• 14 Posts
  • 322 Comments
Joined 2 years ago
cake
Cake day: June 14th, 2023

help-circle

  • It turns out, that for the values we are talking about here, it actually more or less does! A lemon has a pH of around 2.5, while “Flow” has an advertised pH of 8.1. This means roughly that to neutralize 1L of this water you need approximately 0.4mL of lemon juice or about 8 drops/half a gram. It’s hard to tell how much a “spritz” is intended to be, but a single lemon contains about 60mL of juice, so this represents about 0.67% of the total juice inside.

    It’s a surprising consequence of using a logarithmic scale for pH.


  • I am a bit late to this party, but I thought I’d piggy back on your comment to halfway address it using math.

    We want to run data centers cool. This means keeping the center itself as close to 20°C as possible.

    If we lose our convection and conduction then our satellite can only radiate away heat. The formula governing a black body radiator is P = σAT^4. We will neglect radiation received, though this is not actually a negligible amount.

    If we set T = 20°C = 294K. Then we have the relationship of P/A = 423.6 W/m^2

    According to an article I found on the Register from this April:

    According to Google, the larger of the two offered pods will consume roughly 10 megawatts under full load.

    This would imply a surface area of at minimum 23600 m^2 or 5.8 acres of radiator.

    I don’t know how large, physically, such a pod would be. But looking at the satellite view of a google data center in Ohio that I could find, the total footprint area of one of the large building of their data centers is ballpark in that range. I don’t know how many “pods” that building contains.

    So it’s not completely outside of the realm of possibility. It’s probably something that can be engineered with some care, despite my earlier misgivings. But putting things in orbit is very expensive, and latency is also a big factor. I can’t think of any particular practical advantages to putting this stuff into orbit other than getting them out of the jurisdiction of governments. (Not counting the hype and stock song and dance from simply announcing you’re going to set a few billion dollars on fire to put AI into space.)















  • 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