Oh! My bad! I completely missed that the functions were continuous (it isn’t required for 🍊 to be a metric)

Careful ⚠️ there is not guaranteed to be an element such that |🍎(x) - 🍇(x)| is maximized. Consider 🍎 (x) = x if x < 3, 0 otherwise. Let 🍇 (x) = 0, and let the domain be [0, 4]. Clearly, the sup(|🍎 (x) - 🍇 (x)| : x ∈ [0, 4]) = 3, but there is no concrete value of x that will return this result. If you wish to demonstrate this in this manner, you will need to introduce an 🐘 > 0 and do some pedantic limit work.

Anyway, to prove this is a metric we must prove that it satisfies the 4 laws of metrics.

1. The distance from a point to itself is zero. 🍊 (🍎, 🍎) = 0

This can be accomplished by simply observing that |🍎 (x) - 🍎 (x)| = 0 ∀x ∈ [a,b], so its sup = 0.

2. The distance between any two distinct points is non-negative.

If 🍎 ≠ 🍌, then ∃x ∈ [a,b] such that 🍎 (x) ≠ 🍌 (x). Thus for this point |🍎 (x) - 🍌 (x)| > 0 and the sup > 0.

3. 🍊 (🍎, 🍌) = 🍊 (🍌, 🍎) ∀(🍎, 🍌) in our space of functions.

Again, we must simply apply the definition of 🍊 observing that ∀x ∈ [a,b] |🍎 (x) - 🍌 (x)| = |🍌 (x) - 🍎 (x)|, and the sup of two equal sets is equal.

4. Triangle inequality, for any triple of functions (🍎, 🍌, 🍇), 🍊 (🍎, 🍌) + 🍊 (🍌, 🍇) ≥ 🍊 (🍎, 🍇)

For any (🐁, 🐈, 🐕) ∈ ℝ³ it is well known that |🐁 - 🐕| ≤ |🐁 - 🐈| + |🐈 - 🐕|, (triangle inequality of absolute values).

Further, for any two functions 🍍, 🍑 we have sup({🍍 (x) : x ∈ [a, b]}) + sup({🍑 (x) : x ∈ [a, b]}) ≥ sup({🍍 (x) + 🍑 (x) : x ∈ [a, b]})

Letting 🍍 (x) = |🍎 (x) - 🍌 (x)|, and 🍑 (x) = |🍌 (x) - 🍇 (x)|, we have the following chain of implications:

🍊 (🍎, 🍌) + 🍊 (🍌, 🍇) = sup(🍍 (x) : x ∈ [a, b]}) + sup({🍑 (x) : x ∈ [a, b]}) ≥ sup({🍍 (x) + 🍑 (x) : x ∈ [a, b]}) ≥ sup({🍎 (x) - 🍇 (x)| : x ∈ [a, b]) = 🍊 (🍎, 🍇)

Taking the far left and far right side of this chain we have our triangles inequality that we seek.

Because 🍊 satisfies all four requirements it is a metric. QED.

QED stands for 👸⚡💎, naturally

What happened to 🍇 and 🍍?

Johnny Mnemonic

Why is “gives free tech support,” an expectation of being a good person? I spent 7 years of my life doing tech support professionally.

a) I never want to do it again, because honestly fuck computers, but b) If I have to, I’m certainly not going to do it unless I already like you or you are paying me.

Figure your own shit out. I have a life to live and I don’t have infinite free time and energy to help you unfuck whatever stupid bullshit happened to your computer. Especially if you’re being entitled about it.

You might find this interesting: https://citationsneeded.libsyn.com/ep-152-hallmark-christmas-movies-and-the-cozy-conservative-nostalgia-machine

Transcription: https://citationsneeded.medium.com/episode-152-hallmark-christmas-movies-and-the-cozy-conservative-nostalgia-machine-10b6b848ab2a

I mean, by pushing conservative media narratives and moving the Overton window to the right, you could argue that Hallmark is in part responsible for our illegal ICE deportations to El Salvadorian torture prisons, the rise in hate crimes against transpeople, and our future war with Iran.

But maybe an extra white lady got killed too.

Are you kidding? Do you not remember? That is exactly what a pokemon journey is for! Now put on a baseball cap, grab your Bulbasaur, and get out there and be the very best like no one ever was.

To be fair, the last time I saw this image was probably either in the 90s or early 2000s

This is the highest resolution of this picture I’ve ever seen.

Damn dude. Is your life really so pathetic that you are honestly deriving please from bad news being delivered to people who are on the opposite side of a fucking web browser debate? That’s sad. I hope things start looking up for you in the future.

“You mean I hitched my wagon to Grabthor The Wagon Destroyer, and he destroyed my wagon? This is an outrage!”

Bloom County was so great.

Or instead of targeting tiktok specifically, they could have chosen to pass a data privacy law and actually did something worthwhile instead of pointless, unpopular grandstanding. Haha just kidding, they would never do anything to reduce even slightly shareholder value.

NYT has been around since before the civil war.

I’ll let you have a guess what their stance on slavery was. They always have punched left.

deleted by creator

LLMs are bad for the uses they’ve been recently pushed for, yes. But this is legitimately a very good use of them. This is natural language processing, within a narrow scope with a specific intention. This is exactly what it can be good at. Even if does have a high false negative rate, that’s still thousands and thousands of true positive cases that were addressed quickly and cheaply, and that a human auditor no longer needs to touch.

An environmental posadist. Not a stance I’ve normally seen. Imo, if nothing came out of deep water horizon, there’s no oil accident big enough to matter.

Transocean received an early partial insurance settlement for total loss of the Deepwater Horizon of US$401 million about 5 May 2010.[60] Financial analysts noted that the insurance recovery was likely to be more than the value of the rig (although not necessarily its replacement value) and any liabilities – the latter estimated at as much as US$200 million.

If only there were other things that a person could do outside of voting once every four years to participate in the political process.