Search Results (1,685)

This paper studies the existence, regularity, and properties of normalized ground state solutions for the mixed fractional Schrödinger equations. For subcritical cases, we establish the boundedness and Sobolev regularity of solutions, derive Pohozaev identities, and prove the existence of radial, decreasing ground states, while showing nonexistence in the $L^2$-critical case. For $L^2$-supercritical exponents, we identify parameter regimes where ground states exist, characterized by a negative Lagrange multiplier. The analysis combines variational methods, scaling techniques, and the careful study of fibering maps to address challenges posed by competing nonlinearities and nonlocal interactions. Full article

Matrices have an important role in modern engineering problems like artificial intelligence, biomedicine, machine learning, etc. The present paper proposes new algorithms to solve linear problems involving finite matrices as well as operators in infinite dimensions. It is well known that the power method to find an eigenvalue and an eigenvector of a matrix requires the existence of a dominant eigenvalue. This article proposes an iterative method to find eigenvalues of matrices without a dominant eigenvalue. This algorithm is based on a procedure involving averages of the mapping and the independent variable. The second contribution is the computation of an eigenvector associated with a known eigenvalue of linear operators or matrices. Then, a novel numerical method for solving a linear system of equations is studied. The algorithm is especially suitable for cases where the iteration matrix has a norm equal to one or the standard iterative method based on fixed point approximation converges very slowly. These procedures are applied to the resolution of Fredholm integral equations of the first kind with an arbitrary kernel by means of orthogonal polynomials, and in a particular case where the kernel is separable. Regarding the latter case, this paper studies the properties of the associated Fredholm operator. Full article

The analytical solutions for groundwater inflow into tunnels are usually developed under the condition of circular tunnels. However, real-world tunnels often have non-circular cross-sections, such as arched, lens-shaped, or egg-shaped profiles. Accurately assessing water inflow for these diverse tunnel shapes remains challenging. To address this gap, this study developed a closed-form analytical solution for water inflow into a deeply buried arched tunnel using the conformal mapping method. When the tunnel circumference degenerates to a circle, the analytical solution degenerates to the widely used Goodman’s equation. The solution also showed excellent agreement with numerical simulations carried out using COMSOL. Based on the analytical solution, the impact of various factors on water inflow Q was further discussed: (1) Q decreases as the boundary distance $D^{\ast }$ increases. And the boundary inclination angle ($\alpha -\pi /2$) significantly affects Q only when the boundary is close to the tunnel ($D^{\ast }<20$); (2) Q increases quickly with the upper arc radius $r_1^{\ast }$, while it shows minimal variation with the change in the lower arc radius $r_2^{\ast }$. The findings provide a theoretical foundation for characterizing water inflow into arched tunnels, thereby supporting improved tunnel planning and grouting system design. Full article

In this paper, we investigate the generalized Hyers–Ulam stability of bi-homomorphisms, bi-derivations, and bi-isomorphisms in $C^{*}$-ternary algebras. The study of functional equations with a sufficient number of variables can be helpful in solving real-world problems such as artificial intelligence. In this paper, we build on previous research on functional equations with four variables to study functional equations with as many variables as desired. We introduce new bounds for the stability of mappings satisfying generalized bi-additive conditions and demonstrate the uniqueness of approximating bi-isomorphisms. The results contribute to the deeper understanding of ternary algebraic structures and related functional equations, relevant to both pure mathematics and quantum information science. Full article

We develop a symmetry-based variational theory that shows the coarse-grained balance of work inflow to heat outflow in a driven, dissipative system relaxed to the golden ratio. Two order-2 Möbius transformations—a self-dual flip and a self-similar shift—generate a discrete non-abelian subgroup of $\mathrm{PGL}(2,\mathbb{Q}\left(\sqrt{5}\right))$. Requiring any smooth, strictly convex Lyapunov functional to be invariant under both maps enforces a single non-equilibrium fixed point: the golden mean. We confirm this result by (i) a gradient-flow partial-differential equation, (ii) a birth–death Markov chain whose continuum limit is Fokker–Planck, (iii) a Martin–Siggia–Rose field theory, and (iv) exact Ward identities that protect the fixed point against noise. Microscopic kinetics merely set the approach rate; three parameter-free invariants emerge: a $62\%:38\%$ split between entropy production and useful power, an RG-invariant diffusion coefficient linking relaxation time and correlation length ${\mathcal{D}}_{\alpha }={\xi }^z/\tau$, and a $\vartheta ={45}^{\circ }$ eigen-angle that maps to the golden logarithmic spiral. The same dual symmetry underlies scaling laws in rotating turbulence, plant phyllotaxis, cortical avalanches, quantum critical metals, and even de-Sitter cosmology, providing a falsifiable, unifying principle for pattern formation far from equilibrium. Full article

A Stackelberg equilibrium–based Model Reference Adaptive Control (MSE) method is proposed for spacecraft Pursuit–Evasion (PE) games with incomplete information and sequential decision making under a non–zero–sum framework. First, the spacecraft PE dynamics under $J_2$ perturbation are mapped to a dynamic Stackelberg game model. Next, the Riccati equation solves the equilibrium problem, deriving the evader’s optimal control strategy. Finally, a model reference adaptive algorithm enables the pursuer to dynamically adjust its control gains. Simulations show that the MSE strategy outperforms Nash Equilibrium (NE) and Single–step Prediction Stackelberg Equilibrium (SSE) methods, achieving 25.46% faster convergence than SSE and 39.11% lower computational cost than NE. Full article

Digital maps have become important teaching and learning tools in education. However, limited research has examined the factors influencing learners’ acceptance of map-based online learning systems. This study proposes and validates an extended Technology Acceptance Model (TAM) that integrates two psychological constructs—habit and self-efficacy—into the original TAM framework to better explain students’ behavioural intention to use a map-based online learning system (Map-OLS). Structural equation modelling (SEM) was employed to analyse data from 812 participants with prior online learning experience. The results indicated that perceived ease of use (PEoU) and perceived usefulness (PU) had direct positive effects on the behavioural intention to use Map-OLS. PEoU positively affected PU and indirectly influenced behavioural intention to use Map-OLS via PU. Both habit and self-efficacy had significantly positive influences on PEoU and PU. Self-efficacy also directly influenced the behavioural intention to use Map-OLS. This study makes a theoretical contribution by extending and empirically validating TAM in the context of map-based learning environments, while also offering practical insights for designing more engaging and effective online learning systems. Full article

Conventional surface polishing trajectory generation relies on solving the plane trajectory equation in conjunction with the surface equation to obtain a surface polishing trajectory when the surface equation is known. However, in practical polishing processes, there exist equation-free surfaces which cannot be described by equations and the generation of polishing trajectories on equation-free surfaces cannot be realized by an equation-solving strategy. In this paper, a polishing trajectory generation method for equation-free surfaces modeled by meshes based on the trajectory mapping strategy is proposed. As the calculation process for the coverage area of polishing trajectories requires invoking surface curvature information, this paper proposes a mesh surface curvature fitting algorithm. Regarding the problem of non-uniformity in the coverage area of polishing trajectories caused by surface curvature fluctuations, an algorithm for adjusting the position of polishing trajectory points on mesh surfaces is proposed, which enables the mapped trajectory points to be adjusted according to the required overlapping coverage area of the adjacent polishing trajectories. The uniform coverage of multiple polishing trajectories for rotationally symmetric workpieces is achieved by the proposed trajectory point position adjustment algorithm. Through experiment and analysis, it is verified that the proposed algorithm for uniform coverage of polishing trajectories can obtain better polishing results with the same polishing time. Full article

We unify a known technique used for fixed points and coupled, tripled and N-tupled fixed points for weak monotone maps, i.e., maps that exhibit monotone properties for each of their variables. We weaken the classical contractive condition in partially ordered metric spaces by requiring it to hold only for a sequence of successive iterations, generated by the considered map, provided that it is a monotone one. We show that some known results are a direct consequence of the main result. The introduced technique shows that the partial order in the constructed Cartesian space is induced by both the partial order in the considered metric space and by the monotone properties of the investigated maps. We illustrate the main result, which is applied to solve a nonlinear matrix equation, following key ideas from Berzig, Duan & Samet. We present an illustrative example. We comment that a similar approach can be used to solve systems of nonlinear matrix equations. Full article

A hot compression simulation was conducted on the Al-7.62Zn-2.22Mg-0.90Cu-0.30Mn-0.09Er-0.13Zr alloy (7E97) within the temperature range of 300~460 °C and strain rate range of 0.001~10 s⁻¹ using a Gleeble-3500 hot simulator. A flow-stress constitutive equation and hot processing maps were established for the alloy, and the microstructural evolution of the alloy after hot deformation was investigated. It was found that the dominant dynamic softening mechanism of the alloy was dynamic recovery, accompanied by minor dynamic recrystallization. The optimal hot processing window for the alloy was determined to be in the ranges of 0.001~0.05 s⁻¹ and 350~410 °C. Full article

This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article

This paper introduces a novel class of generalized contractions, termed $(\alpha ,\eta ,(Q,h),\mathcal{L})$-contraction mapping, within the context of triple controlled metric-type spaces, extending the framework of fixed point theory in controlled structures. The proposed mapping is defined using $\alpha$-admissible and $\eta$-subadmissible functions, in conjunction with a control pair $(Q,h)$ of upper class of type I, and incorporates Wardowski’s function $\mathcal{L}$-contraction condition. Under suitable hypotheses, we establish both the existence and uniqueness of fixed points for this class of mappings. Several corollaries are derived as special cases of the main result. Moreover, we provide a nontrivial application by analyzing the solvability of a nonlinear equation involving powers of the sine function, thereby illustrating the utility of the developed theory. Full article

During the operation and service of a ship, its power system will affect the stability, reliability, and safety of the ship’s power system and the ship’s vitality if there are typical problems, such as unstable operation and vibration of the shaft system. If the tail bearing is not properly installed, it will lead to increased vibration at its support during operation, which will cause the propulsion system components to come loose and even produce destructive accidents. This paper combines the theory of multi-degree-of-freedom system dynamics to study the propulsion system vibration modeling technology based on the bearing–mounting error, analyze the mapping law between the bearing–mounting error and the shaft system vibration, construct a shaft system vibration model with the bearing–mounting error included, and analyze the influence of the bearing vertical mounting error and lateral mounting error on the vibration performance of the shaft system. This paper establishes the equations of motion of the shaft system with bearing–mounting errors and analyzes the relationship between the bearing vertical mounting errors and lateral mounting errors and the amplitude, speed, and acceleration of the paddle shaft system. The analyzed results show that the vibration response of the shaft system gradually increases with the increase in the bearing–mounting error. With the increase in the bearing vertical mounting error, the increase in vibration amplitude and the transient response of vibration acceleration in the vertical direction is larger than that in the horizontal direction, and the sensitivity of the transient response of vibration acceleration in the vertical direction to the bearing vertical mounting error is larger than that in the horizontal direction. With the increase in the bearing lateral mounting error, the increase in the vibration acceleration transient response value of the paddle shaft system in the horizontal direction is larger than that in the vertical direction, and the sensitivity of the vibration amplitude and vibration acceleration transient response to the bearing lateral mounting error in the horizontal direction is larger than that in the vertical direction. The bearing vertical installation error has a greater effect on the vibration of the paddle shaft system in the vertical direction than in the horizontal direction, and the bearing lateral installation error has a greater effect on the vibration of the paddle shaft system in the horizontal direction than in the vertical direction. The results of this paper can provide a theoretical basis and technical reference for the installation and calibration of ship propulsion system. Full article

In this article, we investigate the 3D additive-type functional equation. Next, we introduce the ternary hom-multiplier in ternary Banach algebras using the concepts of ternary homomorphisms and ternary multipliers. We first establish proof that solutions to the 3D additive-type functional equation are additive mappings. We further demonstrate that these solutions are $\mathbb{C}$-linear mappings. The final portion of our work examines both the stability and hyperstability properties of the 3D additive-type functional equation, ternary hom-multiplier, and ternary Jordan hom-multiplier on ternary Banach algebras. Our analysis employs the fixed-point theorem using control functions developed by Gǎvruta and Rassias. Full article

An analysis is presented of the irreversibility of flow and thermal phenomena in innovative streamlined structured packing of catalytic chemical reactors. The fundamental equations of irreversible thermodynamics defining entropy production as a result of flow friction and heat transport are formulated. The parameters describing the flow and heat transport in these equations are determined using the Computational Fluid Dynamics (CFD) methodology. Local entropy production due to flow friction and heat transport in the channel structure is plotted and compared with flow-temperature maps and relations for flow resistance, pressure gradient, and Nusselt number derived from CFD. The calculations were performed for three gas velocities: 0.3; 2.0, and 6.0 ms⁻¹. It was found that the entropy due to flow friction increases strongly with increasing gas velocity, while the entropy due to heat transport decreases with gas velocity, but to a limited extent. These opposing tendencies mean that there is always a minimum of the total entropy production (including these due to flow friction and heat transport), usually for moderate gas velocity. This minimum constitutes the optimum operating point of the reactor from the thermodynamic point of view. Full article