Minor: n! / (n-2)! is used instead of the simpler, equivalent n*(n-1). The number of passwords of length 2 in their calculation is actually defined (since 0! = 1 by convention); it should say 1040 instead of n/a. The circle-related calculation (95 - 5 + 1)^2... is oversimplified: for instance, a circle of radius 25 cannot have center (5, 5). The necessary modification is tedious, though, and not very significant. Tap reduction calculations are similarly oversimplified.
More serious: I'm unable to determine how the "# of taps" table was generated. The first entry, 270, is approximately 10,000/(3+5+7+7+7+5+3), which makes sense--it uses a 100x100 grid where taps are allowed to be off by a few squares. If subsequent taps are independent, the remaining entries in the table should be approximately 270^n. Instead, the ratio between entries varies between 65 and 101 (non-monotonically, even). The # lines and # circles tables are similarly unclear to me.
The number of unique lines count, 1949, is low, assuming the tap gesture recognition is used independently on each line endpoint. There are about 270 distinct grid positions, leading to about 270^2 = 72900 directed lines (alternatively, (100^2)^2 / 37^2 ~= 73000). Removing lines of length less than 5 is negligible.
I am unable to compute more than the first entry in the "multi-gesture picture password" column. As they say, this should be computed by "summing up the unique gestures for all three gesture types for the specific gesture length n and raise [sic] it to the nth power". The entries should just be 2554^n given the previous data, but they're not--they're significantly smaller.
At least all the holes underestimate the number of distinct gestures in cases I'm able to calculate, though without details on the circle and line recognition systems I can't independently calculate those figures. Everything I didn't mention was correct.
You'd need to do some studies to see how non-uniform combination probabilities are. Asserting without proof that most people will choose easy-to-guess gestures is just as fallacious as just giving the number of unique combinations (which does not change) without discussing the underlying probability distribution.
For those who don't know, the Greek letter chi is normally pronounced like "kai" in English. I'm not sure why the transliteration is unintuitive--perhaps the pronunciation has changed over time.
Here's a random smattering of things in math one might memorize after grade school:
Geometry/Trig: triangle similarity conditions; law of cosines/sines; all the silly little definitions almost no practicing mathematicians use, like alternate interior angles; special values of sin/cos/tan or the underlying right triangles; various area and volume formulae
Basic Calculus: derivatives, integrals, and basic properties of polynomials, (inverse) trig functions, exponentials; basic derivative and integral computation techniques; area/volume-computing techniques
Algebra: definitions of a huge number of algebraic objects, eg. groups, (unital) rings, PID's, integral domains, fields, vector spaces, modules, Dedekind domains,..., together with basic properties for each
Of course, each branch (differential/algebraic geometry, geometric algebra [always makes me chuckle], low-dimensional topology, combinatorics,...) has its own set of definitions and basic results. Each subfield has yet more--there's a lot to memorize. Even in high school there's a bit of memorization, though it's significantly less than later on (because there's so much less content taught at the high school level).
The trick with math, like any tools, is in the understanding of which to use, how to use and when to use - really, it's not difficult
You must be talking about primary and secondary school math (and perhaps some of the easier bits of undergraduate math). A lot of that is mechanical enough that computer algebra systems can perform the necessary operations better than humans. Math really does get hard at some point, though. If you don't agree, please prove the following (assuming you haven't seen it before) within 1 hour, without help:
Let f:R->R be infinitely differentiable. For every x in R, suppose there is some positive integer n(x) such that the n(x)th and higher derivatives of f evaluated at x are 0. Show that f is a polynomial.
(I picked the above since it requires only basic calculus to state. The only proof I've seen also isn't terribly advanced.)
I'm curious, what did you mean? I'm only familiar with the set theory definitions of cardinals and ordinals--roughly, one can say a cardinal is the equivalence class of bijectively equivalent sets, whereas an ordinal is the equivalence class of order-isomorphic well-ordered sets. (As ever with set theory, there are a world of subtleties.) In any case, these seem entirely irrelevant, and after glancing through a few other definitions, they also seem irrelevant.
I re-read my original comment with fresh eyes and discovered, to my surprise, that I didn't communicate what I meant to at the end very well at all. I must have been sleepy when I wrote it. I meant to say that, if we performed the experiment on a single randomly chosen electron, there was a 50% chance we had prepared it the first way, and a 50% chance in the second way. The resulting "system" would be incoherent, since it would be composed of two states.
Now that I understand your objection, I imagine you're trying to say (without appealing to the technical vocabulary) that the density matrix has "nearly" a single eigenvalue, which is what I thought and meant to communicate originally. Oops!
Seriously guys if somebody doesn't understand quantum physics reading the wiki page isn't going to do it.
More than that, the first paragraph of the linked explanation is misleading, and the rest essentially requires an understanding of quite a bit of quantum mechanics to have a chance of following it. I have difficulty imagining somehow who actually understood the concepts involved linking such a poor explanation.
Quantum coherence has to do with multiple particles. If most of the particles are in roughly the same (quantum) state, the system is called coherent. Otherwise, it is not coherent. To give an (oversimplified) example, take a bunch of electrons. Through a clever experiment, we may measure an individual electron's "spin", and the result will either be "up" or "down"--an understanding of spin is immaterial here; feel free to replace "spin" with "mood" and "up"/"down" with "happy"/"sad" if it scares you. The unintuitive part of quantum mechanics is that even if we performed the experiment twice with two indistinguishable electrons, our experiment may well come out differently. The crucial thing, though, is that each outcome has a fixed probability of occurring. Suppose, then, that we've prepared 100 electrons in such a way that if we perform our spin experiment, 30% of the time the electron will have spin up, and 70% of the time it will have spin down. An electron's quantum state for this experiment is (sweeping wavefunctions under the rug...) given by the probability of each outcome. Each of our 100 electrons has the same quantum state as the others, so the system is called (perfectly) coherent. If, however, we prepared 50 of the electrons to come out with the above probabilities and the remaining 50 electrons to come out with 100% spin up, the system is not coherent.
(Disclaimer: I am not a physicist, but rather a mathematician with some interest in quantum physics. Please feel free to correct or supplement the above.)
I did say "To be fair, it does at least encourage some discussion of science, which is a good thing." I just advocate admitting ignorance and being silent in the face of uncertainty. The traditional response, especially with the techie crowd, is to cobble together half-baked ideas and parade them around as fact or well-supported opinion in the hopes of getting approval from others. I hate this response. It's the cause of a large fraction of the garbage that political commentators and 24-hour news networks spew.
There was no input device that I can recall, so Facebook should be out. I suppose it's possible she was looking at something else like the output of an x-ray machine, though those people stand right behind it, and she was definitely not behind an x-ray machine.
I imagine rather than being corrupted, those minds would just become numb to formerly outrageous content. Last time I went through the TSA's full body scanners, I looked at the woman watching the results. There was no hint of any of the usual emotions humans experience when seeing other (semi) naked humans--no excitement, shame, or embarrassment. There was just a blank look. Not even an "ugh, I have to do this all day?" look. At best there was a little buried resentment.
I remember feeling sorry for her. When she goes home to her significant other, I hope she's not reminded of the endless body scans, or of the apathy they've created in her.
I wish I could have responded to both you and the person I was originally responding to simultaneously. In any case, in retrospect, I agree with you about arXiv preprints--my bias is largely unfounded. It's based mostly on a handful of crackpot and retracted papers with extraordinary claims (though Perelman's bit of Poincare was posted on the arXiv, which was both extraordinary and correct) and ignores the bulk of them. My actual bias--and I believe this is the reasonable result of evidence--is against news stories with extraordinary claims that rely on arXiv preprints, and against those preprints themselves. They almost never come to anything.
In retrospect, I agree with you about arXiv preprints--my bias is largely unfounded. It's based mostly on a handful of crackpot and retracted papers with extraordinary claims (though as you mention Perelman's bit of Poincare was posted on the arXiv, which was both extraordinary and correct) and ignores the bulk of them. My actual bias--and I believe this is the reasonable result of evidence--is against news stories with extraordinary claims that rely on arXiv preprints, and against those preprints themselves. They almost never come to anything.
Here we go.... Dark matter is (conjectured to be) involved in much more than just stellar velocity adjustments. There are some posts in these very comments listing some of the other effects. To say that dark matter may be "entirely disposed of" because of this model is then apparently either quite ignorant or prematurely optimistic. That illustrates exactly my point. I could do just what you did, but I'd end up talking out of my ass out of ignorance. Sometimes that's fine, but here there are very few people qualified to correct such garbage. The last few sentences sound rather insulting. I don't mean them to be; sorry. My point is that very few people here have any serious background in dark matter, and reading the Wikipedia article does not count. Without that background, you're really just wasting everyone's time by discussing it.
On a separate note, I'd be surprised if calculations as simple as you've described (involving at the worst some geometric series, apparently) could be of any groundbreaking interest. I have not read the paper. I'm biased against arXiv preprints.
To be honest, almost nobody here is qualified to comment on the actual story and paper. In a more ideal world there would be at most half a dozen comments from the few physicists who post on/. and are in the right specialization to discuss dark matter and galactic velocity curves. I have far more than the average background in physics and especially math, but I won't weigh in on the physics with my random crappy opinions. It seems like the article's author shouldn't have, either, though. The article offers no explanation whatsoever for the incongruity it describes (as quoted in the summary). To be fair, it does at least encourage some discussion of science, which is a good thing.
Add random delay when loading things: probably too much of an impact on loading times to be used
Don't let scripts measure the passage of time while loading: awkward, but at least it has a low impact on other things; you'd also have to randomize the order in which things load to prevent an outside server from acting as a clock
My Bachelor's degree is in "Computer Science and Mathematics". It's not a double major. I just looked through the requirements, and to have gotten a double major I would have needed two more CS courses (I was math-heavy), and I might have needed to write a thesis (the description is unclear if a particular requirement can count for both majors). I'm pursuing a Ph.D. in math after having found out professional CS really isn't for me. One friend of mine (CS-heavy) didn't use the math half of his degree as far as I know. Another person is doing the same thing as me, pursuing a higher degree in pure math. To be fair, my college has a shockingly high rate of people ending up with Ph.D.'s, so it's probably not quite representative in that regard. There are essentially three things you can do with a Ph.D. in math--teach mathematics, do research mathematics (eg. for the NSA), or play the stock market and make obscene amounts of money. This final option might drive up the overall average, depending on how it was calculated.
It would be interesting to hear how more advanced versions of such a course might go. Grading must be automated to handle the volume they talk about, and as difficulty ramps up, grading gets harder. I also wonder if this is the wave of the future--in particular, if extremely polished, low-man-power online courses will start to replace massive intro undergraduate courses. It just seems too efficient to pass up.
Also, a minor point: Stanford actually has excellent financial aid packages. If you can get in, you really should be able to pay for it (possibly with loans, but not unreasonable amounts for a Stanford graduate). See the last paragraph in this section of the Wikipedia article for some details. They have an obscenely large endowment, which helps.
Slashdot Mirror
The math at this MS blog post has a few holes.
Minor: n! / (n-2)! is used instead of the simpler, equivalent n*(n-1). The number of passwords of length 2 in their calculation is actually defined (since 0! = 1 by convention); it should say 1040 instead of n/a. The circle-related calculation (95 - 5 + 1)^2 ... is oversimplified: for instance, a circle of radius 25 cannot have center (5, 5). The necessary modification is tedious, though, and not very significant. Tap reduction calculations are similarly oversimplified.
More serious: I'm unable to determine how the "# of taps" table was generated. The first entry, 270, is approximately 10,000/(3+5+7+7+7+5+3), which makes sense--it uses a 100x100 grid where taps are allowed to be off by a few squares. If subsequent taps are independent, the remaining entries in the table should be approximately 270^n. Instead, the ratio between entries varies between 65 and 101 (non-monotonically, even). The # lines and # circles tables are similarly unclear to me.
The number of unique lines count, 1949, is low, assuming the tap gesture recognition is used independently on each line endpoint. There are about 270 distinct grid positions, leading to about 270^2 = 72900 directed lines (alternatively, (100^2)^2 / 37^2 ~= 73000). Removing lines of length less than 5 is negligible.
I am unable to compute more than the first entry in the "multi-gesture picture password" column. As they say, this should be computed by "summing up the unique gestures for all three gesture types for the specific gesture length n and raise [sic] it to the nth power". The entries should just be 2554^n given the previous data, but they're not--they're significantly smaller.
At least all the holes underestimate the number of distinct gestures in cases I'm able to calculate, though without details on the circle and line recognition systems I can't independently calculate those figures. Everything I didn't mention was correct.
You'd need to do some studies to see how non-uniform combination probabilities are. Asserting without proof that most people will choose easy-to-guess gestures is just as fallacious as just giving the number of unique combinations (which does not change) without discussing the underlying probability distribution.
For those who don't know, the Greek letter chi is normally pronounced like "kai" in English. I'm not sure why the transliteration is unintuitive--perhaps the pronunciation has changed over time.
The summary is actually quite good, in that it captures the essence of the article it's summarizing. It's made mostly of key, near-direct quotes.
I also don't see the problem with using a silly analogy in a popular science context. Even if it annoys you, the video is well worth a silly analogy.
WHAT?! WHY WOULd YOU WANT TO DO THAt?!
FTFY.
Here's a random smattering of things in math one might memorize after grade school:
Geometry/Trig: triangle similarity conditions; law of cosines/sines; all the silly little definitions almost no practicing mathematicians use, like alternate interior angles; special values of sin/cos/tan or the underlying right triangles; various area and volume formulae
Basic Calculus: derivatives, integrals, and basic properties of polynomials, (inverse) trig functions, exponentials; basic derivative and integral computation techniques; area/volume-computing techniques
Algebra: definitions of a huge number of algebraic objects, eg. groups, (unital) rings, PID's, integral domains, fields, vector spaces, modules, Dedekind domains, ..., together with basic properties for each
Of course, each branch (differential/algebraic geometry, geometric algebra [always makes me chuckle], low-dimensional topology, combinatorics, ...) has its own set of definitions and basic results. Each subfield has yet more--there's a lot to memorize. Even in high school there's a bit of memorization, though it's significantly less than later on (because there's so much less content taught at the high school level).
The trick with math, like any tools, is in the understanding of which to use, how to use and when to use - really, it's not difficult
You must be talking about primary and secondary school math (and perhaps some of the easier bits of undergraduate math). A lot of that is mechanical enough that computer algebra systems can perform the necessary operations better than humans. Math really does get hard at some point, though. If you don't agree, please prove the following (assuming you haven't seen it before) within 1 hour, without help:
Let f:R->R be infinitely differentiable. For every x in R, suppose there is some positive integer n(x) such that the n(x)th and higher derivatives of f evaluated at x are 0. Show that f is a polynomial.
(I picked the above since it requires only basic calculus to state. The only proof I've seen also isn't terribly advanced.)
I'm curious, what did you mean? I'm only familiar with the set theory definitions of cardinals and ordinals--roughly, one can say a cardinal is the equivalence class of bijectively equivalent sets, whereas an ordinal is the equivalence class of order-isomorphic well-ordered sets. (As ever with set theory, there are a world of subtleties.) In any case, these seem entirely irrelevant, and after glancing through a few other definitions, they also seem irrelevant.
I re-read my original comment with fresh eyes and discovered, to my surprise, that I didn't communicate what I meant to at the end very well at all. I must have been sleepy when I wrote it. I meant to say that, if we performed the experiment on a single randomly chosen electron, there was a 50% chance we had prepared it the first way, and a 50% chance in the second way. The resulting "system" would be incoherent, since it would be composed of two states.
Now that I understand your objection, I imagine you're trying to say (without appealing to the technical vocabulary) that the density matrix has "nearly" a single eigenvalue, which is what I thought and meant to communicate originally. Oops!
I have difficulty making "it's about the system as a whole having a single quantum state" rigorous. Could you elaborate?
Seriously guys if somebody doesn't understand quantum physics reading the wiki page isn't going to do it.
More than that, the first paragraph of the linked explanation is misleading, and the rest essentially requires an understanding of quite a bit of quantum mechanics to have a chance of following it. I have difficulty imagining somehow who actually understood the concepts involved linking such a poor explanation.
Quantum coherence has to do with multiple particles. If most of the particles are in roughly the same (quantum) state, the system is called coherent. Otherwise, it is not coherent. To give an (oversimplified) example, take a bunch of electrons. Through a clever experiment, we may measure an individual electron's "spin", and the result will either be "up" or "down"--an understanding of spin is immaterial here; feel free to replace "spin" with "mood" and "up"/"down" with "happy"/"sad" if it scares you. The unintuitive part of quantum mechanics is that even if we performed the experiment twice with two indistinguishable electrons, our experiment may well come out differently. The crucial thing, though, is that each outcome has a fixed probability of occurring. Suppose, then, that we've prepared 100 electrons in such a way that if we perform our spin experiment, 30% of the time the electron will have spin up, and 70% of the time it will have spin down. An electron's quantum state for this experiment is (sweeping wavefunctions under the rug...) given by the probability of each outcome. Each of our 100 electrons has the same quantum state as the others, so the system is called (perfectly) coherent. If, however, we prepared 50 of the electrons to come out with the above probabilities and the remaining 50 electrons to come out with 100% spin up, the system is not coherent.
(Disclaimer: I am not a physicist, but rather a mathematician with some interest in quantum physics. Please feel free to correct or supplement the above.)
I did say "To be fair, it does at least encourage some discussion of science, which is a good thing." I just advocate admitting ignorance and being silent in the face of uncertainty. The traditional response, especially with the techie crowd, is to cobble together half-baked ideas and parade them around as fact or well-supported opinion in the hopes of getting approval from others. I hate this response. It's the cause of a large fraction of the garbage that political commentators and 24-hour news networks spew.
There was no input device that I can recall, so Facebook should be out. I suppose it's possible she was looking at something else like the output of an x-ray machine, though those people stand right behind it, and she was definitely not behind an x-ray machine.
Does anyone have experience in the reverse situation, where an Indian/etc. firm outsources some project to an American/etc. firm?
She was standing not far from the machine, looking at some screen I couldn't see. My memory of the surroundings is hazy.
I imagine rather than being corrupted, those minds would just become numb to formerly outrageous content. Last time I went through the TSA's full body scanners, I looked at the woman watching the results. There was no hint of any of the usual emotions humans experience when seeing other (semi) naked humans--no excitement, shame, or embarrassment. There was just a blank look. Not even an "ugh, I have to do this all day?" look. At best there was a little buried resentment.
I remember feeling sorry for her. When she goes home to her significant other, I hope she's not reminded of the endless body scans, or of the apathy they've created in her.
I wish I could have responded to both you and the person I was originally responding to simultaneously. In any case, in retrospect, I agree with you about arXiv preprints--my bias is largely unfounded. It's based mostly on a handful of crackpot and retracted papers with extraordinary claims (though Perelman's bit of Poincare was posted on the arXiv, which was both extraordinary and correct) and ignores the bulk of them. My actual bias--and I believe this is the reasonable result of evidence--is against news stories with extraordinary claims that rely on arXiv preprints, and against those preprints themselves. They almost never come to anything.
In retrospect, I agree with you about arXiv preprints--my bias is largely unfounded. It's based mostly on a handful of crackpot and retracted papers with extraordinary claims (though as you mention Perelman's bit of Poincare was posted on the arXiv, which was both extraordinary and correct) and ignores the bulk of them. My actual bias--and I believe this is the reasonable result of evidence--is against news stories with extraordinary claims that rely on arXiv preprints, and against those preprints themselves. They almost never come to anything.
Here we go.... Dark matter is (conjectured to be) involved in much more than just stellar velocity adjustments. There are some posts in these very comments listing some of the other effects. To say that dark matter may be "entirely disposed of" because of this model is then apparently either quite ignorant or prematurely optimistic. That illustrates exactly my point. I could do just what you did, but I'd end up talking out of my ass out of ignorance. Sometimes that's fine, but here there are very few people qualified to correct such garbage. The last few sentences sound rather insulting. I don't mean them to be; sorry. My point is that very few people here have any serious background in dark matter, and reading the Wikipedia article does not count. Without that background, you're really just wasting everyone's time by discussing it.
On a separate note, I'd be surprised if calculations as simple as you've described (involving at the worst some geometric series, apparently) could be of any groundbreaking interest. I have not read the paper. I'm biased against arXiv preprints.
To be honest, almost nobody here is qualified to comment on the actual story and paper. In a more ideal world there would be at most half a dozen comments from the few physicists who post on /. and are in the right specialization to discuss dark matter and galactic velocity curves. I have far more than the average background in physics and especially math, but I won't weigh in on the physics with my random crappy opinions. It seems like the article's author shouldn't have, either, though. The article offers no explanation whatsoever for the incongruity it describes (as quoted in the summary). To be fair, it does at least encourage some discussion of science, which is a good thing.
Other possible fixes:
do you think they'll want to rejoin the British empire?
That wouldn't be all bad. We'd at least be able to pawn off our debt on someone else.
Now I'm curious. Did you major in philosophy?
My Bachelor's degree is in "Computer Science and Mathematics". It's not a double major. I just looked through the requirements, and to have gotten a double major I would have needed two more CS courses (I was math-heavy), and I might have needed to write a thesis (the description is unclear if a particular requirement can count for both majors). I'm pursuing a Ph.D. in math after having found out professional CS really isn't for me. One friend of mine (CS-heavy) didn't use the math half of his degree as far as I know. Another person is doing the same thing as me, pursuing a higher degree in pure math. To be fair, my college has a shockingly high rate of people ending up with Ph.D.'s, so it's probably not quite representative in that regard. There are essentially three things you can do with a Ph.D. in math--teach mathematics, do research mathematics (eg. for the NSA), or play the stock market and make obscene amounts of money. This final option might drive up the overall average, depending on how it was calculated.
It would be interesting to hear how more advanced versions of such a course might go. Grading must be automated to handle the volume they talk about, and as difficulty ramps up, grading gets harder. I also wonder if this is the wave of the future--in particular, if extremely polished, low-man-power online courses will start to replace massive intro undergraduate courses. It just seems too efficient to pass up.
Also, a minor point: Stanford actually has excellent financial aid packages. If you can get in, you really should be able to pay for it (possibly with loans, but not unreasonable amounts for a Stanford graduate). See the last paragraph in this section of the Wikipedia article for some details. They have an obscenely large endowment, which helps.