On the connection between harmonic maps and the self-dual Yang—Mills and the sine-Gordon equations

This is an algebraic paper which demonstrates some algebraic connections between the Yang—Mills and the sine-Gordon equation.     

Seven kinds of monotone maps

Known as well as new types of monotone and generalized monotone maps are considered. For gradient maps, these generalized monotonicity properties can be related to generalized convexity properties of the underlying function. In this way, pure first-order characterizations of various types of generalized convex functions are obtained.     

Base subsets of symplectic Grassmannians

Let V and V′ be 2n-dimensional vector spaces over fields F and F′. Let also Ω: V× V→ F and Ω′: V′× V′→ F′ be non-degenerate symplectic forms. Denote by Π and Π′ the associated (2n−1)-dimensional projective spaces. The sets of k-dimensional totally isotropic subspaces of Π and Π′ will be denoted by and ${\mathcal G}'_{k}$, respectively. Apartments of the associated buildings intersect and by so-called base subsets. We show that every mapping of to sending base subsets to base subsets is induced by a symplectic embedding of Π to Π′.     

Symplectic spread-based generalized Kerdock codes

Kerdock codes (Kerdock, Inform Control 20:182–187, 1972) are a well-known family of non-linear binary codes with good parameters admitting a linear presentation in terms of codes over the ring (see Nechaev, Diskret Mat 1:123–139, 1989; Hammons et al., IEEE Trans Inform Theory 40:301–319, 1994). These codes have been generalized in different directions: in Calderbank et al. (Proc Lond Math Soc 75:436–480, 1997) a symplectic construction of non-linear binary codes with the same parameters of the Kerdock codes has been given. Such codes are not necessarily equivalent. On the other hand, in Kuzmin and Nechaev (Russ Math Surv 49(5), 1994) the authors give a family of non-linear codes over the finite field F of q = 2 ^l elements, all of them admitting a linear presentation over the Galois Ring R of cardinality q ² and characteristic 2². The aim of this article is to merge both approaches, obtaining in this way new families of non-linear codes over F that can be presented as linear codes over the Galois Ring R. The construction uses symplectic spreads.      

Uniqueness and stability of slowly oscillating periodic solutions of delay equations with bounded nonlinearity

We study slowly oscillating periodic solutions of delay equations with small parameters. When the nonlinearity has finite and nonzero limits at infinities, the appearance of these solutions and their periods can be found though asymptotic analysis. Under further natural assumptions on the nonlinearity, we prove that slowly oscillating periodic solutions are unique and asymptotically stable when parameters are sufficiently small.     

Spatial chaos in boundary layer transition

Experiments have been performed on a laminar flat plate boundary layer undergoing transition to turbulence. Reproducible disturbances were introduced via a loudspeaker embedded at some upstream location and their evolution over the plate measured using hot-wire anemometry. A new technique has been used to estimate the number of nonlinearly independent modes in the reconstructed phase portraits. Nonlinear maps were then fitted that explicitly model the spatial evolution of disturbances. These maps are consistent with classical linear stability theory for small disturbances, and give rise to Smale horse-shoe like behaviour for larger amplitude distrubances. Thus a mechanism for generating chaos has been uncovered.     

小波基方法在波传问题中的应用

基于多尺度分析理论，引入哈密顿体系和插值小波变换，分别构造了适合于求解复杂域波传问题的快速自适应方法——多尺度辛格式和插值小波配点格式，利用小波基的局部性与消失矩等特性改善计算效率，并将插值小波应用到波动方程的多尺度反演问题中。讨论了其优缺点并提出几点展望。     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

集值映射多目标半定规划问题的epsilon-弱有效性

对于集值映射多目标半定规划问题, 在近似锥-次类凸的框架下, 建立了含矩阵和向量的择一性定理, 给出了问题的epsilon-弱有效解的epsilon-Lagrange乘子定理及标量化定理和epsilon-弱鞍点定理．