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Siegel Domain
In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by . They were introduced by in his study of bounded homogeneous domains. Definitions A Siegel domain of the first kind (or first type, or genus 1) is the open subset of C''m'' of elements ''z'' such that :\Im(z)\in V \, where ''V'' is an open convex cone in R''m''. These are special cases of tube domains. An example is the Siegel upper half plane, where ''V''⊂R''k''(''k'' + 1)/2 is the cone of positive definite quadratic forms in R''k'' and ''m'' = ''k''(''k'' + 1)/2. A Siegel domain of the second kind (or second type, or genus 2), also called a Piatetski-Shapiro domain, is the open subset of C''m''×C''n'' of elements (''z'',''w'') such that :\Im(z)-F(w,w)\in V \, where ''V'' is an open convex cone in R''m'' and ''F'' is a ''V''-valued Hermitian form on C''n''. If ''n'' =&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead ''displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Upper Half Plane
In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by . It is the symmetric space associated to the symplectic group . The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case ''g=1''. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix ''Z'' in the Siegel upper half-space in terms of its real and imaginary parts as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tube Domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of complex numbers whose real part lie in a given subset of the real line and whose imaginary part is unconstrained; likewise, a tube is the set of complex vectors whose real part is in some given collection of real vectors, and whose imaginary part is unconstrained. Tube domains are domains of the Laplace transform of a function of several real variables (see multidimensional Laplace transform). Hardy spaces on tubes can be defined in a manner in which a version of the Paley–Wiener theorem from one variable continues to hold, and characterizes the elements of Hardy spaces as the Laplace transforms of functions with appropriate integrability properties. Tubes over convex sets are domains of holomorphy. The Hardy spaces on tubes over convex cones have an especially rich structure, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Symmetric Domain
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Symmetric Space
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space. Harish Chandra showed that each non-compact space can be realized as a bounded symmetric domain in a complex vector space. The simplest case involves the groups SU(2), SU(1,1) and their common complexification SL(2,C). In this case the non-compact space is the unit disk, a homogeneous space for SU(1,1). It is a bounded domain in the complex plane C. The one-point compactific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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