# Correction on my previous blog and some interesting implications …

Andrey Mokhov spotted that to satisfy the actual inverse Pythagorean we need to have alpha=1/2 rather than 2. That’s right. Indeed, what happens is that if we have alpha = 1/2 we would have (1/a)^2=(1/a1)^2+(1/a2)^2. This is what the inverse Pythagorean requires. In that case, for instance if a1=a2=2, then a must be sqrt(2). So the ratio between the individual decay a1=a2 and the collective decay is sqrt(2). For our stack decay under alpha = 2, we would have for a1=a2=2, a=1/2, so the ratio between individual decay and collective decay is 4.

It’s actually quite interesting to look at these relations a bit deeper, and see how the “Pythagorean” (geometric) relationship evolves as we change alpha from something like alpha<=1/2 to alpha>=2.

If we take alpha going to 2 and above, we have the effect of much slower collective decay than 4x compared to the individual decay. Physically this corresponds to the situation when the delay of an inverter in the ring becomes strongly inversely proportional to voltage. Geometrically, this is like contracting the height of the triangle in which sides go further apart than 90 degrees – say the triangle is isosceles for simplicity, and say its angle is say 100 degrees.

The case of alpha = 1/2 corresponds to the case where delay is proportional to the square root of Voltage, and here the stack makes the decay rate to follow the inverse Pythagorean! So this is the case of a triangle with sides being at 90 degrees.

But if alpha goes below 1/2, we have the  effect of the collective decay being closer to individual decays, and geometrically the height of the triangle where sides close up to less than 90 degrees!

Incidentally, Andrey Mokhov suggested we may consider a different physical interpretation for inverse Pythagorean. Instead of looking at lengths a, b and h, one can consider volumes Va, Vb and Vh of 4-D cubes with such side lengths. Then these volumes would relate exactly as in our case of alpha=2, i.e. 1/sqrt (Vh)=1/sqrt(Va)+1/sqrt(Vb).

Cool!

# Charge decay in a stack of two digital circuits follows inverse Pythagorean Law!

My last blog about my talk at HDT 2019 on Stacking Asynchronous Circuits contained a link to my slides. I recommend you having a particular look at slide #21. It talks about an interesting fact that a series (stack) discharge rate follows the law of the inverse Pythagorean!

It looks like mother nature caters for a geometric law of the most economic common between two individual sides.

# My Talk on Stacked Asynchronous Circuits at HDT 2019

I just attended a Second Workshop on Hardware Design Theory, held in Budapest, collocated with 33rd International Symposium on Distributed Computing http://www.disc-conference.org/wp/disc2019/

The HDT’19 workshop was organised by Moti Medina and Andrey Mokhov. It had a number of invited talks, and here is the programme: https://sites.google.com/view/motimedina/hdt-2019

I gave a talk on Stacked Asynchronous circuits.

Here is the abstract: In this talk we will look at digital circuits from the viewpoint of electrical circuit theory, i.e. as loads to power sources. Such circuits, especially when they are asynchronous can be seen as voltage controlled oscillators. Their switching behaviour, including their operating frequency is modulated by the supply voltage. Interestingly, in the reverse direction if they are driven by external event sources, their switching frequency determines their inherent impedance which itself makes them ideal potentiometers or voltage dividers. Such circuits can be stacked like non-linear resistors in series and parallel, and lend themselves to interesting theoretical and practical results, such as RC circuits with hyperbolic capacitor discharges and designs of dynamic frequency mirrors.

Here is the PDF of my slides: https://www.staff.ncl.ac.uk/alex.yakovlev/home.formal/stacked-async-budapest-2019.171019.pdf