Solve Notation Puzzle: Vertical Bars in Equations Homework

[Aaerion]

Homework Statement

I have these two equations from this paper: https://www.scribd.com/doc/299960566/Spiral

Homework Equations

What are the vertical bars next to the drawn red starts supposed to indicate in this context?

I'm trying to implement these equations into a program but I'm totally lost. I have no idea what's up with those vertical bars next to the red stars that I drew. Is it supposed to be the terms evaluated at (j+1-rtheta)? If so, then what variable is it even being plugged into (r or theta)?

The Attempt at a Solution

Googling and asking my professor.

 

[Mark44]

Homework Statement

I have these two equations from this paper: https://www.scribd.com/doc/299960566/Spiral

Homework Equations

What are the vertical bars next to the drawn red starts supposed to indicate in this context?

Vertical bars like that usually mean at what value the quantity to the left is evaluated at. The summation looks screwy to me, though, as the sum runs from j = 1 to j - 1, which doesn't make sense. If it had been written like what's below, that would make more sense to me.
$$\sum_{j = 1}^{J - 1} \dots$$
I.e., not reusing j (lower case) as both the index of the sum and the ending point.

The author of the paper you're citing is an MD, so I suspect that he is not as able as he could be to communicate mathematical equations.

I'm trying to implement these equations into a program but I'm totally lost. I have no idea what's up with those vertical bars next to the red stars that I drew. Is it supposed to be the terms evaluated at (j+1-rtheta)? If so, then what variable is it even being plugged into (r or theta)?

The Attempt at a Solution

Googling and asking my professor.

 

[Aaerion]

I just looked up zero crossing function in general and I think I figured it out. From what I can understand, the summation should actually be from j=1 to J-1. Furthermore, inside the sign functions, the average slope is subtracted from the slope at points j+1 and j respectively. Doing this will allow me to determine how many times the slope crosses its mean during data collection.

 

[Mark44]

I just looked up zero crossing function in general and I think I figured it out. From what I can understand, the summation should actually be from j=1 to J-1.

That makes much more sense.

Furthermore, inside the sign functions, the average slope is subtracted from the slope at points j+1 and j respectively. Doing this will allow me to determine how many times the slope crosses its mean during data collection.