A B S T R A C TIn this paper, we propose a nearly-analytic central difference method, which is an improved version of the central difference method. The new method is fourth-order accurate with respect to both space and time but uses only three grid points in spatial directions. The stability criteria and numerical dispersion for the new scheme are analysed in detail. We also apply the nearly-analytic central difference method to 1D and 2D cases to compute synthetic seismograms. For comparison, the fourthorder Lax-Wendroff correction scheme and the fourth-order staggered-grid finitedifference method are used to model acoustic wavefields. Numerical results indicate that the nearly-analytic central difference method can be used to solve large-scale problems because it effectively suppresses numerical dispersion caused by discretizing the scalar wave equation when too coarse grids are used. Meanwhile, numerical results show that the minimum sampling rate of the nearly-analytic central difference method is about 2.5 points per minimal wavelength for eliminating numerical dispersion, resulting that the nearly-analytic central difference method can save greatly both computational costs and storage space as contrasted to other high-order finitedifference methods such as the fourth-order Lax-Wendroff correction scheme and the fourth-order staggered-grid finite-difference method.
I N T R O D U C T I O NDuring the past several decades, a great increase in computer performance and saving memory have driven wide applications of numerical methods in acoustic phenomena in complex media. The elastic properties of complex media with fractures, cracks or pores show significant spatial variations. Any hope of analysing synthetic seismograms relies heavily on the accurate resolution of a wide range of temporal and spatial scales present in acoustics and seismology. Hence, it is essential to develop efficient numerical methods with high accuracy and fast computational speed. *
In view of the fact that complex signals are often used in the digital processing of certain systems such as digital communication and radar systems, a new complex Duffing equation is proposed. In addition, the dynamical behaviors are analyzed. By calculating the maximal Lyapunov exponent and power spectrum, we prove that the proposed complex differential equation has a chaotic solution or a large-scale periodic one depending on different parameters. Based on the proposed equation, we present a complex chaotic oscillator detection system of the Duffing type. Such a dynamic system is sensitive to the initial conditions and highly immune to complex white Gaussian noise, so it can be used to detect a weak complex signal against a background of strong noise. Results of the Monte-Carlo simulation show that the proposed detection system can effectively detect complex single frequency signals and linear frequency modulation signals with a guaranteed low false alarm rate. Chaos theory is one of the most significant achievements of nonlinear science. Many scholars have researched the basic theoretical issues [1][2][3][4][5][6][7] and practical applications of chaos. Hitherto, chaos theory has been widely applied in the control [8], synchronization [9], prediction [10], communication [11,12], and detection fields, among others. In signal detection, detection of weak signals using a chaotic oscillator in the real domain [13][14][15][16][17][18][19] has been studied by some researchers, with most researchers using a Duffing oscillator or modified version thereof. The basic idea is that a chaotic system is sensitive to the initial conditions and less influenced by noise. Nowadays complex signal processing is being used in many fields of science and engineering including digital communication systems, radar systems, antenna beamforming applications, coherent pulse measurement systems, and so on. As such, it is very meaningful to study how to detect a complex signal against a noisy background.Most of the recent works on complex chaos have focused on solving complex nonlinear differential equations, complex chaos control and synchronization, and so on. For example, Cveticanin [20] developed an approximate analytic approach for solving strong nonlinear differential equations of the Duffing type with a complex-valued function. Furthermore, excellent agreement is found between the analytic results and numerical ones. Li et al.[21] developed a stochastic averaging method for a quasi-Hamilton system to study the stationary solution in a nonlinear stochastically complex dynamical system. Rauh et al. [22] proved by means of a quadratic Lyapunov function that the complex Lorenz model is globally stable and presented an analytic expression for the upper bound on the magnitude of the time-dependent electric field. Mahmoud and Aly [23] illustrated the existence of periodic attractors of complex damped nonlinear systems by constructing Poincaré plots, and investigated the stability properties of the solutions of a complex nonlinear equati...
The title complex, [CaZn(C3H2O4)2(H2O)4]n, is a two-dimensional polymer and consists of CaO8 and ZnO6 polyhedra linked together by malonate ligands. The Ca(II) cation, lying on a twofold axis, is coordinated by two water molecules and six malonate O atoms. The Zn(II) cation, which lies on an inversion center in an octahedral environment, is coordinated by four malonate O atoms in an equatorial arrangement and two water molecules in axial positions. The Zn-O and Ca-O bond lengths are in the ranges 2.0445 (12)-2.1346 (16) and 2.3831 (13)-2.6630 (13) angstroms, respectively. The structure comprises alternating layers along the [101] plane, the shortest Zn...Zn distance being 6.8172 (8) angstroms. The whole three-dimensional structure is maintained and stabilized by the presence of hydrogen bonds.
Three new monosubstituted sucrose fatty acid esters, 1–3, were isolated from Equisetum hiemale L., together with nine known compounds, 4–12. Their structures were elucidated by spectroscopic analyses. Compounds 5, 6, and 10–12 were isolated from the title plant for the first time. All these compounds were evaluated for their cytotoxic activity. However, none of them was cytotoxic.
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