Just something that came out on my feed and looks like a good new resource

Is there a way to make the function

f(w,t)=sin(wt)/w

defined at w = 0?

I want to achieve better numerical stability, and make it continuous everywhere.

Observation: Using limw->0 f(w,t)

I know that at w = 0:

f(0,t) = t

Is it possible to improve this function? I would be really glad to know and why, how. Thank you in advance!

I really do not want to code in smth like: if w < 0.001 { x = t; }

Background notes: My equation also has constant v coefficient, but I removed it for simplicity. f(w,t)=v*sin(wt)/w

If I derived this correctly this should be x component of a motion of a car that rotates with rate w. It goes with const speed and const steer.

These are consts: v - speed, l - base length(dist from rear to front shaft), a - steering angle.

w(a) is defined idk if correctly. w(a) = (vsin(2a))/(2*l) I just logically added proportions, projections, imagining a rear wheel powered car. Put some values I am somewhat confident in, such as: w(0) = 0 w(π/2) = 0

Wikipedia page for the problem. I hadn't thought about this conjecture before hearing the solution, but otoh it seems both interesting and hard.

cross-posted from: https://piefed.world/c/tech/p/1171000/mathematicians-issue-warning-as-ai-rapidly-gains-ground-new-declaration-argues-the-techn

I’ve tutored it from middle school level up to helping with dissertation work. The way it is taught varies so dramatically from class to class and field to field. There’s so many subtle and weird factors in interpretation that it’s honestly the most stressful thing to tutor.

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.