Another option is to plug in AR/VR glasses, which solves the head tilting problem in trains. There are plenty of options around ~500 EUR/USD nowadays, which might be worth it for people with a daily commute.
I'm a doctoral student working in the direction of PDE's and function spaces. I have some colleagues that are using wavelets for numerics. They typically prove that certain wavelet bases are better suited for numerical approximation of certain types of problems. You can for instance think about if you have a signal mainly composed of square waves. Then it would be rather inefficient to decompose this signal in sines and cosines. For certain types of PDE's under certain type of geometrical restrictions, sometimes you can find a much better wavelet bases than just sines and cosines. My research is rather theoretical (so I don't do any numerics), so wavelets don't play a role for me.
What kind of documents are you typing that you wouldn’t want to learn from your mistakes? If you apply more than a handful of “fixes” at once, you cannot see what changed, right? And you also don’t know if the sentences still make sense? Or am I missing something here?
I could see people using this symbol when proving for example that x = 0. Say they first show x \geq 0 and then x \leq 0. Then they've shown x ⪒ 0, and therefore x = 0. But yeah, a symbol like '⪒' for this purpose is maybe natural to write on a blackboard, maybe not so much in electronic form.
In academia people study stochastic versions of PDEs in order to try to answer regularity and existence questions. Think for example about the famous millennium problem of Navier-Stokes. Sometimes the stochastic viewpoint can even give more results about the non-stochastic setting.
I would argue against the statement that all 'truly high talented people' (whatever that may mean) will leave Europe because the salaries are too low. At least in Academia I know plenty of smart people happily working away in Europe. I suppose it may well be different outside of academia, but not every (smart/talented) person in the world has a high salary as their #1 priority. To me, and to many of my friends/acquaintances, quality of life is also extremely important. I would rather, in the long term, live in a economically and politically stable country with good public (health) services and receive mediocre payment, than to receive high payment but with poor quality of life.
I suppose that this differs at how mathematically mature the student is. For undergrads and high school students it is, as you say, probably a good idea to give a lot of motivation and applied examples for the theory being taught. I suppose this article is more geared towards teaching university students, and then it can well be that examples and applications become more abstract. Of course, introductory topics such as Linear algebra and Calculus would lend themselves very well for a wide range of application examples. On the other hand, in more advanced and abstract university courses the 'realistic problems' should already be reasonably clear for the student when starting the course.
I think maybe you mean "don't send your kids to an _American_ school if you can avoid it". Although I have no personal experience with the American school system, I would say that there are many schools in the world that adhere, to some degree, to your 'how not to teach mathematics'. I went to a public school in Europe and have had plenty of enthusiastic teachers that also tried to deliver the beauty of mathematics.
Well, American tech companies are not the only ones using AI/ML for morally questionable activities. Think for example about China's social crediting system. At least it is good that people are thinking about the effects that AI/ML can have on (the freedom) of peoples lives.
Maybe this can be solved on the browser side of things. I can imagine that a browser could communicate the width of the screen as the width of the window to the webpage, even though the width of the window in reality might be made smaller by the user.