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Plurisubharmonic Function
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces. Formal definition A function :f \colon G \to \cup\, with ''domain'' G \subset ^n is called plurisubharmonic if it is upper semi-continuous, and for every complex line :\\subset ^n with a, b \in ^n the function z \mapsto f(a + bz) is a subharmonic function on the set :\. In ''full generality'', the notion can be defined on an arbitrary complex manifold or even a complex analytic space X as follows. An upper semi-continuous function :f \colon X \to \cup \ is said to be plurisubharmonic if and only if for any holomorphic map \varphi\colon\Delta\to X the ... [...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] |
Compact Support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert C
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein Manifold
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Definition Suppose X is a complex manifold of complex dimension n and let \mathcal O(X) denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold: * X is holomorphically convex, i.e. for every compact subset K \subset X, the so-called ''holomorphically convex hull'', ::\bar K = \left \, :is also a ''compact'' subset of X. * X is holomorphically separable, i.e. if x \neq y are two points in X, then there exists f \in \mathcal O(X) such that f(x) \neq f(y). Non-compact Riemann surfaces are Stein manifolds Let ''X'' be a connected, non-compact Riema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy
In mathematics, in the theory of functions of Function of several complex variables, several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be analytic continuation, extended to a bigger domain. Formally, an open set \Omega in the ''n''-dimensional complex space ^n is called a ''domain of holomorphy'' if there do not exist non-empty open sets U \subset \Omega and V \subset ^n where V is connected space, connected, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function f on \Omega there exists a holomorphic function g on V with f = g on U In the n=1 case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulation point, accumulating everywhere on the boundary (topology), boundary of the domain, which must then be a analytic continuation#Natural boundary, natural boundary for a domain of definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoconvexity
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let :G\subset ^n be a domain, that is, an open connected subset. One says that G is ''pseudoconvex'' (or '' Hartogs pseudoconvex'') if there exists a continuous plurisubharmonic function \varphi on G such that the set :\ is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When G has a C^2 (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C^2 boundary, it can be shown t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subharmonic Function
In mathematics, subharmonic and superharmonic functions are important classes of function (mathematics), functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a function, graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball (mathematics), ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the additive inverse, negative of a subharmonic function, and for this rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Superior And Limit Inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. The limit inferior of a sequence x_n is denoted by \liminf_x_n\quad\text\quad \varliminf_x_n. The limit superior of a sequence x_n is denoted by \lims ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a ''positive scalar''. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. In this article, only the case of scalars in an ordered field is considered. Definition A subset ''C'' of a vector space ''V'' over an ordered field ''F'' is a cone (or sometimes called a linear cone) if for each ''x'' in ''C'' and positive scalar ''α'' in ''F'', the product ''αx'' is in ''C''. Note that some authors define co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Lelong
Pierre Lelong (14 March 1912 Paris – 12 October 2011) at the académie des sciences was a French mathematician who introduced the Poincaré–Lelong equation, the Lelong number and the concept of s. Career Lelong earned his doctorate in 1941 from the , under the sup ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
