Circles - } , then d 1 ( ~x, ~ y ) is just the Hamming...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
What shape is a circle? Complex Systems Summer School, Santa Fe Institute Tom Carter http://astarte.csustan.edu/˜ tom/SFI-CSSS June 24, 2006 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
How do we define a circle? We usually define a circle as C = { ~x | || ~x || = 1 } (i.e., the set of all vectors ~x of length 1). Of course, we need to make sense of || ~x || , the length of a vector. Most people start with the definition || ~x || = X i x 2 i 1 2 , but since we are mathematicians, and don’t like our definitions to be too special, we can generalize: || ~x || p = X i | x i | p 1 p . for p > 0. 2
Background image of page 2
We then have the more general definition of the p circle (in dimension n ): C n p = { ~x | || ~x || p = 1 } . What does C n p look like? Let’s work in two dimensions, and leave out the dimension label. C 2 is the familiar circle: 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
C 1 is a diamond: Note that if our vector space is over { 0 , 1 } , then a vector is just a string of zeros and ones, and || ~x || 1 is just the number of ones in the string. We can convert our length measures into a distance measures: d p ( ~x, ~ y ) = || ~x - ~ y || p = X i | x i - y i | p 1 p . 4
Background image of page 4
In particular, if our vector space is over { 0 , 1
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: } , then d 1 ( ~x, ~ y ) is just the Hamming distance between the two vectors (i.e., the number of places in which the two strings dier). We can also dene || ~x || = lim p ( || ~x || p ) . Going through the math, we have that || ~x || = max i ( | x i | ) . We then have that C is a square: 5 This means that for a mathematician, a circle is a circle, is a diamond, is a square (which may explain why I always had trouble with those shape matching tests . . . :-) In general, we have the following sort of picture of various circles: 6 Homework exercises: What happens for 0 < p < 1? What happens if we take the limit as p goes to 0? Show that in the limit as p goes to 0, the corresponding distance d ( ~x, ~ y ) = || ~x-~ y || is a generalized Hamming distance that counts the number of coordinates that are dierent from each other . . . 7...
View Full Document

Page1 / 7

Circles - } , then d 1 ( ~x, ~ y ) is just the Hamming...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online