Also, jogging my memory with Wikipedia's "Cardinality of the Continuum", it appears that indeed space-filling curves are only valid for finite-sized n-dimensional spaces, not infinite n-dimensional spaces.
Aren't space-filling curves valid only for bounded spaces?
That is, the cardinality of a unit line segment is the same as the cardinality of a unit square. However, the cardinality of a line is not the same as the cardinality of a plane.
You're incorrectly imposing the spatial sense of dimensionality on other discussions. What do you mean, "physically not another dimension"? If you mean that spin isn't a spatial dimension, you are correct. Nobody said it was. It is a degree of freedom of an object's state that is completely separable from other components of that state; as such, it's a dimension.
No, calling n-dimensionality by the label n-dimensionality isn't sloppy.
Hilbert spaces are by no means limited to describing the state of a system in space over time; that's falsely imposing the sense that a Hilbert space is redundant with positional space.
Dimensionality is simply the minimum number of variables necessary to completely describe state.
More than a few people told me that there weren't any practical applications of n-dimensional math, where N>5.
These people clearly aren't good physicists, then, or they forget how many dimensions are used in classical mechanics, quantum mechanics, and Hilbert spaces.
Anyway I am mainly talking out of of my ass, so feel free to slap me down.
If you insist.
Surely momentum could be summed up in 4D with time, so where the object is at different positions over time you could deduce the velocity from that, if you want momentum add a 5th dimension for mass.
You're perhaps thinking of, "if I have full functions x(t), y(t), and z(t) describing an object's position, I don't need independent functions for momentum, as the velocity is just the derivative of position" -- which is true. But those full functions is a lot of information. What we're talking about here is the information necessary to describe an object's state at a single point in time.
If you say that the state of an object at time T is determined by its position x(T), y(T), and z(T) and also a later position x(T+d), y(T+d), z(T+d), then you don't need the momentum. (Just let d become infinitesimal, so that you have the derivative of position.) Note though that now you have six bits of information to describe the state of the object at a single point of time -- which is the same number as what I claimed earlier.:-)
True, I was thinking only simple classical mechanics, which is all point objects (no orientation, no internal angular momentum). On the other hand, orientation is only 2D.
Your example is easy to relate to, but there's a problem people should be aware of. While you can refer to the book's location using a 6-dimensional quantity, you *could* do it in 3, by giving its position in space. In a "real" n-dimensional system, you cannot reduce the system to less than n dimensions.
A good, but less-accessible example, is the state of an object in classical mechanics. The position of an object is 3-dimensional. The state, however, is 6-dimensional: your position (3D) and momentum (3D).
Most physicists should be perfectly comfortable applying the term "dimensions" to cases other than spatial dimensions.
Once you're used to infinite-dimensional Hilbert spaces, media articles that mention "five-dimensional storage" are only infinitesimally interesting by comparison.
Yes, because clothing isn't, as far as I recall, part of the definition of pornography. Given the liberties you could take with subjects that are technically still clothed and the variety of entirely reasonable pictures of nude children, that's not too surprising.
Contrary to what the people on Slashdot tell you, every image of a nude person under the age of 18 is not necessarily child porn -- and a millimeter-wave scanner isn't exactly taking a nude photograph.
Booting the Windows install on the hard drive may well be within their bounds, and it's certainly within technical limitations.
Anyway, it's enormously unlikely that using an automated tool to examine all files' data to determine if they are music or not would qualify as "looking at" non-music files.
Okay, instead of the dog example, try the canonical example of a 'ring species': these California salamanders [pbs.org]. In that case, A and C cannot interbreed, and neither can their gametes.
That is a much better example.
I'm saying that going back in time shows how arbitrary the species concept is. At one point, we and chimps shared a common ancestor. Then we went our separate ways, and are now considered two species. 5 million years ago, we were one species. Now, we're two species. Where do you draw the line?
For a lot of speciation (like physical separation of populations), you wouldn't have the data to draw the line if you tried. I don't think it's a problem to say "at one point, there was one species, and now there's two -- somewhere in between, they split". It clearly must be the case for a system with speciation and common ancestors.
* Two same-gendered humans can't make a viable offspring. * Prepubescent children, post-menopausal women, and many other humans are sterile. * Sometimes two "species" could create viable offspring, but they don't. (E.g., different mating dances preclude them mating, but in a lab, sperm A and egg B make a viable offspring.) * Sometimes A can mate with B, and B with C, but A cannot mate with C directly. (A Chihuahua cannot mate with a Great Dane. It's physically impossible.)
3 and 4 are essentially the same, since what is preventing offspring between A and C is a physical problem. Generally, none of these reasons are considered valid for determining species.
* The nontransitivity above (A, B, and C) is generally true of ALL creatures if you're allowed to go back in time. Go back far enough, and our ancestors could mate with chimp ancestors. A little farther and we share ancestors!
Yes, that's what we call "speciation". It's a single species differentiating into two species. I hope you can see why going back in time is not reasonable for determining species.
* What about the poor asexual creatures? How do they have "species"?
Obviously, it's a more complicated problem.
Ability to produce viable offspring is actually only one measure of whether two species are separate, but it's a fairly useful one.
Oddly, if you recharacterize "expensive" from dollars now to impact on human civilization later, your statement works just as well for the opposite group!
Except that the people most likely to know what is really happening sides with one group. Hmm.
They're actually brighter for lower energy use. Also, people rarely buy an N-watt fixture thinking that they don't want to pay for more than an N-watt bulb anyway.
I'm saying it's entirely reasonable to have a limit lower than 0.5% and distortion due to saturation. The function of device is to determine if an arrest is warranted. The legal limit is in the vicinity of 0.08%. Distinctions in values above 0.2% or so aren't meaningful.
If you'll note, I'm not defending anything. I'm just pointing out that your choice of limit is completely arbitrary. It's unreasonable to assume that limit and then make claims about a final sensitivity.
There's no reason to assume that sensitivity. For one, the sensitivity is more likely determined by the sensitivity of the probe rather than some human limit. For another, once you're at n times the legal limit, higher values are completely unnecessary.
I'm not sure how this is insightful. A.app is a directory. While everything is bundled in it (not strictly true, but close enough), they're still separate files, and dynamic linking works just fine. Another thing that works just fine is updaters that replace only some of the files in the.app.
Slashdot Mirror
R is a set. Perhaps you mean |R|.
Also, jogging my memory with Wikipedia's "Cardinality of the Continuum", it appears that indeed space-filling curves are only valid for finite-sized n-dimensional spaces, not infinite n-dimensional spaces.
Aren't space-filling curves valid only for bounded spaces?
That is, the cardinality of a unit line segment is the same as the cardinality of a unit square. However, the cardinality of a line is not the same as the cardinality of a plane.
You're incorrectly imposing the spatial sense of dimensionality on other discussions. What do you mean, "physically not another dimension"? If you mean that spin isn't a spatial dimension, you are correct. Nobody said it was. It is a degree of freedom of an object's state that is completely separable from other components of that state; as such, it's a dimension.
No, calling n-dimensionality by the label n-dimensionality isn't sloppy.
Hilbert spaces are by no means limited to describing the state of a system in space over time; that's falsely imposing the sense that a Hilbert space is redundant with positional space.
Dimensionality is simply the minimum number of variables necessary to completely describe state.
More than a few people told me that there weren't any practical applications of n-dimensional math, where N>5.
These people clearly aren't good physicists, then, or they forget how many dimensions are used in classical mechanics, quantum mechanics, and Hilbert spaces.
Anyway I am mainly talking out of of my ass, so feel free to slap me down.
If you insist.
Surely momentum could be summed up in 4D with time, so where the object is at different positions over time you could deduce the velocity from that, if you want momentum add a 5th dimension for mass.
You're perhaps thinking of, "if I have full functions x(t), y(t), and z(t) describing an object's position, I don't need independent functions for momentum, as the velocity is just the derivative of position" -- which is true. But those full functions is a lot of information. What we're talking about here is the information necessary to describe an object's state at a single point in time.
If you say that the state of an object at time T is determined by its position x(T), y(T), and z(T) and also a later position x(T+d), y(T+d), z(T+d), then you don't need the momentum. (Just let d become infinitesimal, so that you have the derivative of position.) Note though that now you have six bits of information to describe the state of the object at a single point of time -- which is the same number as what I claimed earlier. :-)
True, I was thinking only simple classical mechanics, which is all point objects (no orientation, no internal angular momentum). On the other hand, orientation is only 2D.
Your example is easy to relate to, but there's a problem people should be aware of. While you can refer to the book's location using a 6-dimensional quantity, you *could* do it in 3, by giving its position in space. In a "real" n-dimensional system, you cannot reduce the system to less than n dimensions.
A good, but less-accessible example, is the state of an object in classical mechanics. The position of an object is 3-dimensional. The state, however, is 6-dimensional: your position (3D) and momentum (3D).
Most physicists should be perfectly comfortable applying the term "dimensions" to cases other than spatial dimensions.
Once you're used to infinite-dimensional Hilbert spaces, media articles that mention "five-dimensional storage" are only infinitesimally interesting by comparison.
It's not actually misuse of the word "dimension". You're just used to thinking only of spatial dimensions, and other restricted senses of the term.
Yes, because clothing isn't, as far as I recall, part of the definition of pornography. Given the liberties you could take with subjects that are technically still clothed and the variety of entirely reasonable pictures of nude children, that's not too surprising.
The prosecutor made the decision that it was child porn, and not a judge or jury?
Oh wait, it didn't even go to trial, as the case was dropped.
Contrary to what the people on Slashdot tell you, every image of a nude person under the age of 18 is not necessarily child porn -- and a millimeter-wave scanner isn't exactly taking a nude photograph.
Booting the Windows install on the hard drive may well be within their bounds, and it's certainly within technical limitations.
Anyway, it's enormously unlikely that using an automated tool to examine all files' data to determine if they are music or not would qualify as "looking at" non-music files.
Pound-mass or slug, your choice.
You confuse "innocent" and "unjust".
Okay, instead of the dog example, try the canonical example of a 'ring species': these California salamanders [pbs.org]. In that case, A and C cannot interbreed, and neither can their gametes.
That is a much better example.
I'm saying that going back in time shows how arbitrary the species concept is. At one point, we and chimps shared a common ancestor. Then we went our separate ways, and are now considered two species. 5 million years ago, we were one species. Now, we're two species. Where do you draw the line?
For a lot of speciation (like physical separation of populations), you wouldn't have the data to draw the line if you tried. I don't think it's a problem to say "at one point, there was one species, and now there's two -- somewhere in between, they split". It clearly must be the case for a system with speciation and common ancestors.
The Wikipedia entry on species roughly agrees with what I learned in school.
* Two same-gendered humans can't make a viable offspring.
* Prepubescent children, post-menopausal women, and many other humans are sterile.
* Sometimes two "species" could create viable offspring, but they don't. (E.g., different mating dances preclude them mating, but in a lab, sperm A and egg B make a viable offspring.)
* Sometimes A can mate with B, and B with C, but A cannot mate with C directly. (A Chihuahua cannot mate with a Great Dane. It's physically impossible.)
3 and 4 are essentially the same, since what is preventing offspring between A and C is a physical problem. Generally, none of these reasons are considered valid for determining species.
* The nontransitivity above (A, B, and C) is generally true of ALL creatures if you're allowed to go back in time. Go back far enough, and our ancestors could mate with chimp ancestors. A little farther and we share ancestors!
Yes, that's what we call "speciation". It's a single species differentiating into two species. I hope you can see why going back in time is not reasonable for determining species.
* What about the poor asexual creatures? How do they have "species"?
Obviously, it's a more complicated problem.
Ability to produce viable offspring is actually only one measure of whether two species are separate, but it's a fairly useful one.
Oddly, if you recharacterize "expensive" from dollars now to impact on human civilization later, your statement works just as well for the opposite group!
Except that the people most likely to know what is really happening sides with one group. Hmm.
They're actually brighter for lower energy use. Also, people rarely buy an N-watt fixture thinking that they don't want to pay for more than an N-watt bulb anyway.
Repetitively redundant?
I'm saying it's entirely reasonable to have a limit lower than 0.5% and distortion due to saturation. The function of device is to determine if an arrest is warranted. The legal limit is in the vicinity of 0.08%. Distinctions in values above 0.2% or so aren't meaningful.
If you'll note, I'm not defending anything. I'm just pointing out that your choice of limit is completely arbitrary. It's unreasonable to assume that limit and then make claims about a final sensitivity.
There's no reason to assume that sensitivity. For one, the sensitivity is more likely determined by the sensitivity of the probe rather than some human limit. For another, once you're at n times the legal limit, higher values are completely unnecessary.
It's just "concentration", which is volume per unit volume (or mass per unit mass), not "concentration per unit volume".
http://en.wikipedia.org/wiki/L2_point
I'm not sure how this is insightful. A .app is a directory. While everything is bundled in it (not strictly true, but close enough), they're still separate files, and dynamic linking works just fine. Another thing that works just fine is updaters that replace only some of the files in the .app.