Starting from pointwise gradient estimates for the heat semigroup, we study three characterizations of weak lower curvature bounds on metric graphs. More precisely, we prove the equivalence between a weak notion of the Bakry–Émery curvature condition, a weak Evolutionary Variational Inequality, and a weak form of geodesic convexity. The proof is based on a careful regularization of absolutely continuous curves together with an explicit representation of the Cheeger energy. We conclude with a brief discussion on possible applications to the Schrödinger bridge problem on metric graphs.
This paper presents a class of novel, high-order, A-stable defect correction schemes. These schemes are grounded in a newly developed, more versatile class of BDF approaches, which utilize Taylor expansions at time tn+β with β > 1 being an adjustable parameter. The ranges for the stability parameters of these newly proposed defect correction schemes are specified to ensure A-stability. Additionally, numerical experiments are given to illustrate the precision and robustness of the schemes when addressing stiff problems.
Forecasting systemic financial risks is crucial for effectively preventing and mitigating such risks. This paper introduces an integrated framework for systemic financial risk forecasting, with its core innovation being the novel combination of phase space reconstruction with delay parameterization (RDPM) and a radial basis function neural network (RBFNN). Based on stock trading data from 41 listed companies in the China A-share stock market, a complex financial network is constructed. The RDPM-RBFNN approach reconstructs the underlying dynamical system by embedding a univariate time series into a high-dimensional phase space, thereby offering distinct advantages in characterizing the nonlinear dynamics inherent in financial risk. Its principal merit lies in its interpretability: It enables the estimation of measures including the largest Lyapunov exponent and time delay parameters, which quantitatively capture the system’s chaotic properties and nonlinear evolutionary patterns. On the other hand, phase space reconstruction requires less data and exhibiting resilience to non-stationarity. It provides a rigorous framework for diagnosing chaotic dynamics in financial systems, thereby yielding deeper insights into the complex mechanisms governing risk propagation. Although banks typically carry higher leverage as primary lending institutions and are often assumed to be central to systemic risk, our findings show that in the current financial landscape of China, securities and insurance companies constitute the dominant sources of systemic risk, whereas banks show notable stability. This reversal of conventional risk hierarchies illustrates how systemic risk concentration shifts across financial subsectors along with developmental trajectories. The elevated risk contribution of securities and insurance firms during the sample period, driven by their rapid expansion and deepening network integration, calls for a differentiated, activity-based approach to monitoring, with particular emphasis on systemically significant non-bank financial intermediaries. We think that the proposed framework in this paper is transferable to a broad spectrum of financial and economic systems and can be extended to model systemic interdependence in other complex adaptive systems.
We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin $\varepsilon$-periodic perforated layers wit a thickness of order $\varepsilon$α with $\alpha \in(0,1)$. Therefore the thickness of the layer is large compared with its porosity. The aim is the derivation of effective models for $\varepsilon$ → 0, when the thickness of the layer tends to zero. For the flow problem, we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer. The transport problem is given by reaction–diffusion–advection problem with advective flow governed by the fluid velocity from the Stokes model. Furthermore, we treat different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with the Darcy velocity depending only on the derivative of the Darcy pressure in the vertical direction. The effective equation for the transport problem is again one of the diffusion advection-type including homogenized coefficients, and with advective flow given by the Darcy velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where, in the case of high diffusion in the horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit, we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii operator for thin perforated domains with arbitrary boundary conditions on the top/bottom, and the strong two-scale convergence for the microscopic solution of the transport equation, which is necessary to pass to the limit in the advective term. This strong convergence is established by using a Kolmogorov–Simon compactness argument.
Reliable prediction of financial market movements remains a challenging task due to high volatility, complex interdependencies, and sensitivity to external shocks. This study assessed the performance of advanced machine learning models, including long short-term memory (LSTM), gated recurrent unit (GRU), transformer networks, extreme gradient boosting (XGBoost), and deep multi-layer perceptron (DMLP), as well as proposes their ensemble combinations, in forecasting daily closing prices of five major stock indices (S&P 500, NASDAQ-100, Dow Jones Industrial Average, FTSE 100, and DAX). Results indicate that although all models achieved high predictive accuracy, profitability outcomes varied substantially across models and markets. Among single-model approaches, LSTM generally exhibited more stable positive returns in several indices, while other models showed pronounced variability depending on market conditions. Meanwhile ensemble strategies frequently ranked among the top-performing configurations, often matching or exceeding the performance of adaptive weighting schemes. Performance was strongly index-dependent, with S&P 500 and NASDAQ- 100 exhibiting comparatively stronger profitability, whereas FTSE and Dow Jones showed weaker and less differentiated results. These findings emphasize that statistical accuracy (e.g., RMSE, R2 metrics) alone is insufficient for profitable trading, underscoring the importance of financial performance metrics, such as total return, drawdown, and risk-adjusted measures, when evaluating predictive models.
In this paper, the (4+1)-dimensional Fokas equation is first reduced to a (2+1)-dimensional bilinear form, and then analyzed by an architecture-guided bilinear neural network method (BNNM) within the Hirota framework. Three representative network topologies, namely the “3-2-1”, “3- 3-1”, and “3-2-3-1” configurations, are used to organize trial functions and derive exact solutions through symbolic coefficient matching. As a result, several families of exact wave patterns are obtained, including lump solutions, lump–stripe-type solutions, lump–soliton-type solutions, and selected degenerate/high-amplitude structures. The comparison among different architectures shows that the hidden-layer depth and the activation composition affect the admissible ansatz forms and the morphology of the resulting interaction profiles for the reduced Fokas equation. The derived solutions are interpreted as prototype wave patterns and analytical benchmarks for localization, interaction, and modulation in multidimensional dispersive media, rather than as experimentally calibrated predictions. In addition, an auxiliary Duffing-based modulation diagnostic is used to visualize the irregular modulation of one selected high-amplitude profile; this comparison is qualitative and does not constitute a rigorous proof that the reduced Fokas equation itself is chaotic. These results show that the BNNM framework provides a structured symbolic route to construct and compare the exact interaction solutions of the reduced Fokas equation.
This paper is devoted to the study of mild solutions for the time-fractional Navier-Stokes equations, where the time derivative is interpreted in the Caputo-Hadamard sense. The Caputo-Hadamard derivative, which involves a logarithmic kernel, is particularly suitable for describing anomalous diffusion processes with ultra-slow dynamics. The main contribution of this work is threefold. First, by reformulating the equations as an abstract Cauchy problem and employing the Mittag-Leffler operator representations, we derived an integral formulation of mild solutions. Second, using Banach fixed point theorem with suitable decay estimates of the linear semigroup generated by the Stokes operator, we established the existence and uniqueness of both global (for sufficiently small initial data) and local (for arbitrarily large initial data) mild solutions in critical Lebesgue spaces. Third, we provided explicit logarithmic decay estimates for these solutions, which reflect the ultra-slow dissipation mechanism induced by the Caputo-Hadamard derivative.
Understanding and shaping the dynamics of neural architectures with uncertainty, memory, and spatial interactions is a fundamental problem in neural networks and adaptive systems. In particular, controlled destabilization plays an important role in promoting exploration, adaptability, and non-stationary behavior, yet remains far less studied than stabilization and convergence. In this paper, we investigated destabilizing dynamics in a class of space–time discrete fuzzy multidirectional associative memory (MAM) neural networks with time-varying delays and diffusion effects. Such networks integrate fuzzy rule-based representations, delayed feedback, and spatial coupling, and are relevant to adaptive control, associative memory, and multi-agent dynamical systems. We first established the existence of equilibrium states by using topological degree theory, which provides a rigorous foundation for the subsequent analysis. Then, by designing localized Dirichlet boundary feedback mechanisms and constructing novel discrete Lyapunov–Krasovskii functionals with delay-dependent double-sum terms, we derived verifiable sufficient conditions for global asymptotic and exponential anti-stabilization. These results characterize how diffusion intensity, fuzzy parameters, and self-inhibition coefficients influence destabilizing behavior and determine the rate of divergence from equilibrium. The proposed framework provides new theoretical insights into anti-stabilization dynamics in discrete spatiotemporal fuzzy neural networks. Numerical examples further support the theoretical analysis and demonstrate the effectiveness of the proposed approach.
An enhanced numerical technique for solving the Lane–Emden equation is developed using the spectral homotopy analysis method (SHAM) with Chelyshkov polynomials and Gauss–Lobatto collocation nodes. The numerical results, including absolute errors and comparisons with the exact solution, demonstrate the convergence behavior and accuracy of the proposed approach. The results show that the Chelyshkov-based SHAM provides an effective framework for the numerical treatment of Lane–Emden-type problems.
Temporal metamaterials are artificially manufactured materials with time-dependent material properties that exhibit interesting phenomena when waves propagate through them. The propagation of electromagnetic waves in such time-varying dielectric media is governed by Maxwell’s equations, which lead to wave equations with temporal highly oscillatory coefficients for the electric and magnetic fields. In this study, we analyze the effective behavior of electromagnetic fields in time-varying metamaterials using a formal two-scale asymptotic expansion. We provide a mathematical derivation of the effective equations for the leading-order homogenized solution, as well as for the first- and second- order corrections of the effective solution. While the effective solution and the first-order correction are governed by local material laws, we reveal a nonlocal constitutive relation for the second-order corrections. Special attention is also paid to temporal interface conditions through initial values of the homogenized equations. The results provide a mathematically justified framework for the effective description of wave-type equations of time-varying media, applicable to models in optics, elasticity, and acoustics.
Our aim of this paper was to introduce a new construction of probabilistic Hermite polynomials based on moment generating functions. By using this generating function, we derived several new relations and formulas among the aforementioned polynomials and other types of probabilistic special number sequences and polynomials, such as the probabilistic Stirling numbers of the second kind, probabilistic Bernoulli polynomials of higher order, probabilistic Bernstein polynomials, and probabilistic Euler polynomials of higher order. By selecting special random variables, including Poisson, Uniform, Gamma, Geometric, Exponential, and Normal random variables, we showed that the generating function of probabilistic Hermite polynomials yields distinct and unique generating functions, which lead to new relations among other types of special numbers and polynomials, as presented in the application section of this paper.
Interaction dynamics among species yield to Allee effects, spatial memory, and nonlocal competition. When predators are not subject to the Allee effect, the coexistence equilibrium point remains locally asymptotically stable despite nonlocal competition. However, in the presence of the Allee effect, nonlocal competition leads to the destabilization of the coexistence point. Moreover, the model will undergo stability switches, Hopf bifurcation, and Turing bifurcation, and induced complex dynamics will appear. It is also found that when the memory diffusion is small, it has no effect on the stability switches; as it increases, only the nonlocal model exhibits the spatially inhomogeneous Hopf bifurcation. When it exceeds the maximum threshold, this phenomenon occurs in both models. Furthermore, when the memory diffusion coefficient exceeds the threshold, the stability range of the coexistence equilibrium will decrease in both the local and nonlocal models. Finally, numerical simulations verify the theoretical results.
As financial markets have increasingly exhibited heterogeneity across participants, complex interactions among them, and out-of-equilibrium dynamics, have provided growing interest in agent-based modeling (ABM). Unlike traditional representative-agent models, ABMs generate macro-level market behavior from the bottom up through interactions among adaptive agents. In this paper, we review the development and applications of ABMs in financial markets and provide a structured perspective on the growing literature. Specifically, we reviewed three key design axes-agent heterogeneity, market mechanism fidelity, and interaction topology-and discussed how different modeling assumptions shape the ability of ABMs to reproduce stylized financial market phenomena. We also discussed two emerging domains in which ABMs are increasingly applied: Simulations using LLM-based agents and modeling approaches in green energy finance and climate-related financial systems. Here, we identified the current boundaries of financial ABM research and promising directions for future research.
Usually, convolutional neural networks (CNNs) reduce the features extracted from convolutional layers by adopting the maximum value or arithmetic average method, which is called the pooling process. However, these two pooling methods overlook the spatial dependencies between image features. The Choquet integral is a nonlinear integral, which addresses this limitation by incorporating weighted aggregation via a fuzzy measure, enabling it to effectively model interactions between variables. This capability is essential for processing data with complex dependency structures. Therefore, this paper proposes a novel Choquet integral pooling method for feature extraction in the pooling layer. By conducting experiments on datasets, we demonstrated the effectiveness of the proposed method and compared it with existing techniques. The experimental results indicated that for cases where image features are less prominent and exhibit spatial dependencies, the Choquet integral pooling outperforms traditional pooling methods and demonstrates greater robustness. To ensure the rationality and validity of applying the Choquet integral in CNNs, we conducted a more in-depth study of the Choquet integral based on a copula, that is, the CC−integral, including its averaging, idempotence, translation invariance, and positive homogeneity. This research not only enriches the application fields of the Choquet integral but also provides new theoretical support and technical paths for the research of neural networks.
This paper proposed a multiscale differential principal-feature (MDPF) architecture for portfolio optimization in which the central contribution is an interpretable feature-extraction layer rather than a claim of unconditional benchmark dominance. The model combines the discrete wavelet transform (DWT), ordinary differential equation (ODE) identification, and principal component analysis (PCA) to transform rolling asset-return histories into low-dimensional dynamic signatures that are subsequently mapped into constrained portfolio overlays. In the U.S. mega-cap equity universe, the strongest MDPF overlay obtained a Sharpe ratio of 1.117 with average ℓ1 turnover of 0.132, compared with 1.138 for the equal-weight benchmark and 1.187 for raw mean–variance with substantially higher ℓ1 turnover of 0.464. In the global multi-asset ETF universe, the strongest MDPF overlay obtained a Sharpe ratio of 0.801, close to the equal-weight benchmark at 0.804 but below raw mean–variance at 0.927, again with lower ℓ1 turnover than the raw mean–variance allocation. These results indicated that the proposed architecture is most useful when it is used to extract structured multiscale dynamic information and to express that information as controlled active tilts around a stable base portfolio. Supplementary ablation, statistical, regime, sensitivity, ODE-diagnostic, PCA-interpretability, feature-to-allocation, optimization-refinement, and transaction-cost diagnostics further clarify why the representation is economically meaningful while showing that benchmark dominance is not universal and that higher-order nonlinear components require further refinement. Accordingly, the paper positions MDPF as a transparent feature-extraction framework for dynamic portfolio construction and as a basis for future work on regime-aware multiscale allocation.