普通原因变异和特殊原因变异

吉布斯现象描述了傅里叶级数在跳跃不连续点处的行为。级数的部分和在跳跃点附近会出现一个过冲，并且随着级数项数的增加，这个过冲并不会消失。无论级数项数如何，这个过冲最终都会收敛到一个约为跳跃高度9%的常数值。

该定理指出，在误差均值为零、不相关且方差恒定（同方差性）的线性回归模型中，普通最小二乘法 (OLS) 估计量是最佳线性无偏估计量 (BLUE)。“最佳”意味着它在所有回归系数的线性无偏估计量中方差最小，因此精度最高。

This theorem states that for any continuous function [latex]f[/latex] mapping a compact convex set to itself, there is a point [latex]x_0[/latex] such that [latex]f(x_0) = x_0[/latex]. This point is called a fixed point. Informally, if you take a map of a country, crumple it up, and place it inside the country's borders, there will always be at least one point on the map directly above its corresponding real-world location.

A graphical tool used in SPC to monitor a process variable over time. It plots data points between a central line (CL), representing the process average, and upper (UCL) and lower (LCL) control limits. These limits are typically set at three standard deviations from the mean ([latex]\mu \pm 3\sigma[/latex]), defining the range of expected common cause variation.

One-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent groups. It analyzes the effect of a single categorical independent variable, known as a factor, on a continuous dependent variable. The null hypothesis states that all group means are equal, [latex]H_0: \mu_1 = \mu_2 = \dots = \mu_k[/latex].

The Prime Number Theorem describes the asymptotic distribution of prime numbers among the integers. It states that the prime-counting function [latex]\pi(x)[/latex], which gives the number of primes less than or equal to [latex]x[/latex], is asymptotically equivalent to [latex]x / \ln(x)[/latex]. Formally, [latex]\lim_{x \to \infty} \frac{\pi(x)}{x/\ln(x)} = 1[/latex]. This provides a fundamental link between primes and the natural logarithm.

A statistic indicating the goodness of fit of a model, representing the proportion of the variance in the dependent variable that is predictable from the independent variable(s). An R² of 1 indicates a perfect fit, while 0 indicates no linear relationship. It is calculated as [latex]R^2 \equiv 1 - \frac{SS_{res}}{SS_{tot}}[/latex], where [latex]SS_{res}[/latex] is the residual sum of squares.

The Fredholm index generalizes the rank-nullity theorem to infinite-dimensional spaces like Banach spaces. For a Fredholm operator [latex]T: X \to Y[/latex], its index is defined as [latex]\text{ind}(T) = \dim(\ker(T)) - \dim(\text{coker}(T))[/latex], where the Cokernel's dimension measures how far the image is from being the whole space. This index is a stable integer value under small perturbations of the operator.

A topological space is an ordered pair [latex](X, \tau)[/latex], where [latex]X[/latex] is a set and [latex]\tau[/latex] is a collection of subsets of [latex]X[/latex], called open sets, satisfying three axioms: 1) The empty set [latex]\emptyset[/latex] and [latex]X[/latex] itself are in [latex]\tau[/latex]. 2) The union of any number of sets in [latex]\tau[/latex] is also in [latex]\tau[/latex]. 3) The intersection of any finite number of sets in [latex]\tau[/latex] is also in [latex]\tau[/latex].

策梅洛-弗兰克尔集合论（Zermelo-Fraenkel set theory，简称ZFC，其中ZFC包含选择公理）是当代数学的标准公理系统。它由一系列用一阶逻辑表达的公理组成，这些公理形式化了集合的性质。当今几乎所有数学定理都可以用ZFC来表述和证明。

方差分析 (ANOVA) 是一种统计方法，它将数据集中观察到的总变异性划分为可归因于不同来源的各个部分。其核心思想是比较不同组均值之间的方差与组内方差。如果组间方差显著较大，则表明组均值确实存在差异。

The Courant–Friedrichs–Lewy (CFL) condition is a necessary stability criterion for numerical solutions of hyperbolic partial differential equations using explicit time-integration schemes. It dictates that the time step size must be small enough that information does not travel further than one spatial grid cell per time step. For a 1D case, [latex]C = u \frac{\Delta t}{\Delta x} \le C_{max}[/latex], ensuring numerical stability.

要使方差分析的结果有效，必须满足几个关于数据的关键假设。这些假设包括：(1) 观测值独立，即误差不相关。(2) 正态性，即每组的残差近似呈正态分布。(3) 同方差性，即所有组的残差方差相等。