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Mathematics子网（数学）在拓扑和相关数学领域中，子网是子序列概念对网络情况的概括。网络的“子序列”的类似物是“子网”的概念。该定义并不完全简单，但旨在允许尽可能多的有关子序列的定理推广到网络。 “子网”有三种不同的定义。子网的第一个定义由 John L. Kelley 于 1955 年提出，随后 Stephen Willard 在 1970 年引入了他自己的（非等效）凯利定义的变体。
In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible. There are three non-equivalent definitions of "subnet". The first definition of a subnet was introduced by John L. Kelley in 1955 and later, Stephen Willard introduced his own (non-equivalent) variant of Kelley's definition in 1970.
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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，子铺路是一组不重叠的 R⁺ 盒子。 Rⁿ 的子集 X 可以通过两个子铺路 X⁻ 和 X⁺ 来近似，使得 X⁻ ⊂ X ⊂ X⁺。在 R1 中，方框是线段，在 R² 矩形中，在 Rⁿ 超矩形中。当 R² 底层铺面没有孔时，它也可以是“非规则的矩形瓷砖”。盒子具有很容易被计算机操作的优点，因为它们构成了区间分析的核心。许多间隔算法自然会提供常规子铺路的解决方案。在计算中，R² 中的 subpaaving 的一个著名应用是四叉树数据结构。在图像跟踪上下文和其他应用程序中，将 X⁻ 视为拓扑内部非常重要，如图所示。
In mathematics, a subpaving is a set of nonoverlapping boxes of R⁺. A subset X of Rⁿ can be approximated by two subpavings X⁻ and X⁺ such that X⁻ ⊂ X ⊂ X⁺. In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles. A R² subpaving can be also a "non-regular tiling by rectangles", when it has no holes. Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis. Many interval algorithms naturally provide solutions that are regular subpavings. In computation, a well-known application of subpaving in R² is the Quadtree data structure. In image tracing context and other applications is important to see X⁻ as topological interior, as illustrated.
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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，曲面是曲面这一常见概念的数学模型。它是平面的概括，但是与平面不同，它可以是弯曲的（这类似于曲线概括直线）。非平坦表面的一个例子是球体。有几个更精确的定义，具体取决于研究中使用的上下文和数学工具。最简单的数学曲面是欧几里德 3 空间中的平面和球体。通常，在代数几何中，表面可能会与自身相交（并且可能具有其他奇点），而在拓扑和微分几何中，它可能不会。曲面是二维拓扑空间；这意味着表面上的移动点可以在两个方向上移动（它有两个自由度）。
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved (this is analogous to a curve generalizing a straight line). An example of a non-flat surface is the sphere. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom).
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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 ↗ 在数学的一个分支拓扑学中，拓扑空间X的悬浮是通过将X拉伸成圆柱体然后将两个端面折叠成点来直观地获得的。人们将 X 视为“悬浮”在这些端点之间。 X 的悬置由 SX 或 susp(X) 表示。指向空间的悬架有一种变体，称为简化悬架，用 ΣX 表示。 “通常”悬浮液 SX 有时称为 X 的未还原悬浮液、无基悬浮液或自由悬浮液，以区别于 ΣX。
In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points. The suspension of X is denoted by SX or susp(X). There is a variant of the suspension for a pointed space, which is called the reduced suspension and denoted by ΣX. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.
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View content license ↗ 对称保护拓扑（SPT）序是一种具有对称性和有限能隙的零温量子力学物质状态的序。为了以最不变的方式导出结果，使用重整化群方法（导致对应于某些固定点的等价类）。 SPT 阶具有以下定义属性：(a) 如果变形保持对称性，则具有给定对称性的不同 SPT 状态在没有相变的情况下不能平滑地变形为彼此。 (b) 然而，如果在变形过程中对称性被打破，它们都可以平滑地变形为相同的平凡产物状态，而无需相变。上述定义适用于玻色子系统和费米子系统，从而引出了玻色子SPT阶和费米子SPT阶的概念。
Symmetry-protected topological (SPT) order is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points). The SPT order has the following defining properties: (a) distinct SPT states with a given symmetry cannot be smoothly deformed into each other without a phase transition, if the deformation preserves the symmetry. (b) however, they all can be smoothly deformed into the same trivial product state without a phase transition, if the symmetry is broken during the deformation. The above definition works for both bosonic systems and fermionic systems, which leads to the notions of bosonic SPT order and fermionic SPT order.
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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 ↗ T1过程（或第一类拓扑重排过程）是泡沫或生物组织等细胞材料改变形状的主要过程之一。它涉及四个离散的对象，例如气泡、液滴、细胞等。这四个对象最初按以下方式排列在平面上。物体 A 和 B 接触，物体 C 和 D 位于 AB 组的任一侧，并接触 A 和 B。T1 过程包括打破 A 和 B 之间的接触并建立 C 和 D 之间的接触。当泡沫或组织内部发生大量重排事件（例如具有相似方向的 T1 过程）时，材料相应地会发生变形：它沿邻居离开的方向伸长（此处为 AB），同时沿新邻居对形成的方向收缩（此处为光盘）。
A T1 process (or topological rearrangement process of the first kind) is one of the main processes by which cellular materials such as foams or biological tissues change shape. It involves four discrete objects such as bubbles, drops, cells, etc. The four objects are initially arranged in a plane in the following way. Objects A and B are in contact and objects C and D are on either side of the AB group and touching both A and B. The T1 process consists of breaking the contact between A and B and establishing the contact between C and D. When a significant number of rearrangement events such as T1 processes with similar orientations occur inside a foam or a tissue, the material correspondingly undergoes a deformation: it elongates in the direction in which neighbours depart (here, AB) while it contracts in the direction in which new neighbour pairs form (here, CD).
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View content license ↗ 视觉微积分由 Mamikon Mnatsakanian（简称 Mamikon）发明，是一种解决各种积分微积分问题的方法。许多本来看起来相当困难的问题都可以通过几乎不需要计算的方法来解决。 Mamikon 与 Tom Apostol 合作出版了 2013 年的《几何新视野》一书，描述了这一主题。
Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems. Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation. Mamikon collaborated with Tom Apostol on the 2013 book New Horizons in Geometry describing the subject.
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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，Voorhoeve 指数是与复数上的某些函数相关的非负实数，以 Marc Voorhoeve 的名字命名。它可以用来将罗尔定理从实函数扩展到复函数，其作用与实函数的作用是由函数在一个区间内的零点的数量来实现的。
In mathematics, the Voorhoeve index is a non-negative real number associated with certain functions on the complex numbers, named after Marc Voorhoeve. It may be used to extend Rolle's theorem from real functions to complex functions, taking the role that for real functions is played by the number of zeros of the function in an interval.
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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，代数微分方程是可以用微分代数表示的微分方程。根据所使用的微分代数概念，有几个这样的概念。目的是包括通过微分算子形成的方程，其中系数是变量的有理函数（例如超几何方程）。代数微分方程广泛应用于计算机代数和数论中。一个简单的概念是多项式向量场，换句话说，向量场相对于标准坐标基表示为具有多项式系数的一阶偏导数。这是一阶代数微分算子的类型。
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to include equations formed by means of differential operators, in which the coefficients are rational functions of the variables (e.g. the hypergeometric equation). Algebraic differential equations are widely used in computer algebra and number theory. A simple concept is that of a polynomial vector field, in other words a vector field expressed with respect to a standard co-ordinate basis as the first partial derivatives with polynomial coefficients. This is a type of first-order algebraic differential operator.
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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，微分方程的一个古老解是可以向后外推到所有过去时间的解，没有奇点。也就是说，它是一个“在 (−∞, T) 形式的时间间隔上定义的解”。该术语由理查德·汉密尔顿 (Richard Hamilton) 在其关于里奇流 (Ricci flow) 的著作中引入。此后，它被应用于其他几何流以及其他系统，例如纳维-斯托克斯方程和热方程。
In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on a time interval of the form (−∞, T)." The term was introduced by Richard Hamilton in his work on the Ricci flow. It has since been applied to other geometric flows as well as to other systems such as the Navier–Stokes equations and heat equation.
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View content license ↗ Arnold-Beltrami-Childress (ABC) 流或 Gromeka-Arnold-Beltrami-Childress (GABC) 流是三维不可压缩速度场，是欧拉方程的精确解。它以弗拉基米尔·阿诺德、尤金尼奥·贝尔特拉米和斯蒂芬·柴尔德里斯的名字命名。伊波利特·S·格罗梅卡（Ippolit S. Gromeka，1881）的名字在历史上一直被忽视，尽管大部分讨论都是由他首先完成的。这是一个值得注意的具有混沌轨迹的流体流动的简单示例。
The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional incompressible velocity field which is an exact solution of Euler's equation. It is named after Vladimir Arnold, Eugenio Beltrami, and Stephen Childress. Ippolit S. Gromeka's (1881) name has been historically neglected, though much of the discussion has been done by him first. It is notable as a simple example of a fluid flow that can have chaotic trajectories.
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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 ↗ 屏障证书或屏障函数用于证明给定区域对于给定的常微分方程或混合动力系统是前向不变的。也就是说，屏障函数可用于表明如果解在给定集合中开始，则它不能离开该集合。显示集合是前向不变的是安全性的一个方面，这是保证系统避免指定为不安全集合的障碍的属性。屏障证书将系统状态（例如，其位置）映射到标量值，该标量值的某些范围必须对应于安全或不安全集，其细节因所使用的屏障函数的类型而异。
A barrier certificate or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system. That is, a barrier function can be used to show that if a solution starts in a given set, then it cannot leave that set. Showing that a set is forward invariant is an aspect of safety, which is the property where a system is guaranteed to avoid obstacles specified as an unsafe set. Barrier certificates map the system state (for example, its position) to a scalar value for which certain ranges must correspond to either the safe or unsafe set, the specifics of this vary by the type of barrier function used.
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View content license ↗ 在动力系统的研究中，Biryukov 方程（或 Biryukov 振荡器）以 Vadim Biryukov (1946) 的名字命名，是一个用于模拟阻尼振荡器的非线性二阶微分方程。
In the study of dynamical systems, the Biryukov equation (or Biryukov oscillator), named after Vadim Biryukov (1946), is a non-linear second-order differential equation used to model damped oscillators.
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View content license ↗ 在微分几何中，通气曲面是单参数数学曲面族，对应于正弦戈登方程（理论物理中出现的一种微分方程）的通气解。这些表面具有一个显着的特性，即它们具有恒定的曲率 − 1 {\displaystyle -1} ，其中曲率是明确定义的。这使它们成为广义赝球体的例子。
In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature − 1 {\displaystyle -1} , where the curvature is well-defined. This makes them examples of generalized pseudospheres.
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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，巴克马斯特方程是一个二阶非线性偏微分方程，以约翰·D·巴克马斯特 (John D. Buckmaster) 命名，他于 1977 年推导了该方程。该方程对粘性液体薄片的表面进行了建模。该方程是由 S. H. Smith 和 P Smith 早些时候推导出来的，但这些早期的推导主要集中在方程的稳定版本上。 Buckmaster 方程为 u t = ( u 4 ) x x + λ ( u 3 ) x {\displaystyle u_{t}=(u^{4})_{xx}+\lambda (u^{3})_{x}} 其中 λ {\displaystyle \lambda } 是已知参数。
In mathematics, the Buckmaster equation is a second-order nonlinear partial differential equation, named after John D. Buckmaster, who derived the equation in 1977. The equation models the surface of a thin sheet of viscous liquid. The equation was derived earlier by S. H. Smith and by P Smith, but these earlier derivations focused on the steady version of the equation. The Buckmaster equation is u t = ( u 4 ) x x + λ ( u 3 ) x {\displaystyle u_{t}=(u^{4})_{xx}+\lambda (u^{3})_{x}} where λ {\displaystyle \lambda } is a known parameter.
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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 ↗ 区室模型是一种数学框架，用于模拟人口如何在不同状态或“区室”之间移动。虽然广泛应用于各个领域，但它们对于传染病的数学建模尤其重要。在这些模型中，人群被划分为用简写符号标记的区域——最常见的是 S、I 和 R，代表易感个体、感染个体和康复个体。字母顺序通常表示隔室之间的流动模式；例如，SEIS 模型代表从易感性到暴露于传染性，然后再次回到易感性的进展。这些模型起源于 20 世纪初几位数学家开创性的流行病学工作。
Compartmental models are a mathematical framework used to simulate how populations move between different states or "compartments". While widely applied in various fields, they have become particularly fundamental to the mathematical modelling of infectious diseases. In these models, the population is divided into compartments labeled with shorthand notation – most commonly S, I, and R, representing Susceptible, Infectious, and Recovered individuals. The sequence of letters typically indicates the flow patterns between compartments; for example, an SEIS model represents progression from susceptible to exposed to infectious and then back to susceptible again. These models originated in the early 20th century through pioneering epidemiological work by several mathematicians.
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View content license ↗ 在数学中，如果微分方程满足 M ( x , y ) d x − L ( x , y ) d y + N ( x , y ) ( x d y − y d x ) = 0 {\displaystyle M(x,y)dx-L(x,y)dy+N(x,y)(xdy-ydx)=0} ，则称为达布微分方程。其中 M ( x , y ) {\displaystyle M(x,y)} 、 L ( x , y ) {\displaystyle L(x,y)} 和 N ( x , y ) {\displaystyle N(x,y)} 是 x {\displaystyle x} 和 y {\displaystyle y} 的多项式。
In mathematics, a differential equation is called Darboux differential equation if it satisfies the form M ( x , y ) d x − L ( x , y ) d y + N ( x , y ) ( x d y − y d x ) = 0 {\displaystyle M(x,y)dx-L(x,y)dy+N(x,y)(xdy-ydx)=0} . where M ( x , y ) {\displaystyle M(x,y)} , L ( x , y ) {\displaystyle L(x,y)} and N ( x , y ) {\displaystyle N(x,y)} are polynomials 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. 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，微分伽罗瓦理论是研究微分域的扩张的领域。代数伽罗瓦理论研究代数域的扩展，而微分伽罗瓦理论研究微分域的扩展，即配备导数的域。对于任何场元素 x 和 y，导数满足属性 D(xy) = (Dx)y + xDy 和 D(x + y) = Dx + Dy。 （这些恒等式对应于莱布尼茨乘积法则和导数的线性。）微分伽罗瓦理论的大部分理论类似于代数伽罗瓦理论。两种构造之间的一个区别是，与代数伽罗瓦理论中经常遇到的有限群相比，微分伽罗瓦理论中的伽罗瓦群往往是矩阵李群。大多数微分伽罗瓦理论类似于代数伽罗瓦理论。
In mathematics, differential Galois theory is the field that studies extensions of differential fields. Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation. A derivation satisfies the properties D(xy) = (Dx)y + xDy and D(x + y) = Dx + Dy for any field elements x and y. (These identities correspond to the Leibniz product rule and the linearity of derivatives.) Much of the theory of differential Galois theory is analogous to algebraic Galois theory. One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory. Most of differential Galois theory is analogous to algebraic Galois 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.
View content license ↗ 在数学中，离散最小二乘无网格（DLSM）方法是基于最小二乘概念的无网格方法。该方法基于最小二乘泛函的最小化，最小二乘泛函定义为控制微分方程的残差平方和用于离散化域及其边界的节点处的边界条件。
In mathematics the discrete least squares meshless (DLSM) method is a meshless method based on the least squares concept. The method is based on the minimization of a least squares functional, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries.
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View content license ↗ 在数学中，一个显着的极限是匹配渐近展开方法中使用的适当选择的比例因子。
In mathematics, a distinguished limit is an appropriately chosen scale factor used in the method of matched asymptotic expansions.
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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 ↗ 爱因斯坦-因菲尔德-霍夫曼运动方程由阿尔伯特·爱因斯坦、利奥波德·因菲尔德和巴内什·霍夫曼共同导出，是描述点状质量系统由于相互引力相互作用（包括广义相对论效应）而近似动力学的微分方程。它使用一阶后牛顿展开式，因此在物体速度小于光速且影响它们的引力场相应较弱的情况下是有效的。
The Einstein–Infeld–Hoffmann equations of motion, jointly derived by Albert Einstein, Leopold Infeld and Banesh Hoffmann, are the differential equations describing the approximate dynamics of a system of point-like masses due to their mutual gravitational interactions, including general relativistic effects. It uses a first-order post-Newtonian expansion and thus is valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak.
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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 ↗ 进化动力学是数学进化生物学的一个分支，它是从使用微分方程来模拟遗传和表型变化的研究中发展而来的。因此，它不同于关注遗传变化的群体遗传学或数量遗传学，也不同于描述群体规模随时间变化但不包括遗传变化的群体动态。进化博弈论的相关领域首先由梅纳德·史密斯（Maynard Smith）应用于生物学。乔治·普莱斯通过展示频率依赖选择的重要性，介绍了生态学和进化之间的重要联系，尽管仍然缺乏与种群动态变化的灵活联系。 20 世纪 90 年代，研究人员开始了解使用微分方程将生态模型和遗传模型联系起来的好处，从而产生进化动力学。
Evolutionary dynamics is a branch of mathematical evolutionary biology that developed from research using differential equations to model both genetic and phenotypic change. Thus it differs from population genetics or quantitative genetics that focus on genetic change, and from population dynamics that describes change in population size over time, but does not include genetic change. The related field of evolutionary game theory was first applied to biology by Maynard Smith. George Price introduced an important connection between ecology and evolution by showing the importance of frequency-dependent selection, though a flexible link to population dynamic change was still lacking. In the 1990s, researchers began to understand the benefits of linking ecological and genetic models using differential equations, resulting in evolutionary dynamics.
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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，有限域或伽罗瓦域（为纪念埃瓦里斯特·伽罗瓦而得名）是具有有限数量元素的域。与任何域一样，有限域是定义了乘法、加法、减法和除法运算并满足某些基本规则的集合。有限域最常见的例子是当 p {\displaystyle p} 是素数时的整数 mod p {\displaystyle p} 。有限域的阶是它的元素数量，元素要么是素数，要么是素数幂。对于每个素数 p {\displaystyle p} 和每个正整数 k {\displaystyle k} ，都有 p k {\displaystyle p^{k}} 阶的域。给定阶数的所有有限域都是同构的。
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod p {\displaystyle p} when p {\displaystyle p} is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p {\displaystyle p} and every positive integer k {\displaystyle k} there are fields of order p k {\displaystyle p^{k}} . All finite fields of a given order are isomorphic.
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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，域是定义加法、减法、乘法和除法的集合，其行为与有理数的相应运算相同。域是基本的代数结构，广泛应用于代数、数论和许多其他数学领域。最著名的领域是有理数领域、实数领域和复数领域。许多其他领域，例如有理函数域、代数函数域、代数数域、有限域和 p 进数域，在数学中，特别是在数论和代数几何中，也被广泛使用和研究。场论是证明仅用圆规和直尺无法完成角度三等分和化圆为方的证明的关键要素。
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do. Fields are fundamental algebraic structures that are widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers, and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. The theory of fields is a crucial ingredient in the proofs that angle trisection and squaring the circle cannot be done with a compass and straightedge alone.
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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，模算术是一种整数算术运算系统，与通常的算术运算不同，当达到或超过某个值（称为模）时，数字会“环绕”。使用模算术的现代数论方法由卡尔·弗里德里希·高斯 (Carl Friedrich Gauss) 在其 1801 年出版的《算术研究》一书中提出。模算术以 m 为模，系统地用除以 m 的余数替换加法、乘法和减法的结果。模运算的一个显着特性是，计算的结果并不取决于是否在每次运算后除以 m，而是只在计算结束时执行一次，或者在计算结束时和一些中间结果之后执行（通常是在中间结果变得太大时）。
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching or exceeding a certain value, called the modulus. The modern approach to number theory using modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Modular arithmetic modulo m consists of systematically replacing the results of additions, multiplications, and subtractions by the remainder of the division by m. A remarkable property of modular arithmetic is that the result of a computation does not depend on whether the division by m is performed after each operation, only once at the end of the computation, or at the end of the computation and after some intermediate results—typically when an intermediate result becomes too large.
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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，特别是在算术领域中，整数 a 的模乘逆是整数 x，使得乘积 ax 对于模 m 等于 1。在模算术的标准表示法中，该同余被写为 m 除（均）量 ax − 1 的语句的简写方式，或者换句话说，ax 除以整数 m 后的余数为 1。如果 a 确实有模 m 的逆，则该同余有无限多个解，它们形成关于该模的同余类。此外，任何与 a 同余的整数（即，在 a 的同余类中）都具有 x 的同余类的任何元素作为模乘逆。
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m, then there is an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to a (i.e., in a's congruence class) has any element of x's congruence class as a modular multiplicative inverse.
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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 ↗ 在算术和计算机编程中，扩展欧几里得算法是欧几里得算法的扩展，除了计算整数 a 和 b 的最大公约数 (gcd) 之外，还计算 Bézout 恒等式的系数，它们是整数 x 和 y，使得 a x + b y = gcd ( a , b ) {\displaystyle ax+by=\gcd(a,b)} ;它通常表示为 xgcd ( a , b ) {\displaystyle \operatorname {xgcd} (a,b)} 。这是一个证明算法，因为 gcd 是唯一可以同时满足该方程并除以输入的数字。它还允许人们计算 a 和 b 除以其最大公约数的商，几乎不需要额外的成本。扩展欧几里得算法还指用于计算多项式最大公约数和两个单变量多项式的 Bézout 恒等式系数的非常相似的算法。
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that a x + b y = gcd ( a , b ) {\displaystyle ax+by=\gcd(a,b)} ; it is generally denoted as xgcd ( a , b ) {\displaystyle \operatorname {xgcd} (a,b)} . This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.
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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 ↗ 在数论中，欧拉定理（也称为费马-欧拉定理或欧拉 totient 定理）指出，如果 n 和 a 是互质正整数，则 a φ ( n ) {\displaystyle a^{\varphi (n)}} 与 1 {\displaystyle 1} modulo n 全等，其中 φ {\displaystyle \varphi } 表示欧拉 totient功能；即1736年，莱昂哈德·欧拉发表了费马小定理的证明（由费马无证明地陈述），即欧拉定理对n为素数的情况的限制。随后，欧拉提出了该定理的其他证明，并在 1763 年的论文中达到了顶峰，其中他证明了 n 不是素数情况的推广。欧拉定理的逆命题也成立：如果上述同余成立，则 a {\displaystyle a} 和 n {\displaystyle n} 必定互质。
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, then a φ ( n ) {\displaystyle a^{\varphi (n)}} is congruent to 1 {\displaystyle 1} modulo n, where φ {\displaystyle \varphi } denotes Euler's totient function; that is In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime. The converse of Euler's theorem is also true: if the above congruence is true, then a {\displaystyle a} and n {\displaystyle n} must be coprime.
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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，中国余数定理指出，如果知道一个整数n除以几个整数的欧几里得除法的余数，那么在除数成对互质（没有两个除数有除1之外的公因数）的条件下，可以唯一地确定n除以这些整数的乘积的余数。该定理有时称为孙子定理。该定理的两个名称均指其最早出现在《孙子算经》中的已知陈述，这是一部写于公元 3 至 5 世纪的中国手稿。
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is sometimes called Sunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared in Sunzi Suanjing, a Chinese manuscript written during the 3rd to 5th century CE.
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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，对于给定的实数 a {\displaystyle a} 和 b {\displaystyle b} ，对数 log b ( a ) {\displaystyle \log _{b}(a)} 是一个数 x {\displaystyle x} ，使得 b x = a {\displaystyle b^{x}=a} 。离散对数是群论中的一个类似概念。在任何群 G {\displaystyle G} 中，可以为所有整数 k {\displaystyle k} 定义幂 b k {\displaystyle b^{k}} ，并且离散对数 log b ( a ) {\displaystyle \log _{b}(a)} 是整数 k {\displaystyle k} ，使得 b k = a {\displaystyle b^{k}=a} 。在算术模整数 m {\displaystyle m} 的特殊情况下，更常用的术语是索引：当 b k ≡ a ( mod m ) {\displaystyle b^{k}\equiv 时，可以写 k = ind b a ( mod m ) {\displaystyle k=\operatorname {ind} _{b}a\!\!\!\!{\pmod {m}}} a\!\!\!\!{\pmod {m}}} 。
In mathematics, for given real numbers a {\displaystyle a} and b {\displaystyle b} , the logarithm log b ( a ) {\displaystyle \log _{b}(a)} is a number x {\displaystyle x} such that b x = a {\displaystyle b^{x}=a} . The discrete logarithm is an analogous concept in group theory. In any group G {\displaystyle G} , powers b k {\displaystyle b^{k}} can be defined for all integers k {\displaystyle k} , and the discrete logarithm log b ( a ) {\displaystyle \log _{b}(a)} is an integer k {\displaystyle k} such that b k = a {\displaystyle b^{k}=a} . In the special case of arithmetic modulo an integer m {\displaystyle m} , the more commonly used term is index: One can write k = ind b a ( mod m ) {\displaystyle k=\operatorname {ind} _{b}a\!\!\!\!{\pmod {m}}} when b k ≡ a ( mod m ) {\displaystyle b^{k}\equiv a\!\!\!\!{\pmod {m}}} .
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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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