NON-EXISTENCE OF CONJUGATE-SYMPLECTIC MULTI-STEP METHODS OF ODD ORDER

We prove that any linear multi-step method G1^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（Zk） with odd order u （u≥ 3） cannot be conjugate to a symplectic method G2^T of order w （w 〉 u） via any generalized linear multi-step method G3^T of the form ∑k=0^mαkZk = T∑k=0^mβkJ^-1↓ΔH（∑l=0^mγklZl）. We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G3^T is a more general operator.     

Adjoint classes of functions in the H<Subscript>1</Subscript> sense

Using the concept of the H₁-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.     

The quantitative analysis of enhancement of high-order harmonics in two-color intense laser fields

The time-dependent Schr?dinger equation of the interaction of laser pulse with He⁺ is solved by using the asymptotic boundary condition and symplectic algorithm in fundamental laser-field and two-color laser fields. We find that the conversion efficiency of high-order harmonic generation (HHG) is higher in the two-color laser fields than in the fundamental laser field, especially for the combination of ω ₀ − 19ω ₀. To explain these phenomena, the ionization, the average distance, the probability of first excited sate, and the transition probability are calculated. We give the qualitative and quantitative analysis for the enhancement of conversion efficiency of HHG.     

李群的余切丛上的两种辛群胚结构

本文研究了李群的余切丛上的辛群胚结构.利用李代数胚的对偶丛上有自然诱导的泊松结构,.构造出了同一余切丛上的不同的辛群胚结构,推广了辛群胚的性质.     

A note on the geometry of M 3 and symplectic structures on S 1 × M 3

We investigate the relationship between the geometry of a closed, oriented 3-manifold M and the symplectic structures on S ¹ × M. In most cases the existence of a symplectic structure on S ¹ × M and Thurstonșs geometrization conjecture imply the existence of a geometric structure on M. This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure. This work was supported in part by a GEBIP grant from the Turkish Academy of Sciences and a CAREER grant from the Scientific and Technological Research Council of Turkey.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

Integrable Model of Interacting Elliptic Tops

We suggest a method for constructing a system of interacting elliptic tops. It is integrable and symplectomorphic to the Calogero-Moser model by construction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 55–64, January, 2006.     

The Adjoint Newton Algorithm for Large-Scale Unconstrained Optimization in Meteorology Applications

A new algorithm is presented for carrying out large-scale unconstrained optimization required in variational data assimilation using the Newton method. The algorithm is referred to as the adjoint Newton algorithm. The adjoint Newton algorithm is based on the first- and second-order adjoint techniques allowing us to obtain the Newton line search direction by integrating a tangent linear equations model backwards in time (starting from a final condition with negative time steps). The error present in approximating the Hessian (the matrix of second-order derivatives) of the cost function with respect to the control variables in the quasi-Newton type algorithm is thus completely eliminated, while the storage problem related to the Hessian no longer exists since the explicit Hessian is not required in this algorithm. The adjoint Newton algorithm is applied to three one-dimensional models and to a two-dimensional limited-area shallow water equations model with both model generated and First Global Geophysical Experiment data. We compare the performance of the adjoint Newton algorithm with that of truncated Newton, adjoint truncated Newton, and LBFGS methods. Our numerical tests indicate that the adjoint Newton algorithm is very efficient and could find the minima within three or four iterations for problems tested here. In the case of the two-dimensional shallow water equations model, the adjoint Newton algorithm improves upon the efficiencies of the truncated Newton and LBFGS methods by a factor of at least 14 in terms of the CPU time required to satisfy the same convergence criterion.The Newton, truncated Newton and LBFGS methods are general purpose unconstrained minimization methods. The adjoint Newton algorithm is only useful for optimal control problems where the model equations serve as strong constraints and their corresponding tangent linear model may be integrated backwards in time. When the backwards integration of the tangent linear model is ill-posed in the sense of Hadamard, the adjoint Newton algorithm may not work. Thus, the adjoint Newton algorithm must be used with some caution. A possible solution to avoid the current weakness of the adjoint Newton algorithm is proposed.