## Miscellaneous Science Questions

For the discussion of the sciences. Physics problems, chemistry equations, biology weirdness, it all goes here.

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Pfhorrest
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### Re: Miscellaneous Science Questions

I wasn't sure if space-filling curves actually did what I thought they might do. Thank you for clarifying that they in fact do. (Although I'm still not 100% clear on how exactly they manage that).
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Tub
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### Re: Miscellaneous Science Questions

Hang on.. the discrete iterations of space-filling curves may be bijective, but the limits aren't. Space-filling curves are necessarily self-intersecting.

You can use them to prove that | R² | <= | R |, and it's easy to prove that | R | <= | R² |, but that's just an existence proof. It's not constructive.

If you want a bijection, it can't be continuous (i.e. not a curve).

Bitwise de-interleaving is not a bijection, either, unless you resolve the whole 0.999... = 1.0 ambiguity.

Eebster the Great
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### Re: Miscellaneous Science Questions

That's true, it suffices to prove there is a surjection, but it doesn't explicitly construct a bijection.

Flumble
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### Re: Miscellaneous Science Questions

Tub wrote:Bitwise de-interleaving is not a bijection, either, unless you resolve the whole 0.999... = 1.0 ambiguity.

Oh, bollocks.
Although, does that ambiguity exist since it holds for the components (0.999...,0.999...)=(1,1)=(1,0.999...)=(0.999...,1) too?

Tub
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### Re: Miscellaneous Science Questions

Although, does that ambiguity exist since it holds for the components (0.999...,0.999...)=(1,1)=(1,0.999...)=(0.999...,1) too?

Go ahead and interleave these pairs. If interleaving is a bijection, all four of them must yield the same number.

There is only a countably infinite amount of numbers with two representations, and the whole thing is fixable, but it needs to be fixed if you want your bijection. And after the fix, your bijection will be a lot less beautiful and a lot less simple to use for further proofs.

measure
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### Re: Miscellaneous Science Questions

Tub wrote:There is only a countably infinite amount of numbers with two representations, and the whole thing is fixable, but it needs to be fixed if you want your bijection. And after the fix, your bijection will be a lot less beautiful and a lot less simple to use for further proofs.

Can't you just start by replacing all the terminating decimal representations with the non-terminating equivalents? It shouldn't matter going the other way since both representations correspond to the same real number. All four of the (1,1) pairs would yield 0.9999..... -> 1.

Eebster the Great
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### Re: Miscellaneous Science Questions

Stack exchange has a great answer. Interleaving digits works in the following way:

First you need to exclude terminating representations, so ½ = 0.4999..., not ½ = 0.5000.... Next, instead of interleaving individual digits, you interleave chunks of digits. For every number in R, you separate it into chunks which start with at least zero zeroes and end with a nonzero digit. For instance, the number .1020030004... becomes 1 02 003 0004.... Since we are excluding numbers that end with infinite sequences of zeroes, this is always well-defined. Now, interleaving the chunks of two numbers gives a unique real number. For instance, (½,⅓) maps to 0.4393939... = 29/66, and (1/11,1/13) maps to .0907096099092093... = 907096099092093/9999999999999999. This can map (0,1]2 to (0,1] bijectively. Then you need bijections from (0,1] to R and (0,1]2 to R2, but these are trivial.

The post also mentions that Cantor instead used continued fraction representations to construct a bijection.

Xanthir
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### Re: Miscellaneous Science Questions

Oh yeah, space-filling gives you a bijection, but the nice thing about infinite sets is that you don't need a bijection, just two injections. Mapping the reals to the complex is done trivially via the function `f(x) = x` ^_^
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tomandlu
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### Re: Miscellaneous Science Questions

Yet another relativity question...

At the centre of the Earth, you would be in a state similar to freefall*, yet at the bottom of a gravity well. Does the freefall bit have any impact on relativity or just the gravity well?

* well, identical, provided you mean the sun.
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doogly
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### Re: Miscellaneous Science Questions

yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."
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### Re: Miscellaneous Science Questions

The relativity bit is that being in a box in place at the bottom of a gravity well feels just like being in a box rapidly accelerating towards the Earth's surface.
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tomandlu
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### Re: Miscellaneous Science Questions

doogly wrote:yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."

I think I'm being stupid. Thinking about it, and afaik, freefall has no impact on the effect of a gravitational body with regard to time. e.g. being in freefall towards a black hole isn't going to help you... I guess I had this vague notion that the gravity well might be more of a W shape, rather than V, iyswim.

(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).
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Pfhorrest
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### Re: Miscellaneous Science Questions

tomandlu wrote:(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).

Objects in orbit are in freefall, and objects in orbits at different heights experience time differently, and an object floating in the middle of a hollow at the center of the Earth is essentially just in the lowest possible orbit, and so will experience time differently than any other orbit, or the surface.
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p1t1o
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### Re: Miscellaneous Science Questions

tomandlu wrote:
doogly wrote:yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."

I think I'm being stupid. Thinking about it, and afaik, freefall has no impact on the effect of a gravitational body with regard to time. e.g. being in freefall towards a black hole isn't going to help you... I guess I had this vague notion that the gravity well might be more of a W shape, rather than V, iyswim.

(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).

I believe there is something called "gravitational time dilation":

https://en.wikipedia.org/wiki/Gravitati ... e_dilation

So yes, I think that merely by being in a gravity well, your time will flow differently, regardless of any thrust, acceleration or freefall.

This will be in-addition-to the difference in time dilation caused by relative velocity.

The stronger the gravitational potential, the slower time moves.

Eebster the Great
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### Re: Miscellaneous Science Questions

Wikipedia claims Earth's core is 2.5 years younger than its surface, which gives a time dilation of 5.5×10-10. So it's real, but it's not exactly significant. The difference between the surface of the Earth and the high orbit of GPS satellites (20,180 km) is about 5.2×10-10, or about 45 μs/day, but since the satellites are moving, there is also time dilation due to special relativity of about -7 μs/day, giving a net 38 μs/day correction required to account for special and general relativity.

The effects of gravitational time dilation really don't become significant until you get close to black holes or neutron stars. If you've seen the movie Interstellar, time dilation is what was going on on Miller's planet, and is why most of the cast aged more slowly than Romilly in a higher orbit. If you actually fall into a black hole, things get a lot more complicated, and you can't really even use the same coordinate system, because those coordinates have a singularity at the event horizon.

Sableagle
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### Re: Miscellaneous Science Questions

Is there an SI unit roughly equivalent to a snowball's chance in Hell?

I'm not sure how to answer without using a prefix like "pico" and I'm not sure how to answer with a prefix like "pico."
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cyanyoshi
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### Re: Miscellaneous Science Questions

Sableagle wrote:Is there an SI unit roughly equivalent to a snowball's chance in Hell?

FemtoSnowball.png

I'm not sure how to answer without using a prefix like "pico" and I'm not sure how to answer with a prefix like "pico."

You could always compare the probability to some of the examples here.

gmalivuk
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### Re: Miscellaneous Science Questions

You could always use negative (deci)bels, but then the absolute value of the number conveys its size rather than any prefix.
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Sableagle
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### Re: Miscellaneous Science Questions

"It's likely in the same sense that a Minus Forty on the Richter Scale is powerful."
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p1t1o
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### Re: Miscellaneous Science Questions

The monkeys and typewriters one is an absolute classic

Eebster the Great
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### Re: Miscellaneous Science Questions

Unfortunately, monkeys don't produce particularly random characters on their first try at a typewriter, so the actual probability is likely far lower. (Source: six Sulawesi crested macaques' first try at a computer keyboard.)

Flumble
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### Re: Miscellaneous Science Questions

Can radiation break a torch/flashlight and, if so, how? (It happens in the Chernobyl miniseries and is based on accounts of those people according to the commentary.) Note that these are 80's battery-powered torches.

Pfhorrest
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### Re: Miscellaneous Science Questions

Are modern flashlights no longer battery-powered!?
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KittenKaboodle
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### Re: Miscellaneous Science Questions

I'd imagine a similar mechanism to that which "breaks" people, breaking apart molecules and changing certain chemical reactions etc.., possibly in extreme cases just direct heating which isn't good for batteries. I certainly wouldn't want to observe it myself, and considering the relative fragility of biological systems I'd be skeptical of any "first hand" reports unless made via a Ouija board. Though if you saw it on TV it must be true.

I read they had problems with the robots examining the Fukushima reactors, and while a robot and video cameras are probably much more delicate than a simple battery and incandescent lamp, if the radiation is intense enough I imagine it could damage anything.

Thesh
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### Re: Miscellaneous Science Questions

My first guess is the flashlights didn't get regular battery changes, and so dead batteries were a common occurrence that people didn't notice as much when there wasn't a disaster. I can't find anything on alkaline batteries, but I see an 8.4% loss in capacity for lithium ion after 2.7 million rads (which is fatal for most humans).

https://www.sciencedirect.com/science/a ... 3X1401088X
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Eebster the Great
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### Re: Miscellaneous Science Questions

2.7 Mrad is thousands of times a typical fatal dose of X-radiation. I also have a hard time believing the X- and gamma rays released by the Chernobyl meltdown could have damaged batteries significantly.

A post on Reddit claims the batteries were discharged (not damaged) by the ionizing radiation. I also don't see how that's possible.

p1t1o
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### Re: Miscellaneous Science Questions

Do you guys remember batteries in the 80's? A common-or-garden torch would last about 3 minutes before the light started to go dim, and no recharging, you had to crack open another fresh pack. Most likely explanation is that they were just run down and with the confusion and adrenaline etc.

Its hard to see any mechanism by which nuclear radiation at the levels in question could significantly disrupt a battery powered torch.

I would assume that the lethal dose of radiation for something as simple as a chemical battery and filament lamp, is way waaaaaaay higher than that required to disrupt the delicate and complex biochemistry of a human. If the humans are still standing, so are the torches.

Eebster the Great
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### Re: Miscellaneous Science Questions

I don't remember batteries in the 80s, though the basic technology was the same. On the one hand, it sort of figures that they would have improved over the past 30-40 years, but on the other hand, I don't know what actually changed to increase capacity (or longevity in storage). It's not like lithium ion batteries, where many obvious improvements in density have been made.

ijuin
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### Re: Miscellaneous Science Questions

The greater change in torches/flashlights over the past couple of decades has been the switch from incandescent bulbs to LED lights, resulting in a decrease in power requirement of more than 80%.

That said, as several people in this thread have observed, any radiation that would damage such a simple and robust device as any halfway-decent battery-powered light would be more than enough to kill humans who are not protected with shielding better than any protective suits yet invented.

p1t1o
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### Re: Miscellaneous Science Questions

I think there were numerous incremental developments in battery design over the years, batteries today, even non-Li ones, are much better than they used to be.

It is possible that it is due to reductions in manufacturing cost meaning that batteries may use the same tech, but are higher quality for a lower price (you ever had the misfortune of buying cheap, shitty batteries that last for about 30 seconds?) But I would wager that there have been improvements in design too.