DrRocket

Finite-dimensional Vector Spaces by Halmos But for quantum mechanics you will also need to understand infinite-dimensional spaces, particularly Hilbert spaces. So you will also need something like Introduction to Hilbert Space and the Theory of Spectral Multiplicity, also by Halmos.

No. Difficult questions have simple, easy-to-understand, wrong answers. To understand any theory, and general relativity is no exception, you must invest intellectual capital, and that includes mastering the language in which it is formulated.

Differential geometry. For the details see Gravitation by Charles Misneer, Kip Thorne, and John Archibald Wheeler.

Tau, Yau, C. Fefferman, Perleman, R. Hamilton, Milnor, Thurston, Deligne, Vogan

What is your mathematical baqckground ? And how did you acquire such a task ? i can recommend sources for complex analysis, but I need to know how much you already know, and the nature of your objective. "Complex numbers to Mobius transformations" is a bit obtuse. That is not exactly a linear progression.. A fairly standard undergraduate text. Brown is a relatively recent addition -- the first edition was by Churchill. As with many texts Mobius transformations are called linear fractional transformations in that book (also bilinear transformations).

Nope

This is not really a mathematics question. Unless otheerwise stated explicitly pure mathematicians use exact values and rarely use decimal representations. Engineers and physicists almost always think in terms of "significant figures" and decimal representations are assumed to be rounded using the usual conventions. In very precise work figures are accompanied by explicit error bounds. A "~" in front of a number is interpreted as "approximately equal to" but carries no explicit error bound. As this is a common notation, an alternate use would be likely to lead to confusion. I would suggest sticking with the usual conventions and notation as understood in the engineering and physics community. Otherwise confusion is likely. That is your primary audience anyway. The system works. It isn't broken. There is no need to fix it.

Read the frst entence of my earlier post.

I have yet to see a news story, in print or on TV, about which I had direct knowledge and which was approximately accurate or even recognizable when reported in the media. Not sure why. One anchor made an A in an algebra class that I taught, so the mental ability is adequate in that case. But another anchor drank nearly my entire bottle of whiskey at a party for a departing third reporter who lived across the street. A print reporter screwed up a story after I had pointed out previous errors and he informed me that he was an expert in the subject of the story -- and used a picture of me in the process of hosing things up. None of these guys were Fox reporters, but Fox is no better. On the other hand Fox and the Wall Street Journal are both part of Rupert Murdoch's empire, and the WSJ seems to be as good as anyone for accuracy and depth.

Where in the world did you get that notion ? Sometimes in popularizations the explanation for Hawking radiation is given as the separation of pairs of virtual particle antiparticle pairs very near the event horizon of a black hole. That is an oversimplification and in any case the "gravitational field" near the event horizon of a massive black hole is unremarkable. It can in no way be called "ultra-strong". Gravity might be called "ultra-strong" well into the interior, but not at the event horizon, and it is at the horizon that the action is taking place. In truth, Hawking radiation is predicted by use of quantum field theory on curved spacetime, which is rather exotic and not yet well formulated. The strong force among quarks holds nucleons together. It hardly "tears them apart". Beta decay is mediated by the weak force. You could hardly be farther from reality.

I have not worked this out in detail, but my initial thought is that one might show that, given a,b and n>1 that for sufficiently large x_0,y_0 and x_1,y_1 the difference between ax_0^n + by^n and ax_1^n+by_1^is greater than 1 (so that there are infinitely many "holes" is the set in question). You can reduce this to the case where a,x_0, x_1,y_0, y_1 are positive and b is negative. Details (and verification that this works) are left to the reader.

Just a thought but, in place of opinions from people who have not performed a lick of scientific research, one might inquire as to the opinion of people who have actually made major contributions to fundamental science. http://depts.washington.edu/ssnet/Weinberg_SSN_1_14.pdf http://www.goodreads.com/quotes/show/132077 http://evolvingthoughts.net/2011/09/more-feynman-on-philosophers/ Having no tolerance for discussion of the philosophy of science, in seminars [Linus] Pauling did not hesitate to bring matters to earth with the remark, "I thought chemistry was an experimental science." http://peoplesworld.org/philosophy-is-dead-asserts-stephen-hawking-in-new-book/

It can be divided by a+30, a+50, a+70, any prime number that divides one of those and products of such primes.

And you either learned the theory prior to writing the program or increased the depth of your understanding by writing it. I would not call it cheating. Learned nothing and preserved that state by using the canned program instead of their brain. They were cheating.

Light moves at c. In a gravity well, between atoms in glass, or behind the barn, light always moves at c. Light is confined to the interior of the event horizon of a black hole by the curvature of spacetime. Period.

PHOTON ??

B is symmetric and positive definite, therefore diagonalizable with positive entries on the diagonal. http://en.wikipedia....alizable_matrix You don't have to diagonalize B, but I think it will make life easier. It certainly doesn't hurt and there is nothing lost. Note that you cannot necessarily diagonalize your matrices A and B simultaneously. They need not commute.

They haven't. In a sense "string theory" is intended to convey the impression of a mathematically consistent a la, say, Galois theory. In mathematics "theory" has a somewhat different meaning that in science, since the test of a mathematical theory is logic rather than experiment. Unfortunately string theory is not well-defined in the mathematical sense either. No one can rigorously tell you precisely what string theory is. Nor has it produced a new testable prediction. So when the word "string" is used in a title as an adjective, one requires a noun that it modifies, and "theory" more or less pops up as a knee jerk reaction. It can't be a hypothesis, since there is no hypothesis in evidence to be tested or evaluated. "Fantasy" seem unduly pejorative in this situaation since on the surface the idea has promise. And the practioners have too much personal investment to be willing to call it a "wild ass guess".

Calculators are very useful and appropriate tools in a physics or engineering class. They are useful in mathematics classes to avoid having to spend a lot of time with trig tables or log tables, or tedious arithmetic. They are useful to professional mathematicians for doing tedious calculations and balancing the check book. But calculators and computers are detrimental to learning how to think about mathematics in most situations not listed above. Your struggle with abstract math is a good example of why you in particular ought not use a calculator in a mathematics class. You will not learn to understand abstract mathematics except by struggling with the concepts using nothing but your brain. Plugging and chugging is detrimental to your intellectual health, as is any other mindless reliance on a tool that you don't really understand. So it is not that I have a problem with calculators. You have the problem, and apparently calculators contribute to it. In an industrial setting it doesn't matter how you find a useful answer, so long as it is correct, or at least close enough. But to get to that point you first need to understand the basic theories of mathematics, physics, chemistry and perhaps biology. There is no substitute for thoroughly understanding the fundamentals, and that means theory.

y'z + xy + xz = y'z + xy + xz(y + y') = y'z + xy + xzy + xzy' = (y'z + y'xz) + xy + xzy = y'(z +xz) + (x +xz)y = y'z +xy

You are right. I missed the minus sign. You can do that one using the hint I gave for the second limit.

"Quantum foam" was proposed by John Archibald Wheeler as part of an approach to quantum gravity. The idea, among other things is that spacetime may be multiply-connected (a mathematical condition related to having "holes") at a very small scale. It seems to have a role in tentative theories of quantum loop gravity, but thus far no one really has a clear ideawhat "quantum foam" really is or if the idea has any merit. A photon IS electromagnetic radiation. It does not itself radiate because it carries no charge and does not "feel" the electrmagnetic force. A photon is NOT "the smallest unit of electromagnetic energy". Rather in bound quantum systems, energetic states are discrete, and when electromagnetic energy is emitted it must be emitted in corresponding units. That means that individual photons that are emitted have energies corresponding to those units. But a photon without further qualification or reference to a particular bound quantum system is possible with ANY positive value of energy, and the energy is tied to the "color" or frequency of the photon by Planck's constant. [math]E=h \nu[/math] What Swansont has told you is correct. What Mystery111 has told you is somewhere between confused and just plain wrong.

" "Closer to the speed" of light relative to what ? The nucleus has the same average speed as the electrons relative to an external reference frame so there is no difference in the relative speed between the electrons and the nucleus when the atom is going at near the speed of light relative to some reference frame (the what).

The first one obviously grows without bound as increases. For the second one, rewrite it as something with an infinite limit times something with a zero limit then as a quotient of two expressions with zero limit and apply l'Hopital's rule. Burn your TI-89 and replace it with thought.

Anything that requires student involvement and significant intellectual participation is a step in the right direction. The Moore method, per se, is rather strict. Collaboration and any use of references are strictly forbidden. The student is expected to do all of the work on his own. Many faculty do not like the Moore method. It has the drawback that one cannot cover nearly the amount of material that can be covered if a text is employed, and very difficult material can become esssentially impossible. So there are a lot of less stringent methods that can be successfully employed. For instance you can use a text but require that each student not only read it but come prepared to explain the material to the class. Or you can assign the text as reading and spend class time with student presentations of (often challenging) homework problems. But in my opinion virtually anything beats a straight lecture except for very specialized and advanced subject matter when no good text or readable papers exist. But lectures can provide the platform for some great horse play: We had one algebraist who lectured and also used the students must be ready to lecture themselves technique. He was a great guy, but had a somewhat dour demeanor (which contributes to this story). In any case there were a couple of new Asian students who were advised by one of the older grad students that " When Professor X says something that you don't quite understand, the proper procedure to ask for clarification is to jump up from your desk and shout 'Bullshit !' ". When this happened, Professor X wheeled around, but noticed the older grad student rolling on the floor, unable to breathe, and figured out what was going on.