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It's not just close enough to being zero with regard to computations. Those small masses are upper bounds. Do you understand what that means?

You are misreading the article. It very clearly says that 10-14eV is an upper bound on the mass of the photon. What the article does not explicitly say (and probably should say) is that a mass of zero is completely consistent with those experimental results. There is always some amount of error associated with every scientific experiment. The theoretical value of zero is well within those experimental bounds. In this case, eV, or electron-volts. Strictly speaking, an electron volt is a measure of energy. Since mass is a form of energy, units of energy can be converted to units of mass. 1 eV is about 1.60×10-19 joules. In terms of mass units,that's 1.78×10-36 kilograms or 1.78×10-33 grams. The 10-14 eV upper limit on the photon mass cited in that article is equivalent to 1.78×10-47 grams.

That works. So does saying that |x|=x for x>0, |x|=-x for x<=0. Or a three-way case, [math]|x|=\begin{cases} \phantom{-}x & \text{if~}x>0 \\ -x &\text{if~} x<0\\ \phantom{-}0 & \text{if~}x=0\end{cases}[/math] I like the latter as it explicitly identifies 0 is a special case and because of the symmetry. That's just personal preference. There are many other ways to write it. For example [math]|x|=\sqrt{x^2}[/math]

Just because something sounds unreasonable to you does not mean that it is wrong. Einstein did not just grab his hypothesis on the constancy of the speed of light out of thin air. The constancy of the speed of light is a consequence of Maxwell's equations and it is consistent with all experiments conducted to date. Moderation action: Moved to the trash can.

These data do exist to some extent. That is, in part, the job of the CBO and GAO. That we have both a CBO and a GAO is an example of government inefficiency in action. Devils advocacy: Why not combine them? The argument against combining them is checks and balances. Our government was built from the onset to be inefficient. You want an efficient government? Look to a dictatorship. Bullets are cheap. Our government is inefficient by design, and for a good reason. To quote the person who authored the Declaration of Independence, doubled the size of the US, organized the Lewis & Clark expedition, but nonetheless remains a rather unimportant President, The government which governs best, governs least. One way the founding fathers insured that our government will govern least was to create built-in inefficiencies. Let's look at this example. The USDA gets about 3.9% of the federal budget. So right off the bat this 25% figure is off by almost an order of magnitude. To make matters worse, the vast majority (70%) of the USDA's budget goes to people who don't live on farms. Many of those recipients have never seen a farm. Some can't even spell the word "farm" (illiteracy runs rampant among the urban poor). After taking out food stamps and other such programs, the portion of the federal budget that truly does go to agriculture is a paltry 0.75% of the total. So even if those programs are 100% inefficient (and they aren't; federal subsidies to farmers represents a good chunk of farmers total income), getting rid of the inefficiencies will make a barely perceptible dent in our budget morass. First off: Get real! The people who would have to make campaign receipts taxable are the people elected in large part thanks to those campaign funds. Secondly, campaign contributions are already taxed in a sense. They are not deductible. Suppose, for example, that you gave $500 to Obama's campaign last year. If you declared that $500 as a deduction on your tax form you broke the law. You have to pay taxes on that money.

Dang, Bascule! You aren't just advocating touching the third rail here, you're advocating touching all of the third rails simultaneously, and cross-tieing them too boot. The half-life of the most recently discovered element would exceed yours as a politician. That said, I can't disagree with you much, last item excluded. I do mind paying more taxes, but I also do accept that doing so is inevitable.

A freeze on discretionary spending without touching mandatory spending won't even come close to fixing the problem. Suppose the goal is to just make the budget neutral; neither a surplus nor a deficit. While this won't cure past excesses, it will stop the bleeding. You want no new taxes, and you don't want to touch Social Security, Medicare, or Medicaid. OK then. That means an across the board 40% cut to everything else. There is no easy solution to this problem.

We are, collectively, a bunch of blithering idiots. Almost nobody (7%, which means lunatic fringe numbers) wants to touch Social Security or Medicare. Medicaid garners a paltry 11%. We could zero out *all* discretionary spending (i.e., almost everything on that list), get out of both wars, cut all earmarks, and we will have only have made a tiny dent in our current budget mess. The mandatory spending is where the problem lies -- and any politician who dares attack that will be sent home packing by us blithering idiotic voters.

What do you think? Have you searched the web?

Triclino, instead of arguing (and arguing with an administrator is not very bright), try the hint. Heck, it wasn't a hint. I gave you the answer.

For crying out loud, triclino. Do look things up first. Here: http://en.wikipedia.org/wiki/Floor_and_ceiling_functions. What is the definition of uniform convergence? What is the maximum of the absolute value of your function less the limit of the sequence?

Learn your symbols, triclino. [math]\lceil x\rceil[/math] is the the ceiling function. For example, [math]\lceil \pi \rceil = 4[/math]. Edit I should have said [math]N=\lceil 1/(4\epsilon^2)\rceil+1[/math] to handle the rare case where [math]1/(4\epsilon^2)[/math] is an integer.

You left out your "otherwise" in post #7 from the above, triclino. Furthermore, using A.B for the cross product is very, very bad form. That period looks a lot more like a dot than a cross. This is not a minor detail since there are many products for vectors. For example, the inner or dot product, the cross product for vectors in 3- and 7- space, the outer product, the exterior or wedge product, etc. Each has its own symbol and none of them is denoted with a period.

[math]N=\bigl{\lceil}1/(4\epsilon^2)\bigr{\rceil}[/math]

Is wheelin' and dealin' allowed ("Vote against this next proposal; my turn is next and I'll reward you nicely")? Merged post follows: Consecutive posts mergedWithout wheelin' and dealin', the top person offers a four-way split amongst himself and execs #3, #5, and #7. Let's suppose it goes all the way down to #6. #6's proposal is easy: Everything goes to #6. #6 will vote for this proposal, garnering the needed 50%. Exec #7 will be screwed if it gets down to #6. Before it gets down to #6, #5's proposal needs to be sunk. Unlike #6, #5 needs 1 cohort to go along with his proposal. #7 will go along with anything #5 proposes so long as it is better than nothing. Exec #6 will be screwed if it gets down to #5. Before it gets down to #5, #4's proposal needs to be sunk. #4 also needs 1 cohort to garner 50%, the obvious target being #6. Execs #5 and #7 will be screwed by #4's proposal. Before it gets down to #4, #3's proposal needs to be sunk. #3 needs two cohorts, the two who would be screwed by #5's proposal. #3 offers a three-way split between himself, #5, and #7. Execs #4 and #6 will be screwed if it gets down to #3. Before it gets down to #3, #2's proposal needs to be sunk. #2 also needs two cohorts to get 50%, and these are execs #4 and #6. Execs #3, #5, and #7 will be screwed if it gets down to #2. Before it gets down to #2, #1's proposal needs to be sunk. #1 needs three cohorts: Execs #3, #5, and #7. Exec #1 should offer a four-way split amongst himself and execs #3, #5, and #7.

Don't beat around the bush then! Post your proof. You are the one with the obsession over trivial proofs, triclino. Nobody else really cares.

First of all, you very specifically asked "What is the definition of the empty set," triclino. Mr. Skeptic answered. Secondly, that definition does help. The set {D} is not a subset of the set {A,B,C}. Nor are {A, E} and {A,B,C,F}. Why is that?

That is not a good assumption. It now appears that the massive black holes formed before galaxies did, and they formed before stars. Just a couple of many articles on this topic: http://www.scientificamerican.com/blog/60-second-science/post.cfm?id=which-came-firstgalaxies-or-black-h-2009-01-07 http://news.sciencemag.org/sciencenow/2009/01/07-01.html

This is an easy problem, triclino. I generally solve a problem on which I offer assistance before I offer assistance -- this one included.

THis looks like homework, triclino. We don't do your homework for you. We do give hints.

As stated, this is wrong. What happens at x=0? Eliminate that particular value for x and the statement is true. This suggests a route by which you can prove this conjecture: Split the proof into two parts, one for[math]x\,\in(-1,0)[/math] and the other for [math]x>0[/math].

Classical E&M, as embodied by Maxwell's equations, is not quantum mechanics, Farsight.

Ignore triclino. What he wrote doesn't make sense. Please, people. Learn to use the correct nomenclature. There are two well-defined products for 3-vectors, the scalar product denoted by a center dot, and the cross product denoted by [math]\times[/math]. This doesn't make a lick of sense: [math]\vec a \cdot \vec b \cdot \vec c[/math]. That can only mean triclino was talking about the cross product, and what he wrote isn't correct for that either. The correct condition under which [math]\vec a \times (\vec b \times \vec c) = (\vec a \times \vec b)\times \vec c[/math] is that [math]\vec c[/math] is parallel to [math]\vec a[/math], i.e., [math]\vec c = \alpha \vec a[/math] where [math]\alpha[/math] is some scalar. There is no constraint on [math]\vec b[/math]. If all three are parallel to one another the vector triple product is identically zero for all arrangements of the factors in the product.

Yes, you did. Your continued trolling is starting to get old.

This is starting to look too much like homework, triclino, so for now, I pass.