Evolution of a picture of Hollyhocks

A simple theorem of glance functions

THIS IS NAIVELY WRONG BUT MAY CONTAIN AN IDEA

A little Theorem about glance functions [described in my Hypothesis Testing... article].

Let h_a be the glance function of a simple interval of length a.

If H is the glance function for a collections of intervals separated by gaps, and a new interval of length a is added after a gap of length b, then the glance function of the new collection of intervals is

(*)                                      H -> H – H^b  + H^a + h_b + h_a  - h_a+b

Where the superscript on a function H indicates a term-wise shift of the independent variable by adding that amount, so  H^a(x)= H(a+x).

Corollary: the value of the glance function after the last step equals the number of intervals being glanced.

Proof: It is true for one interval, since h_a equals 1 at it's last step. If true for H, with N final steps, then we note the terms in the above (*) has final step values: +N , -N, +N,  +1, +1, -1. This totals to N+1. Hence the corollary is true by induction.

Update: The correct formula is

(**)                     H à H – H^b  + H^a+b + (h_a + h_b – h_a+b)

The induction argument is the same. We can name the first part the "shifted H" part. The rest is the "self-contained" info about the added segment.

Note: this encourages thinking about re-constructing the original segments, starting from their glance function. Because we know how many segments are involved and the maximum length from first to last end points of the segments.

Compare me to a paremecium

I was reading Feynman about his observing paramecium behavior to be quite diverse and unpredictable versus what the textbook said about paramecium moving and bouncing off of obstructions.

As I sit here, doing what I always did on a happy Saturday morning [except it is Thursday] - sitting with a piece of paper and thinking about abstractions, I realize that sitting at my desk is what I spend almost the entirety of my time doing. I move around a bit. I eat and sleep. But from the pov of an alien observer, they would conclude that my sitting with a piece of paper is what I do. Everything else in my life would be reduced to something not mentioned in the observer's textbook.

June Hollyhocks