高斯-博内定理

A fundamental second-order linear parabolic partial differential equation describing heat distribution or other diffusion processes. Its canonical form is [latex]\frac{partial u}{partial t} = \alpha \nabla^2 u[/latex], where [latex]u(\vec{x},t)[/latex] is temperature, [latex]t[/latex] is time, and [latex]\alpha[/latex] is thermal diffusivity. Solutions model how an initial temperature distribution evolves, smoothing out irregularities over time and approaching a steady state.

The coefficients for the Fourier series of a function [latex]s(x)[/latex] with period [latex]P[/latex] are calculated using integral formulas. The DC component is [latex]a_0 = \frac{2}{P} \int_{P} s(x) , dx[/latex]. The cosine coefficients are [latex]a_n = \frac{2}{P} int_{P} s(x) \cos\left(\frac{2pi n x}{P}\right) , dx[/latex], and the sine coefficients are [latex]b_n = \frac{2}{P} \int_{P} s(x) \sin\left(\frac{2\pi n x}{P}\right) , dx[/latex] for [latex]n ge 1[/latex].

A fundamental solution of a linear partial differential operator [latex]L[/latex] is a solution to the equation [latex]Lu = delta(x)[/latex], where [latex]delta(x)[/latex] is the Dirac delta function. It represents the response of the system to a point source or impulse. Once known, the solution to the inhomogeneous equation [latex]Lu = f(x)[/latex] can be found by convolution: [latex]u(x) = (G * f)(x)[/latex], where [latex]G[/latex] is the fundamental solution.

准确度是指测量值与标准值或已知真值接近的程度。精密度是指两次或多次测量结果彼此接近的程度。一个测量系统可能精密但不准确，也可能准确但不精密，或者两者都不精确，或者两者都精确。在测量理论中，这两个概念是相互独立的。

黎曼几何是微分几何的一个分支，研究黎曼流形——具有黎曼度量的光滑流形。该度量是切空间上内积的集合，其值随点而平滑变化。它允许定义局部几何概念，例如角度、曲线长度、表面积和体积，从而引出广义的曲率概念。

In linear algebra, the rank-nullity theorem states that for any linear map [latex]T: V \to W[/latex] between finite-dimensional vector spaces, the dimension of its domain [latex]V[/latex] is the sum of its rank (the dimension of its image) and its nullity (the dimension of its kernel). The formula is [latex]\dim(V) = \text{rank}(T) + \text{nullity}(T)[/latex].

Bayesian inference is a statistical method where Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. It is a central tenet of Bayesian statistics. The core idea is expressed as: posterior probability is proportional to the product of the prior probability and the likelihood, [latex]p(\theta|D) \propto p(D|\theta)p(\theta)[/latex], where [latex]\theta[/latex] is the parameter and D is the data.

A Fourier series decomposes any periodic function or signal into a sum of simple oscillating functions, namely sines and cosines. For a function [latex]s(x)[/latex] with period [latex]P[/latex], the series is given by [latex]s(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{2\pi n x}{P}\right) + b_n \sin\left(\frac{2\pi n x}{P}\right)\right][/latex]. The terms [latex]a_n[/latex] and [latex]b_n[/latex] are the Fourier coefficients.

Theorema Egregium (Latin for "Remarkable Theorem") states that the Gaussian curvature of a surface is an intrinsic property. This means it depends only on how distances are measured on the surface itself, not on how the surface is embedded in three-dimensional space. A flat sheet of paper can be rolled into a cylinder but not a sphere without stretching.

为了使傅里叶级数收敛到函数值，该函数必须在一个周期内满足狄利克雷条件。这些条件是：（1）该函数必须是绝对可积的；（2）它必须有有限个极值点（极大值和极小值）；（3）它必须有有限个有限间断点。

De Morgan's laws are a pair of transformation rules in Boolean algebra that are fundamental to digital circuit design. The first law states that the negation of a conjunction is the disjunction of the negations: [latex]\neg(P \land Q) \iff (\neg P) \lor (\neg Q)[/latex]. The second states that the negation of a disjunction is the conjunction of the negations: [latex]\neg(P \lor Q) \iff (\neg P) \land (\neg Q)[/latex].

A differentiable manifold is a topological space that is locally similar to Euclidean space, allowing calculus to be applied. Each point has a neighborhood that is homeomorphic to an open subset of [latex]\mathbb{R}^n[/latex]. These local coordinate systems, called charts, are related by smooth transition functions, forming an atlas that defines the manifold's differentiable structure.

数字电子学基于布尔代数，这是一种由乔治·布尔提出的逻辑数学系统。它使用两个值，通常是0和1（或真和假），以及三种基本运算：与（AND）、或（OR）和非（NOT）。这些运算直接对应于构成所有数字电路基本单元的逻辑门。

同胚是指两个拓扑空间之间的连续函数，且该函数存在连续的逆函数。如果这样的函数存在，则称两个拓扑空间是同胚的。从拓扑学的角度来看，同胚空间是相同的。这个概念体现了物体可以被拉伸、弯曲或变形为另一个物体而不会撕裂或粘合的思想，例如将咖啡杯变成甜甜圈。