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1. ### Best 'Advanced Calculus' Book

quite a late reply I write this answer on 2013, so it is quite late but I still hope this be helpful. I know a book that can definitely satisfy you. It is Advanced Calculus by Fitzpatrick. I've recently read Munkres's and Duistermaat's. The former one is very very good. It is...
2. ### About a proof related with connected sum and reference help for algebraic topology

Thanks mathwonk. I will definitely keep in mind your insight and comment when I study this in a more rigorous setting. And thanks Bacle2. Indeed the quotient topology section in Munkres seems quite good; it does not use that fluffy method in the proof.
3. ### About a proof related with connected sum and reference help for algebraic topology

Thanks for the reply mathwonk! I actually thought I wouldn't get any answer, anyway thanks. Okay as for the textbooks that you'v recommneded, I would look them up. But I'm very new to this topic. So obviously I haven't read them at all. Actually I'm not even really studying 'real' algebraic...
4. ### About a proof related with connected sum and reference help for algebraic topology

I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows: Q1) Munkres introduces 'labelling scheme'...
5. ### A measure related with Reimann integral

Thanks for your answer, and as you have told me I should assume that f(x) \geq 0 a.e.. And for the last part you mentioned, I think Dominated Convergence Theorem would suffice, woudn't it? I used it in my solution..
6. ### A measure related with Reimann integral

Homework Statement Let ( \mathbb{R}^k , \mathcal{A} , m_{k} ) be a Lebesgue measurable space, i.e., m_{k}=m is a Lebesgue measure. Let f: \mathbb{R^k} \to \mathbb{R} be a m-integrable function. Define a function \mu : \mathcal{A} \to [0,\infty] by $$\mu(A) := \int_{A} f(x) dx$$ with A \in...
7. ### Error function (defined on the whole complex plane) is entire

Ah.. frankly I'm not sure of how to expand by Taylor series... Is it by using Cauchy's Theorem? Could you give me any reference textbook where I can look up the theorem for this Taylor series expansion?
8. ### Error function (defined on the whole complex plane) is entire

Homework Statement The wiki page says that error function \mbox{erf}(z) = \int_{0}^{z} e^{-t^{2}} dt is entire. But I cannot find anywhere its proof. Could you give me some stcratch proof of this? Homework Equations The Attempt at a Solution I've tried to use Fundamental Theorem...
9. ### Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

Ah... MY BAD! sorry.. what was I thinking... Let me clarify once more: Take x_{n} \in (-n,1-n) . Then \Gamma (x_{n}) \to 0 as n \to \infty . I think I have an idea to solve it without using Gauss's Formula. After I try, I will put on the thread. Anyway thanks for reminding me.
10. ### Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

Ah.. I know what you mean. Maybe I need to modify my problem first. I know it has poles on non-positive integers. But excluding poles, it seems the absolute value of the gamma function tends to zero as x \to - \infty . (http://en.wikipedia.org/wiki/File:Complex_gamma_function_abs.png) May I...
11. ### Gamma Function Limits: Proving \Gamma(x) \to 0 as x \to -\infty

Homework Statement The absolute value of the gamma function \Gamma (x) that is defined on the negative real axis tends to zero as x \to - \infty . Right? But how do I prove it? Homework Equations The Attempt at a Solution I've tried to use Gauss's Formula...
12. ### Proof Error Q. Conway's Functions of One Complex Variable VI

Okay.. I think by your notion it seems very plausible to conclude that \lim_{n\to\infty}\frac{f(z_{0}+h_{n})-f(z_{0})}{h_{n}}=\lim_{h\to0}\frac{f(z_{0}+h)-f(z_{0})}{h} though I'm not using Thm 4.2 in Baby Rudin But what about...
13. ### Proof Error Q. Conway's Functions of One Complex Variable VI

Ah of course yes, baby Rudin does not use sequences in his proof (in his latest edition). And Rudin's proof is very clear whereas Conway's seems not valid. As for the lemma, doesn't that lemma state that 'limit exists iff for an "arbitrary" sequence the sequential limit exists'...
14. ### Proof Error Q. Conway's Functions of One Complex Variable VI

I ask this question only to those who read or have this book: If you have Baby Rudin, it would be even better. On the page 34 of the text Conway's Functions of One Complex Variable Vol 1, it proves the Chain Rule but it seems the proof is not valid: It uses sequences to show the limit is...
15. ### Baby Rudin Problem 2.7

Oh yeah. I think I know what you mean. i_{0} depends on epsilon. Thanks.
16. ### Baby Rudin Problem 2.7

Homework Statement I've proved that if B = \bigcup_{i=1}^{\infty} A_{i} then \overline{B} = \bigcup_{i=1}^{\infty} \overline{A_{i}} but it should not be right. So could you find errors on my reasoning? Homework Equations The Attempt at a Solution Observe x \in \overline{B} iff for...
17. ### Is this correct Baire's Theorem?

Yes the statement that you statetd, i.e., the intersection of open dense subsets is also dense, is equivalent to mine. But is the theorem correct then? (I've learned this from lectures and the lecturer sometimes does not specify everything like a set should not be empty or etc. So I worry...
18. ### Is this correct Baire's Theorem?

Homework Statement Baire's Theorem Let X be a complete metric space. Suppose E \subseteq X and E = \bigcup_{n \in \mathbb{N}} F_{n} where F_{n} \subseteq X is closed in X . If all X \backslash F_{n} are dense then X \backslash E is dense. Homework Equations The Attempt at...
19. ### Factorizing X^4 + 1

Okay. Btw, I've just finished this cousre and am preaparing the upcoming exam for this one. (It is just an undergraduate advanced courese, but personally I think the course is quite fluffy. I don't know... my knowledge is fluffy now, still not sure what I've learned from this course.. the...
20. ### Factorizing X^4 + 1

Okay I think I see what you are trying to say. And indeed I'm taking an algebra course [ring, field, gloais ...]. But the thing is in this course I've never learned a techniqe that is by knowing some roots to generate a polynomial the complete list of the roots of which are the roots known, nor...
21. ### Factorizing X^4 + 1

I've found this post https://www.physicsforums.com/showthread.php?t=371741 but still does not show the 'way' of factorizing.
22. ### Factorizing X^4 + 1

Homework Statement I've seen a factorization of X^4 + 1 like x^{4} + 1 = (X^{2} + \sqrt{2}X + 1)(X^{2} - \sqrt{2}X + 1) Is there an intuitive step-by-step procedure to find this factorization? Homework Equations The Attempt at a Solution NO IDEA..
23. ### About a root in the topic of automorphism and fixed field. HELP

About a root in the topic of automorphism and fixed field. HELP! Homework Statement Let m be a positive integer. Let \xi = \exp (2pi/m) . Then \xi is a primitive m-th root of unity. (I.e., \xi is a solution of \Phi_{m}(X):=(X^m - 1)/(X-1) .) If \phi \in...
24. ### Infimum calculation bizarre for me

I know Bolzano-Weierstrass thm in R^k. But A is a subspace of a general metric space, so I don't think I can use it... It just reminds me now that actually in a general space it may be possible that such a point a_0 may not exist, and for such a_0 to exist, I think it is necessary...
25. ### Infimum calculation bizarre for me

Oh no worries; apology well taken. And yes, that part is wrong I see. Okay, I think the sequence is a very good idea. So if I solve it right now, letting x be arbitrary in X observe for every n in N (the set of natural numbers) there exists a number a(n) in \{ d(x,a) \mid a \in A \} such...
26. ### Infimum calculation bizarre for me

Hey, sorry but I'm not sure what you are talking abt. The thing that if a is in the closure of A then d(a,A)=0 is right. But I'm trying to prove for arbitrary x in X there exists a_0 in the closure of A such that d(x,a_0) <= d(x,A). But I'm stuck...
27. ### Infimum calculation bizarre for me

Thank you very much, I will do that. But if I may aks you, what exactly do you refer to by the last step? you mean the last equality, I guess?
28. ### Infimum calculation bizarre for me

Homework Statement Could you please check the following calculation is right? Let X be a metric space, and A its nonempty subset. Define \inf_{a \in A} d(x,a) = d(x,A) for any x in X We have the following facts (don't have to check this) If a is in the closure of A then d(a,A)=0...
29. ### About the number of irreducible elements in UFD ring

What I can say right now is that for the integer ring, this infinity case is not the case. But I'm not sure if for any infinite ring, the number of irreducible elements is also infinite (possibly regardless of the cardinarlity).
30. ### About the number of irreducible elements in UFD ring

Okay thanks guys. Btw DonAntonio I've thought about what you said. At least among the structures that I know of, the infinite product cannot be intuitively conceived, as you said, except some trivial things like 1 = 11111111... and 0 = 0000000000000... or for a unit u = uuu^-1uu^-1uu^-1...
31. ### Nonzero nonunit non-product-of-irreducibles is reducible?

Thanks a lot!
32. ### About the number of irreducible elements in UFD ring

My argument for that goes like this: Let R' be the set of all irreducible elements in R. Form a product of all of them. Call this product x. Then x is nonzero nonunit. Thus x is represented as c_1...c_n where c_i is irreducible. Then as R is a UFD, the irreducible element in the formation of...
33. ### About the number of irreducible elements in UFD ring

When chracterizing the definition of unique factorization domain ring, the Hungerford's text, for example, states that UFD1 any nonzero nonunit element x is written as x=c_1. . .c_n. Does this mean any nonzero nonunit element is always written as a product of finitely many irreducible...
34. ### Nonzero nonunit non-product-of-irreducibles is reducible?

Homework Statement Claim Let R be an integral domain. Then a nonzero nonunit element in R that is not a product of irreducible elements is reducible. Is this claim true? Homework Equations The Attempt at a Solution By definition, a product of irreducibles is reducible because...
35. ### N distinct points in R^n-1 or higher one can be separated in an equal distance(>0)?

Oh yeah... Thank you indeed!
36. ### Existence of some special kind of isometry in R^n-1

Homework Statement Is there a theorem that states the following? Let P= \{ P_{1}, . . . , P_{n} \} be the set of n distinct points in \mathbb{R}^{n-1} and P'= \{ P'_{1}, . . . , P'_{n} \} also a set of points in \mathbb{R}^{n-1}. If for all i,j |P_{i} - P_{j}|=|P'_{i} - P'_{j}| then there...
37. ### N distinct points in R^n-1 or higher one can be separated in an equal distance(>0)?

Is there a theorem that states that n distinct points in R^n-1 or higher one can be separated in an equal distance as the distance is greater than 0? We know that 4 distinct points in R^2 cannot be positioned in an equal distance>0 but in R^3 it is possible as a pyramid shape. If there is...
38. ### Is boundary of A open?

Thanks HallsofIvy! In this case, if I calculuate the boundary of A I get empty set because A is both open and closed. I've found another example that if A=Q the set of rational numbers, then the boundary of A is R the set of real numbers. It seems that if the boundary is open, which...
39. ### Is boundary of A open?

um... well, I've actullay solved this problem. Anin in my solution, I've proved that \partial A is not open. Could u please check my solution? \partial A is closed: \overline{\partial A}=\overline{\overline{A}\cap\overline{X\backslash A}}\subseteq\overline{\overline{A}} \cap \overline{...
40. ### Is boundary of A open?

Homework Statement Let X be a metric space, and A its nonempty proper subset. Then is \partial A not open? If it is, how do I prove it? Could you give me just some hints, not the whole solution? Homework Equations The Attempt at a Solution I cannot even start..
41. ### Gcd(a,b) unique in Euclidean domain?

gcd(a,b) unique in Euclidean domain?? Homework Statement In Hungerford's Algebra on page 142, the problem 13 describes Euclidean algorithm on a Euclidean domain R to find THE greatest common divisor of a,b in R. My question is that does this THE mean THE UNIUQE? I've heard from my lecturer in...
42. ### How to avoid rounding off in binary representation

I think it's quite enough. Thanks guys.
43. ### How to avoid rounding off in binary representation

I'm not quite sure what you meant. The image of the mapping is still [0,1]. Isn't it?
44. ### How to avoid rounding off in binary representation

Well, yes. Just give me a brief idea of it, please. And also.. as an extra question, well... do you have another idea of making bijection between two sets? This binary representation thing is not the best idea on making bijection between P(N) and [0,1].. right?
45. ### How to avoid rounding off in binary representation

Homework Statement Is the fundtion defined f:P(\mathbb{N}) \to [0,1] by f(X) = 0.a_{1} a_{2} \dots in binary representation where a_{k}=1 if k\in X and otherwise 0 one-to-one? (*note: N does not have 0) If not, can you change bit so that the changed funtion becomes one-to-one...
46. ### The class indexed by real numbers is a set?

You know, the concept of indexing in my mind (in my intuition) is kind of a countable process. But then now the index set is a continuum. So I thought it might not be possible; I mean this kind of indexing might not be possible by ZFC.
47. ### The class indexed by real numbers is a set?

Let \mathcal{S} = \{S_{i}:i \in \mathbb{R} \} where S_{i} is a set. Then \mathcal{S} is a set? Or, can this notation make sense in some way?
48. ### The definition of 'reducible' in Hungerford's Algebra text

Okay, I will take note of that.
49. ### The definition of 'reducible' in Hungerford's Algebra text

He starts using the term 'reducible', as it came out of nowhere, from the page 162 of the text. I know, roughly, what kind of thing he mean by this 'reducible' obejct. (That is that an element is factored into two elements that are not units.) And this should not be a problem if this term is...