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Post by Admin on Oct 12, 2017 14:53:06 GMT
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Gary
GaryVasco
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Post by Gary on Oct 22, 2017 5:25:15 GMT
Vasco, I have read your solution to 21. I notice that you did not explicitly commit to the idea that we can now speak of fractional winding numbers. I wonder what you think about that and whether you think my example is correct. This is a rewrite. nh.ch7.ex21.pdf (142.61 KB) (revised Oct 23, 2017) Gary |
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Post by Admin on Oct 22, 2017 6:08:19 GMT
Gary
No, in my opinion we can only speak of fractional multiplicities. A winding number by definition is an integer.
Explanation:
In the first part of exercise 21 we are still thinking of $p$-points which do not lie on $\Gamma$ and we redefine winding number in terms of net rotation $\mathcal{R}$, and rewrite (17) on page 364 as $\mathcal{R}(\Gamma)=2\pi(N-M)$ or as $\displaystyle\frac{\mathcal{R}(\Gamma)}{2\pi}=N-M$.
Notice that these do not include an explicit reference to winding number.
If we apply these formulae to specific examples of $p$-points and poles which do not lie on $\Gamma$, then we find that, as expected, they are verified.
So we can now write (17) with our new formula in terms of net rotation.
The exercise then asks us to make sense of the statement that the GAP remains valid even if there are some poles and $p$-points on $\Gamma$, provided that we count these points with half their multiplicities. This is what my answer does. Note that I use the new formula, which we have shown to work for the 'normal' situation, and show that it works also for the 'new' situation, with the proviso about multiplicities given in the exercise.
So our new formulation of (17) now works in both cases, where $p$-points or poles are on or off $\Gamma$ with the proviso about multiplicities given in the exercise.
As I pointed out at the beginning, the new formulation avoids references to winding numbers and sticks to net rotations.
To summarise, this new way of writing (17) is a generalisation, since it works for poles and $p$-points inside and on $\Gamma$.
Your rewrite
I have looked at your rewrite and I think that what I have written above applies to it. The exercise requires us to use the new formulation of (17) which does not reference winding numbers. The wording of (17) changes to include the proviso about multiplicities, and the formula at the end makes no reference to winding numbers.
My answer
I will edit the last paragraph of my answer slightly to make it clearer that I am using the new generalised version of (17).
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 22, 2017 23:13:30 GMT
Gary No, in my opinion we can only speak of fractional multiplicities. A winding number by definition is an integer. Explanation:In the first part of exercise 21 we are still thinking of $p$-points which do not lie on $\Gamma$ and we redefine winding number in terms of net rotation $\mathcal{R}$, and rewrite (17) on page 364 as $\mathcal{R}(\Gamma)=2\pi(N-M)$ or as $\displaystyle\frac{\mathcal{R}(\Gamma)}{2\pi}=N-M$. Notice that these do not include an explicit reference to winding number. If we apply these formulae to specific examples of $p$-points and poles which do not lie on $\Gamma$, then we find that, as expected, they are verified. So we can now write (17) with our new formula in terms of net rotation. The exercise then asks us to make sense of the statement that the GAP remains valid even if there are some poles and $p$-points on $\Gamma$, provided that we count these points with half their multiplicities. This is what my answer does. Note that I use the new formula, which we have shown to work for the 'normal' situation, and show that it works also for the 'new' situation, with the proviso about multiplicities given in the exercise. So our new formulation of (17) now works in both cases, where $p$-points or poles are on or off $\Gamma$ with the proviso about multiplicities given in the exercise. As I pointed out at the beginning, the new formulation avoids references to winding numbers and sticks to net rotations. To summarise, this new way of writing (17) is a generalisation, since it works for poles and $p$-points inside and on $\Gamma$. Your rewriteI have looked at your rewrite and I think that what I have written above applies to it. The exercise requires us to use the new formulation of (17) which does not reference winding numbers. The wording of (17) changes to include the proviso about multiplicities, and the formula at the end makes no reference to winding numbers. My answerI will edit the last paragraph of my answer slightly to make it clearer that I am using the new generalised version of (17). Vasco Vasco, I see now that we are instructed to define $\nu[f(\Gamma), p]$ in terms of net rotation of $f(\Gamma)$ round an unspecified point p. Questions remain. Suppose as in my example $f(z) = z (z + 1)$, the algebraic multiplicities of the roots are $\nu(0) = 1$ and $\nu(-1) = 1$. The root (-1) lies on $\Gamma$. Images of both roots lie on $f(\Gamma)$, so the multiplicities must be divided by 2 to give net rotations of $f(\Gamma)$ round $p$. But what is $p$? Should we distinguish loops and the points contained within them, as in [2], p. 339? In that case, $\nu_{exterior - interior} = 1$ and $\nu_{interior} = 2$. We can find the net rotations of $f(\Gamma)$ round $f(-1) = -1$, which I think is $\pi + \pi$, but how does this information relate to the topological numbers such as those in the box in [2]. The net rotation round $f(0) = 0$ appears to be $\pi$. Would we calculate that $N = \frac{1}{2\pi} ( \mathcal{R}_{(-1)} + \mathcal{R}_{(0)} )$? Then, for points inside the interior loop $\nu[f(\Gamma), p] = N = (\frac{1}{2\pi}) (2\pi + \pi) = \frac{3}{2}$ I don't know if this is correct. In the abstract, we didn't have to work with a specific function. I find it difficult to apply the theory to this simple example. Can you shed any light on the application of the GAP? If we just count the 2 poles, each with multiplicity 1, and with that of the pole (-1) divided by 2 because it lies on $\Gamma$, then $\mathcal{R}(\Gamma) = 2 \pi (1 + \frac{1}{2})) = 3\pi$ and $\nu[f(\Gamma), p] = \frac{3}{2}$ (again). But is this meaningful for nested loops? I don't see how we get around fractions when we are taking half of odd multiplicities, and I don't see why it would be a bad thing to speak of fractional winds in situations with p-points and poles lying on $\Gamma$. Gary |
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Gary
GaryVasco
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Post by Gary on Oct 23, 2017 1:09:15 GMT
Vasco,
The format $\nu[L, p]$ and $\nu(L, p)$ have been defined as winding numbers (e.g. pp. 339, 363 bottom, 364, s 6 and equation). The last two occur just prior to use of the format in (17). The format is elsewhere used to refer to "net revolutions" (p.. 338), so it appears that winding numbers and net revolutions are equivalent in practice. In this problem, Needham instructs us to define $\nu[f(\Gamma), p]$ in terms of net revolutions. If "net" means "whole" as in "$2\pi$", then I think we have to disallow fractions. But if not, it appears that the concept of "winding number", or at least the format $\nu[f(\Gamma), p]$, has been generalized along with the GAP to include fractions.
Gary
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Post by Admin on Oct 23, 2017 7:47:43 GMT
Gary It seems to me it's not correct to say that the image points of both roots lie on $f(\Gamma)$. It's true that zero lies on $f(\Gamma)$, but this is not the image point of $z=0$ on $\Gamma$ because $z=0$ does not lie on $\Gamma$. It's OK to say that the image point of $z=-1$ lies on $f(\Gamma)$, because $z=-1$ lies on $\Gamma$. Also we are not concerned with the multiplicities of points on $f(\Gamma)$, but rather the multiplicities of the $p$-points on $\Gamma$. So, in your example, N=1 [for the root at $z=0$] + (1/2) [for the root at $z=-1$]=(3/2), and so $\mathcal{R}=(3/2)\cdot 2\pi=3\pi$ You ask 'what is p?' It is the point $f$ at a root, which is the same for all roots. In this case $p=f(0)=f(-1)=0$. Generally speaking, we can choose any point $p$ and then the solutions (values of $z$) of $f(z)=p$ are the $p$-points. If we then draw any simple loop $\Gamma$ in the '$p$-point plane' then we can apply (17) to the poles and $p$-points inside or on $\Gamma$. Note that (17) specifies that $\Gamma$ is a simple loop. As far as the meaning of 'net' in the phrase 'net revolutions' is concerned, I take it to mean the net revolution of say the white arrow in fig 5 on page 342 about the origin. As $w$ traverses the loop once anticlockwise, the arrow sometimes rotates anticlockwise and sometimes clockwise and finally gets back to its original position with a net revolution equal to an integer, the bits when the arrow waves about cancel each other out. This is how I see the meaning of 'net' in this context. As the arrow point, $w$ traverses the loop once the arrow rotates a number of times round the origin, depending on the complexity of the loop. Here is a link to an animation which shows a complex loop in red and the net revolution in blue. Vasco |
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 23, 2017 15:58:54 GMT
Vasco,
This helps quite a lot. I'll make a few more comments:
I had forgotten that "p-point" is defined as the preimage of a mapping to a point p (p. 345), so that roots are a special kind of p-point, "preimages of 0". In ch 7, ex 21, I note the clause "even if there are some poles and p-points on $\Gamma$", which I think might misled some impressionable person (such as myself) to equate p-points with roots. Your point about f(0) = 0 not being the image point of $z = 0$ on $\Gamma$ is a good one.
This is a useful clarification. There is a potential point of confusion: $\Gamma$ must be a simple loop, but $f(\Gamma)$ need not be a simple loop.
Regarding "net revolution", I agree with your characterization. But it is still the case that in ex. 21, Needham asks us to take the equation containing $\frac{1}{2\pi} \mathcal{R}(L)$ as the definition of $\nu[L (or f(\Gamma)), p]$ and "make sense of the statement that the Generalized Argument Principle (17) remains valid". So if $\mathcal{R} = 3\pi$, it appears that $\nu[f(\Gamma), p] = \frac{1}{2\pi} \mathcal{R}(L) = \frac{1}{2\pi} 3\pi = ...$
Gary
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Post by Admin on Oct 24, 2017 7:00:11 GMT
Gary
We agreed to rewrite this exercise as:
Note that this definition of the winding number
$\displaystyle\nu[f(\Gamma),p]=\frac{1}{2\pi}\mathcal{R}(\Gamma)$.
is only valid when $p$ does not lie on $\Gamma$. (i.e. when there are no $p$-points on $\Gamma$).
So if we reformulate the GAP, result (17) on page 364 for such points, then the wording remains the same but the equation is rewritten as
$\mathcal{R}(\Gamma)=2\pi(N-M)$.
At this point in the proceedings we are still only talking about $p$-points which do not lie on $\Gamma$, and we are still allowed to calculate the winding number.
Then we notice and show that we can use this new equation also for the situation when some of the poles and $p$-points lie on $\Gamma$, with the proviso about multiplicities.
So our new formulation, which now applies in all situations, is a generalisation and as such cannot and does not refer to winding numbers, neither in the mathematics nor in the wording of (17).
In a case where there are no poles or $p$-points on $\Gamma$, then we can apply our formula above and calculate the winding number of $f(\Gamma)$ about $p$.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 26, 2017 3:23:05 GMT
Vasco,
Regarding
What you have done is consistent, so where p-points lie on $\Gamma$, I see why you would limit the characterization to revolutions rather than winding numbers, and I think I might even prefer it in general. I just find it hard to see it as the solution that Needham intended, as he did not explicitly call for a reformulation of the GAP. But the Wikipedia page does present winding numbers as whole numbers, and it seems to be only the physicists who like the fractions.
Gary
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Post by Admin on Oct 26, 2017 5:33:45 GMT
Gary
When in the exercise it says "By taking this formula to be the definition of $\nu$, make sense of...", it seems to me that Needham is asking us to reformulate (17) in terms of $\mathcal{R}$.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 26, 2017 16:11:14 GMT
Gary When in the exercise it says "By taking this formula to be the definition of $\nu$, make sense of...", it seems to me that Needham is asking us to reformulate (17) in terms of $\mathcal{R}$. Vasco Vasco, You are likely correct, as you have been on so many of these issues. I have only persisted because when I read "take something to be the definition of x", I would normally preserve the x along its new definition. I tried to solve it in that spirit. But what is really at stake? The winding number notation $\nu[L, p]$ provides information about the loop and the point it encircles. The rotation notation $\mathcal{R}(L)$ offers information only about the loop. $\nu$ gives complete rotations. $\mathcal{R}$ gives rotations that may be complete or fractional. Even the original restricted definition of $\nu$ implies a value for $\mathcal{R}$. $\nu[L, a]$ is a bit more explicit and perhaps more convenient for problems involving loops around p-points. I profited from the exchange by getting a better understanding of the distinction between $\nu[L, p-point]$ and $\nu[f(L), image-point]$ and the GAP. Gary |
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Post by Admin on Oct 26, 2017 18:32:31 GMT
Gary
I agree that this is probably no big deal in the context of what we are studying - I tend to be a bit too pedantic at times.
Vasco
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Gary
GaryVasco
Posts: 3,352
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Post by Gary on Oct 26, 2017 23:51:37 GMT
Gary I agree that this is probably no big deal in the context of what we are studying - I tend to be a bit too pedantic at times. Vasco Vasco, I think mathematicians must have a zeal for the absolutely correct solution. More than once your follow up has revealed better ways of doing problems that make it easier in the long run. Gary |
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