...postulating black holes well before anyone else.
This isn't really true. The idea of a very massive body from which light cannot escape was proposed in the 18th century, over 100 years before Sidis. Schwartzchild was the first to work out an explicit solution to Einstein's field equations that gave such a body. This page has some historical details.
It seems Sidis also proposed a sort of steady-state model of the universe (see the crankish website linked elsewhere in this thread), but the last 80 years of astronomical observations have made such theories unviable.
Or say you wrap your finite universe into a closed loop, so there's no edge. Except, now you've added dimensions to the finite ones you already had - are they finite or infinite?
Why have I added dimensions? If I have a compact spacetime, there is no mathematical reason for me to embed it into some higher-dimensional space. None of our standard physical theories make use of any such embedding.
If you wrap that up into another loop, you've added more dimensions... and so on and so on infinitely. Infinite dimensions.
When an electron from an accelerator is subject to acceleration by deflecting it by a magnetic or electric field from a non-linear path, it radiates energy called Cerenkov radiation.
No. Cerenkov radiation comes from charged particles travelling through a medium at a speed greater than the speed of light in that medium. You are describing Bremsstrahlung radiation.
It is not known how electrons "know" they are traveling in a curved path as required by the electric fields of an atoms vs when they are deflected by a magnetic or electric field in a vacuum.
This is also wrong. The electrons in orbit about a nucleus are not "moving" in any classical sense of the word. They have kinetic energy, and often they have orbital angular momentum, but there is never any linear velocity relative to the center of mass of the atom, hence there is no centripetal acceleration. The absence of radiation is quite compatible with quantum mechanics. In fact, this is one of the reasons for favoring the quantum mechanical description over the Bohr model.
Zero point energy is the energy left in space that has been cooled to absolute zero temperature.
Why are you unwilling to accept quantum mechanics as a useful description of atomic behavior, but you have no problem with zero-point energy? The latter has no experimental basis whatsoever.
Chandra's limit is all very nice, but I've never heard a compelling explanation as to how matter would be helped across this point?
The Chandrasekhar limit is nothing more than an upper bound on the mass of a stable white dwarf. It doesn't really have much to do with the formation of an event horizon, except in the historical sense.
If you assume the electrons in a white dwarf form a degenerate gas, then in the nonrelativistic limit, the fermionic pressure can form a stable equilibrium with gravitational attraction (a decrease in radius causes the energy levels to rise). Above a certain mass, the highest energy levels are at a substantial fraction of electron rest mass, and relativistic effects cause the equilibrium to vanish. Chandrasekhar proposed that the star above the threshold mass would then collapse to a black hole, but that's because the neutron hadn't been discovered yet, and he didn't know that there could be another stable configuration. The same relativistic problem doesn't happen with the degenerate neutron gas in a neutron star, since neutrons have so much more mass than electrons.
Does the blackhole form in the very middle and expand outwards, or does the whole star just disappear?
If you have a collapsing sphere of uniform density, then the event horizon will form at the surface. This comes from eyeballing the dimensional analysis of Schwartzchild's solution.
why don't protons turn into blackholes from the middle out?
Protons are way too fat to bacome black holes. Something with the mass of a proton would have to be confined to a region much smaller than the Planck length to undergo this sort of gravitational collapse. Also, a black hole with such a small mass would immediately evaporate due to Hawking radiation.
Does time dilation prevent the singularity from ever forming?
I think a more interesting question is whether an event horizon can form in finite time in a distant observer's frame, and no one has ever given me a straight answer.
Or maybe they can't see anything because all light coming off it is so redshifted it can't be distinguished from the background.
Interestingly, this behavior would be consistent with that of a black hole. If you are a distant observer watching a luminous body that collapses to a black hole, you will continue to receive radiation from it, but it will be increasingly redshifted and its luminosity will suffer an exponential decay with relatively short half-life. You will also get black body radiation, but the temperature of a black hole this size is far to low to stand out from the background.
You make a leap from dense matter to singularity which I was questioning.
You don't need a singularity to have a black hole. You just need an event horizon. The reason people claim that the object is a black hole is that you don't need to assume the existence of exotic, unobserved-to-date forms of matter to explain the phenomena, while most alternatives do. Naturally, if you are arguing that no one has definitively established the existence of an event horizon in the center of the galaxy, most people will agree with you. However, I'd say most astrophysicists think a black hole gives the most likely explanation for the observed phenomena.
Just as a planet can orbit a star, or a star can orbit another star, a star can orbit a black hole. It will behave exactly as if it were orbiting a planet of an equal mass, as long as it's going fast enough to maintain orbit.
If the star is too close to the black hole, relativistic effects will mess with the orbit. In praticular, there are regions around black holes with no stable orbits, where a body's trajectory builds up oscillations and it is either flung out or dropped in.
According to MTW, the smallest stable circular orbit around a nonrotating black hole has three times the radius of the black hole. For rotating black holes, this is decreased for orbits in the same direction as rotation, and increased for retrograde orbits.
Spheres orbiting spheres is a time-dependent representation. Note, that electrons still do fly around protons.
You ought to be a bit more precise about what you mean by "fly around," since the s orbital states have zero orbital angular momentum and zero expected linear velocity relative to the nucleus. Also, the only time-dependent behavior in these states is a phase rotation, which doesn't change the probability densities being depicted in the charts. Perhaps it would be more appropriate to say that they sit around, but with some extra undirected kinetic energy.
By the way, you do get time-dependent probability shifts with mixed states, and in the limit of high-energy orbitals, the behavior looks a lot like the normal motion of attracting classical particles.
You're missing a factor of ten in your last division. It should be 80 milliwatts, not 8. To get initial output (about 120 milliwatts) just divide by ln2.
This isn't as bad as it sounds. If you store your tritium as a metal hydride rather than in gaseous form, you can get decent power density. This site indicates that you can get one mole into 3 cubic centimeters without needing any sort of high-pressure containment structure. Even with 10% efficiency, it should be feasible to make a low-power laptop that runs on about 100 cubic centimeters of this, and it would not be particularly heavy.
Nucular has alot of potential but that won't last more than 30 years, the supply of fuel is limited, though lower grade fuels are available at higher cost.
Do you have a source for this information? It is far below any other estimate of nuclear viability I've seen, as they never fall below a few hundred years. In particular, this page suggests that if other sources of energy run short (perhaps in a few centuries), it will become economically reasonable to extract uranium from seawater or granite, and those supplies are virtually limitless.
I was under the impression that information had 0 thermodynamic value. Where'd you hear otherwise?
According to Landauer's principle, if you erase a bit, you must use up at least kT ln2 energy (i.e. that much is liberated as unusable heat). Here, k is Boltzmann's constant, so at room temperature this amounts to about 2.85*10^(-21) Joules per bit. This seems rather small, but trends in computing technology suggest that this hard limit will become a serious problem within the next 20 years.
We can theoretically circumvent this problem by using reversible logic, which is a computing model that does not erase any bits. The MIT Pendulum project did some work on this, and they constructed some defective reversible circuits (info here), but the project seems to have stopped. I haven't heard of any practical implementations, but I'm not a specialist.
The resulting search spaces are about the same size, even though the first example is 9 characters, where the second is 8.
This is clearly false. The power of including punctuation is that a typical keyboard has about 32 punctuation characters:
!@#$%^&*()-_=+\|`~[{]};:'",<.>/?
We can easily calculate the size of password space 2 as (26 + 26 + 10)^8 = 62^8 = 218340105584896 which is about 2x10^14. We can establish a lower bound for password space 1 by counting passwords with no repeated characters:
This is more than 15000 times your password space. Even if we removed one of the "any" spots, making 8-letter passwords, we'd have 4268339712000000 non-repeating possibilities, about 20 times the size of space 1.
A month sounds awfully long for a 486, and it suggests that you were using a rather non-optimal multiplication algorithm.
Judicious use of an FFT-based algorithm should reduce the computation time by some orders of magnitude. There are also fast base conversion algorithms for large integers. These can be found in e.g. Knuth, TAOCP.
Wouldn't a neutron star be spinning at tremendous angular velocities...
You may be thinking of millisecond pulsars. Not all neutron stars spin at high angular velocities, and the slow ones have very little deviation from a ball shape.
...that would give us 20000!, or roughly 1.819e+77337 possibilities.
20000! counts the number of possible permutations, not the number of possible interactions. If you assume that interactions only take place between two genes at a time, and that any pair of genes will interact exactly one way, you get only about 200 million possible interactions. Unfortunately, neither of these assumptions seems to hold, and any meaningful enumeration would require a more rigorous definition of "interaction" anyway.
The big problem with this article is its repeated use of phrases such as "instantly change" and "instantaneously influence" to describe correlations in the results of space-like separated measurements. Simultaneity among separate events is not well-defined under the basic assumptions of relativity, and the use of terms like "instant" is suggestive of an absolute frame of reference.
According to Steven Hawking, they don't actually exist -- the singularity never forms in a quantum mechanical universe.
This may be true, but it doesn't really affect the long-term behavior of anything less than a small distance away from the body.
Bear in mind that a black hole with lunar mass would have a tiny event horizon.
It would be a couple millimeters in radius - quite sufficient to fit atoms and such.
Given the amount of thermal noise in the solar center, it would be very hard for anything to "fall in" without being bumped out first.
That doesn't make any sense. Ultra-dense objects are less susceptible to thermal noise than light things like solar sails. Furthermore, the energy in any sort of "bumping" radiation would get eaten, and added to the mass.
In time, the hole might consume the sun, but my back of the envelope calculations suggest that it's far more likely that the pseudo-singularity would decay in a burst of Hawking radiation long before it consumed anything.
That's interesting. My envelope says this burst is more likely to happen roughly 5*10^48 seconds from the time the moon gets swallowed, and this is somewhat longer than your average power lunch. This page might help your envelope a bit. Remember, the exponent in that 10^71 has no minus sign.
Spacetime diagram doesn't work out for this one, unfortunately...
It works out well if you parse merenguid's use of simultaneity in different frames. The situation is as follows: Alpha Centauri sends a signal to Earth, and this is bounced to the ship. This all happens in the spacelike slice for which Alpha Centauri is motionless and at the time the outcome of the lottery is announced. The ship is traveling rapidly away from Alpha Centauri, so the spacelike slice for which the ship is motionless passes through Alpha Centauri well in the past. Thus, if the ship sends a signal to Alpha Centauri that is instantaneous in the ship's frame, it will arrive before the outcome is announced.
Re:Does NASA have too much money?
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Our Man In Black
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the fact is, they were never able to kill everything; this is a well known fact within that little niche industry. there are simply bacteria that cannot be killed, end of story.
An extended bake at sufficiently high temperatures will destroy all life, and in fact all high-molecular-weight compounds. The problem arises when the equipment you are sterilizing is made of insufficiently thermally robust materials (e.g. sensors on a mars probe).
I'm afraid your scale is off by about a factor of 10, although your assertion that 100km -> 1mm is off by a factor of 100 in the other direction (perhaps you meant 1 micron?). Note that one AU is about 1.5x10^11 meters, so when rescaling by 10^11, Earth should be 1.5 meters from the sun, which should be 13.9mm in diameter. Also note that one light year is about 10^16 meters, so the distance to Alpha Centauri should be 411 km (as opposed to 4200), and the diameter of the Milky Way should be about 10^7 km.
Naturally, your table is essentially correct if instead you are shrinking by a factor of 1e10. This means 10,000 km -> 1mm.
String theory is misnamed. It is more appropriate to call it the 'string method of calculation'. String 'theory' is simply a mathematical metaphor which allows predictions of the behaviour and interactions of sub-atomic entities.
Do you know any useful physical theories which do not boil down to methods of calculation? Any model of reality which predicts behavior and interactions is called a theory. Your semantic hairsplitting is useless.
For example, if you assume that particles are 1-dimensional lines or loops you avoid many of the problems (specifically, singularities and infinities) you get if you assume particles are infinitesimal points.
This sentence is completely pointless. Do you have any evidence that fundamental particles are modeled better by point objects than strings? You never gave any damning evidence against the string metaphor.
Its nonsense to suggest that particles are really small vibrating 'strings' experiencing an tension force, otherwise you get an infinite regress: To explain the behaviour of everyday things (such as real pieces of string which can vibrate), we would be requiring the existence of incredibly small 'strings' which would 'vibrate', which doesn't really get us anywhere!
Where is the infinite regress? Nobody claims that the fundamental strings are actually made of intertwined plant or polymer fiber. The name "vibrating string" is simply evocative of familiar phenomena to make excitations more amenable to study and discussion.
As I understand it, the idea is that the particle and the anti-particle come into being at the same place, moving in different dirrectsion, and the anti-particle is more prone to being pulled in somehow due it its being the opposite of the other mass in the black hole.
Unfortunately for your theory, particles and antiparticles behave essentially identically near a heavy object, so neither type is favored and (assuming the particle description of the mechanism is accurate - which it isn't) you can expect an equal number of particles and antiparticles to fall in. At least, Hawking's calculations didn't take your proposed mechanism into account.
The particle escapes, generating the black-body radiation, and the anti-particle enters the black whole and collides with a corresponding particle, leaving existance as the original particles came into existance - messed up I know.
If an antiparticle were to enter a black hole, it would add to the mass. If it were to collide with a corresponding particle whilst in transit, there would be some kind of radiation released into the hole, conserving total mass-energy. The perturbative explanation people like to give explains the mass theft in terms of the particle-antiparticle pair "borrowing" energy from the vacuum in order to exist, and the event horizon splitting them before they can recombine, causing the energy debt to be furnished by the black hole.
The black hole doesn't gain mass, because the particle that fell in has negative energy. Remember, you can't create energy from nowhere, but you can "borrow" some from the vacuum temporarily... that's where the virtual pairs come from.
The virtual particle approach to Hawking radiation seems to be more of a perturbative approximation that has caught on in the popular press than a reasonable description of reality. It may be more natural to describe the radiation in terms of the Unruh effect, which predicts a thermal spectrum around uniformly accelerating bodies - quasistationary objects near the event horizon are bathed in thermal radiation, and this is gravitationally redshifted as it propagates to the distant observer.
This avoids the rather cumbersome notions of negative energy and virtual particles which people tend to find counterintuitive. I have a recent post here
which gives some relevant information and links.
The heat generated by re-entry is because of the horizontal motion of the craft, but a projectile of this type would only have vertical motion with respect to the atmosphere, and therefore relatively little heat generated.
This is false. The heat is generated by air friction, which is a side-effect of travel through the atmosphere in any direction. The total heat generated between launch and target is minimized if you have a vertical trajectory, but chances are, Professor Richard Garwin crunched a few more numbers than you did. Furthermore, firing directly toward the Earth from geosynchronous orbit will almost guarantee that you miss the planet, because your projectile still has all of the angular momentum associated with the orbit. If you decrease the distance from Earth's center by a factor of 5, you will increase the angular velocity correspondingly.
Given that the military already uses kinetic kill technology (horizontally fired from vehicles, no explosives) that are able to penetrate main battle tank armor, why would dropping a similarly size projectile from orbit (wouldn't the terminal velocity be tremendous) be less than traditional explosives?
Garwin may be ducking the possibility that the rods would be used to penetrate small, hardened targets. However, you can estimate the energy difference yourself: 1 kg of TNT yields 4.184x10^6 Joules (this is a standard unit of explosive power), and higher-quality explosives may do substantially more. To match that energy output, a 1 kg mass needs to strike at about 3 km/s, but it loses some flexibility with respect to how the energy is to be spent on the target. The bottom of this page has an explanation of the structure and function of kinetic anti-tank rounds (section 13.4.5.10). As others mentioned, your use of the term "terminal velocity" is flawed - the speed of the falling projectile is not meant to stabilize.
Why would you loft a laser platform into orbit and fire it through all that atmosphere down to a blimp, when you could just mount it on a large aircraft?
There isn't much atmosphere above the blimp. About half of the atmosphere of the Earth is located below 5000 meters, so unless your ground-based or air-based laser is really close to the blimp, you will be firing through more air molecules than the satellite. You also don't have to spend much fuel keeping a blimp aloft.
Again, from what I've been told, it's not hard to destroy satellites. They are orbiting at ridiculously high speeds. Wouldn't just releasing a cloud of marbles (or even sand!) in their trajectory, orbiting in the opposite direction, easily shred the enemy satellite?
First, it's kind of hard to hit a fast-moving target. That's why our missile defense system doesn't work. Second, there is a lot of space above our heads, and you would have to release your marbles in a very precise trajectory to have any chance of effectiveness - consider the space of all possible orbits, and compare it with the cross-section of your target. The marbles would have no guidance after release, and if you just dump marbles in orbit indiscriminately, the probability that they will wreck your own gear at some point in the future is about as high as that of their taking down your target of choice.
One striking example is classification of finite groups, which is estimated to be spread over around 10000 papers in various journals. Can a single person or even a small group of people read it and say with absolute certainty that there are no errors?
The purpose of publishing a result is to communicate ideas to other mathematicians, not to submit a sequence of logical steps to computerized scrutiny. There is a tacit assumption that the logical arguments given in papers can be rigidly formalized with "absolute certainty", but to do so in every case would slow progress substantially. It would be more accurate to say that a sequence of statements is not a valid proof than to say that a proof does not comunicate an absolute certainty.
Incidentally, while the classification of finite simple groups has its share of problems (e.g. some parts of it are still unpublished), we are not even close to classifying all finite groups (except perhaps up to Jordan-Holder equivalence). It is a good idea to make accurate statements when criticizing others for lack of precision.
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This isn't really true. The idea of a very massive body from which light cannot escape was proposed in the 18th century, over 100 years before Sidis. Schwartzchild was the first to work out an explicit solution to Einstein's field equations that gave such a body. This page has some historical details.
It seems Sidis also proposed a sort of steady-state model of the universe (see the crankish website linked elsewhere in this thread), but the last 80 years of astronomical observations have made such theories unviable.
Or say you wrap your finite universe into a closed loop, so there's no edge. Except, now you've added dimensions to the finite ones you already had - are they finite or infinite?
Why have I added dimensions? If I have a compact spacetime, there is no mathematical reason for me to embed it into some higher-dimensional space. None of our standard physical theories make use of any such embedding.
If you wrap that up into another loop, you've added more dimensions... and so on and so on infinitely. Infinite dimensions.
This is completely unnecessary.
When an electron from an accelerator is subject to acceleration by deflecting it by a magnetic or electric field from a non-linear path, it radiates energy called Cerenkov radiation.
No. Cerenkov radiation comes from charged particles travelling through a medium at a speed greater than the speed of light in that medium. You are describing Bremsstrahlung radiation.
It is not known how electrons "know" they are traveling in a curved path as required by the electric fields of an atoms vs when they are deflected by a magnetic or electric field in a vacuum.
This is also wrong. The electrons in orbit about a nucleus are not "moving" in any classical sense of the word. They have kinetic energy, and often they have orbital angular momentum, but there is never any linear velocity relative to the center of mass of the atom, hence there is no centripetal acceleration. The absence of radiation is quite compatible with quantum mechanics. In fact, this is one of the reasons for favoring the quantum mechanical description over the Bohr model.
Zero point energy is the energy left in space that has been cooled to absolute zero temperature.
Why are you unwilling to accept quantum mechanics as a useful description of atomic behavior, but you have no problem with zero-point energy? The latter has no experimental basis whatsoever.
Chandra's limit is all very nice, but I've never heard a compelling explanation as to how matter would be helped across this point?
The Chandrasekhar limit is nothing more than an upper bound on the mass of a stable white dwarf. It doesn't really have much to do with the formation of an event horizon, except in the historical sense.
If you assume the electrons in a white dwarf form a degenerate gas, then in the nonrelativistic limit, the fermionic pressure can form a stable equilibrium with gravitational attraction (a decrease in radius causes the energy levels to rise). Above a certain mass, the highest energy levels are at a substantial fraction of electron rest mass, and relativistic effects cause the equilibrium to vanish. Chandrasekhar proposed that the star above the threshold mass would then collapse to a black hole, but that's because the neutron hadn't been discovered yet, and he didn't know that there could be another stable configuration. The same relativistic problem doesn't happen with the degenerate neutron gas in a neutron star, since neutrons have so much more mass than electrons.
Does the blackhole form in the very middle and expand outwards, or does the whole star just disappear?
If you have a collapsing sphere of uniform density, then the event horizon will form at the surface. This comes from eyeballing the dimensional analysis of Schwartzchild's solution.
why don't protons turn into blackholes from the middle out?
Protons are way too fat to bacome black holes. Something with the mass of a proton would have to be confined to a region much smaller than the Planck length to undergo this sort of gravitational collapse. Also, a black hole with such a small mass would immediately evaporate due to Hawking radiation.
Does time dilation prevent the singularity from ever forming?
I think a more interesting question is whether an event horizon can form in finite time in a distant observer's frame, and no one has ever given me a straight answer.
Or maybe they can't see anything because all light coming off it is so redshifted it can't be distinguished from the background.
Interestingly, this behavior would be consistent with that of a black hole. If you are a distant observer watching a luminous body that collapses to a black hole, you will continue to receive radiation from it, but it will be increasingly redshifted and its luminosity will suffer an exponential decay with relatively short half-life. You will also get black body radiation, but the temperature of a black hole this size is far to low to stand out from the background.
You make a leap from dense matter to singularity which I was questioning.
You don't need a singularity to have a black hole. You just need an event horizon. The reason people claim that the object is a black hole is that you don't need to assume the existence of exotic, unobserved-to-date forms of matter to explain the phenomena, while most alternatives do. Naturally, if you are arguing that no one has definitively established the existence of an event horizon in the center of the galaxy, most people will agree with you. However, I'd say most astrophysicists think a black hole gives the most likely explanation for the observed phenomena.
Just as a planet can orbit a star, or a star can orbit another star, a star can orbit a black hole. It will behave exactly as if it were orbiting a planet of an equal mass, as long as it's going fast enough to maintain orbit.
If the star is too close to the black hole, relativistic effects will mess with the orbit. In praticular, there are regions around black holes with no stable orbits, where a body's trajectory builds up oscillations and it is either flung out or dropped in.
According to MTW, the smallest stable circular orbit around a nonrotating black hole has three times the radius of the black hole. For rotating black holes, this is decreased for orbits in the same direction as rotation, and increased for retrograde orbits.
Spheres orbiting spheres is a time-dependent representation. Note, that electrons still do fly around protons.
You ought to be a bit more precise about what you mean by "fly around," since the s orbital states have zero orbital angular momentum and zero expected linear velocity relative to the nucleus. Also, the only time-dependent behavior in these states is a phase rotation, which doesn't change the probability densities being depicted in the charts. Perhaps it would be more appropriate to say that they sit around, but with some extra undirected kinetic energy.
By the way, you do get time-dependent probability shifts with mixed states, and in the limit of high-energy orbitals, the behavior looks a lot like the normal motion of attracting classical particles.
You're missing a factor of ten in your last division. It should be 80 milliwatts, not 8. To get initial output (about 120 milliwatts) just divide by ln2.
This isn't as bad as it sounds. If you store your tritium as a metal hydride rather than in gaseous form, you can get decent power density. This site indicates that you can get one mole into 3 cubic centimeters without needing any sort of high-pressure containment structure. Even with 10% efficiency, it should be feasible to make a low-power laptop that runs on about 100 cubic centimeters of this, and it would not be particularly heavy.
Nucular has alot of potential but that won't last more than 30 years, the supply of fuel is limited, though lower grade fuels are available at higher cost.
Do you have a source for this information? It is far below any other estimate of nuclear viability I've seen, as they never fall below a few hundred years. In particular, this page suggests that if other sources of energy run short (perhaps in a few centuries), it will become economically reasonable to extract uranium from seawater or granite, and those supplies are virtually limitless.
I was under the impression that information had 0 thermodynamic value. Where'd you hear otherwise?
According to Landauer's principle, if you erase a bit, you must use up at least kT ln2 energy (i.e. that much is liberated as unusable heat). Here, k is Boltzmann's constant, so at room temperature this amounts to about 2.85*10^(-21) Joules per bit. This seems rather small, but trends in computing technology suggest that this hard limit will become a serious problem within the next 20 years.
We can theoretically circumvent this problem by using reversible logic, which is a computing model that does not erase any bits. The MIT Pendulum project did some work on this, and they constructed some defective reversible circuits (info here), but the project seems to have stopped. I haven't heard of any practical implementations, but I'm not a specialist.
You're right. My mistake.
The resulting search spaces are about the same size, even though the first example is 9 characters, where the second is 8.
This is clearly false. The power of including punctuation is that a typical keyboard has about 32 punctuation characters:
!@#$%^&*()-_=+\|`~[{]};:'",<.>/?
We can easily calculate the size of password space 2 as (26 + 26 + 10)^8 = 62^8 = 218340105584896 which is about 2x10^14. We can establish a lower bound for password space 1 by counting passwords with no repeated characters:
(32 punctuation)*(9 spots) *
* (10*9 numbers) * (8*7 spots) *
* (26*25 lower) * (6*5 spots) *
* (26*25 upper) * (4*3 spots) *
* (87*86 any) * (2*1 spots) = 3303694937088000000
This is more than 15000 times your password space. Even if we removed one of the "any" spots, making 8-letter passwords, we'd have 4268339712000000 non-repeating possibilities, about 20 times the size of space 1.
A month sounds awfully long for a 486, and it suggests that you were using a rather non-optimal multiplication algorithm.
Judicious use of an FFT-based algorithm should reduce the computation time by some orders of magnitude. There are also fast base conversion algorithms for large integers. These can be found in e.g. Knuth, TAOCP.
Wouldn't a neutron star be spinning at tremendous angular velocities...
You may be thinking of millisecond pulsars. Not all neutron stars spin at high angular velocities, and the slow ones have very little deviation from a ball shape.
20000! counts the number of possible permutations, not the number of possible interactions. If you assume that interactions only take place between two genes at a time, and that any pair of genes will interact exactly one way, you get only about 200 million possible interactions. Unfortunately, neither of these assumptions seems to hold, and any meaningful enumeration would require a more rigorous definition of "interaction" anyway.
The big problem with this article is its repeated use of phrases such as "instantly change" and "instantaneously influence" to describe correlations in the results of space-like separated measurements. Simultaneity among separate events is not well-defined under the basic assumptions of relativity, and the use of terms like "instant" is suggestive of an absolute frame of reference.
According to Steven Hawking, they don't actually exist -- the singularity never forms in a quantum mechanical universe.
This may be true, but it doesn't really affect the long-term behavior of anything less than a small distance away from the body.
Bear in mind that a black hole with lunar mass would have a tiny event horizon.
It would be a couple millimeters in radius - quite sufficient to fit atoms and such.
Given the amount of thermal noise in the solar center, it would be very hard for anything to "fall in" without being bumped out first.
That doesn't make any sense. Ultra-dense objects are less susceptible to thermal noise than light things like solar sails. Furthermore, the energy in any sort of "bumping" radiation would get eaten, and added to the mass.
In time, the hole might consume the sun, but my back of the envelope calculations suggest that it's far more likely that the pseudo-singularity would decay in a burst of Hawking radiation long before it consumed anything.
That's interesting. My envelope says this burst is more likely to happen roughly 5*10^48 seconds from the time the moon gets swallowed, and this is somewhat longer than your average power lunch. This page might help your envelope a bit. Remember, the exponent in that 10^71 has no minus sign.
Spacetime diagram doesn't work out for this one, unfortunately...
It works out well if you parse merenguid's use of simultaneity in different frames. The situation is as follows: Alpha Centauri sends a signal to Earth, and this is bounced to the ship. This all happens in the spacelike slice for which Alpha Centauri is motionless and at the time the outcome of the lottery is announced. The ship is traveling rapidly away from Alpha Centauri, so the spacelike slice for which the ship is motionless passes through Alpha Centauri well in the past. Thus, if the ship sends a signal to Alpha Centauri that is instantaneous in the ship's frame, it will arrive before the outcome is announced.
the fact is, they were never able to kill everything; this is a well known fact within that little niche industry. there are simply bacteria that cannot be killed, end of story.
An extended bake at sufficiently high temperatures will destroy all life, and in fact all high-molecular-weight compounds. The problem arises when the equipment you are sterilizing is made of insufficiently thermally robust materials (e.g. sensors on a mars probe).
I'm afraid your scale is off by about a factor of 10, although your assertion that 100km -> 1mm is off by a factor of 100 in the other direction (perhaps you meant 1 micron?). Note that one AU is about 1.5x10^11 meters, so when rescaling by 10^11, Earth should be 1.5 meters from the sun, which should be 13.9mm in diameter. Also note that one light year is about 10^16 meters, so the distance to Alpha Centauri should be 411 km (as opposed to 4200), and the diameter of the Milky Way should be about 10^7 km.
Naturally, your table is essentially correct if instead you are shrinking by a factor of 1e10. This means 10,000 km -> 1mm.
String theory is misnamed. It is more appropriate to call it the 'string method of calculation'. String 'theory' is simply a mathematical metaphor which allows predictions of the behaviour and interactions of sub-atomic entities.
Do you know any useful physical theories which do not boil down to methods of calculation? Any model of reality which predicts behavior and interactions is called a theory. Your semantic hairsplitting is useless.
For example, if you assume that particles are 1-dimensional lines or loops you avoid many of the problems (specifically, singularities and infinities) you get if you assume particles are infinitesimal points.
This sentence is completely pointless. Do you have any evidence that fundamental particles are modeled better by point objects than strings? You never gave any damning evidence against the string metaphor.
Its nonsense to suggest that particles are really small vibrating 'strings' experiencing an tension force, otherwise you get an infinite regress: To explain the behaviour of everyday things (such as real pieces of string which can vibrate), we would be requiring the existence of incredibly small 'strings' which would 'vibrate', which doesn't really get us anywhere!
Where is the infinite regress? Nobody claims that the fundamental strings are actually made of intertwined plant or polymer fiber. The name "vibrating string" is simply evocative of familiar phenomena to make excitations more amenable to study and discussion.
As I understand it, the idea is that the particle and the anti-particle come into being at the same place, moving in different dirrectsion, and the anti-particle is more prone to being pulled in somehow due it its being the opposite of the other mass in the black hole.
Unfortunately for your theory, particles and antiparticles behave essentially identically near a heavy object, so neither type is favored and (assuming the particle description of the mechanism is accurate - which it isn't) you can expect an equal number of particles and antiparticles to fall in. At least, Hawking's calculations didn't take your proposed mechanism into account.
The particle escapes, generating the black-body radiation, and the anti-particle enters the black whole and collides with a corresponding particle, leaving existance as the original particles came into existance - messed up I know.
If an antiparticle were to enter a black hole, it would add to the mass. If it were to collide with a corresponding particle whilst in transit, there would be some kind of radiation released into the hole, conserving total mass-energy. The perturbative explanation people like to give explains the mass theft in terms of the particle-antiparticle pair "borrowing" energy from the vacuum in order to exist, and the event horizon splitting them before they can recombine, causing the energy debt to be furnished by the black hole.
The black hole doesn't gain mass, because the particle that fell in has negative energy. Remember, you can't create energy from nowhere, but you can "borrow" some from the vacuum temporarily ... that's where the virtual pairs come from.
The virtual particle approach to Hawking radiation seems to be more of a perturbative approximation that has caught on in the popular press than a reasonable description of reality. It may be more natural to describe the radiation in terms of the Unruh effect, which predicts a thermal spectrum around uniformly accelerating bodies - quasistationary objects near the event horizon are bathed in thermal radiation, and this is gravitationally redshifted as it propagates to the distant observer.
This avoids the rather cumbersome notions of negative energy and virtual particles which people tend to find counterintuitive. I have a recent post here which gives some relevant information and links.
The heat generated by re-entry is because of the horizontal motion of the craft, but a projectile of this type would only have vertical motion with respect to the atmosphere, and therefore relatively little heat generated.
This is false. The heat is generated by air friction, which is a side-effect of travel through the atmosphere in any direction. The total heat generated between launch and target is minimized if you have a vertical trajectory, but chances are, Professor Richard Garwin crunched a few more numbers than you did. Furthermore, firing directly toward the Earth from geosynchronous orbit will almost guarantee that you miss the planet, because your projectile still has all of the angular momentum associated with the orbit. If you decrease the distance from Earth's center by a factor of 5, you will increase the angular velocity correspondingly.
Given that the military already uses kinetic kill technology (horizontally fired from vehicles, no explosives) that are able to penetrate main battle tank armor, why would dropping a similarly size projectile from orbit (wouldn't the terminal velocity be tremendous) be less than traditional explosives?
Garwin may be ducking the possibility that the rods would be used to penetrate small, hardened targets. However, you can estimate the energy difference yourself: 1 kg of TNT yields 4.184x10^6 Joules (this is a standard unit of explosive power), and higher-quality explosives may do substantially more. To match that energy output, a 1 kg mass needs to strike at about 3 km/s, but it loses some flexibility with respect to how the energy is to be spent on the target. The bottom of this page has an explanation of the structure and function of kinetic anti-tank rounds (section 13.4.5.10). As others mentioned, your use of the term "terminal velocity" is flawed - the speed of the falling projectile is not meant to stabilize.
Why would you loft a laser platform into orbit and fire it through all that atmosphere down to a blimp, when you could just mount it on a large aircraft?
There isn't much atmosphere above the blimp. About half of the atmosphere of the Earth is located below 5000 meters, so unless your ground-based or air-based laser is really close to the blimp, you will be firing through more air molecules than the satellite. You also don't have to spend much fuel keeping a blimp aloft.
Again, from what I've been told, it's not hard to destroy satellites. They are orbiting at ridiculously high speeds. Wouldn't just releasing a cloud of marbles (or even sand!) in their trajectory, orbiting in the opposite direction, easily shred the enemy satellite?
First, it's kind of hard to hit a fast-moving target. That's why our missile defense system doesn't work. Second, there is a lot of space above our heads, and you would have to release your marbles in a very precise trajectory to have any chance of effectiveness - consider the space of all possible orbits, and compare it with the cross-section of your target. The marbles would have no guidance after release, and if you just dump marbles in orbit indiscriminately, the probability that they will wreck your own gear at some point in the future is about as high as that of their taking down your target of choice.
One striking example is classification of finite groups, which is estimated to be spread over around 10000 papers in various journals. Can a single person or even a small group of people read it and say with absolute certainty that there are no errors?
The purpose of publishing a result is to communicate ideas to other mathematicians, not to submit a sequence of logical steps to computerized scrutiny. There is a tacit assumption that the logical arguments given in papers can be rigidly formalized with "absolute certainty", but to do so in every case would slow progress substantially. It would be more accurate to say that a sequence of statements is not a valid proof than to say that a proof does not comunicate an absolute certainty.
Incidentally, while the classification of finite simple groups has its share of problems (e.g. some parts of it are still unpublished), we are not even close to classifying all finite groups (except perhaps up to Jordan-Holder equivalence). It is a good idea to make accurate statements when criticizing others for lack of precision.