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-   -   Problem 2.3c (http://book.caltech.edu/bookforum/showthread.php?t=4632)

sixdegrees 10-12-2015 03:18 PM

Problem 2.3c
 
I am having a bit of trouble with problem 2.3c (page 69). I have read some posts about it, including an exercise of the online course in which positive circles are treated (which I suppose is the 2D dimension of this problem).

I understand that I can transform this R^d problem into an R problem by doing https://latex.codecogs.com/gif.latex...ts+x_d^2} in https://latex.codecogs.com/gif.latex...e;\leq&space;b.

Thus, it turns into an equivalent case to positive intervals. However, I wonder if the growth function should change depending on the number of dimensions...

It is confusing for me to define the +1 region depending on x_1, x_2, ..., x_d. Why do we need d points in R^d? Does this mean that the binomial coefficient of the growth function depends on 'd' and not on 'N'? Does the break point change respect to positive intervals?

It would be great if you could clarify my doubts.

Thanks in advance.

magdon 10-13-2015 12:57 PM

Re: Problem 2.3c
 
Yes, this problem can be reduced to positive intervals for the space [0,Inf) by the transformation you mentioned. This also explains why M(n) does not depend on d.

Quote:

Originally Posted by sixdegrees (Post 12095)
I am having a bit of trouble with problem 2.3c (page 69). I have read some posts about it, including an exercise of the online course in which positive circles are treated (which I suppose is the 2D dimension of this problem).

I understand that I can transform this R^d problem into an R problem by doing https://latex.codecogs.com/gif.latex...ts+x_d^2} in https://latex.codecogs.com/gif.latex...e;\leq&space;b.

Thus, it turns into an equivalent case to positive intervals. However, I wonder if the growth function should change depending on the number of dimensions...

It is confusing for me to define the +1 region depending on x_1, x_2, ..., x_d. Why do we need d points in R^d? Does this mean that the binomial coefficient of the growth function depends on 'd' and not on 'N'? Does the break point change respect to positive intervals?

It would be great if you could clarify my doubts.

Thanks in advance.



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