carstimon

Sorry, this was incorrect. In this integral we're just calculating the geometrical question, what is is the 5d volume of the collection of 3 points whose circumcenter is on {(x, 0) : x in [0, 1]}. So the "overall" probability of the problem has nothing to do with it.

You pointed out one error in your own answer, here's another. I haven't figured out the adjustments, but presumably they cancel.

You've done

∫₀¹ ∫₀¹⁻ᶜ (2π r)³ dr dc

However,

- the `dr` integral is assuming that the radii are uniformly likely in [0, r]

- the `dc` integral is assuming that the centers are uniformly likely in [0, c]

You need to wait these integrals by the conditional probability distributions.

A blog post for it from a prominent c++er https://herbsutter.com/2013/04/05/complex-initialization-for...

You've made the choice of (Locked Door, Open Window) ~ (Good SQL usage, XSS Vulnerability) which seems to be an incorrect rebuttal. Your example doesn't contradict "only as strong as its weakest link", here the weakest link is the XSS Vuln.

The "house analogy" can also support cases where the potential damage is not the same, e.g. if the open window has bars a robber might grab some stuff within reach but not be able to enter.

I think this is in response to

Why should my Dog be able to check out the internal StomachContents of another Dog

This seems to be interpreting the purpose of visibility as controlling the access that actual things represented by objects have. (There could be a non oop-as-real-world-things example here; `UserProfile.equals(UserProfile other)` might access secrets of `UserProfile` that the different actual users are not supposed to see between each other.)

The reason Dog can check the internal details of another Dog is, there's a single file that controls the usage of the fields. If you decide you want to change the semantics of the stomachContents field, you can check all the usages in that file. That's the simple practical reason. The StomachContents thing seems like a confusing analogy, like most that compare OOP to Real World Categorization.

To add on to the defense in another comment,

I don't think the point of the article is a rust-v-python comparison. Sure, unoptimized-python2rust2optimized rust was a part of the clickbait 180,000x title but... - the author explicitly says the python2rust bit is only a factor of 8. - the Python bit is just one section of a pretty long article - the author says they typically start implementing something in Python so... why wouldn't they include their initial implementation?

I don't think this article should be dismissed just for including a simple python implementation- that isn't the point.

I thought the axiom of choice says that it is possible to obtain an [infinite] set of votes from an [infinite] set of voters each with a non-empty set of choices

The axiom of choice does say this but,

1) This is not where we are applying the axiom of choice

2) You don't need the axiom of choice here: you can always just pick the "voting set" where everyone votes for choice # 1.

The axiom of choice says that the product of infinitely many non-empty sets is empty. This is needed only when you might have a different and non-computable set of choices to make at each step.

Where we are applying the axiom of choice here is: the voting scheme is just a rephrasing of an "Ultrafilter on the power set of a set": https://en.wikipedia.org/wiki/Ultrafilter#Ultrafilter_on_the... This is implied by the axiom of choice in a way that requires some working out (and is sketched in wikipedia). But you can roughly see how the axiom of choice comes about if you start trying to construct a voting scheme (the following is just intuition):

1) We know the result if the number of "Nays" or "Yays" is finite.

2) Pick any set of "Yays" so the number of "Nays" and "Yays" is both infinite. Declare this set (arbitrarily) as a victory for either "Nay" or "Yay"

3) The set in (2) implies that a bunch of other sets are either Nay or Yay by the axioms in the article (e.g. flipping all votes flips the outcome, etc)

4) While there are sets we haven't decided on, repeat steps (2) and (3).

Of course (4) is the whole tricky part, since you have to describe with this "looping forever" means rigorously... but you can see how the idea of "making infinitely distinct choices" comes up.

Make A Really QUool EEffect

Undefined behavior is considered worse than crashing, which is typically the alternative when reaching "unreachable" codepaths.

Compare these two blocks similar to the article

   switch (ch) {
   case 'a': do_a(); return;
   case 'd': do_d(); return;
   // ch is guaranteed to be 'a' or 'd' by previous code.
   default: assert(0);
   }

   switch (ch) {
   case 'a': do_a(); return;
   case 'd': do_d(); return;
   default: std::unreachable();
   }

  if (ch == 'a') { do_a(); }
  else { do_d(); }
  return;

N.b. fixed last code example- thanks afiori.

https://aphyr.com/posts/353-rewriting-the-technical-intervie...

Let me rebuke just one part of the post: the example with Newton's laws. Of course everyone knows that Navier-Stokes is just a very nice approximation. That doesn't mean that mathematical results about Navier-Stokes can't improve or understanding of the physics side. For instance, if we find a solution which blows up, that may lead us to a correction to the model or (less likely) it may be actually physically realizable.

I'm pretty ignorant of browser stuff, but couldn't the attack be even worse than mentioned in the article?

If the newly opened website has full control over the Facebook tab, can't the attacker directly modify the html in the opener to pop up a form asking for a password? This would be stronger because the address would not change.

Could the attacker reach into the Facebook tab and pull any information from it?

It seems like the article is missing a possible situation (which I believe is likely). Suppose -The ticket price is the market price and a raise would drive away a lot of buyers -A small portion of buyers do not do the full ticket research and are happy with $265 tickets.

Their conclusion that "Demand and Supply is out of balance" seems to assume that each buyer is the same and has some probability of buying at a given price. Instead, there are a few people who buy at the bigger price without thinking. The fact that these people exist does not mean that raising the price won't decrease the number of buyers.

You can even make it polygonal. Even more interesting, you can make it non symmetrical, contradicting the article's claim that dice must be symmetrical to be fair.

You make an (n-2)-gon cylinder. This has n faces (the sides, the top, and the bottom). If you make the top and bottom very small, the probability of landing on them is smaller than the probability of landing on the sides. If you make the top and bottom huge (compared to the sides) the probability of landing on a side is smaller. Somewhere in the middle, the probabilities are balanced.

In case anyone else is looking, here's the story source for the last picture: http://www.bbc.com/news/business-28212223

Can't we just appreciate that both points of views are helpful? Instead of thinking that one is better, and saying that therefore, the other sucks.

Both of them explain different things. The "it's just a basis transformation" makes invertibility, additivity, etc. obvious. The spinning function things makes it clear why we want to do it and what it means to say, approximate by only the small frequencies. (The basis thing tells you nothing about why looking at just the low coefficients is helpful, sometimes. A basis doesn't care about ordering)

Except for orientability, I guess.

Ok, I can agree that we shouldn't design notation around their limitations. And I do like what they do in SICM. But, I'm just trying to root for the point up-thread:

It seems tempting to have a single unambiguous notation for mathematics. But In constructing such a language, one will quickly realize that doing mathematics becomes an intensely arduous task.

When talking about math, our notation doesn't have to be precise, and that's ok.

No, I don't dispute that blackboards are useful. Just obsolete :P

The great thing is we're not constraining our notation. As reikonomusha said, the standard notation is easier to read. Other notation is better for programming or certain things, and that's what we use there.

Re: your last paragraph. I only went to the end because I knew that there must have been a good example in the questions. If you want I can give you examples from the middle of a talk.

There are two situations which black/white boards are awesome for. -Giving lectures in subjects which desperately need drawing and writing. Look at what happens at 1:16:27 here https://www.youtube.com/watch?v=BPSEpDq6QYc The speaker can just go draw a picture, in response to a question. I know of no alternative which can do something like that nearly as well.

-Collaborating in subjects which need drawing and writing. I can stand next to someone and talk over a problem. They write some formula on the board. I insert some additional bits and pieces that they missed. We draw pictures.

I'd like to hear what you think replaces black/white boards.

The black vs. whiteboard thing is a whole different issue :)

Isn't the conclusion, "all particles have free will"? So, the arguments conclusion is pretty far from its premise. "All numbers are larger than 5" is very far from "some number is larger than 5".

And, whether or not you call this property "free will" is irrelevant to the proof itself; see the final paragraph.

It depends on what you mean by a math problem. It seems a common idea is: A math problem is a question asking for the set of solutions to some mathematical equation, and only the actual solution is useful.

But I would include all of the following as math problems: -Prove that some equation actually has solutions -Find some bounds on solutions to that equation -How can I compute an approximate solution to that equation? -How good is that approximation going to be? -What's a way to try to separate large data sets into clusters?

With regards to the first four: many equations people are interested in simply have no hope of getting a solution you can write down. Simple example: sqrt(2). A more complicated example: solutions to the Navier-Stokes equations.

I've used save several times when I wanted some information on the web page to be accessible offline. Commonly: ticket information. Or, all the information on some table on Wikipedia.

Your edit is correct. Non-transcendentals are algebraic by definition, and the algebraic numbers are countable.

Yeah, I guess those are the only ones I can think of that are more famous than the golden ratio :)

"Although no other algebraic units are as famous as the golden ratio, they are of central importance to algebra."

Arguably more famous algebraic numbers include: 0, 1, The square root of two, the square root of any integer, i, any integer, the nth root of any integer,...

This reminded me of driving the tanks in the video game Grand Theft Auto, Vice City. If you wanted to go faster, a good option was to turn the turret backwards and fire.

Are there any documented cases?

Could that happen? It's not that hard to know that the doors can be locked, and if so, you pretty much give up on trying to get in to the cockpit to take over the plane.

Thanks, edited.

Let me try to explain Schanuel's conjecture a little bit, if you're lost on words like transcendence degree. This turned out longer than I hoped so I hope it's not just a wall of text.

sqrt(2), π, π/2, π^2, and e are all irrational. Now think about π and π/2: these are very closely related. What I mean is:

π - 2(π/2) = 0

So we can add some combination of the two numbers, and get 0.

Now what about π and e? These feel different. After working with them one gets the feeling that aπ + be will never be zero, if a and b are rational[1]. If that's true, then π and e are called "linearly independent over the rationals". We do not know whether π and e are linearly independent over the rationals. (In particular, we don't know whether e+pi is rational.) It turns out that stuff like that is hard.

Now, there's an even stronger thing than being "linearly independent", and that's being "algebraicly independent". When talking about linear indepence we only thought about multiplying by rationals and adding (linear relations). For example, π and π/2 are NOT algebraicly independent. What about π and π^2? Well: (π)^2 - (π^2) = 0 which is rational. This shows that π and π^2 are also not algebraicly independent. (The equation is obvious, but think about what it means: on the left of the minus, I have π, and I'm raising it to a power of 2. on the right of the minus, I have π^2, and I'm raising it to a power of 1). We do not know whether π and e are mutually transcendental.

So you should think: linear independence of rationals is a weak way of saying "these numbers aren't that related." Algebraic independence means there even less related. You can think of both as ways of proving huge amounts of numbers are irrational. If we can prove that e and pi are algebraicly independent then for example we would know e + 5e^3 + e^7 + pi + 4pi^4 is irrational.

Now, you can understand what this conjecture says. Take z_1, z_2, ..., z_n which are complex numbers. If you want, you can just imagine them to be irrational numbers. Suppose they are linearly independent over the rationals. The conjecture is that: at least n of z_1, z_2, ..., z_n, e^(z_1), e^(z_2), ..., e^(z_n) are algebraicly independent.

[1] And not both 0 ;)

Related: If you know some linear algebra, think about this. sqrt(2) is irrational. Think of 1 and sqrt(2) has "vectors" which you can multiply by scalars, but the only scalars you're allowed to multiply by are rational numbers. In this way you get a "vector space over the rationals" which includes things like 3+sqrt(2), 5, and 7sqrt(2). This vector space is also a field: we can multiply things together, and we get another element of the vector space: (3+sqrt(2))(7sqrt(2)) = 3x7sqrt(2) + 7x2 = 14 + 21sqrt(2).

Because sqrt(2) is algebraic over the rationals, this vector space-field thing is finite. But now do it with pi instead of sqrt(2), and you'll get an infinite dimensional vector space with the vectors 1, pi, pi^2, pi^3.... Elements of this field are things like 4+pi+7pi^3.

I prefer that point of view if you want to understand Euler's identity, and I find John Baez does it even better: http://math.ucr.edu/home/baez/trig.html The Baez article does leave it to the reader to convince himself that exp(i*theta) is good notation for a point on a unit circle.

I have problems with the attitude of the article you linked, though. Especially "Therefore, I’d like to complain to the thousands of people who find Euler’s identity stunning and beautiful." followed by a snide list of reasons why someone might find it beautiful. It's very common when doing math that something amazing is obvious an hour later. I believe that we are better served by reminding ourselves that (a) nobody knows everything, and (b) the basics facts are actually very beautiful.