kurlberg

Recently a lean formalized proof of the Collatz conjecture had exploited bugs in the lean kernel. However, the kernel is fairly small so hopefully it'll be bullet proof soonish.

There is a joke saying "a mathematician says X, writes Y on the board and means Z". The really amusing(?) thing is that other mathematicians still (sort of) perfectly understands Z. Once you have enough experience you fill in the blanks automatically.

Math exposition is tricky: too few details and you're just floating in the sky, too many details and the audience loses sight of the forest for all the trees. You can go (more or less) all formal, but it's a pain for the writer and a pain for the experienced reader.

If it's any consolation, the punchline to the joke is that it often is small/big lie: the other mathematicians reads "Y" and goes WTF!? And then 1 minute, 1 hour, 1 day, or one week later says "aaah, that's what he/she meant! I guess it was 'obvious' all along". :-)

Where?

There is a cute argument (I think it is due to Erdos) that, asymptotically, 0% of the integers in [0,n^2] appears in the "n by n multiplication table":

By Erdos-Kac, almost all integers of size about n^2 have about log(log(n^2)) ~ log(log(n)) prime factors. However, almost all integers in the multiplication table have about 2*log(log(n)) prime factors.

Kevin Ford gets much more precise asymptotic estimates.

Convolution alone does not smooth. Eg consider a random variable supported on the pts 0 and 1 (delta masses at 2 pts.) No matter how many convolutions you do, you still have support on integers - not smooth at all. You need appropriate rescaling for a gaussian.

Also, convolving a distribution with itself is NOT a linear operation, hence cannot be described by a matrix multiplication with a fixed matrix.

Read it as a young teenager, can recommend.

I don't think this is entirely due to Wozniak. Early "home" computer systems were based on connecting cards to a bus (eg the S-100 bus), eg. with one card supporting the CPU, another RAM, a third for disk drive, video card etc, etc. The cards where then memory mapped, presumably you controlled the memory mapping by setting jumpers. (I guess you're saying that Apple II managed this automatically?) Of course the full story might be a bit more complicated: 6502 and 6800 used memory mapped I/O, whereas 8080 (and Z80?) had certain I/O pins coming out of the CPU.

Fun historical fact: knot theory got a big boost when lord Kelvin (yeah, that one) proposed understanding atoms by thinking of them as "knotted vortices in the ether".

If you have a child who likes math I highly recommend "Really Big Numbers" by Richard Schwarz. Tons of nice illustrations on how to "take bigger and bigger steps".

"Infinity is farther away than you thought."

I recently got into making some sort of budget hifi setup, and found audiosciencereview.com quite helpful - a good amount of reviewed gadgets with focus on measurements. Ended up with kali lp-6v2 speakers and a SMSL SU-1 dac. Please don't tell me I screwed up. :-)