Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations

We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations. J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS, the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China 2005CB321701.     

Generating functions of multi-symplectic RK methods via DW Hamilton–Jacobi equations

The De Donder–Weyl (DW) Hamilton–Jacobi equation is investigated in this paper, and the connection between the DW Hamilton–Jacobi equation and multi-symplectic Hamiltonian system is established. Based on the DW Hamilton–Jacobi theory, generating functions for multi-symplectic Runge–Kutta (RK) methods and partitioned Runge–Kutta (PRK) methods are presented. The work is supported by the Foundation of ICMSEC, LSEC, AMSS and CAS, the NNSFC (No.10501050, 19971089 and 10371128) and the Special Funds for Major State Basic Research Projects of China (2005CB321701).     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Orthogonal separable Hamiltonian systems on <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript>

In this paper we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T ² for a given metric, and prove that the Hamiltonian flow on any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T ² can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present the families of invariant tori at the same time appearing in such a complicated way. This work is partially supported by the National Natural Science Foundation of China (Grant Nos. 10671123, 10231020), “Dawn” Program of Shanghai Education Comission of China (Grant No. 03SG10) and Program for New Century Excellent Tatents in University of China (Grant No. 050391)     

Expansion of frames to tight frames

We show that every Bessel sequence （and therefore every frame） in a separable Hilbert space can be expanded to a tight frame by adding some elements. The proof is based on a recent generalization of the frame concept, the g-frame, which illustrates that g-frames could be useful in the study of frame theory. As an application, we prove that any Gabor frame can be expanded to a tight frame by adding one window function.     

Hamiltonian decomposition of complete bipartiter-hypergraphs

In [1] the concepts of paths and cycles of a hypergraph were introduced. In this paper, we give the concepts for bipartite hypergraph and Hamiltonian paths and cycles of a hypergraph, and prove that the complete bipartite 3-hypergraph withq vertices in each part is Hamiltonian decomposable whereq is a prime. This research is supported by the National Natural Science Foundation of China (No.19831080).     

Coarse-grained description ofq-deformation top and t-j model

From RTT relations the integrable Hamiltonian of the trigonometric Goryachev-Chaplygin gyrostat is established, which can be reduced to the Hamiltonian of t-j model by using multi-fermion realization ofSU _q(2) algebra and average-field approximation. Project supported in part by the National Natural Science Foundation of China (Grant No. 19377102).     

An Improved Result for Positive Measure Reducibility of Quasi-periodic Linear Systems

In this paper, by the KAM method, under weaker small denominator conditions and nondegeneracy conditions, we prove a positive measure reducibility for quasi-periodic linear systems close to constant： X = （A（λ） ＋ F（ψ, λ））X, ψ=ωwhere the parameter λ∈ （a, b）, w is a fixed Diophantine vector, which is a generalization of jorba ＆ Simo＇s positive measure reducibility result.     

Relations among whitney sets,self-similar arcs and quasi-arcs

We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent. Supported by Special Funds for Major State Basic Research Projects of China, Morningside Center of Mathematics, NSFC (No. 10241003) and ZJNFS (No. 101026).     

On generalized hamiltonian systems

1. PreliminaryIt is well known that{1] a 8ymPlectic form is invariant along the trajectory of a Hamilto-nian system. Based on this fundamental property, certain techniques have been developed.The purpose of this paper is to extend such an approach to a wider class of dynamic systeIns,namely, genera1ized Hamiltonian systems. Our purpose is to investigate a class of dynaInicsystems, which possess a certain "geometric structure".Deflnition 1.1[1'2]. Let M be a tIlallifo1d. w E fl'(M) is call…     

Generalized fibonacci cubes are mostly hamiltonian

The Hamiltonian problem is to determine whether a graph contains a spanning (Hamiltonian) path or cycle. Here we study the Hamiltonian problem for the generalized Fibonacci cubes, which are a new family of graphs that have applications in interconnection topologies [J. Liuand W.-J. Hsu, ?Distributed Algorithms for Shortest-Path, Deadlock-Free Routing and Broadcasting in a Class of Interconnection Topologies,”? International Parallel Processing Symposium (1992)]. We show that each member of this family contains a Hamiltonian path. Furthermore, we also characterize the members of this family that contain a Hamiltonian cycle.     

Hamiltonian path graphs

The Hamiltonian path graph H(G) of a graph G is that graph having the same vertex set as G and in which two vertices u and v are adjacent if and only if G contains a Hamiltonian u-v path. A characterization of Hamiltonian graphs isomorphic to their Hamiltonian path graphs is presented.     

An upper bound on the length of a Hamiltonian walk of a maximal planar graph

A Hamiltonian walk of a connected graph is a shortest closed walk that passes through every vertex at least once, and the length of a Hamiltonian walk is the total number of edges traversed by the walk. We show that every maximal planar graph with p(≥ 3) vertices has a Hamiltonian cycle or a Hamiltonian walk of length ≤ 3(p - 3)/2.     

Hamiltonian structures of the generalized cylindrical KdV equations

Hamiltonian structures of the cylindrical Korteweg-deVries equation and its higher order equations are found. The connection between the generalized C-KdV equation and nonisospectral problem is pointed out.Projects supported by the Science Fund of the Chinese Academy of Sciences.     

Two-dimensional invariant tori in the neighborhood of an elliptic equilibrium of Hamiltonian systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

Periodic solutions of Hamiltonian systems on hypersurfaces in a torus

The existence of periodic solutions to Hamiltonian systems on the symplectic manifold (T ²ⁿ, ω) is studied. We show that on a class of hypersurfaces in the torusT ²ⁿ there is a periodic solution, which generalizes the results due to Long and Zehnder. Supported by NNSF of China     

Well-posedness in the energy space for non-linear system of wave equations with critical growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure{utt-△u=-F1（｜u｜^2,｜v｜^2）u,utt-△u=-F2（｜u｜^2,｜v｜^2）u where there exists a function F（λ,μ） such that δF（λ,μ）/δλ=F1（λ,μ）.δF（λ,μ）/δμ=F2（λ,μ） By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in ＂Shatah and Struwe＇s paper, Ann. of Math. 138, 503-518 （1993）＂.     

控制调幅波耗散方程的整体吸引子的有限维

§ 1　IntroductionIn this paperwe shall study successively the equationsgiven in [1 ,2 ] .Underthe sameconditions(i.e.the initial boundary value and other conditions) as in[1 ,2 ] ,the upperbound of the dimension of their global attractor will be obtained respectively.Theexistence of the global attractors have been proved in[1 ,2 ] .This paper is organized as follows:In§ 2 ,we derive the Frechet differential of the solution operator s(t) for theperturbed system.In§ 3 ,we show the finite di…