Interesting follow-up question: What is the distance between the set of harmonic numbers and the integers? i.e. is there a lower bound on the difference between a given integer and its closest harmonic number? If so, for which integer is this achieved?
For n > 1, there isn’t a lower bound. None of the numbers are integers again (https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#...), and because the difference between successive partial sums goes to zero and the series grows to arbitrary values, you’re bound to find a difference smaller than 1/(2n) somewhere beyond n.
No, because the terms tends monotonically towards zero. Let an integer m with closest harmonic number H_n be given (i.e. n minimizes |H_n-m|). So m exists either between H_n and H_(n+1) or H_n and H_(n-1). Then |H_n-m| < H_(n+1) - H_(n-1) = 1/n + 1/(n+1). We can make that bound arbitrary small by choosing a large enough n.
And node seems to be used only as a dev dependency, to test, benchmark and build/package the project. If you'd be inclined you can use the project's code as-is elsewhere, i.e. in the browser.
Even today it is exceedingly rare to find a still-well-conditioned bottle of wine that has the capability to have aged for 117 years or so. Most often sweet wines are capable of this.
FWIW I was also sceptical, but just tried it from my phone network and it seems indeed blocked. Wouldn't be the first case of different ISPs using different block-lists. c.f. bs.to
Presuming you can create an alloy with the same density as gold, I imagined you could also test it's conductivity. I think performing both tests would be enough.
The only metals with which you could make an alloy with the same density as gold, but cheaper than gold, are uranium and tungsten.
Other metals would require too big additions of expensive rhenium/osmium/iridium/platinum to match the density of gold.
The best choice for matching the density of gold is tungsten, but even with that the cost for an exact match of the density would be high. The tungsten objects that are found easily in commerce have a density significantly lower than gold, because they are made from tungsten powder sintered with nickel, not from pure tungsten, which is hard to melt.
The conductivity test is good, but not easy to perform when the object has a complex form. Surface conductivity is easy to measure on any object, but the object could be plated with pure gold, so surface conductivity would show no difference.
For a gold bar of standard dimensions, it should be easy enough to make a text fixture allowing the measurement of the bulk conductivity.
It is rather expensive to make objects of pure tungsten. The tungsten objects that you see for sale are not pure and they have a density more than 5% lower than gold, which is easy to detect by weighing.
Do not underestimate the urge to procrastinate (by still doing productive things, like learning Mandarin) while pursuing a PhD.
I am not sure if this will be the author's experience too, but pursuing a PhD will often leave you exhausted without any hope of ever finding "the final missing ingredient" to solve the problem you are currently tackling. So turning to entirely unrelated problems, however productive they may seem to outsides, suddenly becomes an attractive alternative in order to procrastinate.
>I find that when someone's taking time to do something right in the present, they're a perfectionist with no ability to prioritize, whereas when someone took time to do something right in the past, they're a master artisan of great foresight.
It is so underrated, that I have been led to put off doing more structured procrastination until I have more time. If more people had told me how great it could be, I would be doing it now!
Probably using a QMK firmware-based keyboard where you can access different layers and shortcuts.
I'm using one right now (though mine runs off ZMK which is similar but wireless) which is a split with just 42 keys. The rest--numbers, symbols, function keys, etc. are all under layers. The layout is dynamic because holding down different keys makes the layout 'change' as you do so. Holding down the left spacebar and pressing 'Z' sends 'F1' to the computer while holding down another key on the right half turns my WER/SDF/XCV keys into a Numpad, etc.
Yes, both keys send the same key code to the computer, however, pabloescobyte said they’re using ZMK, so the left/right space bar distinction is happening on the level of the keyboard controller.
I am currently learning to color grade, am an active bedroom musician, enjoy cooking and learning about food science, and am training for my first half marathon alongside my PhD. The side project thing is definitely real.
I’m not sure it’s procrastination. Years ago, when struggling with maths , I learned juggling ( 5 balls, tricks etc ) and ended up spending quite a bit of time on it every single day.
In practice, it made me feel very good, more relaxed, because I was able to learn something new and make progress rapidly - self confidence was back. The maths soon got unstuck and life became good.
You can get gcc to generate the blend instruction in the example given - you just have to 'force' it to use SIMD more aggressively by, for example, using -march=native.
I recon the general cosensus among mathematicians (as that is what counts) is that the ABC conjecture so far has _not_ been proven. Mochizuki (and his school around him) seem to be the majority of people that believe his proof is correct. As you point out, Scholze has identified a supposed flaw in Mochizuki's argument, but anyone not already at the forefront of IUT/NT/ABC conjecture is probably incapable of telling if this flaw is a true flaw or not. As Mochizuki refuses to elaborate (on this supposed flaw) consensus cannot be reached and thus the ABC conjecture remains open.
I think giving any more details than his three short posts would delve too far into the specific arguments of the paper to be valuable for a wide enough audience, subtracting those that are going to be looking at the pre-print anyhow.
