I think it's more that he wanted to go out undefeated, rather than lacking motivation. Or rather the former driving the latter.
He made 5 title defenses. Two were against the previous generation of players, and he did extremely well. 2 were against players of his generation and were anything but compelling victories. He only won a total of 1 classical game in the 24 played, and that was in a must-win scenario because he had just lost for a final record in these matches of +1 -1 =22. And finally there was his match against Nepo which was looking to be another extremely close match until Nepo lost a critical game, and then went on monkey tilt, as is his reputation - proceeding to play horribly for the rest of the match and get wiped.
In an interview with Rogan, Carlsen stated he felt he peaked a bit before his match against Nepo, and so he probably did not view his chances of success in a world championship match as especially high. So he was going to have to spend months preparing for a match he could very well lose which would certainly tarnish his reputation as the GOAT of chess. I think this is why he couldn't find the motivation.
For instance there were new world records just around the corner. The most successful world title defenses is 6 and that was back in the early 20th century. With one more he could have surpassed Kasparov and at least tied the record.
Magnus has always been unhappy with the format of the WCC cycle. He first skipped it in 2011, when he was already the top-rated player but not yet champion (https://www.chess.com/news/view/carlsen-quits-world-champion...), and very nearly skipped it again in 2013.
Actually he recently stated that he IS still disappointed about that whole incident because nothing changed and is currently backing up hikaru on drama around similar issues.
The motivation issues can stem from poor management :)
yep, also i think while they could have issues with dataset sizes less than 2^k, it's interesting to note their use in accelerating clustering algos like dbscan. they do make neat visualizations though https://marimo.app/?slug=x5fa0x
The BYTE_STREAM_SPLIT is a Z-order (also known as Morton encoding).
As it is better a preserving locality, it usually performs really well compared to classical "orthogonal" orders.
It also is really simple to implement as it boils down to interleaving bits or bytes.
Draw a line of length 1 which makes an angle of p with the positive x axis. The x and y coordinates of a, the point at the end of the line give cos(p) and sin(p).
Now think about what happens if you increase p by a tiny bit. a moves tangentially. (This works very similar to how the tangent to a curve gives the change in y for a change in x.) So the vector of cos'(p), sin'(p) is given by a vector starting from a, at a right angle to 0a, and pointing in the positive direction.
Since the point a moves through 2pi distance while p goes from 0 to 2pi (the definition of measuring angles in radians) the speed of the point is 1, and so the vector of derivatives has length 1.
You can check easily that this makes cos'(p) = -sin(p) and sin'(p) = cos(p).
(If I were trying to present this stuff in a maximally-elegant order without too much regard for what order human brains like to learn things in, the order of things would be: complex numbers, calculus, trigonometry. Then we define something that we might initially call e(t) to satisfy the differential equation de/dt = ie, and observe that having e and e' at right angles means that |e| remains constant, which means that |e'| also remains constant, so if we start with e(0)=1 then we have a point moving at unit speed around the unit circle, etc. Keep the linkage between the geometrical and formal points of view there at all times. But I suspect this wouldn't be great paedagogically for the majority of students.)
I do not understand this consideration:
> By considering a triangle with hypotenuse 1 and a very small “opposite” side, it’s not hard to see geometrically that sin(x)≈x and cos(h)=x when x is small
I fail to see how you can "see" finer than sin(h) -> 0 & cos(h) -> 1
That one line was the part that stood out to me the most as well, but:
If you zoom in sufficiently at x = 0, f(x) = sin(x) looks indistinguishable from f(x) = x, whereas g(x) = cos(x) looks indistinguishable from g(x) = 1.
(also, sin(x) is negative approaching 0 from the left and positive approaching 0 from the right)
> If you zoom in sufficiently at x = 0, f(x) = sin(x) looks indistinguishable from f(x) = x, whereas g(x) = cos(x) looks indistinguishable from g(x) = 1.
You can't use the plot as you only know the triangle definition yet. (And "looks indistinguidable" is rather handwavy).
> (also, sin(x) is negative approaching 0 from the left and positive approaching 0 from the right)
Your limit definition is the same as the part you quoted, so it's not clear what your question is. I also don't see what you are quoting.
Curvature is inverse of radius.
Decreasing angle is equivalent to increasing radius, and this decreasing curvature.
This, as angle decreases, the curve becomes close to a straight line, and that straight line approaches a vertical line.
I quote the second paragraph of the Derivatives section. (which was edited to a better, but not yet enough, sin(h)≈h and cos(h)≈1 when h is close to zero).
I perfectly understand that around 0, sin(x) ~ x and cos(x) = 1 + o(x) but it isn't obvious geometrically, unlike what the article implies.
From my point of view, increasing radius / decreasing curvature only gets you sin(x) -> 0 ; cos(x) -> 1, but that isn't enough to obtain the derivatives.
I found a geometric proof in [1] but that part is the longest and hardest of the page. I was wondering whether the author found a clearer way to express is.
EDIT: after looking at 3B1B's video, the "small" triangle d(sinΘ) by dΘ figure would be a better way to explain the derivative, rather than an "not hard to see geometrically" approximation that isn't enough to conclude.
Came here to say the same thing. But I suppose if you extend cosine into negative values you'll see it has a maximum at zero so its derivative must be zero. Don't know about sin'(0) offhand but you'd think it wouldn't be hard.
The "Jeans" controversy happened during the Rapid championship in December 2024, nearly two years after.
It's universally accepted that his streak Classic Championship ended because of his lack of motivation, not on technicalities.
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