右复合简单函数

本文 Markdown 源码约 \(\rm 1.8\ MiB\),可能需要一定时间加载 \(\rm\LaTeX\) .

首先来个引流内容,最近闲话阅读量有点低:

美图欣赏

太大了博客园不让放,放个 MD5 看看得了:

Code: 68d08969de17e4b78024d044455268d8 image.html .


考察变换在 OGF 上的表现:

\[\begin{aligned}G(z)&=\sum_{i=0}^n\sum_{j=i}^n\dfrac{f_j\cdot z^i}{j+1}\\&=\sum_{j=0}^n\dfrac{f_j}{j+1}\dfrac{1-z^{j+1}}{1-z}\\&=\dfrac1{1-z}\int_z^1F(t)\mathrm dt\\&=\dfrac1{1-z}\int^0_{z-1}\mathsf E^{-1}F(t)\mathrm dt\\&=\mathsf E\left(\dfrac1z\int_0^z\mathsf E^{-1}F(t)\mathrm dt\right)\end{aligned} \]

关于怎么做 \(\mathsf E^{-1}\),展开后就是差卷积 .

时间复杂度 \(\Theta(n\log n)\) .


Try to calculate:

\[\mathcal P(x,y)=\prod_{i=0}^k(1+\mathrm e^{b^iy}x) \]

维护 \(F(y)=(\mathrm e^yx+1)\) 那么相当于要支持两种变换 :

  • \(F(y)\to F(by)\) .
  • \(F(y)\to F(\mathrm e^y)\) .
右复合 exp

\(\displaystyle A(x)=\sum_{i\ge 0}a_ix^i\)

\[\begin{aligned}A(\mathrm e^x)&=\sum_{i\ge 0}a_i(\mathrm e^x)^i\\&=\sum_{i\ge 0}a_i\sum_{j\ge 0}\frac{i^j}{j!}\\&=\sum_{j\ge 0}\frac{1}{j!}\sum_{i\ge 0}a_ii^j\end{aligned} \]

后面就是要算 \(\sum_i\frac{a_i}{1-ix}\) 这样的东西,分治做即可 .

复杂度自己猜吧,我也不知道 . 倍增地做可能会快一点 .


一道经典题:

点击展开题面

今天晚上吃饭回来,看到 yspm 从斜坡上走了下来,转了一圈又走了回去。

紧接着 SoyTony 也模仿 yspm 做了这件事。这是处于模仿有趣的行为,还是将自身带入其中感受转圈的乐趣,我说不上来。

一直在思考一件事情,就是人是不是应该保持「初心」,而这个「初心」究竟指什么。

水群的时候,群友说我变了。但是,人总是在变的,而哪些需要改变,哪些需要保持,是一个值得探讨的问题。

保持「初心」,必然不是保持一成不变,而是保持自己内心的一种目标,在改变之中而不变,才是「初心」吧。

正如现在一样,OI 只是人生中的一部分,而编程的初心,是不会变的。也只有这种初心,才能支持所有人走下去。

不过,即使发生改变,再回过头去看之前的时光,也总是会有一种感慨,有一种回味在里面的。

可能,改变是必然的,而回忆也是必然的,每到一个转折点,回忆正是一种对改变的新认识。

这,又何尝不是在斜坡上转一圈呢?

你仔细想想可以发现这三道题都是一样的 .

把题目翻译成 OGF 就是每次你 \(F(z)\to F((z-1)(z+1))\) .

那么相当于 \(F(z)\to F(z^2-1)\) .

倒过来就是 \(F(z)\to F(\sqrt{1+z})\) .

偶数项就直接 \(2k\to k\) 然后平移即可 .

奇数项就广义二项式定理展一下:

\[(1+z)^{k+\frac 12}=\sum_{i\ge 0}\frac{(2k+1)!!}{2^ii!(2k-2i+1)!!}z^i \]

然后就是卷积了 .

\(\Theta(n\log n)\) 解决!


试看看

习题 1.1 给定多项式 \(F(z)\) 和常数 \(a,b,c,d,e,f,g,h\),求 \(F(\frac{az^3+bz^2+cz+d}{ez^3+fz^2+gz+h})\) .

习题 1.2 给定多项式 \(F(z),G(z)\),求 \(F(G(z))\) .

习题 1.3 给定多项式 \(F(z)\)\(F(\sin z)\) .


最后用我很喜欢的一首诗来结尾吧:

Redo Poem

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posted @ 2023-10-14 11:41  Jijidawang  阅读(155)  评论(0编辑  收藏  举报
😅​