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Mathematics投影值测度在数学中，特别是在泛函分析中，投影值测度是一种映射，其将给定集合的特定子集映射为给定的希尔伯特空间上的一个自伴投影算子。 投影值测度 (projection-valued measure, PVM) 在形式上类似于实值测度，不过其值是自伴投影而不是实数。与普通测度一样，也可以关于PVM进行复值函数的积分；这种积分的结果是给定希尔伯特空间上的线性算子。 投影值测度用于表达谱理论中的结果，例如自伴算子的谱定理，在这种情况下 PVM 有时被称为谱测度。自伴算子的博雷尔函数演算是通过关于 PVM 的积分构造的。在量子力学中，PVM 提供了投影测量的数学表述，它们可推广为正算子值测度（POVM），正如混合态或密度矩阵推广了纯态的概念一样。
In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements.
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View content license ↗ 在线性代数中，正交变换是线性变换的一种。如果对于任意向量 u {\displaystyle \mathbf {u} } 和 v {\displaystyle \mathbf {v} } 其内积等于正交变换后之向量 T ( u ) {\displaystyle T({\displaystyle \mathbf {u} })} 和 T ( v ) {\displaystyle T({\displaystyle \mathbf {v} })} 之内积，则称之为正交变换。
In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have ⟨ u , v ⟩ = ⟨ T u , T v ⟩ . {\displaystyle \langle u,v\rangle =\langle Tu,Tv\rangle \,.} Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases.
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View content license ↗ 极化恒等式（英语：Polarization identity）是一个用范数来计算两个向量的内积的公式。
In linear algebra, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. The norm associated with any inner product space satisfies the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 .
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View content license ↗ 伯努利数 Bn 是一个数学分析中常见的有理数序列。前21个伯努利数的值列于右表。其中上标 ± 在本文中用来区别两种不同的伯努利数定义，这两种定义只对 n = 1 {\displaystyle n=1} 有区别： B 1 − = − 1 / 2 {\displaystyle B_{1}^{-{}}=-1/2} ， B 1 + = + 1 / 2 {\displaystyle B_{1}^{+{}}=+1/2} 。
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table.
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View content license ↗ 在数学中，柯西序列、柯西列、柯西数列（英语：Cauchy sequence），也称为基本列，是指一个元素随着序数的增加而愈发靠近的数列，以数学家奥古斯丁·路易·柯西的名字命名。 柯西列的定义依赖于距离的定义，所以只有在度量空间中柯西列才有意义。在更一般的一致空间中，可以定义更为抽象的柯西滤子和柯西网。 柯西列一个重要性质是，在完备空间中，所有的柯西数列都有极限且极限在这空间里，这就让人们可以在不求出这个极限（如果存在）的情况下，利用柯西列的判别法则证明该数列的极限是存在的。柯西列在构造具有完备性的代数结构的过程中也有重要价值，如构造实数。
In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are less than that given distance from each other. Cauchy sequences are named after Augustin-Louis Cauchy; they may occasionally be known as fundamental sequences. It is not sufficient for each term to become arbitrarily close to the preceding term.
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View content license ↗ 博雷尔集，又称Borel集，是群特殊的子集合，这群子集合的整体是任何内涵某指定的拓扑空间的所有开集中最小的Σ-代数。所以博雷尔集的全体又称为博雷尔代数或者博雷尔σ-代数。博雷尔集是由埃米尔·博雷尔的名字命名的。 博雷尔集在测度论中有着重要的意义，因为任何空间上的开集（或者闭集）上定义的测度，必然可以将定义延拓到空间所有的博雷尔集上。定义在博雷尔集上的测度被称为博雷尔测度。博雷尔集和相关的博雷尔分层在描述集合论中也起着基础性的作用。 某些情况下，博雷尔集定义是由拓扑空间中的紧致集合所构造出来的而不是前面讲的开集合。两个定义在很多良好的空间中是等价的，包括所有 σ-紧的豪斯多夫空间，但是在具有病态性质的空间中两者可能不同。
In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the real numbers might fail to be Lebesgue measurable, every Borel set of reals is universally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named after Émile Borel. The most usual definition goes through the notion of a σ-algebra, which is a collection of subsets of a topological space X {\displaystyle X} that contains both the empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define the Borel σ-algebra over X {\displaystyle X} to be the smallest σ-algebra containing all open sets of X {\displaystyle X} . A Borel subset of X {\displaystyle X} is then simply an element of this σ-algebra.
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View content license ↗ 球（英语：ball）在数学里，是指球面内部的空间。球可以是封闭的（包含球面的边界点，称为闭球），也可以是开放的（不包含边界点，称为开球）。 球的概念不只存在于三维欧氏空间里，亦存在于较低或较高维度，以及一般度量空间里。 n {\displaystyle n\,\!} 维空间里的球称为 n {\displaystyle n\,\!} 维球，且包含于 n − 1 {\displaystyle n-1\,\!} 维球面内。因此，在欧氏平面里，球为一圆盘，包含在圆内。在三维空间里，球则是指在二维球面边界内的空间。
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball.
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View content license ↗ 在拓扑学和有关的数学分支中，分离集合是给定拓扑空间中以特定方式相互关联的一对子集，粗略的说，既不重叠也不接触。两个集合是否分离对于连通空间和拓扑空间的分离公理的概念都很重要。 分离集合不应该与分离空间混淆，它们有些关系但并不相同。而可分离空间则是完全不同的拓扑概念。
In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way: roughly speaking, neither overlapping nor touching. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept.
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View content license ↗ 柯尼斯堡七桥问题（德语：Königsberger Brückenproblem；英语：Seven Bridges of Königsberg）是图论中的著名问题。这个问题是基于一个现实生活中的事例：当时东普鲁士柯尼斯堡（今俄罗斯加里宁格勒）市区跨普列戈利亚河两岸，河中心有两个小岛。小岛与河的两岸有七条桥连接。在所有桥都只能走一遍的前提下，如何才能把这个地方所有的桥都走遍？
The Seven Bridges of Königsberg is a historical puzzle asking for a walking tour through the bridges of the city of Königsberg (now Kaliningrad) where each of the city's bridges is crossed exactly once. Its mathematical formalization and proof of impossibility by Leonhard Euler, in 1736, laid the foundations of graph theory and foreshadowed the idea of topology.
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View content license ↗ 在拓扑学及其相关数学领域，一个商空间（quotient space，也称为等化空间identification space）直观上说是将一个给定空间的一些点等同或“黏合在一起”；由一个等价关系确定哪些点是等同的。这是从给定空间构造新空间的常见方法。
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space. Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.
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View content license ↗ 在数学中，若 X {\displaystyle X} 是一个集合类 C {\displaystyle C} 中并集的子集，则集合类 C {\displaystyle C} 是集合 X {\displaystyle X} 的覆盖。用符号来说，如果 C = { U α } α ∈ A {\displaystyle C=\lbrace U_{\alpha }\rbrace _{\alpha \in A}} 是 X {\displaystyle X} 的子集索引族，则 C {\displaystyle C} 是如下条件下的覆盖（定义可参见: Gamelin 与 Greene 第19页或 Kelly 第49页） X ⊆ ⋃ α ∈ A U α {\displaystyle X\subseteq \bigcup _{\alpha \in A}U_{\alpha }} 。
In mathematics, and more particularly in set theory, a cover (or covering) of a set X {\displaystyle X} is a family of subsets of X {\displaystyle X} whose union is all of X {\displaystyle X} .
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View content license ↗ 数学上，实树，也称为R-树，是指有类似于树的性质的度量空间(M,d)，：对M中任何两点x, y，都有唯一的自x至y的弧，而这条弧是测地线。自x至y的弧，是指从区间[a, b]到M中的拓扑嵌入f，使得f(a)=x，f(b)=y。 一个测地度量空间是实树，当且仅当这空间是δ-双曲空间，且δ=0。 完备实树是单射度量空间。(Kirk 1998) 研究实树上的群作用的理论称为Rips machine，是几何群论的一部分。
In mathematics, real trees (also called R {\displaystyle \mathbb {R} } -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of Gromov hyperbolic spaces.
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View content license ↗ 量子拓扑（英语：Quantum topology）是连接量子力学与低维拓扑的数学分支。 狄拉克符号给出了量子力学的一种观点，被推广成为可以包含拓扑空间相关振幅和空间之间的相关嵌入的框架，例如三维空间中的结和结组。狄拉克符号对右矢和左矢的表示，可以被泛化成映射，表示与允许张量积的拓扑空间有关的向量空间。 涉及结组和辫群的拓扑拓扑纠缠也可以直观地与量子纠缠连接起来。
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology. Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products. Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement.
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View content license ↗ 在拓扑学及相关的数学领域里，通常对于所讨论的拓扑空间加有各种各样的限制条件，分离公理即是指之中的某些限制条件。这些分离公理有时候被叫做吉洪诺夫分离公理，得名于安德烈·尼古拉耶维奇·吉洪诺夫。部分分离公理以字母T开头，是由德文单词“Trennung”而来，意义是分离。 分离公理之所以称为公理，是因为以前定义拓扑空间时，有些人会将其也做为公理来定义，而得出较现在意思狭义的拓扑空间。但在拓扑空间的公理化完成后，那些都成了“各种”的拓扑空间。然而，“分离公理”这一词就这样固定了下来。
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces. The separation axioms are denoted with the letter "T" after the German Trennungsaxiom ("separation axiom"), and increasing numerical subscripts denote stronger and stronger properties. The precise definitions of the separation axioms have varied over time. Especially in older literature, different authors might have different definitions of each condition.
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View content license ↗ 支撑集（英语：support，简称支集），是一个数学概念，它是集合 X {\displaystyle X} 的一个子集，要求对给定的 X {\displaystyle X} 上定义的实值函数 f {\displaystyle f} 在这个子集上恰好非0。最常见的情形是， X {\displaystyle X} 是一个拓扑空间，比如实数轴等等，而函数 f {\displaystyle f} 在此拓扑下连续。此时， f {\displaystyle f} 的支撑集被定义为这样一个闭集 C {\displaystyle C} ： f {\displaystyle f} 在 X ∖ C {\displaystyle X\backslash C} 中为 0 {\displaystyle {0}} ，且不存在 C {\displaystyle C} 的真闭子集也满足这个条件，即， C {\displaystyle C} 是所有这样的子集中最小的一个。拓扑意义上的支撑集是点集意义下支撑集的闭包。 特别地，在概率论中，一个概率分布是随机变量的所有可能值组成的集合的闭包。
In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function's domain consisting of those elements that are not mapped to zero. If the domain of f {\displaystyle f} is a topological space, then the support of f {\displaystyle f} is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis.
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View content license ↗ 在数学分支拓扑学中，特殊化（或规范）预序是在拓扑空间上的自然预序。对在实践中考虑的大多数空间，特别是满足T0 分离公理的那些空间，这个预序甚至是偏序（叫做特殊化序）。在另一方面，对于T1空间这个次序成为平凡的而没有价值。 特殊化序经常在计算机科学应用中考虑，这里的T0空间出现在指称语义中。特殊化序对于识别在偏序集合上合适的拓扑空间是重要的，这在序理论所要做的。
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory.
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View content license ↗ 在拓扑学和数学的其他相关领域里，拓扑空间 X {\displaystyle X} 的子空间是指在 X {\displaystyle X} 中子集 S {\displaystyle S} 及在 S {\displaystyle S} 上赋予的由 X {\displaystyle X} 的拓扑所导出的拓扑。这个导出的拓扑叫做 X {\displaystyle X} 的拓扑在 S {\displaystyle S} 上的子空间拓扑，也称为相对拓扑。导出方式参见 #定义。
In topology and related areas of mathematics, a subspace of a topological space ( X , τ ) {\displaystyle (X,\tau )} is a subset S of X which is equipped with a topology induced from that of τ {\displaystyle \tau } called the subspace topology (or the relative topology, inherited topology, induced topology, or trace topology).
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View content license ↗ 形状理论是拓扑学的一个分支，它是同伦理论的扩展，考虑了特殊局部属性的情形。形状理论将同伦理论扩展至更一般的空间上，比如紧致度量空间，或紧致豪斯多夫空间。
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.
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View content license ↗ 一致连续又称均匀连续，（英语：uniformly continuous），为数学分析的专有名词，大致来讲是描述对于函数 f ( ⋅ ) {\displaystyle f(\cdot )} 我们只要在定义域中让任意两点 x {\displaystyle x} 跟 y {\displaystyle y} 越来越接近，我们就可以让 f ( x ) {\displaystyle f(x)} 跟 f ( y ) {\displaystyle f(y)} 无限靠近，这跟一般的连续函数不同之处在于： f ( x ) {\displaystyle f(x)} 跟 f ( y ) {\displaystyle f(y)} 之间的距离并不依赖 x {\displaystyle x} 跟 y {\displaystyle y} 的位置选择。 一致连续是比连续更苛刻的条件。一个函数在某度量空间上一致连续，则其在此度量空间上必然连续，但反之未必成立。
In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want.
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View content license ↗ 一致收敛，或称均匀收敛，（英语：Uniform convergence），是数学中关于函数序列收敛的一种定义。其概念大致可想成：若函数序列 fn 一致收敛至函数 f，代表对所有定义域中的点 x，fn(x) 收敛至 f(x) 会有（大致）相同的收敛速度。由于它对收敛要求较逐点收敛更强，故能保持一些重要的分析性质，例如连续性、黎曼可积性。
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions ( f n ) {\displaystyle (f_{n})} converges uniformly to a limiting function f {\displaystyle f} if, roughly speaking, they uniformly approximate the function f {\displaystyle f} over the whole domain, meaning that all but finitely many of the functions of the sequence lie in a uniform error bar of the original function. Graphically this means that, given any thin band around the graph of f {\displaystyle f} , the graphs of all but finitely many of the functions f n {\displaystyle f_{n}} lie within that thin band.
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View content license ↗ 在数学里，本迪克森-杜拉克定理（英语：Bendixson–Dulac theorem）说明了对于一个二维的驻定动力系统 d x d t = X ( x , y ) , {\displaystyle {\frac {dx}{dt}}=X(x,y),} d y d t = Y ( x , y ) {\displaystyle {\frac {dy}{dt}}=Y(x,y)} 如果存在 φ ( x , y ) {\displaystyle \varphi (x,y)} 使得 ∂ ( φ X ) ∂ x + ∂ (…
In mathematics, the Bendixson–Dulac theorem is a theorem in dynamical systems that exclude the existence of periodic orbits of two-dimensional flows. Here a flow can be visualized as the surface of a pond. If you drop a leaf into a pond, it will drift according to the currents in the water, and a periodic orbit is when the leaf returns to the same place. Roughly speaking, the theorem states that if it is possible to distort the field of currents by stretching and compressing the pond like a rubber sheet, in such a way that the flow is either always expanding or always contracting, then there is no periodic orbit.
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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 ↗ 阿多米安分解法（Adomian decomposition method，简称：ADM法），是1989年美国籍阿马尼亚数学家George Adomian创建的近似分解法，用以求解非线性偏微分方程 将非线性偏微分方程写成如下形式： L ( u ) + R ( u ) + N L ( u ) = g ( x , t ) {\displaystyle L(u)+R(u)+NL(u)=g(x,t)} 其中 L、R为线性偏微分算子，NL为非线性项。 将反算子 L − 1 = ∫ 0 t ( ) {\displaystyle L^{-1}=\int _{0}^{t}()} .
The Adomian decomposition method (ADM) is a semi-analytical method for solving ordinary and partial nonlinear differential equations. The method was developed from the 1970s to the 1990s by George Adomian, chair of the Center for Applied Mathematics at the University of Georgia. It is further extensible to stochastic systems by using the Ito integral. The aim of this method is towards a unified theory for the solution of partial differential equations (PDE); an aim which has been superseded by the more general theory of the homotopy analysis method. The crucial aspect of the method is employment of the "Adomian polynomials" which allow for solution convergence of the nonlinear portion of the equation, without simply linearizing the system. These polynomials mathematically generalize to a Maclaurin series about an arbitrary external parameter; which gives the solution method more flexibility than direct Taylor series expansion.
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View content license ↗ 在经典力学里，对于一个动力系统，随着时间的演进，所有保持不变的物理量都称为守恒量（conserved quantity），又称为运动常数。由于很多物理定律会表达某种守恒行为，对应的守恒量时常会出现于真实系统。例如，假设在某系统内涉及的作用力是保守力，则此系统的能量是守恒量。假设涉及的作用力是有心力，则此系统的角动量是守恒量。
A conserved quantity is a property or value that remains constant over time in a system even when changes occur in the system. In mathematics, a conserved quantity of a dynamical system is formally defined as a function of the dependent variables, the value of which remains constant along each trajectory of the system. Not all systems have conserved quantities, and conserved quantities are not unique, since one can always produce another such quantity by applying a suitable function, such as adding a constant, to a conserved quantity. Since many laws of physics express some kind of conservation, conserved quantities commonly exist in mathematical models of physical systems. For example, any classical mechanics model will have mechanical energy as a conserved quantity as long as the forces involved are conservative.
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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 ↗ 弗洛凯理论（Floquet theory）是常微分方程理论的一种，讨论有关下列微分方程类型的解答类别， x ˙ = A ( t ) x {\displaystyle {\dot {x}}=A(t)x} , 其中，A(t)是一周期为T的连续周期函数。 弗洛凯理论的主要定理－弗洛凯定理给出了一般线性系统的每个基本解的正规形式。它给定了一坐标转变 y = Q − 1 ( t ) x {\displaystyle y=Q^{-1}(t)x} ，其中 Q ( t + 2 T ) = Q ( t ) {\displaystyle Q(t+2T)=Q(t)} ，用以来转变周期系统至有常数及实系数的传统线性系统。 在固态物理中，其类比的结果(推广至三维)为布洛赫定理。
Given a system in which the forces are periodic—such as a pendulum under a periodic driving force, or an oscillating circuit driven by alternating current—the overall behavior of the system is not necessarily fully periodic. For instance, consider a child being pushed on a swing: although the motion is driven by regular, periodic pushes, the swing can gradually reach greater heights while still oscillating to and fro. This results in a combination of underlying periodicity and growth. Floquet theory provides a way to analyze such systems. Its essential insight is similar to the swing example: the solution can be decomposed into two parts—a periodic component (reflecting the repeated motion) and an exponential factor (reflecting growth, decay, or neutral stability). This decomposition allows for the analysis of long-term behavior and stability in time-periodic systems.
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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、物理学中，变量（variable）又称变数，是表达式或公式中，没有固定的值而可以变动的数或量；该数或量可以是随意的，也可能是未指定或未定的。表示变量的字母，统称为变元、元，即变元是一个用来表示值的符号。在初等数学中，也以未知数、未知量代称变量。 在语义上，变数（变量）相对于常数（常量）。变数（量）强调因果与依存关系；一个（自变）变了，另一个（因变）跟着变，即主从关系。变元强调结构与符号本身，仅为一个占位符，不讨论因果依存。 在其他科学中，英语 variable 亦称变项、变因，是任何欲观测或欲操纵的概念、属性、情况、事物、因素，其在质或量（性质或数量）上可变。 在数学领域中，一个变量可以代表“某个数据”，但也可用以表示：一个数、一个向量、一个矩阵、一个函数、一个函数的参数、一个集合或一个集合的元素等数学符号表达的内容。 变量常见的例子如：一个函数 y = f ( x ) {\displaystyle y=f(x)} 有两个变量（参数 x {\displaystyle x} 和值 y {\displaystyle y} ），当参数“变动”时，值也会相对应地“变动”。
In mathematics, a variable (from Latin variabilis 'changeable') is a symbol, typically a letter, that refers to an unspecified mathematical object. One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set, such as the set of real numbers. The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables p and q and require that the value of the square of p is twice the square of q, which in algebraic notation can be written p2 = 2 q2. A definitive proof that this relationship is impossible to satisfy when p and q are restricted to nonzero integers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.
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View content license ↗ 重力火车是一个仍然处于理论上的运输模式，由行星的一地点，经过地心，到达该行星另一方。 先决条件为隧道已贯穿地核，火车靠地心吸力垂直跌入隧道之中。通过地核之后，火车已加速至极快的速度，后半段的旅程地心吸力会对火车作出反作用，并将之减速。到达行星另一面时，火车刚好停顿。
A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere. In a large body such as a planet, this train could be left to accelerate using just the force of gravity, since during the first half of the trip (from the point of departure until the middle), the downward pull towards the center of gravity would pull it towards the destination. During the second half of the trip, the acceleration would be in the opposite direction relative to the trajectory, but, ignoring the effects of friction, the momentum acquired during the first half of the trajectory would equal this deceleration, and as a result, the train's speed would reach zero at approximately the moment the train reached its destination.
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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 ↗ 近年来，协变经典场论又引起了研究者的兴趣。动力学在这里用有限维空间的在时空中的给定时间点上的场来表述。射流丛现在被认为是这种表述的正确定义域。 本文给出一阶经典场论的协变表述的一些几何结构。
In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that jet bundles and the variational bicomplex are the correct domain for such a description. The Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables with respect to all world coordinates. Non-autonomous mechanics is formulated as covariant classical field theory on fiber bundles over the time axis R {\displaystyle \mathbb {R} } .
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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 ↗ 在数学领域中, 时滞微分方程, 或延时微分方程 (DDE) 是一类微分方程, 其中未知函数的在确定时刻的导数由先前时刻函数所决定. 对于 x ( t ) ∈ R n {\displaystyle x(t)\in R^{n}} , 时滞微分方程方程的一般形式是: d d t x ( t ) = f ( t , x ( t ) , x t ) , {\displaystyle {\frac {d}{dt}}x(t)=f(t,x(t),x_{t}),} 其中 x t = { x ( τ ) : τ ≤ t } {\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}} 表示过去时间的解轨道.
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with a functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs: Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, and communication networks that are now involved in feedback control loops introduce such delays.
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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，柱谐函数是指在柱坐标中，拉普拉斯方程， ∇ 2 V ( ρ , φ , z ) = 0 {\displaystyle \nabla ^{2}V(\rho ,\varphi ,z)=0} ，的一系列的解。每一个柱谐函数 V n , k ( ρ , φ , z ) {\displaystyle V_{n,k}(\rho ,\varphi ,z)} 都是三个函数的积： V n , k ( ρ , φ , z ) = P n , k ( ρ ) Φ n ( φ ) Z k ( z ) {\displaystyle V_{n,k}(\rho…
In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, ∇ 2 V = 0 {\displaystyle \nabla ^{2}V=0} , expressed in cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of three terms, each depending on one coordinate alone. The ρ-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).
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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 ↗ 泛函微分方程是一类具有偏移自变量的微分方程。也就是说，泛函微分方程是包含某个函数及其若干导数在不同自变量取值下的方程。 泛函微分方程常用于数学模型中，这类模型假设某一特定行为或现象不仅取决于系统的当前状态，还取决于其过去状态。换言之，过去的事件会明确地影响未来的结果。因此，与常微分方程相比——后者的未来行为仅隐含地依赖于过去——泛函微分方程具有更广泛的适用性。
A functional differential equation is a differential equation with deviating argument. That is, a functional differential equation is an equation that contains a function and some of its derivatives evaluated at different argument values. Functional differential equations find use in mathematical models that assume a specified behavior or phenomenon depends on the present as well as the past state of a system. In other words, past events explicitly influence future results. For this reason, functional differential equations are more applicable than ordinary differential equations (ODE), in which future behavior only implicitly depends on the past.
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