@iwriteonbananas I wouldn't be intimidated, though admittedly I don't know if they eat students.

Shit me neither. I heard people are pretty weird in that country

@iwriteonbananas where are you planing to go?

To the moon

I built my own spaceship

now? :P

Yeah, after dinner

:P

See you then in tomorrow's sky :)

I got my internet working so will respond properly to emails now.

Sounds good.

@Semiclassical I guess you didn't have time to work on that stuff ...

not really, no

and right now there's something i should put together while my students take their final exam

@Semiclassical my students? Are you a professor (or assistant)?

i'm a TA, but it's just 3 TAs + the prof so i tend to lapse into such language

especially since it's just myself and another TA doing the proctoring

I see.

it's a quantum physics final

@Semiclassical Hope your students will survive to the test you are going to give them ... :-)

heh, yes

well, they're taking it right now. but proctoring such an exam doesn't require laser-like attention

6 problems in 3 hours. their previous quizzes were typically 3-4 problems in an hour, so time-wise this is more generous

@Semiclassical Also some (quantum physics) stuff involving sums, series, integrals?

@iwriteonbananas I sent the email. It took my computer 2 seconds to actually send it.

not really. the most complicated integral they run into on this exam is $\int_0^\infty x^n e^{-x}\,dx$, and the prof gives them the formula

@Semiclassical I agree then you're kind.

well, i'm not the one writing it

and it tests other stuff. not everything is sums and integrals :)

How can show that if a nxn matrix is diagonalizable and has n repeated eigenvalues then it is diagonal?

@Semiclassical Sure, sure, I know. :-)

there is some summation stuff going on under the surface (e.g. how one obtains the Fermi-Dirac distribution) but it's not really needed today

I see.

there's also some integration stuff on the last problem, but not involving anything sophisticated

@Semiclassical They can use calculators?

Nope.

@Semiclassical Why?

$f(x)\in [1,5)\cup [5,6]$ then can i write it as $f(x)\in [1,6]$?

Because the prof said no :p

@Semiclassical OK ...

Oh never mind, I over looked something simple.

main reason one would want a calculator is to speed up the arithmetic, and that can all be done by hand

the numbers are set up so as to not be terrible.

@Semiclassical The idea is to save some time and put the efforts on the core of the problems.

(or not to have the whole problem ruined for a simple mistake)

@Semiclassical Then it's fine.

well, myself i grade as much on their process as their final result.

so if it's something that was obviously a math mistake, i tend to not penalize too much

what i more look for in such cases is whether they can tell in their final result whether they've made an error

That's good.

getting an answer that would imply something like "The car has a length of two seconds," on the other hand...

@Semiclassical Have you found very silly math mistakes in these tests? :-)

some. people make mistakes under pressure

Yeah

a lot of the time i look for whether their answer has the right units, and if they recognize that

I see.

Between two people who give the same wrong answer, I much prefer the one who can tell that their answer is wrong :)

Fair enough.

the main thing I'm not excited about re: this final is that once it finishes (in an hour and a half) i'll have until monday afternoon, more or less, to grade ~150 sheets

not the worst schedule, but not a walk in the park.

@Semiclassical a lot of papers, indeed.

yeah

Anyways, while I'm killing time, here's a question stolen from the main site

Why is $$\int_0^{\pi/2} x\frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}\,dx \approx \frac{77}{333}?$$

i've no idea if there's a nice answer for that, to be clear, but it's an interesting coincidence

@Semiclassical I could create billions of such questions.

fair enough.

@Semiclassical W|A makes tons of such approximations for the integrals you try.

sure.

@Semiclassical What is interesting is to look at that integral without $x$ in front. :-)

well, if there's no $x$ in front, it just integrates to zero since the fraction is antisymmetric under $x\mapsto \pi/2-x$

@Semiclassical Yeap

but for that reason it's a tad silly to write it as $x$. one can just as well replace it to $x-\pi/4$ and get an integrand which is manifestly symmetric under the transformation.

Seems so.

@Semiclassical Have you ever confronted with exhaustion from too much work on math, physics? I experience such days here lately. Hope to finally go to sleep earlier.

Say you wanna go to sleep at 04.00, but can't sleep for another 2, 3 hours. That's when the exhaustion is pronounced.

mostly i experience the exhaustion of putting stuff off. a much more foolish type of exhaustion.

I see.

@Semiclassical Going jogging helps (a lot) but these days are ugly here, rainy days, also some nasty wind.

not a lot of rain here, so i should be more active. did a nice walk this morning, though

lol, I was just taking a look here huffingtonpost.com/leslie-davenport/…

Exorcism of Emotional Vampires

:-)))

the water bit is a good piece of advice

Zen Brain Drain Remedy

i'm not nearly attentive enough to that during the day, and i think i pay a price for that once i get home

@Semiclassical I consume more hot chocolate and milk than water. Just water? Not really.

sure.

This is far better if any interested - webmd.com/balance/guide/get-energy-back

5. Get to bed early.

:-)

lol

simple, but essential. alas, it's a piece of advice i'm terrible at sticking to

Me too. :-)

staying up late is one of those bad decisions that's just so easy to keep making

Indeed.

truly

Looking a bit back I'm so amuzed by those messages with 20 stars (or so) a bit against my saying of reaching the performance of Ramanujan. There was a guy (I forgot his name) saying that after looking over my work my stuff don't seem like that.

In 20 min I'll go to sleep.

@Semiclassical it's not about me, but I would like to hear many people saying they wanna reach the performance of the great figures in mathematics (and do something concrete about that).

Does anyone know how to solve inequality using wavy-curve method?

@Anubhav.K You asked at some point how to show that every oriented covering space of a non-orientable manifold factors through the oriented double cover, yes?

@Semiclassical I say to be nice (with other fellows, of course) in mathematics, but not humble, and why humble? Humble in a church maybe.

I wish you all aim right at the top performance.

Anyway.

Here is an explicit way of seeing this. Let $M$ be the non-oriented manifold, $S$ our oriented covering space of $M$, and $\tilde M$ the oriented double cover. Consider $\tilde M$ as a set to be pairs $(x,\mu_x)$ where $\mu_x$ is an orientation of $T_x M$. Then if $p: S \to M$ is the covering map, define $\tilde p: S \to \tilde M$ by $\tilde p(x) = (p(x),df(\mu_x))$.

Here $dp(\mu_x)$ means the orientation induced on $T_{p(x)}M$ by the orientation $\mu_x$ on $x$ and the isomorphism of vector spaces $dp_x$.

Verify this is a covering map and whatnot; this is straightforward.

@Semiclassical I guess that my thinking is not comfortable at all (for some), but I don't push anyone in such a direction.

I just say I couldn't do math without aiming at the very top (for the simple reason at the very top there is a lot of fun).

I'm out before writing a novel.

@MikeMiller yes, very long time ago, and after that I probably did a solution

why now? do you have any better solution?

boo

Hey all, is there an industry standard approach to find all roots of higher order polynomials having real coefficients? I'm familiar with the Eigenvalue method and have tested several others but there are many different ways. My polynomials are not very high degree, perhaps 7 or 8 at most

Mathematica and other applications appear to use VAS but even after reading about most of these methods, I'm not certain of the advantages of each algorithm (depending on polynomial degree and time complexity etc.)

I also found this discussion which briefly compares Jenkins-Traub to Laguerre and Eigenvalue but there was no obvious approach