DrRocket

No such experiment has yet been performed. However, I am aware of one that is being planned, and may be carried out in the next two or three years. I have no idea how this experiment would be done. Richard Feynman was as good as anyone at producing clear and elementary explanations of physics. Here is what he had to say on the subject: "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

It is really constant. If you want to trce the definition back rigorously to the fundamental Zermelo Fraenkel axioms you can do that and bypass circles and geometry (it is still there if you dig hard enough). This can be easily done: The series [math] \displaystyle \sum_{n=0}^ \infty \frac {(iz)^n}{n!}[/math] can be shown (see e.g. Real and Complex Analysis by Rudin) to define an entire periodic function and [math] \pi [/math] is defined to be one-half of the period. You are free to define [math] \tau = 2 \pi [/math] if you like. The point is that this rigorous definition does not explicitly rely on geometry or trigonometry, so there is no question that pi is just a (constant) number.

No luck invol;ved. No experiments either. Pure mathematics, ajb and I are both mathematicians, so perhaps it was not clear to others that we were talking about mathematics research.

Good idea. Let's arrange for a convention with the Girl Guides of New Zealand, and the Swedish Bikini Team to discuss this issue. Don't forget the beer. I will be happy to represent the AMS, of which I am and have been a member for a long time. IUPAC doesn't get a vote, since this is not a question of chemical nomenclature. I intend to question the GGNZ and SBT at length, uncover the bare essentials and get to the bottom of the issue (and the bottom of the keg).

The origin of the normal distribution was in the work of Gauss. The distribution is also known as the Gaussian distribution. Besides being the limiting density of a sum of independent identically distributed random variables, the central limit theorem, it is also distinguished by the fact that it is it's own Fourier transform and that a convolution produuct of Gaussian densities is again a Gaussian density.

Everyone in industry is familiar with the "pi-tape" which lets you measure the diameter or radius by measuring the circumference. Training technicians to use a "tau-tape" would be costly and pointless, not to mention the need to change the name of the corporation that makes the gage. Not to mention the fact that most experts can multiply by 2.

He is very bright. There was no luck involved. He did what should have been a very good dissertation as a junior, and then did another one later on that counted and was major work in the field. His supervisor didn't do very much except challenge and get out of the way.

There are no formal requirements like that, but it is not uncommon to have done either or both. Conference talks are easy. Publication has become much more difficult. In my case I gave a talk at the invitation of an organizer of a special session at the annual AMS meeting. He was a "big shot", while my advisor was relatively young. He said he thought that my result was a good dissertation, and my advisor agreed. He also said to send it to him for publication in a journal of which he was an editor. So I wrote it up for publication, then adapted it to dissertation format. In short, my sequence was a bit different: talk--->acceptance for publication ---->submittal for publication ----> dissertation ----> degree Note that the guy I mentioned who took three years from the start of his sophomore year, was 4 years out of high school when he received the PhD. That is exceptionally quick. The primary formal requirement is that one write and defend a dissertation in front of an examining committee chaired by the advisor. In practice that means that the advisor must approve. Very few dissertations are rejected by the committee if the advisor approves, and even going to the defense over the objection of the advisor is suicidal.

To add to what swansont already told you, PhD programs are usually quite flexible. The PhD is a research degree and the primary requirement is a significant original contribution to the field. Course requirements are incidental to that. The only requirement that is hard and fast is an acceptable dissertation. I know of one very fine PhD who received the degree in about 4 years after matriculating as a freshman. I know another who took about 10 years in graduate school. Lots never make the grade.

Then by choosing prinhciple axes we can make [math]\tau[/math] go away. So [math]\pi[/math] is universal, but [math]\tau[/math] is just an artifact of the local coordinate system.

The [math]\tau[/math] required for proper TARDIS operation is torsion in the geometry of spacetime, not just [math] 2 \pi [/math] (as any geometer ought to realize). So, get with it and we can still meet for that beer yesterday. BTW Bob Palais recognized that torsion competes for the [math]\tau[/math] moniker in that Intelligencer article linked earlier in the thread.

Only if you can get over here tonight. We can invite Bob Palais and the math department at "The U" -- to this bar, where they used to congregate in the past. http://utah.citysear...ifes_place.html Let me know if you can arrange supersonic transport. Happy hour should start in a couple of hours.

Nothing published in the Ontelligencer ought be taken too seriously. "Real" mathematicians, and I am one and so is Bob Palais, don't worry overmuch about whether one needs to multiply by 2 or not. Back to the beer.

Using the same meaning for the bar produces a rather inscrutable way to write 1/2. And eliminates a perfectly good reason to go get another beer.

I would be happy to. I know that department pretty well. Where has Bob taken a position on this, trivial, issue ? If you mean this tongue-in-cheek piece, then I suggest that you have another beer.

"Tau Day revelers" either a) understand very little real mathematics b) really like beer or c) both.

The first sentence is true but irrelevant. The second is no excuse.

The statement of the problem is a bit odd. But what is going on is this: The second equation constrains x4--x8 in a way that can be used directly in the first equation to find a relation among x1--x3. Those two relations let you find lots of simultaneous solutions of the two equations.

One difficulty with probability theory is that you can never prove anything, except theorems about probability. Nevertheless you can draw inferences and you can become damn suspicious. It is a useful tool in the right hands. It can be badly misused by those who do not understand it. Any event with any non-zero probability (like a string of 100 or even 1,000,000 consecutive heads in coin flips) will occur with probability 1 in an infinite number of trials. The operative word is "infinite" and not just "big". Even probability zero events can occur, just not very often. So, it is possible that in an infinite number of trials with one particular coin, 100 consecutive heads will not occur. It is even possible that only tails will occur. In fact ANY string could occur. Moreover, the probability of any specific infinite sequence is 0, including whatever string is actually produced. This is simply a result of how probability measures on infinite product spaces are built. So, when confronted by an event of extremely low probability you have two choices: 1) You can believe that the event occurred narurally as a result of fluctuations of the cosmos or 2) You can question the underlying assumptions. In my experience 2 is the most profitable course of action. Example: Once I was looking at data from a small but critical component. The leakage current was low, "minus seven sigma". The guy responsible for the component thought this was just dandy since low leakage current is generally good, and "minus seven sigma" events still have a non-zero probability of occurrence. But minus seven sigma events do not occur very often, the world is not really normally distributed, and I thought it much more likely that something was amiss in the manufacturing process. There was a well-publicized and several billion $ project potentially depending on the component working properly. Over the strenuous objections of the other guy, the component was pulled and re-tested. It failed. A lot depends on what is really known about the probability distribution, what is assumed, how many trials are involved and what is known about the associated physics. If you are gambling and a coin has shown 100 consecutive heads, I would suspect a skunk in the wood pile and expect either another head on the next toss or a deft switch of coin by the con man flipping it.

You have two linear equations, x1+x2+x3-x4-x5-x6-x7-x8=0 and x4+x5+x6+x7+x8=0, in eight variables, an undertermined set of equations. The term "solution set" normally refers to the set of all valid solutions to the given equations. One would suppose then that the given task is to show that the solutions that you list belong to that set. Perhaps what is expected is that you will characterize the complete solution set and then show that the given solutions lie within it. Have you tried to do this, by analyzing the two given equations and maybe combinations of them ?

There are some established techniqies for doing cutting and gluing that are, no kidding, called "surgery" and "plumbing". http://en.wikipedia..../Surgery_theory http://mathworld.wol...m/Plumbing.html http://www.maths.ed....rs/kirbysch.pdf Added in edit. John Milnor, who introduced surgery, wrote this brief piece on 3-manifold classification with commments on Perelman's work shortly after Perelman's announcement and before it had been carefully checked by the community. http://www.math.sunysb.edu/~jack/PREPRINTS/tpc.pdf

Time is that which is measured by a clock.

You need to be a bit more clear as to what fields you are talking about. Most theories that involve fields are modeled on ordinary Euclidean space, which may be what you mean by infinite. If so that has nothing to do with expansion. Generally when people speak of "expansion" they are talking about cosmology and the expansion of space, which relates to a model based on general relativity. Current quantum field theories are incompatible with general relativity. So perhaps your question relates to some unified theory that would include quantum gravity. No such theory yet exists, but it is an active area of research. Don't hold your breath. It has been an active area of research for a long time.

It is not true that infinite decimals are unique. There are some exceptions. 1.0000.... = 0.999999.... and variations on that theme. Repeating decimals represent rational numbers and only rational numbers. It is a theorem that a number is rational if and only if the decimal representation is finite or repeating. Irrational numbers have decimal representations that are neither finite nor repeating. Very very little of mathematics is dependent on any number base, so using a different base would not have much impact, and would just be inconvenient. Usually the term "number system", refers to some algebraic structure such as the natural numbers, the integers, the rational numbers, the real numbers, the complex numbers, or even the quaternions, Cayley, numbers, etc. Theree are also some truly abstract systems such as the "non-standard real numbers" and hypercomplex numbers. All of these things have been studied. What might surprise you is that almost all of them are logical consequences of the natural numbers {0,1,2,3,4,...} (some people leave out 0 and add it in later), and basic set theory. This is usually in the form of the Peano Axioms, or in formal set theory the Zermelo-Fraenkel axioms (usually with the axiom of choice added in for more esoteric applications). (The non-standard reals and hypercomplex numbers use a construction that requires the axiom of choice as I recall). For most of mathematics the integers, rationals, reals, complex numbers and things constructed from them with ordinary algebraic constructions (quotients, polynomials, field extensions, etc.) are all that you need.