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The paper presents a medical parallel robot (BR-1), designed for the minimally invasive, targeted, treatment of cancer through brachytherapy. The analysis of Jacobi matrices allows a complete study of the singularities while generating also a set of conditions which, implemented in the control system, enable the safe behaviour of the robot. Brachytherapy is an advanced form of cancer treatment involving the placement of small radioactive seeds directly inside the malignant tumour, allowing a very effective, local treatment of cancer. For this task an innovative parallel robotic system has been developed, having five degrees of freedom, constructed in two versions, BR-1R and BR-1T with a slight variation at the level of a passive joint. The two solutions revealed different accuracy distributions and by selecting the proper variant based on the tumour location, an increased universality degree for the BR-1 robot is obtained. The singularity-free workspace is determined using the assessment of singularities for both robot versions. The control system for both robot versions is simple, robust and intuitive allowing clinicians to have an accurate procedure, having real-time force monitoring during the needle insertion. The experimental data demonstrate that the robotic structure is a worthy candidate for robotic-assisted brachytherapy. [ABSTRACT FROM PUBLISHER]

Full quantum mechanical calculations of vibrational energies of methane and fluoromethane are carried out using a polyspherical description combining Radau and Jacobi coordinates. The Hamiltonian is built in a potential-optimized discrete variable representation, and vibrational energies are solved using an iterative eigensolver. This new approach can be applied to a large variety of molecules. In particular, we show that it is able to accurately and efficiently compute eigenstates for four different molecules : CH4, CHD3, CH2D2, and CH3F. Very good agreement is obtained with the results reported previously in the literature with different approaches and with experimental data. [ABSTRACT FROM AUTHOR]

A linear equation with n variables x1, x2, x3,..... xn as an equation with the form a1 x1 + a2 x2 +..... an xn = b, where a1, a2,.....an and b are real constants. A system of linear equations is a finite collection of linear equations in the variables x1, x2,.... xn. A solution to a system of equations is a sequence of integers s1, s2,....= sn that solves each of the system's equations. The goal of this research is to compare the Jacobi Iteration Method with the Gauss-Seidel Iteration Method in order to find the solution to the Fuzzy Linear Equation System. The Jacobi iteration technique is an indirect method that begins with a rough estimate. A approach for solving a system of linear equations is the Jacobi method. A convergent approach is the Jacobi method. As a result, each equation must be adjusted such that the absolute value coefficients are the highest. The most recently computed values are utilized in all computations in the Gauss-Seidel iteration technique. The next step is to compare the two approaches by looking at the number of iterations and which error value is superior in solving the Fuzzy Linear Equation System after receiving the results of the iteration of the two ways. [ABSTRACT FROM AUTHOR]

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in C, and some objects similar to objects from the theory of spectral problems for Jacobi matrices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the gravitational bending angle due to the Casimir wormholes, which consider the Casimir energy as the source. Furthermore, some of these Casimir wormholes regard Generalized Uncertainty Principle (GUP) corrections of Casimir energy. We use the Ishihara method for the Jacobi metric, which allows us to study the bending angle of light and massive test particles for finite distances. Beyond the uncorrected Casimir source, we consider many GUP corrections, namely, the Kempf, Mangano and Mann (KMM) model, the Detournay, Gabriel and Spindel (DGS) model, and the so-called type II model for the GUP principle. We also find the deflection angle of light and massive particles in the case of the receiver and the source are far away from the lens. In this case, we also compute the optical scalars: convergence and shear for these Casimir wormholes as a gravitational weak lens. Our self-consistent iterative calculations indicate corrections to the bending angle by Casimir wormholes in the previous paper. [ABSTRACT FROM AUTHOR]

Centralized model predictive control is impractical for many complex systems due to communication burden and robustness issues. For these systems, distributed model predictive control (DMPC) is an alternative control strategy. In DMPC, the use of nonlinear first-principle model improves the prediction accuracy. However, it also brings about computational delay due to time-consuming optimization of large, non-convex nonlinear programs, which can then degrade the control performance. In this work, a fully distributed nonlinear model predictive control (DNMPC) algorithm is developed to accelerate control feedback. The input computation procedure contains background and online stages, in which prediction–correction mode is applied. In the background stage, the future state is predicted one step forward based on the nominal plant model. Each controller optimizes its own local input and exchanges latest information with other controllers to improve decision making. After distributed optimization, the local controllers collect optimality information to prepare for future computation. When the true state is available, the state prediction error can be calculated. Each controller formulates its local sensitivity equation based on parametric sensitivity. All the sensitivity equations are solved in parallel with application of the Jacobi iterative method. After solution, the nominal optimum is updated with the correction vector and then implemented to the plant. The theoretical analysis of the proposed method is presented. Four case studies are given to demonstrate the effectiveness of the proposed algorithm. • A fast, fully distributed model predictive control algorithm is proposed for nonlinear systems. • Prediction–correction mode is applied in the proposed method. • The future inputs are computed in background with advanced-step distributed optimization. • The local sensitivity equations are solved in parallel through Jacobi iteration. computation. • The closed-loop system is proved to be input-to-state practical stable. [ABSTRACT FROM AUTHOR]

The paper proves the quadratic convergence of the complex HZ method for solving the positive definite generalized eigenvalue problem. The proof is made for a general cyclic pivot strategy in the case of simple eigenvalues and for any wavefront pivot strategy in the case of simple or double eigenvalues. The proof is valid for the real HZ method. The preliminary numerical tests confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this work we apply a regularized alternating direction method of multiplier of Jacobi type (J‐ADMM) to a frictional multi‐contact (both unilateral and bilateral) problem between an electro‐elastic material and a rigid non conductive foundations. The frictional contact is modelled by the Coulomb friction law. The resulted problem is non symmetric and non coercive. By dissociating the electric potential from the mechanical field, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply a J‐ADMM for the numerical approximation to the resulting perturbed elastic problem. The resolution of obtained sub‐problems is based on convex dualities. Numerical experiments are proposed to illustrate the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

Time-space Jacobi pseudospectral method is constructed to approximate the numerical solutions of the fractional Volterra integro-differential and parabolic Volterra integro-differential equations. We define fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi-Gauss-Lobatto points. The fractional derivative is defined in the modified Atangana-Baleanu derivative defined in the Caputo sense formula at JGL points. Further, we transform the domain of fractional Volterra integro-differential and parabolic Volterra integro-differential equations to the standard interval [ − 1 , 1 ] using variable transformation and function transformation. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The theory of error estimates and convergence analysis for the proposed method is also derived. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity. [ABSTRACT FROM AUTHOR]

The global ground and first three excited electronic state adiabatic as well as the corresponding quasidiabatic potential energy surfaces is reported as a function of nuclear geometries in the Jacobi coordinates (..., ..., γ) using Dunning's cc-pVTZ basis set at the internally contracted multi-reference (single and double) configuration interaction level of accuracy. Nonadiabatic couplings, arising out of relative motion of proton and the vibrational motion of CO, are also reported in terms of coupling potentials. The quasidiabatic potential energy surfaces and the coupling potentials have been obtained using the ab initio procedure [Simah et al., J. Chem. Phys. 111, 4523 (1999)] for the purpose of dynamics studies. [ABSTRACT FROM AUTHOR]

The resonant nonlinear Schrödinger equation (RNLSE) with competing weakly nonlocal nonlinearity and fractional temporal evolution, which describes the propagation of optical solitons along the nonlinear optical fibers, is investigated in this paper. Dynamic behavior of such equation with the parabolic-type nonlinearity is discussed. The relationship between the orbits of the dynamic system and traveling wave solutions is demonstrated. Particularly, a family of hyperbolic curves near the saddle points implies the existence of one family of the braking wave solutions correspondingly. Furthermore, the G ′ / G -expansion method and Jacobi elliptic function rational expansion method are conducted to get the exact traveling wave solutions, such as periodic wave solutions, shock wave solutions, and breaking wave solutions. [ABSTRACT FROM AUTHOR]

Solar wind induces the escape of ions from the ionospheric shell of an unmagnetized planet, Venus. To analyze the dynamics of ions due to the escape process, the modeling of ion-acoustic waves (IAWs) is scrutinized for an interaction between an upper ionospheric plasma of Venus constituting mobile electrons with two kinds of positive ions, viz, H + and O + and the solar-wind which is composed of highly energized electrons and protons. The Sagdeev pseudopotential is examined for wave breaking phenomena. Employing the concept of phase plane analysis, the formulated autonomous dynamical systems are examined for all feasible nonlinear and supernonlinear periodic waves. To support the existence of nonlinear and supernonlinear waves, the graph of the total energy function is plotted in three-dimensional space. A typical set of Venusian plasma parameters are used to discuss the analytical solutions of the dynamical system by the method of Jacobi elliptic function. Effect of traveling wave velocity on the plot of the total energy function and the wave solutions are discussed. [ABSTRACT FROM AUTHOR]

Mild Cognitive Impairment (MCI) is the asymptomatic, preclinical transitional stage among aging and Alzheimer's Disease (AD). Detection of MCI can ensure the timely intervention required to manage the disease's severity. Morphological alterations of Lateral Ventricle (LV) is considered as a significant biomarker for disease diagnosis. This research aims to analyze the shape alterations of the LV region using Fractional Order Jacobi Fourier Moment (FOJFM) features, which are categorized by their generic nature and capabilities to perform time-frequency analysis. T1-weighted transaxial view brain MR images (HC = 92 and MCI = 63) are obtained from publicly available Open Access Series of Imaging Studies (OASIS) database. The LV region is delineated using Weighted Level Set (WLS) segmentation method and results are compared to Ground Truth (GT) images. FOJFM features are employed to characterize the morphometry of LV region. From this segmented region, 200 features are computed by varying the value of order and fractional parameters. Random Forest (RF) and Support Vector Machine (SVM) classifiers are used to differentiate Healthy Control (HC) and MCI subjects. Results show that WLSE is able to delineate the LV structure. The segmented region shows good correlation with the GT area. FOJFM features are observed to be statistically significant in discriminating HC and MCI subjects with p<0.05. For MCI subjects, the feature values show higher variation as compared with HC brain, which might be due to the surface expansion of ventricular area during disease progression. SVM and RF classifiers show high performance F-measure values of 93.14% and 86.24%, respectively, for differentiating MCI conditions. The proposed moment based FOJFM features are able to capture the morphological changes of LV region related to MCI condition. Hence the proposed pipeline of work can be useful for the automated and early diagnosis of diseased conditions [ABSTRACT FROM AUTHOR]

• Integrating the principles of Jacobi Over-Relaxation (JOR) iteration into the Alternating Direction Method of Multipliers (ADMM), we introduce a novel parallel computing framework, termed ADMM-JOR. • The convergence of the proposed ADMM-JOR is rigorously established under an analytical framework of contractive-type methods. • Drawing on the foundational principle of progressively reducing the objective function value, we propose a self-adjustment method to determine the relaxation factor at each iteration for the ADMM-JOR method. • To validate the effectiveness of the proposed ADMM-JOR in addressing the deterministic user equilibrium problem, experiments were conducted on networks of varying scales. Traffic assignment plays a crucial role in transport system analysis, and the user equilibrium model is a very essential tool. It is however highly challenging to efficiently solve the user equilibrium model, especially for large-scale networks. Taking the deterministic user equilibrium (DUE) model as a representative, this paper aims to harness parallel computing to tackle its high computing burden. A novel parallel computing framework is proposed, drawing from the frontier alternating direction method of multipliers (ADMM) method. This algorithmic framework takes a good balance between the convergence rate and the additional communication costs. Specifically, the paper incorporates insights from the Jacobi over-relaxation (JOR) iteration method into the framework of the ADMM method, improving the convergence rate by utilizing more information during the iteration process while striving to reduce the side effects it brings, thereby developing a novel parallel computing framework, named ADMM-JOR. Subsequently, convergence of the proposed algorithm is rigorously proven under an analytic framework of contractive-type methods. Furthermore, we develop an adaptive strategy for adjusting the relaxation factor in ADMM-JOR, guided by the principle of objective function value decline, which aims to further improve the convergence performance of ADMM-JOR, with minimal additional computational cost in a fully parallel setting. Numerical experiments indicate that the proposed ADMM-JOR significantly reduces computation time while retaining the excellent parallel performance of the original ADMM method and significantly improving its convergence rate. [ABSTRACT FROM AUTHOR]

We examine a coupled Radhakrishnan–Kundu–Lakshmanan (cRKL) equation that includes self-phase modulation, third order dispersion, self-steepening, and cross-phase modulation effects. A propagating ultrafast pulse in nonlinear systems is described by the given equation. Utilizing the coupled Radhakrishnan–Kundu–Lakshmanan (cRKL) model, we explored the dynamics of various innovative optical solitons, such as wing-shaped, W-shaped, and brilliant solitons. We used mathematical methods like the Jacobi elliptical sn function method to get the exact analytical solution. Our results demonstrate that the self-phase modulation (SPM), self-steepening (SS), cross-phase modulation (XPM), and third order dispersion (TOD) can be effectively varied to influence the structure of the solitons. It is demonstrated that the precise balance between the self-steepening effect, the self-phase modulation, and the group velocity dispersion forms these novel W-shaped chirp-free dark solitons and dark and bright solitonic structures. By choosing appropriate settings for the physical variables, it is possible to create a graphic representation of the properties of optical solitons. The study and application of nonlinear materials with third order dispersion, cross-phase modulation, self-phase modulation effect, and self-steepening nonlinearity may take new paths as a result of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we propose two numerical approaches for approximating the solution of the following kind of integral equation f (y) − μ ∫ − 1 1 f (x) k (x , y) w (x) d x = g (y) , y ∈ [ − 1 , 1 ] , where f is the unknown solution, μ ∈ R ∖ { 0 } , k , g are given functions not necessarily known in the analytical form, and w is a Jacobi weight. The proposed projection methods are based on the constrained mock-Chebyshev least squares polynomials, and starting from data known at equally spaced points, provide a fine approximation of the solution. Such peculiarity can be helpful in all cases we deal with experimental data, typically measured at equispaced points. We prove the introduced methods are stable and convergent in some Sobolev subspace of C [ − 1 , 1 ]. Several numerical tests confirm the theoretical estimates and numerical effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This study investigates the effect of high dispersion, perturbations, cubic-quintic-septic nonlinearity, and multiplicative white noise on the resonant nonlinear Schrödinger equation in the Bohm potential. The model being studied is novel and requires investigation into its behavior and characteristics. Two integration schemes, namely the unified Riccati expansion method and the Jacobi elliptic function method, are employed in this study to seek soliton solutions for the governing model. Various types of solitons, such as bright, dark, and singular solitons, are observed. Moreover, solutions shaped like Jacobi's elliptic functions are identified. Parametric constraints characterize the obtained soliton solutions. Furthermore, the results are presented comprehensively, accompanied by visual representations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of singularly perturbed differential difference equations (SPDDE) and singularly perturbed convection delayed dominated diffusion equations (SPCDDDE) arising in the modelling of various chemical phenomena. We have approximated the delayed term by Taylor series expansion and then employed the Jacobi wavelet method. We have solved two examples of each SPDDE and SPCDDDE and compared our results with the results of existing methods and found that our method gives better accuracy. The numerical outcomes show that as we increase the values of convergence parameters i.e., degree of Jacobi polynomial (M) or resolution level (k) or both, the approximate solution converges to the exact solution. To the best of our knowledge, the present method is one of the first in literature that makes use of polynomial based wavelet to find the approximate solution of SPDDE and SPCDDDE. [ABSTRACT FROM AUTHOR]

In this paper we present a geometric multigrid method with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid error-propagation operator and show that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization is almost as efficient as when applied to a continuous Galerkin discretization. We furthermore show that multigrid applied to an EDG discretization outperforms multigrid applied to a hybridized discontinuous Galerkin discretization. Numerical examples verify our LFA predictions. [ABSTRACT FROM AUTHOR]

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A (k) converge to zero. However, this fact alone does not necessarily guarantee that A (k) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q (k). We prove that under certain assumptions A (k) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q (k) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q (k) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example. [ABSTRACT FROM AUTHOR]

We study the loss, recovery, and preservation of differentiability of time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time-dependent rate function, both for Glauber dynamics for the Curie-Weiss model and Brownian dynamics in a potential. We extend the variational approach to this problem of time-dependent regularity in order to include Hamiltonian trajectories with a finite lifetime in closed domains with a boundary. This leads to new phenomena, such a recovery of smoothness. We hereby create a new and unifying approach for the study of mean-field Gibbs-non-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi equations. [ABSTRACT FROM AUTHOR]

In this paper, the free vibration behaviors of composite laminated annular and circular plates under complex elastic boundary constraints are investigated. Firstly, Reddy's high-order shear deformation theory (HSDT) and Jacobi polynomial method are effectively combined to establish the unified vibration analysis model of composite laminated annular and circular plates. Secondly, the simulation of complex elastic boundary and coupling boundary is realized by using artificial virtual spring technology. Then, the energy equation of the composite laminated plate is established by using Rayleigh–Ritz energy technology. Finally, the free vibration solution equation of the laminated plate is obtained through the Hamilton differential principle. The fast and uniform convergence of this method and the accuracy of the calculated results are verified by numerical examples and the model experimental method. On this basis, the parameterization study is conducted, and the effects of material parameters, geometric parameters, spring stiffness values, and lamination scheme on the vibration characteristics of the annular or circular plate are fully discussed, which can provide a theoretical basis for future research. [ABSTRACT FROM AUTHOR]

In this study, an analytic method based on the Jacobi elliptic functions has been presented to obtain the exact solutions of time fractional Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation. This equation is reduced to a nonlinear ordinary differential equation. The elementary and elliptic function solutions of this equation are investigated. Many exact solutions containing the rational, complex, trigonometric, and hyperbolic functions are also found. Besides, some of the solutions are demonstrated by the graphics. [ABSTRACT FROM AUTHOR]

The uniform formulation of dynamic vibration analysis of multispan beams is presented by using an efficient domain decomposition method in this paper. Firstly, the structure is divided into several equal sections based on domain decomposition method. Next, the artificial spring is used to simulate complex boundaries and continuity condition of multispan beam. Finally, the admissible displacement functions are expanded through Jacobi orthogonal polynomials, and the free and forced vibration characteristics of multispan beam structures can be obtained by using Rayleigh–Ritz method. Results for various boundary conditions, ratios of thickness to length (h/L), numbers, and stiffness of supporting springs are presented. It is clearly shown that accurate solutions can be obtained by using the proposed method, and this study extends the application range of the Jacobi polynomials-Ritz method. In addition, the research results of this paper can provide data support for engineers such as bridge designers to design multispan bridges. [ABSTRACT FROM AUTHOR]

Recently, Zhang et al. [Applied Mathematics Letters 104 (2020) 106287] proposed a preconditioner to improve the convergence speed of three types of Jacobi iterative methods for solving multi-linear systems. In this paper, we consider the Jacobi-type method which works better than the other two ones and apply a new preconditioner. The convergence of proposed preconditioned iterative method is studied. It is shown that the new approach is superior to the recently examined one in the literature. Numerical experiments illustrate the validity of theoretical results and the efficiency of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

We extend the periodic von Neumann basis to non-Cartesian coordinates. The bound states of two isomerizing triatomic molecules, LiCN/LiNC and HCN/HNC, are calculated using the vibrational Hamiltonian in Jacobi coordinates. The phase space localization of the basis functions leads to a flexible and accurate representation of the Hamiltonian. This results in significant savings compared to a basis localized just in coordinate space. The favorable scaling of the method with dimensionality makes it promising for applications to larger systems. [ABSTRACT FROM AUTHOR]

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two‐dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two‐variable shifted fractional‐order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional‐order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently. [ABSTRACT FROM AUTHOR]

Let K = Q (ζ 8) be the cyclotomic field over Q of the extension degree 4. We give an integral lattice construction on Q (ζ 8) induced from codes over the ring R = F 2 [ u ] / ⟨ u 4 ⟩ . We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a code over R . If C is a Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield k = Q (ζ 8 + ζ 8 - 1) of K. Also, we see that Hilbert–Siegel modular form of weight n and genus g can be seen in terms of the complete joint weight enumerator of codes C j , for 1 ≤ j ≤ g over R . [ABSTRACT FROM AUTHOR]

Sudden vehicle propulsion torque change under tip-in/out maneuver often leads to low-frequency longitudinal vibration due to the flexibility in the half-shaft and tire slip, which greatly affects vehicle drivability. Note that the vibration frequency is between 1 and 10 Hz and is difficult to be absorbed by the vehicle mechanical system. To optimize the vehicle drivability under tip-in maneuver, an Adaptive Linear Quadratic Tracking (ALQT) anti-jerk traction controller is proposed in this paper. Based on the experimental data, a Carsim-Simulink co-simulation model is developed for assessing control performance. A control-oriented model, considering the nonlinear characteristics of the tire-road friction coefficient and slip ratio, is then proposed. A reference model with rigid axle is used to provide the equilibrium points and reference velocity trajectory. Jacobi linearization method is then used to linearize the model along the desired trajectory and a linear deviation model based on equilibrium points is obtained. Finally, the deviation compensation receding horizon LQT controller is designed along with the Kalman state estimation. The effectiveness of the designed controller is assessed via simulation studies under different road surfaces and compared with PID and LQR controllers. The LQT controller is able to track the desired velocity profile with minimum jerk while increasing road safety. Furthermore, the effect of LQT weighting coefficients under different road surfaces are discussed. Simulation results show that the ALQT controller is able to optimize vehicle drivability under different road surfaces and the weighting matrices shall be selected based on the road condition for optimal drivability. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method based on new fractional-order Jacobi polynomials for solving nonlinear fuzzy fractional integro-differential equations. Some operational matrices are used to reduce the problem to the system of algebraic equations. The convergence analysis of the method is provided. The accuracy of the method is illustrated by solving some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper contributes to develop a highly accurate numerical method for solving two-dimensional mass transfer equations during convective air drying of apple slices. The numerical scheme is based on the interpolating the solution of the mentioned equations over the roots of the orthogonal Jacobi polynomials (i.e., the Jacobi-Gauss-Lobatto points) in a nodal form. Moreover, for speeding up the procedure of numerical technique, operational matrices of differentiation are applied to discretize the derivatives of both spatial and temporal variables. After implementing the proposed technique, two-dimensional mass transfer equations would be transferred into the associated systems of linear algebraic equations which can be solved by appropriate iterative solvers such as robust Krylov subspace iterative methods. Some constructed artificial examples are provided to show the efficiency and applicability of the Jacobi pseudo-spectral method for solving two-dimensional mass transfer equations. Finally, a real example is considered and numerical results are validated by the experimental data which confirm the accuracy of the presented numerical approach. [ABSTRACT FROM AUTHOR]

The Fuzzy Comprehensive Evaluation (FCE) and the Principal Component Analysis (PCA) were simulated to assess water quality of the Nansi Lake Basin, China. The membership functions were established via the Nor-Half Sinusoidal Distribution Method, and the weight was calculated via the Exceeding Standard Multiple Method. To enhance the efficiency of extracting principal pollutant, the eigenequation was solved through the Jacobi Method, and the principal components were extracted based on eigenvalue, contribution ratio, accumulating contribution ratio, principal component loading and score. Water quality classification based on "National Surface Water Environmental Quality Standards of China (GB3838-2002) was used to assess the water quality. Considering the difference of the temporal and spatial distribution in average, water quality of Level I was 28.9%, 28.1%, 25.1%, 25.6%, respectively in spring, summer, autumn, and winter, which suggested that water quality in spring and summer was better than in autumn and winter. The order of water quality was Zhaoyang Lake (Level I) > Nanyang Lake (Level I) > Dushan Lake (Level III) > Weishan Lake (Level III and IV). There were four extracted principal components that can replace the fourteen pollutant indexes for assessing water quality. According to the annual mean data of the 1st principal components, the most important pollutions were heavy metals, including As (0.933), Hg (0.931), Cd (0.929), Cr(VI) (0.926), Pb (0.925), and Cu (0.534). It is proved that the combined FCE-PCA model could provide valuable information in the water quality assessment for the Nansi Lake Basin. [ABSTRACT FROM AUTHOR]

Lacking labeled examples of working numerical strategies, adapting an iterative solver to accommodate a numerical issue, e.g., density discontinuities in the pressure Poisson equation, is non-trivial and usually involves a lot of trial and error. Here, we resort to evolutionary neural network. A evolutionary neural network observes the outcome of an action and adapts its strategy accordingly. The process requires no labeled data but only a measure of a network’s performance at a task. Applying neuro-evolution and adapting the Jacobi iterative method for the pressure Poisson equation with density discontinuities, we show that the adapted Jacobi method is able to accommodate density discontinuities.

In this paper, we present a novel indirect convergent Jacobi spectral collocation method for fractional optimal control problems governed by a dynamical system including both classical derivative and Caputo fractional derivative. First, we present some necessary optimality conditions. Then we suggest a new Jacobi spectral collocation method to discretize the obtained conditions. By the proposed method, we get a system of algebraic equations by solving of which we can approximate the optimal solution of the main problem. Finally, we present a convergence analysis for our method and solve three numerical examples to show the efficiency and capability of the method. [ABSTRACT FROM AUTHOR]

The energy matrices of molecules of AB2, A2B2 and A2B3 type have been calculated for three different chemical shifts and several indirect spin-spin coupling coefficients (Jij) to obtain the Nuclear Magnetic Resonance (NMR) hyperfine structure. A computer program implemented in JACOBI method, which is a numerical iterative method for solving linear equation systems or a matrix equation on a matrix that has no zeros among its main diagonal elements, was used to calculate the eigenvalues and eigenvectors of these systems. We have developed a code to obtain the transition probabilities and transition energies. The theoretically calculated spectra has been compared with the experimental spectra and it has been observed a quite acceptable compliance between them. [ABSTRACT FROM AUTHOR]

In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix A of order n we find a constant γ < 1 depending on n, such that S (A ′) ≤ γ S (A) , where A ′ is obtained from A by applying one or more cycles of the Jacobi method and S (⋅) stands for the off-diagonal norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky–Jacobi method for solving the positive definite generalized eigenvalue problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of this nonlinear Volterra integral equations is studied completely. An hp -version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in L 2 -norm. The efficiency of the method is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

For massive multiple-input multiple-output (m-MIMO) uplink, the performancesof thelinearminimummean-square error (MMSE) detector are considered near optimal, and they occupy benchmark place for most linear iterative detectors. However, the MMSE algorithm is known by its load computational complexity due to the implication of large-scale matrix inversions, and in other hand, iterative methods are often preferred in signal detection because of its low complexity. In this paper, we propose a New Damped Jacobi (NDJ) detector in order to improve the performance of the classical Jacobi linear algorithm. Starting from the classical Jacobi technique to our new proposal, we go through thedevelopment of two variants; one uses adamping factor and the other uses a stair-matrix. However, the NDJ incorporates a damping factor in its construction and basing also on stair matrix instead of diagonal matrix. The performances in terms of convergence and low complexity of each Jacobi variant studied in this paper are analyzed. Finally, some simulation examples are giventoillustratetheadvantagesofthenewproposedalgorithm. [ABSTRACT FROM AUTHOR]

A new method is proposed to analyze antenna arrays including mutual coupling (MC), which is based on the concept of multiple scattering, and relates to the iterative Jacobi and Gauss–Seidel methods. The method employs sampled far-field data of an isolated element, which can be obtained by any full-wave simulator and consists of far fields for excitation at the antenna feed and for plane waves with different angles of incidence. Mutual interactions between the array elements are modeled by approximating the incident field as a single dual-polarized plane wave taken from the spherical wave expansion of the scattered field from any other element in the array. The accuracy and run-time performances of the method are evaluated mainly by comparing simulations for several array geometries to method of moments (MoM)-based full-wave solutions. The method is primarily intended as a tool for the fast sequential analysis of arrays while varying the array lattice, particularly in the case of irregular or sparse lattices and complex elements requiring dense meshes in full-wave simulators. The applications of the method may thus range from the systematic analysis of MC to optimization and synthesis. [ABSTRACT FROM AUTHOR]

Performance feature extraction is the primary problem in equipment performance degradation assessment. To handle the problem of high-dimensional performance characterization and complexity of calculating the performance indicators in flexible material roll-to-roll processing, this paper proposes a PCA method for extracting the degradation characteristic of roll shaft. Based on the analysis of the performance influencing factors of flexible material roll-to-roll processing roller, a principal component analysis extraction model was constructed. The original feature parameter matrix composed of 10-dimensional feature parameters such as time domain, frequency domain, and time-frequency domain vibration signal of the roll shaft was established; then, we obtained a new feature parameter matrix Z org ∗ by normalizing the original feature parameter matrix. The correlation measure between every two parameters in the matrix Z org ∗ was used as the eigenvalue to establish the covariance matrix of the performance degradation feature parameters. The Jacobi iteration method was introduced to derive the algorithm for solving eigenvalue and eigenvector of the covariance matrix. Finally, using the eigenvalue cumulative contribution rate as the screening rule, we linearly weighted and fused the eigenvectors and derived the feature principal component matrix F of the processing roller vibration signal. Experiments showed that the initially obtained, 10-dimensional features of the processing rollers' vibration signals, such as average, root mean square, kurtosis index, centroid frequency, root mean square of frequency, standard deviation of frequency, and energy of the intrinsic mode function component, can be expressed by 3-dimensional principal components F 1 , F 2 , and F 3 . The vibration signal features reduction dimension was realized, and F 1 , F 2 , and F 3 contain 98.9% of the original vibration signal data, further illustrating that the method has high precision in feature parameters' extraction and the advantage of eliminating the correlation between feature parameters and reducing the workload selecting feature parameters. [ABSTRACT FROM AUTHOR]

In this article, Jacobi spectral Galerkin method and its iterated version are developed for second kind weakly singular Volterra–Hammerstein integral equations. We transform the domain of integration from [ 0 , t ] to [ − 1 , 1 ] by using variable and function transformations so that the theory of Jacobi polynomials can be applied and obtain the superconvergence results. Further, we enhance these superconvergence results in iterated Jacobi spectral multi-Galerkin method. We verify the theoretical results by numerical examples. [ABSTRACT FROM AUTHOR]

The parameterized generalized inverse eigenvalue problems containing multiplicative and additive inverse eigenvalue problems appear in vibrating systems design, structural design, and inverse Sturm–Liouville problems. In this article, by using the Cholesky factorization and the Jacobi method, we propose two efficient algorithms based on Newton's method and the L-BFGS-B method for solving these problems. To demonstrate the effectiveness of the algorithms, we present three numerical examples. [ABSTRACT FROM AUTHOR]

The one-sided block Jacobi method (OSBJ) has attracted attention as a fast and accurate algorithm for the singular value decomposition (SVD). The computational kernel of OSBJ is orthogonalization of a column block pair, which amounts to computing the SVD of this block pair. Hari proposes three methods for this partial SVD, and we found through numerical experiments that the variant named "V2", which is based on the Cholesky QR method, is the fastest variant and achieves satisfactory accuracy. While it is a good news from a practical viewpoint, it seems strange considering the well-known instability of the Cholesky QR method. In this paper, we perform a detailed error analysis of the V2 variant and explain why and when it can be used to compute the partial SVD accurately. Thus, our results provide a theoretical support for using the V2 variant safely in the OSBJ method. [ABSTRACT FROM AUTHOR]

To solve the problem of high cost and low strength of aerogel-based building materials, expanded perlite filled with aerogel (EPA) has been made by filling the pores in the expanded perlite with aerogel. Adding EPA to cement-based material can reduce the thermal conductivity of the cement-based material. In this study, in order to investigate the thermal conductivity of materials with different content of graded expanded perlite filled with aerogel (GEPA) and non-graded expanded perlite filled with aerogel (NEPA), the experimental and numerical studies have been carried out. The cement-based materials incorporating the EPA (CEPA) have been prepared by replacing different volumes of sand with the NEPA or GEPA, and the thermal conductivity of the CEPA has been measured using the modified transient plane source technique. Based on the law of conservation of energy and the principle of heat transfer, the thermal conductivity models of the CEPA have been developed using the Jacobi iterative method. The results show that the decreasing degree of thermal conductivity of the CEPA is positively correlated with the content of the EPA, and the performance of the GEPA is better than that of the NEPA. As the numerical and experimental results are consistent with each other, the numerical model has been proved to be reasonable and feasible for being widely used to calculate the thermal conductivity of EPA cement-based material, which can be used in sustainable buildings. [ABSTRACT FROM AUTHOR]

At present, numerical analysis provides us with powerful tools to determine the solution of various problems whose mathematical model can be represented by a system of linear equations, these tools correspond to a number of direct and iterative methods, among which are Carl's method. Gustav Jakob Jacobi and the Doolittle and Crout method, which we analyze and compare in this document. To do this we will initially explore the concepts of conditioning the problem to determine how stable is the system from which the model was obtained, until we reach the decomposition of LU arrays proposed in the Doolittle and Crout method. As a result of the analysis and comparison in this document, depending on what is sought when solving a system of equations, either very large or small enough for our computer, we can choose an approximation that will bring a short-term result with an error. Due to the starting point as proposed in the Jacobi method, or it is possible to reach a direct result by implementing fewer iterations as proposed in the Doolittle and Crout method. [ABSTRACT FROM AUTHOR]

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n × n BFSLEs into 4n × 4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs. [ABSTRACT FROM AUTHOR]

The core deformation of a one- and two-proton halo S nucleus was investigated based on experimental data. The Hamiltonian of the three-body systems and the root-mean-square (RMS) matter radii were used to calculate the theoretical value. The calculated theoretical value was analyzed using the relationship of the core deformation parameter () with the binding energy of one- and two-proton halos and the RMS matter radii of nucleus S. The Jacobi method was the primary tool used to describe the motion of the valence proton included in the wave function and was applied to the Hamiltonian of the three-body systems and the RMS matter radius. The calculation was run through MATLAB computational software. The results were shown with the experimental data. The RMS matter radius was large, and the core experienced clear deformation based on the binding energy of one- or two-valence protons ranging from –1.377 to –1.387 MeV. Nucleus S exhibits one- and two-proton halos owing to the low binding energy of the valence nucleon and exhibits either an oblate or prolate shape based on the theoretical binding energy of one- or two-valence protons, respectively. [ABSTRACT FROM AUTHOR]