u/Gabry398

Hi everyone, I have a question regarding the algorithm at the heart of Cantor's diagonal argument. That is we have a list of numbers (It can be finite or infinite) and a new number is added based on the digits of the previous numbers.

I don't care about proving anything about the cardinality of R, I'm interested in the sub-set of R that is actually generated starting from a particular list (be it finite or infinite).

Is there a list of numbers (cardinality N at most obviously) that "generate" all of R starting with this algorithm? More formally, for every r in (0,1) is it true that r is reached by my ever growing list after a certain amount of iterations? I'd intuitively say no, but at this point my intuition starts to break down a little, especially with AC.

The idea behind this was noticing a similarity with span in linear algebra, maybe the sub-set generated by a particular list has some interesting properties, and when I add a new number to the list I'm adding a new class of numbers after a certain amount of iterations.

I'm imagining this can be generalized with any recursive algorithm, so probably there is something similar already out there.

Does anybody know if this question is explored somewhere, and if not how would I approach this?

Hi everyone, I don't really know what I'm doing.

I have an HP laptop, the backlight and various LEDs of my keyboard work as intended except for the one on the mute button. I found out that apparently the LEDs are controlled(?) by "HDA nodes". Through a brute force script I found out what were the HDA nodes that are connected to the LED, sorta. Apparently there are two HDA nodes that respectively turn on and off the thing, but other than that I think that there is a third one that needs to be turned on before being able to do anything else. After the brute force script I could simply use the two commands from the terminal to turn on and off the LED, but when I try it now (after restarting my machine) it doesn't work.

I know there is some way to turn on that damn LED, but I don't know if my hypotesis is correct, what exactly I'm getting wrong and how (if i were able to find the third mysterious HDA node) to use all of this new information to make the LED work as intended. Help is much appreciated, thanks.

Additional info that may or may not help:
Model: HP pavilion 14-ce3xxx

Commands that makes the LED turn on (after the brute force script):
sudo hda-verb /dev/snd/hwC0D0 0x20 0x500 0x08

sudo hda-verb /dev/snd/hwC0D0 0x20 0x400 0x04

Commands that makes the LED turn off (after the brute force script):
sudo hda-verb /dev/snd/hwC0D0 0x20 0x500 0x08

sudo hda-verb /dev/snd/hwC0D0 0x20 0x400 0x08

Codec:ALC295

Also I did use claude to generate the script and commands, I'm just not familiar at all with things closely tied to the hardware, so I'd appreciate someone to clear some confusion I have specifically about how all of this work.