Search

In this work, we present a new mathematical model for the spread of hepatitis C disease in two populations: human population and medical equipment population. Then, we apply the Jacobi wavelets method combined with the decoupling and quasi-linearization technique to solve this set of nonlinear differential equations for numerical simulation. [ABSTRACT FROM AUTHOR]

The paper reports a spectral method for generating an approximate solution for the space-time fractional PDEs with variable coefficients based on the spectral shifted Jacobi collocation method in conjunction with the shifted Jacobi operational matrix of fractional derivatives. The spectral collocation method investigates both temporal and spatial discretizations. By applying the shifted Jacobi collocation method, the problem reduces to a system of algebraic equations, which greatly simplifies the problem. Numerical results are given to establish the validity and accuracy of the presented procedure for space-time fractional PDE. [ABSTRACT FROM AUTHOR]

We extend the so-called Kotani Theory for a particular class of ergodic block Jacobi operators defined in l2(Z;Cl) by the law [Hω u]n:= D∗(Tn-1ω)un-1 +D(Tnω)un+1 + V(Tnω)un, where T: Ω → Ω is an ergodic automorphism in the measure space (Ω,ν), the map D: Ω →GL(l,R) is bounded, and for each ω ∈ Ω, D(ω) is symmetric and D-1(ω) is bounded. Namely, it is shown that for each r ∈{1, . . ., l}, the essential closure of Zr := {x ∈ R | exactly 2r Lyapunov exponents of Az are zero} coincides with σac,2r(Hω), the absolutely continuous spectrum of multiplicity 2r, where Az is a Schr¨odinger-like cocycle induced by Hω . Moreover, if k ∈ {1, . . .,2l} is odd, then σac,k(Hω) =/0 for ν -a.e. ω ∈ Ω. We also provide a Thouless formula for such class of operators. [ABSTRACT FROM AUTHOR]

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman-Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin-Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side. [ABSTRACT FROM AUTHOR]

A recently proposed transition-state wave packet method [R. Welsch, F. Huarte-Larrañaga, and U. Manthe, J. Chem. Phys. 136, 064117 (2012)] provides an efficient and intuitive framework to study reactive quantum scattering at the state-to-state level. It propagates a few transition-state wave packets, defined by the eigenfunctions of the low-rank thermal flux operator located near the transition state, into the asymptotic regions of the reactant and product arrangement channels separately using the corresponding Jacobi coordinates. The entire S-matrix can then be assembled from the corresponding flux-flux cross-correlation functions for all arrangement channels. Since the transition-state wave packets can be defined in a relatively small region, its transformation into either the reactant or product Jacobi coordinates is accurate and efficient. Furthermore, the grid/basis for the propagation, including the maximum helicity quantum number K, is much smaller than that required in conventional wave packet treatments of state-to-state reactive scattering. This approach is implemented for atom-diatom reactions using a time-dependent wave packet method and applied to the H + D2 reaction with all partial waves. Excellent agreement with benchmark integral and differential cross sections is achieved. [ABSTRACT FROM AUTHOR]

Exact quantum dynamics calculations of the eigenstate energy levels and wavefunctions for all bound rovibrational states of the Ne3 trimer (J = 0-18) have been performed using the ScalIT suite of parallel codes. These codes employ a combination of highly efficient methods, including phase-space optimized discrete variable representation, optimal separable basis, and preconditioned inexact spectral transform methods, together with an effective massive parallelization scheme. The Ne3 energy levels and wavefunctions were computed using a pair-wise Lennard-Jones potential. Jacobi coordinates were used for the calculations, but to identify just those states belonging to the totally symmetric irreducible representation of the G12 complete nuclear permutation-inversion group, wavefunctions were plotted in hyperspherical coordinates. 'Horseshoe' states were observed above the isomerization barrier, but the horseshoe localization effect is weaker than in Ar3. The rigid rotor model is found to be applicable for only the ground and first excited vibrational states at low J; fitted rotational constant values are presented. [ABSTRACT FROM AUTHOR]

We present an algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves given by an equation ym = f(x). It relies on rigorous numerical integration of differentials between Weierstrass points, which is done using Gauss method if the curve is hyperelliptic (m = 2) or the Double-Exponential method. The algorithm is implemented and makes it possible to reach thousands of digits accuracy even on large genus curves. [ABSTRACT FROM AUTHOR]

We present new asymptotic series for the Legendre and Jacobi functions of the first and second kinds in terms of Bessel functions with appropriate arguments. The results are useful in the context of scattering problems, improve on known limiting results, and allow the calculation of corrections to the leading Bessel-function approximations for these functions. Our derivations of these series are based on Barnes-type representations of the Legendre, Jacobi, and Bessel functions; our method appears to be new. We use the results, finally, to obtain asymptotic Bessel function expansions for the rotation functions needed to describe the scattering of particles with spin. [ABSTRACT FROM AUTHOR]

Highlights • A modified version of our variable-size batched GJE inversion routine for GPUs that inverts several blocks per warp. • A new variant of the extraction procedure that requires a much smaller amount of shared memory. • A tailored variant of the sparse matrix-vector multiplication that exploits the structure of the preconditioner matrix. Abstract In this work, we address the efficient realization of block-Jacobi preconditioning on graphics processing units (GPUs). This task requires the solution of a collection of small and independent linear systems. To fully realize this implementation, we develop a variable-size batched matrix inversion kernel that uses Gauss-Jordan elimination (GJE) along with a variable-size batched matrix–vector multiplication kernel that transforms the linear systems' right-hand sides into the solution vectors. Our kernels make heavy use of the increased register count and the warp-local communication associated with newer GPU architectures. Moreover, in the matrix inversion, we employ an implicit pivoting strategy that migrates the workload (i.e., operations) to the place where the data resides instead of moving the data to the executing cores. We complement the matrix inversion with extraction and insertion strategies that allow the block-Jacobi preconditioner to be set up rapidly. The experiments on NVIDIA's K40 and P100 architectures reveal that our variable-size batched matrix inversion routine outperforms the CUDA basic linear algebra subroutine (cuBLAS) library functions that provide the same (or even less) functionality. We also show that the preconditioner setup and preconditioner application cost can be somewhat offset by the faster convergence of the iterative solver. [ABSTRACT FROM AUTHOR]

We propose an extension of n -ary Nambu–Poisson bracket to superspace ℝ n | m and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace ℝ n | m . We prove in the case of the superspaces ℝ n | 1 and ℝ n | 2 that our n -ary bracket, defined with the help of superdeterminant, satisfies the conditions for n -ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of n -ary bracket defined with the help of superdeterminant in the case of superspace ℝ n | 2 and show that it is the sum of usual n -ary Nambu–Poisson bracket and a new n -ary bracket, which we call χ -bracket, where χ is the product of two odd degree smooth functions. [ABSTRACT FROM AUTHOR]

We construct L∞ algebras for general “initial data” given by a vector space equipped with an antisymmetric bracket not necessarily satisfying the Jacobi identity. We prove that any such bracket can be extended to a 2-term L∞ algebra on a graded vector space of twice the dimension, with the 3-bracket being related to the Jacobiator. While these L∞ algebras always exist, they generally do not realize a nontrivial symmetry in a field theory. In order to define L∞ algebras with genuine field theory realizations, we prove the significantly more general theorem that if the Jacobiator takes values in the image of any linear map that defines an ideal there is a 3-term L∞ algebra with a generally nontrivial 4-bracket. We discuss special cases such as the commutator algebra of octonions, its contraction to the “R-flux algebra,” and the Courant algebroid. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to establish iterative algorithms for computing the Takagi factorization of complex symmetric matrices. Similar to the classical iterative algorithms of computing the eigenpairs of real symmetric matrices, we derive power-like iterations for computing the Takagi values and associated Takagi vectors of complex symmetric matrices, i.e., the power-like method, the orthogonal-like iteration and the complex symmetric QR-like iteration. We analyze the convergence of these algorithms under some mild conditions. We also investigate the Jacobi-like methods for computing the Takagi factorization of complex symmetric matrices like Jacobi's methods for real symmetric eigenvalue problems. We illustrate our algorithms via numerical examples. [ABSTRACT FROM AUTHOR]

We present a construction of the Jacobi-Maupertuis (JM) principle for an equation of the Liénard type, x + f(x) x² + g(x) = 0; using Jacobi's last multiplier. The JM metric allows us to reformulate the Newtonian equation of motion for a variable mass as a geodesic equation for a Riemannian metric. We illustrate the procedure with examples of Painlevé-Gambier XXI, the Jacobi equation and the Henon-Heiles system. [ABSTRACT FROM AUTHOR]

In this paper, the short-term capture (STC) of an asteroid in the Earth--Moon system is proposed and investigated. First, the space condition of STC is analysed and five subsets of the feasible region are defined and discussed. Then, the time condition of STC is studied by parameter scanning in the Sun--Earth--Moon--asteroid restricted four-body problem. Numerical results indicate that there is a clear association between the distributions of the time probability of STC and the five subsets. Next, the influence of the Jacobi constant on STC is examined using the space and time probabilities of STC. Combining the space and time probabilities of STC, we propose a STC index to evaluate the probability of STC comprehensively. Finally, three potential STC asteroids are found and analysed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O ( p 2 d ) storage and O ( p 3 d ) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O ( p d + 1 ) storage, O ( p d + 1 ) work in two spatial dimensions, and O ( p d + 2 ) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O ( p 9 ) to O ( p 5 ) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p = 30 . For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p . [ABSTRACT FROM AUTHOR]

We investigate Hamilton Jacobi Isaacs equations associated to a two-players zero-sum differential game with incomplete information. The first player has complete information on the initial state of the game while the second player has only information of a – possibly uncountable – probabilistic nature: he knows a probability measure on the initial state. Such differential games with finite type incomplete information can be viewed as a generalization of the famous Aumann–Maschler theory for repeated games. The main goal and novelty of the present work consists in obtaining and investigating a Hamilton Jacobi Isaacs Equation satisfied by the upper and the lower values of the game. Since we obtain a uniqueness result for such Hamilton Jacobi equation, as a byproduct, this gives an alternative proof of the existence of a value of the differential game (which has been already obtained in the literature by different technics). Since the Hamilton Jacobi equation is naturally stated in the space of probability measures, we use the Wasserstein distance and some tools of optimal transport theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cubic Schrödinger equation with a bounded potential, which describes the propagation properties of optical soliton solutions. By employing an ansatz method, we precisely derive the bright and dark soliton solutions of the equation. Moreover, we obtain three classes of analytic periodic wave solutions expressed in terms of the Jacobi's elliptic functions including cn , sn and dn functions. Finally, by using a tan h function method, its complexitons solutions are derived in a very natural way. It is hoped that our results can enrich the nonlinear dynamical behaviors of the cubic Schrödinger equation with a bounded potential. [ABSTRACT FROM AUTHOR]

We prove two new forms of Jacobi-type J-fraction expansions generating the binomial coefficients, (x+n n) and (x n), over all n ≥ 0. Within the article we establish new forms of integer congruences for these binomial coefficient variations modulo any (prime or composite) h ≥ 2 and compare our results with known identities for the binomial coefficients modulo primes p and prime powers pk. We also prove new exact formulas for these binomial coefficient cases from the expansions of the hth convergent functions to the infinite J-fraction series generating these coefficients for all n. [ABSTRACT FROM AUTHOR]

In this paper, we study off-shell currents built from the Jacobi identity of the kinematic numerators of gg → X with $$ X=ss,q\overline{q}, gg $$ . We find that these currents can be schematically written in terms of three-point interaction Feynman rules. This representation allows for a straightforward understanding of the Colour-Kinematics duality as well as for the construction of the building blocks for the generation of higher-multiplicity tree-level and multi-loop numerators. We also provide one-loop integral relations through the Loop-Tree duality formalism with potential applications and advantages for the computation of relevant physical processes at the Large Hadron Collider. We illustrate these integral relations with the explicit examples of QCD one-loop numerators of gg → ss. [ABSTRACT FROM AUTHOR]

The Shortley–Weller method is a standard finite difference method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. In this case, an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial. The extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary. A practical remedy, which we call artificial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system. Our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 3 ) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the artificial perturbation. According to our analysis, the preconditioning not only reduces the eigenvalue ratio from O ( 1 / ( h ⋅ h m i n ) to O ( h − 2 ) , but also keeps the sharp second order convergence. [ABSTRACT FROM AUTHOR]

Currently, community detection in complex networks has become a hot-button topic. In this paper, based on the Spectral Clustering (SC) algorithm, we introduce the idea of Jacobi iteration, and then propose a novel algorithm J-SC for community detection in complex networks. Furthermore, the accuracy and efficiency of this algorithm are tested by some representative real-world networks and several computer-generated networks. The experimental results indicate that the J-SC algorithm can accurately and effectively detect the community structure in these networks. Meanwhile, compared with the state-of-the-art community detecting algorithms SC, SOM, K-means, Walktrap and Fastgreedy, the J-SC algorithm has better performance, reflecting that this new algorithm can acquire higher values of modularity and NMI. Moreover, this new algorithm has faster running time than SOM and Walktrap algorithms. [ABSTRACT FROM AUTHOR]

The β-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at the general order of the correlator via Aomoto's method from the theory of the Selberg integral. We use Aomoto's method to derive the full hierarchy of loop equations for Laguerre and Jacobi β-ensembles and use these to systematically construct the explicit form of the 1/N expansion at low orders. This allows us to give the explicit form of corrections to the global density and allows various moments to be computed, complementing results available in the literature motivated by problems in quantum transport. [ABSTRACT FROM AUTHOR]

In this study, we give the necessary and sufficient conditions for the existence of a solution to the Hochstadt inverse eigenvalue problem (HIEP), i.e., the problem of recovering n unknown parameters of a Jacobi matrix from all n of its eigenvalues and n − 1 known parameters. Effective algorithms are proposed for solving the HIEP. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of enhancing GPU-accelerated Krylov solvers with preconditioners. We consider the BiCGSTAB, CGS, QMR, and IDR( s ) Krylov solvers. For a large set of test matrices, we assess the impact of Jacobi and incomplete factorization preconditioning on the solvers’ numerical stability and time-to-solution performance. We also analyze how the use of a preconditioner impacts the choice of the fastest solver. [ABSTRACT FROM AUTHOR]

In this paper, some of formulations of Hamilton–Jacobi equations for Hamiltonian system and regular reduced Hamiltonian systems are given. At first, an important lemma is proved, and it is a modification for the corresponding result of Abraham and Marsden (1978), such that we can prove two types of geometric Hamilton–Jacobi theorem for a Hamiltonian system on the cotangent bundle of a configuration manifold, by using the symplectic form and dynamical vector field. Then these results are generalized to the regular reducible Hamiltonian system with symmetry and momentum map, by using the reduced symplectic form and the reduced dynamical vector field. The Hamilton–Jacobi theorems are proved and two types of Hamilton–Jacobi equations, for the regular point reduced Hamiltonian system and the regular orbit reduced Hamiltonian system, are obtained. As an application of the theoretical results, the regular point reducible Hamiltonian system on a Lie group is considered, and two types of Lie–Poisson Hamilton–Jacobi equation for the regular point reduced system are given. In particular, the Type I and Type II of Lie–Poisson Hamilton–Jacobi equations for the regular point reduced rigid body and heavy top systems are shown, respectively. [ABSTRACT FROM AUTHOR]

This work studies the evolution of several Distant Retrograde Orbits (DROs) of varying size in the Earth-Moon system over durations up to tens of millennia. This analysis is relevant for missions requiring a completely hands off, long duration quarantine orbit, such as a Mars Sample Return mission or the Asteroid Redirect Mission. Four DROs are selected from four stable size regions and are propagated for up to 30,000 years with an integrator that uses extended precision arithmetic techniques and a high fidelity dynamical model. The evolution of the orbit's size, shape, orientation, period, out-of-plane amplitude, and Jacobi constant are tracked. It has been found that small DROs, with minor axis amplitudes of approximately 45,000 km or less decay in size and period largely due to the Moon's solid tides. Larger DROs (62,000 km and up) are more influenced by the gravity of bodies external to the Earth-Moon system, and remain bound to the Moon for significantly less time. [ABSTRACT FROM AUTHOR]

Evidence theory has strong ability to handle epistemic uncertainties whose precise probability distributions cannot be obtained due to limited information. However, the excessive computational cost produced by repetitively extreme value analysis severely influences the practical application of evidence theory. This paper aims to develop an efficient algorithm for epistemic uncertainty analysis of acoustic problem under evidence theory. Based on the orthogonal polynomial approximation theory, a numerical approach named as the evidence-theory-based Jacobi expansion method (ETJEM) is proposed. In ETJEM, the response of acoustic system with evidence variables is approximated by Jacobi expansion, through which the repetitively extreme value analysis needed in evidence theory can be efficiently performed. The parametric Jacobi polynomial of Jacobi expansion holds a large number of polynomials as special cases, such as the Legendre polynomial and Chebyshev polynomial. Thus, the ETJEM permits a much wider choice of polynomial bases to control the error of approximation than the traditional evidence-theory-based orthogonal polynomial approximation method, in which only the Legendre polynomial is used for approximation. Three numerical examples are employed to demonstrate the effectiveness of the proposed methodology, including a mathematic problem with explicit expression and two engineering applications in acoustic field. In these three numerical examples, efficiency and accuracy are fully studied by comparing with Legendre expansion method as well as Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ignorable coordinate. Some examples are given, demonstrating that these special surfaces can be quite elaborate and varied. [ABSTRACT FROM AUTHOR]

A monic generalized Jacobi matrix $$ \mathfrak{J} $$ is factorized into upper and lower triangular two-diagonal block matrices of special forms so that J = UL. It is shown that such factorization depends on a free real parameter d(∈ ℝ). As the main result, it is shown that the matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)}= LU $$ is also a monic generalized Jacobi matrix. The matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ is called the Darboux transform of $$ \mathfrak{J} $$ with parameter d. An analog of the Geronimus formula for polynomials of the first kind of the matrix $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ is proved, and the relations between m-functions of J and $$ {\mathfrak{J}}^{\left(\mathbf{d}\right)} $$ are found. [ABSTRACT FROM AUTHOR]

We show that many existing elastic body simulation approaches can be interpreted as descent methods, under a nonlinear optimization framework derived from implicit time integration. The key question is how to find an effective descent direction with a low computational cost. Based on this concept, we propose a new gradient descent method using Jacobi preconditioning and Chebyshev acceleration. The convergence rate of this method is comparable to that of L-BFGS or nonlinear conjugate gradient. But unlike other methods, it requires no dot product operation, making it suitable for GPU implementation. To further improve its convergence and performance, we develop a series of step length adjustment, initialization, and invertible model conversion techniques, all of which are compatible with GPU acceleration. Our experiment shows that the resulting simulator is simple, fast, scalable, memory-efficient, and robust against very large time steps and deformations. It can correctly simulate the deformation behaviors of many elastic materials, as long as their energy functions are second-order differentiable and their Hessian matrices can be quickly evaluated. For additional speedups, the method can also serve as a complement to other techniques, such as multi-grid. [ABSTRACT FROM AUTHOR]

In this paper, we study the r-Jacobi-Stirling numbers of the second kind introduced by Gelineau in his Phd thesis. We give, upon using combinatorial and analytic arguments, the ordinary generating function of these numbers, two recurrence relations, their exact expressions and the log-concavity. [ABSTRACT FROM AUTHOR]

We numerically solve the equations of motion (EOM) for two models of circular cosmic string loops with windings in a simply connected internal space. Since the windings cannot be topologically stabilized, stability must be achieved (if at all) dynamically. As toy models for realistic compactifications, we consider windings on a small section of $$\mathbb {R}^2$$ , which is valid as an approximation to any simply connected internal manifold if the winding radius is sufficiently small, and windings on an $$S^2$$ of constant radius $$\mathcal {R}$$ . We then use Kosambi-Cartan-Chern (KCC) theory to analyze the Jacobi stability of the string equations and determine bounds on the physical parameters that ensure dynamical stability of the windings. We find that, for the same initial conditions, the curvature and topology of the internal space have nontrivial effects on the microscopic behavior of the string in the higher dimensions, but that the macroscopic behavior is remarkably insensitive to the details of the motion in the compact space. This suggests that higher-dimensional signatures may be extremely difficult to detect in the effective $$(3+1)$$ -dimensional dynamics of strings compactified on an internal space, even if configurations with nontrivial windings persist over long time periods. [ABSTRACT FROM AUTHOR]

For a large number of real nonlinear equations, either continuous or discrete, integrable or nonintegrable, we show that whenever a real nonlinear equation admits a solution in terms of sech x , it also admits solutions in terms of the PT-invariant combinations sech x ± i tanh ⁡ x . Further, for a number of real nonlinear equations we show that whenever a nonlinear equation admits a solution in terms sech 2 x , it also admits solutions in terms of the PT-invariant combinations sech 2 x ± i sech x tanh ⁡ x . Besides, we show that similar results are also true in the periodic case involving Jacobi elliptic functions. [ABSTRACT FROM AUTHOR]

In Paper II, we presented conjectures of the recurrence relations with constant coefficients for the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper we present a proof for the Laguerre and Jacobi cases. Their bispectral properties are also discussed, which gives a method to obtain the coefficients of the recurrence relations explicitly. This paper extends to the Laguerre and Jacobi cases the bispectral techniques recently introduced by Gómez- Ullate et al. [J. Approx. Theory 204, 1 (2016); e-print arXiv:1506.03651 [math.CA]] to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of Hermite type. [ABSTRACT FROM AUTHOR]

The purpose of this paper is an extension of Jacobi’s criteria for positive definiteness of second variation of the simplest problems of calculus of variations subject to mixed boundary conditions. Both non constrained and isoperimetric problems are discussed. The main result is that if we stipulate conditions (21) and (22) then Jacobi’s condition remains valid also for the mixed boundary conditions. [ABSTRACT FROM AUTHOR]

Many communication algorithms in parallel systems can be efficiently solved by obtaining edge disjoint Hamiltonian cycles in the interconnection topology of the network. The Eisenstein–Jacobi (EJ) network generated by α = a + b ρ , where ρ = ( 1 + i 3 ) / 2 , is a degree six symmetric interconnection network. The hexagonal network is a special case of the EJ network that can be obtained by α = a + ( a + 1 ) ρ . Generating three edge disjoint Hamiltonian cycles in the EJ network with generator α = a + b ρ for gcd ( a , b ) = 1 has been shown before. However, this problem has not been solved when gcd ( a , b ) = d > 1 . In this paper, some results to this problem are given. [ABSTRACT FROM AUTHOR]

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given. [ABSTRACT FROM AUTHOR]

The Hamiltonian of the N-planetary problem is written in the Jacobi coordinates using the second system of Poincare elements. The Hamiltonian is expanded into the Poisson series for the four-planet system. The computer algebra system Piranha is used for analytical transformations. Obtained expansions provide the Hamiltonian expression accuracy up to the third degree of the small parameter for giant planets of the Solar System and up to the second degree of the small parameter for extrasolar planetary systems. The ratio of sums of masses of the planets to the star mass can be selected as a small parameter. [ABSTRACT FROM AUTHOR]

Finding new techniques to accelerate electromagnetic (EM) simulations has become a necessity nowadays due to its frequent usage in industry. As they are mainly based on domain discretization, EM simulations require solving enormous systems of linear equations simultaneously. Available software-based solutions do not scale well with the increasing number of equations to be solved. As a result, hardware accelerators have been utilized to speed up the process. We introduce using hardware emulation as an efficient solution for EM simulation core solvers. Two different scalable architectures are implemented to accelerate the solver part of an EM simulator based on the Gaussian Elimination and the Jacobi iterative methods. Results show that the performance gap between presented solutions and software-based ones increases as the number of equations increases. For example, solving 2,002,000 equations using our Clustered Jacobi design in single floating-point precision achieved a speed-up of 100.88x and 35.24x over pure software implementations represented by MATLAB and the ALGLIB C++ package, respectively. [ABSTRACT FROM AUTHOR]

The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth-Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points $$L_1$$ and $$L_2$$ as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions. [ABSTRACT FROM AUTHOR]

The exact solution to the one-dimensional Poisson-Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically, thus a numerical scheme has been applied. [ABSTRACT FROM AUTHOR]

We find a basis for the space Sk(Γ1(4)) of cusp forms of weight k for the congruence subgroup Γ1(4) in terms of Eisenstein series. As an application, we obtain formulas for r2k(n), the number of ways to represent a non-negative integer n as sums of 2k integer squares. [ABSTRACT FROM AUTHOR]

We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic manifolds of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards along the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that the orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure. [ABSTRACT FROM AUTHOR]

It is common practice in analyses of the configurations of an elastica to use Jacobi's necessary condition to establish conditions for stability. Analyses of this type date to Born's seminal work on the elastica in 1906 and continue to the present day. Legendre developed a treatment of the second variation which predates Jacobi's. The purpose of this paper is to explore Legendre's treatment with the aid of three classical buckling problems for elastic struts. Central to this treatment is the issue of existence of solutions to a Riccati differential equation. We present two different variational formulations for the buckling problems, both of which lead to the same Riccati equation, and we demonstrate that the conclusions from Legendre and Jacobi's treatments are equivalent for some sets of boundary conditions. In addition, the failure of both treatments to classify stable configurations of a free-free strut are contrasted. [ABSTRACT FROM AUTHOR]

In this paper we suggest a direct method for studying local minimizers of one-dimensional variational problems which naturally complements the classical local theory. This method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable. The basis of these results is a regularity theory (a priori estimates and compactness in C) for solutions of obstacle problems with sufficiently close obstacles. In these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory. [ABSTRACT FROM AUTHOR]

The stopping problem with variance as the optimality criterion is introduced. Due to the variance criterion, smooth fit cannot be applied directly. The problem is solved by embedding it into tractable auxiliary optimal stopping problems, where smooth fit is used to obtain explicit, optimal solutions. Optimal strategies are presented in closed form for several examples. A characteristic feature is that the optimal stopping boundaries depend on the initial value of the gain process, i.e. the state space of the gain process does not split into one continuation set and one stopping set. [ABSTRACT FROM AUTHOR]

The article discusses the identity of Jacobi bracket, its differential identities, and its inverse problem for kinetic equations. It determines the natural constraints which show the definite sign of the corresponding quadratic form. It presents the existence theorem for a class of equations. It explains the validity of a theorem on a manifold with affine connection. It notes the application of the tensor rule of summation over the repeated indices. The equations and theorems proving the solution of the problem are also presented.