http://drorbn.net/api.php?action=feedcontributions&user=128.100.68.3&feedformat=atomDrorbn - User contributions [en]2024-03-29T13:24:52ZUser contributionsMediaWiki 1.21.1http://drorbn.net/index.php?title=The_Existence_of_the_Exponential_FunctionThe Existence of the Exponential Function2007-01-18T15:33:55Z<p>128.100.68.3: /* The Scheme */</p>
<hr />
<div>{{Paperlets Navigation}}<br />
<br />
==Introduction==<br />
<br />
The purpose of this [[paperlet]] is to use some homological algebra in order to prove the existence of a power series <math>e(x)</math> (with coefficients in <math>{\mathbb Q}</math>) which satisfies the non-linear equation<br />
<br />
{{Equation|Main|<math>e(x+y)=e(x)e(y)</math>}}<br />
<br />
as well as the initial condition<br />
<br />
{{Equation|Init|<math>e(x)=1+x+</math>''(higher order terms)''.}}<br />
<br />
Alternative proofs of the existence of <math>e(x)</math> are of course available, including the explicit formula <math>e(x)=\sum_{k=0}^\infty\frac{x^k}{k!}</math>. Thus the value of this paperlet is not in the result it proves but rather in the '''allegorical story''' it tells: that there is a technique to solve functional equations such as {{EqRef|Main}} using homology. There are plenty of other examples for the use of that technique, in which the equation replacing {{EqRef|Main}} isn't as easy. Thus the exponential function seems to be the easiest illustration of a general principle and as such it is worthy of documenting.<br />
<br />
Thus below we will pretend not to know the exponential function and/or its relationship with the differential equation <math>e'=e</math>.<br />
<br />
==The Scheme==<br />
<br />
We aim to construct <math>e(x)</math> and solve {{EqRef|Main}} inductively, degree by degree. Equation {{EqRef|Init}} gives <math>e(x)</math> in degrees 0 and 1, and the given formula for <math>e(x)</math> indeed solves {{EqRef|Main}} in degrees 0 and 1. So booting the induction is no problem. Now assume we've found a degree 7 polynomial <math>e_7(x)</math> which solves {{EqRef|Main}} up to and including degree 7, but at this stage of the construction, it may well fail to solve {{EqRef|Main}} in degree 8. Thus modulo degrees 9 and up, we have<br />
<br />
{{Equation|M|<math>e_7(x+y)-e_7(x)e_7(y)=M(x,y)</math>,}}<br />
<br />
where <math>M(x,y)</math> is the "mistake for <math>e_7</math>", a certain homogeneous polynomial of degree 8 in the variables <math>x</math> and <math>y</math>.<br />
<br />
Our hope is to "fix" the mistake <math>M</math> by replacing <math>e_7(x)</math> with <math>e_8(x)=e_7(x)+\epsilon(x)</math>, where <math>\epsilon_8(x)</math> is a degree 8 "correction", a homogeneous polynomial of degree 8 in <math>x</math> (well, in this simple case, just a multiple of <math>x^8</math>).<br />
<br />
{{Begin Side Note|35%}}*1 The terms containing no <math>\epsilon</math>'s make a copy of the left hand side of {{EqRef|M}}. The terms linear in <math>\epsilon</math> are <math>\epsilon(x+y)</math>, <math>-e_7(x)\epsilon(y)</math> and <math>-\epsilon(x)e_7(y)</math>. Note that since the constant term of <math>e_7</math> is 1 and since we only care about degree 8, the last two terms can be replaced by <math>-\epsilon(y)</math> and <math>-\epsilon(x)</math>, respectively. Finally, we don't even need to look at terms higher than linear in <math>\epsilon</math>, for these have degree 16 or more, high in the stratosphere.<br />
{{End Side Note}}<br />
So we substitute <math>e_8(x)=e_7(x)+\epsilon(x)</math> into <math>e(x+y)-e(x)e(y)</math> (a version of {{EqRef|Main}}), expand, and consider only the low degree terms - those below and including degree 8:<sup>*1</sup><br />
<br />
{{Equation*|<math>e_8(x+y)-e_8(x)e_8(y)=M(x,y)-\epsilon(y)+\epsilon(x+y)-\epsilon(x)</math>.}}<br />
<br />
We define a "differential" <math>d:{\mathbb Q}[x]\to{\mathbb Q}[x,y]</math> by <math>(df)(x,y)=f(y)-f(x+y)+f(x)</math>, and the above equation becomes<br />
<br />
{{Equation*|<math>e_8(x+y)-e_8(x)e_8(y)=M(x,y)-(d\epsilon)(x,y)</math>.}}<br />
<br />
{{Begin Side Note|35%}}*2 It is worth noting that in some a priori sense the existence of an exponential function, a solution of <math>e(x+y)=e(x)e(y)</math>, is quite unlikely. For <math>e</math> must be an element of the relatively small space <math>{\mathbb Q}[[x]]</math> of power series in one variable, but the equation it is required to satisfy lives in the much bigger space <math>{\mathbb Q}[[x,y]]</math>. Thus in some sense we have more equations than unknowns and a solution is unlikely. How fortunate we are!<br />
{{End Side Note}}<br />
To continue with our inductive construction we need to have that <math>e_8(x+y)-e_8(x)e_8(y)=0</math>. Hence the existence of the exponential function hinges upon our ability to find an <math>\epsilon</math> for which <math>M=d\epsilon</math>. In other words, we must show that <math>M</math> is in the image of <math>d</math>. This appears hopeless unless we learn more about <math>M</math>, for the domain space of <math>d</math> is much smaller than its target space and thus <math>d</math> cannot be surjective, and if <math>M</math> was in any sense "random", we simply wouldn't be able to find our correction term <math>\epsilon</math>.<sup>*2</sup><br />
<br />
==Computing the Homology==</div>128.100.68.3http://drorbn.net/index.php?title=06-1350/Homework_Assignment_406-1350/Homework Assignment 42006-12-12T20:34:55Z<p>128.100.68.3: </p>
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<div>{{06-1350/Navigation}}<br />
<br />
'''This assignment is due on Tuesday, December 5 2006.'''<br />
<br />
This is an unusual assignment; the task at hand is to do some real research, stuff that to the best of my knowledge had never been done before and most definitely was never written up. Thus the rules will also be a bit different - your work (or at least the accumulation of work on this topic by everyone in class) is meant to be used and useful. So it must be presented in a very readable form (i.e., typed up and with figures) and it must be reliable; in fact, it will be computer verifiable. But some rules will be relaxed, as well.<br />
<br />
The task is a bit technical. But hey, it is a homework assignment, after all!<br />
<br />
==The Task==<br />
<br />
Write all the relations between <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math> in a completely explicit way, both as formulas and as illuminating figures, and then do the same to all the syzygies between these relations. Finally, enter everything you have written into a Mathemmatica script that will verify that for the complex you have created, <math>d^2=0</math>.<br />
<br />
Note that when I write "all relations" or "all syzygies" above I really mean "a complete independent set of relations/syzygies". And while this cannot be formalized, ''the prettier your representatives are, the better!''<br />
<br />
==The Rules==<br />
<br />
The first relation and the first syzygy were written by {{Dror}} (see below or visit [[User:Drorbn/06-1350-HW4]]). You are to copy his work into your user space and complete it there.<br />
* On this wiki create a page named "User:YourUsernameHere/06-1350-HW4" (or simply "User:YourUsernameHere/HW4"). If necessary, go to [[Help:Contents]] to see how this is done.<br />
* Copy [[User:Drorbn/06-1350-HW4]] into your page. The easiest way to do that is to edit [[User:Drorbn/06-1350-HW4]] and copy the source code into your page using copy-paste on your windowing system. Then "preview" or "save" your copy but "cancel" the edit to [[User:Drorbn/06-1350-HW4]].<br />
* Now work on your page...<br />
* Copying is legal! You are allowed, indeed encouraged, to collaborate with others or to simply copy results from other people's pages into yours. The goal is to get something complete. If one of you will start with something incomplete and somebody else will do some other incomplete thing and yet another person will merge the two, we may achieve the goal.<br />
* If you copy, always credit the original source! Likewise, if I will ever use any of the material that will be first produced here, I am committed to crediting the source(s). <br />
* You will get a good though not perfect grade on this assignment for doing anything at all, or even for doing nothing at all but copying on the understanding that by submitting your work you are testifying that you understand it. Perfect grades will go to the people who will make substantial contributions.<br />
* Elegance counts! Beauty counts! A systematic approach counts!<br />
<br />
==A Bonus Question==<br />
<br />
Find the definitive completion of the silliest proof for the existence of exponentials. In other words, find the definitive proof that if <math>M(u,v)</math> is a two-variable power series for which <math>M(y,z)-M(x+y,z)+M(x,y+z)-M(x,y)=0</math>, the there exists a single-variable power series <math>\epsilon(t)</math> for which <math>M(u,v)=\epsilon(v)-\epsilon(u+v)+\epsilon(u)</math>.<br />
<br />
==User:Drorbn/06-1350-HW4==<br />
<br />
{{User:Drorbn/06-1350-HW4}}<br />
<br />
<br />
==Links==<br />
In order to make it easier for us to see each others work, and not all work on the same parts of the assignment I suggest that you can link here to your assignment page. You can also say what you have worked out there and what you are planing on working on.<br />
<br />
My page is [[User:Jana/06-1350-HW4]]<br />
<br />
[[User:Shawkm/06-1350-HW4]]<br />
<br />
[[User:Andy/06-1350-HW4]]<br />
<br />
[[User:Zak/06-1350-HW4]]<br />
<br />
[[User:zsuzsi/HW4]] (worked out some relations and copied Andy's work)<br />
<br />
[[User:Sankaran/06-1350-HW4]]</div>128.100.68.3http://drorbn.net/index.php?title=User:Zsuzsi/HW4User:Zsuzsi/HW42006-12-05T22:28:33Z<p>128.100.68.3: </p>
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<div>===The Generators===<br />
<br />
Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:<br />
{| align=center cellpadding=10 style="border: solid orange 1px"<br />
|- align=center valign=middle<br />
|align=left|Picture<br />
|<br />
|<br />
|<br />
|[[Image:06-1350-BPlus.svg|100px]]<br />
|<br />
|- align=center valign=middle<br />
|align=left|Generator<br />
|<math>T</math><br />
|<math>R</math><br />
|<math>\Phi</math><br />
|<math>B^+</math><br />
|<math>B^-</math><br />
|- align=center valign=middle<br />
|align=left|Perturbation<br />
|<math>t</math><br />
|<math>r</math><br />
|<math>\varphi</math><br />
|<math>b^+</math><br />
|<math>b^-</math><br />
|}<br />
<br />
A low-tech completed version of this chart:<br />
<br />
[[Image:chart.jpg]]<br />
<br />
===The Relations===<br />
<br />
====The Symmetry of B====<br />
<br />
To eliminate the choice involved in placing a B at a crossing, it has to have 180 degrees rotational symmetry. This yields the following picture:<br />
<br />
[[Image:Symm1.jpg]]<br />
<br />
The relation cannot be written in the first notation, as on the right side the chords ending on different red lines could end up on the same pink line.<br />
<br />
In the linearized functional notation though we can express this:<br />
<br />
<math>s_1(x_1,x_2,x_3)=b^+(x_1,x_2,x_3)-b^+(-x_1-x_2-x_3,-x_3,-x_2)</math><br />
<br />
Explanation: on the right side, chords on the first red line can drop off on either the third, second or the first strand, morover, the orders are reversed, hence the minus signs. <br />
<br />
The same picture for B^- yields:<br />
<br />
<math>s_2(x_1,x_2,x_3)=b^-(x_1,x_2,x_3)-b^-(-x_1-x_2-x_3,-x_3,-x_2)</math><br />
<br />
====The symmetry of <math>\Phi</math>====<br />
<br />
<math>\Phi</math> has to have A(4)-symmetry. A(4) is generated by 120 degree rotations around the vertices of the tetrahedron. For example, rotation around the "top" vertex yields the following picture and relation:<br />
<br />
[[Image:Symm3.jpg]]<br />
<br />
The same explanation goes here, and we get the relation:<br />
<br />
<math>s_3(x_1,x_2,x_3)=\varphi(x_1,x_2,x_3)-\varphi(-x_1-x_2,-x_2-x_3,x_2)</math><br />
<br />
<br />
====The Reidemeister move R1====<br />
<br />
[[Image:Reidemeister1.jpg]]<br />
<br />
As with the symmetry relations, we cannot write this one in the first notation either. <br />
<br />
In the linearized functional notation, it looks like this:<br />
<br />
<math> \rho_1(x_1,x_2)=b^-(x_1,x_2,-x_2)</math><br />
<br />
Where the negative sign is because the order of the chords is reversed as we slide them along the little loop.<br />
<br />
====The Reidemeister move R2====<br />
<br />
With three sides of the shielding removed, the picture is:<br />
[[Image:Reidemeister2.jpg]]<br />
<br />
This means:<br />
<math>(123)^\star B^+ (132)^\star B^- = 1</math><br />
<br />
Linearized and in functional form:<br />
<br />
<math>\rho_2(x_1,x_2,x_3)=b^+(x_1,x_2,x_3)+b^-(x_1,x_3,x_2) </math><br />
<br />
And we get the other R2 by switching both crossings, i.e. switching b^+ and b^-:<br />
<br />
<math>\rho_2'(x_1,x_2,x_3)=b^-(x_1,x_2,x_3)+b^+(x_1,x_3,x_2) </math><br />
<br />
<br />
====The Reidemeister Move R3====<br />
The picture (with three sides of the shielding removed) is<br />
[[Image:06-1350-R4.svg|400px|center]]<br />
In formulas, this is<br />
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math><br />
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math><br />
|-<br />
|<br />
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math><br />
|}<br />
<br />
====The Reidemeister Move R4, source:Andy====<br />
First version of R4: <br />
[[Image:06-1350-R4a.png|center]]<br />
In formulas, this is<br />
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math><br />
|}<br />
<br />
Second version: <br />
[[Image:06-1350-R4b.png|center]]<br />
In formulas, this is<br />
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math><br />
|}<br />
<br />
===The Syzygies===<br />
<br />
====The "B around B" Syzygy====<br />
<br />
The picture, with all shielding removed, is<br />
{| align=center<br />
|- align=center<br />
|[[Image:06-1350-BAroundB.svg|center]]<br />
|-<br />
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)<br />
|}<br />
<br />
The functional form of this syzygy is<br />
<br />
{| align=center<br />
|-<br />
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math><br />
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math><br />
|-<br />
|<br />
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math><br />
|-<br />
|<br />
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math><br />
|}<br />
<br />
====The "<math>\Phi</math> around B" Syzygy- I copy-pasted this from Andy, as well as R4====<br />
<br />
The picture, with all shielding (and any other helpful notations) removed, is<br />
{| align=center<br />
|- align=center<br />
|[[Image:06-1350-PhiAroundB.png|center]]<br />
|-<br />
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])<br />
|}<br />
<br />
The functional form of this syzygy is<br />
<br />
{| align=center<br />
|-<br />
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math><br />
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math><br />
|-<br />
|<br />
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math><br />
|-<br />
|<br />
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math><br />
|}<br />
<br />
====The "<math>\Phi</math> around <math>\Phi</math>" Syzygy -also taken from Andy====<br />
<br />
note: I've changed Andy's notation to fit my version of R2.<br />
<br />
The picture is<br />
{| align=center<br />
|- align=center<br />
|[[Image:06-1350-PhiAroundPhi.png|center]]<br />
|-<br />
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])<br />
|}<br />
<br />
The functional form of this syzygy is<br />
<br />
{| align=center<br />
|-<br />
|<math>\Phi\Phi(x_1,x_2,x_3,x_4,x_5) = </math><br />
|<math>-\rho_2'(x_1+x_2,x_3,x_4) - \rho_2'(x_1+x_2+x_4,x_3,x_5) + \rho_{4b}(x_1+x_2,x_4,x_5,x_3)</math><br />
|-<br />
|<br />
|<math>- \rho_2'(x_1,x_2,x_4) - \rho_2'(x_1+x_4,x_2,x_5) + \rho_{4b}(x_1,x_4,x_5,x_2)</math><br />
|-<br />
|<br />
|<math>+ \rho_{4a}(x_1,x_4+x_5,x_2,x_3) - \rho_{4b}(x_1,x_4,x_5,x_2+x_3) - \rho_{4a}(x_1+x_4,x_5,x_2,x_3)</math><br />
|-<br />
|<br />
|<math>+ \rho_2'(x_1+x_4,x_2,x_5) + \rho_2'(x_1+x_2+x_4,x_3,x_5) - \rho_{4a}(x_1,x_4,x_2,x_3)</math><br />
|-<br />
|<br />
|<math>+ \rho_2'(x_1,x_2,x_4) + \rho_2'(x_1+x_2,x_3,x_4)</math><br />
|}<br />
<br />
Note that the first and last terms cancel, as the two steps at the top of the diagram are opposites.<br />
<br />
<br />
===A Mathematica Verification===<br />
<br />
The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!<br />
<br />
{{In|n=1|in=<nowiki>d1 = {<br />
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +<br />
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -<br />
bp[x1 + x4, x2, x3]<br />
};<br />
d2 = {<br />
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + <br />
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -<br />
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -<br />
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +<br />
rho3[x1 + x3, x2, x4, x5]<br />
};</nowiki>}}<br />
<br />
{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]<br />
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]<br />
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]<br />
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}<br />
<br />
{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}</div>128.100.68.3http://drorbn.net/index.php?title=06-1350/Syzygies_in_Asymptote06-1350/Syzygies in Asymptote2006-12-04T21:06:27Z<p>128.100.68.3: More writing.</p>
<hr />
<div>WARNING: These instructions are a work in progress and should be considered highly unreliable.<br />
<br />
To use the syzygy script, first install [http://asymptote.sourceforge.net Asymptote]. Instructions for installing the program on several OSes is given in the documentation at the Asymptote website. The documentation also gives helpful instructions on how to run a script in Asymptote to produce a picture. Once installed, download [http://www.math.utoronto.ca/~andy/syzygy.asy syzygy.asy] and put it in a directory where Asymptote can find it. You should also have (or install) a variant of TeX on your system, such as MiKTeX, so that Asymptote can typeset labels.<br />
<br />
Once installed, we can draw a braid in Asymptote:<br />
<br />
<pre><br />
import syzygy; // Accesses the syzygy module.<br />
Braid b; // Start a new braid.<br />
b.n=3; // The braid has three strands.<br />
// The strands are numbered left to right starting at 0.<br />
b.add(bp,0); // Add a overcrossing component starting at strand 0,<br />
// the leftmost strand.<br />
b.add(bm,1); // Add an undercrossing starting at strand 1.<br />
b.add(phi,0); // Add a trivalent vertex that merges strands 0 and 1.<br />
// Strand 2 is now renumbered as strand 1.<br />
b.draw() // Draw the resulting braid.<br />
</pre><br />
<br />
When saved into an asy file, say <code>mybraid.asy</code> and run with Asymptote, the result is a picture:<br />
<br />
ADD PICTURE<br />
<br />
To define a relation, we first define two braids, and then stick them into a <code>Relation</code> structure. The below script generates an R3 relation.<br />
<br />
<pre><br />
import syzygy; // Access the syzygy module.<br />
Braid l. // Define the left hand side of the relation.<br />
l.n=3; l.add(bp,0); l.add(bp,1); l.add(bp,0);<br />
Braid r. // Define the right hand side of the relation.<br />
r.n=3; r.add(bp,1); l.add(bp,0); l.add(bp,1);<br />
<br />
Relation r3; // Define a relation.<br />
r.lsym="\rho_3"; // Give the relation a name for when it is written in functional form.<br />
r.codename="rho3"; // Give the relation a name to used by Mathematica.<br />
r.lhs=l; r.rhs.r;<br />
r.draw();<br />
</pre><br />
<br />
When saved into an asy file and run, this draws the two sides of the relation. If TeX is installed, Asymptote will also put a lovely equals sign, typeset by TeX, between the two figures.<br />
<br />
ADD ANOTHER PICTURE<br />
<br />
We can also get useful equations out of the relation. The method <code>r3.toFormula()</code> will produce a string that is the formula for the relation.<br />
<br />
<pre><br />
big ugly string<br />
</pre><br />
<br />
This string can be written out to the standard output by <code>write(r3.toFormula())</code>. It can be written to a file by <code>file f=output("filename.txt"); write(f, r3.toFormula())</code>. The string is formatted so it can be put into TeX or a wiki page using math mode:<br />
<br />
ADD EQUATION<br />
<br />
The method <code>r3.toLinear()</code> produces the formula in linear form:<br />
<br />
ADD EQUATION<br />
<br />
and <code>r3.toCode()</code> produces a version of the relation that can be used in Mathematica:<br />
<br />
<pre><br />
ADD CODE<br />
</pre><br />
<br />
A few relations, such as <code>r3</code>, are already defined in <code>syzygy.asy</code> but more should be added.<br />
<br />
Now that we have relations, we can apply them to bigger braids. Let's start with the braid in the <math>\Phi</math> around B syzygy:<br />
<br />
<pre><br />
import syzygy;<br />
Braid b;<br />
b.n=4;<br />
b.add(bp,2);<br />
b.add(bp,0);<br />
b.add(bp,1);<br />
b.add(bp,0);<br />
b.add(bp,2);<br />
b.add(phi,1);<br />
</pre><br />
<br />
ADD PICTURE<br />
<br />
After skipping the lowest knot, we can apply R3 to the next three knots:<br />
<br />
<pre><br />
Braid bb=apply(r3, b, 1, 0);<br />
</pre><br />
<br />
here <code>apply(r, b, k, n)</code> means we are applying the relation <code>r</code> to the braid <code>b</code> at the place in the braid found by counting <code>k</code> components up from the bottom component and <code>n</code> strands in from the leftmost strand. <code>apply</code> does not modify the original braid, but returns the result of applying the relation (stored here as <code>bb</code>):<br />
<br />
ADD PICTURE<br />
<br />
This went from the left hand side of the relation to the right hand side. To apply a relation in reverse, simply prefix it by a minus sign. For example <code>apply(-r3, bb, 1, 0)</code> will yield a braid equivalent to our original. When applying a relation, the script first checks that the one side of the relation matches that portion of the braid, and will give a (somewhat cryptic) error if the relation cannot be applied.<br />
<br />
In our braids, the components are placed from bottom to top in a fixed order. Sometimes when building syzygies, it is neccessary to swap the order that these components occur. This is done by the <code>swap</code> method. For instance, starting from <code>b</code>, we can swap the two bottom crossings:<br />
<br />
<pre><br />
Braid swapped=b.swap(0,1);<br />
</pre><br />
<br />
ADD PICTURE<br />
<br />
Remember that components are ordered from bottom to top, starting at 0. Again, the script checks to make sure the swap is valid (ie. changing the order of the two components, doesn't actually change the knot) and will issue an error if it isn't.<br />
<br />
One could manually apply relations and swaps, and make a whole bunch of braids, but it would be annoying to keep track of them all. Thankfully, the <code>Syzygy</code> structure does that for us. For example, here is the complete code for the <math>\Phi</math> around B syzygy:<br />
<br />
<pre><br />
lots of code<br />
</pre><br />
<br />
Again, like relations, we can use <code>pb.toLinear()</code> and <code>pb.toCode()</code> to give the formulas for the syzygies.</div>128.100.68.3http://drorbn.net/index.php?title=User:Andy/06-1350-HW4User:Andy/06-1350-HW42006-12-04T19:29:14Z<p>128.100.68.3: Added link.</p>
<hr />
<div>===The Generators===<br />
<br />
Our generators are <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>:<br />
{| align=center cellpadding=10 style="border: solid orange 1px"<br />
|- align=center valign=middle<br />
|align=left|Picture<br />
|<br />
|<br />
|<br />
|[[Image:06-1350-BPlus.svg|100px]]<br />
|<br />
|- align=center valign=middle<br />
|align=left|Generator<br />
|<math>T</math><br />
|<math>R</math><br />
|<math>\Phi</math><br />
|<math>B^+</math><br />
|<math>B^-</math><br />
|- align=center valign=middle<br />
|align=left|Perturbation<br />
|<math>t</math><br />
|<math>r</math><br />
|<math>\varphi</math><br />
|<math>b^+</math><br />
|<math>b^-</math><br />
|}<br />
<br />
===The Relations===<br />
<br />
====The Reidemeister Move R3====<br />
The picture (with three sides of the shielding removed) is<br />
[[Image:06-1350-R4.svg|400px|center]]<br />
In formulas, this is<br />
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star B^+ = (1123)^\star B^+ (1203)^\star B^+ (1231)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_3(x_1, x_2, x_3, x_4) = </math><br />
|<math>b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + b^+(x_1,x_3,x_4)</math><br />
|-<br />
|<br />
|<math>- b^+(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_4) - b^+(x_1+x_4,x_2,x_3).</math><br />
|}<br />
<br />
====The Reidemeister Move R4====<br />
To establish the syzygy below, I needed two versions of R4. First: <br />
[[Image:06-1350-R4a.png|center]]<br />
In formulas, this is<br />
<center><math>(1230)^\star B^+ (1213)^\star B^+ (1023)^\star \Phi = (1123)^\star \Phi (1233)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_{4a}(x_1,x_2,x_3,x_4) = b^+(x_1,x_2,x_3) + b^+(x_1+x_3,x_2,x_4) + \phi(x_1,x_3,x_4) - \phi(x_1+x_2,x_3,x_4) - b^+(x_1,x_2,x_3+x_4).</math><br />
|}<br />
<br />
Second: <br />
[[Image:06-1350-R4b.png|center]]<br />
In formulas, this is<br />
<center><math>(1123)^\star B^+ (1203)^\star B^+ (1231)^\star \Phi = (1230)^\star \Phi (1223)^\star B^+</math>.</center><br />
Linearized and written in functional form, this becomes<br />
{| align=center<br />
|-<br />
|<math>\rho_{4b}(x_1,x_2,x_3,x_4) = b^+(x_1+x_2,x_3,x_4) + b^+(x_1,x_2,x_4) + \phi(x_1+x_4,x_2,x_3) - \phi(x_1,x_2,x_3) - b^+(x_1,x_2+x_3,x_4).</math><br />
|}<br />
<br />
Are these independent, or can they be shown to be equivalent using other relations?<br />
<br />
===The Syzygies===<br />
<br />
====The "B around B" Syzygy====<br />
<br />
The picture, with all shielding removed, is<br />
{| align=center<br />
|- align=center<br />
|[[Image:06-1350-BAroundB.svg|center]]<br />
|-<br />
|align=right|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)<br />
|}<br />
<br />
The functional form of this syzygy is<br />
<br />
{| align=center<br />
|-<br />
|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math><br />
|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math><br />
|-<br />
|<br />
|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math><br />
|-<br />
|<br />
|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math><br />
|}<br />
<br />
====The "<math>\Phi</math> around B" Syzygy====<br />
<br />
The picture, with all shielding (and any other helpful notations) removed, is<br />
{| align=center<br />
|- align=center<br />
|[[Image:06-1350-PhiAroundB.png|center]]<br />
|-<br />
|align=right|(Drawn with [http://asymptote.sf.net/ Asymptote], [[06-1350/Syzygies in Asymptote|Syzygies in Asymptote]])<br />
|}<br />
<br />
The functional form of this syzygy is<br />
<br />
{| align=center<br />
|-<br />
|<math>\Phi B(x_1,x_2,x_3,x_4,x_5) = </math><br />
|<math>\rho_3(x_1,x_2,x_3,x_5) + \rho_{4a}(x_1+x_5,x_2,x_3,x_4) + \rho_{4b}(x_1+x_2,x_3,x_4,x_5)</math><br />
|-<br />
|<br />
|<math>- \rho_3(x_1,x_2,x_3+x_4,x_5) - \rho_{4a}(x_1,x_2,x_3,x_4)</math><br />
|-<br />
|<br />
|<math>- \rho_{4b}(x_1,x_3,x_4,x_5) + \rho_3(x_1+x_3,x_2,x_4,x_5).</math><br />
|}<br />
<br />
===A Mathematica Verification===<br />
<br />
The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!<br />
<br />
{{In|n=1|in=<nowiki>d1 = {<br />
rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +<br />
bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -<br />
bp[x1 + x4, x2, x3]<br />
};<br />
d2 = {<br />
BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + <br />
rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -<br />
rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -<br />
rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +<br />
rho3[x1 + x3, x2, x4, x5]<br />
};</nowiki>}}<br />
<br />
{{InOut|n=3|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]<br />
+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]<br />
+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]<br />
+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}<br />
<br />
{{InOut|n=4|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>|out=<nowiki>0</nowiki>}}</div>128.100.68.3http://drorbn.net/index.php?title=06-1350/Class_Photo06-1350/Class Photo2006-10-17T17:44:27Z<p>128.100.68.3: /* Who We Are */</p>
<hr />
<div>Our class on September 28, 2006:<br />
<br />
[[Image:06-1350-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
{{06-1350/Navigation}}<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are===<br />
<br />
{| align=center border=1<br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}<br />
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|}</div>128.100.68.3http://drorbn.net/index.php?title=06-1350/Class_Photo06-1350/Class Photo2006-10-11T22:59:52Z<p>128.100.68.3: /* Who We Are */</p>
<hr />
<div>Our class on September 28, 2006:<br />
<br />
[[Image:06-1350-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
{{06-1350/Navigation}}<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are===<br />
<br />
{| align=center border=1<br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}<br />
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{{Photo Entry|last=Ivrii|first=Oleg|userid=Oivrii|email=oleg@math.toronto.edu|location=last row, middle of the sitting |comments=}}<br />
{{Photo Entry|last=de Jong|first=Michael|userid=michael|email=michael.dejong@utoronto.ca|location=standing in the back, on the left|comments=}}<br />
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{{Photo Entry|last=Shaw|first=Kristin|userid=Shawkm|email=shawkm@math.toronto.edu|location=second row, second from the left|comments=}}<br />
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|}</div>128.100.68.3http://drorbn.net/index.php?title=06-1350/Class_Photo06-1350/Class Photo2006-10-11T22:58:58Z<p>128.100.68.3: /* Who We Are */</p>
<hr />
<div>Our class on September 28, 2006:<br />
<br />
[[Image:06-1350-ClassPhoto.jpg|thumb|center|500px|Class Photo: click to enlarge]]<br />
{{06-1350/Navigation}}<br />
Please identify yourself in this photo! There are two ways to do that:<br />
<br />
* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.<br />
* Send [[User:Drorbn|Dror]] an email message with this information.<br />
<br />
The first option is more fun but less private.<br />
<br />
===Who We Are===<br />
<br />
{| align=center border=1<br />
|-<br />
!First name<br />
!Last name<br />
!UserID<br />
!Email<br />
!In the photo<br />
!Comments<br />
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank}}<br />
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{{Photo Entry|last=de Jong|first=Michael|userid=michael|email=michael.dejong@utoronto.ca|location=standing in the back, on the left|comments=}}<br />
{{Photo Entry|last=Archibald|first=Jana|userid=Jana|email=jfa@ math.toronto.edu|location=Third row, First on right|comments=}}<br />
{{Photo Entry|last=Shaw|first=Kristin|userid=Shawkm|email=shawkm@math.toronto.edu|location=second row, second from the left|comments=}}<br />
{{Photo Entry|last=Antolin Camarena|first=Omar|userid=Oantolin|email=oantolin@math.toronto.edu|location=third row, furthest to the left (mostly obscured by people in front row)|comments=}}<br />
|}</div>128.100.68.3http://drorbn.net/index.php?title=Template:06-1350/NavigationTemplate:06-1350/Navigation2006-10-07T21:17:11Z<p>128.100.68.3: </p>
<hr />
<div>{| cellpadding="0" cellspacing="0" style="clear: right; float: right"<br />
|- align=right<br />
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<div class="NavContent"><br />
{| border="1px" cellpadding="1" cellspacing="0" width="220" style="margin: 0 0 1em 0.5em; font-size: small"<br />
|-<br />
!#<br />
!Week of...<br />
!Links<br />
|-<br />
|align=center|1<br />
|Sep 11<br />
|[[06-1350/About This Class|About]], [[06-1350/Class Notes for Tuesday September 12|Tue]], [[06-1350/Class Notes for Thursday September 14|Thu]]<br />
|-<br />
|align=center|2<br />
|Sep 18<br />
|[[06-1350/Class Notes for Tuesday September 19|Tue]], [[06-1350/Some Equations by Kurlin|Kurlin]], [[06-1350/Class Notes for Thursday September 21|Thu]]<br />
|-<br />
|align=center|3<br />
|Sep 25<br />
|[[06-1350/Class Notes for Tuesday September 26|Tue]], [[06-1350/Class Photo|Photo]], [[06-1350/Class Notes for Thursday September 28|Thu]]<br />
|-<br />
|align=center|4<br />
|Oct 2<br />
|[[06-1350/Homework Assignment 1|HW1]], [[06-1350/Class Notes for Tuesday October 3|Tue]]<br />
|-<br />
|align=center|5<br />
|Oct 9<br />
|[[06-1350/Class Notes for Tuesday October 10|Tue]]<br />
|-<br />
|align=center|6<br />
|Oct 16<br />
|HW2<br />
|-<br />
|align=center|7<br />
|Oct 23<br />
|<br />
|-<br />
|align=center|8<br />
|Oct 30<br />
|HW3<br />
|-<br />
|align=center|9<br />
|Nov 6<br />
|<br />
|-<br />
|align=center|10<br />
|Nov 13<br />
|HW4<br />
|-<br />
|align=center|11<br />
|Nov 20<br />
|<br />
|-<br />
|align=center|12<br />
|Nov 27<br />
|HW5<br />
|-<br />
|align=center|13<br />
|Dec 4<br />
|<br />
|-<br />
|colspan=3 align=left|'''Note.''' HW weeks are tentative.<br />
|-<br />
|colspan=3 align=center|[[Image:06-1350-ClassPhoto.jpg|180px]]<br>[[06-1350/Class Photo|Add your name / see who's in!]]<br />
|}<br />
</div></div><br />
|}</div>128.100.68.3http://drorbn.net/index.php?title=VasCalc_-_A_Vassiliev_Invariants_CalculatorVasCalc - A Vassiliev Invariants Calculator2006-05-15T16:52:14Z<p>128.100.68.3: </p>
<hr />
<div>These pages document our work on VasCalc. Let us start with the project description as appeared in our NSERC proposal:<br />
<br />
<blockquote><br />
Finite type (Vassiliev) invariants stand in the centre of knot theory.<br />
They are known to encompass very many of the invariants pivotal to knot<br />
theory and to low dimensional topology, and thus hundreds of papers<br />
were written about them. Finite type invariants are in principle<br />
algorithmic and computable, yet the computations are a complicated<br />
many-step procedure and there aren't yet coherent computer programs to<br />
carry them out.<br />
<br />
<p>After 15 years of progress regarding finite type invariants, I feel<br />
that finally our understanding of the mathematics is stable enough to<br />
justify and guide a computational effort. I propose that this work be<br />
carried out as a joint NSERC summer research internship by Zavosh<br />
Amir-Khosravi and Siddarth Sankaran, where Zavosh will be writing the<br />
java- or C++-based "inner most loop" while Siddarth will be working on<br />
all the surrounding logic. We will make sure that every piece of the<br />
work will be well documented and will have some "stand alone" value, so<br />
overall, I expect the project to have significant impact on the subject<br />
of finite type invariants.</p><br />
</blockquote><br />
<br />
* [http://www.third-bit.com/swc2/lec/version.html A bit about subversion] and [http://katlas.math.toronto.edu/svn/VasCalc/ our subversion repository].<br />
* First day meeting's blackboard: [[Image:060511-1.jpg|thumb|160px]].<br />
<br />
<br />
J/Link passed the Hello World test. The java class and Mathematica<br />
notebook were commited to the repository trunk. Note the Mathematica file<br />
must be edited in order to specify the path to java and the class.<br />
In order to get it to work I had to tweak a few things:<br />
* The JRE bin directory had to be added to the PATH. The JDK path is not enough because J/Link looks for files like jawt.dll and awt.dll that don't come with JDK.<br />
* HelloWorld.class had to be in one of the dirs in CLASSPATH, because J/Link loads classes only through their full class name.<br />
* In Mathematica, when running InstallJava[], it was necessary to specify the path name for a newer version of java.exe. By default it runs the Windows java, which didn't run what was compiled with a new JDK. <br />
<br />
Ideally, one would like to write a Mathematica notebook without<br />
any hardcoded paths that others can use with no changes. Is this<br />
possible?</div>128.100.68.3