New raid mechanic just dropped guardians go back to your math class
disgusting stuff below.
ykw if the warlock remembered just enough about the relation of fourier series with function inner product space he can indeed do that in his head.
we define our function inner product space C[0, 2π] with a set of functions continuous in range [0, 2π], and the inner product of two functions in this space f, g ∈ C[0, 2π] is defined as ∫f(x)g(x)dx (integrated in [0, 2π]).
note that the functions in set S = {cos(nπ) | ∀ n∈ ℕ} ∪ {sin(nπ) | ∀ n∈ ℕ} are orthogonal to each other in this space, and all of them has squared norm π, so you can just match the terms with the same function and sum them up. in this case only 6cos3x on the left and 5cos3x on the right can match, along with the divide by π, the final answer is 6 × 5 = 30.
another example is ∫ (3cos(x) + 4cos(2x) + 6cos(3x))(-cos(x) + 5cos(3x) + 18cos(5x)) dx (integrated in [0, 2π]) = (6 × 5 + 3 × (-1))π = 27π.
don't actually do that tho it's just a trivia.
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I think it’s interesting in rgu how Utena’s aspiration of being a prince is not only hurtful to Anthy but also serves a sort of double purpose in Utena’s life wherein it allows her to express aspects of her gender expression and sexuality without having to really confront them directly, and a lot of the most important moments in her character arc are moments where she has to confront those things without relying on it.
Utena saying at the end that the only times she was really happy was when she was with Anthy is so important because it’s not just her presence that changes Anthy’s life but the reverse is true as well, and the idea that she is acting solely on some sort of heroic noble mission separate from her own feelings at times allows her to kind of sidestep the vulnerability of admitting that. It’s after Dios has urged her to give up, saying that she’s done all she could and giving her a way out and into complacency that would still leave her ego and dream of keeping her promise to Dios partially intact, but it wasn’t ever really her promise to Dios that mattered but her promise to Anthy both as a child and once she knows her that matters.
It’s also really interesting that the first time she says no to Akio she says it’s because she wants to stay true to her prince—it’s an idea that gives her a way to momentarily reject his advances without acknowledging that her own feelings should be reason enough, and shows how she feels that appealing to (even an abstract) patriarchal figure gives validity to her own feelings that they wouldn’t have on their own.
Episodes 12 and 37 are also really interesting in that they both begin when Utena has realized that she does not truly know or understand Anthy and feels betrayed by her, and she temporarily renounces her role of “protecting” her only to realize that her relationship with Anthy is deeply important to her personally even outside of that dynamic, as is her masculinity/gender expression, but in order to express this realization she ultimately returns to that dynamic (trying to protect Anthy via the dueling system) in both cases because she cannot yet imagine another way. It’s not until she fails at being a prince that she succeeds in expressing her true feelings and escaping the academy/the structure it represents (and inspiring Anthy to do the same) because those things were never truly compatible to begin with.
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