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Mathematics椭圆曲线在数学中，椭圆曲线是平滑的投影代数曲线，其上有一个指定点 O。椭圆曲线在域 K 上定义，并描述 K 中的点（K 与其自身的笛卡尔积）。如果场的特性与2和3不同，则该曲线可以被描述为平面代数曲线，其由以下解（x，y）组成：对于K中的某些系数a和b。该曲线要求是非奇异的，这意味着该曲线没有尖点或自相交。 （这相当于条件 4a + 27b ≠ 0，即 x 中无平方。）通常理解，曲线嵌入在射影平面中，点 O 是无穷远处的唯一点。
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a + 27b ≠ 0, that is, being square-free in x.) It is usually understood that the curve is embedded in the projective plane, with the point O being the unique point at infinity.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在理论计算机科学中，时间复杂度是描述运行算法所需的计算机时间量的计算复杂度。时间复杂度通常通过计算算法执行的基本操作的数量来估计，假设每个基本操作需要固定的时间来执行。因此，算法所花费的时间量和执行的基本运算的数量被认为与常数因子相关。由于算法的运行时间可能因相同大小的不同输入而异，因此通常考虑最坏情况的时间复杂度，即给定大小的输入所需的最大时间量。
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在硬件和软件系统的背景下，形式验证是使用形式数学方法证明或反驳系统在特定形式规范或属性方面的正确性的行为。形式化验证是系统形式化规范的关键激励因素，也是形式化方法的核心。它代表了电子设计自动化中分析和验证的一个重要维度，并且是软件验证的一种方法。使用形式验证可在计算机安全认证通用标准框架内实现最高评估保证级别 (EAL7)。形式验证有助于证明系统的正确性，例如：密码协议、组合电路、带内部存储器的数字电路以及以编程语言表示为源代码的软件。
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents an important dimension of analysis and verification in electronic design automation and is one approach to software verification. The use of formal verification enables the highest Evaluation Assurance Level (EAL7) in the framework of common criteria for computer security certification. Formal verification can be helpful in proving the correctness of systems such as: cryptographic protocols, combinational circuits, digital circuits with internal memory, and software expressed as source code in a programming language.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，排列是一个集合到其自身的双射。它可以被解释为序列或线性顺序的成员的新顺序，或者被解释为改变有序集合的线性顺序的行为或过程。第一种解释的示例是集合 {1, 2, 3} 的六个排列，即六个 3 元组 (1, 2, 3)、(1, 3, 2)、(2, 1, 3)、(2, 3, 1)、(3, 1, 2) 和 (3, 2, 1)。它们对应于将 1、2、3 分别映射到元组的第一个、第二个和第三个成员的六个双射。字母全部不同的单词的字谜词也是排列：字母已经在原始单词中排序，字谜词重新排序它们。有限集排列的研究是组合数学和群论中的一个重要课题。排列几乎用于数学的每个分支和许多其他科学领域。
In mathematics, a permutation is a bijection of a set onto itself. It can be interpreted as a new order of the members of a sequence or linear order, or as the act or process of changing the linear order of an ordered set. An example of the first interpretation is the six permutations of the set {1, 2, 3}, which are the six 3-tuples (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). They correspond to the six bijections that map 1, 2, 3, to the first, the second and the third member of the tuple, respectively. Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 集体智慧 集体行动 自组织临界性 群体心态 相变 基于代理的建模 同步 蚁群优化 粒子群优化 群体行为 社交网络分析 小世界网络 中心性主题 图论 扩展 鲁棒性 系统生物学 动态网络 进化计算 遗传算法 遗传编程 人工生命 机器学习 进化发育生物学 人工智能 进化机器人
Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour Social network analysis Small-world networks Centrality Motifs Graph theory Scaling Robustness Systems biology Dynamic networks Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在常见用法中，随机性是指信息中明显或实际缺乏明确的模式或可预测性。事件、符号或步骤的随机序列通常没有顺序，也不遵循可理解的模式或组合。根据定义，单个随机事件是不可预测的，但如果存在已知的概率分布，则重复事件（或“试验”）中不同结果的频率是可预测的。例如，当掷两个骰子时，任何特定骰子的结果都是不可预测的，但 7 的总和出现的频率往往是 4 的两倍。从这个角度来看，随机性并不是随意性；而是随机性。它是对结果不确定性的衡量。随机性适用于机会、概率和信息熵的概念。
In common usage, randomness is the apparent or actual lack of definite patterns or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在概率论和信息论中，两个随机变量的互信息（mutual Information，MI）度量了两个变量之间相互依赖的程度。具体来说，对于两个随机变量，MI是一个随机变量由于已知另一个随机变量而减少的“信息量”（单位通常为比特）。互信息的概念与随机变量的熵紧密相关，熵是信息论中的基本概念，它量化的是随机变量中所包含的“信息量”。 MI不仅仅是度量实值随机变量和线性相关性(如相关系数)，它更为通用。互信息决定了随机变量 ( X , Y ) {\displaystyle {\displaystyle (X,Y)}} 的联合分布与 X {\displaystyle X} 和 Y {\displaystyle Y} 的边缘分布的乘积之间的差异。MI是点互信息（Pointwise Mutual Information，PMI）的期望。克劳德·香农在他的论文A Mathematical Theory of Communication中定义并分析了这个度量，但是当时他并没有将其称为“互信息”。这个词后来由罗伯特·法诺创造。互信息也称为信息增益。
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" (in units such as shannons (bits), nats or hartleys) obtained about one random variable by observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable. Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair ( X , Y ) {\displaystyle (X,Y)} is from the product of the marginal distributions of X {\displaystyle X} and Y {\displaystyle Y} .
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 卡尔曼滤波（英语：Kalman filter）是一种高效率的递归滤波器（自回归滤波器），它能够从一系列的不完全及包含噪声的测量中，估计动态系统的状态。卡尔曼滤波会根据各测量量在不同时间下的值，考虑各时间下的联合分布，再产生对未知变量的估计，因此会比只以单一测量量为基础的估计方式要准。卡尔曼滤波得名自主要贡献者之一的鲁道夫·卡尔曼。 卡尔曼滤波在技术领域有许多的应用。常见的有飞机及太空船的导引、导航及控制。卡尔曼滤波也广为使用在时间序列的分析中，例如信号处理及计量经济学中。卡尔曼滤波也是机器人运动规划及控制的重要主题之一，有时也包括在轨迹优化。卡尔曼滤波也用在中轴神经系统运动控制的建模中。因为从给与运动命令到收到感觉神经的回授之间有时间差，使用卡尔曼滤波有助于建立符合实际的系统，估计运动系统的目前状态，并且更新命令。 卡尔曼滤波的算法是二步骤的程序。在估计步骤中，卡尔曼滤波会产生有关目前状态的估计，其中也包括不确定性。只要观察到下一个量测（其中一定含有某种程度的误差，包括随机噪声）。会通过加权平均来更新估计值，而确定性越高的量测加权比重也越高。算法是迭代的，可以在实时控制系统中执行，只需要目前的输入量测、以往的计算值以及其不确定性矩阵，不需要其他以往的信息。 使用卡尔曼滤波不用假设误差是正态分布，不过若所有的误差都是正态分布，卡尔曼滤波可以得到正确的条件概率估计。 也发展了一些扩展或是广义的卡尔曼滤波，例如运作在非线性系统的扩展卡尔曼滤波及无迹卡尔曼滤波（英语：unscented Kalman filter）。底层的模型类似隐马尔可夫模型，不过潜在变量的状态空间是连续的，而且所有潜在变量及可观测变量都是正态分布。
In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but also relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán. Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. Kalman filtering is also important for robotic motion planning and control, and can be used for trajectory optimization.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学、尤其是泛函分析中，矢量空间 V {\displaystyle V} 上的自伴算子（self-adjoint operator）是一类特殊的线性算子（自同态），其伴随算子是其自身。根据不同的需要，可以讨论 V {\displaystyle V} 为拓扑矢量空间、赋范矢量空间、巴拿赫空间乃至希尔伯特空间的情况，使得伴随算子、自伴算子可以具有更丰富的性质，一个重要的例子是希尔伯特空间上自伴算子的谱定理。 若 V {\displaystyle V} 是具有规范正交基的有限维复向量空间，其上自伴算子在该基下的矩阵是埃尔米特矩阵——该矩阵等于自身的共轭转置。有限维的谱定理表明，对于一个算子 A {\displaystyle A} ，总能找到 V {\displaystyle V} 上的规范正交基使得 A {\displaystyle A} 在该基下的矩阵是一个对角矩阵，且这些对角元都是实数。 无穷维希尔伯特空间上的自伴算子的谱定理与此类似：一个算子是自伴的，当且仅当其酉等价于一个实值乘法算子。不过，黑林格-特普利茨定理表明了定义于全空间的自伴算子必然是有界的，从而无界算子至多只能定义在全空间的一个稠密子空间上，故对于无界算子须对定义域的问题多加注意。定义域的问题造成了对称算子和自伴算子的区分，而这区分对于谱定理等结论而言是至关重要的。 自伴算子在量子力学中也有重要地位。在量子力学公理的狄拉克-冯诺依曼表述中，位置、动量、角动量和自旋等物理可观测量是由希尔伯特空间上的自伴算子表示。在哈密顿算子的谱（能级）具有重要的物理意义的同时，哈密顿算子中的动能项通常由导数算子构成，而无穷维空间中的导数算子是典型的无界算子。
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is a linear map A {\displaystyle A} (from V {\displaystyle V} to itself) that is its own adjoint. That is, ⟨ A x , y ⟩ = ⟨ x , A y ⟩ {\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle } for all x , y ∈ V {\displaystyle x,y\in V} .
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在泛函分析中，幺正算符（英语：unitary operator，或称酉算符）是定义在希尔伯特空间上的有界线性算符 U : H → H {\displaystyle U:H\rightarrow H} ，满足如下规律： U ∗ U = U U ∗ = I {\displaystyle U^{*}U=UU^{*}=I} 其中 U ∗ {\displaystyle U^{*}} 是 U {\displaystyle U} 的厄米转置, 而 I : H → H {\displaystyle I:H\rightarrow H} 是恒等算符。
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学中，张量积，记为 ⊗ {\displaystyle \otimes } ，可以应用于不同的上下文中如向量、矩阵、张量、向量空间、代数、拓扑向量空间和模。在各种情况下这个符号的意义是同样的:最一般的双线性运算。在某些上下文中也叫做外积。
In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V {\displaystyle V} and W {\displaystyle W} (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\rightarrow V\otimes W} that maps a pair ( v , w ) {\displaystyle (v,w)} , where v ∈ V , w ∈ W {\displaystyle v\in V,w\in W} , to an element of V ⊗ W {\displaystyle V\otimes W} denoted v ⊗ w {\displaystyle v\otimes w} .
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View content license ↗ 在量子力学里，密度算符（英语：density operator）与其对应的密度矩阵（英语：density matrix）专门描述混合态量子系统的物理性质。纯态是一种可以直接用态矢量 | ψ ⟩ {\displaystyle |\psi \rangle } 来描述的量子态，混合态则是由几种纯态依照统计概率组成的量子态。假设一个量子系统处于纯态 | ψ 1 ⟩ {\displaystyle |\psi _{1}\rangle } 、 | ψ 2 ⟩ {\displaystyle |\psi _{2}\rangle } 、 | ψ 3 ⟩ {\displaystyle |\psi _{3}\rangle } 、……的概率分别为 w 1 {\displaystyle w_{1}} 、 w 2 {\displaystyle w_{2}} 、 w…
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed ensembles of states. These arise in quantum mechanics in two different situations: when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of the ensemble of possible preparations; and when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学和数学物理中，泡利矩阵是一组三个2×2的幺正厄米复矩阵，一般都以希腊字母σ来表示，但有时当他们在和同位旋的对称性做链接时，会被写成τ。他们在泡利表像（σz表像）可以写成： σ 1 = σ x = [ 0 1 1 0 ]…
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 {\displaystyle 2\times 2} complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter σ {\displaystyle \sigma } (sigma), and occasionally by τ {\displaystyle \tau } (tau) when used in connection with isospin symmetries.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在量子力学，特别是量子信息和开放量子系统的研究中，迹线距离是密度矩阵空间上的度量，并给出了两种状态之间可区分性的度量。它是经典概率分布的柯尔莫哥洛夫距离的量子推广。
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 阿达马变换（Hadamard transform），或称沃尔什-阿达玛转换，是一种广义傅立叶变换（Fourier transforms），作为变换编码的一种在影片编码当中使用有很久的历史。在近来的影片编码标准中，阿达马变换多被用来计算SATD(一种影片残差信号大小的衡量)。 在数字信号处理大型集成电路算法的领域中，阿达马变换是一种简单且重要的算法之一，主要能针对频谱做快速的分析。
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on a tuple of 2m numbers. The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2. It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (French: [adamaʁ]), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在统计学中，最大似然估计 (MLE) 是一种在给定一些观测数据的情况下估计假设概率分布参数的方法。这是通过最大化似然函数来实现的，以便在假设的统计模型下，观察到的数据是最可能的。参数空间中使似然函数最大化的点称为最大似然估计。最大似然法的逻辑既直观又灵活，因此该方法已成为统计推断的主要手段。如果似然函数可微，则可以应用求最大值的导数检验。
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference. If the likelihood function is differentiable, the derivative test for finding maxima can be applied.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 假发现率（False discovery rate, FDR）完善了对多重假设测试的检验， F D R = Q e = E [ Q ] , {\displaystyle \mathrm {FDR} =Q_{e}=\mathrm {E} \!\left[Q\right],} 其中E表示期望， Q = V / R = V / ( V + S ) {\displaystyle Q=V/R=V/(V+S)} ，V表示错误拒绝零假设的数目，R表示拒绝零假设的数目。R取0时FDR直接取0，写成一句话就是 F D R = E [ V / R | R > 0 ] ⋅ P (…
In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expected proportion of "discoveries" (rejected null hypotheses) that are false (incorrect rejections of the null). Equivalently, the FDR is the expected ratio of the number of false positive classifications (false discoveries) to the total number of positive classifications (rejections of the null). The total number of rejections of the null include both the number of false positives (FP) and true positives (TP). Simply put, FDR = FP / (FP + TP). FDR-controlling procedures provide less stringent control of Type I errors compared to family-wise error rate (FWER) controlling procedures (such as the Bonferroni correction), which control the probability of at least one Type I error. Thus, FDR-controlling procedures have greater power, at the cost of increased numbers of Type I errors.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 系统发育的贝叶斯推理结合了先验和数据似然中的信息来创建所谓的树的后验概率，即给定数据、先验和似然模型时树正确的概率。贝叶斯推理在 20 世纪 90 年代被三个独立的小组引入分子系统发育学：伯克利的 Bruce Rannala 和 Ziheng Yang、麦迪逊的 Bob Mau 以及爱荷华大学的 Shuying Li，最后两位当时是博士生。自 2001 年 MrBayes 软件发布以来，该方法变得非常流行，现在是分子系统发育学中最流行的方法之一。
Bayesian inference of phylogeny combines the information in the prior and in the data likelihood to create the so-called posterior probability of trees, which is the probability that the tree is correct given the data, the prior and the likelihood model. Bayesian inference was introduced into molecular phylogenetics in the 1990s by three independent groups: Bruce Rannala and Ziheng Yang in Berkeley, Bob Mau in Madison, and Shuying Li in University of Iowa, the last two being PhD students at the time. The approach has become very popular since the release of the MrBayes software in 2001, and is now one of the most popular methods in molecular phylogenetics.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在概率论中，隐马尔可夫模型（HMM）是一种马尔可夫模型，其中观测值依赖于潜在（或隐藏）马尔可夫过程（称为 X {\displaystyle X} ）。 HMM 要求存在一个可观察过程 Y {\displaystyle Y}，其结果以已知方式取决于 X {\displaystyle X} 的结果。由于 X {\displaystyle X} 无法直接观察，因此目标是通过观察 Y {\displaystyle Y} 来了解 X {\displaystyle X} 的状态。
In probability theory, a hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent (or hidden) Markov process (referred to as X {\displaystyle X} ). An HMM requires that there be an observable process Y {\displaystyle Y} whose outcomes depend on the outcomes of X {\displaystyle X} in a known way. Since X {\displaystyle X} cannot be observed directly, the goal is to learn about state of X {\displaystyle X} by observing Y {\displaystyle Y} .
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 敏感性分析是研究如何将数学模型或系统（数值或其他）输出的不确定性划分并分配给其输入的不同不确定性来源。这涉及估计敏感度指数，以量化一个输入或一组输入对输出的影响。一个相关的实践是不确定性分析，它更加关注不确定性的量化和不确定性的传播；理想情况下，不确定性和敏感性分析应同时进行。数学模型（例如生物学、气候科学或经济学）可能非常复杂，因此，可能会错误地理解其输入和输出之间的关系。在这种情况下，模型可以被视为黑匣子，即输出是其输入的“不透明”函数。
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system (numerical or otherwise) can be divided and allocated to different sources of uncertainty in its inputs. This involves estimating sensitivity indices that quantify the influence of an input or group of inputs on the output. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem. A mathematical model (for example in biology, climate science, or economics) can be highly complex, and as a result, its relationships between inputs and outputs may be faultily understood. In such cases, the model can be viewed as a black box, i.e. the output is an "opaque" function of its inputs.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 不确定性量化（UQ）是对计算和现实世界应用中的不确定性进行定量表征和估计的科学。它试图确定如果系统的某些方面不完全已知的话某些结果的可能性有多大。一个例子是预测人体与另一辆车正面相撞时的加速度：即使速度是准确已知的，个别汽车制造过程中的微小差异、每个螺栓的拧紧程度等都会导致不同的结果，而这些结果只能在统计意义上进行预测。自然科学和工程学中的许多问题也充满了不确定性。计算机模拟的计算机实验是研究不确定性量化问题的最常见方法。
Uncertainty Quantification (UQ) is the science of quantitative characterization and estimation of uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of the system are not exactly known. An example would be to predict the acceleration of a human body in a head-on crash with another car: even if the speed was exactly known, small differences in the manufacturing of individual cars, how tightly every bolt has been tightened, etc., will lead to different results that can only be predicted in a statistical sense. Many problems in the natural sciences and engineering are also rife with sources of uncertainty. Computer experiments on computer simulations are the most common approach to study problems in uncertainty quantification.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 大地测量学是在一定的时间与空间参考系中，测量和描绘地球形状及其重力场并监测其变化，为人类活动提供地球空间信息的一门学科，属于地球科学的一个分支，也是一切测绘科学技术的基础。传统的大地测量学又称为经典大地测量学，德国大地测量学家赫尔默特将其表述为对地球表面进行测量和描绘的学科。现代大地测量学则以空间测绘技术为主要特征，研究空间精密定位的理论、技术与方法，扩展了经典大地测量学的研究范围，并在空间与时间尺度、实时性、精度和学科融合等各个方面取得了突破。
Geodesy (, jee-OD-iss-ee) or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D space. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems. Geodetic job titles include geodesist and geodetic surveyor. Through highly accurate observations, geodesy provides the scientific basis for mapping, navigation, and positioning, and supports applications such as infrastructure development (including construction), natural resource management, mineral exploration, and geophysics. Its measurements underpin modern geospatial reference frames used in transportation, satellite systems, global trade, and timekeeping.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在工程和科学中，不同物理量的量纲分析是对其物理量纲或数量量纲的分析，定义为识别所涉及的基本量（例如长度、质量、时间等）的幂的数学表达式，并在执行计算或比较时跟踪这些量纲。量纲分析和数量量纲的概念是由约瑟夫·傅里叶于1822年提出的。可通约的物理量具有相同的量纲和同类，因此即使它们以不同的测量单位表示，也可以直接相互比较；例如，米和英尺、克和磅、秒和年。
In engineering and science, dimensional analysis of different physical quantities is the analysis of their physical dimension or quantity dimension, defined as a mathematical expression identifying the powers of the base quantities involved (such as length, mass, time, etc.), and tracking these dimensions as calculations or comparisons are performed. The concepts of dimensional analysis and quantity dimension were introduced by Joseph Fourier in 1822. Commensurable physical quantities have the same dimension and are of the same kind, so they can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 有限元法 (FEM) 是对工程和数学建模中出现的微分方程进行数值求解的常用方法。感兴趣的典型问题领域包括结构分析、传热、流体流动、质量传输和电磁势等传统领域。计算机通常用于执行所需的计算。借助高速超级计算机，可以实现更好的解决方案，并且通常需要解决最大、最复杂的问题。 FEM 是一种用于求解二空间或三空间变量中的偏微分方程（即某些边值问题）的通用数值方法。也有关于使用有限元法解决高维问题的研究。为了解决问题，FEM 将大型系统细分为更小、更简单的部分，称为有限元。
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在物理学中，哈密顿力学是 1833 年出现的拉格朗日力学的重新表述。由 William Rowan Hamilton 爵士提出，哈密顿力学用（广义）动量取代了拉格朗日力学中使用的（广义）速度 q ˙ i {\displaystyle {\dot {q}}^{i}}。两种理论都提供了经典力学的解释并描述了相同的物理现象。哈密顿力学与几何学（特别是辛几何学和泊松结构）有着密切的关系，并且是经典力学和量子力学之间的纽带。
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在理论物理学中，哈密顿场论是经典哈密顿力学的场论模拟。它是经典场论与拉格朗日场论的形式主义。它在量子场论中也有应用。
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
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