标题： 自生成测度与幂律的质心刚性 显示英文标题

标题： Self-Generated Measures and the Centroid Rigidity of Power Laws

摘要： 我们重新审视一个经典的微积分计算：函数在从0到a的区间上的子图的形心，并表明它隐藏了一个刚性定理。 设f在(0, 无穷大)上二次连续可微，取值在(0, 无穷大)，并且满足f(0+) = 0。 定义xbar(a)为从0到a的x f(x) dx的积分除以从0到a的f(x) dx的积分，定义ybar(a)为1/2乘以从0到a的f(x)^2 dx的积分除以从0到a的f(x) dx的积分。 我们证明了几何缩放性质，即对于每个a > 0，恒等式ybar(a) = lambda * f(xbar(a))成立，当且仅当f(x) = A * x^p，其中A > 0且p > 0。 对于这些幂律，最优常数是lambda = (p+1)/(2(2p+1)) * ((p+2)/(p+1))^p。 经过无量纲归一化后，证明是概率性的：在(0, a)上具有与f成比例密度的自生成概率测度，我们有xbar(a)等于Xa的期望值，ybar(a)等于1/2乘以f(Xa)的期望值，因此几何缩放性质在所有截断尺度下的期望中成为等式。 对a求导得到弹性E(x) = x f'(x) / f(x)的加权均值恒等式；第二次求导迫使方差原理消失，使E为常数，因此f为纯幂函数，所述的lambda值随之得出。 该论证不使用渐近分析，并扩展到在(0, 无穷大)上一次连续可微且弹性局部Lipschitz的f。 显示英文摘要

摘要： We revisit a classical calculus computation: the centroid of the subgraph of a function on the interval from 0 to a, and show that it hides a rigidity theorem. Let f be twice continuously differentiable on (0, infinity), take values in (0, infinity), and satisfy f(0+) = 0. Define xbar(a) as (integral from 0 to a of x f(x) dx) divided by (integral from 0 to a of f(x) dx), and define ybar(a) as (1/2) times (integral from 0 to a of f(x)^2 dx) divided by (integral from 0 to a of f(x) dx). We prove that the Geometric Scaling Property, namely the identity ybar(a) = lambda * f(xbar(a)) for every a > 0, holds if and only if f(x) = A * x^p with A > 0 and p > 0. For these power laws the optimal constant is lambda = (p+1)/(2(2p+1)) * ((p+2)/(p+1))^p. After a scale-free normalization, the proof is probabilistic: with the self-generated probability measure on (0, a) having density proportional to f, we have xbar(a) equal to the expected value of Xa and ybar(a) equal to (1/2) times the expected value of f(Xa), so the Geometric Scaling Property becomes an equality in expectation across all truncation scales. Differentiating with respect to a yields a weighted mean identity for the elasticity E(x) = x f'(x) / f(x); a second differentiation forces a vanishing variance principle that makes E constant, hence f a pure power, and the stated value of lambda follows. The argument uses no asymptotics and extends to f that is once continuously differentiable on (0, infinity) with locally Lipschitz elasticity.