Boundedness and continuity of Marcinkiewicz integrals associated to homogeneous mappings on Triebel-Lizorkin spaces

We establish the boundedness and continuity of parametric Marcinkiewicz integrals associated to homogeneous compound mappings on Triebel-Lizorkin spaces and Besov spaces. Here the integral kernels are provided with some rather weak size conditions on the unit sphere and in the radial direction. Some known results are naturally improved and extended to the rough case.     

Revisiting non-degenerate parametric down-conversion

The quantum dynamics of a two-mode non-resonant parametric down-conversion process is studied by recasting the time evolution equations for the basic operators in an equivalent spin equation form with simpler exact solutions for a pump field with harmonic time dependence. Expectation values of suitable operators for studying important features such as squeezing and quantum revivals are presented in simple forms.      

Chaos control and synchronization of the Φ<Superscript>6</Superscript>-Van der Pol system driven by external and parametric excitations

The dynamical behavior of the Φ⁶-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

On the interplay between interior point approximation and parametric sensitivities in optimal control

Infinite-dimensional parameter-dependent optimization problems of the form ‘minJ(u;p) subject to g(u)?0’ are studied, where u is sought in an L_∞ function space, J is a quadratic objective functional, and g represents pointwise linear constraints. This setting covers in particular control constrained optimal control problems. Sensitivities with respect to the parameter p of both, optimal solutions of the original problem, and of its approximation by the classical primal-dual interior point approach are considered. The convergence of the latter to the former is shown as the homotopy parameter μ goes to zero, and error bounds in various Lq norms are derived. Several numerical examples illustrate the results.     

Bootstrap for estimating the MSE of the Spatial EBLUP

This work assumes that the small area quantities of interest follow a Fay–Herriot model with spatially correlated random area effects. Under this model, parametric and nonparametric bootstrap procedures are proposed for estimating the mean squared error of the empirical best linear unbiased predictor (EBLUP). A simulation study based on the Italian Agriculture Census 2000 compares bootstrap and analytical estimates of the MSE and studies their robustness to non-normality. Results indicate lower bias for the non-parametric bootstrap under specific departures from normality.      

基于一族蕴涵算子的对偶三I算法

三I算法是一种新的模糊推理方法,是传统的模糊推理方法的修改和补充. 三I表达式取最小值时的最优解算法(即对偶三I算法)是三I算法思想的延伸和完善.本文针对蕴涵算子族Ip,讨论了FMP和FMT问题的对偶三I算法,给出了相应的计算公式,从而也进一步促进了对三I算法的研究.     

Graphs of transportation polytopes

This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an m×n transportation polytope is a multiple of the greatest common divisor of m and n.