狄利克雷收敛条件

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.

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。

A standard approach for approximating solutions to overdetermined systems by finding model parameters that minimize the sum of the squared differences between observed and predicted values. This sum is known as the sum of squared residuals (SSR). The goal is to find the parameters [latex]\hat{\beta}[/latex] that minimize the function [latex]S(\beta) = \sum_{i=1}^{n} (y_i - x_i^T \beta)^2[/latex].

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.

The Gauss-Bonnet theorem connects the geometry of a compact two-dimensional surface to its topology. It states that the integral of the Gaussian curvature [latex]K[/latex] over the entire surface [latex]M[/latex] is equal to [latex]2\pi[/latex] times the Euler characteristic [latex]\chi(M)[/latex] of the surface. The formula is [latex]\int_M K \, dA = 2\pi \chi(M)[/latex].

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].