The linear instability of double-diffusive convection in a viscoelastic fluid, described by the Navier-Stokes-Voigt (NSV) model, is investigated. The effects of viscoelasticity and diffusive transport parameters on the onset of convection are analyzed for rigid-free, free-free, and rigid-rigid boundary configurations. The stability eigenvalue problem is solved numerically using the Galerkin method and validated against available analytical and numerical results. The analysis shows that viscoelasticity does not influence stationary convection but significantly affects oscillatory instability. The Voigt parameter alters the balance between viscous dissipation, elastic relaxation, and solutal buoyancy. This leads to a decrease in the critical Rayleigh number for small values of the Voigt parameter and an increase for larger values due to dominant viscoelastic damping. A clear stability hierarchy is observed among the boundary configurations: free-free boundaries are the least stable, while rigid-rigid boundaries are the most stable. Increasing the solutal Rayleigh number promotes oscillatory modes and mode transitions. The Prandtl number and Lewis number modify instability thresholds through their influence on thermal and solutal diffusion. Streamline and isotherm patterns at the critical state reveal complex convective structures. These results provide insight into the interaction between viscoelastic stresses and double-diffusive transport, and are relevant to geophysical and industrial processes involving coupled heat and mass transfer.
For a graph G, we call a set D⊆V(G) a total isolating set of G if G[D] has no isolated vertices and G−N[D] contains no edges, where G[D] is the induced subgraph of G on D and N[D] is the closed neighborhood of D. The total isolation number ιt(G) of G is the minimum cardinality of a total isolating set in G. Recently, Boyer, Goddard and Henning proved that if G is a connected graph with n ≥ 4 vertices and G¬≅C7, then ιt(G)≤n2 and the bound is tight. In this paper, we classify all extremal graphs which achieve the bound. We show that excluding an infinite family of graphs, there exist exactly thirteen sporadic extremal graphs. This solves an open problem proposed by Boyer, Goddard and Henning.
This paper presents an efficient stabilized leapfrog finite difference scheme for the ternary Allen–Cahn equations that achieves both computational efficiency and rigorous structure preservation. The scheme treats the nonlinear reaction terms explicitly at the middle time level, yielding a fully linear and decoupled system with a time-independent coefficient matrix. A stabilization technique is incorporated to guarantee the discrete maximum bound principle and energy dissipation. The method is uniquely solvable, and optimal convergence is established in the L∞ norm. Extensive numerical experiments in two- and three-dimensional settings confirm the theoretical results and demonstrate stable long-time performance compared with existing linear schemes.
Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs). However, PINNs often encounter convergence failures when addressing multi-scale or stiff problems. To address this issue, we propose the Order Statistical Adaptive Residual Loss (OS-ARL), a lightweight and implementation-friendly loss modification designed to improve the accuracy of PINNs training. OS-ARL adaptively reweights point-wise residuals based on order statistics of the residual magnitude distribution, enabling the network to emphasize hard-to-fit regions while maintaining numerical stability through a weighting mechanism decoupled from backpropagation. We validate the method across diverse benchmarks, including wave equation, convection equation, diffusion equation, Burgers equation, and Allen–Cahn equation. Experimental results demonstrate that OS-ARL accelerates convergence and improves solution accuracy by one to two orders of magnitude compared with standard PINNs using identical network architectures, and even outperforms complex architectures such as PINNsformer. These results indicate that OS-ARL provides a simple yet promising pathway for PINNs in challenging computational regimes.
This work investigates the impulsive optimal control problem for second-order hybrid systems that combines discrete dynamics with continuous dynamics based on adaptive dynamic programming. An impulsive optimal control protocol is proposed, and a parallel computation method is introduced to reduce computational time. Furthermore, a specialized mechanism is developed to adjust the selection tendency of impulsive intervals. Finally, an illustrative example is provided to demonstrate the feasibility of the proposed method.
An efficient, conservative hybrid weak Galerkin finite element methodology is proposed to simulate multiphase flows. An advection-oriented weak gradient yields a flow-sensitive stabilized weak Galerkin method. The directional stabilization reduces numerical dissipation, improving mass conservation and preserving interface features. High-order numerical experiments for steady advection confirm the theoretical analysis and exhibit the expected optimal convergence behavior. To achieve a balance between computational efficiency and local accuracy in full-domain settings, a hybrid strategy resolves advection with high accuracy near the interface. A Physics-Informed Neural Network (PINN), calibrated with sparse high-fidelity weak Galerkin data from a localized subdomain, provides a global surrogate of the solution guided by these data, with the weak Galerkin approximation retained near the interface through a blending procedure. The framework extends naturally to time-dependent problems, including level-set transport, and is applied to capillarity-dominated interface dynamics in Newtonian fluids. We also introduce an implicit coupling scheme based on a multidimensional generalization of a fifth-order multi-stage Newton-type scheme. Benchmark tests and numerical comparisons demonstrate the reliability and accuracy of the proposed approach.
This study examines the design challenge of a mixed H∞ and passive asynchronous output-feedback controller for discrete-time singular Markov jump systems (SMJSs) characterized by complex transition probabilities (C-TPs). The hidden Markov model (HMM) is utilized to estimate system modes that cannot always be accurately detected. Since complete state information is difficult to be obtained in practical scenarios, an output-feedback control scheme is adopted. Utilizing Lyapunov functional techniques, sufficient conditions are established to ensure that the considered SMJSs are stochastically admissible under the combined H∞ and passive performance γ. Under such circumstances, explicit equations for the controller gain are derived using the augmented matrix method and the formulation of slack matrices. A numerical example and a PWM (Pulse-Width-Modulation)-driven boost converter model are provided as application examples to demonstrate the effectiveness of the proposed controller design methodology.
In the realm of mathematics, the hereditary property is a crucial characteristic. Heredity in a mathematical structure clarifies its internal architecture and drives related research forward. Suppose that D is a nontrivial design with automorphism group G. We have accomplished the classification of all hereditary (G, 3)-flag-transitive designs in this paper, for G is an almost simple group that coincides with its socle.
Fully implicit time-discretization of hyperbolic systems can significantly reduce restrictions on the choice of time steps in numerical methods. The price to be paid may be the resolution of large fully coupled nonlinear algebraic systems to update the numerical solution. Therefore, several efforts have been done recently to reduce the complexity of such algebraic systems. In this work, we present a high-resolution well-balanced implicit scheme to solve numerically hyperbolic systems of balance laws. The scheme has a compact stencil which allows an efficient application of fast algebraic solvers like fast sweeping method. The scheme is formally second-order accurate in time and space, but space and time limiters are added to avoid unphysical oscillations in numerical solutions. The method produces nonlinear algebraic equations whose size is analogous to the first-order accurate fully implicit scheme. Moreover, the methods are well-balanced in the following sense: they preserve exactly all the continuous stationary solutions in the one-dimensional case whenever local equilibria can be obtained analytically and they are able to preserve a family of given stationary solutions in the two-dimensional case. The method is applied to Burgers’ equation with source term and the shallow water equations in 1D and 2D: several numerical experiments confirm these properties of the proposed scheme.
This work is inspired by the method proposed in [1], where planar G2 interpolants based on seventh-degree PH biarcs were constructed to interpolate prescribed points, tangent directions, and curvature values while enforcing a desired arc length. Our aim is to investigate the possibility of achieving analogous results using quintic PH curves. The construction of the quintic biarc model with prescribed arc length is presented, along with the derivation of the associated equations and the strategies used to reduce the degrees of freedom. Numerical experiments are reported to show the validity, limitations, and general applicability of the proposed quintic approach.
A network’s reliability is closely tied to its fault tolerance. When a connected network is fragmented by vertex failures, an increase in the number of small components aggravates network fragmentation and reduces reachability, thereby degrading fault tolerance. Thus, analyzing the abundance and structure of small components provides a useful measure of network reliability. This paper focuses on all possible structures of small components in an arbitrary regular interconnection network G after removing at most κr(G)−1 vertices, where κr(G) is the r-extra connectivity of G. Applying the results, we study the component reliability of (n, k)-bubble-sort network Bn,k and (n, k)-star network Sn,k. Besides, we propose an algorithm for calculating the minimum size of neighborhood of all small components by using Bn,k and Sn,k as application cases. Moreover, we verify the availability and effectiveness of the algorithm by conducting simulation experiments and analyzing their performance.
This research delves into the strategies employed by firms to compete through effective management of their online reputation and sales. We analyze a market comprising multiple firms operating within a platform, with each firm striving to enhance its ratings and sales performance. We transform the problem into a stochastic differential game model that incorporates online ratings and sales in the competition among multiple decision-makers. The resulting system of coupled HJB equations becomes high-dimensional, with the nonlinearity and coupling nature of the equations posing challenges for traditional numerical methods. To address this challenge, we propose a machine learning algorithm, which constitutes one of the main contributions of this paper. We train the neural networks using stochastic gradient descent, where spatiotemporal points are sampled randomly to enforce the partial differential equations together with the prescribed boundary and initial conditions. We reformulate the above problem as a deep learning task by leveraging random sampling to eliminate the need for grid construction. We theoretically demonstrate the convergence of the proposed machine learning algorithm, ensuring the validity and reliability of numerical results. Our results provide valuable insights for platform executives and participating companies regarding equilibrium strategies and investment decisions.
In this paper, we introduce and study the notions of statistical Cesàro summability and statistical deferred Cesàro summability of sequences of functions via the Laplace transform. First, we establish new inclusion theorems between these notions, supported by illustrative examples. We then apply these techniques to obtain the inverse Laplace transforms of sequences of functions. Using this framework, we derive significant results for solving higher-order ordinary differential equations and discuss applications to electrical circuit models, including the RL and LC circuits. Furthermore, we investigate the Laplace transforms of Fourier series by using Cesàro means of a sequence of positive kernel-based Laplace operators and establish some new results. Additionally, based on the proposed approach, we prove two new Korovkin-type theorems with the help of the trigonometric test functions 1, cos t, and sin t. An illustrative example involving a positive linear operator related to a new class of Fejér-type convolution operators is presented to demonstrate the applicability of the theoretical findings. Finally, the convergence behavior of the considered circuit models and operators is demonstrated graphically using MATLAB.
