DrRocket

Simple, I screwed up a long division. Actually, 0.0111......... = 1/90 So, (0.011111...... )^2 = 1/8100 = 0.0001234567900123456790012345679...... The repearing numbers are 0012345679 .

Rubbish [math] 0.111.... = \frac {1}{9}[/math] as any decent high school algebra student can show in 3 lines. So, [math]0.111... ^2 = \frac {1}{81} = 0.011111.....[/math] On the other hand [math]0.012345601234560123456..... = \frac {123456}{9999999}[/math] and [math]0.1234567890123456789..... = \frac {123456789}{999999999}[/math] So, no matter what reasonable interpretation is placed on "0.123456....." you are incorrect.

My mistake. I thought that your objective was to raise your level.

I did get a reply. There are, not surprisingly, some major difficulties with the experimental setup, but the experiment is still planned. the theoretical work for the measurement has been done. Best current estimate is 2-3 years before an announcement. I don't know anything more than this. Assuming that this comes off as planned it will be an impressive display of experimental skill. The result itself is most unlikely to be a surprise.

The effect has been seen in terms of things like the distance traveled by muons, which is also indicative of time dilation. However, as far as I know a direct measurement of length contraction has not been achieved. I am aware of an experiment to attempt just such a measurement. I don't know the details of how that is to be done or current status of the expeeriment. I will PM a physicist involved and see if he responds with any reportable progress.

A good illustration of why in complex variables one must "choose a branch of the logarithm".

Right, and if anything deserves to be called a vacuum as a practical matter, it is "outer space", which is much nearer a perfect vacuum than anything in a laboratory. For purposes of macroscopic stress analysis or gas dynamics, space is a vacuum. Spacecraft hulls, like rocket motors are designed as pressure vessels. That puts the skin in a generally tensile stress state, which allows a relatively lightweight structure, when compared to, say, a submarine hull which is in compression and subject to compressive buckling. A hole in a thin-skinned pressure vessel can result in a marked change in the local stress state, resulting in tearing and enlargement of the hole -- a problem if you are in space. The only diference between the stress in space and at sea level on Earth is the external pressure, 0 in space and about 15 psia on earth. That is not a huge difference. Other structural concerns may dominate, and mitigate against tearing -- not the least such concern would be adequate margins to specifically preclude tearing following micrometeorite impact. When gas flows through an orifice there are two notable flow regimes, subsonic flow and supersonic or choked flow. In choked flow the gas speed is the local speed of sound at the "choke point" and supersonic as the gas expands downstream. The precise choke point varies somewhat with the geometry of the orifice and subtleties of the gas dynamics. Choke flow occurs when the ratio between the internal pressure and the external pressure exceeds a critical value that is determined by the thermodynamic properties of the gas, the ratio of specific heats being a major factor. As I recall, the critical pressure ratio for air is somewhat less than but very roughly 2. Since the pressure outside a spacecraft is essentially 0, the precise critical ratio is unimportant. The flow will be choked. Flow velocity, "wind speed" very near the hole will be nearly mach 1, and one would not want to be caught in this very high speed flow regime. But that regime can be very localized. At points farther removed from the hole the velocity will be much lower, negligible if the hole is small and the volume of the spacecraft is large. Hollywood is more concerned with audience reaction and excitement than with the Navier-Stokes equation..

You are probably familiar with the theorem that shows that the dual space of the continuous functions vanishing at infinity on a locally compact Hausdorff space is the set of regular (signed) Borel measures on that space. That is the Riesz-Markov-Kakutani theorem. The Daniel integral takes that as a starting point to define an integral and works backward to arrive at the idea of a measure. You might know Lou Auslander's book, with McKenzie, on differentiable manifolds. He was a student of S.S. Chern and did a lot of work on the representations of solvable and in particular nilpotent Lie groups. That is the setting in which Kirillov's orbit method works (There is an exposition in Pukansky's Lecons sur les representations des groupes -- a book by a polish mathematician, in Pennsylvania, written in French !!!). I am sure you have run across Pontryagin. He is the Pontryagin of Pontryagin classes in topology, and also the inventor of the idea of the "dual group" of a locally compact abelian group that is the starting point for the general theory of the Fourier transform. Laurent Schwartz is the Schwartz of Schwartz distributions which are critical in harmonic analysis and partial differential equations. He was one of Grothendieck's thesis advisors. I was a bit surprised that you did not name Alain Connes. He has not influenced me, largely because I know nothing about non-commutative geometry -- which is probably why I don't know the people on your list. Excellent.

Common sense is not a good guide. When black holes were first predicted, "common sense" told many people, including Einstein, that they were not physical. Unfortunately there is no substitute for solid theories, backed by empirical evidence, and rigorous deductions from those theories. Nobody knows what the net energy content of the universe is, or if the concept is actually meaningful since there is no universal time in general relativity. The best available theory for cosmology is general relativity (GR), which is a theory of gravity. The other foundations of modern physics, the quantum field theories that comprise the Standard Model of particle physics specifically exclude gravity. GR is a completely deterministic theory. Quantum theories are stochastic. The two are not compatible. GR probably breaks down when quantum effects are important, and the singular nature of spacetime that is predicted in connection with the big bang may not be (I suspect is not) physical. There is a lot of speculation, even informed speculation, put forth in cosmology. but no one really knows what happened prior to about 10^-33 seconds. . Perhaps if and when there is a successful melding of GR and quantum theory more will be understood. Beware of popularizations. They are interesting and entertaining, but not definitive and sometimes inaccurate.

Either of thse likely have merit. However, my take is this: Calculus is a huge philosophical change from elementary algebra. it is no longer "solve the equation and find the number". Rather, there are entirely new concepts, based on estimates and refined estimates. Inequalities are as important or more important than equalities. Limits are a couple of levels of sophistication removed from high school algebra. Calculus requires a change in mind set. The change in mind set is greatly facilitated by classroom discussions and talks with fellow students. This is not available in a self-study environment. Average students (and I suspect that neither of you are in this category) probably cannot really grasp the material without the intangibles that come from these talks and the opportunity to have questions answered and misconceptions corrected in real time. Add to this my prejudice that the (standard) textx from which I have taught the subject are generally poor, and my position that to learn calculus for the first time most students are best served by a traditional class. Books, videos etc. may be useful supplements, but not substitutes for the class.

Lots of things. I am pretty well convinced that there is a problem with CO2 levels and decreasing Ph of the oceans. This could be serious indeed. I am also convinced, based on isotope abundance, that fossil CO2 is a major factor. I am not convinced that the dominant source is other than coal mine fires (China being a huge player), forest fires, and volcanoes. Release of trapped methane due to warming may be a gigantic factor -- we need greater understanding. I am not convinced that the estimated 2mm/yr rise in sea level is real. I have quizzed experts, and they admit that there is no datum against which this is measured. Given just tidal variability, 2mm/yr strikes me as too close to a noise level to be credible. I am not anywhere near convinced that heat transfer models purporting to account for sea currents are at all reliable. I have enough experience with the difficulties associated with large fluid dynamics models to be very skeptical. I am not convinced that we have an accurate, comprehensive heat and mass balance model, or even that such a model is within current technological capability. I am convinced that we understand the physics of CO2 and infrared radiation. I am convinced that the long term trend is toward warming. The glaciers that covered most of North America have receded and the "Little Ice Age" of the 1700's is no longer with us. I am not convinced, nor is at least one emminent and concerned scientist with whom I have spoken, that a modest temperature increase is of necessity bad. I am much more concerned about the oceanic Ph level issue. I am convinced that both sides of the debate have a hidden agenda and that both sides skew the facts to advance their personal interests. I am convinced that qualitative arguments are no substitute for quantitative understanding. CO2 is not all bad. Plants need CO2. Too much is bad. Too little is also bad. We need to understand where the line is and why. There are known ways to reduce CO2 -- iron can encourage a red algae bloom (but we had better know all the ramifications before we do something that might prove to be rash). I am damn tired of either side telling anyone not in complete agreement with them that they are stupid.

I have no idea why you think that curvature of spacetime, as with your torus, would necessitate a violation of conservation of momentum. A torus is not necessarily curved. There is such a thing as a flat torus, and it is seriously being considered as a cosmological model. Take a rectangle. Identify two edges together. That is a cylinder and it is flat, just as flat as the rectangle. Now identify the two ends of the cylinder together. That is a flat 2-torus. You cannot embed it in 3-space. If you identify the ends "with a twist" you get the Klein bottle. You can get a flat 3-torus by starting with a cube and playing a similar game.

"Burden of proof" is a common expression meant to imply which party has the onus. Science is based on evidence, as you describe, not proof. You are correct, that a theory can be falsified, in fact by a single valid empirical observation, or equivalently by disagreement with a body of theory that has been substantiated by empirical data, within its known domain of validity (in essence what is contradicted is the empirical data that supports the theory). That said, the burden to provide evidence lies with anyone advocating a radically new theory. That includes the requirement to clearly and quantitavely define what that theory is. This tends to weed out the wackos and avoid wasting large blocks of time. What in the world is this supposed to mean ? See previous paragraph. The "finiteness in time" that you describe is not really an assumption. It results from an analysis, by Steven Hawking (link to original papers in my post in Cosmo Basics) based on general relativity. The much quoted, and misunderstood singularity of the big bang is a failure to be able to extend timelike geodesics indefinitely into the past. One result is the finite age of the universe. This could be wrong: the hypotheses in Hawking's analysis could be violated, or general relativity could break down in unexpected ways as quantum effects become important. But as it stands there is no meaning to "before" the big bang and no meaning to any causitive factor for the big bang.

One generally does not jump into cutting edge research in any field without some preparation to understand what is already known and the tools and methods used in the field. The topics that you list are at the forefront of research and the tools and methods used are very specialized and very advanced. I suggest that you inquire about programs, graduate or undergraduate as appropriate, at a university where the specialty of a faculty member is of interest to you.

OK, if you have not head much about discontinuities, then you have not seriously studied limits and continuous functions either. Without that background you cannot understand derivatives or integrals beyond my earlier post on tangent lines and areas. If that is all that you need, then there is not much more to be said. If you need something more, then you need to take a real calculus class, preferably from someone who undersatnds the subject. So, my suggestion is to take a class at the university nearest to you. I don't know of any text suitable for self-study at your level.

All mathematics is, virtuallly by definition imaginative. I like ajb's modification to the question. Not being a supermanifolds guy, my list is more conventional, older and more staid (also many of these guys are now deceased) Kakutani, Lou Auslander, Pontryagin, Katznelson, Rudin, Laurent Schwartz, Gelfand, Kirillov, Weiner These guys have made major contributions to harmonic analysis, differential equations, Banach algebras, operator theory and representations of Lie groups. Years ago in grad school I was in an AMS meeting listening to Harry Furstenberg talk about some topic in ergodic theory. He had pretty much lost me. But there was an older oriental gentleman sitting beside me who was not only asking good questions, but seemed to be a step or two ahead of Furstenberg. After the session my advisor came by the room, and I indicated the oriental fellow and asked something like "Who is that masked man ?" His reply -- "Oh, that is Kakuntani !". That explained it all.

There are some very good ones, Anyone who would unequivocably name the number 1 or even the top 10 doesn't understand mathematics very well.

That is precisely the point.

OK, for those who demand the best theorem, rigorously stated, at this level, here are the real fundamental theorems of calculus (there are 2 of them) First fundamental theorem of calculus. Let [math][a,b][/math] be an arbitrary closed interval and let [math]f:[a,b] \rightarrow \mathbb R[/math] be such that[math]f[/math] is bounded and the set of points of discontinuity of [math]f[/math] are of Lebesgue measure 0. Suppose that there exists a function [math]F:(c,d) \rightarrow \mathbb R[/math] where [math]c<a<b<d[/math] such that [math]F'(x)=f(x) \forall x \in [a,b] [/math] Then [math] \int_a^b f(x) \ dx = F(b)-F(a)[/math] Second fundamental theorem of calculus. Let [math][a,b][/math] be an arbitrary closed interval and let [math]f:[a,b] \rightarrow \mathbb R[/math] be such that[math]f[/math] is bounded and the set of points of discontinuity of [math]f[/math] are of Lebesgue measure 0. For [math] x \in [a,b] [/math] set [math]F(x) = \int_a^x f(t) dt [/math] then 1) [math]F[/math] is continuous on [math][a,b][/math] and 2) if [math]f[/math] is continuous at [math]x_0 \in (a,b)[/math] then [math]F[/math] is differentiable at [math]x_0[/math] and [math]F'(x_0)=f(x_0)[/math] This is absolutely correct, and absolutely opaque to the OP who is asking what derviatives and integrals are at the intuitive level. There are two parts to mathematics. 1) The intuitive idea that helps one conceptualize an idea. 2) The rigorous definitions that make the intuitive ideas precise and useable. You need both of them, but often the first element is the most difficult to really understand. (And you don't really have 1vuntil you understand 2, but one must avoid putting the cart before the horse.)

If the link actually states that without further qualification, then the link is wrong. It is true for continuous functions f. It is not true if f has a "jump" discontinuity. Derivatives need not be continuous, but all derivatives satisfy the intermediate value properety. You are being overly pedantic and quite silly.

The fundamental theorem is usually stated in terms of the definite, rather than the indefinite integral. However, the expression as presented by mississippichem does an adequate job of communicating the underlying concept for the pedagogic task at hand. You are splitting hairs needlessly. I would be more precise in a formal class, but for forum purposes, and given the level of the original question, there is no point in full rigor.

The problem that you have identified, the confusion between symbol pushing and understanding is ubiquitous, and is reflective of the distinction between superficial knowledge and the deeper knowledge of true understanding.. I doubt that anything can be done about it at the primary/secondary levels in the foreseeable future because it first needs to be corrected at the university level. So long as authority figures in science and engineering departments encourage superficial application of mathematics -- see the post of BigNose for a clear example -- then students will continue to actively avoid learning real mathematics. The problem is highlighted when those whose understanding of mathematics is clearly deficient are seen as experts. I have seen this carried to the extreme as a proposal for the College of Engineering to teach its own mathematics classes -- blind leading the blind. So long as the attitude exemplified in the post of BigNose is reflected in the attitude of students (engineering students have stated it as "Skip the theory, we just want to plug and chug") the problem that you describe will persist. It is manifested in industry in the form of people who can operate computer codes with great facility as a "black box" but who don't understand the output and who accept nonsensical "answers" from the code. I have seen this even from PhDs and university faculty. In fact, one reason that I left engineering after an MS to pursue a PhD in mathematics was lack of understanding of necessary mathematics by several members of the engineering faculty. To use mathematics effectively in research, or even in non-routine applications, it is necessary to understand it. For instance, one cannot apply the methods of conformal mapping to airfoils if one believes that complex analysis is irrelevant to fluid dynamics -- see the post of BigNose. One sees this problem mollified in the unfortunately rare instances in which educators in using departments have knowledge and appreciation for the body of knowledge that is mathematics. Engineers and physicists like Rudolph Kalman, David Luenberger, Ed Witten, Gene Covert, Charles Desoer, Walter Thirring, Charles Misner, John Archibald Wheeler, Roger Brockett, and Gary Brown have helped to counter the attitude of the adherents of the philosophy represented by the post of BigNose and encourage curiosity and a quest for deep understanding. Ultimately the issue lies in the curiosity of students and their desire to understand. This first and foremost requires that they understand what it means to understand. Faculty and senior practitioners can help or hinder. Some help, others hinder.

That is exactly the problem with many high school calculus courses.

There is nothing wrong with wondering. Wondering is the starting point for good research. Just don't confuse wondering with the more rigorous process of theorizing on the basis of real observation and real mathematics. The cosmological principle could be wrong. But since the observable universe is immense and expansion seems to be accelerating, any counter evidence will have to come from a profound revision in fundamental physics, since otherwise anything beyond the observable universe will forever be causally disconnected from us and our theories cannot address it. There are attempts to develop theories that could do just that, but I am personally skeptical that they will ever reach the necessary level of completion an rigor. I could be wrong too.

Rather than a web site, how about a book or books on Fourier analysis and signal analysis ? What mathematics and engineering classes have you taken ? That information is needed to gauge your background and to get an idea of what books might be suitable. BTW there is nothing weird about Fourier transforms.