Tag: math

Fourier Transform

the Fourier transform tells you how much of each ingredient “note” (sine wave or circle) contributes to the overall wave. Here’s why Fourier’s trick is useful. Imagine you were talking to your friend over the phone and you wanted to get them to draw this squarish wave. The tedious way to do this would be to read out a long list of numbers that represent the height of the wave at every instant in time. With all these numbers, your friend could patiently stitch together the original wave. This is essentially how old audio formats like WAV files worked. But if your friend knew Fourier’s trick, you could do something pretty slick: You could just tell them a handful of numbers—the sizes of the different circles in the picture above. They can then use this circle picture to reconstruct the original wave.

2022-11-14: The nuclear origins of Fast Fourier Transforms

And this trick works even if the signal is composed of a bunch of different frequencies. If the sine waves frequency is one of the components of the signal it will correlate with the signal producing a non-0 area. And the size of this area tells you the relative amplitude of that frequency sine wave in the signal. Repeat this process for all frequencies of sine waves and you get the frequency spectrum. Essentially which frequencies are present and in what proportions. If the signal is a cosine wave, then even if you multiply it by a sine wave of the exact same frequency, the area under the curve will be 0. For each frequency, we need to multiply by a sine wave and a cosine wave and find the amplitudes for each. The ratio of these amplitudes indicates the phase of the signal that is how much it’s shifted to the left or to the right. You can use Euler’s formula so you only need to multiply your signal by one exponential term. Then the real part of the sum is the cosine amplitude and the imaginary part is the sine amplitude.

Prime Numbers Patterns

A new analysis has uncovered patterns in primes that are similar to those found in the positions of atoms inside certain crystal-like materials. The discovery may aid research in both mathematics and materials science. “Prime numbers have beautiful structural properties, including unexpected order, hyperuniformity and effective limit-periodic behavior. The primes teach us about a completely new state of matter.”

Paul Erdős Amphetamine

I began to wonder whether Paul Erdős (who I used as an example of a respected academic who used cognitive enhancers) could actually have been shown to have benefited from his amphetamine use, which began in 1971 according to Hill (2004). One way of investigating is his publication record: how many papers did he produce per year before or after 1971?

The Numbers King

Grotzinger, who was advising, not seeking a job, elegantly guided the group through the challenges of climate modelling. Many of the problems were familiar to the Flatiron staff. “Most of the data actually gets ignored,” Grotzinger explained. And there was a problem of collaboration. He was a specialist in historical climate change—specifically, what had caused the great Permian extinction, during which virtually all species died. To properly assess this cataclysm, you had to understand both the rock record and the ocean’s composition, but, “geologists don’t have a history of interacting with physical oceanographers.” He talked about how his best collaboration had resulted from having had lunch with an oceanographer, and how rare this was. Climate modelling was an intrinsically difficult problem made worse by the structural divisions of academia. “They will grope their way to a solution probably in the next 50 years. But, if you had it all under 1 umbrella, I think it could result in a major breakthrough.” Simons and his team were interested. It seemed Flatiron-ready. The scientists asked Grotzinger how many fellows, and how much computing power, such a group would need. Grotzinger estimated that a division would need at least 50 researchers to be effective. “I would include some programmers”. He hopes to have his 4th division in place by next September. Why stop there? Why not 8 units? Why not Simons University? He had the money, after all. But he insisted that 4 divisions was all he could handle, if he wanted both first-class work and a collaborative atmosphere. He added that he needed to manage it all, with his “light touch.” Simons understood that, whatever structure he set up, it ultimately needed to function well without his supervision. The foundation had signed a 35-year lease on the institute’s building, with an option to renew for 15 more. As long as the tax laws didn’t change dramatically, Simons’s fortune could keep the institute going in perpetuity. But humans, he realized, were not machines. “I’m hoping this is going to last 100 years. But I won’t see it.”

China textbooks

the us can always export textbooks on homeopathy, flat earth, creationism and alternate history:

When primary school administrators in the U.K. choose study materials for the fall semester this year, they will have a new option: math textbooks imported from Shanghai, a city celebrated as a global math power. It is a remarkable admission by British education authorities that their own methods have stumbled, and that Chinese educators – after years of racking up world firsts in math scores – have developed something admirable enough to import in whole cloth.

IEEE floats are broken

IEEE floats are broken. Posits beat floats at their own game:
• superior accuracy, dynamic range, closure
• Bitwise-reproducible answers (at last!)
• Proven better answers with same number of bits
• …or, equally good answers with fewer bits
• Simpler, more elegant design can reduce silicon cost, energy, and latency.

Mathematica 11

i’m always amazed about how powerful mathematica is and then sad about how niche it is. not sure they could do better if they went the R route (open source things) as more and more they’re vertically integrated, but i do wonder what things would be like if it were much more widely used in education and business.

what’s the big new thing in Version 11? Well, it’s not 1 big thing; it’s many big things. To give a sense of scale, there are 555 completely new functions that we’re adding in Version 11—representing a huge amount of new functionality (by comparison, Version 1 had a total of 551 functions altogether). And actually that function count is even an underrepresentation—because it doesn’t include the vast deepening of many existing functions.

Is Infinity Real?

If the physical universe cannot contain infinity (which is a wise default position to have in the absence of extraordinary evidence), there will be limits at which our infinity-laden mathematical models will fail. Our simple problems were meant to explore these limits. We may find that the infinity-based predicted outcome can still hold, but for different reasons (as in the first problem); or that the infinity-based predicted outcome is wrong, and we have to reason differently to reach qualitatively different conclusions (second and third problems).