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Additionally, the extra structure of complex geometry allows, especially in the compact setting, for global analytic results to be proven with great success, including Shing-Tung Yau's proof of the Calabi conjecture, the Hitchin–Kobayashi correspondence, the nonabelian Hodge correspondence, and existence results for Kä... | https://en.wikipedia.org/wiki/Complex_geometry |
It is often a source of examples in other areas of mathematics, including in representation theory where generalized flag varieties may be studied using complex geometry leading to the Borel–Weil–Bott theorem, or in symplectic geometry, where Kähler manifolds are symplectic, in Riemannian geometry where complex manifol... | https://en.wikipedia.org/wiki/Complex_geometry |
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer l... | https://en.wikipedia.org/wiki/Singular_moduli |
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a ... | https://en.wikipedia.org/wiki/Complex_projective_space |
When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by von Staudt (1860) as an instance of what was then known as the "geometry of position", a notion originall... | https://en.wikipedia.org/wiki/Complex_projective_space |
445–446). In modern times, both the topology and geometry of complex projective space are well understood and closely related to that of the sphere. | https://en.wikipedia.org/wiki/Complex_projective_space |
Indeed, in a certain sense the (2n+1)-sphere can be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space of rank 1. Complex projective space has many applic... | https://en.wikipedia.org/wiki/Complex_projective_space |
In algebraic geometry, complex projective space is the home of projective varieties, a well-behaved class of algebraic varieties. In topology, the complex projective space plays an important role as a classifying space for complex line bundles: families of complex lines parametrized by another space. In this context, t... | https://en.wikipedia.org/wiki/Complex_projective_space |
In mathematics, composition operators commonly occur in the study of shift operators, for example, in the Beurling–Lax theorem and the Wold decomposition. Shift operators can be studied as one-dimensional spin lattices. Composition operators appear in the theory of Aleksandrov–Clark measures. The eigenvalue equation of... | https://en.wikipedia.org/wiki/Composition_operator |
In mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory. | https://en.wikipedia.org/wiki/Computable_measure_theory |
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals. The concept of a computable real number was introduced by Emile Bor... | https://en.wikipedia.org/wiki/Computable_real_number |
In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is i... | https://en.wikipedia.org/wiki/Computational_group_theory |
Important algorithms in computational group theory include: the Schreier–Sims algorithm for finding the order of a permutation group the Todd–Coxeter algorithm and Knuth–Bendix algorithm for coset enumeration the product-replacement algorithm for finding random elements of a groupTwo important computer algebra systems ... | https://en.wikipedia.org/wiki/Computational_group_theory |
In mathematics, computer science and digital electronics, a dependency graph is a directed graph representing dependencies of several objects towards each other. It is possible to derive an evaluation order or the absence of an evaluation order that respects the given dependencies from the dependency graph. | https://en.wikipedia.org/wiki/Dependency_diagram |
In mathematics, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is kno... | https://en.wikipedia.org/wiki/Optimal_solution |
In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the distance being used to define this matrix may or may not be a metric. If there ... | https://en.wikipedia.org/wiki/Distance_matrix |
In mathematics, computer science and logic, convergence is the idea that different sequences of transformations come to a conclusion in a finite amount of time (the transformations are terminating), and that the conclusion reached is independent of the path taken to get to it (they are confluent). More formally, a preo... | https://en.wikipedia.org/wiki/Convergence_(logic) |
In mathematics, computer science and logic, overlap, as a property of the reduction rules in term rewriting system, describes a situation where a number of different reduction rules specify potentially contradictory ways of reducing a reducible expression, also known as a redex, within a term.More precisely, if a numbe... | https://en.wikipedia.org/wiki/Overlap_(term_rewriting) |
In mathematics, computer science and network science, network theory is a part of graph theory. It defines networks as graphs where the nodes or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components. | https://en.wikipedia.org/wiki/Network_theory |
Network theory has applications in many disciplines, including statistical physics, particle physics, computer science, electrical engineering, biology, archaeology, linguistics, economics, finance, operations research, climatology, ecology, public health, sociology, psychology, and neuroscience. Applications of networ... | https://en.wikipedia.org/wiki/Network_theory |
In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state. | https://en.wikipedia.org/wiki/Deterministic_system |
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects,... | https://en.wikipedia.org/wiki/Rewrite_system |
One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems ... | https://en.wikipedia.org/wiki/Rewrite_system |
In mathematics, computer science, telecommunication, information theory, and searching theory, error-correcting codes with feedback are error correcting codes designed to work in the presence of feedback from the receiver to the sender. | https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback |
In mathematics, concentration of measure (about a median) is a principle that is applied in measure theory, probability and combinatorics, and has consequences for other fields such as Banach space theory. Informally, it states that "A random variable that depends in a Lipschitz way on many independent variables (but n... | https://en.wikipedia.org/wiki/Concentration_of_measure |
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal tra... | https://en.wikipedia.org/wiki/Conformal_manifold |
In mathematics, conformal welding (sewing or gluing) is a process in geometric function theory for producing a Riemann surface by joining together two Riemann surfaces, each with a disk removed, along their boundary circles. This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk... | https://en.wikipedia.org/wiki/Conformal_welding |
In mathematics, conical functions or Mehler functions are functions which can be expressed in terms of Legendre functions of the first and second kind, P − ( 1 / 2 ) + i λ μ ( x ) {\displaystyle P_{-(1/2)+i\lambda }^{\mu }(x)} and Q − ( 1 / 2 ) + i λ μ ( x ) . {\displaystyle Q_{-(1/2)+i\lambda }^{\mu }(x).} The functio... | https://en.wikipedia.org/wiki/Mehler_function |
Mehler used the notation K μ ( x ) {\displaystyle K^{\mu }(x)} to represent these functions. He obtained integral representation and series of functions representations for them. He also established an addition theorem for the conical functions. Carl Neumann obtained an expansion of the functions K μ ( x ) {\displaysty... | https://en.wikipedia.org/wiki/Mehler_function |
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a comp... | https://en.wikipedia.org/wiki/Connectedness |
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at every point and for every... | https://en.wikipedia.org/wiki/Constant_curvature |
In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc. that are free to be determined, the idea being that in most cases the number of independent choices that can be made is the excess of the latter over the former. For example, i... | https://en.wikipedia.org/wiki/Constraint_counting |
In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically. There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining a suitable measurab... | https://en.wikipedia.org/wiki/Stochastic_dynamics |
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics. | https://en.wikipedia.org/wiki/Constructive_analysis |
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998, 2001). The model of construc... | https://en.wikipedia.org/wiki/Constructive_non-standard_analysis |
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and ... | https://en.wikipedia.org/wiki/Contact_geometry |
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann (1936, 1998), where instead of the dimension of a subspace being in a discrete set 0 , 1 , … , n {\displaystyle 0,1,\dots ,{\textit {n}}} , it can be an element of the unit interval {\displaystyle } . Von Neuman... | https://en.wikipedia.org/wiki/Continuous_geometry |
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to another. However, a discrete symmetry can always be reinterpreted as a subset ... | https://en.wikipedia.org/wiki/Continuous_symmetry |
In mathematics, contour sets generalize and formalize the everyday notions of everything superior to something everything superior or equivalent to something everything inferior to something everything inferior or equivalent to something. | https://en.wikipedia.org/wiki/Contour_set |
In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition... | https://en.wikipedia.org/wiki/Convenient_vector_space |
This leads to a Cartesian closed category of smooth mappings between c ∞ {\displaystyle c^{\infty }} -open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called convenient calculus. It is weaker than any other reasonable notion of differentiability, it is ea... | https://en.wikipedia.org/wiki/Convenient_vector_space |
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} . | https://en.wikipedia.org/wiki/Convergence_test |
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theo... | https://en.wikipedia.org/wiki/Convex_geometry |
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints. Formally, consider a metric space (X, d) and let x and y be two points in X. A point z in X is said to be between x and y if all three point... | https://en.wikipedia.org/wiki/Convex_metric |
In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990. It provides theory for atomic decomposition of a range of Banach spaces of distributions. Among others the well established wavelet transform and the short-time Fourier transform are covered by the theory. The st... | https://en.wikipedia.org/wiki/Coorbit_theory |
Many important transforms are special cases of the transform, e.g. the short-time Fourier transform and the wavelet transform for the Heisenberg group and the affine group respectively. Representation theory yields the reproducing formula V g f = V g f ∗ V g g {\displaystyle V_{g}f=V_{g}f*V_{g}g} . By discretization of... | https://en.wikipedia.org/wiki/Coorbit_theory |
An important aspect of the theory is the derivation of atomic decompositions for Banach spaces. One of the key steps is to define the voice transform for distributions in a natural way. For a given Banach space Y {\displaystyle Y} , the corresponding coorbit space is defined as the set of all distributions such that V ... | https://en.wikipedia.org/wiki/Coorbit_theory |
In mathematics, corank is complementary to the concept of the rank of a mathematical object, and may refer to the dimension of the left nullspace of a matrix, the dimension of the cokernel of a linear transformation of a vector space, or the number of elements of a matroid minus its rank. | https://en.wikipedia.org/wiki/Corank |
In mathematics, coset enumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation. As a by-product, one obtains a permutation representation for G on the cosets of H. If H has a known finite order, coset enumeration gives the order of G as well. For small groups it is... | https://en.wikipedia.org/wiki/Coset_enumeration |
Coset enumeration is usually considered to be one of the fundamental problems in computational group theory. The original algorithm for coset enumeration was invented by John Arthur Todd and H. S. M. Coxeter. Various improvements to the original Todd–Coxeter algorithm have been suggested, notably the classical strategi... | https://en.wikipedia.org/wiki/Coset_enumeration |
A practical implementation of these strategies with refinements is available at the ACE website. The Knuth–Bendix algorithm also can perform coset enumeration, and unlike the Todd–Coxeter algorithm, it can sometimes solve the word problem for infinite groups. The main practical difficulties in producing a coset enumera... | https://en.wikipedia.org/wiki/Coset_enumeration |
If a group is finite, then its coset enumeration must terminate eventually, although it may take arbitrarily long and use an arbitrary amount of memory, even if the group is trivial. Depending on the algorithm used, it may happen that making small changes to the presentation that do not change the group nevertheless ha... | https://en.wikipedia.org/wiki/Coset_enumeration |
In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathem... | https://en.wikipedia.org/wiki/Counterexample |
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974). Crystalline cohomology is partly inspired by the... | https://en.wikipedia.org/wiki/Crystalline_site |
The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic p to characteristic 0 and employing an appropriate version of algebraic de Rham cohomology. Crystalline cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general sc... | https://en.wikipedia.org/wiki/Crystalline_site |
In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Alexander Grothendieck (1966a), who named them crystals because in some sense they are "rigid" and "grow". In particular quasicoherent crystals over the crystalline site are analogous to quasicoherent modules over a s... | https://en.wikipedia.org/wiki/Crystal_(mathematics) |
They are p {\displaystyle p} -adic analogues of Q l {\displaystyle \mathbf {Q} _{l}} -adic étale sheaves, introduced by Grothendieck (1966a) and Berthelot & Ogus (1983) (though the definition of isocrystal only appears in part II of this paper by Ogus (1984)). Convergent isocrystals are a variation of isocrystals that ... | https://en.wikipedia.org/wiki/Crystal_(mathematics) |
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal... | https://en.wikipedia.org/wiki/Signed_curvature |
The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, th... | https://en.wikipedia.org/wiki/Signed_curvature |
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature. For Rieman... | https://en.wikipedia.org/wiki/Signed_curvature |
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. | https://en.wikipedia.org/wiki/Cyclical_monotonicity |
In mathematics, cyclically reduced word is a concept of combinatorial group theory. Let F(X) be a free group. Then a word w in F(X) is said to be cyclically reduced if and only if every cyclic permutation of the word is reduced. | https://en.wikipedia.org/wiki/Cyclically_reduced_word |
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be c... | https://en.wikipedia.org/wiki/Cylindrical_measure |
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set S of polynomials in Rn, a cylindrical algebraic decomposition is a decomposition of Rn into connected semialgebraic sets called cells,... | https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition |
The notion was introduced by George E. Collins in 1975, together with an algorithm for computing it. Collins' algorithm has a computational complexity that is double exponential in n. This is an upper bound, which is reached on most entries. | https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition |
There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity. CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexit... | https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition |
In mathematics, d'Alembert's equation is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as y = x f ( p ) + g ( p ) {\displaystyle y=xf(p)+g(p)} where p = d y / d x {\displaystyle p=dy/dx} . After differentiating once, and rearrang... | https://en.wikipedia.org/wiki/D'Alembert_equation |
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that ( cos x + i sin x ) n = cos n x + i sin n x , {\displaystyle {\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,} where i is the imaginary unit (i2 = −1... | https://en.wikipedia.org/wiki/De_Moivre's_formula |
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It... | https://en.wikipedia.org/wiki/De_Rham_theorem |
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. | https://en.wikipedia.org/wiki/Deconvolution |
Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the signal-to-noise ratio (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem. The foundatio... | https://en.wikipedia.org/wiki/Deconvolution |
In mathematics, definitions are generally not used to describe existing terms, but to describe or characterize a concept. For naming the object of a definition mathematicians can use either a neologism (this was mainly the case in the past) or words or phrases of the common language (this is generally the case in moder... | https://en.wikipedia.org/wiki/Definition |
In some case, the word used can be misleading; for example, a real number has nothing more (or less) real than an imaginary number. Frequently, a definition uses a phrase built with common English words, which has no meaning outside mathematics, such as primitive group or irreducible variety. In first-order logic defin... | https://en.wikipedia.org/wiki/Definition |
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to s... | https://en.wikipedia.org/wiki/Deformation_Theory |
In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution. Pertur... | https://en.wikipedia.org/wiki/Deformation_Theory |
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,... | https://en.wikipedia.org/wiki/Differential-difference_equations |
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. Finally, besides actual delays, time lags are frequently used to simplify very high order models. | https://en.wikipedia.org/wiki/Differential-difference_equations |
Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering. Delay systems are still resistant to many classical controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring eff... | https://en.wikipedia.org/wiki/Differential-difference_equations |
In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations. Voluntary introduction of delays can benefit the control system. In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential... | https://en.wikipedia.org/wiki/Differential-difference_equations |
In mathematics, demonic composition is an operation on binary relations that is similar to the ordinary composition of relations but is robust to refinement of the relations into (partial) functions or injective relations. Unlike ordinary composition of relations, demonic composition is not associative. | https://en.wikipedia.org/wiki/Demonic_composition |
In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations of it. They were introduced to address the deficiencies of derived categories (such as the non-functoriality of the cone construction... | https://en.wikipedia.org/wiki/Derivator |
In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on ... | https://en.wikipedia.org/wiki/Derived_noncommutative_algebraic_geometry |
In mathematics, dianalytic manifolds are possibly non-orientable generalizations of complex analytic manifolds. A dianalytic structure on a manifold is given by an atlas of charts such that the transition maps are either complex analytic maps or complex conjugates of complex analytic maps. Every dianalytic manifold is ... | https://en.wikipedia.org/wiki/Dianalytic_structure |
In mathematics, differential Galois theory studies the Galois groups of differential equations. | https://en.wikipedia.org/wiki/Differential_Galois_theory |
In mathematics, differential algebra is, broadly speaking, the area of mathematics consisting in the study of differential equations and differential operators as algebraic objects in view of deriving properties of differential equations and operators without computing the solutions, similarly as polynomial algebras ar... | https://en.wikipedia.org/wiki/Differential_field |
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.The primary objects of study in differential calculus are the derivative of a... | https://en.wikipedia.org/wiki/Differential_calculus |
Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear appro... | https://en.wikipedia.org/wiki/Differential_calculus |
Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to tim... | https://en.wikipedia.org/wiki/Differential_calculus |
The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories. | https://en.wikipedia.org/wiki/Differential_calculus |
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, d... | https://en.wikipedia.org/wiki/Differential_calculus |
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms (or differentials) without specifyin... | https://en.wikipedia.org/wiki/Differential_forms_on_a_Riemann_surface |
These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1941). This article covers general results on differential forms on a Riemann surface that do not rely on any ... | https://en.wikipedia.org/wiki/Differential_forms_on_a_Riemann_surface |
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f... | https://en.wikipedia.org/wiki/Exterior_calculus |
{\displaystyle \int _{a}^{b}f(x)\,dx.} Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S: ∫ S ( f ( x , y , z ) d x ∧ d y + g ( x , y , z ) d z ∧ d x + h ( x , y , z ) d y ∧ d z ) . {\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(... | https://en.wikipedia.org/wiki/Exterior_calculus |
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over a k-dimensional manifold, and is h... | https://en.wikipedia.org/wiki/Exterior_calculus |
{\displaystyle dx,dy,\ldots .} On an n-dimensional manifold, the top-dimensional form (n-form) is called a volume form. The differential forms form an alternating algebra. | https://en.wikipedia.org/wiki/Exterior_calculus |
This implies that d y ∧ d x = − d x ∧ d y {\displaystyle dy\wedge dx=-dx\wedge dy} and d x ∧ d x = 0. {\displaystyle dx\wedge dx=0.} This alternating property reflects the orientation of the domain of integration. | https://en.wikipedia.org/wiki/Exterior_calculus |
The exterior derivative is an operation on differential forms that, given a k-form φ {\displaystyle \varphi } , produces a (k+1)-form d φ . {\displaystyle d\varphi .} This operation extends the differential of a function (a function can be considered as a 0-form, and its differential is d f ( x ) = f ′ ( x ) d x . | https://en.wikipedia.org/wiki/Exterior_calculus |
{\displaystyle df(x)=f'(x)dx.} ) This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem. Differential 1-forms are naturally dual to vector fields on a differentiable manifold, an... | https://en.wikipedia.org/wiki/Exterior_calculus |
The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in term... | https://en.wikipedia.org/wiki/Exterior_calculus |
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