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Poincaré–Lelong Equation
In mathematics, the Poincaré–Lelong equation, studied by , is the partial differential equation :i\partial\overline\partial u=\rho on a Kähler manifold, where ρ is a positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a positi ... (1,1)-form. References * * Complex manifolds Partial differential equations {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics. Definitions Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Form
In complex geometry, the term ''positive form'' refers to several classes of real differential forms of Hodge type ''(p, p)''. (1,1)-forms Real (''p'',''p'')-forms on a complex manifold ''M'' are forms which are of type (''p'',''p'') and real, that is, lie in the intersection \Lambda^(M)\cap \Lambda^(M,). A real (1,1)-form \omega is called semi-positive (sometimes just ''positive''), respectively, positive (or ''positive definite'') if any of the following equivalent conditions holds: #-\omega is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form. #For some basis dz_1, ... dz_n in the space \Lambda^M of (1,0)-forms, \omega can be written diagonally, as \omega = \sqrt \sum_i \alpha_i dz_i\wedge d\bar z_i, with \alpha_i real and non-negative (respectively, positive). #For any (1,0)-tangent vector v\in T^M, -\sqrt\omega(v, \bar v) \geq 0 (respectively, >0). #For any real tangent vector v\in TM, \omega(v, I(v)) \geq 0 (respectively, >0), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Springer Science+Business Media on behalf of Magnes Press (Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. The journal covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus * ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2022 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 1.0. References External links *{ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifolds
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas of charts to the open unit disc in the complex coordinate space \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) or 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 com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
