A hybrid estimation framework for multivariate ARFIMA (Autoregressive Fractionally Integrated Moving Average) processes with GARCH (Generalized Autoregressive Conditional Heteroskedasticity) innovations is proposed. In the first stage, the ARFIMA parameters are estimated using frequency-domain quasilikelihood methods. In the second stage, the GARCH parameters are estimated in the time domain based on the residuals obtained from the ARFIMA estimation stage. Under suitable regularity conditions, the consistency and asymptotic normality of both estimators are established. The sufficiency of the assumptions required for joint asymptotic inference is also examined.
We develop a delay-memory model for non-stationary solid-propellant combustion. An exponential Volterra kernel is represented by an auxiliary state, yielding a finite-dimensional retarded system. Local well-posedness, global continuation under dissipativity, linear stability, and a Hopf crossing criterion are established. The characteristic equation is derived explicitly, the Lyapunov constants are specified, and the transversality derivative is evaluated numerically. Improved simulations distinguish transient oscillations from a closed post-transient phase portrait.
In this paper, a weighted midpoint method is proposed for the numerical solution of nonlinear equations. The convergence of the iterative process to a root is established, similar to the classical bisection and false position methods. The proposed method is applied to several benchmark problems, and the results are compared with those obtained using conventional methods.
In this work, by introducing a set of nonlinear transformations with four trial functions, we extend the new trial function technique for constructing the explicit and accurate travelling wave solutions of NPDEs developed by Xie to solve NPDE systems. In order to demonstrate the feasibility of this extended technique, we employ it to solve the Whitham-Broer-Kaup equation system and generalized Hirota-Satsuma coupled KdV equation system. Therefore, some explicit and accurate travelling wave solutions of these two NPDE systems are successfully constructed in a systematic and concise manner. The obtained solutions are consistent with those found in the existing references. The procedure used herein can also be applied to solve other NPDE systems. We intend to generalize this strategy to deduce the explicit and accurate travelling wave solutions to other NPDE systems.
This paper presents a mathematical framework for quantifying individual impact in collaborative learning networks. Classroom interactions are modeled as a bidirectional weighted graph and then formulated as a cooperative game. The main methodological contribution is the construction of a graph-based characteristic function that represents learning interactions, together with the systematic use of the Shapley value to compute a unique quantitative impact index for each student based on marginal contributions. Although the Shapley value is a classical concept in cooperative game theory, the novelty of this work lies in its structured integration with interaction networks in educational settings and in the formal translation of collaborative learning mechanisms into a cooperative game framework. Numerical simulations using hypothetical classroom data illustrate the behavior of the proposed index and show consistency between Shapley values and students’ structural roles in the interaction network. The reliance on synthetic data and modeling assumptions is acknowledged as a limitation, and directions for empirical validation, sensitivity analysis, and methodological extensions are outlined. The proposed approach provides a mathematically consistent tool for the fair assessment of individual contributions in collaborative learning environments.
