DrRocket

Note that solutions are congruence classes, as is stated earlier in the text (though you did not quote that part. Since, as you quoted, Hardy and Wright explicitly list the d solutions, I don't understand what it is that you do not understand. There are d of them -- (fixing the typos in the 6th edition) [math] t, t+m', t+2m',...., t+(d-1)m'. [/math]

Try studying. Reading your text is a plus.

In classical Newtonian/Maxwellian physics the two are completely independent, and that is a very good approximation except in extreme circumstances. In general relativity both mass and energy, including electromagnetic energy contribute to gravitation, but it takes a lot of energy to have much real effect. If you throw a magnet through a ring, a lot depends on whether or not the ring is conductive. If the ring is conductive an eddy current will be set up in the ring that will create a magnetic field that opposes the motion of the magnet. This "magnetic damping" is often used in laboratory scales. If the ring is not conductive then nothing unusual will happen -- maybe you score a point depending on the rules of the game.

Any polynomial of degree n with complex coefficients has exactly n complex roots, counting multiplicity. This is the Fundamental Theorem of Algebra, which is usually proved in a class on complex analysis. There exists no "solution by radicals" (i.e. no solution involving just roots addition and multiplication) for general polynomials of degree 5 or higher. This is usually proved in an algebra course on the Galois theory of fields. So, roughly speaking, the roots always exist, but you can't find them exactly.

But I can imagine 3i apples.

The speed of a compression wave in air is NOT the "speed of sound. The speed of such a wave is weakly dependent on the amplitude of the wave, as well as the temperature. The "speed of sound" is the limit of the speed of propagation under given conditions as the amplitude vanishes. Moreover the "speed of sound" is relative to the preferred reference frame in which the air is stationary. A flying bat would not represent that frame. Thus the speed of a compression wave in air does not mimic the axiom of special relativity under which the speed of light in a vacuum is independent of the inertial reference frame , and there would not be any analogous "theory of relativity" for bats.

I lack the patience to go through your calculation in detail, but you should realize that the generalized Stokes theorem (in the form that elementary texts call the divergence theorem) states that [math]\int D \cdot ds[/math] = [math]\int \nabla \cdot Ddv[/math]

Only as an approximation. At the appropriate level of precision, there will be deviations from special relativity. This must be taken into consideration when looking for violations of Lorentz symmetry. Better go into some of those details. What are "two inertial systems that move in parallel with same speed" ? Same speed with respect to what ? Parallel to what ? Do you mean two inertial reference frames that are stationary with respect to one another, with parallel coordinate axes ? If so the transformation is simply translation. The two frames are simply choices of an origin for parallel axes s in a single affine space. This case is utterly trivial.

EMP is over-hyped in the press. It is fairly easy to protect against high altitude EMP (HEMP), which is basically a plane wave pulse of high amplitude.. I once stood in a full-scale threat-level EMP test bed, and had forgotten a cheap credit-card calculator in my shirt pocket. It was exposed to a full threat-level pulse -- with no ill effects. This is not to say that EMP is no threat at all. But if a decision is made to shield from HEMP then protection is relatively easy to implement. Source region EMP is another kettle of fish entirely.

Are you out of your mind ?

I doubt that any amount of estrogen could fix this . Maybe alcohol ?

Wrong. Read my post again. You can ndivide by x^2+9 but you cannot do it synthetically. Synthetic division applies only to linear factors. To divide by X^2+9 you must divide by x-3i and x+3i sequentially. Don't tell your grandmother how to sucfk eggs.

Yes. I have a room full of mathematics, science and engineering books, including almost all of my old text books. Knowing where to look is a great help when refreshing one's memory.

The lion's share of that 90% is taken to be "dark energy" which produces a repulsive effect accelerating the expansion of space. Black holes produce only a conventional gravitational attraction.

Care to elaborate ? If this is an emergency dial 911.

So you can see what the roots are. Since none of them are rational numbers, why would you expect any simplification from the rational root theorem ? If you plug in the obvious roots, which are imaginary, you should get a simplification using synthetic division, or ordinary polynomial division. Remember that synthetic division only works for factors that are first degree polynomials, so you would have to use complex numbers -- you cannot divide synthetically by x^2+9 or x^2+1.

Absolutely. Janus is correct, no matter what "vote" you may have in mind, or what Wiki might say.

If this not a homework problem then it ought to be one.

This appears to be a duplicate of your thread in the homework section. We will not do your work for you. Further in that thread you state an urgency that indicates that this is work for credit. I will not help you cheat.

The fact that some polynomial has no rational roots does not mean that it has no real roots. E.g. [math]X^2-2[/math]

nonsense Wki is not always either clear or accurate. It becomes even muddier when you quote something out of context that you obviously do not understand. Now that you have copied a piece on cardinal numbers, go back and study it until you understand what it says. As an exercise you might test yourself by explaining why this is totally irrelevant to the question of whether the universe is finite or infinite. Hint: to say that the universe is finite is to say that a the decomposition of spacetime as a one-parameter foliation by spacelike hypersurfaces results in those hypersurfaces being compact manifolds. See the thread on cosmo basics. Further hint: Before you challenge mainstream science at least have the good sense to understand what it says and know that which you challenge.

The problem with the stated urgency is that it makes it quite clear that the OP is responsible for turning in or presenting the solutions to these problems very soon, probably tomorrow. Moreover the nature of the questions is that of a set made up to test a relatively broad set of concepts. It is much too broad to be a set of simple homework exercises for a single section of a textbook or set of lecture notes. That makes this not really homework, but more like a take-home test. It clearly carries credit. When I took calculus, and when I taught calculus, the use of help such as is being sought here would have been considered blatant cheating. These problems are not all that hard, but they do take a bit of thought -- rote application of cookbook calculation methods won't do. This makes it more disturbing that the OP has shown zero work of his own and is asking for solutions. I am not inclined to abet cheating.

Wrong. If you had sufficient knowledge to understand what I have said thus far you would realize that I have stated that the universe may be either finite (closed) or infinite (open). Now go read enough to be able to understand an answer when it is presented to you.

You need to learn to read with comprehension. The NASA piece does NOT state that the universe is finite. It states that within the observable region it appears to be nearly flat. That tells one absolutely nothing about the global topology. If you ASSUME homogeneity and isotropy then a very slightly negatively curved or flat universe would be open, while an infitesinally positively curved universe would be closed. If one relaxes that assumption a bit then one can have flat but closed topologies. Now go read some of those references and learn about compact and open manifolds and what curvature really means. The problem with a great many popularizations is that they sacrifice accuracy and truth for easy-to-understand but wrong or misleading explanations. They tend to be interesting if one understands enough to know when the author is stretching the truth or just plain misrepresenting it, but neophytes can get an impression of understanding when they are getting the wool pulled over their eyes. Several recent books by string theorists are particularly egregious. Because of that I am hesitant to recommend popularizations by any other that first-rank physicists -- Feynman, Weinberg, 'tHooft. I have yet to see a good popularization of mathematics or mathematical physics, and know of only two by first-rank mathematicians -- The Road to Reality by Roger Penrose and The Shape of Inner Space by Shing-Tung Yau and Steve Nadi. Those books are really more concerned with physics than mathematics and are very careful to separate conjecture from what is known. It is extremely difficult to separate mathematics from the attendant technicalities -- you lose the substance when you do that. Physicists take out the mathematics when they popularize physics, but you can hardly popularize mathematics by taking out the mathematics. That is the main reason that mathematics is so widely misunderstood outside of the mathematical community itself. The infinite is actually fairly simple. Finite mathematics tends to be more difficult. Halmos's book Naive Set Theory requires essentially no background, just "mathematical maturity". You are right in that the OP probably lacks the wherewithal for the references that were supplied. But one can only dumb down things so far. If one wants to tackle problems at the frontiers of science, one cannot do so without advanced knowledge (and one really needs to go beyond what has appeared in book form). If the OP wants to play with the big boys then he needs to get to that level. One can learn a great deal from simple accounts, but not enough to argue that the mainstream is wrong.

I will not do your work for you. You have been given more than adequate references. There is nothing worthy of debate or even consideration in your assertions. Before you challenge mainstream science you must first understand its content.