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Strawman. Of course we don't declare theories correct based on their simplicity. Neither Ben nor I said that. Physicists first and foremost demand that a theory agree with observations. Physicists use simplicity when two different theories yield identical predictions, e.g. Lorentz ether theory and special relativity. This whole off-topic discussion started with you're complete misunderstanding of several aspects of physics. Sorry to be so blunt, but 'tis the truth. Lucapsa, please shut up about physics. You are great when it comes to discussing biology and evolution. You obviously don't know special from general when it comes to relativity.

That's correct, but the I presume doesn't sound promising. That's correct too, but not quite right in formulation. This statement at the bottom of post #3 is at the bottom is what led you astray: By the fundamental theorem of calculus, [math]\frac d{dx} \int_a^x f(t)dt = f(x)[/math] The Leibniz Integral Rule (see Bignose's post) generalizes the concept of differentiation under the integral sign. Applying this here, [math]f'(x) = \frac d{dx}\int_0^x e^{-\,\frac {t^2} 2} dt = e^{-\,\frac {x^2} 2}[/math] You don't simply substitute zeros for the ts. You substitute xs, and then set x=0 after performing all intermediate steps (i.e., calculating n derivatives). A little more explicitly then, [math]f'(0) = f'(x)|_{x=0} = e^{-\,\frac {x^2} 2}|_{x=0} = 1[/math] You're shorthand started getting you in trouble here: Everything that follows is off as well. Being explicit, [math]f''(x) = \frac d{dx}f'(x) = \frac d{dx}e^{-\,\frac {x^2} 2} = e^{-\,\frac {x^2} 2}\frac d{dx}(-\,\frac {x^2} 2) = -xe^{-\,\frac {x^2} 2}[/math] Thus [math]f''(0) = f''(x)|_{x=0} = 0[/math], not -1/2. It looks like a pattern is developing here! I posit that the nth derivative of f(x) is a product of a polynomial and f'(x): [math]f^{(n)}(x) = p_n(x) e^{-\,\frac {x^2} 2}[/math] This is obviously true for n=1, with p_1(x)=1. If this pattern is true, we should have [math]f^{(n+1)}(x) = p_{n+1}(x) e^{-\,\frac {x^2} 2}\frac d{dx} f^{(n)}(x) = p_n^{\prime}(x) e^{-\,\frac {x^2} 2} - xp_n(x) e^{-\,\frac {x^2} 2} [/math] or [math]p_{n+1}(x) = p_n^{\prime}(x) - xp_n(x)[/math] which gives a recursive relation for the nth polynomial. From above, we have [math]f''(x) = -xe^{-\,\frac {x^2} 2}[/math], from which [math]p_2(x) = -x[/math]. The recursive relation gives the same result. The relation holds for n=1 and n=2, and by induction, for all n>=1. So what is [math]f^{(n)}(0)[/math]? Evaluating the general expression [math]f^{(n)}(x) = p_n(x) e^{-\,\frac {x^2} 2}[/math] at x=0 yields [math]f^{(n)}(0) = p_n(0)[/math]. We only need to evaluate the polynomials. The first few are [math]p_1(x) = 1[/math] [math]p_2(x) = -x[/math] [math]p_3(x) = -1 -x(-x) = -1 + x^2[/math] [math]p_4(x) = 2x -x(-1+x^2) = 3x - x^3[/math] [math]p_5(x) = 3 - 3x^2 - x(3x-x^3) = 3 - 6x^2 + 3x^4[/math] from which [math]f'(0) = p_1(0) = 1[/math] [math]f''(0) = p_2(0) = 0[/math] [math]f'''(0) = p_3(0) = -1[/math] [math]f''''(0) = p_4(0) = 0[/math] [math]f'''''(0) = p_5(0) = 3[/math] The Maclaurin series is [math]f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}[/math] which becomes [math]f(x) = 0 + x - \frac 1 6 x^3 + \frac 1 {40} x^5 + \cdots[/math] How does this compare to the series at mathworld? Using the standard definition of erf(x), you should be able to derive that [math]\int_0^x e^{-\,\frac {t^2} 2} dt = \sqrt{\frac{\pi}2} \mathrm{erf}\left(\frac x {\surd 2}\right)[/math] From mathworld, [math]\mathrm{erf}(x) = \frac 2{\surd\pi} \left(x-\frac 1 3 x^3 + \frac 1 {10} x^5 + \cdots\right)[/math] Evaluating [math]\mathrm{erf}(x/\surd 2)[/math], [math]\mathrm{erf}\left(\frac x {\surd 2}\right) = \frac 2{\surd\pi} \left(\frac 1{\surd 2}x-\frac 1 3 \frac 1 {2\surd 2} x^3 + \frac 1 {10} \frac 1 {4\surd 2} x^5 + \cdots\right)[/math] which simplifies to [math]\mathrm{erf}\left(\frac x {\surd 2}\right) = \sqrt{\frac 2{\pi}}\left(x-\frac 1 6 x^3 + \frac 1 {40} x^5 + \cdots\right)[/math] Multiplying by [math] \surd{\pi}/\surd 2[/math] recovers the series for f(x).

Please. A semantic appeal, and a bad one at that? The word "copyright" does not in any way mean that you have the right to copy anything you wish. It means that the creator of some thing has rights over all copies of that thing. Now I know you are trolling.

That is not the error function. The standard definition is [math]\mathrm{erf}(x) = \frac 2 {\surd \pi} \int_0^x e^{-t^2}dt[/math] The Maclaurin series for erf(x) as reported at mathworld uses the defintion I reported, not the function in your original post. BTW, wikipedia has the same definition for erf(x) as mathworld.

Given the candidate list here in this thread and the candidate list in the corresponding Republican thread, I would say that this fear is already a done deal.

I disagree on the cost-benefits returns from unmanned exploration. Spaceflight, manned or unmanned, remains a fairly poor monetary investment compared to investing in science on the ground. They payoff is too uncertain and too far in the future. Moreover, NASA has already picked the low-lying fruit that do have large tangible benefits: The Earth observation satellites. Most of the Earth observation satellites are now under the purview of Commerce, not NASA. (GPS is of course DoD, and has been from the onset). What is the dollar benefit from Hubble? From the Mars rovers? From setting foot on the Moon? While NASA does have some tangible benefits, for the most part they do not compare favorably to Earth-based investments in technology. It is in the arena of intangible benefits that space exploration excels. We may not know the true financial benefit from Hubble, the Mars rovers, or Apollo for hundreds of years. We certainly can see the intangible benefit right now. As far as cutting the unmanned program, spending on that has ridden the coattails of the human spaceflight program. When human spaceflight was very well funded, so was the unmanned program. When Congress cut human spaceflight at the knees, the unmanned program shortly followed the same dismal path.

Do you know how many people NASA hired between the end of the Apollo program and the start of Shuttle operations? This was the last time NASA faced cuts as draconian as those proposed by Obama.

Doesn't anyone here see the delicious irony in funding education for science and technology by making drastic cuts in an arena where those kids educated in science and technology might have been able to get a job had those drastic funding cuts not been made?

There are also quaternions, which can be viewed as (and were "created" as) an extension of the complex numbers. Representing a quaterion with scalar part [math]q_s[/math] and imaginary vector part [math]\vec q_v[/math] as [math]\bmatrix q_s \\ \vec q_v \endbmatrix[/math], the quaternion product [math] \bmatrix \cos \frac{\theta}2 \\ \sin \frac{\theta}2\hat u \endbmatrix \bmatrix 0 \\ \vec x \endbmatrix \bmatrix \cos \frac{\theta}2 \\ -\sin \frac{\theta}2\hat u \endbmatrix [/math] rotates the three vector [math]\vec x[/math] about the unit vector [math]\hat u[/math] by an angle of [math]\theta[/math]. http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

I know that many advocates of NASA's unmanned efforts decry the bulk of NASA's expenditures on human spaceflight. If only NASA would spend that much on unmanned spaceflight ... You do realize that in the eyes of many, the primary reason we send unmanned probes to other planets is because we might eventually be sending people there? The unmanned program exists in part because it is the forerunner for human space flight. Unmanned spaceflight looks cheap in comparison to human spaceflight, but it looks very expensive when compared to Earth-based science. Canceling just one of those billion dollar probes to Mars would enable funding of many Earth-based science projects. When unmanned spaceflight has to compete on its own, it can't. Don't believe me? The last time that human spaceflight faced massive big budget cuts at the end of the Apollo program, drastic cuts to NASA's unmanned efforts soon followed. Wrong. He is a bit more liberal than I like, but I was not particularly against him until this proposal. This proposal did not just turn me against him, it turned me deadset against him.

There is already a five year lag between the end of the Shuttle program and the first flight of the CEV. To make budget plans work, operations people will have to be laid off during that five year hiatus. This presents a big problem for NASA. Those people are not going to wait around for five years to get their jobs back. They will get jobs elsewhere. The workforce will need to be built up. The five year extension severely exacerbates this problem. That five year holding pattern will have a marked impact on the ones who are spared the pink slips as well. Look at any industry that has suffered a downturn. Corners are cut in an attempt to keep the workforce intact. The sharpest people leave. To this day, NASA is still recovering from the last hiatus between the end of Apollo and the onset of Shuttle operations. NASA's has a significant shortage of middle-aged workers because of this hiatus. Either cut it or fund it fully. Half-assed funding will result in a half-assed program. Even if he rescinds the funding side of this plan, I will never vote for Obama. Even in the case of my personal nightmare tickets, Obama versus Huckabee or Obama versus Paul, I will not vote for Obama. (I won't vote for Huckabee or Paul either. I'll pick some third party candidate.)

Obama completely lost it for me a ten days ago or so with his plan to gut NASA's exploration initiative in favor of space and science education. Some articles and comments at Daily Kos MSNBC's Cosmic Log Wired.com Later again at Daily Kos So where will all the kiddies Obama wants to educate in science and technology find jobs? Pharma is evil; Democratic plans for the pharmacological industry will gut R&D there. Defense is worse than evil; Democrats would like to see a Defense budget smaller than NASA's paltry 0.5% of the federal budget. NASA itself will be gutted thanks to this plan. Overseas? Does Obama foresee a brain drain on the US? Gutting the major sources of R&D work certainly will help facilitate a US brain drain. Moreover, the timing coincides with the pending Social Security disaster. Having our kiddos send Euros, Yen, and Yuan home might be a solution!

A handful of states (Texas and Washington, maybe others?) don't have an income tax. They get by just fine without an income tax. Now if you want to talk about social engineering by the Federal government, the Fed has tried for years to get those rogue states to conform to their ideal, including playing the heavy with taxpayers. State income taxes are deductible expenses with regard to the federal income tax. Sales taxes were not deductible for a long time. (They are deductible once again for people who don't have a state income tax; very recent change.) One of the reasons for making them not deductible (1970s?) was that sales taxes are regressive. Whether that is true or not, THAT IS NONE OF THE FED'S BUSINESS.

Some libertarians (I'm from Ron Paul's home district; I'm surrounded by them) do argue against the Interstate Highway System as an example of something the federal government should not be doing. I thought that is where you were going. As far as the federal government's role in social engineering -- that is a horse of a different color. (Good oops, by the way. You can't quite blame that on a typo. A Freudian slip of the fingers?) Keeping the federal government from meddling in our behavior is one of the best arguments for the "fair tax". The federal government does a lot of social engineering through the tax code. Unfortunately, prior government social engineering is what makes the fair tax dead-on-arrival. What are all the tax benefits regarding home ownership other than social engineering? Those benefits are built into the value of our homes. Switching from the income tax to the fair tax would have drastic consequences on housing values. The fair tax is DOA.

Civil engineering?? Building canals, post-roads, interstate highways? Please! The Constitution supports the notion, Article I, Section 8, Clause 3. Chief Justice Marshall reiterated the support in Gibbons v. Ogden' date=' 22 U.S. 1 (1824), Chief Justice Waite explicitly addressed how new technologies that aid in commerce but were obviously never discussed in the Constitution are captured under the Commerce Clause in Pensacola Telegraph v. Western Union, 96 U.S. 1 (1877),

He certainly is a libertarian. He even ran for President once before as the Libertarian candidate. Ron Paul isn't just a libertarian. His stance is a mix of libertarian, antebellum, and antedilivian ideals. Some are just stupid, some are downright dangerous. I expect intelligent people to look beyond single issues and to inject some realistic into their thinking. Libertarianism is simplistic, shallow idealism. Libertarianism is a knee-jerk response to a bloated and overly intrusive government. Yes, I want the government to leave us alone. There are many things about our government that I don't particularly like. That doesn't mean I think we should throw it out and go back to square one. It means we have to be diligent. That requires thinking. Decimating the federal government is not a realistic solution.

Hey! I voted agin Ron Paul, and I'm from Texas! Not that it matters much. He garnered 70% of the vote or so last time. He appeals to bible thumpers and to engineers and scientists. This neck of the woods is full of both. I just do not get his appeal to supposedly smart people, but I've seen it happen over and over again, right here in his Congressional district and right here at SFN.

Certainly the founding fathers did not believe in a right to steal.

Do you mean factoring? That is an NP problem, and finding a way to solve that in polynomial time is worth big $$$.

Fred Thompson is taller, though. I say we should elect him. We love tall Presidents. They look so ... Presidential. Lincoln, for example, was one ugly SOB, but he was very tall.

Just toying with ya. A trivial polynomial isn't much of much value. I don't think Jone's polynomial has much more than academic interest, either It was an interesting offshoot of the work done on Hilbert's tenth problem. On the other hand, the polynomial-time AKS primality testing algorithm (for-the-masses review and the paper itself) won immediate accolades and awards, including a nomination for the Fields medal.

I just don't get the appeal of Ron Paul, which is quite strong among supposedly intelligent scientists and engineers. He's completely whacked! I vote against him as often as I can, which is every two years since having been gerymandered into his Congressional district. I guess that counts as a valid reason for voting for Ron Paul.

I'm sorry. I thought you might want to add a polynomial that actually does find primes to your spreadsheet. I see now that your spreadsheet is reserved for polynomials that purport to find primes, but fail to achieve that objective. The polynomial f(x)=3 generates a prime for all x.

Rights are acts/conditions protected by law. Taking possession of something that doesn't belong to you (aka stealing) is not a protected act.

Jones, Wada, Sato, and Weins, "Diophantine representation of the set of prime numbers", Amer. Math. Monthly 83(1976), 449-464 Abstract: