I have often argued with Jasper about this, but one of the main ways I chose textbooks was to look at the exercises. Books with crummy exercises don't impress me.
heya Demonark.
I take great pride in a lot of the exercises I've written (in all my books).
Is there an equation that relates time complexity, space complexity, and entropy of the output of a function? It seems to me that there should be a relatively intuitive relationship between the three, something like $E = f(T + S)$ where $f$ is a linear function or something like that. The main argument being that any function can be written as a hash map that takes its input to its output. This hashmap could be compressed based on its entropy.
I have a hunch that the only way to decrease the space an algorithm uses more than that is to increase the time complexity, as if the hashmap is moved onto a time axis instead of a space one. Is this crazy?
@TedShifrin I don't remember if I told you, but Hempel proved a theorem in his book which includes the s-cobordism theorem, once you know the Poincare conjecture.
I eventually did fail it due to attendance, but I did like my combinatorics class many years ago. There were a decent number of problems that expanded my thinking quite a lot.
And then 3 days before the final we got a list of practice problems which our professor meant to assign but forgot about, on spectral measures, Sobolev spaces, and Hille-Yosida
Demonark: One of the most amazing lecturers in the UGA department is terrible about assigning homeworks. When he used my notes for diff geo, he was even too lazy to assign problems enough.
@Ted He proves that a compact 3-manifold (with no 2-sphere boundary) whose pi_1 contains a surface subgroup of finite index is, in fact, an interval bundle over that surface... Up to possible summands of exotic 3-spheres.
@Erico might do so if only because I wanna revisit Sobolev spaces, though there are other things I wanna do in functional analysis which it doesn't really cover so chances are I'll have a different one as my primary source
@Ted I didn't think the stuff in your notes was too bad, I just felt at the time that I couldn't really figure out how to get a handle on the problems that required geometric intuition
So in the psets there were the problems that seemed fun and which I didn't know how to set up, and then there were problems which were "Compute the Christoffel symbol" which I could do but wasn't especially inspired by
I mean I definitely did ask you guys for certain things, I just felt at some point that I was overdoing things. But in any event I've been a bit reinspired by Neves' brief schtick about Yamabe problem the other day, so at some point I'll try it again, perhaps at a more leisurely pace
But we got slowed down so much, sometimes by lectures that got interrupted to hell, other times by lectures where the lecturer didn't seem to learn the stuff as well as they should have/didn't take it too seriously
The interruptions on average were more frequent, and sometimes a bit overdone, but again the real problem was that a few people clearly just didn't actually try to prepare
To do it right, preparing should mean giving the lecture ahead to someone like Eric or the prof who understand things and help you put intuition into the lectures.
i think my year functioned the bst of the three bc we had a smaller group and a prof in the room at all times but still i wouldve cut the content in half
@TedShifrin I told the students i personally supervised to do this last time through and the ones who listened and gave me their lectures beforehand were the best ones imo
The lecture that Schlag hijacked was one I didn't wanna give since I was hoping to do a diffgeo lecture soon afterwards, only did it because people weren't comfortable with lecturing on manifoldsy content, so I signed up on Wednesday afternoon for the Friday lecture and probably spent nearly 20 hours going through and making sure I had things down
One of my colleagues taught G&P once and had the students present the lectures. They got through 1/4 of what I covered and the students only learned the stuff they lectured on, nothing else. I think this is a disaster.
@Ted yeah I felt afterwards that people hadn't absorbed much. Now, I think I was going way too fast that time because I was sorta determined to get to the proof of Van der Waeerden within the hour
@TedShifrin I basically think that a lot of the integral discussion could fit into two weeks. Once you have Green's theorem and do a calculation you get the residue theorem and Cauchy's various theorems.
Yeah my complex analysis course was basically "here's what complex numbers are", "here are the C-R equations", and then just RT for the rest of the course.
@MikeM: But the undergrad complex analysis at UGA had mostly weak students in it who knew almost no analysis and minimal vector calc. The grad course is a different story.
@Daminark What you'll see over time is that it's easy to tell when someone has prepared well, but this does not necessarily correlate to either how much material they get to or how many proofs are presented.
@TedShifrin we have a prereq for complex that includes uniform convergence and in my class first year we STILL didnt use it till 7th week and went through the basics of uniform convergence
But while you had asked maybe one or two notation questions throughout my Skype lecture, I was getting a question every 30 seconds in mine since folk were less familiar with stuff, and as a result I kinda just tripled the speed. I do think I got a bit better at pacing during my probability lecture though
Yeah that was quite something. I feel like the book we used just assumed more stuff, also doing hyperbolic dynamics was a bit of a mistake since people weren't too familiar with Riemannian Geo
I was at the Notre Dame conference for the last two lectures so I didn't see what happened but I heard the one on Wednesday that week was a disaster, the guy apparently didn't know what he was talking about and he was making jokes more than teaching, and nobody signed up for Friday
@Daminark i think the student lecturing thing worked a lot better for the TA cohort for our individual topics bc we were more consistently serious about learning material and had real ass grown ups in the room
Also, have you tried practicing your talks in front of other students (who heckle you with questions)? It's very very difficult to estimate ahead of time how long you'll take, and more importantly how slow you should go. Hint: usually, a lot slower than you think.
By now I have a good feel for timing issues. But only with much much practice and a good dose of failure.
@Jacksoja: I was away from the computer when you uttered your deep words of wisdom. It's sort of interesting — I think I started teaching rather early in life.
@TedShifrin Mike turned me on to a paper of Rafe's w a proof of uniformization through analysis of a curvature PDE that ive been reading for a few days and it is v v v v cool
@TedShifrin old generation profs and teachers are way better than nowdays imo, I saw your lectures and you make this simpler for the audience, not trying to show them that you know better. also you explain pretty well at the right level so anyone can keep up .
Sure---$\Bbb R$ is not compact, because I can cover it by $(n - 1, n + 1)$ where $n$ ranges over the integers, and there's no finite subcover---excluding any interval would exclude the integer $n$.
@TedShifrin Haha, I actually think Daminark is more qualified than I am to make claims about complexity theory (since he is taking a class about it). I'm this awkward blend of software engineering and math. However, cs.stackexchange chat room is probably the best place for a question like that.
honestly, I would look at a comparison between shannon entropy and non-shannon entropy, and try seeing if there is a way to abuse it so there is no relation at all.
please someone confirm this: derivative of function $f:\Bbb R\to\Bbb R$ can't have jump or removable discontinuities, but partial, or directional derivative of $g:\Bbb R^n\to \Bbb R$ can have jump or removable discontinuities, right?
ok, thanks @Astyx, i got my answer, by googling some stuff: which is, if $g:\Bbb R^n\to \Bbb R$ is differentiable everywhere, that is total derivative exists, then no directional derivative of $g$ can have jump discontinuity.
@Astyx, If you help me with this change of basis question, i will be obliged: Let $B_1=\{(1,2),(2,-1)\}$ and $B_2=\{(1,0),(0,1)\}$ be ordered bases of $R^2$. If $T:R^2\to R^2$ is a linear transformation such that $[T]_{B_1,B_2}$, the matrix of $T$ with respect to $B_1$ and $B_2$ is $\begin{pmatrix}4&3\\ 2&-4\end{pmatrix}$ then , $T(5,5)$ is equal to ....
Let mu and nu be sigma finite measures on (X,sigma) such that mu >> nu. from Radon-Nikodym theorem we know that there exists extended integrable function f such that d nu = int f dmu. my question is: assuming mu and nu are both positive measure does the function f have to be also non-negative function?
Could someone help me with the following definition: A bipartite graph G with vertex partition L, R where |L| <= |R| is degree-constrained if deg(l) >= deg(r) for every "l" belongs to L and "r" to R.
