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Journal articles on the topic 'Jacobi equivalence'

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Author: Grafiati

Published: 10 March 2023

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1

Kozhan, Rostyslav. "Equivalence classes of block Jacobi matrices." Proceedings of the American Mathematical Society 139, no. 03 (March 1, 2011): 799. http://dx.doi.org/10.1090/s0002-9939-2010-10582-8.

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2

Ryckman, E. "A spectral equivalence for Jacobi matrices." Journal of Approximation Theory 146, no. 2 (June 2007): 252–66. http://dx.doi.org/10.1016/j.jat.2006.12.005.

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3

Murre, J. P. "Abel-Jacobi equivalence versus incidence equivalence for algebraic cycles of codimension two." Topology 24, no. 3 (1985): 361–67. http://dx.doi.org/10.1016/0040-9383(85)90008-4.

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4

Koornwinder, Tom H. "On the equivalence of two fundamental theta identities." Analysis and Applications 12, no. 06 (October 22, 2014): 711–25. http://dx.doi.org/10.1142/s0219530514500559.

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Two fundamental theta identities, a three-term identity due to Weierstrass and a five-term identity due to Jacobi, both with products of four theta functions as terms, are shown to be equivalent. One half of the equivalence was already proved by R. J. Chapman in 1996. The history and usage of the two identities, and some generalizations are also discussed.

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5

Lekner, John. "Four solutions of a two-cylinder electrostatic problem, and identities resulting from their equivalence." Quarterly Journal of Mechanics and Applied Mathematics 73, no. 3 (August 1, 2020): 251–60. http://dx.doi.org/10.1093/qjmam/hbaa010.

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Summary Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $\theta _1 ,~\theta _2 $ and infinite series over trigonometric and hyperbolic functions.

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6

Faraggi, Alon E., and Marco Matone. "Equivalence principle, Planck length and quantum Hamilton–Jacobi equation." Physics Letters B 445, no. 1-2 (December 1998): 77–81. http://dx.doi.org/10.1016/s0370-2693(98)01484-1.

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7

Faraggi, Alon E. "The Equivalence Postulate of Quantum Mechanics, Dark Energy, and the Intrinsic Curvature of Elementary Particles." Advances in High Energy Physics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/957394.

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The equivalence postulate of quantum mechanics offers an axiomatic approach to quantum field theories and quantum gravity. The equivalence hypothesis can be viewed as adaptation of the classical Hamilton-Jacobi formalism to quantum mechanics. The construction reveals two key identities that underlie the formalism in Euclidean or Minkowski spaces. The first is a cocycle condition, which is invariant underD-dimensional Möbius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of theD-dimensional quantum Hamilton-Jacobi equation. In this approach, the solutions of the associated Schrödinger equation are used to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property of the construction is that the two solutions of the corresponding Schrödinger equation must be retained. The quantum potential, which arises in the formalism, can be interpreted as a curvature term. The author proposes that the quantum potential, which is always nontrivial and is an intrinsic energy term characterising a particle, can be interpreted as dark energy. Numerical estimates of its magnitude show that it is extremely suppressed. In the multiparticle case the quantum potential, as well as the mass, is cumulative.

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8

Xie, Chuanfu. "The equivalence between two jacobi identities for twisted vertex operators." Communications in Algebra 23, no. 7 (January 1995): 2453–67. http://dx.doi.org/10.1080/00927879508825354.

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9

Wang, Jianjun, Chan-Yun Yang, and Shukai Duan. "Approximation Order for Multivariate Durrmeyer Operators with Jacobi Weights." Abstract and Applied Analysis 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/970659.

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Using the equivalence relation betweenK-functional and modulus of smoothness, we establish a strong direct theorem and an inverse theorem of weak type for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex in this paper. We also obtain a characterization for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex. The obtained results not only generalize the corresponding ones for Bernstein-Durrmeyer operators, but also give approximation order of Bernstein-Durrmeyer operators.

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10

Zanelli, Lorenzo. "Hamilton–Jacobi Homogenization and the Isospectral Problem." Symmetry 13, no. 7 (July 2, 2021): 1196. http://dx.doi.org/10.3390/sym13071196.

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We consider the homogenization theory for Hamilton–Jacobi equations on the one-dimensional flat torus in connection to the isospectrality problem of Schrödinger operators. In particular, we link the equivalence of effective Hamiltonians provided by the weak KAM theory with the class of the corresponding operators exhibiting the same spectrum.

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11

Obukhov, Valeriy. "Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)." International Journal of Geometric Methods in Modern Physics 17, no. 14 (October 27, 2020): 2050186. http://dx.doi.org/10.1142/s0219887820501868.

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We can find all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of variables in the Hamilton–Jacobi equation. Separation is carried out using the complete sets of mutually-commuting integrals of motion of type (2.1), whereby in a privileged coordinate system the Hamilton–Jacobi equation turns into a parabolic type equation.

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12

BUCATARU, IOAN, OANA CONSTANTINESCU, and MATIAS F. DAHL. "A GEOMETRIC SETTING FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS." International Journal of Geometric Methods in Modern Physics 08, no. 06 (September 2011): 1291–327. http://dx.doi.org/10.1142/s0219887811005701.

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To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.

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13

El Mfadel, Ali, Said Melliani, and M’hamed Elomari. "New Results on the Equivalence of K -Functionals and Modulus of Continuity of Functions Defined on the Sobolev Space Constructed by the Generalized Jacobi-Dunkl Operator." Advances in Mathematical Physics 2022 (January 20, 2022): 1–6. http://dx.doi.org/10.1155/2022/2835927.

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In this paper, we establish some new generalized results on the equivalence of K -functionals and modulus of continuity of functions defined on the Sobolev space L α , β 2 ℝ , by using the harmonic analysis related to the Jacobi-Dunkl operator Δ α , β , where α ≥ β ≥ − 1 / 2 and α ≠ − 1 / 2 .

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14

VAISMAN, IZU. "COUPLING POISSON AND JACOBI STRUCTURES ON FOLIATED MANIFOLDS." International Journal of Geometric Methods in Modern Physics 01, no. 05 (October 2004): 607–37. http://dx.doi.org/10.1142/s0219887804000307.

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Let M be a differentiable manifold endowed with a foliation ℱ. A Poisson structure P on M is ℱ-coupling if ♯P(ann(Tℱ)) is a normal bundle of the foliation. This notion extends Sternberg's coupling symplectic form of a particle in a Yang–Mills field [11]. In the present paper we extend Vorobiev's theory of coupling Poisson structures [16] from fiber bundles to foliated manifolds and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. We then discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

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15

El Hamma, Mohamed, and Radouan Daher. "Equivalence of K-functionals and modulus of smoothness constructed by generalized Jacobi transform." Integral Transforms and Special Functions 30, no. 12 (July 3, 2019): 1018–24. http://dx.doi.org/10.1080/10652469.2019.1635127.

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16

Gómez-Ullate, David, Yves Grandati, and Robert Milson. "Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials." Journal of Physics A: Mathematical and Theoretical 51, no. 34 (July 16, 2018): 345201. http://dx.doi.org/10.1088/1751-8121/aace4b.

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17

Adsuara, J. E., I. Cordero-Carrión, P. Cerdá-Durán, V. Mewes, and M. A. Aloy. "On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method." Journal of Computational Physics 332 (March 2017): 446–60. http://dx.doi.org/10.1016/j.jcp.2016.12.020.

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18

de León, Manuel, and Manuel Lainz Valcázar. "Singular Lagrangians and precontact Hamiltonian systems." International Journal of Geometric Methods in Modern Physics 16, no. 10 (October 2019): 1950158. http://dx.doi.org/10.1142/s0219887819501585.

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In this paper, we discuss the singular Lagrangian systems on the framework of contact geometry. These systems exhibit a dissipative behavior in contrast with the symplectic scenario. We develop a constraint algorithm similar to the presymplectic one studied by Gotay and Nester (the geometrization of the well-known Dirac–Bergmann algorithm). We also construct the Hamiltonian counterpart and prove the equivalence with the Lagrangian side. A Dirac–Jacobi bracket is constructed similar to the Dirac bracket.

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19

Brezhnev, Yurii V. "What does integrability of finite-gap or soliton potentials mean?" Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (June 27, 2007): 923–45. http://dx.doi.org/10.1098/rsta.2007.2056.

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In the example of the Schrödinger/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field—new explicit Ψ -function, trace formula and the Jacobi problem—provide a complete solution. The Θ -function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.

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20

Soravia, P. "Equivalence between Nonlinear H ∈ fty Control Problems and Existence of Viscosity Solutions of HamiltonJacobiIsaacs Equations." Applied Mathematics and Optimization 39, no. 1 (January 2, 1999): 17–32. http://dx.doi.org/10.1007/s002459900096.

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21

Chen, Cui, Jiahui Hong, and Kai Zhao. "Global propagation of singularities for discounted Hamilton-Jacobi equations." Discrete & Continuous Dynamical Systems 42, no. 4 (2022): 1949. http://dx.doi.org/10.3934/dcds.2021179.

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<p style='text-indent:20px;'>The main purpose of this paper is to study the global propagation of singularities of the viscosity solution to discounted Hamilton-Jacobi equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE333"> \begin{document}$ \begin{align} \lambda v(x)+H( x, Dv(x) ) = 0 , \quad x\in \mathbb{R}^n. \quad\quad\quad (\mathrm{HJ}_{\lambda})\end{align} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with fixed constant <inline-formula><tex-math id="M1">\begin{document}$ \lambda\in \mathbb{R}^+ $\end{document}</tex-math></inline-formula>. We reduce the problem for equation <inline-formula><tex-math id="M2">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> into that for a time-dependent evolutionary Hamilton-Jacobi equation. We prove that the singularities of the viscosity solution of <inline-formula><tex-math id="M3">\begin{document}$(\mathrm{HJ}_{\lambda})$\end{document}</tex-math></inline-formula> propagate along locally Lipschitz singular characteristics <inline-formula><tex-math id="M4">\begin{document}$ {{\bf{x}}}(s):[0,t]\to \mathbb{R}^n $\end{document}</tex-math></inline-formula> and time <inline-formula><tex-math id="M5">\begin{document}$ t $\end{document}</tex-math></inline-formula> can extend to <inline-formula><tex-math id="M6">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>. Essentially, we use <inline-formula><tex-math id="M7">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-compactness of the Euclidean space which is different from the original construction in [<xref ref-type="bibr" rid="b4">4</xref>]. The local Lipschitz issue is a key technical difficulty to study the global result. As a application, we also obtain the homotopy equivalence between the singular locus of <inline-formula><tex-math id="M8">\begin{document}$ u $\end{document}</tex-math></inline-formula> and the complement of Aubry set using the basic idea from [<xref ref-type="bibr" rid="b9">9</xref>].</p>

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22

Karlsen, Kenneth Hvistendahl, and Nils Henrik Risebro. "A note on front tracking and the equivalence between viscosity solutions of Hamilton–Jacobi equations and entropy solutions of scalar conservation laws." Nonlinear Analysis: Theory, Methods & Applications 50, no. 4 (August 2002): 455–69. http://dx.doi.org/10.1016/s0362-546x(01)00753-2.

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23

Bunta, Silviu. "The Likeness of the Image: Adamic Motifs and Anthropoly in Rabbinic Traditions about Jacob's Image Enthroned in Heaven." Journal for the Study of Judaism 37, no. 1 (2006): 55–84. http://dx.doi.org/10.1163/157006306775454497.

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AbstractThe present article analyzes the various texts concerning Jacob's image engraved on the throne of glory. It compares the Jacob texts with previous traditions regarding Adam's special status as the image of God or the equivalent of a cultic representation of an ancient Near Eastern king or of a Roman emperor. The Jacob texts reveal a similar anthropology that emphasizes the dichotomy of humanity. On one hand the earthliness of the functionality of the human body is associated with angelic opposition, and, on the other, the body's divine likeness gives rise to angelic veneration. The investigation of the two traditions demonstrates a conspicuous dependence of the Jacob texts on the Adamic traditions.

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24

Zhao, Jutao, and Pengfei Guo. "A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method." Journal of Mathematics 2021 (December 7, 2021): 1–10. http://dx.doi.org/10.1155/2021/2123897.

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The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

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25

Morita, Shigeyuki. "Families of Jacobian manifolds and characteristic classes of surface bundles. II." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (January 1989): 79–101. http://dx.doi.org/10.1017/s0305004100001389.

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Let Σg be a closed orientable surface of genus g, which will be assumed to be greater than one throughout this paper. In our previous paper [11], we have associated to any oriented ∑g-bundle π: E → X with a cross-section s: X → E a flat T2g-bundle π: J → X and a fibre-preserving embedding j: E → J such that the restriction of j to any fibre Ep = π−1(p)(p ∈ X) is equivalent to the Jacobi mapping of Ep with respect to some conformal structure on it and relative to the base-point s(p) ∈ Ep. There is a canonical oriented S1-bundle over J and the main result of [11] is the identification of the Euler class of the pullback of this S1-bundle by the map j as an element of H2(E, ). In this paper we deal with the case of Σg-bundles without cross-sections. First of all we associate a flat T2g-bundle π′: J′ → X to any oriented Σg-bundle π: E → X and construct a fibre-preserving embedding j′: E → J′ such that the restriction of j′ to any fibre Ep is equivalent to some translation of the Jacobi mapping of it. Although our original motivation for the present work came from a different source, this should be considered as a topological version of Earle's embedding theorem [5] which states that any holomorphic family of compact Riemann surfaces over a complex manifold can be embedded in a certain associated family of the corresponding Jacobian varieties in an essentially unique way.

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26

Shigeta, Yoshinori, Kiyoshi Akama, Hiroshi Mabuchi, and Hidekatsu Koike. "Converting Constraint Handling Rules to Equivalent Transformation Rules." Journal of Advanced Computational Intelligence and Intelligent Informatics 10, no. 3 (May 20, 2006): 339–48. http://dx.doi.org/10.20965/jaciii.2006.p0339.

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We present a way to convert constraint handling rules (CHRs) to equivalent transformation rules (ETRs) and demonstrate the correctness of the conversion in equivalent transformation (ET) theory. In the ET computation model, computation is regarded as equivalent transformations of a description. A description is transformed successively by ETRs. Extensively used in the domain of first-order terms, the ET computation model has also been applied to knowledge processing in such data domains as RDF, UML, and XML. A CHR is a multiheaded guarded rule that rewrites constraints into simpler ones until they are solved. CHRs and ETRs are similar in syntax but they have completely different theoretical bases for the correctness of their computation. CHRs are based on the logical equivalence of logical formulas, while ETRs are based on the set equivalence of descriptions. We convert CHRs to rules used in the ET model and demonstrate converted rules to be correct ETRs, i.e., they preserve meanings of descriptions. We discuss correspondences and differences between CHRs and ETRs in theories, giving examples of correct ETRs that cannot be represented as CHRs.

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27

Abe, Yoshiki, Gou Nishida, Noboru Sakamoto, and Yutaka Yamamoto. "Robust NonlinearH∞Control Design via Stable Manifold Method." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/198380.

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This paper proposes a systematic numerical method for designing robust nonlinearH∞controllers without a priori lower-dimensional approximation with respect to solutions of the Hamilton-Jacobi equations. The method ensures the solutions are globally calculated with arbitrary accuracy in terms of the stable manifold method that is a solver of Hamilton-Jacobi equations in nonlinear optimal control problems. In this realization, the existence of stabilizing solutions of the Hamilton-Jacobi equations can be derived from some properties of the linearized system and the equivalent Hamiltonian system that is obtained from a transformation of the Hamilton-Jacobi equation. A numerical example is shown to validate the design method.

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28

Chen, Ling. "On axiomatic approaches to intertwining operator algebras." Communications in Contemporary Mathematics 18, no. 04 (May 3, 2016): 1550051. http://dx.doi.org/10.1142/s0219199715500510.

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We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [Y.-Z. Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000) 225–267] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [Y.-Z. Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000) 225–267] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and braiding isomorphisms for general intertwining operator algebras and prove that they satisfy the genus-zero Moore–Seiberg equations. We show that the Jacobi identity for intertwining operator algebras is equivalent to generalized rationality, commutativity and associativity properties of intertwining operator algebras. We introduce the locality for intertwining operator algebras and show that the Jacobi identity is equivalent to the locality, assuming that other axioms hold. Moreover, we establish that any two of the three properties, associativity, commutativity and skew-symmetry, imply the other (except that when deriving skew-symmetry from associativity and commutativity, more conditions are needed). Finally, we show that three definitions of intertwining operator algebras are equivalent.

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29

Yang, Ping, and Yao-Lin Jiang. "Truncated model reduction methods for linear time-invariant systems via eigenvalue computation." Transactions of the Institute of Measurement and Control 42, no. 10 (February 3, 2020): 1908–20. http://dx.doi.org/10.1177/0142331219899745.

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This paper provides three model reduction methods for linear time-invariant systems in the view of the Riemannian Newton method and the Jacobi-Davidson method. First, the computation of Hankel singular values is converted into the linear eigenproblem by the similarity transformation. The Riemannian Newton method is used to establish the model reduction method. Besides, we introduce the Jacobi-Davidson method with the block version for the linear eigenproblem and present the corresponding model reduction method, which can be seen as an acceleration of the former method. Both the resulting reduced systems can be equivalent to the reduced system originating from a balancing transformation. Then, the computation of Hankel singular values is transformed into the generalized eigenproblem. The Jacobi-Davidson method is employed to establish the model reduction method, which can also lead to the reduced system equivalent to that resulting from a balancing transformation. This method can also be regarded as an acceleration of a Riemannian Newton method. Moreover, the application for model reduction of nonlinear systems with inhomogeneous conditions is also investigated.

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30

Kuznetsov, V. B., and E. K. Sklyanin. "Eigenproblem for Jacobi matrices: hypergeometric series solution." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (June 22, 2007): 1089–114. http://dx.doi.org/10.1098/rsta.2007.2062.

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We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d . The (small) expansion parameters are the entries of the two diagonals of length d −1 sandwiching the principal diagonal that gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3 d −5 variables in the generic case. To derive the result, we first rewrite the spectral problem for the Jacobi matrix as an equivalent system of algebraic equations, which are then solved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.

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31

Zeidan, Vera. "Local Minimality of a Lipschitz Extremal." Canadian Journal of Mathematics 44, no. 2 (April 1, 1992): 436–48. http://dx.doi.org/10.4153/cjm-1992-028-0.

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AbstractIn this paper the question of weak and strong local optimality of a Lipschitz (as opposed to C1 ) extremal is addressed. We show that the classical Jacobi sufficient conditions can be extended to the case of Lipschitz candidates. The key idea for this achievement lies in proving that the “generalized” strengthened Weierstrass condition is equivalent to the existence of a “feedback control” function at which the maximum in the “true” Hamiltonian is attained. Then the Hamilton-Jacobi approach is pursued in order to conclude the result.

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32

Mishra, T. N., and B. Tiwari. "Stability and Bifurcation Analysis of a Prey–Predator Model." International Journal of Bifurcation and Chaos 31, no. 04 (March 30, 2021): 2150059. http://dx.doi.org/10.1142/s0218127421500590.

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The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.

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33

Hamani, Fatima, and Azedine Rahmoune. "Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method." Tatra Mountains Mathematical Publications 80, no. 3 (December 1, 2021): 35–52. http://dx.doi.org/10.2478/tmmp-2021-0030.

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Abstract In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both L ∞ and weighted L 2 norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.

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Peng, Linyu, Huafei Sun, and Xiao Sun. "Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–26. http://dx.doi.org/10.1155/2011/710274.

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We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, anNdegrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

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35

DE TRAUBENBERG, M. RAUSCH, and C. A. SAVOY. "EQUIVALENT COMPLEX AND REAL FERMIONS IN HETEROTIC SUPERSTRING SOLUTIONS." International Journal of Modern Physics A 06, no. 08 (March 30, 1991): 1301–12. http://dx.doi.org/10.1142/s0217751x9100068x.

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As a first step towards a classification of the heterotic superstring solutions in the fermionic approach, we study a class of solutions characterized by complex fermions with ZN boundary conditions. The same solutions can be obtained by an equivalent modular invariant system of real fermions with Z2 boundary conditions. The proof is supplied by Riemann identities for Jacobi Θ-functions that we derive. The patterns of the gauge groups of the D-dimensional solutions are determined for 4≤D≤10, and their uniqueness in ten dimensions is checked for the different ZN boundary conditions. A construction of the E8×E8 supersymmetric solution based on nine complex fermions with Z3 periodicity is exhibited.

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36

TOH, PEE CHOON. "GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES." International Journal of Number Theory 04, no. 03 (June 2008): 461–74. http://dx.doi.org/10.1142/s1793042108001456.

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We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.

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37

Gersten, Alexander. "Tensor Lagrangians, Lagrangians Equivalent to the Hamilton-Jacobi Equation and Relativistic Dynamics." Foundations of Physics 41, no. 1 (September 29, 2009): 88–98. http://dx.doi.org/10.1007/s10701-009-9352-3.

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38

Goulden, I. P. "Directed Graphs and the Jacobi-Trudi Identity." Canadian Journal of Mathematics 37, no. 6 (December 1, 1985): 1201–10. http://dx.doi.org/10.4153/cjm-1985-065-6.

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Let |aij|n×n denote the n × n determinant with (i, j)-entry aij, and hk = hk(x1, …, xn) denote the kth-homogeneous symmetric function of x1, …, xn defined bywhere the summation is over all m1, …, mn ≧ 0 such that m1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α1 ≧ α2 … ≧ αn ≧ 0, the Jacobi-Trudi identity (see [6], [7]) states thatIn this paper we give a combinatorial proof of an equivalent identity, Theorem 1.1, obtained by moving the denominator on the RHS to the numerator on the LHS.

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39

Platonov, V. P., and G. V. Fedorov. "On S-units for linear valuations and the periodicity of continued fractions of generalized type in hyperelliptic fields." Доклады Академии наук 486, no. 3 (May 30, 2019): 280–86. http://dx.doi.org/10.31857/s0869-56524863280-286.

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This article proves the equivalence theorem for the following conditions: the periodicity of continued fractions of a generalized type for key elements hyperelliptic field L, the existence in the hyperelliptic field L of nontrivial S-units for sets S, consisting two valuations of degree one, and the existence of the torsion of a certain type in the Jacobian variety, associated with the hyperelliptic field L. This theorem allows in practice using continued fractions of a generalized type effectively search for fundamental S-units of hyperelliptic fields. We give an example of the hyperelliptic field of genus 3, showing all three equivalent conditions in the indicated theorem.

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40

YAKUBOVICH, E. I., and D. A. ZENKOVICH. "Matrix approach to Lagrangian fluid dynamics." Journal of Fluid Mechanics 443 (September 25, 2001): 167–96. http://dx.doi.org/10.1017/s0022112001005195.

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A new approach to ideal-fluid hydrodynamics based on the notion of continuous deformation of infinitesimal material elements is proposed. The matrix approach adheres to the Lagrangian (material) view of fluid motion, but instead of Lagrangian particle trajectories, it treats the Jacobi matrix of their derivatives with respect to Lagrangian variables as the fundamental quantity completely describing fluid motion.A closed set of governing matrix equations equivalent to conventional Lagrangian equations is formulated in terms of this Jacobi matrix. The equation of motion is transformed into a nonlinear matrix differential equation in time only, where derivatives with respect to the Lagrangian variables do not appear. The continuity equation that requires constancy of the Jacobi determinant in time takes the form of an algebraic constraint on the Jacobi matrix. An accompanying linear consistency condition, which is responsible for the dependence on spatial variables and does not include time derivatives, ensures completeness of the system and reconstruction of the particle trajectories by the Jacobi matrix.A class of exact solutions to the matrix equations that describes rotational non-stationary three-dimensional motions having no analogues in the conventional formulations is also found and investigated. A distinctive feature of these motions is precession of vortex lines (rectilinear or curvilinear) around a fixed axis in space. Boundary problems for the derived exact solutions including matching of rotational and potential motions across the boundary of a vortex tube are addressed. In particular, for the cylindrical vortex of elliptical cross-section involved in three-dimensional precession, the outer potential flow is constructed and shown to be a non-stationary periodic straining flow at a large distance from the vortex axis.

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41

Blaga, Cristina, Paul Blaga, and Tiberiu Harko. "Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole." Symmetry 15, no. 2 (January 24, 2023): 329. http://dx.doi.org/10.3390/sym15020329.

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We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.

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42

Kondo, Michiro, and Wieslaw A. Dudek. "Topological Structures of Rough Sets Induced by Equivalence Relations." Journal of Advanced Computational Intelligence and Intelligent Informatics 10, no. 5 (September 20, 2006): 621–24. http://dx.doi.org/10.20965/jaciii.2006.p0621.

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In this paper we consider some fundamental topological properties of rough sets induced by equivalence relations and show that 1. Every approximation space is retrieval. 2. For every approximation space <I>X</I>=(<I>X,θ</I>), <I>X</I> is strongly connected if and only if <I>θ</I>=<I>X</I>×<I>X</I>. Moreover we consider topological properties of generalized rough sets.

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43

Melo, Margarida, Antonio Rapagnetta, and Filippo Viviani. "Fourier–Mukai and autoduality for compactified Jacobians. I." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 755 (October 1, 2019): 1–65. http://dx.doi.org/10.1515/crelle-2017-0009.

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AbstractTo every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.

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44

Fu, Baohua, and Fabien Herbaut. "On the tautological ring of a Jacobian modulo rational equivalence." Geometriae Dedicata 129, no. 1 (November 1, 2007): 145–53. http://dx.doi.org/10.1007/s10711-007-9200-6.

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45

Peykrayegan, Narges, Mehdi Ghovatmand, Mohammad Hadi Noori Skandari, and Dumitru Baleanu. "An approximate approach for fractional singular delay integro-differential equations." AIMS Mathematics 7, no. 5 (2022): 9156–71. http://dx.doi.org/10.3934/math.2022507.

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<abstract><p>In this article, we present Jacobi-Gauss collocation method to numerically solve the fractional singular delay integro-differential equations, because such methods have better superiority, capability and applicability than other methods. We first apply a technique to replace the delay function in the considered equation and suggest an equivalent system. We then propose a Jacobi-Gauss collocation approach to discretize the obtained system and to achieve an algebraic system. Having solved the algebraic system, an approximate solution is gained for the original equation. Three numerical examples are solved to show the applicability of presented approximate approach. Obtaining the approximations of the solution and its fractional derivative simultaneously and an acceptable approximation by selecting a small number of collocation points are advantages of the suggested method.</p></abstract>

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46

MOTOYUI, NOBUYUKI, SHOGO TOMINAGA, and MITSURU YAMADA. "HAMILTON–JACOBI SOLUTION TO SOLITON PATHS AND TRIANGULAR MASS RELATION IN TWO-DIMENSIONAL EXTENDED SUPERSYMMETRIC THEORY." Modern Physics Letters A 16, no. 24 (August 10, 2001): 1559–63. http://dx.doi.org/10.1142/s021773230100490x.

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D=2, N=2 generalized Wess–Zumino theory is investigated by the dimensional reduction from D=4, N=1 theory. For each solitonic configuration (i,j), the classical static solution is solved by the Hamilton–Jacobi method of equivalent one-dimensional classical mechanics. It is easily shown that the Bogomol'nyi mass bound is saturated by these solutions and triangular mass inequality [Formula: see text] is automatically satisfied.

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47

Andrews, Lew. "Jacob Burckhardt, Clive Bell and the ‘Equivalents’ of Alfred Stieglitz." History of Photography 27, no. 3 (September 2003): 247–53. http://dx.doi.org/10.1080/03087298.2003.10441250.

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48

Bernis, Julien, and Piernicola Bettiol. "Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 66. http://dx.doi.org/10.1051/cocv/2019041.

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We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.

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49

Lin, Gui Hua, Yan Jun Zhang, Tao Wang, and Yu Ying Wang. "State Estimation of Equivalent Current Measurement Transformation Based on Generalized Tellegen's Theorem." Advanced Materials Research 732-733 (August 2013): 941–47. http://dx.doi.org/10.4028/www.scientific.net/amr.732-733.941.

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One of the most important ways to enhance the speed of state estimation is to establish the constant matrix Jacobian. This essay puts forward the state estimation method of the equivalent current transformation based on the Generalized Tellegen’s Theorem. This estimation method establishes the constant Jacobian matrix without neglecting the secondary factor making use of the Generalized Tellegen’s Theorem, solves the numerical stability problem caused by the establishment of the constant Jacobian matrix in the current state estimation, and has the advantages of a relatively rapid computing rate and an unparalleled astringency. The method put forward in this essay has been verified through IEEE-30 Node System, and the efficiency of it has been fully proved by the example results.

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50

Mikhaylov, A. S., and V. S. Mikhaylov. "On an application of the Boundary control method to classical moment problems." Journal of Physics: Conference Series 2092, no. 1 (December 1, 2021): 012002. http://dx.doi.org/10.1088/1742-6596/2092/1/012002.

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Abstract We establish relationships between the classical moments problems which are problems of a construction of a measure supported on a real line, on a half-line or on an interval from prescribed set of moments with the Boundary control approach to a dynamic inverse problem for a dynamical system with discrete time associated with Jacobi matrices. We show that the solution of corresponding truncated moment problems is equivalent to solving some generalized spectral problems.

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