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Symplectic vector space

About: Symplectic vector space is a research topic. Over the lifetime, 2048 publications have been published within this topic receiving 53456 citations.


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TL;DR: In this paper, it was shown that if ω is a Kahler form on a complex surface (M,J), then ω(M,ω) agrees with the usual holomorphic Kodaira dimension of (m,J).
Abstract: The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.

62 citations

Journal ArticleDOI
TL;DR: In this paper, a symplectic geometry method is proposed to determine the appropriate embedding dimension from a scalar time series, which can keep the essential character of the primary time series unchanged when performing symplectic similar transform.

62 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the manifold dominating the manifold, then their "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states.
Abstract: We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its "purification". As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein–Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its "purification" induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein–Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

62 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method, which produces large classes of integrable rational mappings in two and three dimensions.
Abstract: We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge--Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge--Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.

61 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202221
202113
20208
201910
201818