The fractional Fourier series generalizes the classical Fourier series by introducing a rotation angle alpha in the time-frequency plane, but inherits the Gibbs phenomenon for piecewise smooth functions. Unlike the classical setting, the chirp modulation factor renders the fractional partial sum complex-valued, corrupting both real and imaginary components simultaneously and making direct adaptation of classical remedies insufficient. The Inverse Polynomial Reconstruction Method (IPRM) resolves the Gibbs phenomenon by enforcing that the Fourier coefficients of a Gegenbauer polynomial expansion match the given spectral data, rather than projecting the corrupted partial sum onto a polynomial basis. This paper provides a theoretical and numerical study of IPRM reconstruction from fractional Fourier coefficients. The fractional transformation matrix is derived and its conditioning is shown to be governed by an alpha-independent Gram matrix, which reveals the dependence on the Gegenbauer parameter lambda and the polynomial degree m, while being entirely insensitive to the transform angle. An L-infinity error estimate is established, guaranteeing exponential convergence for analytic functions. Numerical experiments on piecewise analytic test functions demonstrate effective elimination of the Gibbs phenomenon and confirm the theoretical predictions.
This paper presents a theoretical analysis of a fourth-order exponential time differencing Runge-Kutta rational approximation scheme for the Allen-Cahn equation based on real and distinct poles, aiming to establish a theoretical framework for the method combining exponential time differencing with rational approximation. Starting from the scalar linearized equation, we rigorously prove that the scheme is L-stable and that its matrix norm is stable with three types of boundary conditions, revealing that the L-acceptable property of the RDP rational approximation is the key to ensuring stability. Regarding the error analysis, we establish that the fully discrete scheme attains fourth-order temporal convergence, which theoretically confirms that the scheme achieves fourth-order accuracy. Numerical examples in both 2D and 3D verify the accuracy and effectiveness of the method.
This paper raises a membership-function-dependent asymmetric Lyapunov functional for security consensus seeking of leader-following T-S fuzzy multi-agent systems (MASs) vulnerable to random deception attacks. Initially, a generalized dynamic event-triggered mechanism is advanced to optimize inter-agent communication efficiency, which encompasses some existing triggered strategies as exceptional situations and affords flexible state-dependent adjustment of threshold parameters. Subsequently, a security fault-tolerant control (FTC) strategy is designed to resist the actuator failures and random deception attacks, which is depicted by a Bernoulli stochastic variable. Additionally, a new type of membership-function-dependent asymmetric Lyapunov functional is meticulously constructed to obviate the orthodox symmetry and positive-definiteness constraints on matrix variables, thereby resulting in less conservative consensus criteria. Furthermore, this work utilizes three key components: the constructed membership function-dependent fuzzy Lyapunov functional, the time derivative information of fuzzy membership functions (FMFs), and the membership-function-based fuzzy switching strategy. By integrating these approaches, sufficient conditions for synthesizing the fuzzy switching controller are formulated as linearization inequalities. Lastly, two numerical simulations are carried out to verify the accuracy and effectiveness of the proposed approach.
Intertemporal decision-making, i.e., choosing between immediate and delayed rewards is crucial for social functioning, yet social exclusion often biases individuals toward short-term gains. However, it remains unclear which psychological processes most proximally link exclusion to myopic intertemporal choice. Across two complementary experiments, we used psychological network analysis to examine the structural role of self-control resources (SCR) relative to trait self-control, impulsivity, and affect. In Experiment 1 (N = 144), participants reported chronic social exclusion, SCR, trait self-control, impulsivity (BIS subscales), positive and negative affect, and completed a monetary intertemporal choice task. In Experiment 2 (N = 97), we manipulated acute exclusion using the Cyberball paradigm and assessed perceived exclusion alongside the same measures and task. Across both experiments, higher social exclusion was reliably associated with higher degree of temporal discounting, whereas higher SCR was associated with more patient intertemporal choices. Network analyses showed that exclusion and SCR were the only variables that maintained unique connections to the discounting parameter, and SCR emerged as the strongest connector between exclusion and discounting, with impulsivity, trait self control, and affect occupying more peripheral positions. Together, these findings suggest that momentary self control resources form the primary psychological pathway through which social exclusion biases intertemporal decision-making.
