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Lelong Number
In mathematics, the Lelong number is an invariant of a point of a complex analytic variety that in some sense measures the local density at that point. It was introduced by . More generally a closed positive (''p'',''p'') current ''u'' on a complex manifold has a Lelong number ''n''(''u'',''x'') for each point ''x'' of the manifold. Similarly a plurisubharmonic function also has a Lelong number at a point. Definitions The Lelong number of a plurisubharmonic function φ at a point ''x'' of C''n'' is : \liminf_\frac. For a point ''x'' of an analytic subset ''A'' of pure dimension ''k'', the Lelong number ν(''A'',''x'') is the limit of the ratio of the areas of ''A'' ∩ ''B''(''r'',''x'') and a ball of radius ''r'' in C''k'' as the radius tends to zero. (Here ''B''(''r'',''x'') is a ball of radius ''r'' centered at ''x''.) In other words the Lelong number is a sort of measure of the local density of ''A'' near ''x''. If ''x'' is not in the subvariety ''A'' the Lelong n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analytic Variety
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions. Definition Denote the constant sheaf on a topological space with value \mathbb by \underline. A \mathbb-space is a locally ringed space (X, \mathcal_X), whose structure sheaf is an algebra over \underline. Choose an open subset U of some complex affine space \mathbb^n, and fix finitely many holomorphic functions f_1,\dots,f_k in U. Let X=V(f_1,\dots,f_k) be the common vanishing locus of these holomorphic functions, that is, X=\. Define a sheaf of rings on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a ''k''-current in the sense of Georges de Rham is a functional on the space of compactly supported differential ''k''-forms, on a smooth manifold ''M''. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of ''M''. Definition Let \Omega_c^m(M) denote the space of smooth ''m''-forms with compact support on a smooth manifold M. A current is a linear functional on \Omega_c^m(M) which is continuous in the sense of distributions. Thus a linear functional T : \Omega_c^m(M)\to \R is an ''m''-dimensional current if it is continuous in the following sense: If a sequence \omega_k of smooth forms, all supported in the same ... [...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] |
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] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
