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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: The main result of as mentioned in this paper is that a smooth s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary).
Abstract: The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold with b + ≥ 1 from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generalizes Taubes’ theorem “SW ⇒ Gr” to the case of symplectic 4-orbifolds.

26 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the case where G is a fc-dimensional torus T = (S) with the associated Lie algebra t and showed that the image /x(M) of the moment map is a compact convex polytope (cf. Atiyah [1], Guillemin and Sternberg [14]).
Abstract: where $ denotes the Lie algebra of G. This /J, is uniquely determined by the G-action on M as above, and is called the reduced moment map. Let us first consider the case where G is a fc-dimensional torus T = (S)* with the associated Lie algebra t. Then the image /x(M) of the moment map is a compact convex polytope (cf. Atiyah [1], Guillemin and Sternberg [14]). The kernel, denoted by tz, of the exponential map exp : t —> T is called the lattice in t, and points in the lattice are called integral points. They in turn define the dual lattice t| and integral points in t*. By setting iq := tz ^z Q and tq := tj ®z Q? we have rational points in t and t*. A convex polytope in t* is said to be integral or rational, according as all vertices are integral points or rational points, respectively. The fixed point set M of the T-action on M sits in the critical point set for fi, and the image /i(M) is a finite subset of t*. The following proposition, which was originally conjectured by Atiyah [1] in the special case of projective algebraic manifolds, plays a key role in our work:

26 citations

Journal ArticleDOI
TL;DR: In this article, the authors study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS) with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively.
Abstract: We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

26 citations

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