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Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests. The study, published in the New Journal of Physics, shows that physicists pay less attention to theories that are crammed with mathematical details. This suggests there are real and widespread barriers to communicating mathematical work, and that this is not because of poor training in mathematical skills, or because there is a social stigma about doing well in mathematics.
Dr Tim Fawcett and Dr Andrew Higginson, from the University of Exeter, found, using statistical analysis of the number of citations to 2000 articles in a leading physics journal, that articles are less likely to be referenced by other physicists if they have lots of mathematical equations on each page. [...] Dr Higginson said: "We have already showed that biologists are put off by equations but we were surprised by these findings, as physicists are generally skilled in mathematics.
"This is an important issue because it shows there could be a disconnection between mathematical theory and experimental work. This presents a potentially enormous barrier to all kinds of scientific progress."
http://phys.org/news/2016-11-physicists-mathematics.html
[Abstract]: Statistical Analysis of the Effect of Equations on Citations
(Score: 3, Insightful) by mcgrew on Sunday November 13 2016, @08:08PM
Not all of them. Isaac Asimov was a biochemist and wrote a book called "Asimov on Numbers". However, one chapter in that book showed WHY scientists fear math--even the best get it wrong. One chapter in Asimov's book, "A Piece of Pi" got the math right, but he treated a three dimentional object--a big vat the ancient Hebrews made, where the instructions gave both diameter and radius. Asimov held that such a vat would have six sides (I'll have to drag it out and re-read it), forgetting that vats' walls have thickness; a pot's rim is not zero cm thick. So one measurement indicated the outside walls, the other the inside wall.
I argue online with folks who are far better at math than me about the Nyquist equation [wikipedia.org]. Some have come out and said that it says that resulting waveform will be perfect, but it doesn't say that at all. In fact, there is alias distortion. Whether or not it's audible I'm not sure , but at a 44k sampling rate, a 17kHz tone has only three samples. With only three samples it's impossible to discern the difference between a sine wave and a saw tooth wave. I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
So far, no one has done that as far as I know.
mcgrewbooks.com mcgrew.info nooze.org
(Score: 1) by khallow on Sunday November 13 2016, @08:50PM
Whether or not it's audible I'm not sure , but at a 44k sampling rate, a 17kHz tone has only three samples. With only three samples it's impossible to discern the difference between a sine wave and a saw tooth wave. I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
The next harmonic for sawtooth waves after the 17kHz base tone is at 51kHz. Someone might be able to hear that (there apparently are some people who can hear sounds well above the usual range for human hearing), but they're not going to be your average teenager.
(Score: 0) by Anonymous Coward on Sunday November 13 2016, @08:57PM
What about a square wave?
(Score: 2) by rleigh on Sunday November 13 2016, @09:40PM
It doesn't matter. Nyqvist is all about sampling sine waves, so sawtooth/square waves are irrelevant in the context of digital sampling. The reconstruction filter is sinc (or approximating sinc), so you'll get a sine wave back out irrespective of what goes in. If you want to reconstruct or at least approximate a sawtooth or square wave, you need a much higher sampling frequency. And if you take it into 2D with images (Raleigh and Airy discs), it's still all sine waves.
(Score: 2) by mcgrew on Sunday November 13 2016, @11:19PM
Nyqvist is all about sampling sine waves
You think that electric guitar played through a fuzzbox is a sine wave? It sure looked like a sawtooth on the oscilloscope in that physics class (regrettably decades before digital music). If nyquist was only sine waves, you'ld seldom hear Jimmy Page's guitar from a CD or ogg.
Nyquist applies to any shape of waveform.
mcgrewbooks.com mcgrew.info nooze.org
(Score: 2) by jcross on Monday November 14 2016, @02:13AM
A sawtooth wave can be decomposed into a series of sine wave harmonics, with each having a relative amplitude of 1/n, where n is the number of that harmonic. The really high harmonics mostly just make the points of the sawtooth sharper, and beyond the Nyquist limit it's not possible to make it any sharper. If the Nyquist frequency is above your threshold of hearing, you wouldn't be able to tell the difference anyway.
So applying the limit to sine waves doesn't prevent it from being applied to other waveforms as well, it just affects the apparent shape of the wave via its harmonics. If Jimmy Page were to somehow manage to play a 20 kHz sawtooth on his guitar, and we sampled it at 44.1 kHz, the corners of it would be so rounded that it wouldn't be distinguishable from a sine wave anyway. Wave shape just doesn't matter at the edge of the limit because you lose all the higher harmonics that would distinguish one shape from another.
(Score: 2) by rleigh on Monday November 14 2016, @07:32PM
Yes, it applies to any shape of waveform, but that sawtooth is made up of multiple sine waves of varying frequency and amplitude, and that's what's being sampled. Same with a square wave.
http://mathworld.wolfram.com/FourierSeriesSawtoothWave.html [wolfram.com]
http://mathworld.wolfram.com/FourierSeriesSquareWave.html [wolfram.com]
(Score: 1) by khallow on Sunday November 13 2016, @09:53PM
(Score: 2) by tfried on Sunday November 13 2016, @09:29PM
Not all of them.
And wouldn't that be a truly strange world, where an interest in subject A totally precludes an interest in subject B. But one interesting bit along your line of thought may be, that there is a niche for active contributions by a mathematician in - probably - any field of science.
I always propose getting a bunch of teenagers and use analog equipment to generate a 17k sawtooth and a 17k sine and see if the kids can tell the difference.
So far, no one has done that as far as I know.
So, did you? Neither did I, but seriously, the kids would not be able to tell the difference. The ear separates frequencies spatially. Among other things that's important because any single neuron cannot fire more than around 100 per second, so the incoming signal absolutely has to be decomposed. So, to get a pretty good analogy, feed your 17k sawtooth through an FFT. You'll get a 17k sine and a lot of overtones at much higher frequencies. You'll hear the 17k sine component alright (if your ears are young enough), but those higher frequencies will not be representable - neither in a 44khz sample, nor in a human ear.