With a newly discovered mathematical tool, researchers are hoping to gain unprecedented insight into the structure of complex knots.

Saw another few of these today and thought I'd collect related links somewhere. Comment with any that have been useful to you!

• T. Tao on career and writing
• S. Billey's advice collection
• I. Pak's blog
• C. Aten's 'one pager' for all stages in math
• University of michigan grad advice (note: use the dropdown menu at the top; links in main content are broken atm)
• ICERM professional development materials
• AMS profiles of women mathematicians
• I. Musson's LGBT resources links
• Academic Stack exchange on postdocs

not math specific, but grad/research related:

• thesis whisperer
• care and maintenance of your advisor
• (comp-sci inclined) the huge advice collection maintained by Tao Xie and Yuan Xie (I surely duplicate some of their items)

I am so fascinated by the regions where the number of lines to cover the primes levels out, what Brady calls "golden lines". Part 2 is here https://www.youtube.com/watch?v=u-_8wX4cECo

I remember back in 2007 feeling that my TI-89 was almost unfair. Pretty sure I used it for the ACT, and you can solve something like 60% of problems with no effort if you just learn how to graph. And now you can use Desmos on the ACT.

Nowadays, most students in higher level math are equipped with calculators that can just solve things. You don’t need to learn how to convert fractions to decimals, or work with a percentage, or even do algebra (80% of working with calc 1 students can often be “yeah here’s how to put it into solver”).

I’m very torn on this. On one hand, I think that doing it by hand is the only way to develop on understanding of what it all means. There’s patterns to what a base system mean that you start to “get” once you’ve done enough borrows and carries. Small and consistent practice in the small skills adds resonance to the major skills you are building to.

On the other, there are things like dysgraphia that just there’s no reason to not work around. Some people can’t hold onto times tables. There are amazing ways to do multiplication that are slow but work for people (draw a rectangle - 3x4 best for demo purposes - have person count squares, bam, you have now outdone their 2nd grade math teacher.) Why bar someone who can’t memorize things but can understand why things work from further study of math?

I do sorta wish that the SAT kept its no calculator section though. It would be interesting to make a bunch of adults take the modern tests and compare their scores to they got 20+ years ago…

cross-posted from: https://lemmy.world/post/45146291

Many of my clients are in a class that doesn’t teach them, and it’s sad because they are my favorite for chain rule and u-sub.

Last night an old idea came back to me, an idea about a function where all the derivatives start from zero and then grow smoothly. I thought it would be impossible, but then I found some interesting stuff on Wikipedia. So, I learned to use SymPy and wasted a lot of time with it. Here's a report of my (non-)findings.

(UPDATE: I did some numerical differentiation, which showed that h(x) does have negative derivatives. See details in this comment. A disappointment, although perhaps not a surprising one. It doesn't however, necessarily mean the goal is impossible.)

So, if anyone knows whether such a function exists and what it looks like, please tell me.

Sharing this as it's one of my favourite things on the web.

cross-posted from: https://lemmy.bestiver.se/post/852619

Comments

cross-posted from: https://lemmy.bestiver.se/post/847259

Comments

Research claiming that following the taught algorithm is more common for women (and those who agree with 'I want to please the teacher' more broadly), while trying to find shortcuts is more common from men. The latter seems correlated with performance on high stress math tests.

cross-posted from: https://lemmy.bestiver.se/post/774752

Comments

Statistical Tests to determine whether a bit-stream can be considered "random"

If we were to flip a coin 10 times, we would expect to see roughly 5 heads and 5 tails. Let's assign 00 to heads and 11 to tails. Therefore, we might see a sequence like this...

Backpropagation is one of the most important concepts in neural networks, however, it is challenging for learners to understand its concept because it is the most notation heavy part

cross-posted from: https://sh.itjust.works/post/48633930

Normally, we use a place-value system. This uses exponentials and multiplication.

1234
^^^^
||||
|||└ 4 * 10^0 = 4
||└ 3 * 10^1 = 30
|└ 2 * 10^2 = 200
1 * 10^3 = 1000

1000 + 200 + 30 + 4 = 1234

More generally, let d be the value of the digit, and n be the digit's position. So the value of the digit is d * 10^n^ if you're using base 10; or d * B^n^ where B is the base.

1234
^^^^
||||
|||└ d = 4, n = 0
||└ d = 3, n = 1
|└ d = 2, n = 2
d = 1, n = 3

---

What I came up with was a base system that was polynomial, and a system that was purely exponential, no multiplication.

In the polynomial system, each digit is d^n^. We will start n at 1.

polynomial:
1234
^^^^
||||
|||└ 4^1 = 4 in Place-Value Decimal (PVD)
||└ 3^2 = 9 PVD
|└ 2^3 = 8 PVD
1^4 = 1 PVD

1234 poly = 1 + 8 + 9 + 4 PVD = 22 PVD

This runs into some weird stuff, for example:

• Small digits in high positions can have a lower magnitude than large digits in low positions
• 1 in any place will always equal 1
• Numbers with differing digits being equal!

202 poly = 31 poly
PVD: 2^3 + 2^1 = 3^2 + 1^1
8 + 2 = 9 + 1 = 10

---

In the purely exponential system, each digit is n^d^. This is a bit more similar to place value, and it is kind of like a mixed-base system.

1234
^^^^
||||
|||└ 1^4 = 1
||└ 2^3 = 8
|└ 3^2 = 9
4^1 = 4

1234 exp = 4 + 8 + 9 + 1 PVD = 22 PVD

However it still runs into some of the same problems as the polynomial one.

• Small digits in high positions can have a lower magnitude than large digits in low positions (especially if the digit is 1)
• The digit in the ones place will always equal 1
• Numbers with differing digits can still be equal

200 exp = 31 exp
PVD: 3^2 = 2^3 + 1^1
9 = 8 + 1

---

So there you have it. Is it useful? Probably not. Is it interesting? Of course!

Crossposted from https://piefed.social/c/science/p/1406456/first-shape-found-that-cant-pass-through-itself