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Biholomorphic Map
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal definition Formally, a ''biholomorphic function'' is a function \phi defined on an open subset ''U'' of the n-dimensional complex space C''n'' with values in C''n'' which is holomorphic and one-to-one, such that its image is an open set V in C''n'' and the inverse \phi^:V\to U is also holomorphic. More generally, ''U'' and ''V'' can be complex manifolds. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11). If there exists a biholomorphism \phi \colon U \to V, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biholomorphism Illustration
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal definition Formally, a ''biholomorphic function'' is a function \phi defined on an open subset ''U'' of the n-dimensional complex space C''n'' with values in C''n'' which is holomorphic and one-to-one, such that its image is an open set V in C''n'' and the inverse \phi^:V\to U is also holomorphic. More generally, ''U'' and ''V'' can be complex manifolds. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11). If there exists a biholomorphism \phi \colon U \to V, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Map
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing definitions of a "proper function". Some authors call a function f : X \to Y between two topological spaces if the preimage of every compact set in Y is compact in X. Other authors call a map f if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff. Let f : X \to Y be a closed map, such that f^(y) is compact (in X) for all y \in Y. Let K be a compact subset of Y. It remains to show that f^(K) is compact. Let \left\ be an open cover of f^(K). Then for all k \in K this is also an open cover of f^(k). Since the latter is assumed to be compact, it has a finite subcover. In other words, for every k \in K, there exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs. More specifically, if we denote by D(z,r) the open disc of center ''z'' and radius ''r'' in the complex plane, then an open polydisc is a set of the form :D(z_1,r_1) \times \dots \times D(z_n,r_n). It can be equivalently written as :\. One should not confuse the polydisc with the open ball in Cn, which is defined as :\. Here, the norm is the Euclidean distance in Cn. When n > 1, open balls and open polydiscs are ''not'' biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.Poincare, H,Les fonctions analytiques de deux variables et la r?epresentation conforme, Rend. Circ. Mat. Palermo23 (1907), 185-220 When n=2 the term ''bidisc'' is sometimes used. A polydisc is an example of logarithmicall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Ball
Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album), 1997 album by the Australian band Regurgitator * The Units, a synthpunk band Television * ''The Unit'', an American television series * '' The Unit: Idol Rebooting Project'', South Korean reality TV survival show Business * Stock keeping unit, a discrete inventory management construct * Strategic business unit, a profit center which focuses on product offering and market segment * Unit of account, a monetary unit of measurement * Unit coin, a small coin or medallion (usually military), bearing an organization's insignia or emblem * Work unit, the name given to a place of employment in the People's Republic of China Science and technology Science and medicine * Unit, a vessel or section of a chemical plant * Blood unit, a measuremen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Mapping Theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy, biholomorphic mapping ''f'' (i.e. a bijective function, bijective holomorphic function, holomorphic mapping whose inverse is also holomorphic) from ''U'' onto the open unit disk :D = \. This mapping is known as a Riemann mapping. Intuitively, the condition that ''U'' be simply connected means that ''U'' does not contain any “holes”. The fact that ''f'' is biholomorphic implies that it is a conformal map and therefore angle-preserving. Such a map may be interpreted as preserving the shape of any sufficiently small figure, while possibly rotating and scaling (but not reflecting) it. Henri Poincaré proved that the map ''f'' is essentially unique: if ''z''0 is an element of ''U'' and φ is an arbitrary angle, then there exists precis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Disc
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose distance from ''P'' is less than or equal to one: :\bar D_1(P)=\.\, Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term ''unit disk'' is used for the open unit disk about the origin, D_1(0), with respect to the standard Euclidean metric. It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted \mathbb. The open unit disk, the plane, and the upper half-plane The function :f(z)=\frac is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a complex manifold in the sense above (which can be specified as an integrable complex manifold), and an almost complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be embedded as a smooth submanifold of R2''n'', whereas it is "rare" for a complex manifold to have a holomorphic embedding into C''n''. Consider for example any compact connected complex manifold ''M'': any holomorphic function on it is cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
