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1.
Aubry and Chartier introduced (1998) the concept of pseudo-symplecticness in order to construct explicit Runge-Kutta methods,
which mimic symplectic ones. Of particular interest are methods of order (p, 2p), i.e., of orderp and pseudo-symplecticness order 2p, for which the growth of the global error remains linear. The aim of this note is to show that the lower bound for the minimal
number of stages can be achieved forp=4 andp=5. 相似文献
2.
Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim
of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the
global error is concerned. Such method of orderp have to bepseudo-symplectic of pseudosymplecticness order2p, i.e. to preserve the symplectic form to within ⊗(h
2p
)-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the
performances of the new methods are illustrated on several test problems. 相似文献
3.
Joseph P. S. Kung 《Annals of Combinatorics》1997,1(1):159-172
Just as matroids abstract the algebraic properties of determinants in a vector space, Pfaffian structures abstract the algebraic properties of Pfaffians or skew-symmetric determinants in a symplectic space (that is, a vector space with an alternating bilinear form). This is done using an exchange-augmentation axiom which is a combinatorial version of a Laplace expansion or straightening identity for Pfaffians. Using Pfaffian structures, we study a symplectic analogue of the classical critical problem: given a setS of non-zero vectors in a non-singular symplectic spaceV of dimension2m, find its symplectic critical exponent, that is, the minimum of the set {m?dim(U):U∩S=0}, whereU ranges over all the (totally) isotropic subspaces disjoint fromS. In particular, we derive a formula for the number of isotropic subspaces of a given dimension disjoint from the setS by Möbius inversion over the order ideal of isotropic flats in the lattice of flats of the matroid onS given by linear dependence. This formula implies that the symplectic critical exponent ofS depends only on its matroid and Pfaffian structure; however, it may depend on the dimension of the symplectic spaceV. 相似文献
4.
François Laudenbach 《Commentarii Mathematici Helvetici》1995,70(1):558-614
Let }L
t{,t ∈ [0, 1], be a path of exact Lagrangian submanifolds in an exact symplectic manifold that is convex at infinity and of dimension
≥6. Under some homotopy conditions, an engulfing problem is solved: the given path }L
t{ is conjugate to a path of exact submanifolds inT
*Lo. This impliesL
t must intersectL
o at as many points as known by the generating function theory. Our Engulfing theorem depends deeply on a new flexibility property
of symplectic structures which is stated in the first part of this work.
相似文献
5.
We are concerned with Runge-Kutta-Nyström methods for the integration of second order systems of the special formd
2
y/dt
2=f(y). If the functionf is the gradient of a scalar field, then the system is Hamiltonian and it may be advantageous to integrate it by a so-called canonical Runge-Kutta-Nyström formula. We show that the equations that must be imposed on the coefficients of the method to ensure canonicity are simplifying assumptions that lower the number of independent order conditions. We count the number of order conditions, both for general and for canonical Runge-Kutta-Nyström formulae.This research has been supported by Junta de Castilla y León under project 1031-89 and by Dirección General de Investigación Científica y Técnica under project PB89-0351. 相似文献
6.
Alessandro De Paris Alexandre M. Vinogradov 《Central European Journal of Mathematics》2011,9(4):731-751
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect
to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants
is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce
a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants
of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an
example we study equations of the form u
xy
+ f(x, y, u
x
, u
y
) = 0 and in particular find a simple linearization criterion. 相似文献
7.
Dan Mangoubi 《Mathematische Annalen》2008,341(1):1-13
We consider Riemannian metrics compatible with the natural symplectic structure on T
2 × M, where T
2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive
eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is
that the same is true for any symplectic manifold of dimension ≥ 4. We reduce the general conjecture to a purely symplectic
question. 相似文献
8.
Matroids admitting an odd ear-decomposition can be viewed as natural generalizations of factor-critical graphs. We prove that
a matroid representable over a field of characteristic 2 admits an odd ear-decomposition if and only if it can be represented
by some space on which the induced scalar product is a non-degenerate symplectic form. We also show that, for a matroid representable
over a field of characteristic 2, the independent sets whose contraction admits an odd ear-decomposition form the family of
feasible sets of a representable Δ-matroid. 相似文献
9.
Peter Albers Urs Frauenfelder Felix Schlenk 《Journal of Differential Equations》2019,266(5):2466-2492
According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations. 相似文献
10.
Amy N. Myers 《Annals of Combinatorics》2007,11(3-4):507-517
It is well-known that the number of permutations of n letters that avoid a pattern τ of 3 letters is independent of τ. In this note we provide bijective proof that the same result
holds for permutations of a multiset.
Received August 16, 2006 相似文献
11.
We use cohomological methods to study the existence of symplectic structures on nilmanifolds associated to two-step nilpotent
Lie groups. We construct a new family of symplectic nilmanifolds with building blocks the quaternionic analogue of the Heisenberg
group, determining the dimension of the space of all left invariant symplectic structures. Such structures can not be K?hlerian.
Also, we prove that the nilmanifolds associated to H type groups are not symplectic unless they correspond to the classical Heisenberg groups.
Received: 26 May 1999 / Revised version: 10 April 2000 相似文献
12.
Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer). 相似文献
13.
Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures. 相似文献
14.
We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied
by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions
to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism
of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior derivative is a Hamiltonian graded vector field. 相似文献
15.
Leonid Polterovich 《Journal of the European Mathematical Society》1999,1(1):87-107
Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S1→G is called strictly ergodic if for some irrational number α the associated skew product map T:S1×Y→S1×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements
of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic
loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on
the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove
that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology.
Received July 7, 1998 / final version received September 14, 1998 相似文献
16.
Grothendieck polynomials, introduced by Lascoux and Schützenberger, are certain K-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the K-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the K-theoretic Schur P-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain “Grassmannian” orbit closures. 相似文献
17.
Aiping Wang 《Journal of Functional Analysis》2008,255(6):1554-1573
There are three basic types of self-adjoint regular and singular boundary conditions: separated, coupled, and mixed. For even order problems with real coefficients, one regular endpoint and arbitrary deficiency index d, we give a construction for each type and determine the number of possible conditions of each type under the assumption that there are d linearly independent square-integrable solutions for some real value of the spectral parameter. In the separated case our construction yields non-real conditions for all orders greater than two. It is well known that no such conditions exist in the second order case. Our construction gives a direct alternative to the recent construction of Everitt and Markus which uses the theory of symplectic spaces. We believe our construction will prove useful in the spectral analysis of these operators and in obtaining canonical forms of self-adjoint boundary conditions. Such forms are known only in the second order, i.e. Sturm-Liouville, case. Even for regular problems of order four no such forms are available. 相似文献
18.
Chun-Gen Liu 《Journal of Differential Equations》2005,209(1):57-76
In this paper, we consider the Arnold conjecture on the Lagrangian intersections of some closed Lagrangian submanifold of a closed symplectic manifold with its image of a Hamiltonian diffeomorphism. We prove that if the Hofer's symplectic energy of the Hamiltonian diffeomorphism is less than a topology number defined by the Lagrangian submanifold, then the Arnold conjecture is true in the degenerated (nontransversal) case. 相似文献
19.
Alvaro Pelayo 《Topology and its Applications》2006,153(18):3633-3644
We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion.In order to do this we first describe a problem in geometric-combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory.Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section. 相似文献
20.
KAM theorem of symplectic algorithms for Hamiltonian systems 总被引:5,自引:0,他引:5
Zai-jiu Shang 《Numerische Mathematik》1999,83(3):477-496
Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel
(1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence
of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable
Hamiltonian system if the system is analytic and the time-step size of the algorithm is s
ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system,
possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in
the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical
invariant tori of the algorithm approximating the exact ones of the system are also given.
Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999 相似文献