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This article presents a new approach of shifted Jacobi spectral Galerkin methods to solve weakly singular Fredholm integral equations with non-smooth solutions. We have incorporated the singular part of the kernel into a single Jacobi weight function, by dividing the integration into two parts and using a simple variable transformation. Taking advantage of orthogonal projection operator and weighted inner product with respect to that same Jacobi weight function, we are able to obtain improved convergence rate for iterated shifted Jacobi spectral Galerkin method (SJSGM) and iterated shifted Jacobi spectral multi-Galerkin method (SJSMGM) in both weighted and infinity norms. Further, we obtain improved superconvergence rate for iterated SJSGM and iterated SJSMGM, by improving the regularity of exact solution, using smoothing transformation. Increasing the value of the smoothing parameter we can improve the regularity of the exact solution upto the desired degree. Numerical results with a comparative study of pre and post smoothing transformation are given to illustrate the theoretical results and efficiency of our proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, a reliable Jacobi Galerkin method is developed and analyzed to solve a particular class of cordial Volterra integral equations. Existence and uniqueness theorems approve that these kinds of equations possess smooth solutions versus smooth data, so representing their Galerkin solutions based on suitable polynomial basis produces spectrally accurate approximations. Moreover, in order to control condition number growth, we designed a reliable approach that calculates the approximate solutions recursively without solving any high-conditioning systems. Indeed, this approach deals with solving equations in a long integration domain with even high oscillatory solutions. The convergence analysis of the presented approach is established, and the familiar spectral accuracy is justified in L ∞ -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. Finally, we provide an application of this method to approximate solution of a two-dimensional case. [ABSTRACT FROM AUTHOR]

In this study, the travelling wave solutions of the general Drinfel'd–Sokolov–Wilson (DSW)-system, introduced as a model of water waves, are obtained and wave dynamics are investigated. Jacobi elliptic functions are valuable mathematical tools that can be used to various aspects of physics engineering. The method used is effective in generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different, and some new periodic solutions can be obtained. To examine the behaviour of the solutions obtained for appropriate values of the parameters, three-dimensional simulations were conducted using Maple TM . Additionally, 2D simulations are presented to physicists for easy observation of wave motion. The usage of these simulations aids in better understanding and predicting the behaviour of waves, tidal forces and other phenomena, ultimately contributing to the development of safer and more efficient structures and systems. The stability property of the obtained solutions was tested to demonstrate their reliability. The novelty of this study is that the rich solution group obtained with Jacobi elliptic functions is different from the solutions in the literature and thus provides new physical interpretations. The findings are expected to contribute to the development of new analytical and numerical tools for solving other nonlinear systems in the future, containing a large number of travelling wave solutions crucial for explaining certain scientific phenomena in fluid media. [ABSTRACT FROM AUTHOR]

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations. [ABSTRACT FROM AUTHOR]

The two-sided Jacobi method for singular value decomposition (SVD) has the advantage of obtaining singular vectors quickly and accurately. In previous research, fast and accurate implementations of the two-sided Jacobi method have been achieved for real matrices. In this study, we implemented SVD for complex matrices using the two-sided Jacobi method. In SVD, given rectangular matrices can be converted into upper-triangular matrices by conducting QR decomposition as a preprocessing step. Then, the upper-triangular matrices are decomposed. In the case where the given matrices are complex, the upper-triangular matrices are complex. Two implementations are proposed: the first requires SVD for complex matrices, whereas the second requires SVD for only real matrices. [ABSTRACT FROM AUTHOR]

In this paper, we employ Tikhonov regularization method with the projection methods using Jacobi polynomial bases to the first kind of Fredholm integral equations to find the approximate solution. We discuss the convergence analysis and obtain the convergence rates in L w α , β 2 norm under a priori parameter choice strategy. We also consider the Engl-type discrepancy principle as a posteriori parameter strategy for finding the regularization parameter and also evaluate the convergence rate which is of optimal order. Finally, we provide the numerical experiments to justify the theoretical results. [ABSTRACT FROM AUTHOR]

The use of optical fiber for communication has grown at a rapid pace due to the demands of the modern information era. In this paper, the modified extended direct algebraic method is implemented to explore optical solitons and other exact wave solutions for the coupled system of perturbed highly dispersive complex Ginzburg–Landau equation with polynomial nonlinearity law which describe the transmission of solitons in birefringent fibers. The obtained solutions include bright solitons, dark solitons, singular solitons and combo dark-singular solitons. Additionally, Jacobi elliptic function solutions, exponential solutions, and singular periodic solutions are also offered. To ensure the existence of the obtained soliton solutions, we set some constraints on the parameters. In addition, some selected solutions are visually shown to demonstrate the physical properties of the exact solutions. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation are studied in light transmission, optical fiber communication, and nonlinear effects in optics. The auxiliary equation method is not only suitable for solving specific types of nonlinear partial differential equations, but also has strong applicability to all kinds of different types of equations. It helps us to deduce the exact solution of the equation faster and analyzes the dynamic behavior of the system further. The extended fourth Jacobi elliptic equation is used in this paper to seek different types of exact solutions, which include bright soliton solutions, kink solutions, periodic wave solutions, and singular traveling wave solutions via selecting appropriate parameters.The characteristics of some solutions are graphically presented using two- and three-dimensional graphs such as the real part, the imaginary part, and their modulus via providing suitable values to arbitrary parameters. Compared to other methods, the method is more direct and easier for calculations. Kindly check and confirm the edit made in the title. The edit made in the title is correct. The word "extended" in the title can be deleted if necessary. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

This study addresses the integrable Akbota equation which is a Heisenberg ferromagnet-type equation and has a significant study of the curve and surface geometry. The conservation laws of the model are computed using the multipliers approach. The Jacobi elliptic function solutions for the model are derived by utilizing the new extended auxiliary equation method. The Jacobi elliptic functions solutions are degenerated to dark, bright, singular and singular periodic solitons in their limit conditions. Further, the modulation instability analysis of the equation is studied. Physical interpretations of the obtained results are represented graphically. To the best of our knowledge, the obtained results in this article are novel and have not been reported before. [ABSTRACT FROM AUTHOR]

The solution of the Poisson equation raised from large-scale problems requires iterative techniques. Multilevel and Krylov subspace methods have been proven to be the most efficient among various iterative methods. Recently, the scheduled relaxation Jacobi technique has been developed, which can be used as a stand-alone iterative solver as well as a preconditioner to the Krylov subspace methods. The relaxed Jacobi (RJ) method has also been implemented as a smoother in the multigrid method, with no-post smoothening. Here, the focus is to test the performance of relaxed Jacobi when applied to the large scale Poisson problems. Multigrid methods with both post-smoothening and pre-smoothening operations have been conducted in this paper. The methodology is applied to a rectangular domain using structured Cartesian grids. Also, multigrid methods with relaxed Jacobi smoothers are used as preconditioners to Krylov subspace solvers, and their performances are compared with the red-black Gauss-Seidel smoothers. Similar to the results in the literature, RJ smoothers to multigrid techniques provide better results than Gauss-Seidel smoothers when no-post smoothening operations are carried out. However, the use of post-smoothening operations changed the entire dynamic of the RJ smoothers. [ABSTRACT FROM AUTHOR]

We consider the Alternating Direction Implicit (ADI) method to compute the numerical solution of a continuous Sylvester equation A X + X B = C , based on the recently developed inexact ADI iteration, and we propose classical acceleration techniques to enhance its convergence rate. An extrapolated variant (EADI) and a block successive overrelaxation variant (block SOR-ADI) of the ADI iterative method are described. These relaxation approaches are similar to what is used in Gauss-Seidel and Jacobi methods for linear systems, and, to our knowledge, novel, especially the block SOR-ADI scheme. Convergence properties of these two relaxed variants are analyzed when the matrix A is positive definite and the matrix B is positive semi-definite (not necessarily Hermitian matrices), or conversely. Our numerical experiments suggest that these new schemes are computationally attractive. The convergence rate of the ADI method is usually increased, particularly with the block SOR-ADI variant. A comparison with the well-known Hermitian and skew-Hermitian splitting (HSS) method emphasizes the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

Hadamard type fractional calculus involves logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenge in numerical treatment. In this paper we present a spectral collocation method with mapped Jacobi log orthogonal functions (MJLOFs) as basis functions and obtain an efficient algorithm to solve Hadamard type fractional differential equations. We develop basic approximation theory for the MJLOFs and derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e, the fractional Helmholtz equation, and the fractional Burgers equation. [ABSTRACT FROM AUTHOR]

A previous study by Mantas and Vecil (Int J High Perform Comput Appl 34(1): 81–102, 2019) describes an efficient and accurate solver for nanoscale DG MOSFETs through a deterministic Boltzmann-Schrödinger-Poisson model with seven electron–phonon scattering mechanisms on a hybrid parallel CPU/GPU platform. The transport computational phase, i.e. the time integration of the Boltzmann equations, was ported to the GPU using CUDA extensions, but the computation of the system's eigenstates, i.e. the solution of the Schrödinger-Poisson block, was parallelized only using OpenMP due to its complexity. This work fills the gap by describing a port to GPU for the solver of the Schrödinger-Poisson block. This new proposal implements on GPU a Scheduled Relaxation Jacobi method to solve the sparse linear systems which arise in the 2D Poisson equation. The 1D Schrödinger equation is solved on GPU by adapting a multi-section iteration and the Newton-Raphson algorithm to approximate the energy levels, and the Inverse Power Iterative Method is used to approximate the wave vectors. We want to stress that this solver for the Schrödinger-Poisson block can be thought as a module independent of the transport phase (Boltzmann) and can be used for solvers using different levels of description for the electrons; therefore, it is of particular interest because it can be adapted to other macroscopic, hence faster, solvers for confined devices exploited at industrial level. [ABSTRACT FROM AUTHOR]

The fundamental formula in an optical system is Rayleigh diffraction integral. In practice, we deal with Fresnel diffraction integral as approximate diffraction formula. Fresnel transforms and inverse Fresnel transforms have been formulated systematically and defined as a bounded additive, a unitary operator in Hilbert space. To investigate the band-limited effect in Fresnel transform plane, we seek the function that its total power in finite Fresnel transform plane is maximized, on condition that an input signal is zero outside the bounded region. This problem is a variational one with an accessory condition. This leads to the eigenvalue problems of Fredholm integral equation of the first kind. The kernel of the integral equation is of Hermitian conjugate and positive definite. Moreover, we prove that the eigenfunctions corresponding to distinct eigenvalues have dual orthogonal property. By discretizing the kernel and integral range to seek the approximate solutions, the eigenvalue problems of the integral equation can depend on a one of the Hermitian matrix in finite-dimensional complex value vector space. We use the Jacobi method to compute all eigenvalues and eigenvectors of the matrix. We consider the application of the eigenvectors to the problems of the approximating any functions. We show the validity and limitation of the eigenvectors in computer simulations. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the dynamical behavior of (1 + 1) -dimensional Kudryashov's equation with generalized anti-cubic nonlinearity by extracting the exact closed form solitons and other traveling wave solutions. This nonlinear mathematical model explains the propagation of pulses though optical fibers. It is based on the well-known nonlinear Schrödinger equation with an arbitrary power nonlinearity. In this paper, two techniques are used to extract the optical solitons and other solitary wave solutions of Kudryashov's equation. One is the general projective Riccati equation method and the other is the Jacobi elliptic method. The proposed techniques retrieve the explicit solution expressions by calculating the coefficients of a truncated power series. The exact solutions are obtained in terms of hyperbolic, trigonometric and Jacobi elliptic functions. The constructed solitons include kink, combo dark-bright and dark solitons. Periodic traveling wave solutions are also reported. The derived solutions are shown as three-dimensional graphs for the appropriate choice of parametric parameters. The corresponding two dimensional contour graphs are included to understand dynamical behavior of obtained results. [ABSTRACT FROM AUTHOR]

We consider strongly convex distributed consensus optimization over connected networks. EFIX, the proposed method, is derived using quadratic penalty approach. In more detail, we use the standard reformulation—transforming the original problem into a constrained problem in a higher dimensional space—to define a sequence of suitable quadratic penalty subproblems with increasing penalty parameters. For quadratic objectives, the corresponding sequence consists of quadratic penalty subproblems. For generic strongly convex case, the objective function is approximated with a quadratic model and hence the sequence of the resulting penalty subproblems is again quadratic. EFIX is then derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver, e.g., Jacobi Over-Relaxation method. The exact convergence is proved as well as the worst case complexity of order O (ϵ - 1) for the quadratic case. In the case of strongly convex generic functions, the standard result for penalty methods is obtained. Numerical results indicate that the method is highly competitive with state-of-the-art exact first order methods, requires smaller computational and communication effort, and is robust to the choice of algorithm parameters. [ABSTRACT FROM AUTHOR]

The paper revisits the topic of block-Jacobi algorithms for the symmetric eigenvalue problem by proposing a few alternative versions. The main advantage of a block Jacobi method is that it is built entirely from computations with small dense matrices. The proposed mehod is based on a sequence of subspace rotations whose determination requires to solve small Riccati-like correction equation. The paper discusses theoretical and algorithmic aspects of the algorithm, and illustrates its behavior on a few simple examples. [ABSTRACT FROM AUTHOR]

Solving weakly singular equations has wide applications in engineering and science. The delay term makes it difficult for most existing solvers to deal with. Therefore, constructing an efficient and accurate solver is a challenge. In this paper, a Jacobi spectral method is developed for the delay calculus equation with a weakly singular kernel. The error analysis is also provided to justify the spectral rate of convergence for the error of approximate solution and approximate derivative decay exponentially in both the infinity norm and the weighted L 2 norm. In the end, numerical results are displayed to confirm the effectiveness of the Jacobi spectral analysis. [ABSTRACT FROM AUTHOR]

The eikonal equation in an attenuating medium has the form of a complex—valued Hamilton—Jacobi equation and must be solved in terms of the complex—valued travel time. A very suitable approximate method for calculating the complex—valued travel time right in real space is represented by the perturbation from the reference travel time calculated along the real—valued reference rays to the complex—valued travel time defined by the complex—valued Hamilton—Jacobi equation. The real—valued reference rays are calculated using the reference Hamiltonian function. The reference Hamiltonian function is constructed using the complex—valued Hamiltonian function corresponding to a given complex—valued Hamilton—Jacobi equation. The ray tracing equations and the corresponding equations of geodesic deviation are often formulated in terms of the eigenvectors of the Christoffel matrix. Unfortunately, a complex—valued Christoffel matrix need not have all three eigenvectors at an S—wave singularity. We thus formulate the ray tracing equations and the corresponding equations of geodesic deviation using the eigenvalues of a complex—valued Christoffel matrix, without the eigenvectors of the Christoffel matrix. The resulting equations for the real—valued reference P—wave rays and the real—valued reference common S—wave rays are applicable everywhere, including S—wave singularities. [ABSTRACT FROM AUTHOR]

We propose a novel method for the computation of Jacobi sets in 2D domains. The Jacobi set is a topological descriptor based on Morse theory that captures gradient alignments among multiple scalar fields, which is useful for multi-field visualization. Previous Jacobi set computations use piecewise linear approximations on triangulations that result in discretization artifacts like zig-zag patterns. In this paper, we utilize a local bilinear method to obtain a more precise approximation of Jacobi sets by preserving the topology and improving the geometry. Consequently, zig-zag patterns on edges are avoided, resulting in a smoother Jacobi set representation. Our experiments show a better convergence with increasing resolution compared to the piecewise linear method. We utilize this advantage with an efficient local subdivision scheme. Finally, our approach is evaluated qualitatively and quantitatively in comparison with previous methods for different mesh resolutions and across a number of synthetic and real-world examples. [ABSTRACT FROM AUTHOR]

We present an inherently parallel method using a Taylor polynomial as an estimator for several numerical examples of strictly diagonally dominant and banded triangular Toeplitz systems. We demonstrate the effectiveness of the method using multiple precisions for each of four test cases, and compare the results versus classical Jacobi and Newton methods. We also show that the residual norms can be driven down to any arbitrary precision with less hardware requirements than existing methods. Finally, we use truncated Taylor series as initial estimates for the Newton, and in each case, we obtain equal or better accuracy than the Newton method alone, while using less vector computations. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the projected Jacobi and the projected Gauss–Seidel methods to vertical linear complementarity problems (VLCPs) characterized by matrices with positive diagonal entries. First, we formulate the methods and show that the subproblems that must be solved at each iteration have an explicit solution, which is easy to compute. Then, we prove the convergence of the proposed procedures when the matrices of the problem satisfy some assumptions of strict or irreducible diagonal dominance. In this context, for simplicity, we first analyze the convergence in the special case of VLCPs of dimension 2 n × n , and we then generalize the results to VLCPs of an arbitrary dimension ℓ n × n . Finally, we provide several numerical experiments (involving both full and sparse matrices) that show the effectiveness of the proposed approaches. In this context, our methods are compared with existing solution methods for VLCPs. A parallel implementation of the projected Jacobi method in CUDA is also presented and analyzed. [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]

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]

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]

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]

Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L2- and L ∞ - norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper. [ABSTRACT FROM AUTHOR]

In this paper, the collocation method is applied on two-dimensional integral equations of the second kind on non-rectangular domains. Since the domains of these equations are non-rectangular and so directly applying the collocation method for them is difficult, at first, the integral equations are converted to equivalent integral equations on rectangular domains. Then, two-dimensional Jacobi collocation method is applied. Furthermore, an error estimate for the method is investigated and several examples demonstrate the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak gravitational deflection of relativistic massive particles for a receiver and source at finite distance from the lens in stationary, axisymmetric and asymptotically flat spacetimes. For this purpose, we extend the generalized optical metric method to the generalized Jacobi metric method by using the Jacobi–Maupertuis Randers–Finsler metric. More specifically, we apply the Gauss–Bonnet theorem to the generalized Jacobi metric space and then obtain an expression for calculating the deflection angle, which is related to Gaussian curvature of generalized optical metric and geodesic curvature of particles orbit. In particular, the finite-distance correction to the deflection angle of signal with general velocity in the the Kerr black hole and Teo wormhole spacetimes are considered. Our results cover the previous work of the deflection angle of light, as well as the deflection angle of massive particles in the limit for the receiver and source at infinite distance from the lens object. In Kerr black hole spacetime, we compared the effects due to the black hole spin, the finite-distance of source or receiver, and the relativistic velocity in microlensings and lensing by galaxies. It is found in these cases, the effect of black hole spin is usually a few orders larger than that of the finite-distance and relativistic velocity, while the relative size of the latter two could vary according to the particle velocity, source or observer distance and other lensing parameters. [ABSTRACT FROM AUTHOR]

In this paper we present a paradigmatic example of the use in knowledge management of techniques from other fields, namely mathematical analysis. We also highlight that the Jacobi method presented here takes precedence over the better known Hungarian method. Finally, we signify that the Jacobi method represents the first known or recognized case of serendipity in both knowledge management and operational research. This paper thus demonstrates the intersection between knowledge management, mathematical analysis and operational research and how taking historical perspectives are important for recognising future applications of knowledge management. This results in a better understanding of knowledge management and how it can be applied in the future. It has been shown how knowledge management relates to historical mathematical principles. [ABSTRACT FROM AUTHOR]

Herein, we have proposed a scheme for numerically solving hyperbolic partial differential equations (HPDEs) with given initial conditions. The operational matrix of differentiation for exponential Jacobi functions was derived, and then a collocation method was used to transform the given HPDE into a linear system of equations. The preferences of using the exponential Jacobi spectral collocation method over other techniques were discussed. The convergence and error analyses were discussed in detail. The validity and accuracy of the proposed method are investigated and checked through numerical experiments. [ABSTRACT FROM AUTHOR]

Aircraft panel automatic drilling and riveting system is the key equipment to realize aircraft panel assembly with high efficiency and quality. In practical application, automatic drilling and riveting system should bear cyclic riveting force with large load, which means such systems must have enough stiffness to deal with this situation. The stiffness of each components in automatic drilling and riveting system reflects as the end stiffness finally; in the meanwhile, end stiffness and riveting ability of the system will be affected when the posture of the system changes. According to this situation, stiffness matrixes related to mechanics characteristic of motion axes, transmission parts, inner-side working head, and outer-side working head of automatic horizontal dual-machine cooperative drilling and riveting system are established, in which the Jacobi matrix method, point transformation matrix method, and finite element analysis are applied. Finally, the static end stiffness model is proposed based on multi-body small deflection theory. The end stiffness model has been validated through experimental measurement, and the stiffness relationship of dual-machine tool system was obtained, which applied important data for structure optimization and control system design. [ABSTRACT FROM AUTHOR]

Asynchronous algorithms may increase the performance of parallel applications on large-scale HPC platforms due to decreased dependence among processing elements. This work investigates strategies for implementing asynchronous hybrid parallel MPI–OpenMP iterative solvers. Seven different implementations are considered, and results show that striking a balance between communication and computation that balances the number of iterations in each processing element benefits performance and solution quality. A predictive performance model that utilizes kernel density estimation to model the underlying probability density function to the collected data is then developed to optimize execution parameters for a given problem. For the majority of iteration executions, the performance model matches within about 6% of the empirical data. The different hybrid parallel implementations are examined further to find optimal parametric distributions whose parameters can be tuned to the problem at hand. The generalized extreme value distribution was selected based on a combination of quantitative and qualitative comparisons, and for the most of the iteration executions, the model matches the data within about 6.1%. Results from the parametric distribution model are examined along with results of the model on related problems, and possible further extensions to the predictive model are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we propose two splitting methods for solving horizontal linear complementarity problems characterized by matrices with positive diagonal elements. The proposed procedures are based on the Jacobi and on the Gauss-Seidel iterations and differ from existing techniques in that they act directly and simultaneously on both matrices of the problem. We prove the convergence of the methods under some assumptions on the diagonal dominance of the matrices of the problem. Several numerical experiments, including large-scale problems of practical interest, demonstrate the capabilities of the proposed methods in various situations. [ABSTRACT FROM AUTHOR]

A recent generalization of the "Kleinian sigma function" involves the choice of a point P of a Riemann surface X, namely a "pointed curve" (X, P). This paper concludes our explicit calculation of the sigma function for curves cyclic trigonal at P. We exhibit the Riemann constant for a Weierstrass semigroup at P with minimal set of generators {3,2r+s,2s+r}, r

We are interested in the phenomenon of the essential spectrum instability for a class of unbounded (block) Jacobi matrices. We give a series of sufficient conditions for the matrices from certain classes to have a discrete spectrum on a half-axis of a real line. An extensive list of examples showing the sharpness of obtained results is provided. [ABSTRACT FROM AUTHOR]

We study the asymptotics of the global fluctuations for the difference between two adjacent levels in the β-Jacobi corners process (multilevel and general β extension of the classical Jacobi ensemble of random matrices). The limit is identified with the derivative of the 2d Gaussian free field. Our main tools are integral forms for the (Macdonald-type) difference operators originating from the shuffle algebra. [ABSTRACT FROM AUTHOR]

We study Jacobi matrices on trees with one end at infinity. We show that the defect indices cannot be greater than 1 and give criteria for essential self-adjointness. [ABSTRACT FROM AUTHOR]

We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalized eigenvectors and conditions implying complete indeterminacy are also provided. [ABSTRACT FROM AUTHOR]

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem Ax = λBx, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begović Kovač (Trans. Numer. Anal. (ETNA) 47, 107-147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of ΔAAΔAand ΔBBΔBare small, for some nonsingular diagonal matrices ΔAand ΔB. [ABSTRACT FROM AUTHOR]

In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate. [ABSTRACT FROM AUTHOR]

The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A[t+3]) ≤ γS(A[t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A[t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method. [ABSTRACT FROM AUTHOR]

In this paper, we construct a generalization of the Jacobi iterative method adapted to a system of linear bounded operators. This new method is used with product integration method to solve a linear Fredholm integral equation of the second kind with a weakly singular kernel. The convergence analysis is proved. The numerical tests developed show its effectiveness. [ABSTRACT FROM AUTHOR]

This paper presents an efficient method to solve the approximation problem of the rational Bézier curve by continuous piecewise polynomial curve in L2-norm. For this purpose, extended Jacobi wavelets together with the Gauss-Jacobi quadrature rules are employed. The proposed technique has some advantages such as: simplicity, high accuracy and fast computation. Also, the method in the paper performs multi-degree reduction at one time and does not need the stepwise computing. Several examples are given to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log-returns admits a Gram-Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical study, we show that option prices can be accurately and efficiently approximated by truncating their series representations. [ABSTRACT FROM AUTHOR]

We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted. [ABSTRACT FROM AUTHOR]

We disprove a conjecture of Breuer-Last-Simon (Breuer et al. in Constr Approx 32(2):221-254, 2010) concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an ℓ2 bounded variation condition with step q. We prove existence of a.c. spectrum on a smaller set than that specified by the conjecture and prove that our result is optimal. [ABSTRACT FROM AUTHOR]

The directed acyclic graph (DAG) associated with a parallel algorithm captures the partial order in which separate local computations are completed and how their outputs are subsequently used in further computations. Unlike in a synchronous parallel algorithm, the DAG associated with an asynchronous parallel algorithm is not predetermined. Instead, it is a product of the asynchronous timing dynamics of the machine and cannot be known in advance, as such it is best thought of as a pseudorandom variable. In this paper, we present a formalism for analyzing the performance of asynchronous parallel Jacobi's method in terms of its DAG. We use this approach to prove error bounds and bounds on the rate of convergence. The rate of convergence bounds is based on the statistical properties of the DAG and is valid for systems with a non-negative iteration matrix. We support our theoretical results with a suit of numerical examples, where we compare the performance of synchronous and asynchronous parallel Jacobi to certain statistical properties of the DAGs associated with the computations. We also present some examples of small matrices with elements of mixed sign, which demonstrate that determining whether a system will converge under asynchronous iteration in this more general setting is a far more difficult problem. [ABSTRACT FROM AUTHOR]

Following the history of mechanics (Dugas in A history of mechanics, Dover, New York, 1988), we read that Poisson's theorem figured in the 1811 edition of the celebrated Mécanique Analytique. Indeed, as it can be observed in the classical textbooks, Poisson brackets formulation is one of the cardinal chapters of analytical mechanics. Given the natural motivation to accommodate variable-mass systems at the level of classical analytical mechanics, we will herein direct our attention toward Poisson brackets formulation. To wit, considering the case of a position-dependent mass particle, in which we will assume that the absolute velocity of mass ejection or aggregation is a linear function of the generalized velocity, we will endeavor to provide such position-dependent mass problems with an appropriate Poisson brackets formulation. To our very best knowledge, this means an original contribution to the research field of the analytical mechanics of variable-mass systems. We will start establishing the Poisson brackets definition for the dynamics of a position-dependent mass particle, which will be posited in harmony with the classical mathematical portrait of analytical mechanics. Therefrom, we will demonstrate consequent results which give rise to the required formulation, namely, Jacobi's identity, canonical equations expressed by means of such Poisson brackets and Poisson's theorem. Last, we will apply Poisson's theorem to evaluate the relationship between the conservation laws which are at our disposal in the domain of position-dependent mass problems. [ABSTRACT FROM AUTHOR]