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You don't need a proof for everything, triclino. Suppose a mathematician, after some arduous calculations in a math paper, arrived at [math]x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 0[/math] Nobody would blink an eye if the next sentence in the paper was "and thus the only solution in the reals is the trivial solution." If you need a proof, first prove that The sum of a finite set of non-negative numbers is non-negative. The sum of a finite set of non-negative numbers, at least one of which is positive, is positive. Note that I said a finite set. Whether [math]1+1+1+\cdots = \sum_{n=1}^{\infty}1 = -1/2[/math] or [math]1+2+3+\cdots = \sum_{n=1}^{\infty} n = -1/12[/math] are different questions.

Now that we're in the proper forum, You need to show some work before we can help you, triclino.

shyvera chose to look at the problem from the perspective of finding the values of a that make the inequality true given some particular value of b. You can certainly look at it from the alternative perspective, finding the values of b that make the inequality true given some particular value of a. Both points of view yield the same set, just described in two different ways.

Exactly. Adding the same quantity to both sides of an inequality does not change the inequality.

Oh, please. [math]a^2+ab+b^2 = a^2 + 2ab + b^2 - ab = (a+b)^2 - ab[/math]

If you complete the square you should get [math](a+b)^2 \ge ab[/math] Yet another way to prove this is to find a 2x2 matrix A such that [math]a^2 + ab + b^2 = \bmatrix a & b \endbmatrix \boldsymbol A \bmatrix a \\ b \endbmatrix[/math] There is only one such matrix, and it is positive definite.

There are no other solutions if x, y, and z are restricted to the reals.

If x, y, and z are limited to the reals, what is the only solution to equation (2)?

Hint: You should be able to readily prove that this is the case when ab<0. The cases where ab=0 are even easier. That leaves ab>0. Complete the square.

No. While you are correct that [math]q\wedge\neg q[/math] is a contradiction, it is not the only thing that qualifies as a contradiction. A contradiction is any statement that is unsatisfiable. x=0 and x=1 is unsatisfiable. It is a contradiction.

The contradiction here is x=0 and x=1. Think about it.

No you can't, triclino. x=0 and x=1 implies absolutely nothing. x=0 and x=1 is a contradiction. The only conclusion that can drawn from (1) P -> Q and (2) ~P is that P is false.

Sure you did. You asked for "governing body restricting the supply of oil." Why the bleep do some people insist on poking their noses in other people's bedrooms? Some want to peep at our sex lives, you want to peep at our thermostat settings. Get out of our bedrooms, the whole lot of you!

Isn't that rather obvious?

For once I agree with you, bascule. That is a much more apt comparison. Those are two sad clowns. A much more apt comparison is Cooper to Hannity.

The basic concept at play here is conservation of momentum. No. Think of it this way. When you jump off the floor of a train speeding along at 60 miles per hour you don't suddenly go flying toward the back of the train at 60 miles per hour relative to the train. Similarly, when you jump off the ground at the equator you do not suddenly go flying toward the west at more than 1000 miles per hour relative to the earth. If that did happen, you would feel a 1000+ mph wind and during a two foot jump you would travel over 1000 feet (ignoring that 1000+ mph wind). The reason this doesn't happen is conservation of momentum.

Oh. Then welcome to our world.

To be brutally honest, if you don't understand the concept of an inner product then higher dimensional geometry really is not something you should be looking into yet. You need to learn to walk before you can learn to run ...

Back up a second, everyone. Bignose and the tree, you two are talking about the inner product. Mukilab appears to be talking about an extension of solid angle to higher dimensions. Those are completely different things. Edit Here's an article on computing solid angle in higher dimensions: http://karthikshekhar.wordpress.com/2009/05/23/computing-the-solid-angle-for-a-d-dimensional-sphere/.

I am so glad you chaps got things straightened out because that means one of the best comedy shows on the telly, Prime Minister's Questions, is back from reruns.

Almost. The derivative of F(x) is equal to f(x) almost everywhere. Click on the 'almost everywhere' in either my previous post, or in your quote of my post. Or google the term "almost everywhere".

The first one, obviously not. The latter, it depends on what the Supreme Court decides. That is, assuming that (a) Congress makes Hilary's wish list into law, and (b) someone appeals to the Supreme Court (which would almost be a certainty if (a) occurs.) Merged post follows: Consecutive posts merged I am not arguing for this. I am just avoiding joining the band wagon in decrying it as inherently evil.

That's asking a bit much of the Constitution, bascule. Where in the Constitution does it say that it is acceptable for the federal government to have an Air Force? The Constitution is intentionally a high-level, and somewhat vague, document. The details are in Supreme Court's interpretation of the Constitution.

From the wikipedia article on Afroyim v. Rusk, The decision did not change these other aspects of U.S. immigration law: Someone who committed fraud in the naturalization process (for example, by lying about themselves to U.S. immigration services) could still have their naturalization voided on the grounds that they had never truly been naturalized as U.S. citizens in the first place. Naturalization applicants could still be (and indeed, as of 2008, still are) required to make a statement under oath or affirmation, renouncing any prior allegiance to any foreign country or ruler, upon becoming a U.S. citizen. A naturalized citizen who behaved in a manner inconsistent with this oath (for example, by continuing to use the passport of their previous nationality) might—at least in theory—still be liable to loss of U.S. citizenship on the grounds that the oath had not been taken in good faith (and hence that the naturalization was fraudulent). However, U.S. State Department policy since 19902 has been not to pursue such cases. Also important is Vance v. Terrazas, http://en.wikipedia.org/wiki/Vance_v._Terrazas Note this ruling occurred after Afroyim v. Rusk. The issue at hand is whether committing a terrorist act Constitutes fraud in the naturalization process (and hence the person never really was a citizen), or Suffices as evidence of intent to relinquish citizenship (and hence Vance v. Terrazas applies).

How does Clinton's statment run contrary to the 14th amendment? That amendment restricts states, but not the federal government. To make bascule's day just a tad better, http://www.google.com/hostednews/ap/article/ALeqM5jAz5HTbaFcgiGhmbogdxlE2QCDzQD9FJFQLO0 In the wake of the Times Square bombing plot, the Obama administration said on Sunday it wants to work with Congress on possible limitations of the constitutional rights afforded terrorism suspects — even for American citizens. Note that this comes from the AP, not the dreaded FoxNews.