markisus

The author describes a process called "ontological remodeling", which is when a change in viewpoint radically simplifies a previously intractable problem. This is the story, not just of Sudoku, but all of mathematics.

See the demo page for example applications of inside out tracking and virtual metrology.

https://github.com/markisus/pytagmapper/blob/main/DEMO.md

The projection viewpoint for a rotation matrix using dot products is totally valid, but I think the column viewpoint is more intuitive.

A 3x3 matrix can be viewed as something that acts on the standard unit basis vectors. The action on the standard unit basis vectors can be read out by looking at the columns. (A matrix times the ith standard basis vector is just the ith column.) Since the transform is linear, all the other points in space are carried along with the action as well. This is the "linear map" viewpoint of a rotation matrix.

I show an example here https://biro.ai/we-learned-the-wrong-way-to-matrix-multiply/ where this viewpoint lets you instantly see what a rotation matrix does by looking at the columns.

I really like the animations and graphs. Anyone know what library if any was used for those?

Source: https://github.com/markisus/aces.io

Spawn a room by visiting http://www.aces.io/your-room-name

Features: open source, no registration, anyone with the link can join, fake money, copy/pastable auto-updating ledger

If anyone wants to play around with the geometric algebra over R^3 in C++, I wrote a little library a while back which handles vectors (3d), quaternions (4d), bivectors (3d, quaternions without the scalar part), multivectors (8d)

https://github.com/markisus/g3

Maybe an example will help me understand. Let's say I wanted to have the leading digits 921, and I start with a guess of X=5.

What do I do now?

Okay I think I buy it. Great proof, thanks!

I'm not sure you only need that many steps. As you start out, each step gives you a lot of distance. But log gets flatter and flatter the more steps you take.

If you knew the number of steps, I think you could only make a lower bound on spacing, not an upper bound. Since at worst all the steps could be clustered really close, except for the last one.

But now I have to think more because it seems like it should actually help the argument that the spacing gets finer and finer as you step, because that is what we want anyway. And is the number of steps actually important to the proof?

What does "multiplying by N" mean? Like if my current guess is X! and that doesn't work, I should try (N*M)! ?

What does "increase by at least 0.5" mean? If my leading digits are 999, how do you increase them at all?

The approximation holds only for b << 10^k. Then you don't get to take O(10^k) steps to work you way around the unit interval, do you? After the approximation breaks down, you don't know that the spacing is approximately uniform anymore.

I'm also not getting your proof. What does it mean every time you hit a power of 10? The sequence n! does not hit powers of 10 since all terms of the sequence after n=3 are at least divisible by 3.

I agree though, density in (0,1) of the fracPart(log10(n!)) is a good thing to try to prove, since 10^fracPart(log10(n!)) should have the same digits as 10^intPart(log10(n!))*10^fracPart(log10(n!))=10^log10(n!)=n!

How do you know these prices? I generally find these companies do not publish them on their websites

And it can get even worse because your adversary can choose A and B as close together as they want.

I think I drew a picture that could explain the effect. In the picture, player 2's lower number is A, and their higher number is B.

https://imgur.com/a/DKCtH81/

Red is the region where you lose, and Green is the region where you win.

I was able to find this page through Google search: https://chronicdata.cdc.gov/Behavioral-Risk-Factors/Behavior...

and I found the survey question by searching with the filter Question > contains > "firearm."

At this point I'm tempted to say that the author's complaints with respect to data suppression are exaggerated.

I wonder what the author means when he says the CDC did not report their results? In the final paragraph he says, "Regardless of how the decision was made, it was a disservice to the American people, who paid for the survey and the information it yielded, but who were not allowed to see it and judge its worth for themselves." Indeed, it's wrong for government agencies to suppress publicly funded data sets, especially when suppression is due to political affiliations. But then how did the author obtain the data?

I agree with the author that we should avoid feelings of resentment towards people who are successful. However it bothers me that the author classifies trace of a matrix as math trivia. I think sometimes people will categorize their current knowledge as "the important stuff", and everything else is "just trivia". This can be a dangerous trap that prevents personal improvement.

What about cowboy Indian? All I see is face looking right. And for mouse man I can't find the mouse

It seems that this subpixel convolution layer is equivalent to what is known in the neural net community as the "deconvolution layer" but it is much more memory and computation efficient. The interlacing rainbow picture was a bit hard to understand until I read this https://export.arxiv.org/ftp/arxiv/papers/1609/1609.07009.pd...

These questions are indeed atrocious. The author's point would have been better served with these real examples. Over-exaggerating is like jumping the shark, and makes it difficult to entertain the premise.

If the IQ test anecdote about the horse is real I'd like to see a source. It's extremely hard to believe that even a malicious scientist could convince himself that a horse cannot be ridden into town.

> One question in the early test asks, “A man comes into town but his feet do not touch the ground. What is he riding?” The story goes that a young Native American boy responded, “A horse” to which the tester yelled, “Wrong! It’s a bicycle!”

I think the author was trying to say that there seems to be no simple pattern. The original sequences are meant to demonstrate how hard it is to see a pattern. However, a relatively recent empirical discovery found very simple statistical biases between consecutive primes. Mathematicians were shocked by these results because they were so simple. This article demonstrates and further explores this surprising phenomenon.

I think there is a bug in the program. Dropping 900,000 needles gave pi = 3.139 which seems pretty bad

The Lebesgue measure could be a candidate for the measure you are asking for. Creating a bounding box around the convex set in question and then scaling, we get a probability measure (measure of the entire set is 1). Some sorts of random walks inside this bounding box will converge to this scaled Lebesgue measure. I think this is the main idea behind the convex volume estimation algorithms.

The oracle you mention is the indicator function of the convex set. Instead of integrating this indicator function directly over Lebesgue measure to get volume, we can use MCMC to get the "expectation" of the oracle.

It's not obvious if we can realize the Riemann zeta function as an integral over a finite measure space, but if we could, I think MCMC could be used to evaluate it.

The author draws a distinction between computing expected value of a random variable and computing the volume of a convex set, but from the point of measure theory, both are computations of integrals. So maybe the definition given from the encyclopedia wasn't so bad, given the right prior knowledge.

I don't understand how the author's estimate is justified. He wants to take the probability that the margin is within 2 million and then divide by two million? Maybe he is appealing to some limiting argument? But we are presumably dealing with a discrete random variable.

It must be that, once you re-arrange the pieces, you transform the rectangle with a hole into a shape that looks like a rectangle with no hole but is slightly caved in.

Can you please explain what it means to "be a sock puppet?"