THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.

TWO NEW CONSTRUCTIONS OF CARTESIAN AUTHENTICATION CODES FROM SYMPLECTIC GEOMETRY

In this paper, two new constructions of Cartesian authentication codes from symplectic geometry are presented and their size parameters are computed.

New key pre-distribution scheme using symplectic geometry over finite fields for wireless sensor networks

To achieve secure communication in wireless sensor networks （WSNs）, where sensor nodes with limited computation capability are randomly scattered over a hostile territory, various key pre-distribution schemes （KPSs） have been proposed. In this paper, a new KPS is proposed based on symplectic geometry over finite fields. A fixed dimensional subspace in a symplectic space represents a node, all 1-dimensional subspaces represent keys and every pair of nodes has shared keys. But this naive mapping does not guarantee a good network resiliency. Therefore, it is proposed an enhanced KPS where two nodes have to compute a pairwise key, only if they share at least q common keys. This approach enhances the resilience against nodes capture attacks. Compared with the existence of solution, the results show that new approach enhances the network scalability considerably, and achieves good connectivity and good overall performance.

Construction of Authentication Codes with Double Arbiters over Symplectic Geometry

In a common authentication code with arbitration, the dishonest arbiter may make a threat to the security of authentication system. In this paper, an authentication code with double arbiters over symplectic geometry is constructed, and the relevant parameters and the probabilities of successful attacks are calculated. The model not only prevents deception from the opponent and members of the system, but also effectively limits the attacks of single arbiter. Moreover, the collusion attacks from arbiters and participators are difficult to succeed.

Fault Diagnosis Method of Rolling Bearing Based on ESGMD-CC and AFSA-ELM

Incomplete fault signal characteristics and ease of noise contamination are issues with the current rolling bearing early fault diagnostic methods,making it challenging to ensure the fault diagnosis accuracy and reliability.A novel approach integrating enhanced Symplectic geometry mode decomposition with cosine difference limitation and calculus operator(ESGMD-CC)and artificial fish swarm algorithm(AFSA)optimized extreme learning machine(ELM)is proposed in this paper to enhance the extraction capability of fault features and thus improve the accuracy of fault diagnosis.Firstly,SGMD decomposes the raw vibration signal into multiple Symplectic geometry components(SGCs).Secondly,the iterations are reset by the cosine difference limitation to effectively separate the redundant components from the representative components.Additionally,the calculus operator is performed to strengthen weak fault features and make them easier to extract,and the singular value decomposition(SVD)weighted by power spectrum entropy(PSE)can be utilized as the sample feature representation.Finally,AFSA iteratively optimized ELM is adopted as the optimized classifier for fault identification.The superior performance of the proposed method has been validated by various experiments.

SYMPLECTIC SOLUTION SYSTEM FOR REISSNER PLATE BENDING

Based on the Hellinger_Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamiltonian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction_vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigenfunction vectors for zero eigenvalue are formed.It is showed that the all eigensolutions for zero eigenvalue are basic solutions of the Saint_Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint_Venant theorem.The symplectic solution method is not the same as the classical semi_inverse method and breaks through the limit of the traditional semi_inverse solution.The symplectic solution method will have vast application.

Symplectic solution for three dimensional couple stress problem and its variational principle

A new state vector is presented for symplectic solution to three dimensional couple stress problem. Without relying on the analogy relationship, the dual PDEs of couple stress problem are derived by a new state vector. The duality solution methodology in a new form is thus extended to three dimensional couple stress. A new symplectic orthonormality relationship is proved. The symplectic solution to couple stress theory based a new state vector is more accordant with the custom of classical elasticity and is more convenient to process boundary conditions. A Hamilton mixed energy variational principle is derived by the integral method.

SYMPLECTIC DUALITY SYSTEM ON PLANE MAGNETOELECTROELASTIC SOLIDS

By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of original variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and the eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were shown clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain.

HAMILTONIAN SYMPLECTIC SEMI-ANALYTICAL METHOD AND ITS APPLICATION IN INHOMOGENEOUS ELECTROMAGNETIC WAVEGUIDES

Dual vectors are applied in Hamilton system of applied mechanics. Electric and magnetic field vectors are the dual vectors in electromagnetic field. The Hamilton system method is introduced into the analysis of electromagnetism waveguide with inhomogeneous materials. The transverse electric and magnetic fields are regarded as the dual. The basic equations are solved in Hamilton system and symplectic geometry. With the Hamilton variational principle, the symplectic semi_analytical equations are derived and preserve their symplectic structures. The given numerical example demonstrates the solution of LSE (Longitudinal Section Electric) mode in a dielectric waveguide. (

NUMERICAL METHOD BASED ON HAMILTON SYSTEM AND SYMPLECTIC ALGORITHM TO DIFFERENTIAL GAMES

The resolution of differential games often concerns the difficult problem of two points border value （TPBV）, then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.

Analytic solution for Reissner plate bending based on new symplectic approach

This paper presents the analytic solution for Reissner plate bending derived by the symplectic geometry approach.Firstly,the basic equations for Reissner plate are transferred into Hamilton canonical equations.And then the whole state variables are separated.Finally,the solution is obtained according to the method of eigenfunction expansion in the symplectic geometry.Only the basic elasticity equations of Reissner plate are used in the present study and the pre-selection of the deformation function is abandoned,which is requisite in classical solution methods.Therefore,the utilized approach is completely reasonable and theoretical.To verify the accuracy and validity of the formulations derived,the numerical results are presented to compare with those available in the open literatures.

Connections with Symplectic Structures

A charming feature of symplectic geometry is that it is at the crossroad of many other mathematical disciplines. In this article we review the basic notions with examples of symplectic structures and show the connections of symplectic geometry with the various branches of differential geometry using important theorems.

ON THE ASYMPTOTIC BEHAVIOR OF HOLOMORPHIC ISOMETRIES OF THE POINCAR DISK INTO BOUNDED SYMMETRIC DOMAINS

In this article we bounded symmetric domains study holomorphic isometries of the Poincare disk into Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometrics up to normalizing constants with respect to the Bergman metric, showing in particular that the graph 170 of any germ of holomorphic isometry of the Poincar6 disk A into an irreducible bounded symmetric domain Ω belong to C^N in its Harish-Chandra realization must extend to an affinealgebraic subvariety V belong to C × C^N = C^N＋1, and that the irreducible component of V ∩ （△ × Ω） containing V0 is the graph of a proper holomorphic isometric embedding F ： A→ Ω. In this article we study holomorphie isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δ△. Starting with the structural equation for holomorphic isometrics arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form σ of the holomorphie isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ‖σ‖ must vanish at a general boundary point either to the order 1 or to the order 1/2, called a holomorphie isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

Variational Principles and Hamiltonian Formulation for Nonlinear Water Waves

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The Hamilton System and Hamilton Type Generalized Variational Principle for the Laminated Composite Plates and Shells

By introducing the Hamilton theory and algorithms into the problems of laminated composite plates andshells, the Hamiltion type generalized variational principle is established, and the Hamilton canonical equations andthe boundary conditions for the static and elastoplastic analysis of composite plates are presented. With thetransformation of phase variables, the Hamilton canonical equations and their boundary conditions for thecylindrical shells and doubly curved shells in the curvilinear coordinate are given.

Uniruled Symplectic Divisors

In this article,we consider the problem of lifting the GW theory of a symplectic divisor to that of the ambient manifold in the context of symplectic birational geometry.In particular,we generalizeMaulik-Pandharipande’s relative/absolute correspondence to relative-divisor/absolute correspondence.Then,we use it to lift a minimal uniruled invariant of a divisor to that of the ambient manifold.

Periodic Orbits on AF-Symplectic Manifolds

The method by McDuff is used to get the existence of the J holomorphic sphere in some symplectic Manifolds and then the J holomorphic sphere is perturbed to split a periodic solution of Hamiltonian systems in these sympletic manifolds. As a result,the Weinstein conjecture is proved in the asymptotically manifolds.

Analysis and geometry of Floer theory of Landau-Ginzburg model on C^n

This article studies the Floer theory of Landau-Ginzburg （LG） model on C^n. We perturb the Kahler form within a fixed Kahler class to guarantee the transversal intersection of Lefschetz thimbles. The C^0 estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.

Symplectic Blowing Down in Dimension Six

We establish a blowing down criterion in the context of birational symplectic geometry in dimension 6.

CONTACT ALGORITHMS FOR CONTACT DYNAMICAL SYSTEMS

In this paper, we develop a general way to construct contact algorithms for contact dynamical systems. Such an algorithm requires the corresponding step transition map preserve the contact structure of the underlying contact phase space. The constructions are based on the correspondence between the contact geometry of R2n+1 and the conic symplectic one of R2n+2 and therefore, the algorithms are derived naturally from the symplectic algorithms of Hamiltonian systems.