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-   -   Concentric circles in Q10 (http://book.caltech.edu/bookforum/showthread.php?t=926)

ESRogs 07-28-2012 10:11 AM

Concentric circles in Q10
 
The inequality a^2 <= x1^2 + x2^2 <= b^2 in question 10 implies

1) that the points have to be outside of the smaller circle and inside the larger circle, and
2) that the circles have to be centered at the origin.

Is that correct?

yaser 07-28-2012 01:45 PM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by ESRogs (Post 3724)
The inequality a^2 <= x1^2 + x2^2 <= b^2 in question 10 implies

1) that the points have to be outside of the smaller circle and inside the larger circle, and
2) that the circles have to be centered at the origin.

Is that correct?

Correct. To be exact about inside and and outside, the perimeters of the inner circle and the outer circle are included in the region.

ESRogs 07-28-2012 05:01 PM

Re: Concentric circles in Q10
 
Thanks!

ilya239 07-31-2012 06:54 AM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by yaser (Post 3725)
Correct. To be exact about inside and and outside, the perimeters of the inner circle and the outer circle are included in the region.

So once the set of N points is fixed, and we pick different subsets of the N points and try to cover them with "donuts" -- all donuts must be centered at the same point? We can't use a donut centered at p1 to cover {x1,x2} but a donut centered at a different point p2 to cover {x3,x4}? In other words, after picking N points, we pick ONE origin, and _then_ can pick donuts centered at that origin only?

yaser 07-31-2012 07:08 AM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by ilya239 (Post 3757)
So once the set of N points is fixed, and we pick different subsets of the N points and try to cover them with "donuts" -- all donuts must be centered at the same point? We can't use a donut centered at p1 to cover {x1,x2} but a donut centered at a different point p2 to cover {x3,x4}? In other words, after picking N points, we pick ONE origin, and _then_ can pick donuts centered at that origin only?

Correct. You can choose any N points you want to work with (with a view to maximizing the number of dichotomies), so the origin is effectively arbitrary, but fixed, for a given set of points.

doris 09-21-2012 07:14 PM

Re: Concentric circles in Q10
 
In order to test the effective number of hypotheses in H, how can we fix the center of the circles? by fixing the center we restrict ourselves to one hypothesis. So, I am more confused by the last comment. My thought process is find a set of N points and look through all possible concentric circles, so all radii and centers, that will give me each dichotomy possible on N. And this is how H shatters N, not each single hypothesis. Is this correct?

What is a correct strategy to approach this problem? can we reduce it to 1D with an interior interval an exterior interval (to infinity) =-1 and the 2 in-between regions (between the 2 circles) =+1

then it becomes a more complex version of the 2.3 c, the positive, negative intervals?

doris 09-22-2012 02:57 PM

Re: Concentric circles in Q10
 
the last comment confused me a little bit.
For a given set of N points, we should change the center of the sphere to get as many dichotomies as we can, thus measuring the effective number of hypotheses (spheres) in this hypothesis set.

Does it make sense to move project the spheres from 3D to 1D and look at the problem as intervals of +1 for a<=x<=b and -a>=x>=b?

geekoftheweek 01-25-2013 03:13 PM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by ilya239 (Post 3757)
So once the set of N points is fixed, and we pick different subsets of the N points and try to cover them with "donuts" -- all donuts must be centered at the same point? We can't use a donut centered at p1 to cover {x1,x2} but a donut centered at a different point p2 to cover {x3,x4}? In other words, after picking N points, we pick ONE origin, and _then_ can pick donuts centered at that origin only?

This approach confuses my understanding of the problem. Why doesn't the symmetry of the problem reduce it to 1-D (r instead of x) with two parameters, i.e. the "positive intervals" case in example 2.2.2?
Thanks

Suhas Patil 01-26-2013 12:14 AM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by geekoftheweek (Post 8995)
This approach confuses my understanding of the problem. Why doesn't the symmetry of the problem reduce it to 1-D (r instead of x) with two parameters, i.e. the "positive intervals" case in example 2.2.2?
Thanks

Sounds logical...the linear scale representing radius can range from 0 (center of the concentric circles) to infinity with positive interval a-b contained within.

Elroch 04-17-2013 06:18 PM

Re: Concentric circles in Q10
 
When I skimmed this question first of all, I started thinking about annuli with arbitrary centres, which is an interesting more powerful hypothesis set.

IsidroHidalgo 04-21-2013 08:11 AM

Re: Concentric circles in Q10
 
But I finally don't have a clear idea of the center of our hypothesis here: is always x_1=0 and x_2=0 or not?

Elroch 04-21-2013 08:56 AM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by IsidroHidalgo (Post 10529)
But I finally don't have a clear idea of the center of our hypothesis here: is always x_1=0 and x_2=0 or not?

Well, x_1 and x_1 are the variables, but the centre is at (0,0) as indicated in yaser's post #2.

marek 04-21-2013 09:56 AM

Re: Concentric circles in Q10
 
You can choose your points however you want. So fix any layout of points by drawing them on a piece of piece of paper. If you translate all the points by moving that piece of paper around, the new positions are another set of points which you could have chosen.

So in a sense you can consider any layout of points and then start the circles where ever you wanted. The trick is that once you pick a spot for the centers of the circles you have to keep that same center for all the dichotomies you're generating.

And that does also mean that you can assume one point is in the center if you want.

IsidroHidalgo 04-21-2013 10:26 AM

Re: Concentric circles in Q10
 
But he says to in #5 "so the origin is effectively arbitrary", so there's a little confusion here for me...

marek 04-21-2013 11:06 AM

Re: Concentric circles in Q10
 
Quote:

Originally Posted by IsidroHidalgo (Post 10534)
But he says to in #5 "so the origin is effectively arbitrary", so there's a little confusion here for me...

That's what I'm trying to address in my above post.

Let me make my analogy a little more substantial. Let's say your grid and origin are fixed on your desk. The paper you draw your points on is transparent. You can slide around that transparency as you want, since your choice of points is up to you. But from the perspective of the points on the transparency, the points are fixed but the origin is sliding around. That is what he's referring to by saying the origin is effectively arbitrary.

Once you pick your points, you can put your origin anywhere you want (though the actual mechanic is that you're making new set of points that are appropriately translated with respect to the origin and using them instead)


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