2023.5.27 闲话

昨天 APJ 给我发的歌词：

Give a Little - LeGrand

Everyone projecting all their dreams on me
A million life plans of what they want me to be
Saying, give a little, give a little
Give a little, give a little
Piece by piece i'm slipping away
Hey
Be kind, be patient
Be hard to please
It seems contradicting, but do it honestly
Just give a little, give a little
Give a little, give a little
Until you disintegrate
Put your hands up if you've lied awake at 3 A.M
Promising yourself all the things
That you think that you'll never do again
Put your hands up
If you have a dream about running away
Somewhere you can get a little wild for a night
And the next day no one remembers your name
Put your hands up if you've been manipulated
By the masses
By the people who just want to complicate 'ya
By the ones who only want to see you crash and burn
Or tell me honestly
The things that you yearn for!
Do you want the voices to stop?
In and out of your head
Making your mouth go pop
Can I get an Amen? (Amen!)
Why don't you forget it, and dance till you drop
Three, two, one
All the hypocrites
With slivers in their eyes
I'll scream that I'm sick of it
I can't see the beams in mine
Obscene expectations
Is the most toxic form of pride
So why should I deserve to do more with my life?
Everyone projecting all their dreams on me
A million life plans of what they want me to be
Saying, give a little, give a little
Give a little, give a little
Piece by piece I'm slipping away
Hey
Be kind, be patient
Be hard to please
It seems contradicting, but do it honestly
Just give a little, give a little
Give a little, give a little
Until you disintegrate

World's Smallest Violen - AJR

My grandpa fought in World War II
He was such a noble dude
I can't even finish school
Missed my mom and left too soon
His dad was a fireman
Who fought fires so violent
I think I bored my therapist
While playing him my violin
(Oh my God) That's so insane
(Oh my God) That's such a shame
Next to them, my shit don't feel so grand
But I can't help myself from feeling bad
I kinda feel like two things can be sad
The world's smallest violin
Really needs an audience
So if I do not find somebody soon (That's right, that's right)
I'll blow up into smithereens
And spew my tiny symphony
Just let me play my violin for you, you, you, you
My grandpa fought in World War II
He was such a noble dude
Man, I feel like such a fool
I got so much left to prove
All my friends have vaping friends
They're so good at making friends
I'm so scared of caving in
Is that entertaining yet?
(Oh my God) That's so insane
(Oh my God) That's such a shame
Next to them, my shit don't feel so grand
But I can't help myself from feeling bad
I kinda feel like two things can be sad
The world's smallest violin
Really needs an audience
So if I do not find somebody soon (That's right, that's right)
I'll blow up into smithereens
And spew my tiny symphony
Just let me play my violin for you, you, you, you
Somewhere in the universe
Somewhere someone's got it worse
Wish that made it easier
Wish I didn't feel the hurt
The world's smallest violin
Really needs an audience
So if I do not find somebody soon
I'll blow up into smithereens
And spew my tiny symphony
All up and down a city street
While tryna put my mind at ease
Like finishing this melody
This feels like a necessity
So this could be the death of me
Or maybe just a better me
Now come in with the timpanis
And take a shot of Hennessy
I know I'm not there mentally
But you could be the remedy
So let me play my violin for you

推歌：Small Cloud Sugar Candy / テヅカ x Aoi feat.桃雛なの .

---

看的 Alpha1022 的结论，Link .

\[f*g=h\iff\sum_{i=1}^nh(i)=\sum_{i=1}^nf(i)\sum_{j=1}^{\lfloor\frac ni\rfloor}g(j) \]

证明：

\[\begin{aligned}\sum_{i=1}^nh(i)&=\sum_{i=1}^n\sum_{d\mid i}f(d)g(i/d)\\&=\sum_{d=1}^nf(d)\sum_{i=1}^{\lfloor\frac nd\rfloor}g(i)\end{aligned} \]

Q.E.D.

这之后，为了方便，让

\[(f\mathop\square g)(n)=\sum_{i=1}^nf(i)g\left(\left\lfloor\dfrac ni\right\rfloor\right) \]

那么原命题就是 \(f*g=h\iff f\mathop\square \sum g=\sum h\) .

那么可以导出推论 \(f=g\mathop\square h\iff h=g^{-1}\mathop\square f\) .

具体就是根据 Möbius 反演：

\[f=g\mathop\square h\iff g*\Delta h=\Delta f\iff g^{-1}*\Delta f=\Delta h\iff h=g^{-1}\mathop\square f \]

那么展开来就是：

\[f(n)=\sum_{i=1}^ng(i)h\left(\left\lfloor\dfrac ni\right\rfloor\right)\iff h(n)=\sum_{i=1}^ng^{-1}(i)f\left(\left\lfloor\dfrac ni\right\rfloor\right) \]

阿，其实就是 整除 Möbius 反演 罢，你又在炒冷饭了！

其实也不是那么的炒冷饭，毕竟这个证法要简洁许多，而且和 Koishi 的数学题那题的推导技巧（即 \(\lfloor\frac ni\rfloor=\lfloor\frac {n-1}i\rfloor\iff i\mid n\)）基本没关系了，还是挺有意思的是不是！（试图狡辩）