No... if I wanted to solve that DE, I would integrate both sides with respect to $x$. On the right, the integral is easy enough to compute. On the left, it comes down to an application of the fundamental theorem of calculus. The Leibniz notation is convenient in this case, since it lets you treat the differential as a number, but the usual exposition relies on actual theorems which justify this kind of manipulation. It should also be noted that Abraham Robinson went over this in the 60s...
Shinichi Mochizuki has a solid history of producing good mathematics. While it is possible that he is trying to pull a fast one, that seems quite unlikely, given his reputation. The most charitable explanation is that he has invented a new branch of mathematics (the "inter-universal Teichmüller theory") in order to resolve the ABC Conjecture, and that the "newness" of this approach is causing difficulty for outsiders.
This is not at all true. First off, mathematics itself has many different highly specialized sub-fields, many of which don't communicate effectively between each other. An complex analyst and a homotopy theorist speak very different mathematical languages, and may have difficulty communicating their ideas to each other. It is reasonable to suggest that this represents a different "cultural" background (as per Tylor's definition, these differences are differences in knowledge and belief, as well as differences in language).
Additionally, mathematicians from different parts of the world conduct mathematics differently. The internet and the wide-spread adoption of English as the de facto language of discourse has ameliorated this problem some in recent history, but there are still very significant cultural differences between American, European, Russian, and Japanese mathematics (for example). The approach that one takes in tackling a mathematical problem does depend quite a bit on where one learns it. As a historic example, Ramanujan was interesting to Hardy not just because he was producing interesting results, but because he was producing these results in an idiosyncratic way which differed immensely from the British approach.
Hence the original question
is it obtuse because he's trying to pull a fast one, or does it appear obtuse because he's form a different cultural background than?
There is where you are wrong. Wonder Woman is clearly historical fiction, grounded in reality. It is neither science fiction nor fantasy, because everything in that movie is based well documented historical fact. Or are you calling Herodotus a liar?
My bachelor's degree, from the University of Nevada Reno is a Mathematics BA (Statistics emphasis). The difference between that degree and a Mathematics BS (Statistics emphasis) is that the BA requires a foreign language, does not require any CS, and is not required to take numerical modeling (which is essentially a CS class). Otherwise, the requirements for the two degrees are identical. The same structure exists for the other possible emphases in mathematics: the BA requires a foreign language and does not require any CS.
I'm not disputing anything that Mandelbrot said. He himself walked back from a formal definition of a fractal, and ultimately chose not to give a precise mathematical definition. From the second edition of Mandelbrot's book (on page 459):
...to leave the term "fractal" without a pedantic definition, to use "fractal dimension" as a generic term applicable to all the variants in Chapter 39, and to use in each specific case whichever definition is the most appropriate.
In the first edition of the book, Mandelbrot suggested that a fractal should be any set with Hausdorff dimension strictly larger than its topological dimension (note that this does not mean that the Hausdorff dimension is non-integer; only that it is larger than its topological dimension). However, it was pointed out that there are lots of sets that we intuit as fractals, but which fail to satisfy this property (the Devil's staircase, for example, has both topological and Hausdorff dimension equal to 1 (it is a rectifiable curve)).
Falconer, who (in my opinion) is a much better source for a rigorous mathematical discussion of fractals, also chooses not to give a formal definition. From the introduction Fractal geometry. Mathematical foundations and applications (pp. xx--xxi):
In his original essay, Mandelbrot defined a fractal to be a set with Hausdorff dimension strictly greater than its topological dimension... This definition proved to be unsatisfactory in that it excluded a number of sets that clearly ought to be regarded as fractals. Various other definitions have been proposed, by they all seem to have this same drawback.
My personal feeling is that the definition of 'fractal' should be regarded in the same way as the biologist regards the definition of 'life'. There is no hard and fast definition, but just a list of properties characteristic of a living thing...
When we refer to a set F as a fractal, therefore, we will typically have the following in mind.
F has a fine structure, i.e. detail on arbitrarily small scales.
F is too irregular to be described in traditional geometric language, both locally and globally.
Often F has some form of self-similarity, perhaps approximate or statistical.
Usually, the 'fractal dimension' of F (defined in some way) is greater than its topological dimension.
In most cases of interest F is defined in a very simple way, perhaps recursively.
Where in the constitution of WSFS (the organization that gives the awards) is it written that Hugos must be given to works that are strictly science fiction? Indeed, the constitution states
3.2.1: Unless otherwise specified, Hugo Awards are given for work in the field of science fiction or fantasy appearing for the first time during the previous calendar year.
To the contrary, in a recent interview (maybe on the Jist or Marketplace? I can't recall exactly where I heard this...) the author mentioned that the distinction between "nigga" (a common lyric) and "ni**er" (not a common lyric) made it easier to distinguish potentially racist searches from others. On the flip side, the author ran into trouble when trying to study sexist/misogynistic searches, as many of those are people looking for porn.
It should also be noted that the punchline is not "people who search offensive phrases are racist." The punchline is that seemingly racist searches correlate (i.e. the effect is statistical, rather than individual) with other variables (such as regions where Obama underperformed when compared to other Democrat candidates and/or polling) that seem to indicate some underlying racism.
The actual book appears to be pretty nuanced. The Vox interview linked above is also appears to be relatively nuanced. The Slashdot summary and the paragraphs preceding the interview on Vox are sensationalist, click-baity claptrap.
To anyone who knows what they are doing, both pi and tau are simply constants. Absorb either into some constant term, or carry it around with you wherever you go---it doesn't matter. Real engineers, mathematicians, scientists, etc can handle either constant without difficulty. The real advantage that tau has over pi is pedagogical. It is much easier to communicate the relation between angle measure and arc length with tau. Since trig functions deal with lengths more readily than area, it makes sense to teach people trigonometry using a constant that is more closely related to length.
This is not the paper described in the summary, but rather an older paper with some of the same authors. The paper referenced in the summary was published online yesterday in Nature Climate Change. I'm sorry that I can't give a direct link to a.pdf (yay for paywalls keeping all of the non-ivory tower plebs out! huzzah!), but for those with access, the paper can be found at Influence of high-latitude atmospheric circulation changes on summertime Arctic sea ice. For those without access to an academic library, the first author provides an email contact. One presumes that a polite request would yield the full text of the paper.
The problem isn't really the textbooks---the books themselves are often relatively cheap (for example, a 9th edition of Sullivan's Precalculus can be had for $30 or $40 if you don't mind being an edition out of date). The problem is that students are also required to buy access to the publisher's website in order to do their homework. One alternative is to hire advanced undergraduates to grade papers, or (better yet) hire more expensive graduate students, or even (heaven forbid) tenure track lecturers to teach smaller sections and/or grade papers. There is basically no money to do that, so it isn't going to happen. Another alternative is to use something like MAA's WeBWorK for homework. This might be quite feasible in the future as WeBWorK is improved (or another, better free, open source system comes along), and my department is doing as much as it can via WeBWorK, but the system is still not all there---there are simply things that, as bad as it is, MathXL can do much better than WeBWorK.
This might be evidence of my own lack of creativity, but I just don't see many other alternatives, and none of them are going to be any cheaper at the end of the day.
*Mathematics* isn't science. It is more properly a branch of philosophy that happens to be really, really useful for the sciences. The difference is that sciences are empirical---ideally, scientists observe the world, form explanations of their observations, then test those explanations with further observations. Mathematics is not empirical---mathematicians start from a set of fundamental assumptions, then use logic to deduce the consequences of those assumptions.
From the summary, it seems that the criticism is that economists are behaving more like mathematicians than like scientists---that is, they are making assumptions about how the world works, then using logic to determine the consequences of those assumptions. Instead, they should be making observations, then using the tools of mathematics analyze data taken from the real world and test their explanations.
I don't know about that. A couple of back-of-the-envelope computations make me think that 10 years is not a long enough timeframe to make such a camera anywhere near common. Consider, for instance, the 3 ton weight. Suppose that technology develops such that an equivalent sensor halves in weight every year. Ten years then represents halving the weight 10 times, giving a weight of approximately 6 lbs. That definitely isn't iPhone weight, and comes from a pretty optimistic assumption about how quickly the technology will develop. The computation, for completeness: (3 tons) / 2^10) ~= 5.9 lbs
Or we could look at pixel counts. The summary claims that the camera will capture 3.2 gigapixel images. Apple claims that the iPhone 6 has a 8 mega pixel camera. So the telescope camera will capture 400 times as much data. Assuming that the iPhone camera doubles its pixel count every year, it would take almost 9 years to get to 3.2 gigapixels. Even if we assume that the iPhone is used to take panoramas, where a panorama can have up to about 2^3 the pixel count of a non-panorama (again, see Apple's claims), this represents 6 years of doubling every year, which is, again, pretty optimistic.
Long story short: yes, technology marches forward, but this is likely to be a pretty impressive instrument even 10-15 years in the future.
Change (1) to "Following too close for the given conditions," and both (3) and (4) are dealt with. If conditions are such that your stopping distance will be increased, you are responsible for leaving a correspondingly larger amount of space between yourself and the car in front of you. There is, perhaps, an edge case in freeway driving: someone changes lanes in front of you, then slams on their brakes, but that isn't really relevant to approaching a stop sign.
Slashdot Mirror
No... if I wanted to solve that DE, I would integrate both sides with respect to $x$. On the right, the integral is easy enough to compute. On the left, it comes down to an application of the fundamental theorem of calculus. The Leibniz notation is convenient in this case, since it lets you treat the differential as a number, but the usual exposition relies on actual theorems which justify this kind of manipulation. It should also be noted that Abraham Robinson went over this in the 60s...
Shinichi Mochizuki has a solid history of producing good mathematics. While it is possible that he is trying to pull a fast one, that seems quite unlikely, given his reputation. The most charitable explanation is that he has invented a new branch of mathematics (the "inter-universal Teichmüller theory") in order to resolve the ABC Conjecture, and that the "newness" of this approach is causing difficulty for outsiders.
This is not at all true. First off, mathematics itself has many different highly specialized sub-fields, many of which don't communicate effectively between each other. An complex analyst and a homotopy theorist speak very different mathematical languages, and may have difficulty communicating their ideas to each other. It is reasonable to suggest that this represents a different "cultural" background (as per Tylor's definition, these differences are differences in knowledge and belief, as well as differences in language).
Additionally, mathematicians from different parts of the world conduct mathematics differently. The internet and the wide-spread adoption of English as the de facto language of discourse has ameliorated this problem some in recent history, but there are still very significant cultural differences between American, European, Russian, and Japanese mathematics (for example). The approach that one takes in tackling a mathematical problem does depend quite a bit on where one learns it. As a historic example, Ramanujan was interesting to Hardy not just because he was producing interesting results, but because he was producing these results in an idiosyncratic way which differed immensely from the British approach.
Hence the original question
is entirely reasonable.
There is where you are wrong. Wonder Woman is clearly historical fiction, grounded in reality. It is neither science fiction nor fantasy, because everything in that movie is based well documented historical fact. Or are you calling Herodotus a liar?
My bachelor's degree, from the University of Nevada Reno is a Mathematics BA (Statistics emphasis). The difference between that degree and a Mathematics BS (Statistics emphasis) is that the BA requires a foreign language, does not require any CS, and is not required to take numerical modeling (which is essentially a CS class). Otherwise, the requirements for the two degrees are identical. The same structure exists for the other possible emphases in mathematics: the BA requires a foreign language and does not require any CS.
I'm not disputing anything that Mandelbrot said. He himself walked back from a formal definition of a fractal, and ultimately chose not to give a precise mathematical definition. From the second edition of Mandelbrot's book (on page 459):
In the first edition of the book, Mandelbrot suggested that a fractal should be any set with Hausdorff dimension strictly larger than its topological dimension (note that this does not mean that the Hausdorff dimension is non-integer; only that it is larger than its topological dimension). However, it was pointed out that there are lots of sets that we intuit as fractals, but which fail to satisfy this property (the Devil's staircase, for example, has both topological and Hausdorff dimension equal to 1 (it is a rectifiable curve)).
Falconer, who (in my opinion) is a much better source for a rigorous mathematical discussion of fractals, also chooses not to give a formal definition. From the introduction Fractal geometry. Mathematical foundations and applications (pp. xx--xxi):
It all depends on what you mean by "fractal." The term has no precise mathematical meaning.
On the other hand, car theft is almost a non-issue anymore, thanks in large part to the technology that you decry.
Where in the constitution of WSFS (the organization that gives the awards) is it written that Hugos must be given to works that are strictly science fiction? Indeed, the constitution states
This specifically includes fantasy.
To the contrary, in a recent interview (maybe on the Jist or Marketplace? I can't recall exactly where I heard this...) the author mentioned that the distinction between "nigga" (a common lyric) and "ni**er" (not a common lyric) made it easier to distinguish potentially racist searches from others. On the flip side, the author ran into trouble when trying to study sexist/misogynistic searches, as many of those are people looking for porn.
It should also be noted that the punchline is not "people who search offensive phrases are racist." The punchline is that seemingly racist searches correlate (i.e. the effect is statistical, rather than individual) with other variables (such as regions where Obama underperformed when compared to other Democrat candidates and/or polling) that seem to indicate some underlying racism.
The actual book appears to be pretty nuanced. The Vox interview linked above is also appears to be relatively nuanced. The Slashdot summary and the paragraphs preceding the interview on Vox are sensationalist, click-baity claptrap.
I lived in Reno for over a decade, and referred to the mountains as ``the Sierras'' for years. I am so embarrassed that no one ever corrected me. :(
To anyone who knows what they are doing, both pi and tau are simply constants. Absorb either into some constant term, or carry it around with you wherever you go---it doesn't matter. Real engineers, mathematicians, scientists, etc can handle either constant without difficulty. The real advantage that tau has over pi is pedagogical. It is much easier to communicate the relation between angle measure and arc length with tau. Since trig functions deal with lengths more readily than area, it makes sense to teach people trigonometry using a constant that is more closely related to length.
This is not the paper described in the summary, but rather an older paper with some of the same authors. The paper referenced in the summary was published online yesterday in Nature Climate Change. I'm sorry that I can't give a direct link to a .pdf (yay for paywalls keeping all of the non-ivory tower plebs out! huzzah!), but for those with access, the paper can be found at Influence of high-latitude atmospheric circulation changes on summertime Arctic sea ice. For those without access to an academic library, the first author provides an email contact. One presumes that a polite request would yield the full text of the paper.
You might be willing to compromise on this for the sake of practicality, but I am not.
What is there to compromise if I don't share you veneration of dead bone?
His manners have made the universe a more pleasant place for me to live, so, on balance, I'd call it a win for the AC.
You mean something like the Sieve of Eratosthenes?
You mean 2, right? I mean, it is the only even prime number, which makes it rather odd among primes...
The problem isn't really the textbooks---the books themselves are often relatively cheap (for example, a 9th edition of Sullivan's Precalculus can be had for $30 or $40 if you don't mind being an edition out of date). The problem is that students are also required to buy access to the publisher's website in order to do their homework. One alternative is to hire advanced undergraduates to grade papers, or (better yet) hire more expensive graduate students, or even (heaven forbid) tenure track lecturers to teach smaller sections and/or grade papers. There is basically no money to do that, so it isn't going to happen. Another alternative is to use something like MAA's WeBWorK for homework. This might be quite feasible in the future as WeBWorK is improved (or another, better free, open source system comes along), and my department is doing as much as it can via WeBWorK, but the system is still not all there---there are simply things that, as bad as it is, MathXL can do much better than WeBWorK.
This might be evidence of my own lack of creativity, but I just don't see many other alternatives, and none of them are going to be any cheaper at the end of the day.
*Mathematics* isn't science. It is more properly a branch of philosophy that happens to be really, really useful for the sciences. The difference is that sciences are empirical---ideally, scientists observe the world, form explanations of their observations, then test those explanations with further observations. Mathematics is not empirical---mathematicians start from a set of fundamental assumptions, then use logic to deduce the consequences of those assumptions.
From the summary, it seems that the criticism is that economists are behaving more like mathematicians than like scientists---that is, they are making assumptions about how the world works, then using logic to determine the consequences of those assumptions. Instead, they should be making observations, then using the tools of mathematics analyze data taken from the real world and test their explanations.
I don't know about that. A couple of back-of-the-envelope computations make me think that 10 years is not a long enough timeframe to make such a camera anywhere near common. Consider, for instance, the 3 ton weight. Suppose that technology develops such that an equivalent sensor halves in weight every year. Ten years then represents halving the weight 10 times, giving a weight of approximately 6 lbs. That definitely isn't iPhone weight, and comes from a pretty optimistic assumption about how quickly the technology will develop. The computation, for completeness: (3 tons) / 2^10) ~= 5.9 lbs
Or we could look at pixel counts. The summary claims that the camera will capture 3.2 gigapixel images. Apple claims that the iPhone 6 has a 8 mega pixel camera. So the telescope camera will capture 400 times as much data. Assuming that the iPhone camera doubles its pixel count every year, it would take almost 9 years to get to 3.2 gigapixels. Even if we assume that the iPhone is used to take panoramas, where a panorama can have up to about 2^3 the pixel count of a non-panorama (again, see Apple's claims), this represents 6 years of doubling every year, which is, again, pretty optimistic.
Long story short: yes, technology marches forward, but this is likely to be a pretty impressive instrument even 10-15 years in the future.
I would still consider "fraud" to be an edge case.
Change (1) to "Following too close for the given conditions," and both (3) and (4) are dealt with. If conditions are such that your stopping distance will be increased, you are responsible for leaving a correspondingly larger amount of space between yourself and the car in front of you. There is, perhaps, an edge case in freeway driving: someone changes lanes in front of you, then slams on their brakes, but that isn't really relevant to approaching a stop sign.
2001 called!? Did you warn them about the airplanes?
Yeah, because Slashdot never, ever covered general politics, popular culture, or economic issues prior to the Dice takeover.
You're not the boss of me!