Related Equations Research Articles

Application of Differential Equations on the Ricci Curvature of Contact CR-Warped Product Submanifolds of S2n+1(1) with Semi-Symmetric Metric Connection

In this paper, we explore the uses of Obata’s differential equation in relation to the Ricci curvature of an odd-dimensional sphere that possesses a semi-symmetric metric connection. Specifically, we establish that, given certain conditions, the underlying submanifold can be identified as an isometric sphere. Additionally, we investigate the impact of specific differential equations on these submanifolds and demonstrate that, when certain geometric conditions are met, the base submanifold can be characterized as a special type of warped product.

Enhanced Oscillation Criteria for Non-Canonical Second-Order Advanced Dynamic Equations on Time Scales

This study aims to establish novel iterative oscillation criteria for second-order half-linear advanced dynamic equations in non-canonical form. The results extend and enhance recently established criteria for this type of equation by various authors and also encompass the classical criteria for related ordinary differential equations. Our methodology involves transforming the non-canonical equation into its corresponding canonical form. The inherent symmetry of these canonical forms plays a pivotal role in deriving our new criteria. By employing techniques from the theory of symmetric differential equations and utilizing symmetric functions, we establish precise conditions for oscillation. Several illustrative examples highlight the accuracy, applicability, and versatility of our results.

Analytical Second-Order Extended Kalman Filter for Satellite Relative Orbit Estimation

This study considers a relative orbit estimation problem wherein an observing spacecraft navigates with respect to a target space object at a large separation distance (several kilometers) using only the bearing angles obtained by a single onboard camera. Generally, the extended Kalman filter (EKF), which is based on linear relative motion equations such as the Clohessy–Wiltshire equation, is used for the relative navigation of satellites. The EKF linearizes the estimation error around the current estimate and applies the Kalman filter equations to this linearized system. However, it has been shown that nonlinearities of the orbit determination problem can make the linearization assumption insufficient to represent the actual uncertainty. Therefore, an analytical second-order extended Kalman filter (ASEKF) for relative orbit estimation is proposed in this study. The ASEKF, to sequentially estimate the relative states of satellites and their associated uncertainties, is formulated based on a second-order analytic relative-motion equation under J2-perturbtation, which can overcome the deficiencies of existing approaches that mainly focus on applications in two-body, near-circular, and linearized orbit dynamics. Numerical results show that the proposed method provides superior robustness and mean-square error performance compared to linear estimators under the conditions considered.

Synchronous flying-around control for uncontrolled tumbling spacecraft

The problem of synchronous flying-around control for uncontrolled tumbling target spacecraft is investigated in the paper. Firstly, the relative motion equations of the chaser and the target is established based on the line of sight (LOS) coordinate system, and the scenario of the chaser flying-around uncontrolled tumbling target is analyzed. Secondly, the attitude motion model of the uncontrolled tumbling target is established, and the attitude of the target is described by the Modified Rodriguez Parameters (MRPs). The synchronous flying-around problem is transformed into a two-dimensional control problem in the instantaneous rotation plane of LOS (IRPL). The sliding mode controller is designed to control the LOS direction and vertical LOS direction of the chaser in the IRPL, and the stability of the controller is analyzed. In order to verify the effectiveness of the synchronous flying-around control method, three simulation scenarios are designed. The simulation results verify the effectiveness of the proposed method.

Counterfactuals in fuzzy relational models

Given the pressing need for explainability in Machine Learning systems, the studies on counterfactual explanations have gained significant interest. This research delves into this timely problem cast in a unique context of relational systems described by fuzzy relational equations. We develop a comprehensive solution to the counterfactual problems encountered in this setting, which is a novel contribution to the field. An underlying optimization problem is formulated, and its gradient-based solution is constructed. We demonstrate that the non-uniqueness of the derived solution is conveniently formalized and quantified by admitting a result coming in the form of information granules of a higher type, namely type-2 or interval-valued fuzzy set. The construction of the solution in this format is realized by invoking the principle of justifiable granularity, another innovative aspect of our research. We also discuss ways of designing fuzzy relations and elaborate on methods of carrying out counterfactual explanations in rule-based models. Illustrative examples are included to present the performance of the method and interpret the obtained results.

Persistent Summer North Atlantic Jet Variability: Dynamical Feedbacks and Model‐Observation Discrepancies

AbstractPersistent fluctuations in the latitudinal position of the North Atlantic (NATL) jet stream are closely linked to extreme weather anomalies, particularly over Europe. Accurately representing jet stream persistence in climate models is essential for enhancing the reliability of future regional extreme weather projections. This study investigates the summer persistence of the NATL jet stream latitudinal fluctuations and the Summer North Atlantic Oscillation (SNAO) using data from CMIP6 models and ERA5 reanalysis. Our findings reveal that CMIP6 models generally overestimate persistence compared to ERA5 during the historical period. Utilizing the relative vorticity tendency equation, we assess the strength of the eddy‐mean flow feedback and identify a positive relationship between persistence and feedback strength across models. We also found that a large fraction of the intermodel spread in precipitation persistence is associated with differences in the persistence of jet fluctuations.

Development of Potential Slip Surface Identification Model for Active Deep-Seated Landslide Sites: A Case Study in Taiwan

Identifying locations of landslide slip surfaces provides critical information for understanding the volume of landslides and the scale of disasters, both of which are essential for formulating disaster preparedness and mitigation strategies. Based on hydrogeological survey data from 24 deep-seated landslide-prone sites in Taiwan’s mountainous regions, this study developed the hydraulic conductivity potential index (HCPI) using principal component analysis to quantify the hydraulic properties of disturbed rock formations with six geological factors. Then, regression analysis was performed to construct a permeability estimation model for the geological environment of landslides. Finally, the established model was utilized to develop an identification method for potential slip depths in landslide-prone sites. Results indicated a strong relation between HCPI and hydraulic conductivity with a determination coefficient of 0.895. The relation equation confirmed that the data it generated concerning the depths of significant changes in hydraulic conductivity could be used to identify potential slip surfaces. Additionally, this study successfully established a rule for identifying potential slip zones by summarizing data concerning the generated hydraulic conductivity profiles, stratigraphic lithology, existing inclinometer slip depth records, and groundwater level of landslide sites. This identification method was then applied to predict potential slip depths for ten landslide sites where slip surfaces have not yet occurred. These findings offer a new alternative to having early information on potential sliding depths for timely disaster management and control implementation.

Repercussion of quadratic density variation on nonlinear mixed convective Darcy–Forchhimer Al2O3, Cu\\EG flow over an inclined plate with exponential heat source and Arrhenius equation

Generating and controlling energy at high temperatures is extremely challenging in solar energy applications with hybrid nanofluids (HNFs). This challenge is tackled with nonlinear changes in density with temperature and concentration. Hence, a nonlinear Boussinesq approximation is taken on Darcy – Forchhimer hybrid flow containing aluminium oxide ( A l 2 O 3 ) and copper ( Cu ) nanoparticles with base fluid Ethylene Glycol (EG) over an inclined plate at an angle of ξ = π 6 . The physical model is supported by a space-based exponential heat source (EHS) along with a temperature-based heat source (THS) in the energy equation and Arrhenius equation in diffusion relation with suction and blowing on the wall. The regulating equations are transformed into standard differential equations using the similarity conversions, and these equations are subsequently solved in MAPLE 2022. The thermal field is affected positively by EHS and THS parameters, which fulfil the purpose of including these parameters in the flow model. The rate of heat transport increment for Q E parameter by 1.04% for HNF whereas 1.06% for NF when Su > 0 . Also, Su < 0 heat transfer rate increases by 3.68% for HNF whereas 3.75% for NF. As A l 2 O 3 , Cu ∖EG HNF move from suction to injection, the Nusselt number reduces by more than 50%.

Tailoring intermetallic compound formation within laser weld brazed joints using a novel heat input approach

The challenge of intermetallic compound (IMC) embrittlement, resulting from thick IMC layers at the braze/substrate interface, and the lack of a clear strategy to manipulate IMC formation, has hindered the development of reliable laser weld brazing (LWB) processes. This study addresses these challenges by introducing a novel approach to IMC manipulation during LWB of thin-gauge Zn-coated steel with Si-bronze filler on a double-flanged lap joint. By shifting IMC formation from the interface towards the interior region of the braze, the research mitigates embrittlement by developing a new IMC category, termed surrounded interface-IMCs (SI-IMCs), distinct from traditional interface-IMCs (I-IMCs). The study proposes a Wire-adjusted heat input strategy to optimize brazing conditions, introducing a relative heat input equation (HIRelative) that correlates with various brazing defects and IMC formation. The generic scientific contribution of this work lies in identifying a critical HIRelative value of 32 J/mm for defect-free brazing, with an additional threshold of 12.44 J/mm above this level to promote a high density of SI-IMCs, occupying up to 38.2 ± 16.9 % of the braze cross-sectional area. These SI-IMCs, characterized by a shell-like Fe-Si layer and a bulky (Fe-rich)-Cu eutectic phase, enhance the mechanical performance of the brazed joints. Furthermore, this study reveals the novel role of Mn segregation in creating diffusion channels for Fe-Si IMC development, advancing the scientific understanding of IMC formation. Visualization through digital image correlation (DIC) during tensile testing showed that increasing the SI-IMC area fraction from 1.2 ± 2.4 % to 38.2 ± 16.9 % resulted in a 14 % increase in tensile peak load and a 350 % increase in ductility. This highlights the critical role of SI-IMCs in improving the strength and ductility of LWB joints, offering a new pathway for enhancing the performance of brazed structures.

A recursive algorithm for dynamics of planar multibody systems with frictional unilateral constraints

The contact impact problem in planar multibody systems can be efficiently solved by formulating it as the linear complementarity problem, which requires a complex modeling process. To simplify the process, a recursive algorithm for the dynamics of planar multibody systems with frictional unilateral constraints is proposed based on the reduced multibody system transfer matrix method. Firstly, the contact forces of frictional unilateral constraints are integrated into the recurrence relations of system components using Lagrange multipliers. Subsequently, the relative motion equations of the contact positions are discretized in time, which are then utilized to describe the linear complementarity problem for systems. The dynamics of the system is solved by the Moreau time-stepping method with the recursive method. Finally, the proposed algorithm was validated using the woodpecker toy and used to model a slider-crank mechanism with clearance, which shows its characteristics of facilitating modeling, universal, and highly programmable. This recursive algorithm provides an effective tool for solving non-smooth planar multibody systems while extending the application of the multibody system transfer matrix method.

Study on the influence of heave plate on energy capture performance of central pipe oscillating water column wave energy converter

This paper proposes a novel central pipe oscillating water column (OWC) wave energy converter (WEC) with a heave plate and investigates the influence of heave plate on energy capture performance of the OWC WEC through numerical methods. First, dynamic equations are established for floater and water column. Then, energy capture performance of the WEC is computed by solving relative motion equations using numerical methods. Wave tank experiments are conducted to validate the proposed numerical models. Finally, the influences of heave plate opening diameter and connecting column length on energy capture performance are investigated based on the proposed numerical models. The results indicate that increasing heave plate opening diameter decreases both the maximum capture width ratio (CWR) and response period. And increasing connecting column length widens the response bandwidth. When the heave plate opening diameter decreases from 350 mm to 0, the maximum CWR increases from 30.05% to 41.68% and the response period increases from 1.28 s to 1.72 s. When the connecting column length increases from 225 mm to 600 mm, the response band extends by 0.1 s. The proposed device and the obtained analysis results are of great significance for broadband and efficient wave energy generation.

Minimal solutions of fuzzy relation equations via maximal independent elements

Fuzzy relation equations (FRE) are a useful formalism with a broad number of applications in different computer science areas. Testing if a solution exists and, if so, computing the unique greatest solution is straightforward. In contrast, the computation of minimal solutions is more complex. In particular, even in FRE with a very simple structure, the number of minimal solutions can increase exponentially. However, minimal solutions are immensely useful since, under mild conditions, they (together with the greatest solution) allow one to describe the entire space of solutions to an FRE. The main result of this work is a new method for enumerating the set of minimal solutions. It works by establishing a relationship between coverings of FRE and maximal independent elements of (hyper-)boxes. We can thus make efficient enumeration methods for maximal independent elements of (hyper-)boxes applicable also to our setting of FRE, where the operator considered in the composition of fuzzy relations only needs to preserve suprema of arbitrary subsets and infima of non-empty subsets. More specifically, we thus show that the enumeration of the minimal solutions of an FRE can be done with incremental quasi-polynomial delay.

A Quantization for the Toy Model of Black Hole and a Suggestion to the Unification Theory

In my previous paper about the correlations between quantum mechanics and the four different natural forces, I suggested that there may be a fifth force or even the existence of a new particle. However, it remains a mystery to physicists, even after nearly a hundred years, that they cannot unify these four natural forces. This may be because there are indeed too many variables for them to consider. Also, there is a possibility that quantum mechanics may coexist with quantum field theory. In the present paper, this author proposes that there may be a misconception in the computational equation for relative time or gaps in the system for measuring time between quantum mechanics and general relativity. Hence, one may still not be able to unify these natural forces. This author suggests that we may need to rewrite parts of quantum mechanics – the Schrödinger equation – or even the general relativity equation. Additionally, this author proposes that there may be a bridge equation converting between quantum mechanics and general relativity. Moreover, this author has employed the non-trivial zeta zeros to simulate the black hole or the so-called black hole toy model. In practice, there may be an electromagnetic field surrounding the boundary of the black hole, as well as the existence of a continuum along the boundary contour. This author hopes that, in such a case, we may decode those high-frequency electromagnetic waves into useful information and take a further step toward verifying Stephen Hawking’s famous theory on black hole radiation and information entropy. In fact, this author has used the HKLam statistical model theory to express the electromagnetic field energy-stress tensor (with the possibility of quantization) to analogically establish a quantized model for the Einstein Gravitational Field Equation. Hence, the problem of quantum gravity may then be solved. Last but not least, this author also expects that humanity may finally find a way to unify quantum mechanics and general relativity through modifications to the current quantum gravity theory, such as my proposed bridge-converting equation, etc.

Coupling Dynamics Study on Multi-Body Separation Process of Underwater Vehicles

Based on the Newton-Euler method, a coupling rigid-body dynamics model of a Trans-Medium Vehicle (TMV) separating from an Unmanned Underwater Vehicle (UUV) has been established. The modeling is based on the “holistic method” and “Kane” ideas respectively, so that most of the equations can be derived without considering the internal forces between the two bodies. The separation propulsion force, which is an internal force, only appears in the relative glide dynamics equation of the TMV along the axis of the separation tube that is installed on the UUV. This greatly reduces the workload of modeling and derivation. The UUV works entirely underwater, while the hydrodynamic shape of the TMV changes continuously during the process of the TMV separating from the UUV. Therefore, accurate hydrodynamic calculations for the UUV and TMV are the basis of numerical resolution for the two rigid bodies’ coupling dynamics model in water. A large number of numerical simulations was conducted using CFD methods to investigate the hydrodynamic performance of the UUV and TMV under various conditions. These simulations aim to establish a hydrodynamic database, and accurate hydrodynamic models were developed through fitting methods and online interpolation. In the process of solving the coupling dynamics of two bodies, the hydrodynamic model is used to calculate the hydrodynamic force experienced by the UUV and TMV. This balances the accuracy and efficiency of a numerical simulation. Finally, numerous simulations and comparative analyses were conducted under various operational conditions and separation parameters. The simulation results indicate that the impact of TMV separation on the motion state of the UUV becomes more prominent with smaller UUV to TMV mass ratios or deeper TMV separation depths. This effect can further influence the stability control of the UUV. The coupling rigid body dynamics analysis method established in this paper provides a fast and effective prediction method for use during the scheme design and separation safety evaluation phases of creating UUV-TMV systems.

Research on the Stability of Salt Cavern Hydrogen Storage and Natural Gas Storage under Long-Term Storage Conditions

The stability of salt cavern storage during prolonged operation is a crucial indicator of its safety. This study focuses on an operational underground gas storage facility, conducting comparative numerical simulations for the storage of natural gas and hydrogen. We investigated the evolution of stability for natural gas and hydrogen storage under long-term storage conditions. The main conclusions are as follows: (1) A new equation for stress equilibrium and constitutive relations are derived. (2) At the same storage pressure, the effective stress at the same position in the interlayer is greater for hydrogen storage compared to natural gas storage, signifying a higher level of danger. (3) At the same storage pressure, the displacement at the cavity top for hydrogen storage is greater than that for natural gas storage. The displacement difference between the two is greatest at 9 MPa, amounting to 0.026 m. (4) Due to hydrogen’s lower dynamic viscosity and higher permeability, the depth and extent of the plastic zones within the interlayers are greater compared to natural gas. When the storage pressure is 15 MPa, the depth of the plastic zone within the interlayer can be up to 2.1 m greater than when storing natural gas, occurring in the third interlayer from the top. These research findings may serve as a valuable reference for determining the operational parameters of on-site salt cavern hydrogen storage facilities.

Finite-frequency model order reduction of linear and bilinear systems via low-rank approximation

In this paper, we first investigate the finite-frequency model order reduction for linear systems based on low-rank Gramian approximations. An efficient algorithm for computing low-rank approximations of the finite-frequency and frequency-dependent Gramians based on Laguerre functions is proposed. The approach constructs the low-rank decomposition factors of the finite-frequency Gramians or frequency-dependent Gramians through a recursive formula of Laguerre functions expansion coefficient vectors and then combines the low-rank square root method and frequency-dependent balanced truncation method to obtain the reduced-order models. In this process, it avoids dealing with the matrix-valued functions and solving the related (generalized) Lyapunov matrix equations directly, making them computationally efficient. Furthermore, the above method is successfully extended to bilinear systems, and a corresponding efficient computation method for low-rank approximations of the finite-frequency Gramians of bilinear systems is derived. Finally, some numerical simulations are provided to illustrate the effectiveness of our proposed algorithms.

Stability and bifurcation of a delayed Aron–May model for malaria transmission

In this paper, the dynamical behavior in a delayed Aron–May model for malaria transmission is investigated. The basic reproduction number is defined. The global stability of the malaria-free equilibrium (MFE) is established. By using the Bendixson theorem, a sufficient condition for the global stability of the delay-free equilibrium (DFE) is also established. Furthermore, to deal with the local stability of endemic equilibrium (EE), by means of the stability switches analysis method proposed in [E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33(5) (2002) 1144–1165.] the related characteristic equation at EE is investigated, and the occurrence of Hopf bifurcation is discussed by using the incubation period in mosquito as a bifurcation parameter. Last, the simulation analysis is also performed to verify the dynamical behavior of the model.

Research on the Control Method of a 2DOF Parallel Platform Based on Electromagnetic Drive

In this paper, a spatial two-degree-of-freedom (2DOF) parallel platform based on electromagnetic redundant drive and its control method are investigated. The platform is redundantly driven by three electromagnetic-spring conforming branched chains, and the design provides better flexibility and responsiveness than conventional parallel structures. The introduction of the electromagnetic drive alleviates the stresses within the conventional rigid redundant drive structure and reduces the detrimental effects associated with rigid redundancy. In this paper, the structure and equivalent SPU model of the platform are first introduced, with S referring to the kinematic sub, P to the spherical sub, and U to the universal joint. The degrees of freedom of the platform are analyzed, and the inverse kinematic model and velocity Jacobi matrix are derived, so as to derive the relationship between the pitch, roll angles, and length of the gimbal chain, and the relational equation between the angle and the current is further established to realize the electromagnetic control of the parallel redundant platform. The control part is realized as follows. Firstly, the angle information of the platform is obtained from the gyroscope to the microcontroller, the filtered angle is derived through the Untraceable Kalman Filter (UKF), and the angle value can be fused with data by both the mathematical model and PID algorithm to introduce the current value required to achieve the balance and realize the balance. In the simulation part, this paper uses Simulink and Simscape in MATLAB for joint simulation, and by giving the simulated trajectory and the desired trajectory of the joints, the driving force diagrams of the three branched chains based on the Least-Second Paradigm method are derived, and the trajectory error and driving force error are given to validate the reliability of the method of this paper.

Early-time resonances in the three-dimensional wall-bounded axisymmetric Euler and related equations

We investigate the complex-time analytic structure of solutions of the three-dimensional (3D)-axisymmetric, wall-bounded, incompressible Euler equations, by starting with the initial data proposed in Luo and Hou [“Potentially singular solutions of the 3D axisymmetric Euler equations,” Proc. Natl. Acad. Sci. U. S. A. 111(36), 12968–12973 (2014)], to study a possible finite-time singularity. We use our pseudospectral Fourier–Chebyshev method [Kolluru et al., “Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)], with quadruple-precision arithmetic, to compute the time-Taylor-series coefficients of the flow fields, up to a high order. We show that the resulting approximations display early-time resonances; the initial spatial location of these structures is different from that for the tygers, which we have obtained in Kolluru et al. [“Insights from a pseudospectral study of a potentially singular solution of the three-dimensional axisymmetric incompressible Euler equation,” Phys. Rev. E 105(6), 065107 (2022)]. We then perform asymptotic analysis of the Taylor-series coefficients, by using generalized ratio methods, to extract the location and nature of the convergence-limiting singularities and demonstrate that these singularities are distributed around the origin, in the complex-t2 plane, along two curves that resemble the shape of an eye. We obtain similar results for the one-dimensional (1D) wall-approximation (of the full 3D-axisymmetric Euler equation) called the 1D Hou–Luo model, for which we use Fourier-pseudospectral methods to compute the time-Taylor-series coefficients of the flow fields. Our work examines the link between tygers, in Galerkin-truncated pseudospectral studies, and early-time resonances, in truncated time-Taylor expansions of solutions of partial differential equations, such as those we consider.

Suppression of Azimuthal Combustion Instability Using Perforated Liner Under Symmetry Breaking

Symmetry breaking occurring in annular combustors shows different stability patterns due to the formation of nondegenerate modes with different frequencies. This also brings about more profound complexity for control methodology. This paper presents a three-dimensional analytical model to investigate how to control nondegenerate azimuthal modes by modifying the wall boundary conditions of a multiburner annular combustor. The stability of the system can be evaluated by solving the eigenvalue of the dispersion relation equation derived in this study. Results show that changes in the parameters of the perforated liner cause different performance in nondegenerate modes due to frequency splitting. Meanwhile, the asymmetric perforated liner alters the original asymmetry of the annular combustor, inducing a shift in the nature of nondegenerate modes when significant symmetry breaking exists. Furthermore, variations of pressure amplitude in the backing cavity lead to diverse acoustic energy dissipation and even distinct stability patterns under two types of azimuthally nonuniformly perforated liner. Overall, this paper presents a new methodology for suppressing nondegenerate azimuthal modes under symmetry breaking by utilizing a perforated liner in an annular combustor. Additionally, a clear understanding of the evolution behavior of nondegenerate modes is also provided by studying their stability behavior and energy dissipation.