OmnipotentEntity

Our right to speak out was core to our merger deal 25 years ago, but now it’s being silenced. Same execs, same censorship, new name and logo.

From the article. Sometimes a buy out comes with conditions and obligations.

My brother is a crank because ChatGPT enables him.

"Socialism is when the government does stuff, and the more stuff it does the more socialister it is."

-Carlos Marks

I have a framework. I'm not aware of any others though. But I do remember that physical wifi switches were common on HPs back about 15-20 years ago.

Get a laptop with hardware switches for the camera and microphone, tbh

The way that arsonist was able to melt the weird pointy top into something more appealing was really a thing of beauty.

Bouba > Kiki

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 🍍?

but can I say he's wrong? I don't know.

Not a cult. Not a cult. Not a cult. Not a cult.

Anakin brought balance to the force by slaughtering all of the Jedi.

Before, thousands of Jedi, like 4(?) sith.

After, 2 Jedi, 2 sith. Perfectly balanced.