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Journal articles on the topic 'Jacobi equivalence'

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 (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 (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 (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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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 (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 (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 t

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8

Xie, Chuanfu. "The equivalence between two jacobi identities for twisted vertex operators." Communications in Algebra 23, no. 7 (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 (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 (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 (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 moti

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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 (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 co

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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 (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 (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 (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 (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-

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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 (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">\be

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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 (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 inv

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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

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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 (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 Eul

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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 (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

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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

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28

Chen, Ling. "On axiomatic approaches to intertwining operator algebras." Communications in Contemporary Mathematics 18, no. 04 (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 “Jaco

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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 (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 reduc

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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 (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 applic

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31

Zeidan, Vera. "Local Minimality of a Lipschitz Extremal." Canadian Journal of Mathematics 44, no. 2 (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 (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

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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 (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

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34

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, anNdeg

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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 (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 construct

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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 (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 (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 (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 (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

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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 non

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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 (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

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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 (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 (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

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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 (2007): 145–53. http://dx.doi.org/10.1007/s10711-007-9200-6.

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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.

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

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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 ter

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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 rat

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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 (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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