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This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods. [ABSTRACT FROM AUTHOR]

Through the First National Comprehensive Risk Survey of Natural Disasters, the Three Data of Historical Disasters, Disaster Bearing Bodies and Disaster Reduction Capabilities of 31 provinces (cities, autonomous regions and municipalities directly under the central government) in China have been studied. Excluding regional factors, the statistical indicators of the Three Data and their functional relationship are studied by using Origin software. It is found that the three data do not meet the normal distribution, but the Growth/Sigmoidal Functional relationship is satisfied between Two and Three Data, i.e., S-type Growth Function, and the functional equation between the three is obtained. [ABSTRACT FROM AUTHOR]

The author, together with M. Veerapazham, S. Wang and W. Zhang, considered the existence and uniqueness of continuous solutions to an iterative equation involving the multiplication of iterates in Gopalakrishna et al. (Sci. China Math., to appear). In this paper we continue to investigate this equation for differentiable solutions. Similarly to continuous solutions until Gopalakrishna et al. (Sci. China Math., to appear), no result on differentiable solutions of this equation on non-compact intervals of R has been obtained until now. Although our strategy here is to use the logarithmic conjugation to reduce the equation to the well-known polynomial-like iterative equation, all known results on differentiable solutions of the latter are given on compact intervals. We revisit polynomial-like iterative equations on the whole of R and use the Banach contraction principle to prove the existence and uniqueness of differentiable solutions to our equation on R + . We then technically extend our results on R + to R - and show that none of the pairs of solutions obtained on R + and R - can be combined at the origin to obtain even a continuous solution of the equation on the whole R . [ABSTRACT FROM AUTHOR]

Microseismic monitoring is a rock breakdown monitoring technology, and it has become a major technical tool for underground disaster warning and prevention. However, the massive amount of data involved in multipoint monitoring raises the difficulty for microseismic research when the research object involves underground large-scale complex engineering on time and space scales. The risk predictions of rockburst based on microseismic parameters rely only on the experience of researchers and subjective factors by the human factor. The dependent variable cannot be related to the acoustic and energy signals in the form of a functional equation. Therefore, we collected a large amount of microseismic data obtained from the working faces of Shoushan Mine of Pingdingshan Coal Group in China and filtered the data, then we constructed a prediction model of microseismic data based on underground spatial three-dimensional coordinates using genetic programming (GP), which can realize real-time monitoring and disaster warning of microseismic signals. We established the magnitude and energy prediction formulas by GP and obtained the real-time evolution of each parameter of the optimal individual. The results showed that the actual seismic and energy data of the working faces exhibited good agreement with those predicted by the prediction formulas. The high energy and seismic distribution were caused by the mining stress at the edge of working faces due to the excavation and unloading effect of the surface, and the energy and seismic evolution predicted by GP could also show this phenomenon well. [ABSTRACT FROM AUTHOR]

We investigate the order of exponential sums involving the coefficients of general L -functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for G L m (ℤ) , Sci. China Math.58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math.57 (2017) 185–204]. [ABSTRACT FROM AUTHOR]