0708-1300/Class notes for Tuesday, October 16 - Revision history

Trefor: /* Second Hour */

2007-11-07T22:06:07Z

Second Hour

← Older revision		Revision as of 18:06, 7 November 2007
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	''Proof''		''Proof''

	W.L.O.G (without loss of generality) we can assume n=1. Intuitively this is reasonable as in lower dimensions the theorem is harder to prove; indeed, a set of size <math>\epsilon</math> in 1D becomes a smaller set of size <math>\epsilon^2</math> in 2D etc. More precisely, for <math>f = (f_1,\ldots,f_n)</math>, <math>f(~~D_n~~)\subset f_1(D_m)\times{R}^{n-1}</math>. Applying the proposition that if A is of measure zero in <math>\mathbb{R}</math> then <math>A\times\mathbb{R}^{n-1}</math> is measure zero in <math>\mathbb{R}^n</math> we now see that assuming n=1 is justified.		W.L.O.G (without loss of generality) we can assume n=1. Intuitively this is reasonable as in lower dimensions the theorem is harder to prove; indeed, a set of size <math>\epsilon</math> in 1D becomes a smaller set of size <math>\epsilon^2</math> in 2D etc. More precisely, for <math>f = (f_1,\ldots,f_n)</math>, <math>f(D_m)\subset f_1(D_m)\times{R}^{n-1}</math>. Applying the proposition that if A is of measure zero in <math>\mathbb{R}</math> then <math>A\times\mathbb{R}^{n-1}</math> is measure zero in <math>\mathbb{R}^n</math> we now see that assuming n=1 is justified.

Megan: /* Second Hour */

2007-11-05T02:18:57Z

Second Hour

← Older revision		Revision as of 22:18, 4 November 2007
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	For notational convenience let us say <math>f =(f_1,f_{rest})</math>.		For notational convenience let us say <math>f =(f_1,f_{rest})</math>.

	Now define <math>U_t = \{t\} \times\mathbb{R}^{m-1}</math>		Now define <math>U_t = \{t\} \times\mathbb{R}^{m-1}</math>.
	Also ~~lets~~ denote "critical points of f'' by CP(f) and "critical values of f" by CV(f)		Also, let us denote "critical points of f" by CP(f) and "critical values of f" by CV(f).

	'''Claim 4'''		'''Claim 4'''

Megan: /* Second Hour */

2007-11-05T02:07:10Z

Second Hour

← Older revision		Revision as of 22:07, 4 November 2007
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	Recall <math>D_0</math> is defined differently from the <math>D_k</math> and so requires a different technique to prove.		Recall <math>D_0</math> is defined differently from the <math>D_k</math> and so requires a different technique to prove.

	W.L.O.G. ~~lets~~ assume that <math>D_1</math> is the empty set. So, some derivative of f is not zero. W.L.O.G. <math>\frac{\partial f_1}{\partial x_1}</math> is non zero near some point p. We can simply move to a coordinate system where this is true.		W.L.O.G. assume that <math>D_1</math> is the empty set. So, some derivative of f is not zero. W.L.O.G. <math>\frac{\partial f_1}{\partial x_1}</math> is non zero near some point p. We can simply move to a coordinate system where this is true.

	The idea here is to prove that the intersection with any "slice" has measure zero where we will then invoke a theorem that will claim everything has measure zero.		The idea here is to prove that the intersection with any "slice" has measure zero where we will then invoke a theorem that will claim everything has measure zero.

Megan: /* First Hour */

2007-11-05T00:59:12Z

First Hour

← Older revision		Revision as of 20:59, 4 November 2007
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	Firstly, it is enough to argue locally.		Firstly, it is enough to argue locally, since a manifold is second countable (that is, a manifold has a countable basis) and a countable union of sets of measure zero has measure zero.

	Further, the technical assumption about manifolds that until now has been largely ignored is that our M must be second countable. Recall that this means that there is a countable basis for the topology on M.		Further, the technical assumption about manifolds that until now has been largely ignored is that our M must be second countable. Recall that this means that there is a countable basis for the topology on M.
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	Now, set <math>D_k = \{p \|</math> all partial derivities of f of order <math>\leq k\}</math> for <math>k\geq 1</math>		Now, set <math>D_k = \{p \|</math> all partial derivities of f of order <math>\leq k </math> vanish} for <math>k\geq 1</math>

	Set <math>D_0 = \{p\ \|\ df_p</math> is not onto }. This is just the critical points.		Set <math>D_0 = \{p\ \|\ df_p</math> is not onto }. This is just the critical points.
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	''Comment:''		''Comment:''

	We have not actually defined the measure <math>\mu</math>. We use it merely to denote that <math>F(D_k))</math> has measure zero, a concept that we DID define.		We have not actually defined the measure <math>\mu</math>. We use it merely to denote that <math>F(D_k))</math> has measure zero, a concept that we DID define.


	===Second Hour===		===Second Hour===

Aptikuis: /* Second Hour */

2007-10-31T12:52:58Z

Second Hour

← Older revision		Revision as of 08:52, 31 October 2007
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	So <math>D_k\subset g^{-1}(0)</math> but <math>g^{-1}(0)</math> is at least locally a manifold of dimension 1 less. So, <math>f(D_k)\subset f(D_k\cap g^{-1}(0)</math> which has measure zero due to our induction hypothesis.		So <math>D_k\subset g^{-1}(0)</math> but <math>g^{-1}(0)</math> is at least locally a manifold of dimension 1 less. So, <math>f(D_k)\subset f(D_k\cap g^{-1}(0))</math> which has measure zero due to our induction hypothesis.

	''Q.E.D for Claim 2''		''Q.E.D for Claim 2''
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	For notational convenience let us say <math>f =(f_1,f_{rest})</math>.		For notational convenience let us say <math>f =(f_1,f_{rest})</math>.

	Now define <math>U_t = \{t\times\mathbb{R}^{m-1}\}</math>		Now define <math>U_t = \{t\} \times\mathbb{R}^{m-1}</math>
	Also lets denote "critical points of f'' by CP(f) and "critical values of f" by CV(f)		Also lets denote "critical points of f'' by CP(f) and "critical values of f" by CV(f)

Drorbn at 13:48, 23 October 2007

2007-10-23T13:48:15Z

← Older revision		Revision as of 09:48, 23 October 2007
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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}
	==Dror's Computer Program for C+<math>\alpha</math> C==		==Dror's Computer Program for C+<math>\alpha</math> C==


	Our handout today is a printout of a Mathematica notebook that computes the measure of the projection of <math>C\times C</math> in a direction <math>t\in[0,\pi/2]</math> (where <math>C</math> is the standard Cantor set). Here's the {{Home Link\|classes/0708/GeomAndTop/CCShadow.nb\|notebook}}, and here's a {{Home Link\|classes/0708/GeomAndTop/CCShadow.pdf\|PDF}} version. Also, here's the main picture on that notebook:		Our handout today is a printout of a Mathematica notebook that computes the measure of the projection of <math>C\times C</math> in a direction <math>t\in[0,\pi/2]</math> (where <math>C</math> is the standard Cantor set). Here's the {{Home Link\|classes/0708/GeomAndTop/CCShadow.nb\|notebook}}, and here's a {{Home Link\|classes/0708/GeomAndTop/CCShadow.pdf\|PDF}} version. Also, here's the main picture on that notebook:

Trefor at 17:49, 16 October 2007

2007-10-16T17:49:57Z

← Older revision		Revision as of 13:49, 16 October 2007
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	==Typed Class Notes==		==Typed Class Notes==

			<span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span>

	=== First Hour===		=== First Hour===

Trefor at 17:47, 16 October 2007

2007-10-16T17:47:38Z

← Older revision		Revision as of 13:47, 16 October 2007
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	[[Image:0708-1300-CCShadow.png\|center\|540px]]		[[Image:0708-1300-CCShadow.png\|center\|540px]]

	==Typed Class Notes ~~- First Hour~~==		==Typed Class Notes==
			=== First Hour===

	'''Today's Agenda:'''		'''Today's Agenda:'''
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	We have not actually defined the measure <math>\mu</math>. We use it merely to denote that <math>F(D_k))</math> has measure zero, a concept that we DID define.		We have not actually defined the measure <math>\mu</math>. We use it merely to denote that <math>F(D_k))</math> has measure zero, a concept that we DID define.


			===Second Hour===

Trefor at 17:44, 16 October 2007

2007-10-16T17:44:51Z

Show changes

Drorbn at 23:02, 15 October 2007

2007-10-15T23:02:38Z

← Older revision		Revision as of 19:02, 15 October 2007
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	{{0708-1300/Navigation}}		{{0708-1300/Navigation}}

	Our handout today is a printout of a Mathematica notebook that computes the measure of the projection of <math>C\times C</math> in a direction <math>t\in[0,\pi/2]</math>. Here's the {{Home Link\|classes/0708/GeomAndTop/CCShadow.nb\|notebook}}, and here's a {{Home Link\|classes/0708/GeomAndTop/CCShadow.pdf\|PDF}} version. Also, here's the main picture on that notebook:		Our handout today is a printout of a Mathematica notebook that computes the measure of the projection of <math>C\times C</math> in a direction <math>t\in[0,\pi/2]</math> (where <math>C</math> is the standard Cantor set). Here's the {{Home Link\|classes/0708/GeomAndTop/CCShadow.nb\|notebook}}, and here's a {{Home Link\|classes/0708/GeomAndTop/CCShadow.pdf\|PDF}} version. Also, here's the main picture on that notebook:

	[[Image:0708-1300-CCShadow.png\|center\|540px]]		[[Image:0708-1300-CCShadow.png\|center\|540px]]