• 6 Posts
  • 69 Comments
Joined 1 year ago
Cake day: June 14th, 2023
  • OmnipotentEntity@beehaw.orgtoSpace@beehaw.org•Highest-resolution black hole images ever taken usher in a new era
    3·
    12 days ago

    Oh, I’ll try to describe Euler’s formula in a way that is intuitive, and maybe you could have come up with it too.

    So one way to think about complex numbers, and perhaps an intuitive one, is as a generalization of “positiveness” and “negativeness” from a binary to a continuous thing. Notice that if we multiply -1 with -1 we get 1, so we might think that maybe we don’t have a straight line of positiveness and negativeness, but perhaps it is periodic in some manner.

    We can envision that perhaps the imaginary unit, i, is “halfway between” positive and negative, because if we think about what √(-1) could possibly be, the only thing that makes sense is it’s some form of 1 where you have to use it twice to make something negative instead of just once. Then it stands to reason that √i is “halfway between” i and 1 in this scale of positive and negative.

    If we figure out what number √i we get √2/2 + √2/2 i

    (We can find this by saying (a + bi)^(2) = i, which gives us (a^(2) - b^(2) = 0 and 2ab = 1) we get a = b from the first, and a^(2) = 1/2)

    The keen eyed observer might notice that this value is also equal to sin(45°) and we start to get some ideas about how all of the complex numbers with radius 1 might be somewhat special and carry their own amount of “positiveness” or “negativeness” that is somehow unique to it.

    So let’s represent these values with R ∠ θ where the θ represents the amount of positiveness or negativeness in some way.

    Since we’ve observed that √i is located at the point 45° from the positive real axis, and i is on the imaginary axis, 90° from the positive real axis, and -1 is 180° from the positive real axis, and if we examine each of these we find that if we use cos to represent the real axis and sin to represent the imaginary axis. That’s really neat. It means we can represent any complex number as R ∠ θ = cos θ + i sin θ.

    What happens if we multiply two complex numbers in this form? Well, it turns out if you remember your trigonometry, you exactly get the angle addition formulas for sin and cos. So R ∠ θ * S ∠ φ = RS ∠ θ + φ. But wait a second. That’s turning multiplication into an addition? Where have we seen something like this before? Exponent rules.

    We have a^(n) * a^(m) = a^(n+m) what if, somehow, this angle formula is also an exponent in disguise?

    Then you’re learning calculus and you come across Taylor Series and you learn a funny thing, the Taylor series of e^x looks a lot like the Taylor series of sine and cosine.

    And actually, if we look at the Taylor series for e^(ix) is exactly matches the Taylor series for cos x + i sin x. So our supposition was correct, it was an exponent in disguise. How wild. Finally we get:

    R ∠ θ = Re^(iθ) = cos θ + i sin θ

  • OmnipotentEntity@beehaw.orgtoNeurodivergence@beehaw.org•Why Anti-Authoritarians Are Diagnosed as Mentally IllEnglish
    11·
    13 days ago

    It’s not a mental disorder to simply be anti-authority, but it can be pathological. Not to take away from your point or anything, but I have a true story about a kid who went to college with my wife.

    He was a real prat who didn’t like being told what to do, and he seemed to take perverse pleasure in antagonizing authority figures who couldn’t directly punish him and who he considered to be beneath him. For instance, he would frequently leave his messy plates out at the dining hall, because he knew there would be no consequences for him, and he wanted the staff to have to clean up after him.

    Or this one time where the RA in the dorm was explaining how to choose a room for next year because everyone had to move out, he had a zippo lighter and was just throwing it up and catching it, and occasionally letting it fall to the ground and make a loud noise. He ignored instructions to stop doing that because it’s obnoxious, because the RA was an authority with no power, so was beneath him.

    All in all, cowardly behavior, and while I’m not a psychiatrist, and I cannot diagnose him, it certainly sounds like ODD to me.

    Anyway, this piece of shit’s name is Stephen Miller.

  • OmnipotentEntity@beehaw.orgtoProgramming@beehaw.org•Malicious VSCode extensions with millions of installs discovered
    8·
    3 months ago

    Not quite. Their “malicious” extension only got a few hundred installs. Using the data gathered by that extension and via other means they were able to locate other actually malicious extensions. Those total in the millions of installations.

    Through this process, they have found the following:

    1,283 with known malicious code (229 million installs).
8,161 communicating with hardcoded IP addresses.
1,452 running unknown executables.
2,304 that are using another publisher's Github repo, indicating they are a copycat.
  • OmnipotentEntity@beehaw.orgtoTechnology@beehaw.org•Adobe roofies all of their customers (Louis Rossmann's reaction)
    1·
    3 months ago

    To add, let’s do some math!

    Let s be the total annual salary of every employee using Adobe. Our goal is to find the productivity ratio r such that changing to Gimp and open source more generally is a net positive from the standpoint of productivity and labor.

    s/r will be the total annual salary after changing over, because (for instance) if r = 0.8 then LTT will need to either hire or work his existing hires 1/0.8 times longer, giving (at best, ignoring overtime and so on) s/r as the new labor cost.

    We then subtract the current labor cost to get the switching cost s/r - s, and if this is greater than $10,000 then the switch is not worth it.

    For instance, let’s say LTT employs 1 person at $50k/year. He’s a bit of a skinflint. We solve for r and arrive at a ratio of 5/6 or 83.33%.

    If we have a different world where LTT hires 10 people and pays each of them $100k, we solve for r and get about 99%.

    In other words, the switch is worth it only if the labor cost is small, so the extra labor is not very expensive, or the difference between the two software is negligible.