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Generally, it is difficult to obtain convergent chaotic solution in an arbitrarily given finite interval of time. Some researchers even believe that all chaotic responses are simply numerical noise and have nothing to do with solutions of differential equations. However, using 1200 CPUs of the National Supercomputer TH-A1 at Tianjin and a parallel integration algorithm of the so-called "Clean Numerical Simulation" (CNS) based on the 3500th-order Taylor expansion and data in 4180-digit multiple precision, one can obtain reliable convergent chaotic solution of the Lorenz equation in a rather long time interval [0,10 000]. This supports Lorenz's optimistic viewpoint [Lorenz 2008] that "numerical approximations can converge to a chaotic true solution throughout any finite range of time". It also supports Tucker's proof [Tucker 1999, 2002] for the famous Smale's 14th problem that the strange attractor of the Lorenz equation indeed exists.

The aim of this study is to investigate whether the adoption of convergent-International Financial Reporting Standards (IFRS) in China affects the audit fees of initial public offerings (IPO) firms. An empirical regression analysis using panel data for 1,094 nonfinancial IPOs (excluding season equity offers) of A-shares listed on the Shanghai and Shenzhen Stock Exchanges between 2003 and 2012 is adopted. The results reveal that audit fees increase following convergent-IFRS adoption in China and additionally suggest that convergent-IFRS adoption eases the intense price competition that previously existed in China’s audit market and thus has important policy implications for regulators. To the best of the authors’ knowledge, this study represents the first reported attempt to adopt the IPO setting to examine the effects of convergent-IFRS adoption on audit fees and fills the gap in literature. Using a setting of IPOs enables this paper to further exclude the influence of quasi-rents derived from low-balling after initial audit engagement when testing audit fees.

This paper focuses on the computation issue of portfolio optimization with scenario-based mean-average value at Risk (AVaR) in uncertain environment. The portfolio optimization problem is designed in two cases: risk-taker and risk-averse models. The main idea is to replace the portfolio selection models with linear programming (LP) problems. Since the computing time required for solving LP greatly depends on the dimension and the structure of the problem, the conventional numerical methods are usually less effective in real-time applications. One promising approach to handle online applications is to employ recurrent neural networks based on circuit implementation. Hence, according to the convex optimization theory and some concepts of ordinary differential equations, a neural network model for solving the LP problems related to portfolio selection problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns. Some illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

Given x ∈ (0, 1), let [a₁(x), a₂(x), a₃(x),…] be the continued fraction expansion of x and be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {a_n(x), n ≥ 1} and proved that for any β > 0, the set

The design and optimization procedure for a loudspeaker driven 10-W cooling power thermoacoustic refrigerator components with a temperature difference of 120 K has been discussed using the linear thermoacoustic theory. The resonator losses are proportional to surface area and the optimum diameter ratio of small and large resonator tubes for minimum heat loss for a quarter-wavelength hemispherical ended resonator design is discussed. The hemispherical ended resonator design is further analytically optimized to increase COP, cooling effect at cold heat exchanger and power density by decreasing total resonator surface area and volume. An alternate convergent-divergent resonator design is proposed which is found to be more efficient compared to previous published designs. Resonator designs are tested with DeltaEC software, which predicts a lowest temperature of -48°C and -47°C for the improved hemispherical and convergent-divergent resonator designs, respectively. Theoretical results are in good agreement with DeltaEC results.