HN user
It's complicated. :-)
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/
"Learning styles" might be a myth. Eg, see
Learning Styles: VAK Doesn't Exist (Here's What Research Actually Shows)
https://www.structural-learning.com/post/learning-styles-myt...
Belief in Learning Styles Myth May Be Detrimental (by American Psychological Association)
https://www.apa.org/news/press/releases/2019/05/learning-sty...
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."
More details at
https://terrytao.wordpress.com/2018/12/29/jean-bourgain/
(In particular, see the "???" in the Tao's annotated copy of Bourgain's paper.)
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.
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
https://www.theregister.com/2026/05/01/canonical_confirms_ub...
"Canonical says its web infrastructure is under attack after a pro-Iran hacktivist group instructed its members to target the open source giant."
Perhaps more to do with extortion rather than activism. (I have no idea how accurate theregister is on this story.)
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.
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".
"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. :-)
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).
1:
https://www.reddit.com/r/openwrt/comments/1cr1lvp/is_the_asu...
2:
https://github.com/openwrt/openwrt/issues/14192#issuecomment...
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
https://hal.science/hal-02070778v2/document
A pop-sci description can be found at
https://theconversation.com/weve-found-a-quicker-way-to-mult...
There is a nice picture of the "best" 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/
BTW, I like Bernstein's survey of different multiplication algorithms at
https://cr.yp.to/papers/m3.pdf
(there is a unifying theme about using ring isomorphisms to explain many of the "standard" routines.)
Check out "Diamond Age" by Neal Stephenson.
It's very inefficient, both on terms of runtime and in terms wasted entropy.
This has been discussed on HN some times before. User xornot looked at the zfs source code and debunked it, for more details see
Yes, it has uses. E.g., see Kahan's "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit", copy available at
https://people.freebsd.org/~das/kahan86branch.pdf
(he gives an example regarding complex branch cuts and fluid dynamics applications.)
Years ago I saw it at:
https://www.truenas.com/community/threads/ecc-vs-non-ecc-ram...
(the gist of the scary story is that faulty ram while scrubbing might kill "everything".) However, in the end ECC appears to NOT be so important, e.g., see
There is an extremely nice "Digital show and tell" video by Monty at
https://xiph.org/video/vid2.shtml
(part 1 at https://xiph.org/video/vid1.shtml is also very well worth watching.)
I lived in the states for a while and missed Bregott. "Land o Lakes butter with canola oil" is a pretty good substitute.
According to
https://www.servethehome.com/hpe-proliant-microserver-gen10-...
the 10+ seems to run freenas fine.
Thanks! I've got a new alias cut out for me: duh! :-)
I'm overall very happy with my kinesis advantage, but there is something (subtle?) to be aware of: you tend to use your thumbs quite a bit to hit control/alt/return/space (especially with emacs), and after some time my thumb joints started hurting. Thumbonitis? :-)
Another option is
du | sort -n
(I've got it aliased to "dun".)
It's not only in the abstract - "category" occurs all over the document. Possibly he's poking fun at category theory/theorists.