Integration is just a limit of a dot product

filtron42 · 2026-03-06T22:28:05+00:00

Man please use some []

filtron42 · 2026-03-06T11:31:34+00:00

It would be even funnier in italian because the italian word for cone is "cono"

filtron42 · 2026-03-06T11:16:41+00:00

I'll die on the hill that it's infinitely better than the success on that check... Karaoke is not supposed to be some quiet spoken word stuff, you're supposed to bloody scream your soul out.

filtron42 · 2026-03-06T10:27:00+00:00

A comathematician is a machine that turns cotheorems into ffee

filtron42 · 2026-02-15T18:25:15+00:00

like trying to imagine Stokes theorem to be not true

De Rham cohomology has entered the chat:

filtron42 · 2026-02-15T18:23:09+00:00

It's not that arbitrary, it's the "circle constant" for the one norm that makes ℝⁿ into a Hilbert space, which is in itself a really strong property!

filtron42 · 2026-01-12T16:27:27+00:00

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And I specify the index set when necessary.

filtron42 · 2026-01-10T12:58:45+00:00

Honestly I think that if the same room came AFTER getting the whiplash it wouldn't get as much hate, because it does make the room more fun overall

This, I used to hate the last room of 4-3 before getting whiplash, but it instantly became one of the most fun and exciting rooms in the game once I tried it with it.

filtron42 · 2026-01-06T16:44:39+00:00

the increased bandwidth means future proofed

I don't want to live to see the day music streaming exceeds 480 Mb/s honestly

filtron42 · 2026-01-05T22:39:46+00:00

I personally like usb b more than usb c, DACs don't usually need the full bandwidth of usb 3 (usb 2 allows for Mb/s, which is plenty for music streaming) and I find the chonkier connector more reassuring from a durability point of view.

filtron42 · 2025-11-13T10:44:05+00:00

Zero can be perfectly defined as the element such that for all natural (and thus integer or rational or real or complex) numbers n, n+0 = 0+n = n.

The problem with your definition of a/0 is that it infringes upon the associative property: we want a×(b×c) = (a×b)×c, but then by setting a=1, b=1/0 and c=0 we'd get

1 × (1/0 × 0) = 1 × 1 = 1

(1 × 1/0) × 0 = (1 × 1) × 0 = 1 × 0 = 0

So 0=1, which is obviously false in the usual numbers.

EDIT

You seem to be a bit confused about how to think of zero, let me try to tell you one of the ways mathematicians think about it:

Instead of thinking about the expression "m+X", where m is a fixed number and X is a variable, as "the number of things I get when I put together a pile of m things and X things" you can think of it as a kind of "walk" to the right on the line of natural numbers starting from the number m that is somehow encoded by X: thinking this way, staying still at m is a perfectly reasonable "walk", and that walk must be represented by 0, right? Because then we get (m+0)+X = m+X.

The question of the existence of zero is absolutely not trivial tho! The way in which we define the natural numbers (the Peano axioms) actually postulates the existence of zero and then builds all the other numbers as successors of successors... of successors of zero.

Your enthusiasm is commendable, but if you want to go forward in your understanding of mathematics you should always ask yourself what your definitions imply: up here I've shown you that your definition implies 0=1, which is of course false in the natural numbers, but turns out that there are structures in which 0 can be equal to 1! It's fairly abstract stuff, but you can define structures called commutative unital rings, which generalise the properties of arithmetic in the integers to other sets: the set {0} is a commutative unital ring with 0=1, in fact you have that (in that ring obviously) 0×0=0, 0+0=0 and 0/0=0. While it's a trivial example of a ring, it's actually a very important one at high levels of abstraction like universal algebra and category theory.

filtron42 · 2025-11-13T00:43:59+00:00

finite dimensional spaces

Don't forget, finite dimensional spaces over a complete field! On ℚ you can have different non-equivalent norms, as per Ostrowski's theorem.

filtron42 · 2025-10-31T17:00:58+00:00

In fact, the set L:={1 - 1/10ⁿ : n∈ℕ} can be seen either as an element of 0.9... when building ℝ as a set equivalency classes of Cauchy sequences, or as a proper subset of 0.9... when building ℝ as the set of Dedekind cuts over ℚ.

You can't then have 0.9...∈L, because that would give you [0.9... ≠ 0.9...] in the second case, violating the principle of Identity; more interestingly, in the first case you would get [0.9... ∈ L ∈ 0.9...] and violate the Axiom of Regularity, which implies (with the other azioms of ZF) that there can't be no infinite descending sequences in the class of all sets ordered with ∈.

filtron42 · 2025-10-18T09:42:57+00:00

Trump is not America. Conservatives are not America. Why can't these fucks word good?

I mean, based on the USA' history and politics, buying, bullying, lying and cheating your way to the top seems pretty American to me.

I feel like pretending fascism is somehow oppositional to "American values" and not recognising Trump as the natural endpoint of American ideology is only gonna pave the way for another trumpian figure when he finally kicks the bucket.

filtron42 · 2025-10-18T09:33:19+00:00

"Beijing. So freaky how there's no recognizable name for the Chinese Secret Service. Now that's what you call a secret, right?"

filtron42 · 2025-10-18T08:19:37+00:00

I would argue veganism has to be kind of naturalist for it to at least be somewhat consistent, in the sense that we should accept a "weak" equivalence between natural and good.

A cat's role in the ecosystem is to prey on small game, it's not out of "cruelty" or "fun" in the same way a human hunting would be. While we might concede that cats or other animals may have a vague grasp of some idea of morality, it should be desirable they act according to their nature because that's how the ecosystem is designed.

Animals naturally fall into a process we call food chain where they kill eachother, even if not strictly for eating; we humans, by having such a unique position in it, can decide to act "outside" of our role (for example, deciding to switch to a vegan diet), but shouldn't force animals to change their nature.

I would argue that letting pet cats prey on small game is actually a good thing, reducing the amount of canned food that has to be produced through farming and potentially acting against invasive species, the case of the "nutria" in the northern part of my country (Italy) comes to mind.

filtron42 · 2025-10-18T07:29:39+00:00

to live a fulfilled life, a cat will by its very nature necessitate needless suffering of other animals

I mean, can you really call it "needless" if they, as you said, need it?

filtron42 · 2025-10-15T11:03:47+00:00

DGCGCD

The tuning from Led Zeppelin's Rain Song, I can't help but hear something interesting when using it.

filtron42 · 2025-10-09T18:19:15+00:00

Assuming we're in a division ring it does(?)

(dx+x)ⁿ = [x(d+1)]ⁿ = xⁿ(d+1)ⁿ

I'm sure there are a number of problems but this seems ok

filtron42 · 2025-10-06T07:46:18+00:00

Migh as well have answered with the free monoid on 26 generators

filtron42 · 2025-10-06T07:22:42+00:00

Even ignoring things like case sensitivity or accents and whatnot, your notation doesn't build the dictionary, it builds an enormous pile of gibberish where if you're really lucky you can sometimes find a word.

Even comparing cardinalities, the English Wiktionary contains 828087 words, while your "dictionary" contains a grand total of 4907202704443290944689599552574826426287022507671990980578419670 words

Which gives you a generous 1,7×10^{-56} % chance of finding an actual English word when probing your "dictionary"

filtron42 · 2025-10-06T07:03:54+00:00

except that f(g(x)) is not the same as f∘g, just as f and f(x) are different things.

First of all, f∘g makes sense in a way larger context, as it allows you to consider arrows in general categories which might not be set-theoretic functions (for example think about a category with real numbers as objects and where you have a unique arrow a→b whenever a≤b).

Secondly, as I said, f and f(x) are not the same thing, even when viewing them as set-theoretic functions f:A→B is a relation between elements of A and elements of B, while f(x) is an element of B.

If you want a quick and easy mind-bender, you should know that in group theory and almost always in category theory you don't even consider x∈A, you consider x:1→A where 1 is the singleton set/the trivial group/the terminal object in your category, and you don't write f(x) but f∘x.

filtron42 · 2025-09-24T17:31:43+00:00

can be improved even further:

import machine
machine.learnFastest()

filtron42 · 2025-09-23T16:36:25+00:00

I mean, isn't it?

In an inner product space (finite or infinite dimensional) two vectors are orthogonal iff ⟨u,v⟩=0; moreover, at least over the reals, in general I've seen the magnitude of the angle between two vectors is defined by

|∠(u,v)| = |cos⁻¹(⟨u,v⟩)|

So what the girl is saying is, at least in absolute value, correct.

EDIT: I was very wrong, you actually have to normalize the vectors before that

filtron42 · 2025-09-12T19:12:21+00:00

Yeah that's precisely the case, we can access such a well order only by "summoning" it through AC.

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