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wasoxygen  ·  52 days ago  ·  link  ·    ·  parent  ·  post: Autograms: Self-enumerating Sentences

Related:

(Hey Devac, when I run the code in a sandbox, then copy the output into the source, the output becomes static.)

wasoxygen  ·  52 days ago  ·  link  ·    ·  parent  ·  post: Patterns That Eventually Fail

Hmm, good question. That's really high up, far higher than aircraft travel. I would guess that there's an interaction with solar radiation of some kind. Is it where the magnetic field deflects the "solar wind" perhaps? With the atmosphere that thin and pressure so low, I do think most heat energy absorbed near the Earth's surface would be dissipated, continuing the pattern seen on mountaintops.

Thinking about this abstractly, I realized I didn't know why the Earth's core is hot. If it was formed by a lot of space dust accreting, it would seem more likely to be very cold. And if the sun is the source of the heat, it should heat the surface more than the core. So I have to guess that the same mechanism is at work: the intense pressure due to gravity is enough to heat the core beyond iron's melting point.

Devac  ·  48 days ago  ·  link  ·  

Let's focus on one problem at a time. Also, let's not forget that every step was simplified.

You can simplify the initial problem to the following: you have a source of energy (Sun), a black body screen (Earth's surface) and a transfer medium between the two that can act as a buffer.

Since both of them radiate, there should be a point where energy going in, is equal to the energy going out. If you'd have two bodies at T1 and T2 (T1 < T2) and connect them with a strip of copper, you could reasonably expect there's going to be a point on the strip where T = (T1 + T2) / 2. That's your basic concept of heat transfer.

Now, time for some facts regarding our buffer:

The density and pressure of the atmosphere change with the distance from the Earth's surface.

The amount of energy per volume that air can retain is proportional to the number of particles enclosed in said volume (and their energies).

The average velocity of those particles is proportional to temperature and kinetic energy. Mass is another important factor, pertinent to the original question.

Heavier gasses stay close to the surface while lighter ones go up. It affects the distribution of gasses throughout the atmosphere. All gasses in the atmosphere (at least up to at least 80-90 km) are composed of more than one atom. N2, O2, CO, CO2, H2O, O3, Ar, CH4, NOx. Those are the most important ones. I'm going to use the word 'particle' when referring to them.

The amount of energy a particle can retain, and how hard or easy it is to change its temperature, is related to its heat capacity and specific heat. This has three components, each related to the type of motion a particle can make. First is your basic translational motion, which is to say "how a particle travels in 3D space". Up, down, left, right, forwards, backwards and their combinations. Second is the way a particle can rotate. Because we have only three axes (and three moments of inertia associated with them), it can only spin around X, Y or Z axis (or a combination of them). The third way a particle can move is the way it can vibrate. Imagine balls connected by a spring, and you have the basic idea. For an n-atom particle, there are (n - 1) ways it can vibrate.

The more ways a particle can move, the number of degrees of freedom, gives it a broader range of energy it can absorb. Each of those plays their part due to the equipartition of energy. Broadly speaking, each degree of freedom gets proportional for its type share of energy. Meaning: they have more ways to absorb and emit radiation.

FYI: Wikipedia has a good article on rotational-vibrational spectroscopy.

I told you to consider what happens at various heights of the atmosphere. I did it to point you toward things like clouds and water vapour in general. They scatter light and absorb tremendous amounts of energy on their own. Combine it with all the greenhouse gasses, some mentioned above, that are most excited in the infrared spectrum, and you get even more buffer properties. They act like a one-way mirror, absorbing energy radiated from the ground and radiate almost all of it it back towards the ground. That's part of the reason (if not the sole reason, I'm not certain about the specifics) why satelite images of clouds are done in IR spectrum.

Now, let's bring those facts together. Close to Earth's surface, you have more of those massive, multi-atom particles. This skews the point of energy transfer equilibrium closer to the surface.

Wait, so where's the role of pressure? It's talked about indirectly. Pressure and temperature are related to a collective force given by all the particles hitting surfaces (translational motion). The higher you go and the fewer of those particles are present, the less energy they carry on average.

There's a second part to this post that I'm still writing. It's more about convective processes and why/how they don't matter nearly as much as you might be thinking. My goal is to write it from the first principles approach like this one, but it's going to take me a while.

flagamuffin  ·  45 days ago  ·  link  ·  

    Let's focus on one problem at a time.

nah i think the other problem is more interesting than gases. one at a time is so '90s. witness:

-it seems intuitive that random space dust which became the core is cold because space is cold... but remember space is cold because it's empty-ish and the core is very much not empty, so lots of heat got trapped in the core. there was, at the time, lots of jostling. jostling is heat.

-as mr. devac pointed out, we can figure out how old rocks'n'stuff is sometimes by measuring radiation, which is just more heat

-also the layers of the earth trap heat pretty well right, i mean there's a ton of magma down there and not many access points on the surface

and the sheer magnitude of energy is unnerving--

temp of sun at surface 6000 kelvin. temp of earth, at same, a few hundred. massive difference in energy, so a sort of "arbitrage" occurs (obviously not really, and not forever, pace second law) and order springs from disorder. because order is just spending energy to create a temporary equilibrium. anyway, i don't know a damn thing about the thermosphere but the point is we don't really blast that much heat out. we trap it, and have since day 1, and we also accidentally used it to create a bunch of neat shit. https://en.wikipedia.org/wiki/Autocatalysis#Creation_of_order

borrowed time, but then we're in good company -- the whole universe is borrowing too

flagamuffin  ·  45 days ago  ·  link  ·  

hubski paragraph spacing still sucks

wasoxygen  ·  52 days ago  ·  link  ·    ·  parent  ·  post: LaGuardia Flights Halted as Shutdown Hits Air-Traffic Staffing

How Will the Shutdown End? Time for Some Game Theory

    January 17, 2019

    The real power here is held by government employees, especially those in critical jobs. Let’s say that more TSA screeners decided to walk off the job. It’s already the case that the TSA absentee rate has gone up to 7.6 percent, from 3.2 percent a year ago. It is possible to imagine screeners staying home in much greater numbers, thus crippling the entire nation. That could either force President Donald Trump’s hand or lead to a congressional override of a potential presidential veto.

    (You might think the nation can do without TSA workers, but I doubt that is true. Even if they don’t make air travel safer, it is hard to imagine airports and airlines going about their business, in a litigious society, without TSA assistance.)

wasoxygen  ·  53 days ago  ·  link  ·    ·  parent  ·  post: Patterns That Eventually Fail

    Potassium–argon dating, abbreviated K–Ar dating, is a radiometric dating method used in geochronology and archaeology. It is based on measurement of the product of the radioactive decay of an isotope of potassium (K) into argon (Ar). Potassium is a common element found in many materials, such as micas, clay minerals, tephra, and evaporites. In these materials, the decay product 40Ar is able to escape the liquid (molten) rock, but starts to accumulate when the rock solidifies (recrystallizes).

Crazy.

I left out the cold mountain tops. This seems like an easy one. Hot air rises because it is less dense than the surrounding air, and the density is related to the ambient pressure. As the hot air rises, the pressure decreases, so the temperature of the rising air will decrease according to Boyle's or Charles' law.

That explains why a column of hot air would cool off as it rises, but not why the atmosphere is typically colder at elevation. I suppose what heat there is in the atmosphere comes from the sun warming the Earth's surface, heating the air by conduction, so temperatures drop as you get farther from the surface. If that's true, then we would expect high-elevation places like Denver to have similar temperatures to low-altitude places at a similar latitude, but higher places appear to have lower temperatures even when they are not mountain tops.

Maybe this one is not so easy. Ideas?

Devac  ·  53 days ago  ·  link  ·  

Then how would you explain thermosphere with only the facts you mention? It starts just below the Kármán line at 90km above the sea level, where the pressure is much lower than even the peak of Mt. Everest, and yet the temperature range within is between 500 and 2500 °C.

You aren't saying anything that's wrong on its own, but it's not the full picture of the problem. Think of what happens in the atmosphere at different heights, its composition and how it affects the flow of energy.

wasoxygen  ·  52 days ago  ·  link  ·  

Hmm, good question. That's really high up, far higher than aircraft travel. I would guess that there's an interaction with solar radiation of some kind. Is it where the magnetic field deflects the "solar wind" perhaps? With the atmosphere that thin and pressure so low, I do think most heat energy absorbed near the Earth's surface would be dissipated, continuing the pattern seen on mountaintops.

Thinking about this abstractly, I realized I didn't know why the Earth's core is hot. If it was formed by a lot of space dust accreting, it would seem more likely to be very cold. And if the sun is the source of the heat, it should heat the surface more than the core. So I have to guess that the same mechanism is at work: the intense pressure due to gravity is enough to heat the core beyond iron's melting point.

Devac  ·  48 days ago  ·  link  ·  

Let's focus on one problem at a time. Also, let's not forget that every step was simplified.

You can simplify the initial problem to the following: you have a source of energy (Sun), a black body screen (Earth's surface) and a transfer medium between the two that can act as a buffer.

Since both of them radiate, there should be a point where energy going in, is equal to the energy going out. If you'd have two bodies at T1 and T2 (T1 < T2) and connect them with a strip of copper, you could reasonably expect there's going to be a point on the strip where T = (T1 + T2) / 2. That's your basic concept of heat transfer.

Now, time for some facts regarding our buffer:

The density and pressure of the atmosphere change with the distance from the Earth's surface.

The amount of energy per volume that air can retain is proportional to the number of particles enclosed in said volume (and their energies).

The average velocity of those particles is proportional to temperature and kinetic energy. Mass is another important factor, pertinent to the original question.

Heavier gasses stay close to the surface while lighter ones go up. It affects the distribution of gasses throughout the atmosphere. All gasses in the atmosphere (at least up to at least 80-90 km) are composed of more than one atom. N2, O2, CO, CO2, H2O, O3, Ar, CH4, NOx. Those are the most important ones. I'm going to use the word 'particle' when referring to them.

The amount of energy a particle can retain, and how hard or easy it is to change its temperature, is related to its heat capacity and specific heat. This has three components, each related to the type of motion a particle can make. First is your basic translational motion, which is to say "how a particle travels in 3D space". Up, down, left, right, forwards, backwards and their combinations. Second is the way a particle can rotate. Because we have only three axes (and three moments of inertia associated with them), it can only spin around X, Y or Z axis (or a combination of them). The third way a particle can move is the way it can vibrate. Imagine balls connected by a spring, and you have the basic idea. For an n-atom particle, there are (n - 1) ways it can vibrate.

The more ways a particle can move, the number of degrees of freedom, gives it a broader range of energy it can absorb. Each of those plays their part due to the equipartition of energy. Broadly speaking, each degree of freedom gets proportional for its type share of energy. Meaning: they have more ways to absorb and emit radiation.

FYI: Wikipedia has a good article on rotational-vibrational spectroscopy.

I told you to consider what happens at various heights of the atmosphere. I did it to point you toward things like clouds and water vapour in general. They scatter light and absorb tremendous amounts of energy on their own. Combine it with all the greenhouse gasses, some mentioned above, that are most excited in the infrared spectrum, and you get even more buffer properties. They act like a one-way mirror, absorbing energy radiated from the ground and radiate almost all of it it back towards the ground. That's part of the reason (if not the sole reason, I'm not certain about the specifics) why satelite images of clouds are done in IR spectrum.

Now, let's bring those facts together. Close to Earth's surface, you have more of those massive, multi-atom particles. This skews the point of energy transfer equilibrium closer to the surface.

Wait, so where's the role of pressure? It's talked about indirectly. Pressure and temperature are related to a collective force given by all the particles hitting surfaces (translational motion). The higher you go and the fewer of those particles are present, the less energy they carry on average.

There's a second part to this post that I'm still writing. It's more about convective processes and why/how they don't matter nearly as much as you might be thinking. My goal is to write it from the first principles approach like this one, but it's going to take me a while.

flagamuffin  ·  45 days ago  ·  link  ·  

    Let's focus on one problem at a time.

nah i think the other problem is more interesting than gases. one at a time is so '90s. witness:

-it seems intuitive that random space dust which became the core is cold because space is cold... but remember space is cold because it's empty-ish and the core is very much not empty, so lots of heat got trapped in the core. there was, at the time, lots of jostling. jostling is heat.

-as mr. devac pointed out, we can figure out how old rocks'n'stuff is sometimes by measuring radiation, which is just more heat

-also the layers of the earth trap heat pretty well right, i mean there's a ton of magma down there and not many access points on the surface

and the sheer magnitude of energy is unnerving--

temp of sun at surface 6000 kelvin. temp of earth, at same, a few hundred. massive difference in energy, so a sort of "arbitrage" occurs (obviously not really, and not forever, pace second law) and order springs from disorder. because order is just spending energy to create a temporary equilibrium. anyway, i don't know a damn thing about the thermosphere but the point is we don't really blast that much heat out. we trap it, and have since day 1, and we also accidentally used it to create a bunch of neat shit. https://en.wikipedia.org/wiki/Autocatalysis#Creation_of_order

borrowed time, but then we're in good company -- the whole universe is borrowing too

flagamuffin  ·  45 days ago  ·  link  ·  

hubski paragraph spacing still sucks

wasoxygen  ·  54 days ago  ·  link  ·    ·  parent  ·  post: Patterns That Eventually Fail

Related:

It's big numbers all the way down. I would like to understand this stuff better than I do. But I enjoyed the author's comments:

    I’m sometimes intimidated by young mathematicians who know (∞,1)-categories and the like better than I do… until I say something about other branches of math or physics and discover they are completely clueless about many basic facts. Then I’m reassured that my life hasn’t actually been wasted.

    For example, I remember explaining to some mathematicians why the Moon rises a bit later each day. I can easily imagine not remembering whether it’s earlier or later. But this is something one can work out from first principles if one knows the Moon orbits the same way the Earth turns. Given that fact, a good mathematician should be able to figure out pretty quickly about much later the Moon rises each day. If they can’t do that, no amount of (∞,1)-categorical expertise will impress me.

    I also remember stumping people with the question “if a solar eclipse happens when the Moon comes between the Sun and the Earth, why isn’t there one every month?”

    A more significant challenge: “Since all rocks come from material that formed the Earth about 3 or 4 billion years ago, how can we use radioactive dating to measure the age of rocks and get different answers for different rocks?”

    And: “If hot air rises, why is it colder on mountain tops?”

Confusion over the movement of the Moon caused me to miss an occultation of the Pleiades. Experience tells me that the Moon rises about an hour later each day. But my first intuition on seeing this question is that the Moon's orbit should put it ahead, not late, when my position on the Earth's surface repeats after 24 hours. Of course, that's right, and that's why I need some more time to catch up with the Moon: the observer is late. How much more time? Well, after about 28 days the catching-up periods should add up to a full revolution, so I suppose it's about 1/28 of a day, 0.86 hours, or 51 minutes. I'll make some observations if my memory and the weather hold up (both poor prospects).

For solar eclipses, I'm not sure it is so far off to say there is about one (at least partial) eclipse every month, somewhere on Earth. The tilt of the Earth's axis should not be a factor in eclipse frequency, but if the Moon's orbit is tilted with respect to the Earth's orbit around the Sun, that would make alignments less frequent.

For the rocks I have no idea. I think with once-living material, the different carbon isotopes are kept in balance by metabolism, and when the creature dies there is a predictable shift as the less-stable isotope decays. Rocks should be pretty uniform to start with. Maybe it's a clue in the question that the rock-forming "material" is uniform, but different rocks were formed at different times. If they are sedimentary, like limestone, they might contain material from living creatures. I can't remember the difference between igneous and metamorphic, but one of them is something about lava cooling down. I'll guess that something about hot lava prevents some kind of radioactive decay from occurring, but I can't imagine how.

Devac  ·  54 days ago  ·  link  ·  

    I'll guess that something about hot lava prevents some kind of radioactive decay from occurring, but I can't imagine how.

That's because it doesn't influence decay rates. The age of stones is measured by effects of decaying nuclei or other forms of radiation on them, from within (Uranium-Lead or other forms of isotope ratio dating) and without ((thermo)luminescence). Once the rock is formed, it stops releasing (some) products of decay or replenishing less stable isotopes, which is not unlike removing a living organism from C14 cycle after it dies.

EDIT: Do keep in mind that the above explanation is somewhat idealised. Also, wikipedia has a nice overview of chronological dating methods if you are interested.

wasoxygen  ·  53 days ago  ·  link  ·  

    Potassium–argon dating, abbreviated K–Ar dating, is a radiometric dating method used in geochronology and archaeology. It is based on measurement of the product of the radioactive decay of an isotope of potassium (K) into argon (Ar). Potassium is a common element found in many materials, such as micas, clay minerals, tephra, and evaporites. In these materials, the decay product 40Ar is able to escape the liquid (molten) rock, but starts to accumulate when the rock solidifies (recrystallizes).

Crazy.

I left out the cold mountain tops. This seems like an easy one. Hot air rises because it is less dense than the surrounding air, and the density is related to the ambient pressure. As the hot air rises, the pressure decreases, so the temperature of the rising air will decrease according to Boyle's or Charles' law.

That explains why a column of hot air would cool off as it rises, but not why the atmosphere is typically colder at elevation. I suppose what heat there is in the atmosphere comes from the sun warming the Earth's surface, heating the air by conduction, so temperatures drop as you get farther from the surface. If that's true, then we would expect high-elevation places like Denver to have similar temperatures to low-altitude places at a similar latitude, but higher places appear to have lower temperatures even when they are not mountain tops.

Maybe this one is not so easy. Ideas?

Devac  ·  53 days ago  ·  link  ·  

Then how would you explain thermosphere with only the facts you mention? It starts just below the Kármán line at 90km above the sea level, where the pressure is much lower than even the peak of Mt. Everest, and yet the temperature range within is between 500 and 2500 °C.

You aren't saying anything that's wrong on its own, but it's not the full picture of the problem. Think of what happens in the atmosphere at different heights, its composition and how it affects the flow of energy.

wasoxygen  ·  52 days ago  ·  link  ·  

Hmm, good question. That's really high up, far higher than aircraft travel. I would guess that there's an interaction with solar radiation of some kind. Is it where the magnetic field deflects the "solar wind" perhaps? With the atmosphere that thin and pressure so low, I do think most heat energy absorbed near the Earth's surface would be dissipated, continuing the pattern seen on mountaintops.

Thinking about this abstractly, I realized I didn't know why the Earth's core is hot. If it was formed by a lot of space dust accreting, it would seem more likely to be very cold. And if the sun is the source of the heat, it should heat the surface more than the core. So I have to guess that the same mechanism is at work: the intense pressure due to gravity is enough to heat the core beyond iron's melting point.

Devac  ·  48 days ago  ·  link  ·  

Let's focus on one problem at a time. Also, let's not forget that every step was simplified.

You can simplify the initial problem to the following: you have a source of energy (Sun), a black body screen (Earth's surface) and a transfer medium between the two that can act as a buffer.

Since both of them radiate, there should be a point where energy going in, is equal to the energy going out. If you'd have two bodies at T1 and T2 (T1 < T2) and connect them with a strip of copper, you could reasonably expect there's going to be a point on the strip where T = (T1 + T2) / 2. That's your basic concept of heat transfer.

Now, time for some facts regarding our buffer:

The density and pressure of the atmosphere change with the distance from the Earth's surface.

The amount of energy per volume that air can retain is proportional to the number of particles enclosed in said volume (and their energies).

The average velocity of those particles is proportional to temperature and kinetic energy. Mass is another important factor, pertinent to the original question.

Heavier gasses stay close to the surface while lighter ones go up. It affects the distribution of gasses throughout the atmosphere. All gasses in the atmosphere (at least up to at least 80-90 km) are composed of more than one atom. N2, O2, CO, CO2, H2O, O3, Ar, CH4, NOx. Those are the most important ones. I'm going to use the word 'particle' when referring to them.

The amount of energy a particle can retain, and how hard or easy it is to change its temperature, is related to its heat capacity and specific heat. This has three components, each related to the type of motion a particle can make. First is your basic translational motion, which is to say "how a particle travels in 3D space". Up, down, left, right, forwards, backwards and their combinations. Second is the way a particle can rotate. Because we have only three axes (and three moments of inertia associated with them), it can only spin around X, Y or Z axis (or a combination of them). The third way a particle can move is the way it can vibrate. Imagine balls connected by a spring, and you have the basic idea. For an n-atom particle, there are (n - 1) ways it can vibrate.

The more ways a particle can move, the number of degrees of freedom, gives it a broader range of energy it can absorb. Each of those plays their part due to the equipartition of energy. Broadly speaking, each degree of freedom gets proportional for its type share of energy. Meaning: they have more ways to absorb and emit radiation.

FYI: Wikipedia has a good article on rotational-vibrational spectroscopy.

I told you to consider what happens at various heights of the atmosphere. I did it to point you toward things like clouds and water vapour in general. They scatter light and absorb tremendous amounts of energy on their own. Combine it with all the greenhouse gasses, some mentioned above, that are most excited in the infrared spectrum, and you get even more buffer properties. They act like a one-way mirror, absorbing energy radiated from the ground and radiate almost all of it it back towards the ground. That's part of the reason (if not the sole reason, I'm not certain about the specifics) why satelite images of clouds are done in IR spectrum.

Now, let's bring those facts together. Close to Earth's surface, you have more of those massive, multi-atom particles. This skews the point of energy transfer equilibrium closer to the surface.

Wait, so where's the role of pressure? It's talked about indirectly. Pressure and temperature are related to a collective force given by all the particles hitting surfaces (translational motion). The higher you go and the fewer of those particles are present, the less energy they carry on average.

There's a second part to this post that I'm still writing. It's more about convective processes and why/how they don't matter nearly as much as you might be thinking. My goal is to write it from the first principles approach like this one, but it's going to take me a while.

flagamuffin  ·  45 days ago  ·  link  ·  

    Let's focus on one problem at a time.

nah i think the other problem is more interesting than gases. one at a time is so '90s. witness:

-it seems intuitive that random space dust which became the core is cold because space is cold... but remember space is cold because it's empty-ish and the core is very much not empty, so lots of heat got trapped in the core. there was, at the time, lots of jostling. jostling is heat.

-as mr. devac pointed out, we can figure out how old rocks'n'stuff is sometimes by measuring radiation, which is just more heat

-also the layers of the earth trap heat pretty well right, i mean there's a ton of magma down there and not many access points on the surface

and the sheer magnitude of energy is unnerving--

temp of sun at surface 6000 kelvin. temp of earth, at same, a few hundred. massive difference in energy, so a sort of "arbitrage" occurs (obviously not really, and not forever, pace second law) and order springs from disorder. because order is just spending energy to create a temporary equilibrium. anyway, i don't know a damn thing about the thermosphere but the point is we don't really blast that much heat out. we trap it, and have since day 1, and we also accidentally used it to create a bunch of neat shit. https://en.wikipedia.org/wiki/Autocatalysis#Creation_of_order

borrowed time, but then we're in good company -- the whole universe is borrowing too

flagamuffin  ·  45 days ago  ·  link  ·  

hubski paragraph spacing still sucks

wasoxygen  ·  68 days ago  ·  link  ·    ·  parent  ·  post: Weekly Photo Challenge: Natural Light

A friend of a friend snapped this photo of some guy near the Burmese border in Thailand.

blackbootz  ·  66 days ago  ·  link  ·  

Some guy is a tiny bit embarrassed for attaching this photo to a completely different thread by accident. But is glad his digital mess is tiny.

wasoxygen  ·  69 days ago  ·  link  ·    ·  parent  ·  post: The Strong Law of Small Numbers

And in 2019, I just entered the phrase "is 4294967297 prime" into a search engine. That level of effort might explain why I only got two out of three of the examples above correct (no better than chance!) before looking at the solutions.

But #1 stumped Fermat too! I didn't know that this problem sparked Euler's interest in number theory, according to "How Euler Did It" (4-page PDF). It was one of the many problems left over from the famous Fermat-Descartes correspondence.

    Fermat and Descartes did not like each other very much. In fact, some people describe their relationship as a “feud,” but it seems that Descartes resented Fermat more than Fermat disliked Descartes. They probably never met.

I figured Euler must have scribbled out a lot of long division problems to crack the Fermat number conjecture. But apparently he found a shortcut.

[SPOILER]

    Euler’s mentor in St. Petersburg, Christian Goldbach, alerted Euler to the conjecture in 1729. Euler responded almost immediately that he could make no progress on the problem, but by 1732, close to a hundred years after Fermat had originally made the conjecture, Euler had a solution: Fermat was wrong. In Euler’s first paper on number theory [E26] Euler announced that 641 divides 4,294,967,297.... What Euler did not tell us in E26 was how he thought to try to divide 4,294,967,297 by 641. He hadn’t simply been dividing by prime numbers until he got to 641. He had a much better way, but he waited about fifteen years, until E134, to reveal that secret.
wasoxygen  ·  70 days ago  ·  link  ·    ·  parent  ·  post: The Strong Law of Small Numbers
am_Unition  ·  70 days ago  ·  link  ·  

Yes, I think my statement needs to be amended to say:

    ...no known set of functions, mappings, transformations etc. that generates ALL prime numbers.

Edit: Actually I wasn't sure there exists even a single algorithm capable of producing only prime numbers as n -> infinity, even if the results for n below a "really large" were a subset of all primes. But Rowland (from wasoxygen's wiki article above) apparently managed to do exactly that.

Can you imagine being a reviewer for this article? It would be incredibly punishing to go through every problem and substitute increasing n until you saw whether or not it continued to meet the criteria after leaving the "small n" domain. In fact, a reviewer wouldn't do that, they'd write code to do it for them, even in 1988.

bfv  ·  70 days ago  ·  link  ·  

Generating all primes is just as easy as testing for primes:

    import math

def is_prime(n):

for m in range(2,int(math.sqrt(n))+1):

if n%m == 0:

return False

return True

def primes():

n = 2

while True:

if is_prime(n):

yield n

n += 1

for p in primes():

print p

it's just that we don't have useful ways to generate primes.

wasoxygen  ·  69 days ago  ·  link  ·  

And in 2019, I just entered the phrase "is 4294967297 prime" into a search engine. That level of effort might explain why I only got two out of three of the examples above correct (no better than chance!) before looking at the solutions.

But #1 stumped Fermat too! I didn't know that this problem sparked Euler's interest in number theory, according to "How Euler Did It" (4-page PDF). It was one of the many problems left over from the famous Fermat-Descartes correspondence.

    Fermat and Descartes did not like each other very much. In fact, some people describe their relationship as a “feud,” but it seems that Descartes resented Fermat more than Fermat disliked Descartes. They probably never met.

I figured Euler must have scribbled out a lot of long division problems to crack the Fermat number conjecture. But apparently he found a shortcut.

[SPOILER]

    Euler’s mentor in St. Petersburg, Christian Goldbach, alerted Euler to the conjecture in 1729. Euler responded almost immediately that he could make no progress on the problem, but by 1732, close to a hundred years after Fermat had originally made the conjecture, Euler had a solution: Fermat was wrong. In Euler’s first paper on number theory [E26] Euler announced that 641 divides 4,294,967,297.... What Euler did not tell us in E26 was how he thought to try to divide 4,294,967,297 by 641. He hadn’t simply been dividing by prime numbers until he got to 641. He had a much better way, but he waited about fifteen years, until E134, to reveal that secret.
wasoxygen  ·  73 days ago  ·  link  ·    ·  parent  ·  post: No one makes a living on Patreon

    Of those creators, only 1,393 — 2 percent — make the equivalent of federal minimum wage of $7.25 an hour, or $1,160 a month, in October 2017.

— assuming each creator worked full time, 160 hours, on their Patreon projects in October.

wasoxygen  ·  78 days ago  ·  link  ·    ·  parent  ·  post: One day, a mighty oak.

Recipe?

I dunked a healthy-looking acorn in the dirt of a potted plant and kept it watered a while but nothing came up.

johan  ·  78 days ago  ·  link  ·  

Disclaimer: I only know this works for European cold-temperate oaks. The Quercus genus has hundreds of species with Mexico having the most species.

I don’t have any direct experience sprouting acorns, but you might have just gotten a dud or, depending on the species, you need to put it in the fridge for a while to mimick the cold period it would experience in nature to wake it from hibernation. Bonus tip for growing your mighty oak a little bit faster: plant it together with a nurse tree of a faster growing species so it gets some competition for light, as well as a bit of protection from the wind. Around here birch or larch are commonly planted together with oak like this, the nurse tree gets removed after a couple of years. This also reduces the risk of multiple trunks, if that’s something you want to avoid.

mk  ·  77 days ago  ·  link  ·  

johan is correct. I put it in the fridge for a month.

I am experimenting with other seeds as well.

WanderingEng  ·  76 days ago  ·  link  ·  

In the refrigerator dry or wet? When I sprouted apple seeds I put them in the refrigerator in a plastic bag with damp paper towels. They sprouted fine.

mk  ·  71 days ago  ·  link  ·  

Wet. I dampened the soil. Funny thing, the leave in there is another plant. I just discovered so when re-potting it. However, the real thing is now sprouting.

WanderingEng  ·  78 days ago  ·  link  ·  

I planted some apple seeds which sprouted and then died. Now I have some healthy pepper plants instead. I might buy some pine tree seeds next. They're on my Amazon wish list.

cgod  ·  77 days ago  ·  link  ·  

My father was an infamous planter of trees.

He never started any by seed but bought saplings from nurseries. He planted thousands of flowers by seed but never a tree.

Once a year or every other year fifty saplings would show up and he'd get to work, planting a tree anywhere he thought would be nice, peoples yards, empty lots, the park and nature preserves. He did all of this under the cover of night, fueled by whimsy and malt liquor with permission never asked or granted.

The cops came by to tell him to cut it out, people aren't allowed to plant trees on public land or something but knowing it was him and proving it was evidently a crime that they weren't ready to assign a detective over. He never did cut it out.

Anyway, I'm sure pops would tell you to quit screwing around with seeds and carefully pick a few saplings of trees that you love.

WanderingEng  ·  77 days ago  ·  link  ·  
This comment has been deleted.
mk  ·  71 days ago  ·  link  ·  

If you put a colored plastic ring around the bottom, workers are more likely to assume the planting wasn't rogue.

steve  ·  77 days ago  ·  link  ·  

I love the stories of your dad’s rogue tree shenanigans.

wasoxygen  ·  89 days ago  ·  link  ·    ·  parent  ·  post: It's been too long! Book thread time. What have you been reading?

With time to spare, I am in sight of ten thousand pages for the year, a goal I don't often achieve. E-books made it possible, contributing about two-thirds of the total. A lot of these were free downloads, starting with a bootleg copy of Orwell's amazing The Road to Wigan Pier, which paired well with The People of the Abyss, but mostly Gutenberg classics.

The Adventures of Tom Sawyer and Huckleberry Finn were good reading; the first made me wonder what all the fuss over content was about, the second relieved my confusion. Additional hits were The Autobiography of Benjamin Franklin and The Adventures of Sherlock Holmes.

    Sherlock Holmes sat moodily at one side of the fireplace cross-indexing his records of crime, while I at the other was deep in one of Clark Russell's fine sea-stories...

William Clark Russell was a great discovery. I much enjoyed The Wreck of the Grosvenor, The Death Ship, and The Frozen Pirate (though this last title had some passages that made Mark Twain's racial insensitivity seem mild).

Reamde was a $3 selection on Bookperk, and with a November repeat of Quicksilver gave me nearly 2000 pages from the author who dominates a word cloud based on my reading records.

Most of my more serious reading is still on paper, where I can scribble notes. The News: A User’s Manual was a reminder to focus on things of lasting importance, mostly unheeded, with some memorable exceptions.

And one more Alain de Botton title, thanks to a Hubski tip.

Vaclav Smil's Making the Modern World was relatively small and densely-packed, like the landfills he says are the endpoint for much of Europe's recycling.

    Collection rates of post-consumer plastic waste are impressive, more than 90% in eight EU countries and in Switzerland, and an average of 43% (25.1 Mt) in 2011 for the EU-27; but more than 40% (10.3 Mt) of collected plastics were then landfilled and of the 14.9 Mt (60%) that were recovered nearly 9 Mt were incinerated (energy recovery) and only about 6 Mt (or close to 10% of annual production) were recycled (Plastics Europe, 2019).
wasoxygen  ·  89 days ago  ·  link  ·    ·  parent  ·  post: Times Literary Supplement on Jack London's People of the Abyss

Updated link: Grape stems

by the same author: Jack London, a century on