How far sovereignty has moved between different branches of government | 主权在政府不同分支间移动了多远

📚 How far sovereignty has moved between different branches of government | 主权在政府不同分支间移动了多远

In A Level Mathematics, the question of how far sovereignty has moved between different branches of government can be explored by treating sovereignty as a measurable quantity and applying functions, calculus, probability and statistical reasoning. This article builds a quantitative model of the executive, legislature and judiciary to assess the distance and rate of sovereignty transfer.

在 A Level 数学中，关于主权在政府不同分支之间移动了多远的问题，可以通过将主权视为一个可测量的量，并运用函数、微积分、概率和统计推理来探讨。本文将行政、立法和司法三个分支建立定量模型，以评估主权转移的距离和速率。

---

1. Sovereignty as a Measurable Variable | 将主权视为可测量变量

To answer ‘how far’ sovereignty has moved, we first define sovereignty S(t) as a continuous variable representing the share of total constitutional power held by a branch at time t. S(t) can be measured on a scale from 0 to 1, or equivalently as a percentage from 0% to 100%.

为了回答主权移动了多远，我们首先将主权 S(t) 定义为一个连续变量，表示某一分支在时间 t 所持有的宪制权力份额。S(t) 可以在 0 到 1 的范围内测量，或者等价地用百分比表示为 0% 到 100%。

Using a mathematical definition allows us to compare changes over time, calculate rates of change, and estimate total movement. A value of 0 means the branch has no sovereignty, while 1 means it holds all sovereignty.

使用数学定义使我们能够比较随时间的变化、计算变化率并估计总移动量。数值为 0 表示该分支没有主权，而数值为 1 表示该分支拥有全部主权。

• S(t) ∈ [0, 1] for each branch.
• S(t) ∈ [0, 1] 对每个分支成立。
• The sum of sovereignty across branches equals 1 at any time t.
• 所有分支的主权在任意时间 t 的总和等于 1。

---

2. Representing Branches of Government | 表示政府的不同分支

Let E(t), L(t) and J(t) denote the sovereignty shares of the executive, legislature and judiciary respectively. The constraint E(t) + L(t) + J(t) = 1 must hold at all times, because the total constitutional power is fixed.

设 E(t)、L(t) 和 J(t) 分别表示行政、立法和司法分支的主权份额。由于宪制权力总量固定，任何时刻都必须满足 E(t) + L(t) + J(t) = 1。

A simple data table can show how sovereignty has shifted between the three branches over selected years.

一个简单的数据表可以展示主权在这三个分支之间在选定年份如何转移。

Year (t)	Executive E(t)	Legislature L(t)	Judiciary J(t)
0	0.45	0.40	0.15
5	0.53	0.32	0.15
10	0.60	0.25	0.15

This table indicates that sovereignty has moved from the legislature to the executive, while the judiciary has remained stable.

该表表明主权从立法分支转移到了行政分支，而司法分支保持稳定。

---

3. Modelling the Shift with Linear Functions | 用线性函数对转移建模

A linear model is the simplest way to describe a steady transfer of sovereignty. Suppose the executive share increases at a constant rate and the legislature decreases at a constant rate, while the judiciary remains unchanged.

线性模型是描述主权稳定转移的最简单方法。假设行政分支的份额以恒定速率增加，立法分支以恒定速率减少，而司法分支保持不变。

E(t) = 0.45 + 0.015t, L(t) = 0.40 – 0.015t, J(t) = 0.15

At any time t, the sum E(t) + L(t) + J(t) = 0.45 + 0.40 + 0.15 = 1, so the model preserves total sovereignty. The coefficient 0.015 means that the executive gains 1.5 percentage points of sovereignty per year.

在任何时间 t，E(t) + L(t) + J(t) = 0.45 + 0.40 + 0.15 = 1，因此该模型保持了主权总量。系数 0.015 意味着行政分支每年获得 1.5 个百分点的主权。

Linear functions are useful for short-term analysis because they provide clear, constant gradients that can be interpreted directly as rates of sovereignty movement.

线性函数对于短期分析很有用，因为它们提供了清晰且恒定的梯度，可以直接解释为主权移动的速率。

---

4. Calculus: Rate of Change | 微积分：变化率

Differentiation gives the instantaneous rate at which sovereignty moves from one branch to another. For the linear model above, the derivatives are:

微分可以给出主权从一个分支转移到另一个分支的瞬时速率。对于上述线性模型，其导数为：

dE/dt = 0.015, dL/dt = -0.015, dJ/dt = 0

The negative derivative dL/dt = -0.015 shows that legislative sovereignty is decreasing at 0.015 per year. The absolute values of the derivatives sum to 0.03, but because one gain and one loss are two sides of the same transfer, the net movement rate is half of this sum, namely 0.015 per year.

负导数 dL/dt = -0.015 表明立法主权每年减少 0.015。导数绝对值之和为 0.03，但由于一个增加和一个减少是同一转移的两个方面，净移动速率是这一总和的一半，即每年 0.015。

In a more general model, the total rate of sovereignty movement v(t) between branches can be defined as:

在更一般的模型中，分支之间主权移动的总速率 v(t) 可以定义为：

v(t) = ½(|dE/dt| + |dL/dt| + |dJ/dt|)

This formula avoids double counting the transfer, since a move of 0.015 from the legislature to the executive is a single shift, not two separate shifts.

该公式避免了对转移的重复计算，因为从立法分支到行政分支的 0.015 移动是一次单一转移，而不是两次独立转移。

---

5. Integration: Total Distance Moved | 积分：移动的总距离

To find the total amount of sovereignty that has moved between the branches over a time interval [0, T], we integrate the rate of movement. The total distance D is:

为了求在时间区间 [0, T] 内分支之间主权移动的总量，我们对移动速率进行积分。总距离 D 为：

D = ∫₀ᵀ v(t) dt = ½ ∫₀ᵀ (|dE/dt| + |dL/dt| + |dJ/dt|) dt

For the linear model over 10 years, v(t) is constant at 0.015, so the integral is:

对于 10 年期的线性模型，v(t) 恒为 0.015，因此积分为：

D = 0.015 × 10 = 0.15

This means that 15 percentage points of sovereignty have moved from the legislature to the executive over the ten-year period. The integral approach is powerful because it works even when the rate of movement changes with time.

这意味着在十年期间有 15 个百分点的主权从立法分支转移到了行政分支。积分方法非常有效，因为即使移动速率随时间变化，它也能适用。

---

6. Exponential and Logarithmic Models | 指数与对数模型

Sovereignty transfer is not always linear. In many political systems, the executive may gain power rapidly at first and then more slowly, or vice versa. An exponential model captures such behaviour.

主权转移并不总是线性的。在许多政治体系中，行政分支可能起初权力增长迅速，随后变慢，或者相反。指数模型可以捕捉这种行为。

E(t) = E₀e^(kt)

Here E₀ is the initial executive sovereignty and k is the continuous growth rate. If k > 0, executive sovereignty grows exponentially. Taking natural logarithms gives a linear relationship:

其中 E₀ 是初始行政主权，k 是连续增长率。如果 k > 0，行政主权按指数增长。取自然对数可得到线性关系：

ln E(t) = ln E₀ + kt

This means that a plot of ln E(t) against t should produce a straight line with gradient k. The value of k can be estimated from data by using two points and the formula:

这意味着 ln E(t) 对 t 作图应产生一条梯度为 k 的直线。k 的值可以通过两个数据点用以下公式估计：

k = (ln E₂ – ln E₁) / (t₂ – t₁)

Exponential models reveal whether sovereignty movement is accelerating or decelerating, which is central to the question of how far it has moved.

指数模型揭示了主权移动是在加速还是减速，这对于主权移动了多远的问题至关重要。

---

7. Probability and Voting Power | 概率与投票权

Another mathematical perspective treats sovereignty movement as a random process. Suppose the transfer of sovereignty from the legislature to the executive in a given year is a random variable X with a normal distribution X ~ N(μ, σ²).

另一种数学视角将主权移动视为一个随机过程。假设某一年主权从立法分支向行政分支的转移量是一个随机变量 X，服从正态分布 X ~ N(μ, σ²)。

The mean μ represents the expected annual shift, while the standard deviation σ measures uncertainty. We can calculate the probability that the executive gains more than a certain threshold amount in a given year.

均值 μ 表示预期的年度转移量，而标准差 σ 衡量不确定性。我们可以计算行政分支在某一年获得超过某个阈值主权量的概率。

P(X > a) = 1 – Φ((a – μ)/σ)

Here Φ is the cumulative distribution function of the standard normal distribution. If μ = 0.015 and σ = 0.005, the probability that the executive gains more than 0.02 in a year is P(X > 0.02) = 1 – Φ(1) = 0.1587.

其中 Φ 是标准正态分布的累积分布函数。如果 μ = 0.015 且 σ = 0.005，则行政分支在一年内获得超过 0.02 的主权的概率为 P(X > 0.02) = 1 – Φ(1) = 0.1587。

This probabilistic model allows us to attach confidence levels to statements about how far sovereignty has moved.

这个概率模型使我们能够为关于主权移动了多远的陈述附加置信水平。

---

8. Game Theory and Institutional Equilibrium | 博弈论与制度均衡

Game theory explains why sovereignty moves between branches as a result of strategic interaction. The executive and legislature can be modelled as players choosing between ‘expand power’ and ‘accept limits’.

博弈论解释了主权为何因战略互动而在分支之间移动。行政和立法分支可以被建模为在“扩张权力”和“接受限制”之间进行选择的参与者。

A payoff matrix can show the net sovereignty gain for the executive under each combination of strategies.

一个收益矩阵可以显示在每个策略组合下行政分支的净主权收益。

Executive \ Legislature	Accept limits	Expand power
Expand power	+0.03	-0.01
Accept limits	0	-0.02

The Nash equilibrium occurs when neither player can improve by changing strategy unilaterally. In many such models, both branches choose to expand power, leading to a gradual shift of sovereignty toward the branch with greater institutional resources.

当任何一方都无法通过单方面改变策略而改善结果时，就达到了纳什均衡。在许多此类模型中，两个分支都会选择扩张权力，导致主权逐渐转向拥有更多制度资源的分支。

Game theory therefore provides a causal mechanism for the movement measured by calculus and statistics.

因此，博弈论为微积分和统计所测量的主权移动提供了因果机制。

---

9. Statistical Correlation between Branches | 分支之间的统计相关性

To assess whether sovereignty moves as a zero-sum trade-off between branches, we can calculate the Pearson correlation coefficient r between the executive and legislative sovereignty over time.

为了评估主权是否在分支之间以零和方式转移，我们可以计算行政和立法主权随时间的皮尔逊相关系数 r。

r = Sxy / (√Sxx × √Syy)

Here Sxy is the covariance of the two variables, and Sxx, Syy are the sums of squared deviations. If r = -1, every gain by the executive is exactly matched by a loss by the legislature, confirming a perfect zero-sum transfer.

其中 Sxy 是两个变量的协方差，Sxx、Syy 是离差平方和。如果 r = -1，则行政分支的每一次增加都恰好对应立法分支的等量减少，证实了完美的零和转移。

A value of r close to -0.9 would indicate that most, but not all, sovereignty movement is a direct transfer, with some sovereignty moving to or from the judiciary or external bodies.

r 接近 -0.9 的值表明大多数但并非全部主权移动是直接转移，部分主权会移向或移出司法分支或外部机构。

Correlation analysis therefore quantifies how closely the branches are linked in their sovereignty changes.

因此，相关性分析量化了各分支在主权变化中的关联程度。

---

10. Evaluating How Far: Limits and Error Bounds | 评估多远：极限与误差界

Mathematical limits can describe the furthest possible movement under a given model. For the exponential model E(t) = E₀e^(kt) with k > 0, the limit as t tends to infinity is:

数学极限可以描述在给定模型下可能的最大移动。对于 E(t) = E₀e^(kt) 且 k > 0 的指数模型，当 t 趋于无穷大时，极限为：

lim(t→∞) E(t) = ∞

This is not realistic for sovereignty, because S(t) cannot exceed 1. A logistic model is more appropriate when there is an upper bound:

这对于主权来说并不现实，因为 S(t) 不能超过 1。当存在上限时，逻辑斯蒂模型更为合适：

E(t) = K / (1 + ae^(-bt))

Here K is the carrying capacity, or maximum possible sovereignty share. As t → ∞, E(t) approaches K, so the furthest possible movement is K – E₀.

其中 K 是承载容量，即主权份额的最大可能值。当 t → ∞ 时，E(t) 趋近于 K，因此可能的最大移动量为 K – E₀。

Error bounds from regression analysis or confidence intervals can also state how precise our estimate of movement is. A 95% confidence interval for D might be [0.12, 0.18], meaning the true sovereignty shift is likely between 12 and 18 percentage points.

来自回归分析或置信区间的误差界也可以说明我们对移动量估计的精确程度。D 的 95% 置信区间可能是 [0.12, 0.18]，意味着主权转移的真实值很可能在 12 到 18 个百分点之间。

---

11. Conclusion: A Mathematical Answer to a Political Question | 结论：政治问题的数学答案

The mathematical analysis shows that the question ‘How far has sovereignty moved between different branches of government?’ can be answered using functions, derivatives, integrals, probability, game theory and statistical correlation. In our worked example, sovereignty moved 15 percentage points from the legislature to the executive over ten years, with a rate of 1.5 percentage points per year and a correlation of approximately -1 between the two branches.

数学分析表明，“主权在政府不同分支之间移动了多远”这一问题可以通过函数、导数、积分、概率、博弈论和统计相关性

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）