Even Terry Tao struggled at times: "When I was a graduate student in Princeton, Tom Wolff came and gave a course on recent progress on the restriction and Kakeya conjectures, starting from the breakthrough work of Jean Bourgain in a now famous 1991 paper in Geom. Func. Anal.. I struggled with that paper for many months; it was by far the most difficult paper I had to read as a graduate student, as Jean would focus on the most essential components of an argument, treating more secondary details (such as rigorously formalising the uncertainty principle) in very brief sentences."
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". :-)
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.
I had the same impulse (or at least copy.fail inducing many to upgrade at the same time.) However, it might be a "pro-Iran hacktivist group" according to
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.
This has been discussed on HN some times before. User xornot looked at the zfs source code and debunked "faulty ram corrupts more and more on scrub", for more details see
https://news.ycombinator.com/item?id=14207520
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".
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. :-)
I have used merlin for quite a while, mostly happy (except for some security holes...) However, once asus drops support for older devices (e.g. rt-ac68u and rt-ac86u), merlin might also drop it. For now rt-ac68u is dropped by merlin, but ac86u is fine for now (at least until the end of the year.)
Upshot: if you care about very long term support, openwrt is nice.
Based on reddit [1] and other some other recommendations I got an asus ax4200 and put openwrt on it. I'm fairly happy, but some people have run into connection dropping (possibly due to ISP power saving resulting in link dropping down to 10 mbs, and something then goes wrong.) With forum help [2] I found a workaround: either turn off auto negotiation (works) or using a lan port as a wan port (have not tried).
PS: if you're interested in multiplying "ludicrously large numbers", Harvey and van der Hoeven had a nice breakthrough and got multiplication down to "FFT speed" (n*log(n)), see
There is a nice picture of the "best" choice for different ranges of sizes of numbers to be multiplied at http://gmplib.org/devel/log.i7.1024.png
More context and explanation can be found at: http://gmplib.org/devel/