Jacobian Matrix Research Articles

Fired Heaters Optimization by Estimating Real-Time Combustion Products Using Numerical Methods

Fired heaters upstream of distillation towers, despite their optimal thermal efficiency, often suffer from performance decline due to fluctuations in fuel composition and unpredictable operational parameters. These heaters have high energy consumption, as fuel properties vary depending on the source of the crude oil. This study aims to optimize the combustion process of a three-gas mixture, mainly refinery gas, by incorporating more stable fuels such as natural gas and liquefied petroleum gas (LPG) to improve energy efficiency and reduce LPG consumption. Using real-time gas chromatography-mass spectrometry (GC-MS) data, we accurately calculate the mass fractions of individual compounds, allowing for more precise burner flow rate determinations. Thermochemical data are used to calculate equilibrium constants as a function of temperature, with the least squares method, while the Newton–Raphson method solves the resulting nonlinear equations. Four key variables (X4,X6,X8, and X11), representing H2,CO,O2, and N2, respectively, are defined, and a Jacobian matrix is constructed to ensure convergence within a tolerance of 1 ×10−6 over a maximum of 200 iterations, implemented via Python 3.10.4 and the scipy.optimize library. The optimization resulted in a reduction in LPG consumption by over 50%. By tailoring the fuel supply to the specific thermal needs of each processing unit, we achieved substantial energy savings. For instance, furnaces in the hydrocracking unit, which handle cleaner subproducts and benefit from hydrogen’s adiabatic reactions, require much less energy than those in the primary distillation unit, where high-impurity crude oil is processed.

Electronic circuit and image encryption using a novel simple 4D hyperchaotic system

Abstract A new simple 4D autonomous hyperchaotic system with seven terms is introduced. This system was inspired by an unusual 3D chaotic Liu system with six terms. The proposed system has two unstable saddle and saddle-foci points. Theoretical and numerical analyses are conducted to investigate various dynamical features of the system, including its equilibria, Jacobian matrix, Lyapunov exponents, Lyapunov dimension (Kaplan-Yorke), and multistability. The proposed system demonstrates multistability, enhancing its potential for various applications. An electronic circuit implementation using NI Multisim software 14.3 validates the system’s practical feasibility. A novel image encryption algorithm has been developed based on the system’s hyperchaotic properties. Experimental results confirm the algorithm’s robustness in both encryption accuracy and computational efficiency compared to existing methods. As well as, correlation analysis of adjacent pixels in encrypted images yields near-zero or negative values, indicating adequate randomization. The NIST SP800–22 statistical tests confirm the randomness of generated sequences with p-values consistently above 0.01. Information entropy analysis of encrypted images approaches the ideal value. These results demonstrate the proposed system’s effectiveness in secure image encryption, offering a promising solution for multimedia security applications.

Nonlinear Optimal and Multi-Loop Flatness-Basedcontrol of Omnidirectional 3-Wheel Mobile Robots

In this article, the control problem for omnidirectional 3-wheel autonomous mobile robots is solved with the use of (i) a nonlinear optimal control method (ii) a flatness-based control approach which is implemented in successive loops. To apply method (i) that is nonlinear optimal control, the dynamic model of the omnidirectional 3-wheel autonomous mobile robots undergoes approximate linearization at each sampling instant with the use of first-order Taylor series expansion and through the computation of the associated Jacobian matrix. The linearization point is defined by the present value of the system's state vector and by the last sampled value of the control inputs vector. To compute the feedback gains of the optimal controller an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. The global stability properties of the nonlinear optimal control method are proven through Lyapunov analysis. To implement control method (ii), that is flatness-based control in successive loops, the state-space model of the omnidirectional 3-wheel autonomous mobile robot is separated into chained subsystems, which are connected in cascading loops. Each one of these subsystems can be viewed independently as a differentially flat system and control about it can be performed with inversion of its dynamics as in the case of input-output linearized flat systems. The state variables of the preceding (i-th) subsystem become virtual control inputs for the subsequent (i+1-th) subsystem. In turn, exogenous control inputs are applied to the last subsystem. The whole control method is implemented in successive loops and its global stability properties are also proven through Lyapunov stability analysis. The proposed method achieves trajectory tracking and autonomous navigation for the omnidirectional 3-wheel autonomous mobile robots without the need of diffeomorphisms and complicated state-space model transformations.

Stability Analysis of SIR Mathematics Model in Shopeepay Later Addiction Case

This research discusses the case of Shopeepay Later addiction using the SIR mathematics model. Therefore, the purpose of this research is to build, analyze and find out the basic reproduction number (????0) and simulation of the SIR model of Shopeepay Later addiction. The research begins by building the assumptions of the SIR model of Shopeepay Later addiction, finding the equilibrium point, analyzing the stability of the equilibrium point using the jacobian matrix, finding the basic reproduction number (????0), and doing a simulation using Google Colab. The parameters used in this SIR model include the transmission rate parameter (????), the recovery rate parameter (????) the self-control parameter (????1), the promotion parameter (????2), and the application stop parameter (????3). The result of this study obtained the SIR model of Shopeepay Later addiction, one endemic equilibrium point is stable and the basic reproduction number ????0 = −37,29 that is, there is no transmission. Then to get expected conditions from the simulation result, namely given parameter values ???? = 0.5, ???? = 0.4, and ????1, ????2, ????3 = 0.9.

Epidemiological Implications of Vertical Transmission and Nonlinear Treatment in Lassa fever: A Mathematical Study

Lassa fever is a viral hemorrhagic disease primarily transmitted through contact with food or household items contaminated by urine or feces of infected rodents. The disease poses a significant health risk in endemic regions, yet the effect of vertical transmission and non-linear treatment on its spread has not been thoroughly explored. To address this, a mathematical model was constructed to assess the effect of vertical transmission and non-linear treatment in the transmission dynamics of Lassa fever. The model was validated through the theory of positivity and boundedness, ensuring that its solution remains biologically meaningful over time. The existence of equilibrium points was examined and the basic reproduction number was calculated using the next generation matrix operator. Bifurcation analysis is performed using the Center Manifold Theory. The Local and global stability of the model around the Lassa fever free equilibrium is investigated using Jacobian Matrix method and the theorem proposed by Castillo-Chavez. The effect of the parameters of the basic reproduction number was investigated using normalized forward sensitivity index. Furthermore, it was inferred that decrease in the contact rate and the rate of vertical transmission were instrumental to curtailing the spread of Lassa fever in the population.

Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning

The deep operator network (DeepONet) has shown remarkable potential in solving partial differential equations (PDEs) by mapping between infinite-dimensional function spaces using labeled datasets. However, in scenarios lacking labeled data, the physics-informed DeepONet (PI-DeepONet) approach, which utilizes the residual loss of the governing PDE to optimize the network parameters, faces significant computational challenges, particularly due to the curse of dimensionality. This limitation has hindered its application to high-dimensional problems, making even standard 3D spatial with 1D temporal problems computationally prohibitive. Additionally, the computational requirement increases exponentially with the discretization density of the domain. To address these challenges and enhance scalability for high-dimensional PDEs, we introduce the Separable physics-informed DeepONet (Sep-PI-DeepONet). This framework employs a factorization technique, utilizing sub-networks for individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix required for gradient computations. By incorporating forward-mode automatic differentiation (AD), we further optimize computational efficiency, achieving linear scaling of computational cost with discretization density and dimensionality, making our approach highly suitable for high-dimensional PDEs. We demonstrate the effectiveness of Sep-PI-DeepONet through three benchmark PDE models: the viscous Burgers’ equation, Biot’s consolidation theory, and a parametrized heat equation. Our framework maintains accuracy comparable to the conventional PI-DeepONet while reducing training time by two orders of magnitude. Notably, for the heat equation solved as a 4D problem, the conventional PI-DeepONet was computationally infeasible (estimated 289.35 h), while the Sep-PI-DeepONet completed training in just 2.5 h. These results underscore the potential of Sep-PI-DeepONet in efficiently solving complex, high-dimensional PDEs, marking a significant advancement in physics-informed machine learning.

Research Progress and Key Technology Analysis of Lower Limb Rehabilitation Robots

In China, there is a large number of patients with lower limb movement disorders and limited medical resources, while the market for rehabilitation technologies is growing rapidly. The technology of lower limb rehabilitation exoskeleton robots has been successfully applied and shows significant development potential. Both domestic and international research on lower limb rehabilitation exoskeleton robots focuses on the design aspects of comfort, bionics, lightweight construction, and safety. Notable domestic and international companies and universities are committed to enhancing personalization and human-machine interaction experiences, reducing costs, and expanding application fields. A preliminary analysis of the forward and inverse kinematics of lower limb rehabilitation exoskeleton robots has been conducted to derive the kinematic equations and velocity Jacobian matrices, facilitating better optimization of the robot's control algorithms. Existing control algorithms are diverse, including position-based, force information-based, bioelectric signal-based, and intelligent control strategies, each with its own advantages and disadvantages. Among these, control algorithms that integrate artificial intelligence and other cutting-edge technologies hold the most promise. As the aging population continues to grow, this technology is expected not only to play a role in rehabilitation therapy but also to expand into other areas, such as assistive walking, along with the optimization of mechanisms and the evolution of control algorithms.

Element-wise and recursive solutions for the power spectral density of biological stochastic dynamical systems at fixed points

Stochasticity plays a central role in nearly every biological process, and the noise power spectral density (PSD) is a critical tool for understanding variability and information processing in living systems. In steady state, many such processes can be described by stochastic linear time-invariant (LTI) systems driven by Gaussian white noise, whose PSD is a complex rational function of the frequency that can be concisely expressed in terms of their Jacobian, dispersion, and diffusion matrices, fully defining the statistical properties of the system's dynamics at steady state. Here, we arrive at compact, element-wise solutions of the rational function coefficients for the auto- and cross-spectrum that enable the explicit analytical computation of the PSD in dimensions n=2,3,4. We further present a recursive Leverrier-Faddeev-type algorithm for the exact computation of the rational function coefficients. Crucially, both solutions are free of matrix inverses. We illustrate our element-wise and recursive solutions by considering the stochastic dynamics of neural systems models, namely, FitzHugh-Nagumo (n=2), Hindmarsh-Rose (n=3), Wilson-Cowan (n=4), and the stabilized supralinear network (n=22), as well as an evolutionary game-theoretic model with mutations (n=5,31). We extend our approach to derive a recursive method for calculating the coefficients in the power-series expansion of the integrated covariance matrix for interacting spiking neurons modeled as Hawkes processes on arbitrary directed graphs. Published by the American Physical Society 2024

Partition Differential Equations and Some Combinatorial Algebraic Structures

Let Λ be the algebra of symmetric functions. We introduce Stirling partitions, factorial partition polynomials, partition differential equations and their corresponding partitions, and partition primitive functions. Most importantly, this investigation provides a new combinatorial coalgebra structure on Λ, and it characterizes the primitive elements in Λ using the Jacobian determinants of partition primitive functions.

Dynamical Complexity of Modified Leslie–Gower Predator–Prey Model Incorporating Double Allee Effect and Fear Effect

This contribution concerns studying a realistic predator–prey interaction, which was achieved by virtue of formulating a modified Leslie–Gower predator–prey model under the influence of the double Allee effect and fear effect in the prey species. The initial theoretical work sheds light on the relevant properties of the solution, presence, and local stability of the equilibria. Both analytic and numerical approaches were used to address the emergence of diverse bifurcations, like saddle-node, Hopf, and Bogdanov–Takens bifurcations. It is noteworthy that while making the assumption that the characteristic equation of the Jacobian matrix J has a pair of imaginary roots C(ρ)±ιD(ρ), it is sufficient to consider only C(ρ)+ιD(ρ) due to symmetry. The impact of the fear effect on the proposed model is discussed. Numerical simulation results are provided to back up all the theoretical analysis. From the findings, it was established that the initial condition of the population, as well as the phenomena (fear effect) introduced, played a crucial role in determining the stability of the proposed model.

A novel numerical method for mixed-frame multigroup radiation-hydrodynamics with GPU acceleration implemented in the quokka code

Abstract Mixed-frame formulations of radiation-hydrodynamics (RHD), where the radiation quantities are computed in an inertial frame but matter quantities are in a comoving frame, are advantageous because they admit algorithms that conserve energy and momentum to machine precision and combine more naturally with adaptive mesh techniques, since unlike pure comoving-frame methods they do not face the problem that radiation quantities must change frame every time a cell is refined or coarsened. However, implementing multigroup RHD in a mixed-frame formulation presents challenges due to the complexity of handling frequency-dependent interactions and the Doppler shift of radiation boundaries. In this paper, we introduce a novel method for multigroup RHD that integrates a mixed-frame formulation with a piecewise powerlaw approximation for frequency dependence within groups. This approach ensures the exact conservation of total energy and momentum while effectively managing the Lorentz transformation of group boundaries and evaluation of group-averaged opacities. Our method takes advantage of the locality of matter-radiation coupling, allowing the source term for Ng frequency groups to be handled with simple equations with a sparse Jacobian matrix of size Ng + 1, which can be inverted with O(Ng) complexity. This results in a computational complexity that scales linearly with Ng and requires no more communication than a pure hydrodynamics update, making it highly efficient for massively parallel and GPU-based systems. We implement our method in the GPU-accelerated RHD code quokka and demonstrate that it passes a wide range of numerical tests, including preserving the asymptotic diffusion limit. We demonstrate that the piecewise powerlaw method shows significant advantages over traditional opacity averaging methods for handling rapidly variable opacities with modest frequency resolution.

A mathematical model to study the role of dystrophin protein in tumor micro-environment

In this research work, the authors have developed a mathematical model to examine the interaction between dystrophin protein and tumor. The authors formulated a system of ordinary differential equations to describe the dynamics of the dystrophin-tumor interaction system. Jacobian matrix and Routh–Hurwitz stability techniques were used to determine equilibrium points, perform stability and bifurcation analysis, and establish the conditions required for the stability of the proposed model. Numerical simulations are performed using Euler’s method to investigate the temporal evolution of the proposed model under different parameter values, such as tumor growth rate and feedback strength of dystrophin protein. The numerical results are presented in tables, and corresponding to each table, a graphical analysis is done. The graphical analysis includes creating phase portraits to visually represent stability regions around the equilibrium points, bifurcation diagrams to identify critical points, and time series analysis to highlight the behavior of the proposed model. The authors explore how variations in dystrophin expression impact tumor progression, identifying potential therapeutic implications of maintaining higher dystrophin levels. This comprehensive analysis enhances our understanding of the dystrophin-tumor interaction, providing a basis for further experimental validation and potential therapeutic strategies.

Nonlinear Dynamics of an Electromagnetically Actuated Cantilever Beam Under Harmonic External Excitation

The present work is devoted to the study of nonlinear vibrations of an electromagnetically actuated cantilever beam subject to harmonic external excitation. The soft actuator that controls the vibratory motion of such components of a robotic structure led to a strongly nonlinear governing differential equation, which was solved in this work by using a highly accurate technique, namely the Optimal Auxiliary Functions Method. Comparisons between the results obtained using our original approach with those of numerical integration show the efficiency and reliability of our procedure, which can be applied to give an explicit analytical approximate solution in two cases: the nonresonant case and the nearly primary resonance. Our technique is effective, simple, easy to use, and very accurate by means of only the first iteration. On the other hand, we present an analysis of the local stability of the model using Routh–Hurwitz criteria and the eigenvalues of the Jacobian matrix. Global stability is analyzed by means of Lyapunov’s direct method and LaSalle’s invariance principle. For the first time, the Lyapunov function depends on the approximate solution obtained using OAFM. Also, Pontryagin’s principle with respect to the control variable is applied in the construction of the Lyapunov function.

Game Evolution Research on the Choice of Training Strategies between Clinical Departments and Doctors under the Background of Hospital-Provided Training

Objective: This study aims to explore the dynamic game process of training participation strategies among hospitals, clinical departments, and newly recruited doctors, as well as the impact of hospital intervention on this process. By applying game evolution theory, a comprehensive model is constructed to analyze the decision-making logic of training participation and propose new training strategies to promote the professional growth of newly recruited doctors and improve hospital service quality. Methods: The stability of dynamic game equilibrium points was analyzed by constructing Jacobian matrices, considering nine different game scenarios with and without hospital intervention. An empirical study from a hospital was introduced to verify the theoretical model's predictions. Results: The study found that without hospital intervention, clinical departments and newly recruited doctors tend to choose not to participate in training. After hospital intervention, through the provision of resources and incentives, the participation and effectiveness of training significantly improved. Conclusion: The study reveals the crucial role of hospital intervention in promoting the participation of newly recruited doctors in training and proposes a training strategy that combines theory and practice. This strategy not only helps new doctors quickly adapt to the hospital environment but also improves the quality of medical services, providing talent support for the hospital's continuous development. It offers theory-based decision support for hospital managers and provides new ideas for the improvement of future medical education and practice.

Efficient non-probabilistic parallel model updating based on analytical correlation propagation formula and derivative-aware deep neural network metamodel

Non-probabilistic convex models are powerful tools for structural model updating with uncertain‑but-bounded parameters. However, existing non-probabilistic model updating (NPMU) methods often struggle with detecting parameter correlation due to limited prior information. Worth still, the unique core steps of NPMU, involving nested inner layer forward uncertainty propagation and outer layer inverse parameter updating, present challenges in efficiency and convergence. In response to these challenges, a novel and flexible NPMU scheme is introduced, integrating analytical correlation propagation and parallel interval bounds prediction to enable automatic detection of parameter correlations. In the forward uncertainty propagation phase, a linear coordinate transformation is applied to map the original parameter space to a standard hypercube space, simplifying correlation-involved bounds prediction into conventional interval bounds prediction. Moreover, an analytical correlation propagation formula is derived using a second-order response approximation to sidestep the complexities of geometry-based correlation calculations. To expedite forward propagation, a derivative-aware neural network model is employed to replace the physical solver, facilitating improved fitting capabilities and automatic differentiation, including the calculation of Jacobian and Hessian matrices essential for correlation propagation. The neural network's inherent parallelism accelerates interval bounds prediction through parallel computation of samples. In the inverse parameter updating phase, the block coordinate descent algorithm is embraced to narrow the search space and boost convergence capabilities, while the perturbation method is utilized to determine the optimal starting point for optimization. Two numerical examples illustrate the efficacy of the proposed method in updating structural models while considering correlations.

Analysis of Force Distribution and Motion Characteristic of Hybrid-Driven Tensegrity Parallel Mechanism

Abstract Tensegrity parallel mechanisms is a novel spatial structure composed of cable-based and rigid-based chains, which is characterized by lightweight, multi-stability, high precision, good stiffness, and high load-bearing capacity. This paper focuses on the six-degree-of-freedom tensegrity parallel mechanism, analyzing its static equilibrium and presenting the force equilibrium equations. A model for cable tension distribution optimization is established, utilizing different P-norm objective functions to optimize the distribution of tension in cables and rigid links, discussing the reasons for unreasonable tensions in the system. Under given constraints on the motion pairs of the mechanism, the workspace of the mechanism is plotted, and factors influencing the size of the workspace are analyzed. Finally, the singularity of the mechanism is addressed based on the Jacobian matrix, demonstrating the feasibility of reaching the space volume by the mechanism. It has certain reference significance for the configuration theory and analysis methods of tensegrity parallel mechanisms and rigid-flexible coupling mechanisms, and provides a reference for the further promotion and application of mechanisms.

Compositional reservoir simulation with a high-resolution compact stencil adaptive implicit method

The adaptive implicit method (AIM) is the mainstream approach for compositional reservoir simulation. In its standard form, AIM uses single-point upwinding to reconstruct the fluxes across the interfaces of the control volume, and this leads to substantial numerical diffusion and loss of accuracy. Previous efforts to improve the accuracy of AIM focused on using high-order schemes to reconstruct the interfacial fluxes; those schemes introduced additional numerical nonlinearities to the system of equations or compromised the accuracy of the Jacobian by neglecting the high-order terms in its construction. In this work, we describe a high-resolution compact-stencil (HRCS) AIM. The new scheme is applied to compositional reservoir simulation. In addition to the mixed implicit/explicit time discretization of standard AIM, the HRCS AIM scheme uses a mixed time and space discretization. Specifically, we blend low- and high-order fluxes according to a well defined rule that uses a high-order reconstruction in the explicit regions of the domain and a low-order reconstruction in the implicit regions. This strategy ensures that additional nonlinearities introduced by the high-order reconstruction do not impact the Jacobian matrix, thus preserving the algebraic structure of standard AIM. The HRCS AIM method is demonstrated using several compositional problems. The results indicate substantial gains in accuracy with a small additional computational cost compared with standard AIM. Additionally, HRCS AIM is more robust and has a smaller computational cost compared with its full high-resolution counterpart.

Application of simplifying jacobian matrix and introducing iterative threshold to improve newton-raphson method in three-phase power flow calculation

Abstract With the expansion of the scale and complexity of the power system, traditional three-phase power flow calculation methods are facing dual challenges of computational efficiency and accuracy. Power flow calculation is a mathematical method for analyzing the power flow of an electrical system. And the Newton-Raphson (NR) method is an effective algorithm for solving nonlinear problems. However, updating the Jacobian matrix in the NR method can take a lot of time. Based on the asymmetry in three-phase power systems and the time-consuming nature of the NR method, an enhanced version of the NR method is proposed in this paper that improves the efficiency of power flow calculations by simplifying the Jacobian matrix and optimizing the iteration speed. Furthermore, in order to minimize the number of iterations in the NR method, it is proposed to use the PO method to determine the ideal iteration threshold.

Synthesis and Analysis of 3-CPC Space Mobile Parallel Mechanism

Abstract The degree of freedom of 3-CPC space mobile parallel mechanism is calculated according to the screw theory, and the kinematics model of 3-CPC space mobile parallel mechanism is established by coordinate transformation method. By analyzing the forward and inverse kinematics, the Jacobian matrix 3-CPC space mobile parallel mechanism is derived. The workspace is obtained, and the kinematics performance is simulated. The results show that the forward and inverse kinematics of 3-CPC space mobile parallel mechanism is simple to solve, the workspace is large, and the mechanism has good kinematics performance and strong practical value. Finally, the 3-CPC space mobile mechanism is applied to the vibrating screen for material screening, and the motion trajectory of the moving platform is obtained. The kinematic performance analysis 3-CPC space mobile parallel mechanism provides the theoretical basis for the application of the mechanism.

Data-Driven Robust Finite-Iteration Learning Control for MIMO Nonrepetitive Uncertain Systems.

This work considers three main problems related to fast finite-iteration convergence (FIC), nonrepetitive uncertainty, and data-driven design. A data-driven robust finite-iteration learning control (DDRFILC) is proposed for a multiple-input-multiple-output (MIMO) nonrepetitive uncertain system. The proposed learning control has a tunable learning gain computed through the solution of a set of linear matrix inequalities (LMIs). It warrants a bounded convergence within the predesignated finite iterations. In the proposed DDRFILC, not only can the tracking error bound be determined in advance but also the convergence iteration number can be designated beforehand. To deal with nonrepetitive uncertainty, the MIMO uncertain system is reformulated as an iterative incremental linear model by defining a pseudo partitioned Jacobian matrix (PPJM), which is estimated iteratively by using a projection algorithm. Further, both the PPJM estimation and its estimation error bound are included in the LMIs to restrain their effects on the control performance. The proposed DDRFILC can guarantee both the iterative asymptotic convergence with increasing iterations and the FIC within the prespecified iteration number. Simulation results verify the proposed algorithm.