c_wInf's blog

\[ \newcommand{\ci}{\mathrm{i}} \newcommand{\e}{\mathrm{e}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\eps}{\varepsilon} \newcommand{\dsum}{\displaystyle\sum} \]

Abstract

This article develops the exponential and trigonometric functions from first principles using infinite series, avoiding reliance on geometric intuition or pre-existing notions of angle. Motivated by foundational concerns raised in early twentieth-century analysis, the construction is carried out directly on the complex plane. After establishing the algebraic structure and completeness of \(\C\) as a normed vector space, basic results on complex series are proved, including absolute convergence and a Fubini-type theorem. The exponential function is then defined by its power series, shown to be well-defined on \(\C\), and its fundamental properties—such as the exponent law, positivity on \(\R\), and monotonicity—are derived. Trigonometric functions are introduced via the complex exponential, leading naturally to Euler’s formula, addition formulas, and the Pythagorean identity. This approach demonstrates that the classical properties of exponential and trigonometric functions arise purely from analytic and algebraic considerations, without geometric assumptions.

Introduction

The definition of exponential function and trigonometric functions originates from some elementary perspectives at first. Exponential function was defined as an extension of exponentiation with rational exponents, and trigonometric arose from geometric connections between angles and lengths.

However, in modern maths, these intuitive definitions aren’t well enough. G. H. Hardy noted in his 1908 work A Course of Pure Mathematics that the definition of the trigonometric functions in terms of the unit circle is not satisfactory, because it depends implicitly on a notion of angle that can be measured by a real number. Exponential function extended from rational exponents are also hard to achieve some of the basic properties easily.

Therefore, modern definitions express exponential function and trigonometric functions as infinite series or as solutions of differential equations. This paper will discuss about the former in detail.

This paper is going to define these functions directly on \(\C\), so the first part is some properties of complex series, followed by definitions and properties of exponential and trigonometric functions. We assume some of the basic properties of \(\R\) (it’s a complete field with an Archimedean total order), and will build everything else from scratch.

Complex Series

Complex Numbers

We define

\[ \C = \{(a, b) : a, b \in \R\} \]

With addition

\[ \begin{array}{rrl} \cdot + \cdot : & \C \times \C & \to \C \\ & (a, b), (c, d) & \mapsto (a+c, b+d) \end{array} \]

And multiplication

\[ \begin{array}{rrl} \cdot \cdot \cdot : & \C \times \C & \to \C \\ & (a, b), (c, d) & \mapsto (ac-bd, ad+bc) \end{array} \]

We denote \(1 = (1, 0), \ci = (0, 1)\), then \((a, b)\) will be denoted as \(a + \ci b\).

We define

\[ \begin{array}{rrl} \Re : & \C & \to \R \\ & a + \ci b & \mapsto a \end{array} \]

\[ \begin{array}{rrl} \Im : & \C & \to \R \\ & a + \ci b & \mapsto b \end{array} \]

Then \(\forall z \in \C, z = \Re(z) + \ci \Im(z)\).

Completeness of \(\C\)

We see \(\C\) as a vector space on \(\R\) and define the norm on \(\C\) as

\[ \begin{array}{rrl} | \cdot | : & \C & \to \R \\ & a+\ci b & \mapsto \sqrt{a^2+b^2} \end{array} \]

Theorem (Completeness of \(\C\)) \(\C\) is complete under this norm.

Proof. For any Cauchy sequence on \(\C\), that is, \(\{z_k = x_k + \ci y_k\} \subset \C\) that satisfies

\[ \forall \eps \in \R, \exists N \in \N, \forall n, m > N, |z_n - z_m| < \eps \]

Since

\[ |x_n - x_m|^2 + |y_n - y_m|^2 = |z_n - z_m|^2 \]

We have

\[ |x_n - x_m| < |z_n - z_m| < \eps \]

And \(\{x_k\}\) constructs a Cauchy sequence on \(\R\). Say

\[ x = \lim_{k \to \infty} x_k \]

Similarly, we assume that

\[ y = \lim_{k \to \infty} y_k \]

Then, \(\forall \eps \in \R\), choose \(N \in \N\) such that

\[ \forall k > N, |x_k - x| < \frac{\eps}{2} \]

Choose \(M \in \N\) such that

\[ \forall k > M, |y_k - y| < \frac{\eps}{2} \]

Let \(z = x + \ci y\). Then

\[ \forall k > \max\{N, M\}, |z_k - z| = \sqrt{|x_k - x|^2 + |y_k - y|^2} < \sqrt{\left(\frac{\eps}{2}\right)^2 + \left(\frac{\eps}{2}\right)^2} < \eps \]

Thus \(\{z_k\}\) converges to \(z\).

Since \(\C\) is a complete normed vector space (Banach space) over \(\R\), we naturally have

Theorem (Absolute Convergence of Complex Series) If a complex series converges absolutely, it converges.

Proof. Let \(\dsum_{k=0}^{\infty} |z_k|\) converge. Then, since it’s a Cauchy sequence, \(\forall \eps > 0, \exists N \in \N, \forall n, m > N\),

\[ \dsum_{k=n}^{m} |z_k| < \eps \]

For sequence \(\{z_k\}\), we have

\[ \left|\dsum_{k=n}^{m} z_k\right| \le \dsum_{i=n}^{m} |z_k| < \eps \]

Thus \(\{z_k\}\) is a Cauchy sequence. Then it converges.

Theorem (Fubini, for real series) If \(\{a_{m, n}\}_{m, n \in \N} \subset \R\) is a double-indexed sequence of real numbers and \(\dsum_{(m, n) \in \N \times \N} a_{m, n}\) is absolutely convergent, then

\[ \dsum_{(m, n) \in \N \times \N} a_{m, n} = \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \]

Proof. By Riemann’s rearrangement theorem, the sum doesn’t change whatever the order of a non-negative sequence is, which indicates Fubini’s theorem for non-negative series. For an arbitrary absolutely convergent sequence, let

\[ b_{m, n} = \max\{a_{m, n}, 0\}, c_{m, n} = \max\{-a_{m, n}, 0\} \]

Then we guarantee that \(b_{m, n}\) and \(c_{m, n}\) are convergent, non-negative sequences that satisfy

\[ \forall m, n \in \N, a_{m, n} = b_{m, n} - c_{m, n} \]

Then

\[ \begin{aligned} \dsum_{(m, n) \in \N \times \N} a_{m, n} =& \dsum_{(m, n) \in \N \times \N} b_{m, n} - \dsum_{(m, n) \in \N \times \N} c_{m, n} \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} b_{m, n} - \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} c_{m, n} \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \end{aligned} \]

Theorem (Fubini, for complex series) If \(\{a_{m, n}\}_{m, n \in \N} \subset \C\) is a double-indexed sequence of complex numbers and \(\dsum_{(m, n) \in \N \times \N} a_{m, n}\) is absolutely convergent, then

\[ \dsum_{(m, n) \in \N \times \N} a_{m, n} = \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \]

Proof. Since \(\{\Re(a_{m, n})\}\) and \(\{\Im(a_{m, n})\}\) are also absolutely convergent, we have

\[ \begin{aligned} \dsum_{(m, n) \in \N \times \N} a_{m, n} =& \dsum_{(m, n) \in \N \times \N} \Re(a_{m, n}) + \ci \dsum_{(m, n) \in \N \times \N} \Im(a_{m, n}) \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} \Re(a_{m, n}) + \ci \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} \Im(a_{m, n}) \\ =& \dsum_{m=0}^{\infty} \dsum_{n=0}^{\infty} a_{m, n} \\ \end{aligned} \]

Exponential Function

Definition (Exponential Function)

\[ \begin{array}{rrl} \exp : & \C & \to \C \\ & z & \mapsto \e^z = \dsum_{k=0}^{\infty} \frac{z^k}{k!} \end{array} \]

Theorem (Well-definition of \(\exp\) on \(\R_{\ge 0}\)) \(\forall x \ge 0, \exp x\) converges.

Proof. Let \(a_k = \frac{x^k}{k!}\). Choose \(M \in \N, M > 2x\). Then,

\[ \forall k \ge M, \frac{a_{k+1}}{a_k} = \frac{x}{k+1} < \frac{x}{M} < \frac{1}{2} \]

By induction, we have

\[ \forall k \ge M, a_k < \frac{a_M}{2^{k-M}} \]

Then

\[ \dsum_{k=M}^{\infty} \frac{x^k}{k!} < \dsum_{k=M}^{\infty} \frac{x^M}{M! 2^{k-M}} = \frac{2x^M}{M!} < +\infty \]

Thus \(\dsum_{k=0}^{\infty} \frac{x^k}{k!}\) converges.

Theorem (Well-definition of \(\exp\)) \(\forall z \in \C, \exp z\) converges.

Proof. The absolute series of \(\exp z\)

\[ \dsum_{k=0}^{\infty} \left| \frac{z^k}{k!} \right| = \dsum_{k=0}^{\infty} \frac{|z|^k}{k!} = \exp |z| < +\infty \]

Converges. Therefore \(\exp z\) converges.

Theorem (Exponent Law)

\[ \forall z_1, z_2 \in C, \e^{z_1 + z_2} = \e^{z_1} \cdot \e^{z_2} \]

Specifically, \(\forall z \in \C\), \(\e^z \neq 0\); \(\forall x \in \R\), \(\e^x \in \R_{>0}\).

Proof.

\[ \begin{aligned} \e^{z_1} \cdot \e^{z_2} =& \dsum_{k=0}^{\infty} \frac{z_1^k}{k!} \dsum_{k=0}^{\infty} \frac{z_2^k}{k!} \\ =& \dsum_{i=0}^{\infty} \dsum_{j=0}^{\infty} \frac{z_1^i z_2^j}{i! j!} \\ =& \dsum_{k=0}^{\infty} \dsum_{i+j=k} \frac{z_1^i z_2^j}{i! j!} \\ =& \dsum_{k=0}^{\infty} \frac{1}{k!} \dsum_{i+j=k} \begin{pmatrix}k \\ i\end{pmatrix} z_1^i z_2^j \\ =& \dsum_{k=0}^{\infty} \frac{1}{k!} (z_1 + z_2)^k \\ =& \e^{z_1 + z_2} \\ \end{aligned} \]

By the definition of \(\exp\), \(\e^0 = 1\).

Since \(\forall z \in C, \e^z \e^{-z} = 1\), \(\e^z \neq 0\).

By the definition of \(\exp\),

\[ \forall x \in \R_{>0}, \e^x = \dsum_{k=0}^{\infty} \frac{x^k}{k!} > \frac{x^0}{0!} = 1 \]

Moreover, \(\forall x < 0, \e^x = (\e^{-x})^{-1} \in (0, 1)\).

Theorem (Monotonicity of \(\exp\) on \(\R\))

\[ \forall x_1, x_2 \in \R, x_1 < x_2 \Leftrightarrow \e^{x_1} < \e^{x_2} \]

Proof.

\[ x_1 < x_2 \Leftrightarrow x_2 - x_1 > 0 \Leftrightarrow \e^{x_2 - x_1} > 1 \Leftrightarrow \e^{x_2} > \e^{x_1} \]

Trigonometric Functions

Definition (Trigonometric Functions)

\[ \begin{array}{rrl} \cos : & \C & \to \C \\ & z & \mapsto \displaystyle\frac{\e^{\ci z} + \e^{-\ci z}}{2} = \dsum_{k=0}^{\infty} \frac{(-1)^k z^{2k}}{(2k)!} \end{array} \]

\[ \begin{array}{rrl} \sin : & \C & \to \C \\ & z & \mapsto \displaystyle\frac{\e^{\ci z} - \e^{-\ci z}}{2\ci} = \dsum_{k=0}^{\infty} \frac{(-1)^k z^{2k+1}}{(2k+1)!} \end{array} \]

The definitions indicate that \(\sin\) is an odd function and \(\cos\) is an even function.

Theorem (Euler’s Formula) \(\forall z \in \C\),

\[ \e^{\ci z} = \cos z + \ci \sin z \]

Proof.

\[ \cos z + \ci \sin z = \frac{\e^{\ci z} + \e^{-\ci z}}{2} + \ci \frac{\e^{\ci z} - \e^{-\ci z}}{2\ci} = \e^{\ci z} \]

Theorem (Sum Formulas) \(\forall z_1, z_2 \in \C\),

\[ \begin{cases} \cos(z_1 + z_2) = \cos z_1 \cos z_2 - \sin z_1 \sin z_2 \\ \sin(z_1 + z_2) = \sin z_1 \cos z_2 + \cos z_1 \sin z_2 \end{cases} \]

Proof.

\[ \begin{aligned} \cos(z_1 + z_2) =& \frac{\e^{\ci (z_1 + z_2)} + \e^{-\ci (z_1 + z_2)}}{2} \\ =& \frac{\e^{\ci z_1} \e^{\ci z_2} + \e^{-\ci z_1} \e^{-\ci z_2}}{2} \\ \end{aligned} \]

\[ \begin{aligned} \cos z_1 \cos z_2 - \sin z_1 \sin z_2 =& \frac{\e^{\ci z_1} + \e^{-\ci z_1}}{2} \frac{\e^{\ci z_2} + \e^{-\ci z_2}}{2} - \frac{\e^{\ci z_1} - \e^{-\ci z_1}}{2\ci} \frac{\e^{\ci z_2} - \e^{-\ci z_2}}{2\ci} \\ =& \frac{1}{4}((\e^{\ci z_1} + \e^{-\ci z_1})(\e^{\ci z_2} + \e^{-\ci z_2}) + (\e^{\ci z_1} - \e^{-\ci z_1})(\e^{\ci z_2} - \e^{-\ci z_2})) \\ =& \frac{1}{4}(2\e^{\ci z_1} \e^{\ci z_2} + 2\e^{-\ci z_1} \e^{-\ci z_2}) \\ =& \cos(z_1 + z_2) \end{aligned} \]

\[ \begin{aligned} \sin(z_1 + z_2) =& \frac{\e^{\ci (z_1 + z_2)} - \e^{-\ci (z_1 + z_2)}}{2\ci} \\ =& \frac{\e^{\ci z_1} \e^{\ci z_2} - \e^{-\ci z_1} \e^{-\ci z_2}}{2\ci} \\ \end{aligned} \]

\[ \begin{aligned} \sin z_1 \cos z_2 + \cos z_1 \sin z_2 =& \frac{\e^{\ci z_1} - \e^{-\ci z_1}}{2\ci} \frac{\e^{\ci z_2} + \e^{-\ci z_2}}{2} + \frac{\e^{\ci z_1} + \e^{-\ci z_1}}{2} \frac{\e^{\ci z_2} - \e^{-\ci z_2}}{2\ci} \\ =& \frac{1}{4\ci}((\e^{\ci z_1} - \e^{-\ci z_1})(\e^{\ci z_2} + \e^{-\ci z_2}) + (\e^{\ci z_1} + \e^{-\ci z_1})(\e^{\ci z_2} - \e^{-\ci z_2})) \\ =& \frac{1}{4\ci}(2\e^{\ci z_1} \e^{\ci z_2} - 2\e^{-\ci z_1} \e^{-\ci z_2}) \\ =& \sin(z_1 + z_2) \end{aligned} \]

Theorem (Pythagorean’s Identity) \(\forall z \in \C\),

\[ \cos^2 z + \sin^2 z = 1 \]

Proof. \(1 = \cos 0 = \cos(z-z) = \cos z \cos(-z) - \sin z \sin(-z) = \cos^2 z + \sin^2 z\)

Conclusion

By defining the exponential and trigonometric functions through power series on the complex numbers, this paper provides a self-contained and conceptually rigorous foundation for these central objects of analysis. The completeness of \(\C\) ensures the convergence of the relevant series and justifies algebraic manipulations such as termwise addition and rearrangement. From these definitions, the familiar laws of exponentiation, Euler’s formula, and the fundamental identities of trigonometry follow naturally and transparently. This series-based construction not only resolves the foundational issues associated with geometric or rational-exponent definitions, but also highlights the unity of exponential and trigonometric functions within complex analysis. As a result, the classical real-variable theory emerges as a special case of a broader and more coherent complex-analytic framework.

finite ordinals (\(0\) ~ \(\mathrm{FTO} = \omega\))

one-column BMS (\(\omega\) ~ \(\mathrm{SCO} = \varepsilon_0\))

\(\omega\) ~ \(\omega^2\)

\(\omega^2\) ~ \(\omega^\omega\)

\(\omega^\omega\) ~ \(\omega^{\omega^2}\)

Using \(1 = \psi(0)\) in BOCF.

\(\omega^{\omega^2}\) ~ \(\mathrm{SCO} = \varepsilon_0\)

Using \(\omega= \psi(1)\) in BOCF.

two-column BMS (\(\varepsilon_0\) ~ \(\mathrm{BO} = \psi(\Omega_\omega)\))

\(\varepsilon_0\) ~ \(\mathrm{CO} = \zeta_0\)

\(\varepsilon_0\) ~ \(\varepsilon_1\)

Using \(\forall \alpha< \varepsilon_0, \omega^\alpha= \psi(\alpha)\) in BOCF.

\(\varepsilon_1\) ~ \(\varepsilon_{\varepsilon_0}\)

\(\varepsilon_{\varepsilon_0}\) ~ \(\mathrm{CO} = \zeta_0\)

\(\zeta_0\) ~ \(\mathrm{FSO} = \Gamma_0 = \varphi(1,0,0)\)

\(\zeta_0\) ~ \(\mathrm{HCO} = \varphi(\omega,0)\)

Using \(\forall \alpha< \psi_1(\psi_2(0)) = \varepsilon_{\Omega+1}, \omega^{\Omega+\alpha} = \psi_1(\alpha)\) in BOCF.

\(\varphi(\omega,0)\) ~ \(\mathrm{FSO} = \Gamma_0 = \varphi(1,0,0)\)

\(\Gamma_0\) ~ \(\mathrm{SVO} = \varphi(1@\omega)\)

\(\varphi(1@\omega)\) ~ \(\mathrm{LVO} = \varphi(1@(1,0))\)

\(\varphi(1@(1,0))\) ~ \(\mathrm{BHO} = \psi(\Omega_2)\)

\(\psi(\Omega_2)\) ~ \(\mathrm{BO} = \psi(\Omega_\omega)\)

To be continued …

ACMOJ链接

形式化题意

给定长度为 \(n\) 的字母序列 \(S\) ，和长度为 \(m\) 的字母序列 \(T\) ，求 \(S\) 中和 \(T\) 相似的子串有多少个，相似代表离散化后一致，即每一个对应位置字符的序关系保持一致。

切入

我们联想到KMP算法：KMP中的匹配是字串每一个字符都相等，而这里的匹配条件更加宽松，只要离散化后相等即可。考虑到KMP的想法是每次判断下一个字符是否匹配，匹配就继续，不匹配就跳转，容易想到这里我们只需要修改每次匹配下一个字符时匹配的逻辑即可。

从某一个位置开始， \(S\) 的长度为 \(i\) 的子串已经和 \(T\) 的长度为 \(i\) 的子串相似。那么我们只需要找到在此基础上，让长度为 \(i+1\) 的子串匹配的条件即可。

不难发现前 \(i\) 个字符已经满足对应的序关系，只需要让第 \(i+1\) 个字符满足和前 \(i\) 个字符相容的序关系即可。即，在 \(S\) 的子串和 \(T\) 中，都满足比第 \(i+1\) 个字符大的最小字符和比其小的最大字符是同一个字符即可，分别记作 \(up\) 和 \(down\) 。（当然，在这里，这样的字符可能有多个，它们也会是一一对应的。）换言之，第 \(i+1\) 个字符与 \(up\) 和 \(down\) 保持了相同的序关系。

复杂度化简

我们考虑在KMP预处理时同时预处理 \(up\) 和 \(down\) 的位置。对于此题，我们只需要维护一个数组记录哪些字符已经出现过即可。复杂度 \(O (k^2)\) ，其中 \(k\) 是字符种类数，此题中为 \(26\) 。对于字符种类数较大的情形，这部分复杂度可以通过维护一个树状数组/线段树/平衡树来降低复杂度到 \(O (k \log k)\)

匹配时，只需要用 \(up\) 和 \(down\) 给出的限制条件替换KMP中的字符相等条件，然后在匹配失败时用 \(next\) 进行跳转即可，复杂度线性。

在预处理KMP中 \(next\) 数组时，也沿用KMP的思想，通过前面已经预处理好的部分 \(next\) 数组和 \(up\) 与 \(down\) 来将预处理复杂度减少到和匹配一样，即线性。

代码

为防止抄袭，仅展示kmp部分，匹配部分其实已经蕴含在kmp部分了。

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constexpr unsigned K = 26;

bool single_match (int* ps, int up, int low) {
    if (up == low) return ps[-up] == *ps;
    if (up) {
        if (*ps >= ps[-up]) return false;
    }
    if (low) {
        if (*ps <= ps[-low]) return false;
    }
    return true;
}
/*************
ps => the pointer to the current matching position
up, low => the matching limit, indicating the relative position of the character to be compared

ps[-low] < *ps, *ps < ps[-up]

special cases:
up == 0 => no upper limit
low == 0 => no lower limit
up == low => ps[-low] == *ps, *ps == ps[-up]
*************/

void kmp (int* next, int* up, int* low, int* s, int m) {
    up[0] = low[0] = 0;

    int last_pos[K] = {};
    for (int i = 1; i <= m; ++i) {
        if (last_pos[s[i]]) {
            up[i] = low[i] = i - last_pos[s[i]];
        } else {
            int upx = s[i];
            for (; upx < K && !last_pos[upx]; ++upx) continue;
            if (upx == K)
                up[i] = 0;
            else
                up[i] = i - last_pos[upx];

            int lowx = s[i];
            for (; lowx >= 0 && !last_pos[lowx]; --lowx) continue;
            if (lowx == -1)
                low[i] = 0;
            else
                low[i] = i - last_pos[lowx];
        }

        last_pos[s[i]] = i;
    }

    next[0] = next[1] = 0;
    for (int i = 2, j = 0; i <= m; ++i) {
        for (; j > 0 && !single_match (s + i, up[j + 1], low[j + 1]); j = next[j]) continue;
        next[i] = ++j;
    }
}

原知乎回答

前言

两年前一位姚班大神学长回到高中母校并出了这样一道题：抢红包。

\(10\) 个人抢价值 \(10\) 元的红包，手气最佳同学抢到的红包的期望是多少？

注意这里是最大值，而最大值问题其实要比最小值问题难。当时我硬用积分和一些奇技淫巧给出了最小值的解答—— \(\dfrac{1}{10}\) 。而后大神给出了一个非常漂亮的解答，在这个解答中给出了任意排名的红包期望值。

为了尽量严谨化题意，他给出了一个等价的问题：

一段长为 \(n\) 的环形绳子，随机地切 \(n\) 刀，最长的一段长度的期望是多少？

“红包”的题面有一定歧义，因为我们没法很好定义怎么让 \(n\) 个人均匀地抢；而这个问题是良定义的，因为我可以良定义每一刀的均匀性——只要这一刀的概率密度在整根绳上均匀分布就可以了。

题主的问题相较于绳子版本的问题，相当于已经切好了第一刀——无论第一刀怎么切，环状绳子都会变成一段线状绳子，而这就和“玻璃棒”是等价的了。我们只需要一个额外假设——玻璃棒每一处断裂的可能性（在某种测度意义下）都是均等的——就可以说，题主的问题和绳子问题是等价的了。

题外话：我猜这个问题是受毕导一篇关于抢红包的文章的启发而来的。都是清华的大佬，想必在学术问题上也一定有很多交流吧（？

形式化题意

我们令 \(x_i\) 代表玻璃棒碎片的第 \(i\) 段的长度是多少。

概率密度函数在概率空间

\[ \Delta = \left\lbrace x \in \mathbb{R}^n : \sum x_i = 1, x_i \ge 0 \right\rbrace \]

给出的 \((n-1)\) 维多面体（如果读者熟悉的话，也就是单纯形）中是均匀分布的。从而，我们要研究的就变成了 \(\textbf {E} [\min \lbrace x_i\rbrace]\) 的值是多少。

至于为什么是均匀分布的，这里加以不太严谨地简单说明：

我们令 \(b_i\) 代表在环状绳子上切的第 \(i\) 刀的位置，那么， \(b \in \mathbb{T}^n\) 就是均匀分布的。我们令 \(l : \mathbb{T}^n \rightarrow \Delta_{n-1}\) 代表从 \(b_1\) 开始顺时针排列下每一段的绳长的计算函数。那么不难验证，这个函数是分 \(n!\) 段线性的，而每一段恰好对应一种 \(b_1\) 到 \(b_n\) 的大小顺序关系（也就是每一段绳子的起点和终点到底对应第几刀），且每一段几乎都是一个双射。而线性函数总是将均匀分布映射到均匀分布。

漂亮的构造

令

\[ S = \lbrace x \in \Delta : x_1 \ge \dots \ge x_n \rbrace \]

这是 \(\Delta\) 中保证 \(x_i\) 大小顺序的一个子多面体。显然，对于 \(x_i\) 之间大小顺序的 \(n!\) 种情况都是对称的，所以我们只要考虑在 \(S\) 中 \(\textbf {E}_{x \in S} [\min \lbrace x_i \rbrace] = \textbf {E}_{x \in S} [x_n]\) 的值是多少。

最关键的一步：构造线性双射：

\[ f : \Delta \rightarrow S, t \mapsto \left( \displaystyle\sum_{k=i}^n \frac{t_k}{k} \right)_i = \left( t_1 + \dfrac{1}{2} t_2 + \dots + \dfrac{1}{n} t_n, \dfrac{1}{2} t_2 + \dots + \dfrac{1}{n} t_n, \dots, \dfrac{1}{n-1} t_{n-1} + \dfrac{1}{n} t_n, \dfrac{1}{n} t_n \right) \]

用矩阵来表达，就是

\[ f = \begin{pmatrix} 1 & \dfrac{1}{2} & \dfrac{1}{3} & \cdots & \dfrac{1}{n} \\[1.5ex] 0 & \dfrac{1}{2} & \dfrac{1}{3} & \cdots & \dfrac{1}{n} \\[1.5ex] 0 & 0 & \dfrac{1}{3} & \cdots & \dfrac{1}{n} \\[1.5ex] \vdots & \vdots & \vdots & \ddots & \vdots \\[1.5ex] 0 & 0 & 0 & \cdots & \dfrac{1}{n} \end{pmatrix} \]

容易验证 \(f\) 双射的性质。

我们发现，随机变量 \(x\) 在这里可以用随机变量 \(t\) 来表示： \(x=f(t)\)

那么，对于题主最小值的问题，有

\[ \textbf {E}_{x \in S} [x_n] = \textbf {E}_{t \in \Delta} \left[ \dfrac{1}{n} t_n \right] = \dfrac{1}{n} \textbf {E}_{t \in \Delta} [t_n] = \dfrac{1}{n} \cdot \dfrac{1}{n} = \dfrac{1}{n^2} \]

而对于原题最大值的问题，有

\[ \textbf {E}_{x \in S} [x_1] = \textbf {E}_{t \in \Delta} \left[ \displaystyle\sum_{i=1}^{n}\dfrac{t_i}{i} \right] = \displaystyle\sum_{i=1}^{n} \dfrac{1}{i} \textbf {E}_{t \in \Delta} [t_i] = \displaystyle\sum_{i=1}^{n} \dfrac{1}{i} \cdot \dfrac{1}{n} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \dfrac{1}{i} \]

但是要注意原题红包总额是 \(n = 10\) ，所以手气最佳期望能抢到 \(1 + \dfrac{1}{2} + \cdots + \dfrac{1}{10}\) 元。

更进一步

这个构造十分精妙。但是，读者也许会不禁疑惑：这个构造看起来很随意，它是唯一的吗？

\(f\) 需要是一个从 \(\Delta\) 到 \(S\) 的线性双射；几何意义下，它们都是 \((n-1)\) 维的单形，也就是这样的双射，只需给定每个顶点是怎样被映射的，就可唯一确定。也就是说， \(f\) 几乎是唯一构造。

我们从代数角度说明为什么它是几乎唯一的。

令 \(f_i\) 代表 \(f\) 的第 \(i\) 行的行向量（也就是作为映射， \(x_i\) 作为 \(t_i\) 的线性组合的 \(n\) 个系数）。我们发现 \(f\) 必须满足三个性质：

• \(\sum f_i = (1)_j = (1, \dots , 1)\)
• \(f_i\) 非负且单调不增（需要每一个变元都单调不增）
• \(|\det f| = \dfrac{1}{n!}\)

第一条性质保证了 \(f\) 会从 \(\Delta\) 映射到 \(\Delta\) ；第二条性质保证了 \(f\) 会映射到 \(\Delta\) 中 \(S\) 的部分；第三条性质保证了 \(f\) 会正确地对应概率密度函数（再从几何直观上理解， \(f\) 的像集既在 \(S\) 之内，又和 \(S\) 的体积是一样的，那么只能恰好为 \(S\) 了）。

由性质二，不妨设 \(f_{n+1}=0\) ，并取 \(f\) 的一个差分矩阵（每行差分）：

\(M_i = f_i-f_{i+1}\)

从而三条性质等价地变化为

• \(\sum iM_i = (1)_j\)
• \(M_i \ge 0\) （指每个矩阵元非负）
• \(|\det M| = \dfrac{1}{n!}\)

则

\[ \begin{align*} |\det M| &\le \prod_{i=1}^n \|M_i\|_2 \\ &= \frac{1}{n!} \prod_{i=1}^n \|i M_i\|_2 \\ &\le \frac{1}{n!} \left( \frac{\sum_{i=1}^n \|i M_i\|_2}{n} \right)^n \\ &\le \frac{1}{n!} \left( \frac{\sum_{i=1}^n \sum_{j=1}^n i M_{ij}}{n} \right)^n \\ &= \frac{1}{n!} \left( \frac{\sum_{j=1}^n 1}{n} \right)^n \\ &= \frac{1}{n!} \left( \frac{n}{n} \right)^n \\ &= \frac{1}{n!} \end{align*} \]

第一个不等号对应Hadamard不等式，第二个不等号对应均值不等式。取等条件由三个不等号的取等条件共同给出：第三个不等号告诉我们每一行只有一个非零元，第一个不等号告诉我们每一行都是正交的向量，即每行的非零元对应的列都不一样，第二个不等号告诉我们第 \(i\) 行的非零元满足 \(M_{i, \sigma(i)} \cdot i\) 是一个常量，从而易知 \(M_{i, \sigma(i)} = \dfrac{1}{i}\) （因为现在行列式的绝对值等于每个非零元的乘积）。也就是 \(M\) 必须是我们给出的例子 \(f\) 的差分矩阵只交换列得到的。从而每一个满足要求的 \(f\) 都可以通过我们给出的例子交换一些列来得到。这刚好对应了几何直观： \((n-1)\) 维单纯形的 \(n\) 个顶点的排列。