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1.
Peter Šepitka 《Journal of Difference Equations and Applications》2013,19(12):1894-1934
ABSTRACTIn this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445–459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431–444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Do?lý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268–1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. ?epitka and R. ?imon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243–275; P. ?epitka and R. ?imon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657–698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system. 相似文献
2.
Darko Milinkovic 《Transactions of the American Mathematical Society》1999,351(10):3953-3974
The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional' symplectic invariants constructed via generating functions to the ``infinite dimensional' ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional. I, J. Diff. Geom. 46 (1997), 499-577.
3.
Normal Forms of Symplectic Matrices 总被引:1,自引:0,他引:1
Abstract
In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P
−1
MP is a symplectic matrix of the normal forms defined in this paper.
Partially supported by the NSF, MCSEC of China, and the Qiu Shi Sci. Tech. Foundation
* Associate Member of the ICTP 相似文献
4.
M.-C. Arnaud 《Publications Mathématiques de L'IHéS》2009,109(1):1-17
A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus
is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2).
Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain
that the graph is even C
1 (Theorem 3).
Then we consider the case of the C
0 integrable symplectic twist maps and we prove that for such a map, there exists a dense G
δ
subset of the set of its invariant curves such that every curve of this G
δ
subset is C
1 (Theorem 4). 相似文献
5.
Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ0 with the induced Poisson structure Λ0 is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic
integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability
([2], [6], [11], [17], [28]).
The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability
of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will
provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s
program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s
third theorem in the infinite dimensional case.
Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90) 相似文献
Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90) 相似文献
6.
Yong-Geun Oh 《纯数学与应用数学通讯》1992,45(1):121-139
We give a proof of the theorem of removing isolated singularities of pseudo-holomorphic curves with Lagrangian boundary conditions and bounded symplectic area. The proof is a combination of some Lp-type estimates, standard techniques of geometric P.D.E., and some ideas from symplectic geometry and calibration theory. 相似文献
7.
8.
Yu. V. Eliseeva 《Differential Equations》2010,46(9):1339-1352
In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference
analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of
conjoined bases of two symplectic systems with matrices W
i
and $
\hat W_i
$
\hat W_i
as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider
applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number
of eigenvalues λ of a symplectic boundary value problem on the interval (λ
1, λ
2]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined
bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes
the well-known reciprocity principle for discrete Hamiltonian systems. 相似文献
9.
Heike Fassbender 《BIT Numerical Mathematics》2000,40(3):471-496
A rounding error analysis of the symplectic Lanczos algorithm for the symplectic eigenvalue problem is given. It is applicable when no break down occurs and shows that the restriction of preserving the symplectic structure does not destroy the characteristic feature of nonsymmetric Lanczos processes. An analog of Paige's theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. As to be expected, it follows that (under certain assumptions) the computed J-orthogonal Lanczos vectors loose J-orthogonality when some Ritz values begin to converge. 相似文献
10.
Paul Biran 《Inventiones Mathematicae》1999,136(1):123-155
We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H
2(M, Q) there exists a number N
0 such that for every N≥N
0, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N
0. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of
Gromov invariants.
Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999 相似文献
11.
Abstract We prove in details the higher codimensional version of Theorem 1.1 [11]. This provides a complete proof of Fefferman’s SAK
Principle for a class of PDO’s with symplectic characteristic manifold.
Keywords: A priori estimates, General theory of PDO’s 相似文献
12.
We introduce new invariants associated with collections of compact subsets of a symplectic manifold. They are defined through
an elementary-looking variational problem involving Poisson brackets. The proof of the non-triviality of these invariants
involves various flavors of Floer theory, including the μ
3-operation in Donaldson-Fukaya category. We present applications to approximation theory on symplectic manifolds and to Hamiltonian
dynamics. 相似文献
13.
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
14.
Let (M, ω, Φ) be a Hamiltonian T-space and let H í T{H\subseteq T} be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion
in the integral full orbifold
K-theory of the orbifold symplectic quotient [M//H]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the
moment map of a Hamiltonian T-space M are Morse-Bott functions on M. As first applications, we conclude that a large class of symplectic toric orbifolds, as well as certain S
1-quotients of GKM spaces, have integral full orbifold K-theory that is free of additive torsion. Finally, we introduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses of the main theorem hold. We illustrate our
results using low-rank coadjoint orbits of type A and B. 相似文献
15.
Andreas Floer 《纯数学与应用数学通讯》1988,41(6):775-813
The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories. 相似文献
16.
Maria Luiza Lapa de Souza 《K-Theory》1998,13(4):347-361
The purpose of this paper is, first to define the Maslov invariant in the U-theory, the intermediate theory between real K-theory and complex K-theory, whose vanishing is a necessary and sufficient condition for the stable transversality of two Lagrangian sub-bundles of a symplectic fiber bundle. The second purpose is to show, in conformity with Bott periodicity, that Chern, Pontryand Stiefel-Whitney classes are precise enough to play the same role as the Maslov classes when one replaces the base space X of the symplectic bundle, by S
7
X
+the seventh topological suspension of X
+the compactification of X. 相似文献
17.
Katsuhiko Kuribayashi 《Differential Geometry and its Applications》2011,29(6):801-815
Let F be a fibration on a simply-connected base with symplectic fiber (M,ω). Assume that the fiber is nilpotent and T2k-separable for some integer k or a nilmanifold. Then our main theorem, Theorem 1.8, gives a necessary and sufficient condition for the cohomology class [ω] to extend to a cohomology class of the total space of F. This allows us to describe Thurston?s criterion for a symplectic fibration to admit a compatible symplectic form in terms of the classifying map for the underlying fibration. The obstruction due to Lalond and McDuff for a symplectic bundle to be Hamiltonian is also rephrased in the same vein. Furthermore, with the aid of the main theorem, we discuss a global nature of the set of the homotopy equivalence classes of fibrations with symplectic fiber in which the class [ω] is extendable. 相似文献
18.
Steven H. Weintraub 《Journal of Geometry》2007,86(1-2):165-180
Let V be a vector space of dimension 2n, n even, over a field F, equipped with a nonsingular symplectic form. We define a new algebraic/combinatorial structure, a spread of nonsingular
pairs, or nsp-spread, on V and show that nsp-spreads exist in considerable generality. We further examine in detail some particular cases. 相似文献
19.
Morris Newman 《Linear and Multilinear Algebra》2013,61(1-2):95-98
The principal results are that if A is an integral matrix such that AAT is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic. 相似文献
20.
A. Iserles 《Foundations of Computational Mathematics》2001,1(2):129-160
In this paper we develop in a systematic manner the theory of time-stepping methods based on the Cayley transform. Such methods
can be applied to discretize differential equations that evolve in some Lie groups, in particular in the orthogonal group
and the symplectic group. Unlike many other Lie-group solvers, they do not require the evaluation of matrix exponentials.
Similarly to the theory of Magnus expansions in [13], we identify terms in a Cayley expansion with rooted trees, which can be constructed recursively. Each such term is an integral over a polytope but all such integrals
can be evaluated to high order by using special quadrature formulas similar to the construction in [13]. Truncated Cayley expansions (with exact integrals) need not be time-symmetric, hence the method does not display the usual
advantages associated with time symmetry, e.g., even order of approximation. However, time symmetry (with its attendant benefits)
is attained when exact integrals are replaced by certain quadrature formulas.
March 7, 2000. Final version received: August 10, 2000. Online publication: January 2, 2001. 相似文献