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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 Kähler–Einstein metrics and constant scalar curvature Kähler metrics. These results often feed back into complex algebraic geometry, and for example recently the classification of Fano manifolds using K-stability has benefited tremendously both from techniques in analysis and in pure birational geometry. Complex geometry has significant applications to theoretical physics, where it is essential in understanding conformal field theory, string theory, and mirror symmetry.
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https://en.wikipedia.org/wiki/Complex_geometry
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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 manifolds provide examples of exotic metric structures such as Calabi–Yau manifolds and hyperkähler manifolds, and in gauge theory, where holomorphic vector bundles often admit solutions to important differential equations arising out of physics such as the Yang–Mills equations. Complex geometry additionally is impactful in pure algebraic geometry, where analytic results in the complex setting such as Hodge theory of Kähler manifolds inspire understanding of Hodge structures for varieties and schemes as well as p-adic Hodge theory, deformation theory for complex manifolds inspires understanding of the deformation theory of schemes, and results about the cohomology of complex manifolds inspired the formulation of the Weil conjectures and Grothendieck's standard conjectures. On the other hand, results and techniques from many of these fields often feed back into complex geometry, and for example developments in the mathematics of string theory and mirror symmetry have revealed much about the nature of Calabi–Yau manifolds, which string theorists predict should have the structure of Lagrangian fibrations through the SYZ conjecture, and the development of Gromov–Witten theory of symplectic manifolds has led to advances in enumerative geometry of complex varieties. The Hodge conjecture, one of the millennium prize problems, is a problem in complex geometry.
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https://en.wikipedia.org/wiki/Complex_geometry
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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 lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.There is also the higher-dimensional complex multiplication theory of abelian varieties A having enough endomorphisms in a certain precise sense, roughly that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules.
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https://en.wikipedia.org/wiki/Singular_moduli
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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 complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn.
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https://en.wikipedia.org/wiki/Complex_projective_space
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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 originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Subsequently, near the turn of the 20th century it became clear to the Italian school of algebraic geometry that the complex projective spaces were the most natural domains in which to consider the solutions of polynomial equations – algebraic varieties (Grattan-Guinness 2005, pp.
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https://en.wikipedia.org/wiki/Complex_projective_space
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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.
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https://en.wikipedia.org/wiki/Complex_projective_space
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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 applications in both mathematics and quantum physics.
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https://en.wikipedia.org/wiki/Complex_projective_space
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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, the infinite union of projective spaces (direct limit), denoted CP∞, is the classifying space K(Z,2). In quantum physics, the wave function associated to a pure state of a quantum mechanical system is a probability amplitude, meaning that it has unit norm, and has an inessential overall phase: that is, the wave function of a pure state is naturally a point in the projective Hilbert space of the state space.
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https://en.wikipedia.org/wiki/Complex_projective_space
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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 the composition operator is Schröder's equation, and the principal eigenfunction f ( x ) {\displaystyle f(x)} is often called Schröder's function or Koenigs function. The composition operator has been used in data-driven techniques for dynamical systems in the context of dynamic mode decomposition algorithms, which approximate the modes and eigenvalues of the composition operator.
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https://en.wikipedia.org/wiki/Composition_operator
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In mathematics, computable measure theory is the part of computable analysis that deals with effective versions of measure theory.
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https://en.wikipedia.org/wiki/Computable_measure_theory
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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 Borel in 1912, using the intuitive notion of computability available at the time.Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms. The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
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https://en.wikipedia.org/wiki/Computable_real_number
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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 impractical to perform calculations by hand.
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https://en.wikipedia.org/wiki/Computational_group_theory
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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 (CAS) used for group theory are GAP and Magma. Historically, other systems such as CAS (for character theory) and Cayley (a predecessor of Magma) were important. Some achievements of the field include: complete enumeration of all finite groups of order less than 2000 computation of representations for all the sporadic groups
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https://en.wikipedia.org/wiki/Computational_group_theory
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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.
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https://en.wikipedia.org/wiki/Dependency_diagram
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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 known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems.
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https://en.wikipedia.org/wiki/Optimal_solution
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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 are N elements, this matrix will have size N×N. In graph-theoretic applications, the elements are more often referred to as points, nodes or vertices.
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https://en.wikipedia.org/wiki/Distance_matrix
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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 preordered set of term rewriting transformations are said to be convergent if they are confluent and terminating.
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https://en.wikipedia.org/wiki/Convergence_(logic)
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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 number of different reduction rules share function symbols on the left-hand side, overlap can occur. Often we do not consider trivial overlap with a redex and itself.
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https://en.wikipedia.org/wiki/Overlap_(term_rewriting)
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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.
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https://en.wikipedia.org/wiki/Network_theory
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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 network theory include logistical networks, the World Wide Web, Internet, gene regulatory networks, metabolic networks, social networks, epistemological networks, etc.; see List of network theory topics for more examples. Euler's solution of the Seven Bridges of Königsberg problem is considered to be the first true proof in the theory of networks.
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https://en.wikipedia.org/wiki/Network_theory
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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.
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https://en.wikipedia.org/wiki/Deterministic_system
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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, plus relations on how to transform those objects. Rewriting can be non-deterministic.
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https://en.wikipedia.org/wiki/Rewrite_system
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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 can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting.
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https://en.wikipedia.org/wiki/Rewrite_system
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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.
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https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback
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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 not too much on any of them) is essentially constant".The concentration of measure phenomenon was put forth in the early 1970s by Vitali Milman in his works on the local theory of Banach spaces, extending an idea going back to the work of Paul Lévy. It was further developed in the works of Milman and Gromov, Maurey, Pisier, Schechtman, Talagrand, Ledoux, and others.
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https://en.wikipedia.org/wiki/Concentration_of_measure
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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 transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry.
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https://en.wikipedia.org/wiki/Conformal_manifold
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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 and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domains, such that the images are complementary Jordan domains and such that on the unit circle they differ by a given quasisymmetric homeomorphism. Several proofs are known using a variety of techniques, including the Beltrami equation, the Hilbert transform on the circle and elementary approximation techniques. Sharon & Mumford (2006) describe the first two methods of conformal welding as well as providing numerical computations and applications to the analysis of shapes in the plane.
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https://en.wikipedia.org/wiki/Conformal_welding
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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 functions P − ( 1 / 2 ) + i λ μ ( x ) {\displaystyle P_{-(1/2)+i\lambda }^{\mu }(x)} were introduced by Gustav Ferdinand Mehler, in 1868, when expanding in series the distance of a point on the axis of a cone to a point located on the surface of the cone.
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https://en.wikipedia.org/wiki/Mehler_function
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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 ) {\displaystyle K^{\mu }(x)} in terms of the Legendre polynomials in 1881. Leonhardt introduced for the conical functions the equivalent of the spherical harmonics in 1882.
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https://en.wikipedia.org/wiki/Mehler_function
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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 component (or connected component).
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https://en.wikipedia.org/wiki/Connectedness
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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 two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature.
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https://en.wikipedia.org/wiki/Constant_curvature
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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, in linear algebra if the number of constraints (independent equations) in a system of linear equations equals the number of unknowns then precisely one solution exists; if there are fewer independent equations than unknowns, an infinite number of solutions exist; and if the number of independent equations exceeds the number of unknowns, then no solutions exist. In the context of partial differential equations, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation.
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https://en.wikipedia.org/wiki/Constraint_counting
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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 measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists. This theorem, which is an existence theorem for measures on infinite product spaces, says that if any finite-dimensional distributions satisfy two conditions, known as consistency conditions, then there exists a stochastic process with those finite-dimensional distributions.
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https://en.wikipedia.org/wiki/Stochastic_dynamics
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In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.
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https://en.wikipedia.org/wiki/Constructive_analysis
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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 constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic.
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https://en.wikipedia.org/wiki/Constructive_non-standard_analysis
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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 the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or constant-energy hypersurface, which, being codimension one, has odd dimension.
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https://en.wikipedia.org/wiki/Contact_geometry
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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 Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
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https://en.wikipedia.org/wiki/Continuous_geometry
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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 of some higher-dimensional continuous symmetry, e.g. reflection of a 2 dimensional object in 3 dimensional space can be achieved by continuously rotating that object 180 degrees across a non-parallel plane.
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https://en.wikipedia.org/wiki/Continuous_symmetry
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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.
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https://en.wikipedia.org/wiki/Contour_set
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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 of continuous linear mappings stop being jointly continuous at the level of Banach spaces, for any compatible topology on the spaces of continuous linear mappings. Mappings between convenient vector spaces are smooth or C ∞ {\displaystyle C^{\infty }} if they map smooth curves to smooth curves.
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https://en.wikipedia.org/wiki/Convenient_vector_space
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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 easy to apply, but there are smooth mappings which are not continuous (see Note 1). This type of calculus alone is not useful in solving equations.
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https://en.wikipedia.org/wiki/Convenient_vector_space
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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}} .
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https://en.wikipedia.org/wiki/Convergence_test
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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 theory, etc.
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https://en.wikipedia.org/wiki/Convex_geometry
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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 points are distinct, and d ( x , z ) + d ( z , y ) = d ( x , y ) , {\displaystyle d(x,z)+d(z,y)=d(x,y),\,} that is, the triangle inequality becomes an equality. A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y. Metric convexity: does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers) nor does it imply path-connectedness (see the example of the rational numbers) nor does it imply geodesic convexity for Riemannian manifolds (consider, for example, the Euclidean plane with a closed disc removed).
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https://en.wikipedia.org/wiki/Convex_metric
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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 starting point is a square integrable representation π {\displaystyle \pi } of a locally compact group G {\displaystyle {\mathcal {G}}} on a Hilbert space H {\displaystyle {\mathcal {H}}} , with which one can define a transform of a function f ∈ H {\displaystyle f\in {\mathcal {H}}} with respect to g ∈ H {\displaystyle g\in {\mathcal {H}}} by V g f ( x ) = ⟨ f , π ( x ) g ⟩ {\displaystyle V_{g}f(x)=\langle f,\pi (x)g\rangle } .
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https://en.wikipedia.org/wiki/Coorbit_theory
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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 this continuous convolution integral it can be shown that by sufficiently dense sampling in phase space the corresponding functions will span a frame for the Hilbert space.
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https://en.wikipedia.org/wiki/Coorbit_theory
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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 g f ∈ Y {\displaystyle V_{g}f\in Y} . The reproducing formula is true also in this case and therefore it is possible to obtain atomic decompositions for coorbit spaces. == References ==
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https://en.wikipedia.org/wiki/Coorbit_theory
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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.
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https://en.wikipedia.org/wiki/Corank
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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 sometimes possible to perform a coset enumeration by hand. However, for large groups it is time-consuming and error-prone, so it is usually carried out by computer.
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https://en.wikipedia.org/wiki/Coset_enumeration
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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 strategies of V. Felsch and HLT (Haselgrove, Leech and Trotter).
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https://en.wikipedia.org/wiki/Coset_enumeration
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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 enumerator are that it is difficult or impossible to predict how much memory or time will be needed to complete the process.
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https://en.wikipedia.org/wiki/Coset_enumeration
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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 have a large impact on the amount of time or memory needed to complete the enumeration. These behaviours are a consequence of the unsolvability of the word problem for groups. A gentle introduction to coset enumeration is given in Rotman's text on group theory. More detailed information on correctness, efficiency, and practical implementation can be found in the books by Sims and Holt et al. == References ==
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https://en.wikipedia.org/wiki/Coset_enumeration
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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 mathematical development consists primarily in finding (and proving) theorems and counterexamples.
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https://en.wikipedia.org/wiki/Counterexample
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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 p-adic proof in Dwork (1960) of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p (after taking into account higher Tors). The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures.
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https://en.wikipedia.org/wiki/Crystalline_site
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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 schemes.
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https://en.wikipedia.org/wiki/Crystalline_site
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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 scheme. An isocrystal is a crystal up to isogeny.
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https://en.wikipedia.org/wiki/Crystal_(mathematics)
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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 work better over non-perfect fields, and overconvergent isocrystals are another variation related to overconvergent cohomology theories. A Dieudonné crystal is a crystal with Verschiebung and Frobenius maps. An F-crystal is a structure in semilinear algebra somewhat related to crystals.
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https://en.wikipedia.org/wiki/Crystal_(mathematics)
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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 of its radius. Smaller circles bend more sharply, and hence have higher curvature.
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https://en.wikipedia.org/wiki/Signed_curvature
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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, that is, it is expressed by a single real number.
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https://en.wikipedia.org/wiki/Signed_curvature
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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 Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.
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https://en.wikipedia.org/wiki/Signed_curvature
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In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function.
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https://en.wikipedia.org/wiki/Cyclical_monotonicity
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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.
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https://en.wikipedia.org/wiki/Cyclically_reduced_word
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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 countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
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https://en.wikipedia.org/wiki/Cylindrical_measure
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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, on which each polynomial has constant sign, either +, − or 0. To be cylindrical, this decomposition must satisfy the following condition: If 1 ≤ k < n and π is the projection from Rn onto Rn−k consisting in removing the last k coordinates, then for every pair of cells c and d, one has either π(c) = π(d) or π(c) ∩ π(d) = ∅. This implies that the images by π of the cells define a cylindrical decomposition of Rn−k.
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https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
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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.
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https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
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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 complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research.
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https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
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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 rearranging we have d x d p + x f ′ ( p ) + g ′ ( p ) f ( p ) − p = 0 {\displaystyle {\frac {dx}{dp}}+{\frac {xf'(p)+g'(p)}{f(p)-p}}=0} The above equation is linear. When f ( p ) = p {\displaystyle f(p)=p} , d'Alembert's equation is reduced to Clairaut's equation. == References ==
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https://en.wikipedia.org/wiki/D'Alembert_equation
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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). The formula is named after Abraham de Moivre, although he never stated it in his works. The expression cos x + i sin x is sometimes abbreviated to cis x. The formula is important because it connects complex numbers and trigonometry. By expanding the left hand side and then comparing the real and imaginary parts under the assumption that x is real, it is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x. As written, the formula is not valid for non-integer powers n. However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the nth roots of unity, that is, complex numbers z such that zn = 1.
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https://en.wikipedia.org/wiki/De_Moivre's_formula
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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 is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
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https://en.wikipedia.org/wiki/De_Rham_theorem
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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.
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https://en.wikipedia.org/wiki/Deconvolution
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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 foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949). The book was based on work Wiener had done during World War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of weather forecasting and economics.
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https://en.wikipedia.org/wiki/Deconvolution
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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 modern mathematics). The precise meaning of a term given by a mathematical definition is often different from the English definition of the word used, which can lead to confusion, particularly when the meanings are close. For example a set is not exactly the same thing in mathematics and in common language.
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https://en.wikipedia.org/wiki/Definition
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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 definitions are usually introduced using extension by definition (so using a metalogic). On the other hand, lambda-calculi are a kind of logic where the definitions are included as the feature of the formal system itself.
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https://en.wikipedia.org/wiki/Definition
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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 solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations.
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https://en.wikipedia.org/wiki/Deformation_Theory
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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. Perturbation theory also looks at deformations, in general of operators.
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https://en.wikipedia.org/wiki/Deformation_Theory
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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 the 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.
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https://en.wikipedia.org/wiki/Differential-difference_equations
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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.
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https://en.wikipedia.org/wiki/Differential-difference_equations
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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 effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design.
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https://en.wikipedia.org/wiki/Differential-difference_equations
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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 equations (PDEs).A general form of the time-delay differential equation for x ( t ) ∈ R n {\displaystyle x(t)\in \mathbb {R} ^{n}} is where x t = { x ( τ ): τ ≤ t } {\displaystyle x_{t}=\{x(\tau ):\tau \leq t\}} represents the trajectory of the solution in the past. In this equation, f {\displaystyle f} is a functional operator from R × R n × C 1 ( R , R n ) {\displaystyle \mathbb {R} \times \mathbb {R} ^{n}\times C^{1}(\mathbb {R} ,\mathbb {R} ^{n})} to R n . {\displaystyle \mathbb {R} ^{n}.}
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https://en.wikipedia.org/wiki/Differential-difference_equations
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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.
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https://en.wikipedia.org/wiki/Demonic_composition
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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) and provide at the same time a language for homotopical algebra. Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages. Essentially the same concept was introduced (apparently independently) by Alex Heller.The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller, Franke, Keller and Groth.
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https://en.wikipedia.org/wiki/Derivator
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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 a smooth variety, D b ( X ) {\displaystyle D^{b}(X)} , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted D perf ( X ) {\displaystyle D_{\operatorname {perf} }(X)} . For instance, the derived category of coherent sheaves D b ( X ) {\displaystyle D^{b}(X)} on a smooth projective variety can be used as an invariant of the underlying variety for many cases (if X {\displaystyle X} has an ample (anti-)canonical sheaf). Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.
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https://en.wikipedia.org/wiki/Derived_noncommutative_algebraic_geometry
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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 given by the quotient of an analytic manifold (possibly non-connected) by a fixed-point-free involution changing the complex structure to its complex conjugate structure. Dianalytic manifolds were introduced by Klein (1882), and dianalytic manifolds of 1 complex dimension are sometimes called Klein surfaces.
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https://en.wikipedia.org/wiki/Dianalytic_structure
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In mathematics, differential Galois theory studies the Galois groups of differential equations.
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https://en.wikipedia.org/wiki/Differential_Galois_theory
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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 are used for the study of algebraic varieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may be considered as belonging to differential algebra. More specifically, differential algebra refers to the theory introduced by Joseph Ritt in 1950, in which differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations.A natural example of a differential field is the field of rational functions in one variable over the complex numbers, C ( t ) , {\displaystyle \mathbb {C} (t),} where the derivation is differentiation with respect to t . {\displaystyle t.} More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions appearing in the equation.
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https://en.wikipedia.org/wiki/Differential_field
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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 function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation.
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https://en.wikipedia.org/wiki/Differential_calculus
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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 approximation to the function at that point. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
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https://en.wikipedia.org/wiki/Differential_calculus
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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 time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion.
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https://en.wikipedia.org/wiki/Differential_calculus
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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.
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https://en.wikipedia.org/wiki/Differential_calculus
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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, differential geometry, measure theory, and abstract algebra.
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https://en.wikipedia.org/wiki/Differential_calculus
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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 specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces.
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https://en.wikipedia.org/wiki/Differential_forms_on_a_Riemann_surface
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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 choice of Riemannian structure.
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https://en.wikipedia.org/wiki/Differential_forms_on_a_Riemann_surface
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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(x) dx is an example of a 1-form, and can be integrated over an interval contained in the domain of f: ∫ a b f ( x ) d x .
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https://en.wikipedia.org/wiki/Exterior_calculus
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{\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(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).}
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https://en.wikipedia.org/wiki/Exterior_calculus
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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 homogeneous of degree k in the coordinate differentials d x , d y , … .
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https://en.wikipedia.org/wiki/Exterior_calculus
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{\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.
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https://en.wikipedia.org/wiki/Exterior_calculus
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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.
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https://en.wikipedia.org/wiki/Exterior_calculus
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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 .
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https://en.wikipedia.org/wiki/Exterior_calculus
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{\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, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the interior product.
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https://en.wikipedia.org/wiki/Exterior_calculus
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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 terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
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https://en.wikipedia.org/wiki/Exterior_calculus
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