Showing papers on "Symplectic vector space published in 2003"
TL;DR: The connection between closed Newton-Cotes differential methods and symplectic integrators is considered in this article, where the authors apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum.
161 citations
TL;DR: In this paper, the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, were discussed.
Abstract: We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon [9].
113 citations
TL;DR: In this paper, it was shown that a Gromov invariant counting pseudoholomorphic sections of an associated bundle of symmetric products is non-zero and that the Poincare dual of KX may be represented by an embedded symplectic submanifold.
85 citations
TL;DR: In this paper, mean-square and weak quasi-symplectic methods for Langevin type equations with separable Hamiltonians and additive noise were constructed. And the methods derived are based on symplectic schemes for stochastic Hamiltonian systems.
Abstract: Langevin type equations are an important and fairly large class of systems close to Hamiltonian ones. The constructed mean-square and weak quasi-symplectic methods for such systems degenerate to symplectic methods when a system degenerates to a stochastic Hamiltonian one. In addition, quasi-symplectic methods' law of phase volume contractivity is close to the exact law. The methods derived are based on symplectic schemes for stochastic Hamiltonian systems. Mean-square symplectic methods were obtained in Milstein et al. (2002, SIAM J. Numer. Anal., 39, 2066-2088; 2003, SIAM J. Numer. Anal., 40, 1583-1604) while symplectic methods in the weak sense are constructed in this paper. Special attention is paid to Hamiltonian systems with separable Hamiltonians and with additive noise. Some numerical tests of both symplectic and quasi-symplectic methods are presented. They demonstrate superiority of the proposed methods in comparison with standard ones.
69 citations
TL;DR: In this paper, the authors solved the topological Poisson Sigma model for a Poisson-Lie group G and its dual G*, showing that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target.
Abstract: We solve the topological Poisson Sigma model for a Poisson-Lie group G and its dual G*. We show that the gauge symmetry for each model is given by its dual group that acts by dressing transformations on the target. The resolution of both models in the open geometry reveals that there exists a map from the reduced phase space of each model (P and P*) to the main symplectic leaf of the Heisenberg double (D0) such that the symplectic forms on P, P* are obtained as the pull-back by those maps of the symplectic structure on D0. This uncovers a duality between P and P* under the exchange of bulk degrees of freedom of one model with boundary degrees of freedom of the other one. We finally solve the Poisson Sigma model for the Poisson structure on G given by a pair of r-matrices that generalizes the Poisson-Lie case. The hamiltonian analysis of the theory requires the introduction of a deformation of the Heisenberg double.
34 citations
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TL;DR: In this paper, it was shown that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice and the product decomposition is compatible with natural Poisson structures.
Abstract: We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice. The transversal slice is also a symplectic singularity, and the product decomposition is compatible with natural Poisson structures. Moreover, we prove that the transversal slice admits a $C^*$-action dilating the symplectic form.
33 citations
01 Jan 2003
TL;DR: In this article, a collection of new and old proofs showing that the determinant of any symplectic matrix is +1 is presented, and a constructive derivation of the symplectic analogue of the classical analogue is given.
Abstract: A collection of new and old proofs showing that the determinant of any symplectic matrix is +1 is presented. Structured factorizations of symplectic matrices play a key role in several arguments. A constructive derivation of the symplectic analogue of the
29 citations
TL;DR: In this paper, the authors extend the Balian-low theorem to functions of several variables and prove it for differential operators associated with a symplectic basis for the symplectic form on R2d.
Abstract: In this paper we extend the Balian–Low theorem, which is a version of the uncertainty principle for Gabor (Weyl–Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian–Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian–Low theorem for differential operators associated with a symplectic basis for the symplectic form on R2d.
28 citations
TL;DR: In this paper, small contractions of (holomorphically) symplectic 4-folds are classified as follows: 14E30, 14J35, 14E31.
Abstract: We classify small contractions of (holomorphically) symplectic 4-folds. AMS MSC: 14E30, 14J35.
27 citations
TL;DR: In this article, the notion of strict quantization of a symplectic manifold was introduced and its existence under a topological condition was shown. But it was not shown in practice.
Abstract: We introduce the notion of a strict quantization of a symplectic manifold and show its existence under a topological condition
27 citations
TL;DR: In this article, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points, and some of the implications are discussed.
Abstract: For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
TL;DR: In this article, the authors present a review of Hilbert and symplectic space theory for elliptic differential operators with respect to global boundary conditions and modified Laplace operators, as well as a list of symbols and notations.
Abstract: Introduction: Organization of results Review of Hilbert and symplectic space theory GKN-theory for elliptic differential operators Examples of the general theory Global boundary conditions: Modified Laplace operators Appendix A. List of symbols and notations Bibliography Index.
TL;DR: In this paper, a universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics.
Abstract: A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian system by using the generating function method.
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TL;DR: In this article, the authors introduced the concept of Hofer-Zehnder $G$-semicapacity and proved that given a geometrically bounded symplectic manifold and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $\phi$, $N$ has bounded Hofer Zehnder capacity, where π is the subgroup generated by the homotopy class of the orbits of π.
Abstract: We introduce the concept of Hofer-Zehnder $G$-semicapacity (or $G$-sensitive Hofer-Zehnder capacity) and prove that given a geometrically bounded symplectic manifold $(M,\omega)$ and an open subset $N \subset M$ endowed with a Hamiltonian free circle action $\phi$ then $N$ has bounded Hofer-Zehnder $G_\phi$-semicapacity, where $G_\phi \subset \pi_1(N)$ is the subgroup generated by the homotopy class of the orbits of $\phi$. In particular, $N$ has bounded Hofer-Zehnder capacity.
We give two types of applications of the main result. Firstly, we prove that the cotangent bundle of a compact manifold endowed with a free circle action has bounded Hofer-Zehnder capacity. In particular, the cotangent bundle $T^*G$ of any compact Lie group $G$ has bounded Hofer-Zehnder capacity. Secondly, we consider Hamiltonian circle actions given by symplectic submanifolds. For instance, we prove the following generalization of a recent result of Ginzburg-G\"urel: almost all low levels of a function on a geometrically bounded symplectic manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold.
TL;DR: In this article, a 10-dimensional moduli space of torsion sheaves on the cubic 4-fold in the 5-dimensional projective space is constructed, which is closely related to the quasi-symplectic structure induced by the Yoneda pairing.
Abstract: We construct a 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold in the 5-dimensional projective space. It parametrizes the stable rank 2 vector bundles on hyperplane sections of the cubic 4-fold which are obtained by the Serre construction from normal elliptic quintics. The natural projection of this moduli space onto the dual projective 5-space is a Lagrangian fibration. The symplectic structure is closely related (and conjecturally equal) to the quasi-symplectic structure induced by the Yoneda pairing on the moduli space.
01 Jan 2003
TL;DR: In this article, it was shown that a translation plane of even order is symplectic if and only if it admits a completely regular line oval and a geometric characterization of line ovals related to certain symmetric designs is given.
Abstract: A symplectic spread of a 2n-dimensional vector space V over GFðqÞ is a set of q n þ 1 totally isotropic n-subspaces inducing a partition of the points of the underlying projective space. The corresponding translation plane is called symplectic. We prove that a translation plane of even order is symplectic if and only if it admits a completely regular line oval. Also, a geometric characterization of completely regular line ovals, related to certain symmetric designs S 1 ð2d Þ, is given. These results give a complete solution to a problem set by W. M.
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TL;DR: In this paper, the trace map and the gluing map are considered as rational maps on Lagrangian Grassmannians, and the singularities of the symplectic reduction are studied.
Abstract: The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the "trace map" and the "gluing map") and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of self-similar lattices. In this text, we show that this renormalization map can be expressed in terms of symplectic reductions and that some of its key properties are direct consequences of general properties of symplectic reductions (and the singularities of the symplectic reduction play an important role in relation with the spectral properties of our operator). We also present new examples where we can compute the renormalization map.
TL;DR: In this article, the authors studied the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid and showed that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.
Abstract: We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionrn2≤2r12 the symplectic ellipsoidE(r1,…,rn)with radiir1≤…≤rndoes not symplectically embed into a ball of radius strictly smaller thanrn.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.
TL;DR: In this article, the geometrical nature of fiber space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations is introduced. But the structure of the fiber space is not described.
Abstract: We introduce the geometrical nature of fibre space structures of an irreducible symplectic manifold and holomorphic Lagrangian fibrations.
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TL;DR: In this paper, the authors studied isomorphism classes of symplectic dual pairs P P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres.
Abstract: We study isomorphism classes of symplectic dual pairs P P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P, these Morita self-equivalences of P form a group Pic(P) under a natural ``tensor product'' operation. We discuss this group in several examples and study variants of this construction for rings (the origin of the notion of Picard group), Lie groupoids, and symplectic groupoids.
TL;DR: In this article, an invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost's cohomological invariant of degree 3 for torsors under symplectic groups.
Abstract: An invariant for symplectic involutions on central simple algebras of degree divisible by 4 over fields of characteristic different from 2 is defined on the basis of Rost's cohomological invariant of degree 3 for torsors under symplectic groups. We relate this invariant to trace forms and show how its triviality yields a decomposability criterion for algebras of degree 8 with symplectic involution.
TL;DR: Fu as mentioned in this paper showed that given two projective resolutions Z 1 and Z 2 of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z 1 is deformation equivalent to Z 2.
TL;DR: In this article, explicit volume-preserving and symplectic integrators for generalized trigonometric polynomial flows are introduced, and computational trials are conducted using a 4-dimensional vector field.
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TL;DR: In this paper, the authors reinterpreted the Poisson structure of the Hitchin-type system in cohomological terms and identified the Riemann bilinear form and the symplectic structure.
Abstract: In this paper we reinterpret the Poisson structure of the Hitchin-type system in cohomological terms The principal ingredient of a new interpretation in the case of the Beauville system is the meromorphic cohomology of the spectral curve, and the main result is the identification of the Riemann bilinear form and the symplectic structure of the model Eventual perspectives of this approach lie in the quantization domain
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TL;DR: In this paper, it was shown that all Lagrangian spheres in S 2 x S 2 are Hamiltonian isotopic, using various properties of holomorphic curves in symplectic manifolds with cylindrical ends which were developed in connection with the Symplectic Field Theory.
Abstract: We prove that all Lagrangian spheres in S^2 x S^2 are Hamiltonian isotopic. The proof uses various properties of holomorphic curves in symplectic manifolds with cylindrical ends which were recently developed in connection with the Symplectic Field Theory.
TL;DR: In this article, the anticommutator algebra for the fermionic coordinates of open superstrings attached to branes with antisymmetric tensor fields was built up using both Dirac quantization and the symplectic Faddeev-Jackiw approach.
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TL;DR: In this paper, a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003 are presented.
Abstract: This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the connection to Gromov-Witten invariants of symplectic 4-manifolds (following Smith) and to Fukaya categories (following Seidel). In the last lecture, we offer an alternative description of symplectic 4-manifolds by viewing them as branched covers of the complex projective plane; the corresponding monodromy invariants and their potential applications are discussed.
01 Jun 2003
TL;DR: In this article, the generalization to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developed by Witten, Zuckerman, and others in the context of field theory is presented.
Abstract: This paper treats the generalization to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developed by Witten, Zuckerman, and others in the context of field theory. After a general presentation, including a review of the relationships between the various (Lagrangian, Eulerian, and other) relevant kinds of variation, the procedure is illustrated by application to the particularly simple case of branes of the Dirac—Goto—Nambu type, in which internal fields are absent.