DrRocket

Uniform motion between inertisl reference frames in SR results in differing measurements of length, along the direction of motion. This is a coordinate effect which has nothing to do with "warping space" or "contracting space". The spacetime interval, determined by the Minkowski metric is invariant. SR is definitely a "special case". It can be viewed as either the case of general relativity in the absence of gravity or as a localization of general relativity (i.e. general relativity on the tangent space).

Right. http://www.stephenjaygould.org/library/gould_noma.html

Causility would be violated in special relativity if it were possible to send "information" at superluminal speeds. That is the reason that relativists proscribe such possibilities. But if a particle can travel at superluminal speed, then the logical foundation for special relativity is violated, and all bets are off. You then have a great deal of empirical evidence, but the theoretical explanation is no longer valid. There is no such causality problen in Newtonian mechanics. The problem is that Newtonian mechanics disagrees with experiment when high speeds are involved. IF neutrinos really traveled at speed greater than c we will now have a massive disconnect between relativity and experiment. That is a BIG deal.

If you were to interpret [math] 1^ \infty[/math] as a cardinal number [math]1^K[/math] where [math]K[/math] is an infinite cardinal then you would be talking about the cardinality of an infinite cartesian product of the singleton set. That cardinality is 1. As you note, if you try to interpret the expression in the context of real numbers, you can make all sorts of interpretations and get about any answer that strikes your fancy. "There's no sense in being precise when you don't even know what you're talking about." -- John von Neumann

There is a direct experiment being planned. It is at least a couple of years out. I do not have any details, but have been in intermittent contact with one participant on another forum. The group is not saying much.

Wrong. If you look at the Lorentz transformations you will find that superluminal speeds result in imaginary time dilation. What happens is that if superluminal information transfer were possible then the order of events is not preserved in all reference frames, resulting in effect preceding cause in some reference frames. The following argument is taken from Essential Relativity by Rindler. Fix points P and Q in spacetime and assume that a signal is sent from P to Q at some speed U>c. Choose a reference frame S such that the spatial coordinates of each lie on the x-axis. Select a second frame S' in standard orientation with respect to S moving relative to it at some velocity [math]0<v<c[/math]. Then in S' [math]\Delta t' = \gamma (\Delta t- \frac {v \Delta x}{c^2}) = \gamma \Delta t (1- \frac{vU}{c^2})[/math] For values of [math]v [/math] with [math] \frac {c^2}{v}<v<c[/math] one then has [math]\Delta t' <0 [/math] which means that in S' the signal is received before it is sent (i.e. P precedes Q in S but Q precedes P in S').

One more time. The fundamental Theorem of Algebra, usually proved using complex analysis, shows that any polynomial with complex coefficients has a complex root (the complex numbers are algebraically complete). It then follows from simple division of polynomials (Euclidean algorithm) that an nth degree polynomial with complex coefficients has n roots, counting multiplicity. The Abel-Ruffini, now usually proved using Galois theory shows that in general a polynomial of degree 5 or higher is not solvable by radicals -- i.e. there is no formula in terms of addition, subtraction, multiplication division and nth roots. This does not mean that some particularly simple polynomials of degree 5 or higher are not exactly solvable, but there is no generally applicable solution for quintic or higher degree polynomials. Numerical methods will allow you to approximate roots as closely as you wish, but will not yield exact solutions. If the polynomial has rational coefficients you can find a polynomial with integer coefficients having the same roots, and the rational root theorem will let you find any rational roots by checking all possibilities. But there may not be any rational roots.

Macroscopic matter certainly has volume. But elementary particles are currently understood as points. Then Paulimexclusion principle prevents two feermions from having the same quantum state state. It applies only to fermions, not to bosons. It does not prevent fermions from simply occupying the same position.

Partical accelerators are not an attempt to "reproduce conditions sililar to those that caused the big bang". They are simply an attempt to investigate the interaction of elementary particles at high energy. That is a LONG way from big bang conditions. "Cause of the big bang" is an oxymoron in our current best theory. Black holes, so far as is understood, are in now way analogs of particle accelerators. It is a GIGANTIC stretch of the imagination -- a hallucination -- to think otherwise. Maybe you won't be surprised, but you should be astounded, if all the m,ass and energy of the universe came from black holes.

A few points. 1. The discipline of probability theory starts with a probability space -- that is with events and their probabilities being known. Probability theory does not involve the determination of the probabilities of elementary events. 2. The estimation of probabilities is the province of statistics. Statistics quite often relies on assumptions that are convenient but not rigorously justifiable. The results are at best approximate, and when the assumptions are grossly violated can be just plain wrong. 3. What can be proved rigorously is that given an event of non-zero probability, in infinitely many independent trials that event will occur infinitely often with probability one (aka almost surely). So if enough vehicles pull up beside you it is highly likely that eventually one of them will have a dalmation dog in it. 4. Probability theory is one of the most misunderstood and misapplied branches of mathematics. All sorts of nonsense is "proved" or justified based on misapplications of probabilitry theory. 5. Just today I saw a car with the license plate VZK 9025. Just imagine the odds against that!

No, you simply need to know the point as a function of t, then differentiate. You do not need to find y as a function of x. In fact the parameterized curve need not be the graph of a function -- for instance it can have closed loops.

No. [math]\sqrt 2 [/math] is exact even if it is not representable by a finite decimal expression. For the general case of polynomials of degree 5 or larger no solution as an algebraic expression in terms of nth roots exists. It is not that no one has found such a solution. It has been proved (see a text on Galois theory) that no such solution exists. The fundamental theorem of algebra (see a text on complex analysis) shows that roots for any polynomial exist, at least as complex numbers. But Galois theory shows that you cannot them, in general, exactly (as radical expressions. This does not mean thjat some particularly simple 5th degree polynomials cannot be solved exactly, but no formula exists for an arbitrary 5th degree or higher polynomial. "e" is not only irrational, but is also transcendental. That means that it is not a root of any polynomial with rational coefficients.

That is one way. A more geometric approach involves finding the tangent vector to the parameterized curve. If [math]\gamma (t) = (x(t), y(t)) [/math] is your curve then the tangent vector (velocity if [math]\gamma[/math] represents position) is just [math]\gamma '(t) = (x'(t),y'(t))[/math]. The slope of the tangent line at some point [math]\gamma (t_0)[/math] is then just [math]\frac{y'(t_0)}{x'(t_0)}[/math] This generalizes immediately to curves in 3 or more dimensions, except that "slope" of a tangent is not a defined notion.

I have taught enough calculus classes to be aware of the capabilities of the average calculus student. Noting that a decimal expansion has countably many instances of the set of symbols 0,1,2,3,4,5,6,7,8,9 is well within those capabilities. That is what "decimal places" are all about. There is no overestimation involved. In point of fact any decent high school algebra student can handle this easily.

Read what you have written, perhaps after reading the books suggested. If you cannot find several examples quickly then further discussion would be futile.

This is more subtle that you might think. Start by reading a book on special relativity. Rindler's An Introduction to Special Relativity is a good one. Pay attention to the "relativity of simultaneity". One observer's "now" need not agree with the "now" of an observer in relative motion. If that is not enough, then look into general relativity and the notion of a Cauchy surface.

I am sure such a theory could be devised if anyone could figure out what "the matrix of energy fluxuations, accelerations, and particle collisions surrounding the event horizon of the black hole" means. You might Google "acretion disk".

The proof is the trivial fact that any real number can be represented as an infinite decimal. You ought to be able to find this in any elementary text on real analysis. For a nice accessibe account of the theory of cardinal numbers read Naive Set Theory by Paul Halmos. A reading shopuld correct your many misconceptions. Wrong. Go read the referent book by Halmos. The fact that you are all wet is pretty obvious to anyone who understands simple set theory. You have made several misstatements regarding well-known elementary facts from the theory of cardinal numbers. Wrong again. There is no such "number". Read my post to see that the cardinality of the real numbers is [math]2^{\aleph _0}[/math]. If you are unfamiliar with the real numbers then you really need to study some remedial mathematics. This is both true and well-known to even a freshman calculus student. And most certainly your original post makes no sense.

Some people have 40 years of experience. Some people have 1 year of experience 40 times.

1. Your answer is correct. 2. The answer in the text is nonsensical --- a) it is not any kind of equation for a line, just a number for any fiven value of t, and b), it is not defined, even as a simple scalar, for t=0. I don't have that book. You present an excellent case against acquiring, or using, it.

Getting a patent involves some expense ($5K-$20K or so) in conducting searches and properly phrasing the application. A good patent attorney can patent almost anything. Having the patent upheld in court in the face of a challenge is another (expensive, sometimes $ millions)) kettle of fish.

This makes no sense. Your reasoning is nothing but errors. Any real number can be represented as an infinite decimal -- so with at most countably many ([math]\aleph _0[/math] symbols. But the cardinality of the real numbers is strictly greater than [math] \aleph _0[/math] I have no idea what "anumber who has 2^C symbols to represent it" would be, andneither do you. Whatever it might possibly be, it most certainly is not a real number. [math] \aleph _0 [/math] is the cardinality of the natural numbers, and any set of cardinality [math] \aleph _0 [/math] is called "countably infinite". If a set has cardinality [math]K[/math] then the "power set", the set of all subsets of that set has cardinality [math] 2^K[/math] -- this is easily seen to be true if K is finite and is the definition of [math]2^K[/math] if [math]K[/math] is infinite. It can be proved that [math]2^k>K[/math] for any cardinal number [math]K[/math]. If [math]c[/math] denotes the cardinality of the real numbers then [math]2^{\aleph _0} = c [/math]. The continuum hypothesis asserts the non-existence of any cardinal [math]K[/math] such that [math] \aleph _0 < K < 2^{\aleph _0}[/math]. It has been proved that the continuum hypothesis is independent of the usual Zermelo Fraenkel axioms of set theory.

The latest estimate seems to be that it is less than 145 Gev. http://www.technewsworld.com/story/73450.html http://www.sciencenews.org/view/feature/id/334164/title/Last_Words http://news.sciencemag.org/sciencenow/2011/08/hints-of-higgs-boson-appear-weaker.html

It is BS and Anderson is a nut.

Be careful. Apparently typos were introduced when the 6th editions was typeset. This one, at least, does not occur in the 4th edition.