Indeed. Of course, I did not design Desmos; I just made it to work with bitwise logic so that I can see what it outputs because it does not support boolean algebra natively. :D
Interesting. As is not too incredibly surprising, each graduation in the graph corresponds to the values in the range defined by $a = \lfloor\log_2(t)\rfloor ; 2^{a} \leq t \lt 2^{a+1}$ it would seem.
And then that is just multiplied by a power of two constant. The graph remains the same overall shape and the y value halves as $m$ increases.
following a standard math curriculum, do people eventually get good enough at real analysis that, say, thrown any theorem of the single variable material from Baby Rudin, they're able to prove it without any other reference than required definitions?
shin: i dunno if you'd get that only from a 'standard math curriculum', at least at the undergraduate level. even good analysis students tend not to be that proficient the first time around. some of those exercises are quite difficult.
i'd expect that most graduate students could handle it, though. at least those with some interest in analysis. after a second or third trip through that material it becomes very clear.
same with your average calculus book, frankly. the level of proficiency you need to get a perfect grade in calculus is somewhat lower than 'do any exercise in the textbook.' but not unattainable with further experience.
@Semiclassical Something interesting I noticed is that you also get a Sierpinski triangle if you do $x \land x^2$ which, you probably weren't here when I mentioned it, but I noticed that the high bits of $x^2$ appeared related to $x - \neg x$ and multiplied by a power of two, but this means we can square things somehow using even simpler bit hacks and faster than O(M(n)).
others may have a different view. i think the ideal for an undergraduate textbook is that it be challenging on a first read-through, but manageable or even easy on a second or third encounter, perhaps after other courses.
parts of rudin are too challenging to be ideal in this sense.
rudin tends not to tell you where an argument comes from. i don't like 'chatty' textbooks or textbooks with piles of redundant examples, but one can assist the reader far more than rudin does without doing that.
when i first met rudin, i think two people in a class of about 30 went on to graduate school in math. most people were CS oriented or K12 education oriented and could not get past the expository style.
i think at some schools the math major is so small, and the student body sufficiently self-selecting, that it can be assumed a larger fraction of math majors will go to grad school in math.
when i was at berkeley, the computer science major was very selective due to high demand. you had to apply to get in. sometimes people would fail to get in and major in math simply because they'd taken a lot of the prerequisites already, and could still use programming in applied math classes.
sometimes the reason they did not get into the computer science major was low math grades. i knew several people who basically had the following: "oh crap, i got some bad grades in math. now i guess i have to major in math."
makes for odd encounters with rudin, to say the least.
@leslietownes At most schools with small programs, they don’t get into much in the way of top grad programs. Williams, on the other hand, with star teachers and star students has (relative to school size) a huge major.
i was thinking of places like yale and princeton. are there many people who major in math there who aren't mini-mathematicians?
i don't think of those places as powerhouses of applied math or engineering or anything else that someone might take math for. but i know next to nothing about their undergrad curricula.
when i graduated there were about 2x applied math majors as math majors. at time people interested in applications tended to just go major in the other thing.
even then they were the clear majority of math majors.
while i student i had no conception of how big some of the majors were. when i went to my wife's graduation in sociology, i could not stop thinking, "how many people are going to be walking across the stage? when is this over? i'm hungry." while hundreds of people got diplomas.
some schools have very strong actuarial specific tracks. uiowa did when i taught there. berkeley did not.
@shintuku I am not a mathematician, so take with a grain of salt. I imagine a better goal would be to be able to prove new things as needed rather than proof-on-demand for well known material. Clearly you need familiarity with the techniques used, but spend energy going forward, not dealing with history.
i did it from the first part of Gamelin Introduction to topology, but can't speak about the book for topology on other topological spaces that aren't metric. the parts i did were nice
In an exam I took recently, they asked this question -Let $C$ be a subset of $R$ endowed with subspace topology. If every continuous real valued function on $C$ is bounded, then $C$ is compact.
And I skipped it the moment I saw subspace topology
I discussed this with Leslie and he told me that subspace topology here is not to be confused and that consider $C$ as any subset of R with usual metric. And I think he's right. Leslie gave a hint to solve the problem and I solved it using that.
I'd like to report one issue of LaTeX in chat bookmark: I'm using opera on macbook pro now. I noted that when I remove the bookmark bar and click on the bookmark, then the bookmark doesn't work. It only works when clicked from bookmark bar.
that is, I have to keep the bookmark bar and this occupies screen
I was thinking to try my hand on it (actually it is for a purpose) looking at assembly makes me feel like it's something hard to understand I have no idea
I don't think I learn assembly for any other purpose; I use js and c++ most of the time I don't have too much experience in low level languages, but I am learning
i think he means that he chooses 10 cards from the deck, leaving them in the deck, and then someone else chooses 5 cards from the deck, leaving them in the deck
@shintuku I see, thanks. I always pronounced Dirichlet as "dirishlet" (without t) but then I heard someone pronouncing it as "diriklet" that's why I asked
#1 for all $\varepsilon$ and for all $n$, $x<\delta \implies |f(x) - f(\frac{x}{2^n})| < 2\varepsilon x$ #2 $\lim \limits_{n \to \infty}|f(x) - f(\frac{x}{2^n})| = |f(x)|$
intuitively, I interpret this to mean: if I pick a big enough $n$, and suppose $x<\delta$, I get that $|f(x)| < 2\varepsilon x$. How do I make this rigorous?
I ask because for an epsilon-delta proof I can't just take the limit as $n$ goes to infinity in the middle of the proof, all I have is specific choices of $n$
states 1 and 3 are transient. 2 and 4 form a single (recurrent) communicating class and a bipartite graph -- draw this and write out the transition matrix. Hence $p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$ — user867530911 hours ago
I just quickly wanted to ask, if I have f(x)=1/x for x unequal to 0, is it still appropriate to say that x=0 is an infinite discontinuity while also concluding that f is continuous over its domain?
I learnt the concept of the period of state i for DTMC and I'm wondering
why we have $p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$?
if $d(i)=p$ then $p_{ii}^{(pn)}=1$ and $p_{ii}^{(pn+1)}=0$?
I learnt the concept of the period of state i for DTMC and I'm wondering
why we have $p_{ii}^{(2n)}=1$ and $p_{ii}^{(2n+1)}=0$?
if $d(i)=p$ then $p_{ii}^{(pn)}=1$ and $p_{ii}^{(pn+1)}=0$?
