Search Results (182)

The gradual developments in mathematical analysis have increased the demand for improving the efficiency of constraints and their validation, which have a significant contribution to resolving many real-world problems. There are many techniques that are used to modify the fractional inequalities, but all approaches are analytical. The adoption of Machine learning (ML) models is one practical approach to optimize the bounds of inequalities and their numerical validations. In this paper, we implement ML models to modify the bounds of Hermite–Hadamard-type inequalities, which consider the h-Godunova–Levin as a weight function. The classical development of inequalities by generalized fractional operators was never systematically studied to determine which value of the weight gives the best bound possible. We prove a sharp lower bound for the Hermite–Hadamard gap uniform over all admissible weights, show that it is achieved exactly by the classical convex weight, and characterize the optimal weights. To complement this analytical result, a parameterization of the weight function by a feedforward neural network is introduced, which is assumed to be admissible in the h-Godunova–Levin framework, and the validity of the resulting inequalities is proved. A Lipschitz-type error analysis is used to relate the approximation accuracy of the network to the tightness of the bound, and it is shown in numerical experiments that the network learns the optimal weight without any knowledge of the analytical weight’s form. The framework thus ensures the rigor of fractional convexity theory while offering a data-driven method of determining the optimal weight functions in cases where they are not explicitly known. The inherent symmetry properties of the fractional operators and the symmetric structure of the Hermite-Hadamard inequalities are preserved, while the neural network framework introduces a symmetry-breaking mechanism that enables the discovery of optimal weights. This dual perspective on symmetry—both preserving and breaking—provides a comprehensive understanding of the underlying mathematical structures and aligns perfectly with the scope of the journal Symmetry. This practical approach opens a new horizon for researchers and yields better results in the field of analysis. Full article

This study obtained the energy levels and examined the partition function (Z) of a quantum system described by an Eckart-like potential model. By adopting the Greene–Aldrich approximation scheme for the centrifugal term, the radial Schrödinger equation (SE) is solved and the analytic expression of the energy eigenvalues is obtained. The ro-vibrational Z is computed by explicitly incorporating the rotational quantum number, a feature often neglected or misapplied in many studies. This result is used to evaluate the key thermodynamic properties (TP), including the Gibbs free energy (G), entropy (S), and enthalpy (H). Numerical analysis reveals that the Z increases monotonically with temperature, while the G decreases in accordance with statistical thermodynamics. The S exhibits saturation-like behaviour at higher temperatures, while the H displays convex growth with increasing thermal energy. Parametric studies demonstrate that the Eckart-like potential allows for the controlled tuning of TP, with variations in the potential parameters, including the screening parameter, having distinct effects. The results generalise existing models, reproduce the Hulthén potential under specific conditions, show the effect of the rotational quantum number of TP, and provide new insights into the ro-vibrational statistical mechanics of exponential-type potentials. Full article

This paper establishes new Fejér and Hermite–Hadamard-type inequalities for functions of two variables whose mixed second-order partial derivatives satisfy coordinated GA-convexity or coordinated GA-quasi-convexity on a rectangle in the positive quadrant. Our main results are formulated for non-negative continuous weight functions that are not necessarily symmetric with respect to the geometric means of the interval endpoints, thereby extending the classical framework to genuinely asymmetric weights. However, to obtain explicit and sharp integral bounds in certain cases, we also employ a technical lemma that assumes a special symmetric setting where the weight function is symmetric on each coordinate with respect to $\sqrt{h_1{h}_2}$ and $\sqrt{k_1{k}_2}$. We clearly distinguish which theorems hold for general asymmetric weights and which depend on this symmetry condition. Our findings unify and extend numerous previously known results for both symmetric and non-symmetric weight functions. Full article

This paper establishes a weighted Simpson-type identity on the parameter domain associated with quantum integral operators. Using this identity together with Hölder’s inequality and the power mean inequality, we derive new estimates for classes of functions whose associated parameter-domain q-derivatives satisfy h-convexity assumptions. Additional bounds are obtained under boundedness and Lipschitz conditions. Applications to the s-moment of a random variable and to several special means are derived in the classical limit $q\to 1^{-}$. Full article

In this paper, we establish new Milne–Mercer-type inequalities via Atangana–Baleanu conformable fractional integral operators for differentiable functions whose derivatives in absolute value are h-convex. First, we derive a novel identity involving the Atangana–Baleanu conformable fractional integral operators. Then, by employing the properties of h-convex functions and fractional integral operators, several new inequalities of the Milne–Mercer type are obtained. The results presented in this paper extend and generalize various previously known inequalities, including classical Milne inequalities, Riemann–Liouville fractional integral inequalities, and conformable fractional integral inequalities. Moreover, several special cases are discussed to demonstrate the generality and applicability of the obtained results. The findings provide new refinements in the theory of fractional integral inequalities and contribute to the development of convex analysis within fractional frameworks. Full article

Robust disturbance rejection in voice coil motor (VCM) motion stages is often limited by model uncertainties and the difficulty of obtaining accurate plant inverses. To address this issue, this paper develops a data-based Youla parameterization method for designing a robust disturbance observer (DOB) without relying on an analytical plant model. Frequency response data from the VCM stage are measured directly under multiple operating conditions. The Youla parameter Q is expanded using a Laguerre orthogonal basis, and its coefficients are optimized by solving a convex problem that enforces H∞ robust stability and H₂ average tracking error constraints on a finite frequency grid. Experiments on a VCM motion stage demonstrate that the optimized Q filter effectively estimates and rejects electromagnetic noise and other disturbances. A total of 30 groups of data covering the full range of operating conditions were used for optimization, and 10 randomly designed experiments were conducted to validate the controller, with the maximum average error below 0.05%. Repetitive tests were carried out to verify the tracking performance for 1 Hz sinusoidal and triangular signals. The results show that the average RMSEs of the proposed method is 0.87% and 0.59%, respectively, which are lower than those of the ITAE-PID, ADRC and K₀ controllers. Finally, the robustness of the proposed method is further verified by analyzing the sensitivity function of the closed-loop system. Full article

Background: Some widely used wildland fire behavior models, like the Fire Area Simulator (FARSITE), propagate fire fronts by computing the front-normal velocity (spread rate) as a function of local inputs and the front-normal direction. Such models are sometimes observed to cause the collapse of crown fires into sharp wedge shapes that eliminate heading fire behavior. Aims: We set out to document this phenomenon and, more generally, understand the relationships between fire shapes and spread rate functions. Methods: The phenomenon is studied both mathematically and through simulation experiments. Non-smooth fire fronts are theorized mathematically by an Eikonal partial differential equation ($H(x,\tau ,D\tau )=1$), where the unknown $\tau \left(x\right)$ is the time-of-arrival function and the Hamiltonian $H(x,t,p)$ is positively homogeneous and possibly non-convex in p; convex analysis is used to study viscosity solutions in constant conditions. Results: We show that a fire spread model preserves the smoothness of fire fronts if and only if it is equivalent to using the Huygens principle. Nontrivially, this is equivalent to a convexity criterion on the inverse spread rate profile, which is then the polar dual of the Huygens wavelet; this corresponds to Hamiltonian–Lagrangian duality. The relevance of smoothness-destroying models to crown fire is debated. Exact analytical formulas are derived for fire growth in constant conditions. Conclusions: Our understanding of fire spread models is improved by solving the spread equations in more general ways than previously known. In particular, the collapse of heading crown fires into sharp shapes is now explained. Smoothness-destroying spread models cannot be simulated by algorithms based on travel time like cellular automata; their general well-definedness remains an open question. Fire modelers can use these findings to guide their search for improved crown fire models, and more generally to verify the accuracy of numerical implementations. Full article

In this paper, certain subclasses of meromorphic harmonic functions which are formulated using a q-differential operator are meticulously analyzed. Initially, two new subclasses $W_H^q(k;E,F)$ and $W_{\eta }^q(k;E,F)$ associated with the Janowski function with relevance to the idea of weak subordination are defined. These classes are further studied through their various analytical and geometric properties. Some of these explored properties include the necessary and sufficient coefficient condition, the radii of starlikeness, characterizations of extreme points, distortion estimation, closeness under convolution, and convex combination features. Additionally, the asymptotic behavior of the coefficients is also examined, and to express the findings, the Big-O, little-o, and asymptotic equivalency notations are used. These findings significantly represent the interaction between the growth, dominant terms, and limiting behavior of functions within these subclasses. Full article

Adversarial perturbations measured by ${\mathsf{\ell }}_p$ norms do not reflect key structural constraints in tabular security data, including anisotropic geometry, feature dependence, and distributional plausibility. We introduce a composite attacker cost functional $C_{\mathrm{atk}}(\mathit{x},{\mathit{x}}^{\prime })=\tau max\{0,m\left({\mathit{x}}^{\prime }\right)\}+{\lambda }_1{\mathit{\delta }}^{\top }{\mathit{G}}_{\perp }^{\left(\gamma \right)}\left(\mathit{x}\right)\mathit{\delta }+\lambda {\displaystyle \sum _j}{\omega }_j|{\delta }_j|+{\lambda }_2{\mathit{\delta }}^{\top }{\mathit{L}}_H\mathit{\delta }+{\lambda }_3\left[log{\widehat{f}}_1^{\left(ϵ\right)}\left(\mathit{x}\right)-log{\widehat{f}}_1^{\left(ϵ\right)}\left({\mathit{x}}^{\prime }\right)\right]+{\displaystyle \sum _{j\in supp\left(\mathit{\delta }\right)}}{c}_j+\beta {\left|\mathcal{M}(supp\left(\mathit{\delta }\right))\right|}^{\nu },$ which integrates a spectrally truncated geometric term, a graph-based coherence penalty, a smooth copula density barrier, and a superlinear module-spread term. Under spectral degeneracy of the legitimate-class covariance, we establish nonnegativity under density dominance, exact zero self-cost, lower semicontinuity, and ${\lambda }_3{\kappa }_K$-weak convexity of the continuous component on compact convex sets, for both affine and ${\rho }_m$-weakly convex scoring functions. These properties yield existence of constrained minimizers. The continuous component is locally Lipschitz, whereas the full functional is not due to the support-counting term. A component feasibility result shows that each term eliminates a distinct class of degenerate perturbations. Limiting regimes and refined evasion cost bounds are derived. An empirical instantiation on PHIUSIIL indicates that perturbations with identical ${\mathsf{\ell }}_2$ norm can incur costs differing by an order of magnitude. Full article

Uninterrupted traffic flow monitoring is a prerequisite for optimal resource allocation and minimizing vehicular emissions in smart cities. However, centralized traffic management architectures are highly vulnerable to single points of failure. When structural sensor malfunctions occur, the resulting network unobservability paralyzes dynamic signalization, triggering cascading traffic congestion, extended idling times, and severe greenhouse gas emissions. To address this cyber-ecological vulnerability, we propose the Hybrid Multi-Agent State Estimation (H-MASE) protocol, a fully decentralized decision-support framework designed from an applied systems reliability engineering perspective. By deploying PSAs and VLAs directly onto IoT-enabled edge devices at smart intersections, H-MASE leverages a hop-by-hop edge computing topology to collaboratively track macroscopic route flow dynamics. Mathematically, this distributed estimation process is formulated as a network-wide least-squares convex optimization problem, where local projection operators function as exact Distributed Gradient Descent steps to minimize the global residual sum of squares. The distributed consensus mechanism acts as a spatial variance reduction tool, effectively dampening measurement noise and stochastic demand fluctuations. Furthermore, we introduce an autonomous anomaly detection logic that isolates severe structural faults rapidly, which is mathematically structured to prevent false alarms under bounded disturbance conditions. Numerical simulations demonstrate that the protocol yields a highly resilient optimality gap (e.g., a Root Mean Square Error of merely 0.81 vehicles per estimated state) even under catastrophic hardware failures. Ultimately, H-MASE provides a robust, fail-safe data foundation for sustainable urban logistics and green-wave signalization, ensuring that smart cities maintain ecological resilience and optimal resource utilization under severe structural disruptions. Full article

The increasing penetration of photovoltaic (PV) systems in distribution networks has introduced new challenges in voltage regulation and energy loss mitigation, particularly under time-varying loading conditions. This paper presents a constrained multi-objective mathematical optimization framework for the optimal allocation and sizing of PV-STATCOM devices in radial distribution systems. The problem is formulated as a nonlinear optimization model that minimizes the daily energy losses over a 24 h operating horizon while satisfying network operational constraints, inverter capacity limits, and renewable penetration restrictions. To efficiently solve the resulting non-convex optimization problem, a metaheuristic algorithm based on the weighted mean of vectors (WMV) is employed. The WMV method integrates wavelet-based weighting mechanisms, mean-driven update rules, vector combination strategies, and a local refinement operator to balance global exploration and local exploitation within the feasible search domain. Constraint violations are handled through a penalty-based mathematical transformation of the objective function. The proposed framework is validated on the IEEE 33-bus and IEEE 69-bus distribution systems under realistic daily load variations. The numerical results demonstrate significant reductions in daily energy losses compared to differential evolution, particle swarm optimization, artificial rabbits optimization, and golden search optimization algorithms. Furthermore, convergence analysis confirms the robustness and computational efficiency of the WMV approach in solving large-scale constrained power system optimization problems. Full article

Some inequalities for products of Struve functions $H_{\nu }\left(x\right)$ when $\left|\nu \right|\le {\displaystyle \frac{1}{2}}$ are established using their infinite product formula as well as the arithmetic–geometric mean inequality. When these results are combined with some previously established results in the literature, they lead to interesting inequalities between $H_{\nu }\left(x\right)$ and $J_{\nu }\left(x\right)$, where $J_{\nu }\left(x\right)$ is the Bessel function of the first kind. Analogous results are also derived for $H_{\nu }^{\prime }\left(x\right)$ when $0\le \nu \le {\displaystyle \frac{1}{2}}$. Moreover, results concerning the log-convexity or log-concavity of certain functions involving $H_{\nu }^{\left(m\right)}\left(x\right)$, $m=0,1$ are also established. Full article

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type ($H$-type) inclusions are established for fuzzy-number-valued convex functions ($F$·$N\cdot V$-functions) and $F$·$N\cdot V$-coordinated convex functions. These results are obtained by employing $F$·$N\cdot V$-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known $H\cdot H$-type inequalities previously established for real-valued and interval-valued functions ($I\cdot V$-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted $H\cdot H$-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity. Full article

In this study, we examine a length preserving geodesic curvature difference flow for smooth strictly horocyclically convex simple closed curves in the hyperbolic plane ${\mathbb{H}}^2$. Given an initial curve ${\gamma }_1$ and a target curve ${\gamma }_2$ of the same hyperbolic length, we evolve ${\gamma }_1$ by a normal speed given by the difference of the reciprocals of geodesic curvatures evaluated at points with the same outward unit normal, together with a time-dependent scalar term $\mathrm{\Gamma }\left(t\right)$ chosen to preserve the hyperbolic length. Using Leichtweiβ’s hyperbolic support function and Howe’s curvature formula, the flow is reformulated as a quasilinear uniformly parabolic equation on ${\mathbb{S}}^1$ with a nonlocal term $\mathrm{\Gamma }\left(t\right)$. We prove short-time existence, uniqueness, and preservation of strict horocyclic convexity. Linearizing the support function equation at the target support function yields a uniformly elliptic operator whose kernel contains the infinitesimal isometry directions. Under a spectral gap assumption on a normalized slice transverse to the isometry orbit, we prove global existence and exponential convergence for initial data sufficiently close to the target curve. In the last section, this assumption is verified explicitly when the target curve is a geodesic circle. Full article

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak gradients on each convex quadrilateral element, while a one-to-one mapping is used to establish the approximation space for weak functions. Subsequently, based on the weak Galerkin spectral element approximation space defined on convex quadrilateral meshes, a Galerkin approximation scheme is formulated, and its well-posedness is then analyzed. Furthermore, numerical experiments are performed on arbitrary convex quadrilateral meshes of the square and L-shaped domains to explore the upper- and lower-bound approximations of numerical eigenvalues. Numerical findings indicate that the presented method not only obtains optimal orders of convergence with respect to both the mesh size and the polynomial degree, but also provides upper- and lower-bound approximations for the reference eigenvalues by proper choices of polynomial degrees in approximation spaces and parameters of the approximation scheme in both h-version and p-version weak Galerkin spectral element methods. This study offers new perspectives and methodologies for the high-precision numerical solution of eigenvalue problems in elliptic equations. Full article