Talk:Brouwer Fixed Point Theorem

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Great things about the page

  • Really clear description of what the theorem says, and well-described examples.
  • Nice integration of technical mathematical reasoning with statements that amount to, "and isn't this really weird!"
  • Great use of images to break up and clarify text without superfluous use of images
  • Clear development of proof. Also, nice job asking the non-technical reader to engage in some heavy mathematical reasoning.
Abram, 6/15

The coffee cup example is feeling less and less accurate

Liquids just don't move continuously. For example, if you put a layer of dense blue liquid on top of a layer of less dense red liquid, everything will swap places by the end. (Abram, 6/28)

I made lots of small wording changes

A few changes were meant to clean up the mathematical precision or accuracy (see mathematical precision details, below). Others were meant to make the presentation just a bit clearer. (Abram, 6/28)

Refer the reader to images (and number them)

  • In the one-dimensional proof, it might help to refer the reader to the image (e.g. add "As can be seen in Figure 1" to the beginning of some sentences). You'll have to note that in this case we are assuming a = 0 and b = 1.
  • The three-dimensional proof keeps referring to "the figure". It might help to number/anchor it.
Abram, 7/5
  • Ok. I added these and put anchors on all the images that I ever refer to. Rebecca 15:05, 8 July 2010 (UTC)
  • Your description of how 2.2 is created by folding and sliding doesn't refer the reader to the currently unanchored image that shows this folding and sliding. Maybe it should? (Abram, 7/8)
  • I decided not to put links over Image 2.1 2.2 and 2.3 because I want there to be consistency as far as those three go. The picture is right next to the description and it's still labeled, and people can pretty much see what I'm talking about so links might just be distracting.

Put the balloon over "continuous" back in the main text.

This sentence doesn't seem like one we want people to miss, and lots of people will skip over the balloon if they already know what the word "continuous" means. (Abram, 6/28)

Looks like it has been taken care of. (Abram, 7/5)

Small Layout Issues

  • In the "generalizing to all dimensions" section, there is a lot of black text. You've done a good job spacing it out, but I still think that it's too easy to have eyes glaze over it. I've got a couple of ideas for making it better, and I encourage you to spend 15 minutes playing with it. I don't know what will look best, and if any of these suggestions will actually make things worse or not.
    • Idea #1 is to make your pictures bigger.
    • Idea #2 change the colors of parts of the text or just the equations. Anna 6/25
Thanks Anna! I played with this issue, but I couldn't find a great solution either. I decided to change the warning about the proof being non-rigorous (in the beginning) to green and I added a mouseover for a little more color. Also, I increased this size of the pictures a little, but it was hard to do without screwing up the spacing etc. Hopefully these changes are an improvement. Rebecca 18:10, 25 June 2010 (UTC)

Slightly change description of fixed points in first paragraph and one dimensional case

In the one-dimensional case, for example, you say that "there will be a point on the top string has not changed its horizontal position with respect to the bottom string!" You can remove "with respect to the bottom string". Instead, you can explain that the color represents the initial horizontal position of each string, and that after deforming the string, the place where the colors continue to line up represents the fixed point. (Abram, 6/15)

I fixed this issue. 6/15 Becky
Nicely done. Maybe expand a bit on the part that says "the place where the colors continue to line up represents the fixed point"to hammer this point a little more, but I'm not sure how. (Abram, 6/16)
I ended up expanding this part anyway to make "place where the colors line up" more precise so I didn't add any more. 6/16
Yeah, looks good. (Abram, 6/25)

Outline the explanation of figure 2 (in the 2-d section) a bit more clearly

At the end of this explanation, you do a nice job stating what the two main facts are that you have established. Instead of doing this at the end, do it at the beginning, and as you go, make it clear which part you are working on. (Abram, 6/15)

I reorganized this section. 6/15 Becky
Nice reorganization. I have a couple of small wording things that can wait until the final pass. (Abram, 6/16)
OK, there are three ways in which the explanation of this example could be more precise or your argument could be more thorough.
  • Saying the red disk was "replaced in the center of the blue disk" is a little bit vague, though I know what you are saying. How about the red disk is "replaced so that it covers the center point of the blue disk."
  • You may want to give an argument for why the rotation you describe puts the red disk in the top right quarter of the blue disk (let me know if you decide to include this argument but want help figuring it out).
  • Saying that because one of the red *layers* is in its original position, that one of the red *points* must be in its original position is a bit of a slide. If that reasoning were acceptable, we wouldn't have any need for the fixed point theorem! See if you can think of a valid reason for your conclusion. If you get stuck, I can help.
Abram, 6/23
We changed the example so that this "proof" actually works and decided not to bother with the first argument above. Nicely done fixing the second issue, though I'll have a slight wording suggestion in the final pass-through. (Abram, 6/25)

Do something about the main picture for the 2-d proof

Your picture of the rectangle W is a bit problematic, because the reader can't tell what the two curves represent until quite a bit further down, and you never actually explicitly say what each curve represents while referring to *this* picture. This is a tricky organizational issue that you can work on or I can generate some ideas. (Abram, 6/15)

I moved the picture down, but I'm not sure if this is the best solution. We can talk tomorrow. 6/16
It seems like we're reasonably satisfied with how this has worked out. Anna and Becca, do you concur? (Abram, 7/5)
Yeah. Looks good to me.

Flush out the economics examples

For example, in the Nash equilibrium description, what does the function with the fixed point represent, and what do the points in "W" represent? And similarly for the supply/demand description. (Abram, 6/15)

I added more detail to the Nash equilibrium part, but I'm having a lot of difficulty finding more info about this. I will contact Prof. Bayer who teaches game theory to see if she can answer some of my questions.
You've already done a nice job effectively describing that "W" from the proof is the set of all possible strategy combinations. You just need to describe what "f" is (I'm guessing f maps a given combination of strategies, p to the combination of strategies that responds to p). Also, you may want to check with Prof. Bayer to check that W really is what you are saying it is (unless you're sure about this). Finally, can you describe what W and f are for the supply/demand scenario? (Abram, 6/16)
I e-mailed Prof. Bayer, but she hasn't responded. 6/16
Nicely done in this section, with one slight notation problem in the way you use "f(x)" (see the mathematical precision conversation). (Abram, 7/5)

Define "convex"

Include a mouse-over definition for convex somewhere. (Abram, 6/15)

Changed this 6/15
Nice definition. It looks like you have included a picture, but for some reason, it's rendering as simply text saying [[Image:convex1.jpg]] instead of as the actual image. I can point you to pages that have actually included images in balloons if you can't figure out the right syntax at WikiTricks, or somewhere else. (Abram, 6/16)
I fixed the pictures and spacing of the bubble and I think it looks much better. 6/16

Be more clear about what parts of the intuitive proof are hard to make rigorous.

The disclaimer at the beginning of the proof stating that this proof is only informal is great. Readers may find it interesting to know where you are saying things that can be made more rigorous without too much difficulty and where you are saying things that are really hard to make rigorous. (Abram, 6/15)

Changed this 6/15. I added a part about the continuity of the curve of fixed points being difficult to prove. I didn't add anything after the part about the extended f being continuous though. Essentially, adding something to this part would amount to saying "Proving the continuity of the extended f is a point that should be addressed in more rigorous proofs, but is just outside the reach of this explanation." I'm not sure how I can say this without making readers wonder why i didn't just prove it if it isn't so hard. Any ideas? Or do you think it is OK to just note where the really hard part is?
I think it's probably fine. I had in mind saying something like, "Actually proving this requires techniques from algebraic topology" about the continuity of the curve of fixed points, and something like, "Making this proof rigorous is not difficult using the formal definition of continuity" for the function extension. I'm not sure, though. Some readers would find that really interesting, while some would find it alienating. (Abram, 6/16)

A couple of mathematical precision issues

  • In the 1-dimensional case towards the end you write, "the position of this point didn't change throughout the process." That's not actually true. The position may have changed a lot throughout the process -- it just ended up where it started.
Thanks, I changed this 6/16
  • Also in the 1-dimensional case, replace "place where the colors line up" with more precise language.
Ok, I changed this a little. I struggled to think of a better way to word it. I think the way I changed it makes the section a little redundant, but I'm not sure what I should remove. Maybe you can address this when you do the more detailed read-through. 6/16
  • At the beginning of the figure 2 description for the 2-dimensional case, you say, "red disk was ... replaced in the exact center of the blue disk." What do you mean by "in the exact center of the blue disk"? I think for this proof to work it's just that *some* point on the folded red disk has to be sitting directly on top of the center of the blue disk.
You're right. The textbook I read decided said that they placed it in the exact middle. But I agree, this is confusing. I didn't want to leave out the word and risk falsifying the statement. When I thought about it though, my explanation will work as long as it is rotated around the midpoint. I decided to take out the word exactly because it is confusing and unnecessary. 6/16
  • In the intermediate value theorem proof of the 1-dimensional case, you write, "First, we must know that f(a) ≠ a and f(b) ≠ b because if f(a) = a or f(b) = b that point will already be a fixed point." It's slightly more accurate to say we can *assume* that f(a) = a and f(b) = b. Saying we "know" implies that it's not possible that f(a) = a or f(b) = b, whereas saying "assume" means we don't need to worry about situations in which f(a) = a and f(b) = b. Abram, 6/16
I changed this 6/16
I also changed yellow line to yellow curve as you suggested.

  • I'm not seeing a definition of  x_0 ,  x_1,  y_0 , and  y_1 before you define g and h. You can do this by saying that W has corners  (x_0, y_0), (x_1,y_0) , etc. a Alternatively, you could label them on your picture and direct the reader to look at it when you first mention them. Anna 6/25
  • I just edited the page directly, making a bunch of small changes. There were a couple places where your wording implied that B was true because of A, while in fact A and B were true independently, and together led to conclusion C. I changed the wording to make this more clear. (Abram, 6/28).
We still have this problem where sentence structure doesn't mirror logical flow where you say "If the point P is in the same position both when the red disk is deformed, when it is being rotated, and when it has returned to its initial position, then it is a fixed point." The actual logical organization is that after the rotation, one of the four red points located at P is at its original location. But the rotation didn't actually move this point, THEREFORE it was in its original location even before the rotation, but after the deformation. The function f was the deformation, so it held this point fixed. (Abram, 7/5)
  • I also fixed a function notation problem. Somewhere, you said that g holds y-values constant. Technically, g doesn't hold y-values constant. Instead, it takes in an x-value and a y-value, and spits out a new x-value, and doesn't spit out any y-value. (Abram, 6/28).
I realized that my italics formatting doesn't work because the subscripts look different with my italics formatting than with your math formatting. Can you put these sentences that define g into math formatting? Sorry to ask you to do this -- if it feels like unfair grunt work, I'll take care of it. I'm just trying to give myself more time to read pages, so that I'll actually be helpful! (Abram, 7/5)
  • Where you refer to the "function f(x)" in the economics section, this is technically incorrect. The name of the function is simply f, while f(x) means the output of f, given an input of x. For example, the sentence "The function f(x) gives a new set of strategies, one for each player, where each player's strategy is his best move against his opponent's strategies in x" could be rewritten as "If x is a point in W, representing a combination of strategies, then f(x) is the set of strategies the players form in response to x." This way, the notation f(x) refers to an output rather than the name of the function. A similar change can be made for the supply/demand statement. (Abram, 7/5)
  • Maybe clarify that in the animation, the point P is the black dot, since it isn't labeled there.
Cool... I've made all the changes suggested above.
Maybe in your definition of x_i and y_i, write "The corners of W are A = (x_0, y_0), etc." and refer the reader to the image showing W. (Abram, 7/8)
I did change this...

  • Let's talk about what a tuple is. (Abram, 6/28).
We talked, and decided that use of the terminology tuple is totally unnecessary. Nice job explaining what W is using simpler wording. (Abram, 7/5)

Reframe the "you are here" map section

The economics section really shows the fixed point theorem being used to prove the existence of something that, a priori, may or may not exist. In this section, the fixed point theorem is proving something that is so obviously true, it's hard to see it really as an application. This example is an interesting lens on the theorem / a cool example of how the theorem can manifest itself, so maybe it should be framed that way instead of as an application. (Abram, 6/16)

OK.. Changed 6/16
Nicely done. (Abram, 6/25)

Add a science application?

Adding yet another application may seem crazy, but one of the things about this theorem is that its applications are so diverse. Maybe there's a science application you can add. Resources for learning these applications include: Don Shimamoto (because he's a topologist), folks from other natural science departments (I think a lot of the physicists are really receptive to these kinds of questions; Lynne Molter in engineering would be), maybe other people I haven't thought of yet. (Abram, 6/16)

You're right. I've been itching to add a section of Brouwer and heart attacks as it is... the problem is that I can barely find any material on this subject. I know it will be worthwhile if I can get enough to make a section because what I have seems VERY interesting. I'll have to fill you in when you're here tomorrow. 6/16

Smaurer1 02:29, 5 June 2010 (UTC)

Rather than have another conference with Becky where I go over a hardcopy I have marked, I have simply edited the More Math section online. I think Becky has stretched herself nicely to understand this proof, but for her to get it all right, and to learn to use all the language correctly, by hardcopy conferences would take several more meetings, and I think it is time to go on to something else. I have put in several comments as to why I made certain changes. I hope these are helpful. Of course, you should now take these comments out. I have said to Becky that she should come talk to me if she doesn't understand why I made certain changes. This too will be a good learning process.

In our group meeting today, we had a discussion about whether this article could be written without using math symbols. The thought this afternoon was to use text italic. I now think that was a bad idea, not because it looks bad, but because it is more of a pain to write that writing math formulas in this wiki. Every letter needs its own double quotes, because you don't want anything like ( + = to be italic. Also, the wiki formatter will break text-italic expressions in unallowed places, like breaking f(x) into f on one line and (x) on the other! Don't take out all this italic now that it is there, but try ordinary roman on your next article.

I haven't reviewed the other sections.

I think this is becoming a nice article.

Becky 6/3

Hi Anna! Thanks for your help, and I'll be working on your suggestions... Just a heads up though, you might want to hold off on looking at my page again until Monday because I'm going to be reworking a lot of the math with Dr. Maurer tomorrow.

Anna 6/3

Hi Becky, I'll work on giving this a more thorough look through in the next couple of days, but I wanted to go ahead and give a few quick comments.

1. I'd encourage you to go ahead and post a list of references or places to find other proofs

2. When you define variables, try to not do that in the middle of a line. An example of that is when you define "f" in the middle of a paragraph. It would be better to start a new paragraph then (after all, remember how some of the hardest parts of understanding the proofs was understanding how they had defined functions?).

3. When you're talking about specific numbers, don't just use x and y. Any time that it's a particular x and y, try to call it x',  x_0 ,  x_1 , etc. When people see x, they think variable. When they see  x_p they think some particular number.

Nice work so far, and I'll come back to it soon!