@TedShifrin i have taught a few classes, but mostly at the university level. but luckily i never encountered the modern mindset. generally folks were more or less engaged.
there was a berkeley hs lad on here recently, way above my mathematical level, i put him in contact with some local group that deals with such precociousness.
@TedShifrin I am reviewing linear algebra so the plane is solution of system that looks like a flat surface on which a straight line joining any two points on it would wholly lie from 3rd dimension but we don't deduce it using normal vector,dot product and more advance concepts. Also i apologize for not using logical quantifier which kinda confused you all.
@user863565 You have to come to terms with what is expected in the problem. I have explained that you can show that one of the vectors can be written as a combination of the other two, and those two either span a plane or they are collinear.
@copper Over my career, I taught a handful of students who were truly impressive and I knew would far exceed my accomplishments. One has been tenured and in his second round of being department head at Stanford.
The Stanford guy and I are close friends, and I was visiting him and his family annually until the fires hit two years ago. Then COVID. I'm hoping to make a trek to the Bay Area soon, although I'm worried my neck and back won't make the 8-hour drive.
say $f\colon \mathbb{R}^m \to \mathbb{R}^n$ is differentiable at $U \subseteq \mathbb{R}^n$. should I think of $f''$ as a function $U \to \mathrm{Hom}(\mathbb{R}^m, \mathrm{Hom}(\mathbb{R}^m, \mathbb{R}^n))$?
@LucasHenrique much of my earlier life was in design with continuous parameter optimisation. if one could even approximate the Hessian it could be used to great effect.
so @Ted, he told us that a tensor is simply an element of $M\otimes N$. but everywhere that geometry comes up, a tensor is something like an element of $\underbrace{V\otimes\cdots\otimes V}_\text{p times} \otimes \underbrace{V^*\otimes \cdots \otimes V^*}_\text{q times}$. why?
the difference is that in engineering each individual compnent of a tensor has its own name and physical significance. to a mathematician they are all just elements.
i myself get tensor and tensor as every day passes by.
a tensor is either a normal abstract algebra thing, or it's like twenty million different numbers that transform according to some law. no in between.
i was talking about von neumann tensor products this morning. quickest route to stuff where it seems like you might need a matrix of matrices. often very short and simple proofs, you just need to spend months understanding the formalism first.
if you're ever reviewing or editing a paper in mathematical physics, and they do stuff with tensors of operators, you need to find out right away if there is a coordinate free transformation on the underlying space that implements their formulas, and if there's not, you should begin to get very nervous.
one time i wrote a referee report which was "this paper seems very good except it assumes that this operator-valued function is continuous and i have no idea if it is or not, which is necessary to lemmas 2 and 3. [several reasons why the obvious approaches to proving continuity didn't work]." it ended up being published in a physics journal and not a math journal.
it's just very easy to write down infinite series that look 'nice enough.' it was a case of that. it was nice, i just don't know if it was nice enough.
what bothers me about it is that half of the time their formulas turn out to be true. then the egg is really on our face.
i remember thinking when reviewing the paper, did the author just never take a class on operator theory, or are they the next nobel prize winner? and i still don't know.
also the most annoying technical questions when i was trying to get my thesis approved so i could graduate were from the physicist on my committee. really, really good questions.
i wanted to say "you understand that the outside committee member is just supposed to sign this thing, right?" but his questions were too good. he even caught a small mistake that i and my advisor completely missed.
i don't think he'd even spent a lot of time with my thesis. he hadn't read any of the definitions.
in mid career we had (sold) a tool that checked if two designs (FSMs) were the same. they were used to avoid expensive simulation. i found a bug where it assumed some unconnected input was in fact connected to zero.
the chip was already on the way to the fab. i called the application engineer and asked, as nonchalantly as i could, hey eric, were there umm any floating inputs? he replied yeah, i meant to ask you about that but i tied it to ground (zero) just in case, why do you ask? oh, nothing really just wondering. (it could have been an unmitigated disaster otherwise.) that is what i lose sleep over.
we did a case once with the guy who invented PCR. really eccentric person, made a wreck i think partially by the fact that about a billion people made a billion dollars off of his work and he didn't.
there was too long of a gap between when it was scalable/fully realized and when it was invented, to benefit the inventor. pharma has a longstanding version of this problem.
if a drug takes longer than the patent term for anybody to prove that it's better than other stuff, it will not be developed, even if it's better than other stuff. the industry can't wait and you need the IP protection.
it's crazy to me that PCR was invented when it was invented. people didn't even have commercial thermocyclers. you'd move your stuff from one batch of water at one temperature, manually, to another batch of water. and you'd have to add in new polymerase every cycle.
there are some cool newer methods of amplification that are isothermal and not only isothermal but isothermal around room temperature and the reagents don't need special storage. could be very cool for the developing world.
no it's rare where something isn't a combination of 20 million other things. PCR just popped out of one person's head. very cool. i am often skeptical about invention in my line of work, but that one counts.
