Estimating Orthant Probabilities of High-Dimensional Gaussian Vectors with An Application to Set Estimation

The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low-dimensional term and a remainder. While the low-dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art techniques in a numerical study, which also calls attention to the advantages and drawbacks of the procedure. Finally, the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption. Supplementary material for this article is available online.     

Unboundedness of the large solutions of some asymmetric oscillators at resonance

In this paper, we consider the unboundedness of solutions of the following differential equation (φ_p(x′))′ + (p ? 1)[αφ_p(x⁺) ? βφ_p(x^?)] = f(x)x′ + g(x) + h(x) + e(t) where φ_p(u) = |u|^{p? 2} u, p > 1, x^± = max {±x, 0}, α and β are positive constants satisfying with m, n ∈ N and (m, n) = 1, f and g are continuous and bounded functions such that lim_x→±∞g(x) ? g(±∞) exists and h has a sublinear primitive, e(t) is 2π_p‐periodic and continuous. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

n维耦合振子格点系统的同步性

本文通过在相空间中引入新范数的方法研究了Dirichlet,Neumann,周期三个不同边界条件下带有周期外力的n维二阶耦合振子格点系统的同步性.在Dirichlet边界条件下,如果系统非线性项的一阶偏导数有界,则当耦合系数足够大时,系统是有界耗散的并且任意两个解之间是同步的.在Neumann与周期边界条件下,如果不同子系统的外力之间的差和不同子系统的非线性项之间的差都比较小,并且系统是有界耗散的,则当耦合系数足够大时,系统任意一个解的任意两个分量之间是渐近同步的.这两种情况下,当耦合系数c_1→+∞,c_2→+∞时,系统任意一个解的任意两个分量之间是同步的.     

Hybrid projective synchronization of complex Duffing–Holmes oscillators with application to image encryption

Many works on hybrid projective synchronization (or simply ‘HPS’ for short) of nonlinear real dynamic systems have been performed, while the HPS of chaotic complex systems and its application have not been extensively studied. In this paper, the HPS of complex Duffing–Holmes oscillators with known and unknown parameters is separately investigated via nonlinear control. The adaptive control methods and explicit expressions are derived for controllers and parameters estimation law, which are respectively used to achieve HPS. These expressions on controllers are tested numerically, which are in excellent agreement with theory analysis. The proposed synchronization scheme is applied to image encryption with exclusive or (or simply ‘XOR’ for short). The related security analysis shows the high security of the encryption scheme. Concerning the complex Duffing–Holmes oscillator, we also discuss its chaotic properties via the maximum Lyapunov exponent. Copyright © 2017 John Wiley & Sons, Ltd.     

Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d²/dx²+Bx²+Ax⁻²+λx^−α (B>0, A?0) in L₂(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L₂(0,∞), namely C₀^∞(0,∞) and D(T_2,F)∩D(M_λ,α), where the latter is a subspace of the Sobolev space W_2,2(0,∞). Adjoints of these differential operators on C₀^∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) in L₂(0,∞) lead to the other adjoints. Further density properties C₀^∞(0,∞) in D(T_2,F)∩D(M_λ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T_2,F)∩D(M_λ,α).     

守恒型扩散方程非线性离散格式的性质分析和快速求解

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.     

A conservative extension of the method of characteristics for 1-D shallow flows

The method of characteristics (MOC) has been used for a long time in open channels and pipes flows. It is based on non-conservative equations, and hence it cannot be used directly for solving discontinuous shallow flows. In this paper we develop a conservative version of the MOC scheme for 1-D shallow flows by imposing the conservation law at the interpolation step. The conservation property of the scheme ensures the production of an accurate shock modeling and enables the MOC scheme to simulate dam-break type flows. By using a proper interpolation function, the proposed method can also produce quite accurate low-oscillatory results. A number of challenging test cases show considerable improvement compared to the traditional non-conservative MOC scheme in the case of dam-break type and trans-critical flow simulations.     

模糊关系几何规划

提出一类目标函数为正项式,约束是模糊关系方程的优化问题。阐述模糊关系方程解集的结构以及求解的方法,基于目标函数中每个单项式的指数取值情况讨论最优解,并且给出解决此类优化问题的一个程序,为了说明该方法的有效性给出具体例子。     

The behaviour of nonlinear oscillators subjected to ramped forcing

This paper considers the behaviour of a mechanical oscillator with cubic non-linearity subjected to a forcing excitation whose frequency remains constant while the amplitude is ramped, growing until it reaches a predetermined value. We concentrate on the nature of the basins of attraction whose size indicates the stability of the system, in a structural sense. The reduced level of forecing at the initial stages of ramping produces a delay in bifurcational events when compared to the constant sinusoidally forced counterpart. Preliminary results show that for some parameter values the area of basin does not increase monotonically as the length of ramping is varied. A preliminary version was presented at BIFCHAOS '92, Workshop held at L'Aquila, Italy, on 20–21 May 1992.     

Note on Chaos in Three Degree of Freedom Dynamical System with Double Pendulum

The nonlinear response of a three degree of freedom vibratory system with double pendulum in the neighbourhood internal and external resonances is investigated. The equations of motion have bean solved numerically. In this type system one mode of vibration may excite or damp another one, and for except different kinds of periodic vibration there may also appear chaotic vibration. To prove the character of this vibration and to realise the analysis of transitions from periodic regular motion to quasi-periodic and chaotic, the following have been constructed: bifurcation diagrams and time histories, phase plane portraits, power spectral densities, Poincaré maps and exponents of Lyapunov. These bifurcation diagrams show many sudden qualitative changes, that is, many bifurcations in the chaotic attractor as well as in the periodic orbits.