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100 Years to Solve an Integral (2020)

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Re: 100 Years to Solve an Integral (2020)

#21

Post by 2b3a51 »

anthk wrote: post #20
331c8c71 wrote: post #18

Earlier quoted context omitted.

Wolfram engine was taking integrals just fine way before LLMs were even a thing.

And Lisps too, fitting a sector from a disk: https://justine.lol/sectorlisp2/ And probably a small forth too, with a dictionary defining every math word, something not so different to Lisp. LLM's? 4GB of RAM? Your grampa's 486 with 16MB of RAM can do calculus too.

Derive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample.

You can do a lot of numerical maths just with a noddy spreadsheet of course.

Re: 100 Years to Solve an Integral (2020)

#22

Post by mppm »

ziofill wrote: post #3

I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.

Learned something new today, thank you!

If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that regard?

Re: 100 Years to Solve an Integral (2020)

#23

Post by anthk »

2b3a51 wrote: post #21
anthk wrote: post #20

Earlier quoted context omitted.

And Lisps too, fitting a sector from a disk: https://justine.lol/sectorlisp2/ And probably a small forth too, with a dictionary defining every math word, something not so different to Lisp. LLM's? 4GB of RAM? Your grampa's 486 with 16MB of RAM can do calculus too.

Derive 2 for Dos. Green Screen 286 I think or 386 computers in a small side room. Later Windows version was better. Then there was the DOS version of Minitab 5 I think that came as floppy disks in the back of a spiral bound book which I used to generate data sets for students to process for homework so everyone got a slightly different sample. You can do a lot of numerical maths just with a noddy spreadsheet of cours…

Macsyma, PDP10 + ITS under Maclisp.

https://en.m.wikipedia.org/wiki/PDP-10

https://en.m.wikipedia.org/wiki/Incompatible_Timesharing_Sys...

https://en.m.wikipedia.org/wiki/Macsyma

Fun fact: old Macsyma's math code still runs at is on modern Linux'/BSD's with Maxima. Even plots work the same, albeit in a different output format.

A 386 it's far more powerful than this.

Re: 100 Years to Solve an Integral (2020)

#24

Post by ljsprague »

>[the Mercator projection] unnecessarily distorts shapes and in particular makes the Americas and Europe look much larger than they actually are. This has been linked, not without rational, to colonialism and racism.

The fact that on many maps Europe is much smaller that it appears should just make you all the more impressed by its achievements.

Re: 100 Years to Solve an Integral (2020)

#25

Post by Sesse__ »

cl3misch wrote: post #11

Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?

It's a US thing. Europeans just write 1/cos(x) instead of treating it as a special thing with its own name. The Americans have sec, csc, and a bunch of others I never bothered to learn. It doesn't seem to add all that much to me? (Of course, it's a bit hypocritical since I gladly use tan(x).)

Re: 100 Years to Solve an Integral (2020)

#26

Post by rini17 »

LegionMammal978 wrote: post #4

If we're playing the map-projection-advocacy game, I'd say the Mollweide projection is underrated among equal-area maps [0]. (For local maps, use whatever you want, appropriately centered.) Sure, it distorts shapes away from the central meridian, but locally it only adds a simple horizontal skew. I'm not a big fan of how many equal-area 'compromise' projections lie about how long the lines of latitude are. [0] https:…

You probably don't live in New Zealand? (Yes I know it's there. Barely.)

Re: 100 Years to Solve an Integral (2020)

#27

Post by 867-5309 »

seanhunter wrote: post #17
cl3misch wrote: post #11

Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?

Trig is full of functions that fall into disuse and are forgotten. For example "versine" versin theta = 1-cos theta. There is also "haversine" which is (1-cos theta)/2. Which is used in navigation apparently https://en.wikipedia.org/wiki/Versine

iirc, haversine is useful for transforming 2-d "as the crow flies" coords to their 3-d equivalents. at longer distances a body's curvature is really noticeable and often overlooked

Re: 100 Years to Solve an Integral (2020)

#28

Post by fxj »

mppm wrote: post #22
ziofill wrote: post #3

I know the article is about sec(x) but I want to share this tidbit about its cousin, the hyperbolic secant: sech(x) is its own Fourier transform (modulo rescalings). That’s right, exp(-x^2) is not the only one.

Learned something new today, thank you! If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that…

Yes the dirac comb for example. Actually there are infinitely many.

https://en.wikipedia.org/wiki/Dirac_comb

and for other:

http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...

Re: 100 Years to Solve an Integral (2020)

#29

Post by RichardLake »

Sesse__ wrote: post #25
cl3misch wrote: post #11

Neither in (German) high school nor in the many math courses of a physics B.Sc. have I ever used the secant function. I am surprised the article does not explain it in the beginning. I assume for other people it must be a common function?

It's a US thing. Europeans just write 1/cos(x) instead of treating it as a special thing with its own name. The Americans have sec, csc, and a bunch of others I never bothered to learn. It doesn't seem to add all that much to me? (Of course, it's a bit hypocritical since I gladly use tan(x).)

I imagine it was more useful when using tables to lookup/approximate the values before calculators with trig support were a thing.

Re: 100 Years to Solve an Integral (2020)

#30

Post by grandempire »

anyfoo wrote: post #15
asplake wrote: post #14

Earlier quoted context omitted.

The secant is the reciprocal of a cosine – the hypotenuse over the adjacent

That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.

It is a function. What do you mean?
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