -- the Riemann hypothesis is one of the original "Hilbert Problems", the solution of any one of which would get you great recognition in the mathematical community. It is the only one of the original Hilbert Problems that is also a Milenium Problem, the solution to which would carry a $1 million prize. If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven? -- David Hilbert

Rotation is not commutative. To get back where you started you need to apply the inverse rotations in the reverse order.

The key to understanding mathematics is being able to read and understand it and do the proofs for yourself. That way, by applying rigor and logic, you know what is right without having to rely on anyone, professors included, to tell you what is right. In short you figure out what is right for yourself -- based on what you can prove, not what you "think" or "believe".

What is your question ?

Actually there is a perfectly valid theory that does admit an ether. It is the "Lorentz Ether Theory" and is it completely equivalent to special relativity. It is not usually taught simply because there is no good reason to teach it. It produces exactly the same predictions as special relativity. To get the Lorentz Ether theory you simply select some inertial reference frame and, by fiat, call it the "ether frame". You then, also by fiat, demand that the relationship between distance and time in one inertial frame be related to distance and time in any other inertial frame by the associated Lorentz transformation. Since the Lorentz transformation group is what really determines special relativity you automatically have a theory that is equivalent to ordinary special relativity. The significant difference between special relativity and the Lorentz Ether theory is that special relativity is based on two straightforward assumptions: 1) that light propagates at the same speed in all reference frames and 2) the laws of the physics are the same in all inertial reference frames. While these tenets are also true in Lorentz Ether theory, they are derived from the less satisfying axiom that the Lorentz transformations relate time and space in different frames, while in special relativity it is the Lorentz transformations themselves that are derived. Before you dismiss this out of hand, remember that in quantum electrodynamics the quantum vacuum (which is NOT simply "nothing") essentially plays the role of the ether. One ought not be too dogmatic about viewpoints in physics -- quite often a twist on an old viewpoint turns out to be very productive. A very good example is the original corpuscular theory of light of Newton which gave way to Huygens wave theory of light which in turn gave way to the modern quantum theory which is again corpuscular (photons).