This paper studied the characterization and existence of solutions to complex Hessian equations associated with a given weight. We provided a complete characterization of the Radon measures that could be represented as the complex Hessian measure of an Fm degrees chi (S2)-function, where chi was a decreasing weight function. Our main results provided both global and local characterizations of the range of the complex Hessian operator acting on these classes. Specifically, we demonstrated that the solvability of the equation Hm(cp) = & micro; in the class Fm degrees chi(S2) is equivalent to a certain functional inequality involving the measure & micro; and the weighted energy (m degrees chi. Furthermore, we demonstrate that this global condition could be localized: a solution existed globally if, and only if, for every point in the closure S2 of the domain, a solution existed in some neighborhood of that point.
We establish Lyapunov-type inequalities for Dirichlet boundary value problems driven by the modified discrete Helmholtz operator on finite balls of the integer lattice Z. This yields weighted inequalities for scalar problems and a spectral-radius criterion for coupled systems, as well as sharpness results. As an application, we study a weighted eigenvalue problem and obtain explicit two-sided bounds for the first eigenvalue by combining a Lyapunov-type lower estimate with a variational upper bound. Numerical illustrations are provided for localized weights.
We develop an objective Bayesian framework for estimating the common standardized mean difference in meta-analysis. To construct noninformative priors, we derive both probability matching and reference priors. Our analysis reveals that although general second-order matching prior does not exist, a valid version can be obtained when the sample sizes of the two arms in each study are equal. Among the reference priors evaluated, we find that both one-at-a-time and two-group reference priors satisfy the first-order matching criterion, whereas Jeffreys' prior does not. Simulation studies demonstrate that the proposed matching prior and the one-at-a-time reference prior yield accurate frequentist coverage probabilities, consistently outperforming Jeffreys' prior. Finally, the practical utility of this framework is validated through two real-world meta-analytic applications, underscoring its effectiveness for robust objective Bayesian inference in standardized mean difference models.
In many practical situations, distance measurements are affected by unavoidable inaccuracies due to instrumental limitations and external factors. Although such errors are often small, their accumulation may significantly impact the validity of mathematical models. This motivates the use of perturbed metric spaces as a natural framework to incorporate such imperfections into fixed point theory. In the current study, many theorems are established for three-point mappings contracting the perimeters of triangles under the structure of triple perturbed metric spaces. The findings presented here extend and consolidate numerous known results in fixed point theory by combining three-point contraction techniques with different perturbed metric structures. The use of three distinct perturbed metrics provides a more flexible and generalized contractive framework. Some examples are given showing that these satisfy the proposed conditions, while existing results cannot be applied to these examples, highlighting the wider applicability of the present results. Finally, we derived rigorous existence and uniqueness conditions that guarantee solutions for fractional differential equations and illustrated their relevance in modeling population dynamics, including factors such as memory effects and mortality in rabbit growth. A complementary numerical example further validates the rigorous results and demonstrates the practical applications of the iterative approximation scheme.
We study a class of nonconvex and nonsmooth optimization problems arising in sparse recovery and related applications, which are often addressed using iteratively reweighted [1 (IRL1)-type algorithms. Classical IRL1 methods are typically developed under a Lipschitz gradient assumption, which may limit their applicability. In this paper, we propose a generalized iteratively reweighted [1 algorithm with inertial extrapolation (GIRL1E), where the generalization is based on a co-coercivity condition imposed on the smooth component, thereby allowing a broader class of problems to be treated. By integrating inertial extrapolation into the generalized IRL1 framework, we establish global convergence of the proposed algorithm to a critical point of the objective function. In particular, we prove a sufficient descent property of an associated Lyapunov function and the convergence of the entire sequence without assuming convexity. Numerical experiments on compressive sensing-based signal recovery and image deblurring demonstrated that GIRL1E consistently achieves improved practical performance compared with existing IRL1 methods.
This paper analyzes the global dynamics of an Human Immunodeficiency Virus (HIV) delay model which incorporates cytokine enhancement and three saturated incidence rates. Based on the distinct thresholds of two reproduction numbers, we establish the existence of both immunityinactivated and immunity-activated equilibria. Furthermore, the global attractivity of all three equilibria is rigorously established through constructing Lyapunov functionals. Our simulations demonstrate the following: (i) enhanced virus-cell saturation demonstrates a superior efficacy over the saturation effect of inflammatory cytokines in driving systemic parameters toward Acquired HIV defense mechanisms; (iii) increasing the virus-cell saturation levels significantly neutralizes the adverse effects of immune saturation on disease progression; and (iv) time delays exhibit therapeutic benefits within an optimal range, with diminishing returns beyond this threshold. These results suggest both saturation parameters and time delays represent potential therapeutic targets for HIV treatment.
The increasing integration of renewable energy resources and active distribution networks has significantly increased the complexity of optimal power flow (OPF) problems in integrated transmission-distribution (T&D) power systems. To address these challenges, this paper proposes a novel tri-swarm adaptive hybrid optimizer (TAHO) that integrates particle swarm optimization (PSO), grey wolf optimizer (GWO), and jellyfish search (JS) within a unified adaptive optimization framework. The proposed method effectively balances exploration and exploitation to improve convergence stability and optimization accuracy. A multi-objective OPF model is developed to minimize generation cost, power loss, and voltage deviation under operational constraints. Experimental results on integrated IEEE 30-bus and IEEE 33-bus systems demonstrate that the proposed TAHO achieves superior performance with the minimum fitness value of 0.0008, faster convergence within 75 iterations, and the lowest standard deviation of 0.0005 compared with PSO, GWO, and JS. Benchmark evaluations further confirm the robustness and strong global search capability of the proposed framework for renewable-integrated smart grid optimization and real-time OPF applications.
The Dutch windmill graph Dqp is formed by q cycles of length p sharing a common vertex v0. In this paper, we derive closed-form expressions for the characteristic polynomials-specifically, the adjacency polynomial Phi A(Dqp, lambda), the Laplacian polynomial Phi L(Dqp, & micro;), and the signless Laplacian polynomial Phi L+(Dqp, v) -of this family of graphs. As a direct consequence, we compute the exact values of the graph energy, Laplacian energy, and signless Laplacian energy of Dutch windmill graphs.
The standardized precipitation index (SPI) is a foundational tool for drought assessment. However, its application is constrained by complex probability distribution selection and mandatory normal transformation. To overcome these limitations, this study introduces a systematic framework that uses the Anderson-Darling test to optimize distribution selection. This approach refines drought evaluation through the actual precipitation index (API). By using untransformed precipitation data, the API enables a more direct climatic assessment. Results demonstrated strong consistency between the API and SPI across upper northern Thailand. Both indices successfully detected the 2011 regional floods and the 2015-2016 El Ni & ntilde;o drought. Comparatively, the API demonstrated superior sensitivity to moisture saturation, particularly in Chiang Mai, Nan, and Phayao. Furthermore, spatial dynamic analysis using regular vine (R-vine) copulas identified Lampang as the primary regional hub. Lampang governs the central cluster (Chiang Mai, Lamphun, and Phrae) and mediates dependencies between Phayao and Chiang Rai. Nevertheless, localized geographic interactions create substantial concurrent extreme rainfall risks for the Chiang Mai-Lamphun pair. Because it preserves physical rainfall units, the API facilitates more actionable risk management than the SPI. Consequently, integrating R-vine copulas within the API framework is strongly recommended. This integration enhances spatial rainfall modeling, early warning systems, and adaptive water resource management, ultimately supporting climate resilience and sustainable development in vulnerable regions.
Digital image processing is essential in fields such as medical imaging, satellite imagery, and autonomous vehicles. Besides its applications, texture enhancement is vital for improving image visibility while preserving geometric structure. To improve visual clarity and image texture, we introduced a novel algorithm for texture enhancement. Initially, we determined coefficient inequalities of S & lowast;(phi H) which is a subclass of lambda- generalized Sakaguchi type functions, and examined upper bounds of second and third order Hankel determinants and obtained sharp results. We proposed a texture enhancement algorithm that utilized convolution masks derived from Hankel determinants with the pixels of segmented image. This approach provided a mathematical tool for computer-aided dermatological analysis, which was used to enhance lesion boundaries and structural visibility in dermascopic images. Image quality was evaluated using different quality metrics like contrast, correlation, energy, homogeneity, and entropy. The experimental results demonstrated uniform texture enhancement and improved edge preservation in all directions. Comparative analysis showed the efficacy of our proposed algorithm compared to existing methods reported in this study, proving it suitable to enhance image quality.
crater prediction is critical for optimizing charge design and ensuring safety in engineering practice; however, traditional theoretical models suffer from idealized assumptions, numerical simulations are hindered by empirical parameter calibration and high computational cost, and physical experiments are constrained by scalability and repeatability. To address these challenges, a novel Wavelet-Augmented Physics-Informed Neural Networks (WA-PINNs) framework for accurately and efficiently predicting the diameter and volume of blasting craters induced by cylindrical charges is proposed. By leveraging Starfield's superposition method to equivalently model cylindrical charges as a series of spherical sources, a partial differential equation (PDE) system governing the tensile stress field responsible for crater formation was established. Comprehensive validation against high-fidelity numerical simulations and field experiments at Baima Iron Mine demonstrated that the WA-PINNs (Dog) achieves superior accuracy in predicting crater diameter and volume compared to conventional PINNs, with relative errors as low as 0.01 and markedly reduced training time. The results confirmed that the proposed WA-PINNs (Dog) is a robust, efficient, and physics-consistent intelligent solution for complex blasting problems, offering significant potential for real-world blasting design optimization.
Capturing hesitation and uncertainty in decision-making (DM) problems remains a critical challenge in many real-world applications. The picture fuzzy set (P-FS) model significantly represents positive, neutral, and negative information, and interval-valued PFSs (IVP-FSs) support additional flexibility by enabling interval-based evaluations. However, these approaches may still lack the ability to effectively model multilayered uncertainty and fluctuations of expert assessments. To handle such situations, the proposed study established an innovative fuzzy approach that combines P-FSs, IVP-FSs, and a radius parameter within a unified framework. The proposed model was termed cubic circular picture fuzzy sets (CuCP-FSs). In this proposed approach, the combination of P-FSs and IVP-FSs was crucial for capturing multilayered uncertainty, and the radius parameter represents the degree of reliability or fluctuations around the P-FS information. Several fundamental set-theoretic operations were demonstrated for the proposed CuCP-FSs framework. Moreover, Bonferroni operational laws were presented, and the corresponding aggregation operators (AOs) were designed to efficiently integrate assessment information. The essential characteristics of the proposed AOs are also investigated to ensure their validity. A multi-criteria decision-making (MCDM) technique was constructed based on these developments within the environment of CuCP-FSs. The proposed technique was employed for applications in the selection of quantum computing technology, presenting the effectiveness and practicality of the proposed MCDM approach. Furthermore, a comparative analysis with several existing fuzzy approaches was examined, which reflects the robustness and reliability of the newly defined CuCP-FS model in presenting complex uncertain information.
In this work, an explicit formula in terms of the Hadamard product was derived for the solutions to fractional differential Stein matrix equations (FDSMEs), assuming that the coefficient matrices are separately diagonalizable. To begin with, we gave a general formula using the Kronecker product notation, which does not need any diagonalizability assumption to show the existence and uniqueness of the solution. In the case of diagonalizable coefficient matrices, the matrix equation became decoupled scalar Caputo fractional differential equations. Solutions to these differential equations were given explicitly in terms of two-parameter Mittag-Leffler functions and put together using the Hadamard product. The results for integer-order differential equations are obtained as special cases. Finally, we give three examples.
This work presents two new numerical methods for finding the zeros of nonlinear equations, based on the Adomian decomposition approach and the quadrature rule. Using a predictor-corrector scheme, the proposed iterative methods provide a high order of convergence and do not require second derivatives, thereby reducing computational time. The methods are tested on several functions, and their efficiency is demonstrated through comparisons with various established techniques. To support the numerical findings, graphical analyses are provided. Furthermore, a study of the basins of attraction is conducted to validate the stability of the proposed methods.
From the perspective of the dominated splitting, under certain assumptions, there existed finitely many closed sets A(1), A(2),. .., A(m) (with A(i) not equal A(j) for i not equal j) such that the support of any ergodic geometric measure coincided with one of them.
We developed a physics-informed neural network (PINN) framework for solving integrodifferential equations (IDEs), with particular emphasis on Volterra-type problems with exponentially decaying kernels. While PINNs provide a flexible approach for incorporating physical laws without mesh-based discretization, the treatment of convolution integral terms remains computationally demanding. To address this issue, we introduced internal variables for exponential kernels, transforming the original IDE into an equivalent system of differential equations, eliminating the need for explicit quadrature, and significantly reducing computational cost and memory requirements. The proposed method incorporates both differential and integral operators within the PINN framework. Numerical results demonstrate that the method maintains accuracy while significantly improving computational efficiency. It also extends naturally to inverse problems, where viscoelastic parameters are accurately identified from sparse observations.
We develop a Hilfer-adapted Taylor-type framework that is compatible with the natural initial trace of the Hilfer fractional derivative. For 0 < alpha < 1 and beta is an element of [0, 1], we introduce a shifted (alpha,beta)-fractional power series (FPS) with delta = alpha(1-beta) + beta - 1 and define Hilfer-Taylor coefficients via the regularized trace T-n(f) = (I(1-beta)(1-alpha)(D-alpha,D-beta)(n)f)(0+). This yields an explicit coefficient formula and a Taylor-type expansion in the normalized basis t(n alpha+delta)/Gamma(n alpha + delta + 1). Using the associated Mittag-Leffler eigenfunction kernel G(alpha,delta)(t, x) = x(delta)E(alpha,delta+1)(t(alpha)x(alpha)), we define fractional Appell-type sequences through a Hilfer-adapted generating identity and establish their main operational properties, including a lowering relation under D-x(alpha,beta) . As an application, we introduce Bernoulli-type objects and derive a convolution recurrence for the corresponding fractional Bernoulli numbers, recovering the classical case when (alpha,beta) = (1, 1).
In this paper, we use the minimax method and certain analysis techniques to establish the uniqueness of solutions for a class of (strongly) singular Kirchhoff equations with the Kohn-Laplace operator in the n-Heisenberg group and further deduce that the solution is cylindrically symmetric under some necessary structural conditions for a Kohn-Laplace equation with singularity.
This paper studies a coupled suspension bridge system featuring a unilateral nonlinear coupling of positive-part type and distributed delay feedback acting on the velocity components. Under appropriate assumptions linking the instantaneous damping coefficients and the delay kernels, we prove well-posedness and exponential stability via a carefully designed Lyapunov functional and multiplier technique. This work extends the existing theory by treating, for the first time in this setting, the combined effect of nonlinear unilateral coupling and distributed delay feedback.
This paper focuses on analyzing the dynamic model and the stability of the hydraulic servo system, where the random impulsive disturbances are fully considered. To address these issues and achieve the tracking control of the hydraulic support pushing system subject to random impulsive disturbances, we present the mathematical model of the hydraulic support pushing system and define the error variables to complete the design of the controller. Based on the Lyapunov backstepping method, the designed controller can guarantee the error variables converge to zero exponentially and defend random impulsive disturbances effectively, which enhances the robustness of the hydraulic support pushing system. Finally, the validity of the proposed control method is verified by two simulation examples.
