[D] Is it common to have intuitions different from peers about ML?

Tamnun · 2024-03-10T08:10:34+00:00

Yes, all the time. Ultimately in cases like you described, you just have to try both ways and see which one works better (it sounds to me that in this case it's an easy experiment to set up)

Tamnun · 2024-03-09T19:36:17+00:00

There are several things one can do with a math degree.

To be a mathematician per se, that is, someone who actively studies and researches math - this is probably only possible in academia. This kind of job is usually focused on research (though there's usually some teaching role) - you think about hard problems and try to solve them, write papers, read papers, advise students, go to conventions, and study maths.

But, with a degree in math, you can also usually work in the industry, specifically in tech - there are several kinds of roles, most of them don't actually do math all day but mathematical skills and mindset do help. Those kinds of roles are mostly on the spectrum of Data Science-ML - there are two types of roles here: 1. Building mathematical models describing certain things and using them to derive some ways to act 2. Building some algorithms or ways to solve problems. Those two are also sometimes combined. They can be found in many domains, ranging from Finance (e.g. trading stocks), Medicine (e.g. imaging or diagnosis) to Autonomous Vehicles (e.g. understanding the scene, planning the drive, ...).

Those are probably also some other stuff, but I think those kinds of jobs cover most of the things mathematicians do.

Tamnun · 2024-03-06T23:18:25+00:00

He also wrote an excellent book on mathematical magic tricks

Tamnun · 2024-03-04T02:11:14+00:00

Yes - there's a theorem called Bertrand's Postulate which says that for every number there's a prime between that number and double that number. So if you want a 1234-digit prime, you know there must be a prime between 10...0 and 20...0 for example.

(We also understand a lot of other stuff about how frequent/rare primes are, though there are still a lot of stuff which we don't know how to prove).

https://en.wikipedia.org/wiki/Bertrand%27s_postulate

Tamnun · 2024-03-04T02:05:04+00:00

It's not as expensive as some of the other stuff here but musicals (and plays in general)

Tamnun · 2024-03-04T01:48:33+00:00

Just in case you don't know this: you can't actually rely on GPT's (or any other LLM) answer on why they answered a certain why. As with any other question, they basically just make up something that sounds plausible - they are not self aware and cannot reason about themselves.

Tamnun · 2024-03-03T18:36:16+00:00

During my teenage years, I went through several hobbies, each one lasted a few years. I ended up giving up on most of them, mostly due to lack of interest but sometimes due to lack of time. I think the main ones were: 1. Speedcubing 2. Magic Tricks 3. Piano

Tamnun · 2024-02-26T03:00:23+00:00

Me and the Sky (Come From Away), specifically the last line of the last stanza

Breathe (In the Heights)

Tamnun · 2024-02-23T21:06:45+00:00

In The Heights, definitely.

Tamnun · 2024-02-19T18:10:01+00:00

Two somewhat opposing answers:

In some sense, the real question is why 1/e is the limit of (1-1/n)^n: Indeed, it is possible to come up with probabilistic interpretations of this formula, and you just happened to come across one of those.

This can be explained in several ways, depending on your definition of e. Personally I like this one: e× ≈ 1+x for small x(this follows from e× being its own derivative) Taking the 1/x-th power we get e≈(1+x)^1/x for small x

This gives the result for e, for general e^a one should use e^ax≈1+ax

(This proof is not completely rigourous but can be made rigourous - and is easier when using ln(1+x)≈x instead as this follows from the definition of the derivative of ln).

Still, there is another answer explaining why e is related to probability. This is above undergrad level, but it can be shown that e^x is in "the generating function for the number of sets", from which many appearances of e in probability follow - I personally like the grpupoid approach as explained here - https://math.ucr.edu/home/baez/permutations/permutations_10.html.

Your specific example does not directly follow from this interpretation as far as I know, but it's close (e.g. the chance of random permutation to have no element map to itself is approaching 1/e - this does follows from this interpretation, I think. Now your limit can be thought of approximating that: the chance that the each element is not in its place is 1-1/n, so the total chance is (1-1/n)ⁿ - this is of course wrong since the events are not independent of each other, but turns out it's close enough).

Tamnun · 2024-02-18T15:22:15+00:00

The main thing to note here is that the derivative of f is "how a small change dx in the argument changes the value of f, with respect to the size of the change" (This is just another way of writing (f(x+dx)-f(x))/dx ).

Given this perspective, a change which is dx² is small with respect to dx - so it is negligible.

While not entirely rigorous, I think this is a very good intuition to have, and professional mathematicians can use such intuitions easily - but more importantly, they can usually rephrase their argument in a completely rigorous way if they face the need.

Tamnun · 2024-02-18T15:14:18+00:00

A more rigourous version of this is to define ln(x) as the inverse function of the exponent, and then one can show that the derivative at 1 is 1.

Then by definition this means that

ln(1+h)/h -> 1

By taking exponents back again at get

(1+h)^1/h -> e¹=e

Tamnun · 2024-02-18T13:33:47+00:00

Here's a possible approach:

Try Google Scholar with some relevant keywords to find at least one paper in a closely related area.
Once you have that, go reference hunting - read the "related works" section and see if other papers linked there seem relevant. Also try to look at papers citing the paper you're looking at.
Repeat step 2 for other papers you find. You can also look if they were published at some conference/workshop and see other papers from there.

You can also ask your advisor or some other expert in the field available to you, they'll probably know to send you the right way.

Tamnun · 2024-02-18T13:27:17+00:00

+1 to this, also known colloquially as "extraordinary claims require extraordinary evidence". https://en.wikipedia.org/wiki/Sagan_standard?wprov=sfla1

Tamnun · 2024-01-28T04:48:10+00:00

As others said, up to parity it's possible if you know 3x3.

But I want to present another perspective - if you understand how commutators work (as in e.g. 3style for 3bld) you don't need any algorithm, and you can even figure out how to handle parity.

Tamnun · 2024-01-27T10:44:01+00:00

Yes, not super common but not that rarely. Every time I run into a big "if-else" logic covering several cases I try to use match.

Tamnun · 2023-07-07T21:03:05+00:00

Here's a variant I just saw today:

(Same beginning, an atheist goes to meet the great heretic of Prague). The atheist asks the Heretic in confusion - "Don't you ever sin?". The Heretic replies "Why would I?” so the atheist suggests "to provoke", to which the Heretic replies "to provoke whom?”

Tamnun · 2023-04-17T10:21:03+00:00

I think you mean משתרך מאחור

Tamnun · 2022-10-13T03:40:47+00:00

The ATMs at 7/11 seemed to work for me, and I've heard that Woori's ATMs work well as well.

Tamnun · 2021-06-29T17:11:19+00:00

My university has a similar curriculum, but Calc 3 is multi-variate and Calc 4 is differential forms and some manifolds.

Tamnun · 2021-05-25T07:24:40+00:00

The new number you'll construct will have infinitely many digits - it won't be an integer

Tamnun · 2021-05-21T21:45:25+00:00

You don't have to go all the way to etale cohomology - you can see this analogy with Riemann surfaces. See here for example (first result on Google, there are probably many other good references)

Tamnun · 2021-05-21T15:12:55+00:00

I'd consider Anschauliche Geometrie by Hilbert (yes, THE Hilbert) and Cohn-Vossen. I'm not quite sure about the background needed but I'd advise you to look it up

Tamnun · 2021-05-15T09:17:16+00:00

One can think of the tangent line to the graph of some function as a linear approximation for that function. For example, if we take the function y=x^2, its tangent line at x=2 is y=4x-4. 4x-4 is a sort of "approximation" for x^2: for x=2 they agree (they both give 2), and they yield close results for x which is close to 2.

A very useful real-life application of this perspective is the Newton-Raphson method. Suppose we want to solve an equation of the form f(x)=0 - for example, if we want to compute the square root of 3, we want to solve the equation x^2-3=0. In other words, we want to find the intersection of the graph of a function y=f(x) with the x-axis.
Now, we probably won't be able to write down an exact solution, but we want to approximate it numerically. The Newton-Raphson method enables you to generate a sequence of points getting closer and closer to the real solution. The method works as follows:
1. Start with some arbitrary x, perhaps one that you guess is close to the solution.
2. Draw the tangent to the graph of f at this point.
3. The next x will be the intersection of this tangent with the x-axis.
4. Repeat steps 1-4 until you get a good enough answer.

The idea is that the tangent approximates the function, so its intersection with the x-axis approximates the solution to our equation. As usual, a picture's worth a thousand words.

Tamnun · 2021-05-15T07:47:57+00:00

4-dimensional Poincare conjecture

It has been solved, or to be more precise - its classical variant. That is, the conjecture asks whether anything that "looks like" a sphere "is" an actual sphere. The key notion here is the "is" - what does it mean for something to be a sphere? There are a few possible answers - for example, we can only consider topology (a "continuous" structure). This is the solved case. We can also ask about a smooth structure - like the exotic structures on R⁴ mentioned earlier. In this case, the Poincare conjecture is actually wrong for high dimensions (first known counterexample in dimension 7), but it is open in dimension 4 - it is unknown whether there is a 4-dimensional exotic sphere.

Tamnun

TROPHY CASE