Search Results (67)

Monitoring critical transmission sections is essential for ensuring the operational security of power grids. This paper proposes a systematic method to identify critical transmission sections using the maximum flow–minimum cut theorem. The approach begins by representing the power grid as an undirected graph and identifying its hanging nodes. The network is then partitioned into several undirected subgraphs based on identified cut points. Each subgraph is transformed into a flow network according to actual power flow data. An efficient minimum cut set search algorithm is developed to locate potential transmission sections. To assess the risk under extreme conditions, a mixed-integer optimization model is formulated to select sections that are vulnerable to overload-induced tripping during N-K line outages caused by natural disasters. Simulation results on the IEEE RTS 24-bus and IEEE 39-bus systems validate the effectiveness and applicability of the proposed method. Full article

In this investigation, we introduce a new meromorphic operator defined by meromorphic univalent functions. A new class of meromorphic functions is introduced by this operator, which can generate several distinct subclasses depending on the values of its parameters. Within the framework of this class of functions, we obtain several significant algebraic and geometric properties, including coefficient estimates, distortion theorems, the radius of starlikeness, convex combination closure, extreme point characterization, and neighborhood structure. Our findings are sharp, offering accurate and significant insights into the mathematical structure and behavior of these functions. In addition, we present several applications of these results in fluid mechanics, like identifying stagnation points in vortex flows, predicting velocity decline in source/sink systems, and determining stability thresholds that protect crucial streamlines from perturbations, which demonstrates that the introduced operator and class characterize critical properties of 2D potential flows. Full article

This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive and bounded solutions through the application of the generalized mean-value theorem and Banach fixed-point theory methods. The fractional-order model is shown to be Ulam–Hyers stable, ensuring the model’s resilience to small errors. By employing the normalized forward sensitivity method, we identify critical parameters that profoundly influence the transmission dynamics of the fractional-order virus model. Additionally, the framework of optimal control theory is used to explore the characterization of optimal adaptive immune responses, encompassing antibodies and cytotoxic T lymphocytes (CTL). To assess the influence of memory effects, we utilize the generalized forward–backward sweep technique to simulate the fractional-order virus dynamics. This study contributes to the existing body of knowledge by providing insights into how the interaction between virus-to-cell and cell-to-cell dynamics within the host is affected by memory effects in the presence of optimal control, reinforcing the invaluable synergy between fractional calculus and optimal control theory in modeling within-host virus dynamics, and paving the way for potential control strategies rooted in adaptive immunity and fractional-order modeling. Full article

In this paper, we are interested in a class of critical fractional Choquard–Kirchhoff equations with p-Laplacian on the Heisenberg group. By employing several critical point theorems, we obtain the existence and multiplicity of nontrivial solutions under different perturbation terms. Due to the critical convolution term, the compactness condition may fail. To overcome this, we apply the concentration-compactness principle. The results in this paper can be viewed as complementary to the previous results under the conditions of $s=1$, $p=2$, and in the subcritical case. Full article

In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results. Full article

During offshore operations of jack-up platforms, the spudcan may experience sudden punch-through failure when penetrating from an overlying stiff clay layer into the underlying soft clay, posing significant risks to platform safety. Conventional punch-through prediction methods, which rely on predetermined soil parameters, exhibit limited accuracy as they fail to account for uncertainties in seabed stratigraphy and soil properties. To address this limitation, based on a database of centrifuge model tests, a probabilistic prediction framework for the peak resistance and corresponding depth is developed by integrating empirical prediction formulas based on Bayes’ theorem. The proposed Bayesian methodology effectively refines prediction accuracy by quantifying uncertainties in soil parameters, spudcan geometry, and computational models. Specifically, it establishes prior probability distributions of peak resistance and depth through Monte Carlo simulations, then updates these distributions in real time using field monitoring data during spudcan penetration. The results demonstrate that both the recommended method specified in ISO 19905-1 and an existing deterministic model tend to yield conservative estimates. This approach can significantly improve the predicted accuracy of the peak resistance compared with deterministic methods. Additionally, it shows that the most probable failure zone converges toward the actual punch-through point as more monitoring data is incorporated. The enhanced prediction capability provides critical decision support for mitigating punch-through potential during offshore jack-up operations, thereby advancing the safety and reliability of marine engineering practices. Full article

This paper is concerned with the non-existence of a global solution to the initial value problem of the inhomogeneous damped wave equation with a nonlinear memory term and a nonlinear gradient term. The critical exponent and formation of singularity of solution are closely related to symmetry in the study of blow-up dynamics for nonlinear wave equations, which provides a profound mathematical tool for analyzing the explosion of solutions within finite time. The proofs of blow-up results of solutions are based on the test function method, where the test function is variable separated. The influences of two types of damping terms, two types of nonlinearities, and an inhomogeneous term on exponents of the problem in blow-up cases are explained. It is worth pointing out that the inhomogeneous term in the problem is discussed with respect to the exponent $\sigma$ in three cases (namely, $\sigma =0$, $-1<\sigma <0$, and $\sigma >0$). As far as we know, the results in Theorems 1–4 are new. Full article

This paper is devoted to a class of general nonlocal fourth-order elliptic equation with concave–convex nonlinearities. First, using the $Z_2$-mountain pass theorem in critical point theory, we obtain the existence of infinitely many large energy solutions. Then, using the dual fountain theorem, we prove that the equation has infinitely many negative energy solutions, whose energy converges at 0. Our results extend and complement existing findings in the literature. Full article

Estimating Remaining Useful Life (RUL) is crucial in modern Prognostic and Health Management (PHM) systems providing valuable information for planning the maintenance strategy of critical components in complex systems such as aircraft engines. Deep Learning (DL) models have shown great performance in the accurate prediction of RUL, building hierarchical representations by the stacking of multiple explicit neural layers. In the current research paper, we follow a different approach presenting a Deep Equilibrium Model (DEM) that effectively captures the spatial and temporal information of the sequential sensor. The DEM, which incorporates convolutional layers and a novel dual-input interconnection mechanism to capture sensor information effectively, estimates the degradation representation implicitly as the equilibrium solution of an equation, rather than explicitly computing it through multiple layer passes. The convergence representation of the DEM is estimated by a fixed-point equation solver while the computation of the gradients in the backward pass is made using the Implicit Function Theorem (IFT). The Monte Carlo Dropout (MCD) technique under calibration is the final key component of the framework that enhances regularization and performance providing a confidence interval for each prediction, contributing to a more robust and reliable outcome. Simulation experiments on the widely used NASA Turbofan Jet Engine Data Set show consistent improvements, with the proposed framework offering a competitive alternative for RUL prediction under diverse conditions. Full article

In this paper, we investigate the existence of at least one weak solution for a nonlinear fourth-order elliptic system involving variable exponent biharmonic and Laplacian operators. The problem is set in a bounded domain $\mathcal{D}\subset {\mathbb{R}}^N$ ($N\ge 3$) with homogeneous Dirichlet boundary conditions. A key feature of the system is the presence of a Hardy-type singular term with a variable exponent, where $\delta \left(x\right)$ represents the distance from x to the boundary $\partial \mathcal{D}$. By employing a critical point theorem in the framework of variable exponent Sobolev spaces, we establish the existence of a weak solution whose norm vanishes at zero. Full article

This study explores the existence and multiplicity of weak solutions for a double-phase elliptic problem with nonlocal interactions, formulated as a Dirichlet boundary value problem. The associated differential operator exhibits two distinct phases governed by exponents p and q, which satisfy a prescribed structural condition. By employing critical point theory, we establish the existence of at least one weak solution and, under appropriate assumptions, demonstrate the existence of three distinct solutions. The analysis is based on abstract variational methods, with a particular focus on the critical point theorems of Bonanno and Bonanno–Marano. Full article

In this work, we develop and analyze a novel fractional-order framework to investigate the interactions among oxygen, phytoplankton, and zooplankton under changing climatic conditions. Unlike standard integer-order formulations, our model incorporates a Proportional–Caputo $\left(\mathbb{PC}\right)$ fractional derivative, allowing the system dynamics to capture non-local influences and memory effects over time. Initially, we rigorously verify that a unique solution exists by suitable fixed-point theorems, demonstrating that the proposed fractional system is both well-defined and robust. We then derive stability criteria to ensure Ulam–Hyers stability (UHS), confirming that small perturbations in initial states lead to bounded variations in long-term behavior. Additionally, we explore extended UHS to assess sensitivity against time-varying parameters. Numerical simulations illustrate the role of fractional-order parameters in shaping oxygen availability and plankton populations, highlighting critical shifts in system trajectories as the order of differentiation approaches unity. Full article

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ (zeros different from $s=-2m$, $m\in \mathbb{N}$) lie on the critical line $\sigma =1/2$. In this paper, combining the universality property of $\zeta \left(s\right)$ with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts $\zeta (s+i{t}_{\tau })$ approximating the function $\zeta \left(s\right)$. Here, $t_{\tau }$ denotes the Gram function, which is a continuous extension of the Gram points. Full article

This work proposes a novel multi-camera system-based method for relative pose estimation and virtual–physical collision detection for anchor digging equipment. It is dedicated to addressing the critical challenges of achieving accurate relative pose estimation and reliable collision detection between multiple devices during collaborative operations in coal mines. The key innovation is that the multi-camera multi-target system is established to collect images, and the relative pose estimation is completed by the EPNP (Efficient Perspective N-Point) algorithm based on multiple infrared LED targets. At the same time, combined with the characteristics of a roadheader and anchor drilling machine, AABB (Axis Alignment Bounding Box) with a simple structure and convex hull with a strong wrapping are selected to create the mixed hierarchical bounding box, and the collision detection is carried out by combining SAT (Split Axis Theorem) and GJK (Gilbert–Johnson–Keerthi) algorithms. The experimental results show that the relative pose estimation error of the multi-camera system is within 20 mm, with an angular error within 1.002°. The position error in the X-axis direction is within 1.160 mm, and the maximum deviation in the Y-axis direction is within 0.957 mm in the virtual–physical space. Compared with the existing methods, our method integrates digital twin technology, and has a simple system structure, which can meet the requirements of relative attitude estimation and collision detection between equipment in the process of heading face operation, and at the same time improve the system performance. Full article