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0708-1300/Class notes for Tuesday, October 2

2007-10-08T07:06:09Z

Kuramay: /* First Hour */

{{0708-1300/Navigation}}
==English Spelling==
Many interesting rules about [[0708-1300/English Spelling]]
==Class Notes==
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

'''General class comments'''

1) The class photo is up, please add yourself

2) A questionnaire was passed out in class

3) Homework one is due on thursday

===First Hour===

Today's Theme: Locally a function looks like its differential

'''Pushforward/Pullback'''

Let <math>\theta:X\rightarrow Y</math> be a smooth map.

We consider various objects, defined with respect to X or Y, and see in which direction it makes sense to consider corresponding objects on the other space. In general <math>\theta_*</math> will denote the push forward, and <math>\theta^*</math> will denote the pullback.

1) points ''pushforward'' <math>x\mapsto\theta_*(x) := \theta(x)</math>

2) Paths <math>\gamma:\mathbb{R}\rightarrow X</math>, ie a bunch of points, ''pushforward'', <math>\gamma\rightarrow \theta_*(\gamma):=\theta\circ\gamma</math>

3) Sets <math>B\subset Y</math> ''pullback'' via <math>B\rightarrow \theta^*(B):=\theta^{-1}(B)</math>
Note that if one tried to pushforward sets A in X, the set operations compliment and intersection would not commute appropriately with the map <math>\theta</math>

4) A measures <math>\mu</math> ''pushforward'' via <math>\mu\rightarrow (\theta_*\mu)(B) :=\mu(\theta^*B)</math>

5)In some sense, we consider functions, "dual" to points and thus should go in the opposite direction of points, namely <math>\theta^*f = f\circ\theta</math>

6) Tangent vectors, defined in the sense of equivalence classes of paths, [<math>\gamma</math>] ''pushforward'' as we would expect since each path pushes forward. <math>[\gamma]\rightarrow \theta_*[\gamma]:=[\theta_*\gamma] = [\theta\circ\gamma]</math>

CHECK: This definition is well defined, that is, independent of the representative choice of <math>\gamma</math>

7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, ''pushforward'' via <math>D\rightarrow (\theta_*D)(f):= D(\theta^*f)</math>

CHECK: This definition satisfies linearity and Liebnitz property.

'''Theorem 1'''

The two definitions for the pushforward of a tangent vector coincide.

''Proof:''

Given a <math>\gamma</math> we can construct <math>\theta_{*}\gamma</math> as above. However from both <math>\gamma</math> and <math>\theta_*\gamma</math> we can also construct <math>D_{\gamma}f</math> and <math>D_{\theta_*\gamma}f</math> because we have previously shown our two definitions for the tangent vector are equivalent. We can then ''pushforward'' <math>D_{\gamma}f</math> to get <math>\theta_*D_{\gamma}f</math>. The theorem is reduced to the claim that:

<math>\theta_*D_{\gamma}f = D_{\theta_*\gamma}f</math>

for functions <math>f:Y\rightarrow \mathbb{R}</math>

Now, <math>D_{\theta_*\gamma}f = \frac{d}{dt}f\circ(\theta_*\gamma)|_{t=0} = \frac{d}{dt}f\circ(\theta\circ\gamma)|_{t=0} = \frac{d}{dt}(f\circ\theta)\circ\gamma |_{t=0} = D_{\gamma}(f\circ\theta) =\theta_*D_{\gamma}f</math>

''Q.E.D''

'''Functorality'''

let <math>\theta:X\rightarrow Y, \lambda:Y\rightarrow Z</math>

Consider some "object" s defined with respect to X and some "object u" defined with respect to Z. Something has the property of functorality if

<math>\lambda_*(\theta_*s) = (\lambda\circ\theta)_*s</math>

and

<math>\theta^*(\lambda^*u) = (\lambda\circ\theta)^*u</math>

Claim: All the classes we considered previously have the functorality property; in particular, the pushforward of tangent vectors does.

Let us consider <math>\theta_*</math> on <math>T_pM</math> given a <math>\theta:M\rightarrow N</math>

We can arrange for charts <math>\varphi</math> on a subset of M into <math>\mathbb{R}^m</math> (with coordinates denoted <math>(x_1,\dots,x_m)</math>)and <math>\psi</math> on a subset of N into <math>\mathbb{R}^n</math> (with coordinates denoted <math>(y_1,\dots,y_n)</math>)such that <math>\varphi(p) = 0</math> and <math>\psi(\theta(p))=0</math>

Define <math>\theta^o = \psi\circ\theta\circ\varphi^{-1} = (\theta_1(x_1,\dots,x_m),\dots,\theta_n(x_1,\dots,x_m))</math>

Now, for a <math>D\in T_pM</math> we can write <math>D=\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}</math>

So, <math>(\theta_*D)(f) = \sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\varphi) ///\sum_{i=1}^m a_i\frac{\partial}{\partial x_i}(f\circ\theta)should be?? ///= \sum_{i=1}^m a_i \sum_{j=1}^n\frac{\partial f}{\partial y_j}\frac{\partial\theta_j}{\partial x_i}=</math>
<math>=\begin{bmatrix}
\frac{\partial f}{\partial y_1} & \cdots & \frac{\partial f}{\partial y_n}\\
\end{bmatrix}
\begin{bmatrix}

\frac{\partial \theta_1}{\partial x_1} & \cdots & \frac{\partial \theta_1}{\partial x_m}\\
\vdots& & \vdots\\
\frac{\partial \theta_n}{\partial x_1} & \cdots & \frac{\partial \theta_n}{\partial x_m}\\
\end{bmatrix}
\begin{bmatrix}
a_1\\
\vdots\\
a_m\\
\end{bmatrix}</math>

Now, we want to write <math>\theta_*D = \sum b_j\frac{\partial}{\partial y_j}</math>

and so, <math>b_k = (\theta_*D)y_k =\begin{bmatrix}
0&\cdots & 1 & \cdots &0\\
\end{bmatrix}
\begin{bmatrix}

\frac{\partial \theta_1}{\partial x_1} & \cdots & \frac{\partial \theta_1}{\partial x_m}\\
\vdots& & \vdots\\
\frac{\partial \theta_n}{\partial x_1} & \cdots & \frac{\partial \theta_n}{\partial x_m}\\
\end{bmatrix}
\begin{bmatrix}
a_1\\
\vdots\\
a_m\\
\end{bmatrix}</math>

where the 1 is at the kth location.

So, <math>\theta_* = d\theta_p</math>, i.e., <math>\theta_*</math> is the differential of <math>\theta</math> at p

We can check the functorality, <math>(\lambda\circ\theta)_* = \lambda_*\circ\theta_*</math>, then <math>d(\lambda\circ\theta) = d\lambda\circ d\theta</math>
This is just the chain rule.

===Second Hour===

'''Defintion 1'''

An ''immersion'' is a (smooth) map <math>\theta:M^m\rightarrow N^n</math> such that <math>\theta_*</math> of tangent vectors is 1:1. More precisely, <math>d\theta_p: T_pM\rightarrow T_{\theta(p)}N</math> is 1:1 <math>\forall p\in M</math>

'''Example 1'''

Consider the canonical immersion, for m<n given by <math>\iota:(x_1,...,x_m)\mapsto (x_1,...,x_m,0,...,0)</math> with n-m zeros.

'''Example 2'''

This is the map from <math>\mathbb{R}</math> to <math>\mathbb{R}^2</math> that looks like a loop-de-loop on a roller coaster (but squashed into the plane of course!) The map <math>\theta</math> itself is NOT 1:1 (consider the crossover point) however <math>\theta_*</math> IS 1:1, hence an immersion.

'''Example 3'''

Consider the map from <math>\mathbb{R}</math> to <math>\mathbb{R}^2</math> that looks like a check mark. While this map itself is 1:1, <math>\theta_*</math> is NOT 1:1 (at the cusp in the check mark) and hence is not an immersion.

'''Example 4'''

Can there be objects, such as the graph of |x| that are NOT an immersion, but are constructed from a smooth function?

Consider the function <math>\lambda(x) = e^{-1/x^2}</math> for x>0 and 0 otherwise.

Then the map <math>x\mapsto \begin{bmatrix}
(\lambda(x),\lambda(x))& x>0\\
(0,0)& x=0\\
(-\lambda(-x),\lambda(-x)) & x<0\\
\end{bmatrix}</math>

is a smooth mapping with the graph of |x| as its image, but is NOT an immersion.

'''Example 5'''

The torus, as a subset of <math>\mathbb{R}^3</math> is an immersion

Now, consider the 1:1 linear map <math>T:V\rightarrow W</math> where V,W are vector spaces that takes <math>(v_1,...,v_m)\mapsto (Tv_1,...,Tv_m) = (w_1,..,w_m,w_{m+1},...,w_n)</math>

From linear algebra we know that we can choose a basis such that T is represented by a matrix with 1's along the first m diagonal locations and zeros elsewhere.

'''Theorem 2'''

Locally, every immersion looks like the inclusion <math>\iota:\mathbb{R}^m\rightarrow \mathbb{R}^n</math>.

More precisely, if <math>\theta:M^m\rightarrow N^n</math> and <math>d\theta_p</math> is 1:1 then there exists charts <math>\varphi</math> acting on <math>U\subset M</math> and <math>\phi</math> acting on <math>V\subset N</math> such that for <math>p\in U, \phi(p) = \psi(\theta(p)) = 0</math> such that the following diagram commutes:

<math>\begin{matrix}
U&\rightarrow^{\phi}&U'\subset \mathbb{R}^m\\
\downarrow_{\theta} &&\downarrow_{\iota} \\
V& \rightarrow^{\psi}& V'\subset \mathbb{R}^n\\
\end{matrix}</math>

that is, <math>\iota\circ\varphi = \psi\circ\theta</math>

'''Definition 2'''

M is a ''submanifold'' of N if there exists a mapping <math>\theta:M\rightarrow N</math> such that <math>\theta</math> is a 1:1 immersion.

'''Example 6'''

Our previous example of the graph of a "loop-de-loop", while an immersion, the function is not 1:1 and hence the graph is not a sub manifold.

'''Example 6'''

The torus is a submanifold as the natural immersion into <math>\mathbb{R}^3</math> is 1:1

'''Definition 3'''

The map <math>\theta:M\rightarrow N</math> is an embedding if the subset topology on <math>\theta(M)</math> coincides with the topology induced from the original topology of M.

'''Example 7'''

Consider the map from <math>\mathbb{R}\rightarrow \mathbb{R}^2</math> whose graph looks like the open interval whose two ends have been wrapped around until they just touch (would intersect at one point if they were closed) the points 1/3 and 2/3rds of the way along the interval respectively.
The map is both 1:1 and an immersion. However, any neighborhood about the endpoints of the interval will ALSO include points near the 1/3rd and 2/3rd spots on the line, i.e., the topology is different and hence this is not an embedding.

'''Corollary 1 to Theorem 2'''

The functional structure on an embedded manifold induced by the functional structure on the containing manifold is equal to its original functional structure.

Indeed, for all smooth <math>f:M\rightarrow \mathbb{R}</math> and <math>\forall p\in M</math> there exists a neighborhood V of <math>\theta(p)</math> and a smooth <math>g:N\rightarrow \mathbb{R}</math> such that <math>g|_{\theta(M)\bigcap U} = f|_U</math>

''Proof of Corollary 1''

Loosely (and a sketch is most useful to see this!) we consider the embedded submanifold M in N and consider its image, under the appropriate charts, to a subset of <math>\mathbb{R}^m\subset \mathbb{R}^n</math>. We then consider some function defined on M, and hence on the subset in <math>\mathbb{R}^n</math> which we can extend canonically as a constant function in the "vertical" directions. Now simply pullback into N to get the extended member of the functional structure on N.

''Proof of Theorem 2''

We start with the normal situation of <math>\theta:M\rightarrow N</math> with M,N manifolds with atlases containing <math>(\varphi_0,U_)0)</math> and <math>(\psi_0, V_0)</math> respectively. We also expect that for <math>p\in U_0, \varphi_0(p) = \psi_0(\theta(p)) = 0</math>. I will first draw the diagram and will subsequently justify the relevant parts. The proof reduces to showing a certain part of the diagram commutes appropriately.

<math>
\begin{matrix}
M\supset U_0 & \rightarrow^{\varphi_0} & U_1\subset \mathbb{R}^m & \rightarrow^{Id} & U_2 = U_1 \\
\downarrow_{\theta} & &\downarrow_{\theta_1} & &\downarrow_{\iota}\\
N\supset V_0 & \rightarrow^{\psi_0} & V_1\subset \mathbb{R}^n & \leftarrow^{\xi} & V_2\\
\end{matrix}
</math>

It is very important to note that the <math>\varphi_0</math> and <math>\psi_0</math> are NOT the charts we are looking for , they are merely one of the ones that happen to act about the point p.

In the diagram above, <math>\theta_1 = \psi_0\circ\theta\circ\varphi^{-1}</math>. So, <math>\theta_1(0) = 0</math> and <math>(d\theta_1)_0 = i</math>. Note the <math>\theta_1</math>, being merely the normal composition with the appropriate charts, does not fundamentally add anything. What makes this theorem work is the function <math>\xi</math>

We consider the map <math>\xi:V_2\rightarrow V_1</math> given <math>(x,y)\mapsto \theta_1(x) + (0,y)</math>. We note that <math>\xi</math> corresponds with the idea of "lifting" a flattened image back to its original height.

Claims:

1) <math>\xi</math> is invertible near zero. Indeed, computing <math>d\xi_0 = I</math> which is invertible as a matrix and hence <math>\xi</math> is invertible as a function near zero.

2) Take an <math>x\in U_2</math>. There are two routes to get to <math>V_1</math> and upon computing both ways yields the same result. Hence, the diagram commutes.

Hence, our immersion looks (locally) like the standard immersion between real spaces given by <math>\iota</math> and the charts are the compositions going between <math>U_0</math> to <math>U_2</math> and <math>V_0</math> to <math>V_2</math>

''Q.E.D''

0708-1300/Class Photo

2007-09-29T01:45:55Z

Kuramay:

{{0708-1300/Navigation}}

Our class on September 27, 2007:

[[Image:0708-1300-ClassPhoto.jpg|thumb|centre|600px|Class Photo: click to enlarge]]

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{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}}
{{Photo Entry|last=Bazett|first=Trefor|userid=Trefor|email=trefor.bazett @ toronto.ca|location=tallest person a little right of center in a beige shirt|comments=}}
{{Photo Entry|last=Bjorndahl|first=Adam|userid=ABjorndahl|email=adam.bjorndahl @ utoronto.ca|location=back row, fifth from the left, under the "f(tp)dt"|comments=Looking forward to a great year!}}
{{Photo Entry|last=Chow|first=Aaron|userid=aaron.chow|email=aaron @ utoronto.ca|location=Third from right, in a black shirt.|comments=Hope we have a good year together!}}
{{Photo Entry|last=Isgur|first=Abraham|userid=Abisgu|email=abraham.isgur@ math.toronto.edu|location=2nd person in the back row, from the right, the one with the beard and long hair|comments=}}
{{Photo Entry|last=Vera Pacheco|first=Franklin|userid=Franklin|email=franklin.vp @ gmail.com|location=Xth from left to right|comments=To find me you must first go to [[http://www.deathball.net/notpron/]] solve the first 4 pages. Once this done you will know how to find me. Once this done go back to NOTPRON an solve the rest of the puzzle}}
{{Photo Entry|last=Wong|first=Silian|userid=kuramay|email=kurama_y @ hotmail.com|location=One of the Asian-looking girls...with sparkling teeth(??)|comments=I'll write up some comments after their existences}}
|}

0708-1300/Class Photo

2007-09-29T01:45:36Z

Kuramay:

{{0708-1300/Navigation}}

Our class on September 27, 2007:

[[Image:0708-1300-ClassPhoto.jpg|thumb|centre|600px|Class Photo: click to enlarge]]

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{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}}
{{Photo Entry|last=Bazett|first=Trefor|userid=Trefor|email=trefor.bazett @ toronto.ca|location=tallest person a little right of center in a beige shirt|comments=}}
{{Photo Entry|last=Bjorndahl|first=Adam|userid=ABjorndahl|email=adam.bjorndahl @ utoronto.ca|location=back row, fifth from the left, under the "f(tp)dt"|comments=Looking forward to a great year!}}
{{Photo Entry|last=Chow|first=Aaron|userid=aaron.chow|email=aaron @ utoronto.ca|location=Third from right, in a black shirt.|comments=Hope we have a good year together!}}
{{Photo Entry|last=Isgur|first=Abraham|userid=Abisgu|email=abraham.isgur@ math.toronto.edu|location=2nd person in the back row, from the right, the one with the beard and long hair|comments=}}
{{Photo Entry|last=Vera Pacheco|first=Franklin|userid=Franklin|email=franklin.vp @ gmail.com|location=Xth from left to right|comments=To find me you must first go to [[http://www.deathball.net/notpron/]] solve the first 4 pages. Once this done you will know how to find me. Once this done go back to NOTPRON an solve the rest of the puzzle}}
{{Photo Entry|last=Wong|first=Silian|userid=kuramay|email=kurama_y@hotmail.com|location=One of the Asian-looking girls...with sparkling teeth(??)|comments=I'll write up some comments after their existences}}
|}

0708-1300/Class Photo

2007-09-29T01:44:45Z

Kuramay:

{{0708-1300/Navigation}}

Our class on September 27, 2007:

[[Image:0708-1300-ClassPhoto.jpg|thumb|centre|600px|Class Photo: click to enlarge]]

Please identify yourself in this photo! There are two ways to do that:

* [[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.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{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}}
{{Photo Entry|last=Bazett|first=Trefor|userid=Trefor|email=trefor.bazett @ toronto.ca|location=tallest person a little right of center in a beige shirt|comments=}}
{{Photo Entry|last=Bjorndahl|first=Adam|userid=ABjorndahl|email=adam.bjorndahl @ utoronto.ca|location=back row, fifth from the left, under the "f(tp)dt"|comments=Looking forward to a great year!}}
{{Photo Entry|last=Chow|first=Aaron|userid=aaron.chow|email=aaron @ utoronto.ca|location=Third from right, in a black shirt.|comments=Hope we have a good year together!}}
{{Photo Entry|last=Isgur|first=Abraham|userid=Abisgu|email=abraham.isgur@ math.toronto.edu|location=2nd person in the back row, from the right, the one with the beard and long hair|comments=}}
{{Photo Entry|last=Vera Pacheco|first=Franklin|userid=Franklin|email=franklin.vp @ gmail.com|location=Xth from left to right|comments=To find me you must first go to [[http://www.deathball.net/notpron/]] solve the first 4 pages. Once this done you will know how to find me. Once this done go back to NOTPRON an solve the rest of the puzzle}}
{{Photo Entry|last=Wong|first=Silian|userid=kuramay|email=kurama_y@homail.com|location=One of the Asian-looking girls...with sparkling teeth(??)|comments=I'll write up some comments after their existences}}
|}

0708-1300/Class notes for Tuesday, September 18

2007-09-26T19:06:17Z

Kuramay:

{{0708-1300/Navigation}}
==Dror's Note==
My office hours today will take place at 3:30-4:30 instead of the usual 12:30-1:30.

===Exercise===

[[Image:0708-1300torus.jpeg|thumb|left|200px|Finding the hidden body]]

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
{{0708-1300/Class Notes}}

==Scanned lecture notes for Thurs Sept 18 by Kuramay:==

[[Image:0708-1300-Sept18_01.jpg|200px|]]
[[Image:0708-1300-Sept18_02.jpg|200px]]
[[Image:0708-1300-Sept18_03.jpg|200px]]
[[Image:0708-1300-Sept18_04.jpg|200px]]
[[Image:0708-1300-Sept18_05.jpg|200px]]

File:0708-1300-Sept18 05.jpg

2007-09-26T18:56:35Z

Kuramay:

File:0708-1300-Sept18 04.jpg

2007-09-26T18:56:15Z

Kuramay:

File:0708-1300-Sept18 03.jpg

2007-09-26T18:55:54Z

Kuramay:

File:0708-1300-Sept18 02.jpg

2007-09-26T18:55:34Z

Kuramay:

File:0708-1300-Sept18 01.jpg

2007-09-26T18:54:59Z

Kuramay:

0708-1300/Class notes for Thursday, September 13

2007-09-24T19:24:01Z

Kuramay:

{{0708-1300/Navigation}}
==A Message From Ed Barbeau==
The Putnam Competition is written by undergraduates in North
America. It is now about 70 years old and University of Toronto
students have competed with distinction.

There is no registration fee to write the competition. It is open
to all university undergraduates who have written the examination
no more than three times in the past.

It is generally held on the first Saturday of December. There are
two three-hour papers, each with six questions, written at 10 am
and 3 pm.

Any student interested in writing the Putnam Competition should
send an email message to Ed Barbeau at barbeau@math.utoronto.ca.
It would help to know (a) your year, (b) program of study,
(c) any successes that you have had in past competitions at the
secondary level.

----

== Scanned lecture notes for Thurs Sept 13 by Kuramay: ==
[[Image:0708-1300-Sept13_01.jpg]]
[[Image:0708-1300-Sept13_02.jpg]]
[[Image:0708-1300-Sept13_03.jpg]]

User talk:Kuramay

2007-09-23T15:13:02Z

Kuramay:

Sorry, I was trying to type in "0708-1300/" before the name of the images, and it didn't work.
(Only the "0708-1300-" works well). Professor, please kindly delete all the duliputed pages.

Dear Kuramay,

I've deleted the images that does not begin with "0708-1300-". Two further notes -

# The scans are of a rather low quality. Scanners can do a lot better and it's worthwhile to learn how to make them work right!
# You did not link the images to the relevant class page ([[0708-1300/Class notes for Thursday, September 13]], I suppose), so as they stand, they are not useful. Please link them soon; I routinely remove unlinked images from the wiki.

--[[User:Drorbn|Drorbn]] 14:00, 21 September 2007 (EDT)

Professor:
I do have question regarding your further notes:

# the scans I've made are in resolution 1000 x 1200 already, which made the files larger than the recommanded upload limit. May I know what should I do to make them in higher quality?
# I am still investigating how to use the wiki system. where should I go to link the images?

User talk:Kuramay

2007-09-21T01:44:04Z

Kuramay:

Sorry, I was trying to type in "0708-1300/" before the name of the images, and it didn't work.
(Only the "0708-1300-" works well). Professor, please kindly delete all the duliputed pages.

File:0708-1300-Sept13 01.jpg

2007-09-21T01:38:44Z

Kuramay: Class note for last Thurs: Sept 13

Class note for last Thurs: Sept 13

File:0708-1300-Sept13 02.jpg

2007-09-21T01:37:37Z

Kuramay: Class note for last Thurs: Sept 13

Class note for last Thurs: Sept 13

File:0708-1300-Sept13 03.jpg

2007-09-21T01:35:10Z

Kuramay: