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Post by Admin on Feb 25, 2017 12:18:17 GMT
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Gary
GaryVasco
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Post by Gary on Feb 28, 2017 23:10:01 GMT
Vasco, Regarding p. 2, sentence 5 It might be helpful to distinguish "the white dot" from the white dot at A, which is more prominent and competes for the starting place. In the rotation of $\Phi$, you described two stages. There appear to be three: (1) increase up to A, (2) decrease up to - Pi (approximate), (3) increase to the starting point at 2Pi. Gary |
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Post by Admin on Feb 28, 2017 23:31:03 GMT
Gary
I suggest you reread the second part of your post. I think there is a typo but I can't work out where.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 1, 2017 1:23:15 GMT
Gary I suggest you reread the second part of your post. I think there is a typo but I can't work out where. Vasco I meant to say "you describe two stages", not three. Sorry. |
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Post by Admin on Mar 1, 2017 7:49:03 GMT
Gary
I agree and I have edited my document to make things clearer I hope.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 16, 2017 4:24:18 GMT
Vasco, It seemed to me that some of the difficulties in understanding the chapter and exercises 2 and 3 have to do with the notation in Section II, Hopf's Degree Theorem, much of it on pp. 342, 343. The linked document contains a list of relevant notation with definitions. There are times when a list is more useful than a discussion. nh.ch7.notes.notation.pdf (76.95 KB) Gary |
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Post by Admin on Mar 16, 2017 6:38:08 GMT
Vasco, It seemed to me that some of the difficulties in understanding the chapter and exercises 2 and 3 have to do with the notation in Section II, Hopf's Degree Theorem, much of it on pp. 342, 343. The linked document contains a list of relevant notation with definitions. There are times when a list is more useful than a discussion. View AttachmentGary Gary I initially got bogged down in the detail of section II and eventually realised that a lot of it (especially subsection 3 The Explanation) is only there to enable us to prove Hopf's Theorem (easy to write but not easy to say). It is essentially about real functions of a single variable, and topological ideas of loops rather than complex analysis, which seems incidental to this section. The whole proof of the theorem could be done without using complex analysis it seems to me and is only needed in later chapters of the book - especially the second-last section in chapter 8. This is why Needham uses "L" in its various forms all the way through this section - "L" for Loops. Section III and later sections go back to complex analysis again and that is why Needham uses $\Gamma$ for loops in this section because they are now sitting firmly in the complex plane again and points not on the loops are meaningful points which they weren't in section II. In this sense section II is a little bit like a fish out of water in this chapter, and a lot of the detail seems not to be relevant to the rest of the chapter, although the result - the Theorem of Hopf (easier to say) is taken as read in many places (eg subsection 2 of section IV on pages 347-8). It's just that all that detail in subsection 2 of section II can be forgotten - just remember Herr Hopf and his important theorem! Your list is a very useful summary when reading section II. Just a few observations: $\nu(a)$ is also the topological multiplicity of $a$. Your 9th definition: the transformation transforms a point $z$ on $C$ to a point $\widehat{w}$ on $\widehat{L}$. This transformation is the composition of transformations from $C$ to $L$ to $\widehat{L}$. Your second to last definition: Not just to a zero point, but also any other point (see page 345 paragraph 4). Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 16, 2017 21:16:22 GMT
Vasco,
Thank you for the comments. Will incorporate them later.
Gary
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 16, 2017 23:58:00 GMT
Vasco,
I believe you are referring to $\mathcal{L}_s(z) = w + s(\widehat{w} - w)$. It does show z as an argument, but z does not appear in the RHS, so I can only see it as a typo that should have been $\mathcal{L}_s(w)$, which gives us the radial deformation of L (where a point is $w$). Then we have the circumferential or angular deformation on p. 344.
Gary
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Post by Admin on Mar 17, 2017 5:24:37 GMT
Vasco, I believe you are referring to $\mathcal{L}_s(z) = w + s(\widehat{w} - w)$. It does show z as an argument, but z does not appear in the RHS, so I can only see it as a typo that should have been $\mathcal{L}_s(w)$, which gives us the radial deformation of L (where a point is $w$). Then we have the circumferential or angular deformation on p. 344. Gary Gary Yes, that's the one, but it's perfectly OK the way it's written because on page 342 we have $w=\mathcal{L}(e^{i\theta})=\mathcal{L}(z)$ and so we can write $\mathcal{L}_s(z)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=w+s(\widehat{w}-w)$, $z$ is mapped to $\widehat{w}$ via $w$. Needham writes just after (3) on page 343: "As $s$ varies from $0$ to $1$, $\mathcal{L}_s(C)$ changes from $L$ to $\widehat{L}$." Note the $C$ (ie $z$). Another way of looking at it is to introduce $\Lambda_s(w)=w+s(\widehat{w}-w)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=\mathcal{L}_s(z)$ In other words $\mathcal{L}_s$ can be thought of as mapping $z$ or $w$. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 17, 2017 22:45:19 GMT
Vasco, I believe you are referring to $\mathcal{L}_s(z) = w + s(\widehat{w} - w)$. It does show z as an argument, but z does not appear in the RHS, so I can only see it as a typo that should have been $\mathcal{L}_s(w)$, which gives us the radial deformation of L (where a point is $w$). Then we have the circumferential or angular deformation on p. 344. Gary Gary Yes, that's the one, but it's perfectly OK the way it's written because on page 342 we have $w=\mathcal{L}(e^{i\theta})=\mathcal{L}(z)$ and so we can write $\mathcal{L}_s(z)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=w+s(\widehat{w}-w)$, $z$ is mapped to $\widehat{w}$ via $w$. Needham writes just after (3) on page 343: "As $s$ varies from $0$ to $1$, $\mathcal{L}_s(C)$ changes from $L$ to $\widehat{L}$." Note the $C$ (ie $z$). Another way of looking at it is to introduce $\Lambda_s(w)=w+s(\widehat{w}-w)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=\mathcal{L}_s(z)$ In other words $\mathcal{L}_s$ can be thought of as mapping $z$ or $w$. Vasco Vasco, What you say is all true, but it seems somewhat indirect. This section seems to be starting with a loop L and showing how we can analyze it by reducing it to a standard. A first step is to shrink it down to unit circle size. I don't see the point to writing the equation as a transformation of the unit circle into L (or z to w) followed by the shrinkage of L to C size (or w to z size). Gary |
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Post by Admin on Mar 17, 2017 22:56:31 GMT
Gary Yes, that's the one, but it's perfectly OK the way it's written because on page 342 we have $w=\mathcal{L}(e^{i\theta})=\mathcal{L}(z)$ and so we can write $\mathcal{L}_s(z)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=w+s(\widehat{w}-w)$, $z$ is mapped to $\widehat{w}$ via $w$. Needham writes just after (3) on page 343: "As $s$ varies from $0$ to $1$, $\mathcal{L}_s(C)$ changes from $L$ to $\widehat{L}$." Note the $C$ (ie $z$). Another way of looking at it is to introduce $\Lambda_s(w)=w+s(\widehat{w}-w)=\mathcal{L}(z)+s(\widehat{w}-\mathcal{L}(z))=\mathcal{L}_s(z)$ In other words $\mathcal{L}_s$ can be thought of as mapping $z$ or $w$. Vasco Vasco, What you say is all true, but it seems somewhat indirect. This section seems to be starting with a loop L and showing how we can analyze it by reducing it to a standard. A first step is to shrink it down to unit circle size. I don't see the point to writing the equation as a transformation of the unit circle into L (or z to w) followed by the shrinkage of L to C size (or w to z size). Gary Gary It's all explained in the last paragraph of subsection 3 of section II on page 344: it enables us to demonstrate (2) on page 341- Hopf. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Mar 17, 2017 23:18:57 GMT
Vasco, What you say is all true, but it seems somewhat indirect. This section seems to be starting with a loop L and showing how we can analyze it by reducing it to a standard. A first step is to shrink it down to unit circle size. I don't see the point to writing the equation as a transformation of the unit circle into L (or z to w) followed by the shrinkage of L to C size (or w to z size). Gary Gary It's all explained in the last paragraph of subsection 3 of section II on page 344: it enables us to demonstrate (2) on page 341- Hopf. Vasco Vasco, I reread it and I still don't see why $\mathcal{L}_s(z)$ appearing in the argument at that particular point contributes to demonstration of (2), but I can tell that you have a better feeling for it, so I will accept it and await for enlightenment to catch up with me. Gary |
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