Although Shannon mutual information has been widely used, its effective calculation is often difficult for many practical problems, including those in neural population coding. Asymptotic formulas based on Fisher information sometimes provide accurate approximations to the mutual information but this approach is restricted to continuous variables because the calculation of Fisher information requires derivatives with respect to the encoded variables. In this paper, we consider information-theoretic bounds and approximations of the mutual information based on Kullback--Leibler divergence and Rényi divergence. We propose several information metrics to approximate Shannon mutual information in the context of neural population coding. While our asymptotic formulas all work for discrete variables, one of them has consistent performance and high accuracy regardless of whether the encoded variables are discrete or continuous. We performed numerical simulations and confirmed that our approximation formulas were highly accurate for approximating the mutual information between the stimuli and the responses of a large neural population. These approximation formulas may potentially bring convenience to the applications of information theory to many practical and theoretical problems.

It is imperative to conduct experimental studies on the seepage behavior of gas-bearing coal under cyclic dynamic loading conditions. This paper focuses on the evolution of coal permeability under the combined effects of dynamic loading, static loading, and gas adsorption. The principal conclusions are as follows. (1) As the frequency and amplitude of dynamic loading increase, the development of pore and fissure structures within the coal body becomes increasingly pronounced during dynamic loading cycles, resulting in a gradual rise in permeability. Notably, as the coal approaches its yielding stage, the permeability can increase by up to 47%. (2) The permeability curve is divided into four regions: the compaction reduction zone, the oscillation zone, the gradual recovery zone, and the abrupt increase zone of failure. Ultimately, in the failure phase, the permeability surges dramatically, potentially reaching 4 to 5 times the initial permeability. (3) When the static load stage is constant, the rate of change in permeability of the coal under dynamic loading decreases with increasing adsorption amount. When the adsorption amount is constant, the rate of change in permeability of the coal under dynamic loading increases with the increase in static load stress stage, with the maximum increase reaching 75.2%. It can be concluded from the rate of change in permeability and the dynamic loading sensitivity coefficient that the permeability of the coal is highly sensitive to cyclic dynamic loading, with increased sensitivity associated with larger static load stages and decreased sensitivity with greater adsorption amounts.

This research focuses on the Hopf bifurcation occurring at a singular point with high-order degeneracy in a class of three-dimensional systems. Based on center manifold theorem, by expanding the technique for determining singular point values of degenerate singularities in two-dimensional systems, the formal series method to determine the singular point values at the high-order degenerate critical point is discussed. Furthermore, a class of specific three-dimensional differential systems with high-order degenerate singularities is explored. By computing the singular point values and determining center conditions, we proved that a minimum of 5 small-amplitude limit cycles in the vicinity of the origin. This research provides new perspectives on the center problem and limit cycle bifurcation in high-dimensional systems with high-order degenerate singular points.

Session-based recommendations which aim to predict subsequent user-item interaction based on historical user behavior during anonymous sessions can be challenging tasks. Two main challenges need to be addressed and improved: (1) How to analyze these sessions to accurately and completely capture users’ preferences, and (2) how to identify and eliminate any interference caused by noisy behavior. Existing methods have not adequately addressed these issues since they either neglected the valuable insight that can be gained from analyzing consecutive groups of items or failed to take these noisy data in sessions seriously and handled them properly, which can jointly impede recommendation systems from capturing users’ real intentions. To address these two problems, we designed a multi-order semantic denoising (MSD) model for session-based recommendations. Specifically, we grouped items of different lengths as varying multi-order semantic units to mine the user’s primary intention from multiple dimensions. Meanwhile, a novel denoising network was designed to alleviate the interference of noisy behavior and provide a more precise session representation. The result of extensive experiments on three real-world datasets demonstrated that the proposed MSD model exhibited improved performance compared with existing state-of-the-art methods in session-based recommendation.