Mr Skeptic

It seems the best idea for moving asteroids (other than bumping it) is to shine mirrors on it, creating a spot hot enough to vaporize. The vapor then pushes the asteroid, smoothly and gently so it won't break. At this point, we can hardly move a rock as big as Apophis, so I don't see us catching it. We're likely to send a tag-along though.

Except for things like glass which as they cool smoothly transition from a somewhat viscous liquid to a liquid so viscous it seems solid. Or to a solid. I wasn't ever very clear on that.

It may be a somewhat reasonable thing to do, so long as they are not talking nitrogen fertilizers. I think I read about iron being the limiting mineral in some areas, and you could get a huge algal bloom with a bit of iron. I think this is only for certain parts of the ocean. This may be a good idea, or it may make a worse mess.

Maybe you have a problem with the bacteria in your intestine (or lack thereof). If that is the case, drinking non-pasteurized yogurt may solve the problem. Did you take some strong antibiotics or other medicine 10 months ago?

Very true. Units are a simple, easy to use sanity check. Now if you correct the units, [math]E = \frac{h^2f^2}{pv}[/math], where f is frequency, you get an equation that gives the energy of a photon (but not for anything else). Of course, that is a coincidence and you cannot count on it. In this case, it is because of two different equations for a photon's energy, [math]E_{photon} =hf[/math] and [math]E_{photon} = pc[/math]. So as swansont said, make sure the units match.

I'm glad you're enjoying this as much as I do. I notice we (kind of) share avatars, which might hint at similar personalities. Be warned that I can be a bit crazy at times! (I'm sane enough to realize it though.) Are you saying that rather than emit a photon of such unholy energy, the particles/whatever involved would turn into a black hole instead? I guess that would make more sense than somehow emitting such a photon and the photon collapsing afterward. The other problem with the photon is that its energy would depend on frame of reference, so if it could collapse into a black hole, that would be rather unsettling in a different reference frame, I too did some rough estimating. Is one Planck energy more than the rest mass of every known particle? Yes, a photon's gravity was also bubbling around in my head but I wasn't sure where to ask about it. So I'll put it here: If gravity moves at c, and a photon moves at c, and a photon has gravity, does that mean that a photon has a gravitational "sonic boom"? Also, how would such a "sonic boom" affect a photon's interactions?

Best ask it here, so folks like me who are quietly listening in can learn too. I appreciate your perseverance and patience with this, Bignose. Me too. I understand that the field lines were originally modeled as fluids as way back then too. In any case, it is a useful analogy and allows visualization of the field and allows for [some] "common sense" like you showed.

The wikipedia article on black holes talks about a singularity (due to the breakdown of relativity equations) though also says that quantum mechanics would forbid that and that it would just be a superdense point. According to the wikipedia article, you could survive passing the event horizon, provided the black hole was really really big, but you would then inevitably fall into a singularity farther in? That makes no sense to me. I thought that anything past the event horizon was not understood?

OK, t[math]\hbar[/math]ank you. Would you know if the situation I described, a photon with enough energy that its wavelength would be smaller than its Schwarzschild radius (treating [math]\frac{E}{c^2}[/math] as mass), become a black hole? Or does that only work with rest mass?

Ah, so more or less in asking physically significant question I ended up with Planck units. And because it does not involve other non-dimensional constants like the mass ratios or the fine-structure constant, it is easily recognizable as Planck units. Or something like that. I used h instead of hbar because I had already used h, and wasn't sure how to put hbar in LaTeX. OK That seems interesting. Thanks, that will help me remember the name. I'm pretty bad with names. I thought it was something similar to the [iceland?] thing of adding -son to someone's name. I noticed the other thread is also mistitled. It should be illegal to have such a long name with only two vowels

I'm splitting this off to a new thread. Also, I got confused with the reduced planck's constant, so it is actually off by [math]\sqrt{4\pi}[/math] or [math]\sqrt{\pi}[/math]

Suppose that you had a photon with a wavelength equivalent to the Schwarzchild radius of a mass of equivalent energy. Then [math]\lambda = \frac{2Gm}{c^2} = \frac{2GE}{c^4} = \frac{2Gfh}{c^4} = \frac{1}{\lambda}\frac{2Gh}{c^3} = \sqrt{\frac{2Gh}{c^3}} = 5.72891594 X 10^{-35} meters[/math] [math]f = \frac{c}{\lambda} = \sqrt{\frac{c^5}{2Gh}} = 5.23297009 X 10^{42} Hertz[/math] [math]E = fh = \sqrt{\frac{hc^5}{2G}} = 3.46740157 X 10^9 Joules[/math] Damn! But for the [math]\sqrt{2}[/math], those are Plank units! That can't be a coincidence! This originally from the Schwarzchild radius thread, but I decided to move it to a new thread because it seems important. Coincidences in physics should be thoroughly investigated. I'm way over my head here, though. Anyone have an idea of what is going on here? I'm so excited! Actually, the relevant Plank units are [math]L_P = \sqrt{\frac{Gh}{2 \pi c^3}}[/math] [math]\omega_P = \sqrt{\frac{2 \pi c^5}{Gh}}[/math] [math]E_P = \sqrt{\frac{hc^5}{2 \pi G}}[/math] I got confused by the reduced Plank's constant.

Suppose that you had a photon with a wavelength equivalent to the Schwarzchild radius of a mass of equivalent energy. Then [math]\lambda = \frac{2Gm}{c^2} = \frac{2GE}{c^4} = \frac{2Gfh}{c^4} = \frac{1}{\lambda}\frac{2Gh}{c^3} = \sqrt{\frac{2Gh}{c^3}} = 5.72891594 X 10^{-35} meters[/math] [math]f = \frac{c}{\lambda} = \sqrt{\frac{c^5}{2Gh}} = 5.23297009 X 10^{42} Hertz[/math] [math]E = fh = \sqrt{\frac{hc^5}{2G}} = 3.46740157 X 10^9 Joules[/math] Could such an object be a black hole? It would seem like it should be illegal in various respects, what with having a black hole traveling at c, a photon with almost as much energy as a nuke, etc, but I don't see how it would violate a law of physics. Is there anything preventing a photon from having such high energy? Damn! But for the [math]\sqrt{2}[/math], those are Plank units! That can't be a coincidence!

It seems that most of this disagreement is related to whether or not the natural numbers are finite. Finite seems to mean having an end, but there is no limit to how many digits a natural number can have, nor is there a limit to their size. There is no last or biggest natural number. How then can they be finite? How can an infinite set be composed of finite, unique numbers? PS: the wikipedia article about natural numbers does not seem to say they are finite PPS: Should this be split to a new thread, why are natural numbers finite?

The new number need only have as many digits as there are numbers in the list, in this case 4. Also, it would be 1100, not 1110. Fine, then. They are all finite but have a number of digits that keeps getting bigger and bigger without end, and the number itself gets bigger and bigger without end. Do you have a link for that? Eh? So then [math] \sum_{n=0}^{\infty}\frac{1}{2^n} = 2[/math] shows that 2 is irrational? Then the sum [math]\sum_{n=0}^{\infty}1[/math] is finite. Since that sum is usually considered to be infinite, then perhaps finite = infinite according to your proof? You say that "clearly" adding 1 to any finite number results in another finite number, but it is the "clear" things that are hardest to prove

Making a blank config file does not sound like a good idea. Most source files contain a file called configure. Change to the directory this file is in, and if it is not executable, make it so chmod u+x configure, then execute it ./configure That should generate a config file

Huh. I missed that. I'd cast my vote for whoever was in charge of the Human Genome Project. I think gene sequencing is the future for evolution study.

I will clarify some more According to the theory of evolution, species, not individuals evolve. According to the Bible, Adam and Eve are the ancestors of all humans (as are Noah and his wife). Evolution would require many many others almost identical to Adam and Eve. I didn't bother to mention the different timescales, because I thought it was common knowledge. In the Bible, God created everything in 6 days, not a few billion. Adam in particular was formed "from the dust of the earth". People attempting to reconcile Christianity and evolution will usually sacrifice the first few chapters of Genesis, but in doing so would lose Adam, a figure about as important as Christ himself. It is not completely clear that it would apply to non-humans, but after Adam sinned, the ground was cursed. This gives the impression that things such as thorns would not have existed beforehand, whereas evolution says they should have. Also has other changes, such as birds before land animals, that directly conflicts with evolution. Also, I doubt plants could evolve without sunlight. These are (some) of the reasons that you can't just eliminate Adam. These conflict with the random aspect of evolution. Often the reconciliation attempt claims that God "guided" evolution. But evolution is supposed to be random. If the genealogies are not to be taken literally, most of the bible also shouldn't be taken literally. But lets not get too far off topic here. Just pointing out that the conflict between evolution and Christianity is why the issue is so politicized.

The way you phrase it it sounds like you are restricting the field of study to things relating to evolution. Did you simply mean who is the most influential biologist?

Hm, but consider the binary numbers, 0001 0010 0011 0100 ... 1100 is not on the list Then you can take the diagonal from the top right to the bottom left, and invert each of the digits, to get a different number not on the list. Would this not show that the natural numbers (in binary) are just as uncountable? Is countable by definition different from uncountable? But does that make it finite? There is no maximum number of digits a natural number can have. I think it was clear what he meant, about its decimal expansion Depends on what you mean by countable. Finite sets are not isomorphic to N. Yes, but what about the 3 at the start? In my example, the decimal expansion keeps going toward the left. Yes, that is much less elegant because it diverges instead of converging, but it has no end on that side. Yes, what is troubling about this is that it is just as impossible to say what an infinitely long natural number "starts" with as it is to say what an irrational number "ends" with. It should be writable as an infinite sum though. Just like an irrational number is written as an infinite sum, but you can't actually write it as a decimal expansion either. This is probably what is wrong with my argument. However, I cannot see how there can be an infinite number of numbers, each with a different number of digits, without the number of digits also being infinite. Which of the Peano axioms wouldn't hold with infinitely many digits? Countably infinite? I think that all irrational numbers are also calculated by taking a limit to infinity. This does seem to be different in the naturals and reals though. Need to think a bit more on this one. I did a proof by negation. So I assumed that there was a maximum finite number of digits n for the naturals, then showed that there would be a number with n+1 digits. Then the assumption is false, meaning that there is not a maximum finite number of digits for [math] n \in \mathbb{N}[/math] --- Is there anything wrong with saying there are an infinite number of zeros to the left of any finite number?

By different infinities. For example [math]F(1/3) = F(.33333...) = \sum^{\infty}_{n=0}3*10^n[/math] = ...33333, an infinite number consisting of 3's. This is much less elegant than the infinite expansion 0.33333... because it diverges so you can't approximate it. How many digits?

Merging Christianity and evolution would destroy the vary basis of one or both. Consider: 1) the world was perfect and without death before Adam sinned. No natural selection. 2) the order of creation, eg the plants before the sun, the earth before the stars. 3) Original Sin -- as by one man (Adam) sin and death came into the world, by one man (Christ) salvation 4) Jesus' genealogy traced down to Noah then to Adam. In fact, every single genealogy in the Bible. Or were those figurative? 5) God having rights over us as our maker. 6) God having a plan for us before the creation of the world vs chance 7) Evolution being due to random mutations and natural selection 8) if evolution is true, then God has no business there (Occam's Razor) So yeah, it would require some modification. And that's just off the top of my head. Basically, it would be "the Bible is not literal wherever it conflicts with science" also known as "ad hoc hypothesis". The genealogies, if nothing else, are not the type of thing that can be taken poetically.

I want to count the reals on between 0 and 1, never mind that they're supposed to be uncountable. Let F(x) be a function that mirrors each of the digits of a number across the decimal point; that is, every digit [math]a 10^b [/math] of the original number gets converted to [math]a 10^{-b}[/math]. F(x) is its own inverse. Since F(x) has an inverse, it is a bijection. If you restrict the domain of F(x) to the natural numbers, then its range is the set of all numbers from 0 to 1 with as many digits as the natural number. 1 <--> .1 2 <--> .2 ... 10 <--> .01 11 <--> .11 ... 3256 <--> .6523 ... 1234567 <--> .7654321 ... Now the real numbers from 0 to 1 can be represented as an infinite decimal expansion 0.xxxxxxxxxxxxxx... where each x is a digit. Since the natural numbers can have infinitely many digits*, then this would be a bijection from the naturals to the reals from 0 to 1, thus counting the reals from 0 to 1. *Suppose that the naturals could have only n digits. Then take a number such as [math]10^n[/math] with n digits and multiply it by 10, resulting in [math]10^{n+1}[/math] digits. This number belongs to the naturals because the naturals are a closed set, and has n+1 digits. So the premise that the naturals could have only n digits must be false. Therefore, the naturals must be allowed to have an infinite number of digits. Am I doing something wrong?

I guess we mostly agree then. I got carried away being the devil's advocate so I forgot to mention it also takes faith (of a slightly different kind) to believe in God. Merging Christianity and evolution would destroy the vary basis of one or both. Consider: 1) the world was perfect and without death before Adam sinned. No natural selection. 2) the order of creation, eg the plants before the sun, the earth before the stars. 3) Original Sin -- as by one man (Adam) sin and death came into the world, by one man (Christ) salvation 4) Jesus' genealogy traced down to Noah then to Adam. In fact, every single genealogy in the Bible. Or were those figurative? 5) God having rights over us as our maker. 6) God having a plan for us before the creation of the world vs chance 7) Evolution being due to random mutations and natural selection 8) if evolution is true, then God has no business there (Occam's Razor) So yeah, it would require some modification. And that's just off the top of my head. Basically, it would be "the Bible is not literal wherever it conflicts with science" also known as "ad hoc hypothesis". You can still love your neighbor as yourself though

Then consider [math]v = \frac{dx}{dt}[/math], the instantaneous velocity. This is opposed to Zeno's idea that at any instant in time, an object is motionless.