@Chris'ssis cute one! :) .. btw I got a nice way around the alternating Au-Yeung series $\displaystyle \sum\limits_{n=1}^{\infty} (-1)^{n-1}\frac{H_n^2}{n^2}$ :)
I will be overjoyed if I see a solution that just uses series manipulation :D (I couldn't get rid of le integrals :( .. )
so the book I'm looking at right now has about 25 problems at the end of each chapter, plus exercises in the chapters as you go, and in the preface it says that it'd be a bit much to expect the student to go through all the problems
the question I have is, if I'm reading on my own, how do I decide which problems to do?
I read the first two chapters a while ago, and the first one had no actual work associated with it, so I'm coming back maybe a week and a half later and trying to do the chapter 2 problems and I'm finding them a little harder than I might like
but this is probably due to a number of factors and I feel like I do learn a lot when I actually do solve them
I've so far been able to do the first 5 and I'm on the 6th so if you think it's worth it then I suppose it just confirms my fear that I'll need to spend a couple nights getting through all of these problems
I am not familiar with the book, but it might mean that in a typical class it might be unreasonable to have students do every problem for homework, say over a semester or year, not necessary that the problems are too difficult for a grad student to do.
@SamuelYusim: A few of these look somewhat tedious and unnecessary, but in the large, these are all standard things worth doing. Some important examples get developed etc.
I think they all should be doable with some effort. The real worry is less that you can't do it, and more that you'll get sick of it. I think one was like "Here's six properties of maps, find examples of maps with and without each" or something like that
that's a lot of things to check, I would get sick of it pretty quick
but the things, IIRC, were reasonable to find examples of, and it helps to build your collection of examples
I really am getting the feeling from his exposition that point-set topology is just a somewhat uninteresting tool that's really useful for solving very interesting problems
it's just very very long and he explains the most trivial details of proofs; on the one hand this is helpful when you're working on something you're having a tough time grappling with, and on the other hand for the chapters where you've really got it down it's so long and unnecessary
it's been a while so i couldn't give any examples but i left with the feeling that you could cut the size by, say, 30% without substantially changing the content or feel of the book
The table of contents look interesting. I really don't know enough topology to even guess what the purpose of "Topological Manifolds" would be. To get started classifying "nice" spaces?
Samuel, you might consider trying to find a copy of Berger's Geometry, if you have an algebraic/group theory background at all. I haven't gotten far, but it's definitely some grade A geometry
So, if a region is revolved around the x axis. The disks (calculus, integrating) perpendicular to the axis of revolution and with respect to x. But shell method shells are also called perpendicular to the axis of revolution?
and shell integration then done with respect to y
no wait, the shells are called parallel to the axis of revolution sigh
ok i read chapter 4. other than paracompactness, which is kinda technical and not really necessary for a basic level of understanding (it's important so that you can do an important construction called partitions of unity; that's pretty much literally it), i guess there's nothing i wouldn't do, so i'm not sure how i would change it
i'm not sure he uses paracompactness anywhere else in the book.
if i were teaching a course based on the book, i would not cover paracompactness
same thing with normal spaces (this is certain to get more flak from others). they're important to know eventually. they're not important to know now. he never uses the main theorem (Urysohn's lemma) anywhere else in the book.
I'm having difficulty figuring out why if two finite sets have the same cardinality then there is a bijection between them; is this provable? Is it just a matter of definition (the definition of equinumerosity)?
I haven't been using a rigorous definition; I'm just looking at {1,2,3} and {star, heart, diamond} and trying to figure out how I would rigorously prove that there is a bijection between them
The basic idea is to carry around a "canonical" set of a given size. Usually for a set of size $n$, it's convenient to use $[n] = \{1, 2, \ldots, n\}$. Then, if your set has size $n$, it's in bijection with $[n]$.
But then to generalize, for arbitrary n \in \mathbb{N}, if a set has cardinality n then there is a bijection between it and the set {1,2,...n}; then I would need to use pjs36's idea?
Well just after my initial posting I discovered/thought that by noting time is independent of space and the same time is used for all points it means I can just remove the subscript there. Which cleaned up the lot.
Hmm this is probably turning out to a matter of asking the question is answering it
@TobiasKildetoft what background do you require for real and complex analysis?? Is calculus,number theory,a bit of linear algebra,abstract algebra enough??
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God knows...See i dont pay attention to what he says i keep on doing other maths stuff..But when i heard the person sitting beside me saying "isnt that obvious that N is uncountable" i asked my prof can he revise the definition of countability and after that what he said shocked me....@TobiasKildetoft
And worst part about him is he just writes the formulas how you reach there is none of our business he says....
He says derivative of sin(x) is cos(x) and that is a fundamental result .... I asked him how did you reach here and he is like its just a result So i said i think you woke up in the morning and found cos(x) sitting on your dining table saying"I am the derivative of sin(x)!!!!!!"
@Rememberme Also, remember that if proofs of the statements are not part of the curriculum, then the rest of the class might not appreciate you wanting to see them all the time
This is just an institution which makes you ready for an engineering entrance exam whose scores are important for most of the mathematical institutions in India@TobiasKildetoft
I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How do you represent a group geometrically in a space?" Is there any way of representing it?
@Rememberme 'engineering entrance exam whose scores are important for most of the mathematical institutions in India' .. I don't follow, what has mathematical institutions have to do with scores of an engineering entrance exam?
@Rememberme Also, any group can be represented as a group of linear transformations on some vector space (but it might be infinite dimensional, so not necessarily as matrices)
@TobiasKildetoft So this implies i can represent linear transformation geometrically which would indirectly mean that i am representing groups geometrically
I mean, if you take the identity matrix and delete columns from it, you get a nonsquare matrix that deletes the corresponding entry of the vectors it's applied to. that's what's going on here.
@Rememberme When I study complex representations, I fix the group and study all the possible representations of that group. The "interesting" question are then no longer about the group but about those representations as abstract objects
your comment is very muddled. in what sense do graphs "contain complex elements"? does that make a graph into a vector space? the word "represent" has a very precise meaning, it is not just some sloppy informal word we're using.
Can we possibly compute the following integral in terms of known constants?
$$\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$$
Some progress was already done here http://integralsandseries.prophpbb.com/topic279.html but still we have a hypergeometric function. What's your thoughts on it?
UPDA...
@r9m then here is another question I created $$\int_0^{\pi/4} \frac{\cos (2 x) }{1+\sin ^2(2 x)}\log (\cos (x)) \, dx$$
@r9m maybe it's not that hard as you expected but it gives you a very important lesson in case you don't know how to do it. I'm seriously thinking to add it to my book too.
I cant solve questions in it......First it just writes down the formulas how you get to them no idea second it defies logic how you reach conclusions it@Chris'ssis
@r9m I'm preparing a new question for American Monthly, perhaps one of the most beautiful questions they received this year. :-) It might not look friendly, but it can be finished without pen and paper.
@PaulPlummer interestingly, no, it doesn't, and that's what's terrifying us. actually, these earthquakes are generating on Nepal for about a month, but I guess this is the greatest earthquake we have ever seen.
Nepal has been completely destroyed with millions of people dead. I wonder if we'll get hit by bigger ones in coming days.
Can we possibly compute the following integral in terms of known constants?
$$\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$$
Some progress was already done here http://integralsandseries.prophpbb.com/topic279.html but still we have a hypergeometric function. What's your thoughts on it?
UPDA...
I am pretty sure you need $X$ to be a CW-complex, @iwriteonbananas, but I can't think of a counterexample right now. Does this work with the Hawaiian earring? I haven't thought about it.
If I just answer the first two, since those are the only ones I have a "real" opinion about I get Joseph H. Silverman's The Arithmetic of Elliptic Curves.
@Balarka: Your chapter counting reminds me of the way they number floors in Europe. The first floor is the first floor above ground level. Always confuses Americans.
@TedShifrin Indeed I do, and the meeting went swimmingly last week. He said my work is fantastic, but it wasn't his specialty. However, he did give me the names of two people who DO specialize in analysis
@TedShifrin I don't know, it looks like a pretty strong result. I don't believe it should be true if my manifold doesn't have a differentiable structure.
No, the point is that you have to wrestle with what parallel ought to mean in general. How do you move a tangent vector from one point to another (along a particular path) so that a resident of the surface never sees it turn as it moves along (i.e., all turning occurs normal to the surface).
I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How do you represent a group geometrically in a space?" Is there any way of representing it?
Can we possibly compute the following integral in terms of known constants?
$$\int_0^1 \int_0^1 \frac{dx\,dy}{1-xy(1-x)(1-y)}$$
Some progress was already done here http://integralsandseries.prophpbb.com/topic279.html but still we have a hypergeometric function. What's your thoughts on it?
UPDA...
