Academic literature on the topic 'Differential and difference algebra'

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Journal articles on the topic "Differential and difference algebra"

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Mikhalev, A. V., and E. V. Pankrat'ev. "Differential and difference algebra." Journal of Soviet Mathematics 45, no. 1 (1989): 912–55. http://dx.doi.org/10.1007/bf01094866.

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Zhang, Yang, and Xiangui Zhao. "Gelfand–Kirillov dimension of differential difference algebras." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 485–95. http://dx.doi.org/10.1112/s1461157014000102.

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AbstractDifferential difference algebras, introduced by Mansfield and Szanto, arose naturally from differential difference equations. In this paper, we investigate the Gelfand–Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand–Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand–Kirillov dimension under some specific conditions and construct an example to show that this upper bound cannot be sharpened any further.
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Zhao, Xiangui, and Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras." Algebra Colloquium 23, no. 04 (2016): 701–20. http://dx.doi.org/10.1142/s1005386716000596.

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Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-K
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Li, Wei, and Chun-Ming Yuan. "Elimination Theory in Differential and Difference Algebra." Journal of Systems Science and Complexity 32, no. 1 (2019): 287–316. http://dx.doi.org/10.1007/s11424-019-8367-x.

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Hooker, John W. "Some differences between difference equations and differential equations." Journal of Difference Equations and Applications 2, no. 2 (1996): 219–25. http://dx.doi.org/10.1080/10236199608808056.

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DOBREV, V. K., H. D. DOEBNER, and C. MRUGALLA. "DIFFERENCE ANALOGUES OF THE FREE SCHRÖDINGER EQUATION." Modern Physics Letters A 14, no. 17 (1999): 1113–22. http://dx.doi.org/10.1142/s021773239900119x.

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We propose an infinite family of difference equations, which are derived from the first principle that they are invariant with respect to the Schrödinger algebra. The first member of this family is a difference analogue of the free Schrödinger equation. These equations are obtained via a purely algebraic construction from a corresponding family of singular vectors in Verma modules over the Schrödinger algebra. The crucial moment in the construction is the realization of the Schrödinger algebra through additive difference vector fields, i.e. vector fields with difference operators instead of di
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Kakei, Saburo, and Yasuhiro Ohta. "A differential-difference system related to toroidal Lie algebra." Journal of Physics A: Mathematical and General 34, no. 48 (2001): 10585–92. http://dx.doi.org/10.1088/0305-4470/34/48/322.

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Levin, Alexander. "Reduced Gröbner Bases, Free Difference–Differential Modules and Difference–Differential Dimension Polynomials." Journal of Symbolic Computation 30, no. 4 (2000): 357–82. http://dx.doi.org/10.1006/jsco.1999.0412.

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Dunkl, Charles F. "Computing with Differential-difference Operators." Journal of Symbolic Computation 28, no. 6 (1999): 819–26. http://dx.doi.org/10.1006/jsco.1997.0341.

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SMIRNOV, YURI, and ALEXANDER TURBINER. "LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS." Modern Physics Letters A 10, no. 24 (1995): 1795–802. http://dx.doi.org/10.1142/s0217732395001927.

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A certain representation for the Heisenberg algebra in finite difference operators is established. The Lie algebraic procedure of discretization of differential equations with isospectral property is proposed. Using sl 2-algebra based approach, (quasi)-exactly-solvable finite difference equations are described. It is shown that the operators having the Hahn, Charlier and Meissner polynomials as the eigenfunctions are reproduced in the present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is intro
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Dissertations / Theses on the topic "Differential and difference algebra"

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Hendriks, Peter Anne. "Algebraic aspects of linear differential and difference equations." [S.l. : [Groningen] : s.n.] ; [University Library Groningen] [Host], 1996. http://irs.ub.rug.nl/ppn/153769580.

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Zhao, Xiangui. "Groebner-Shirshov bases in some noncommutative algebras." London Mathematical Society, 2014. http://hdl.handle.net/1993/24315.

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Groebner-Shirshov bases, introduced independently by Shirshov in 1962 and Buchberger in 1965, are powerful computational tools in mathematics, science, engineering, and computer science. This thesis focuses on the theories, algorithms, and applications of Groebner-Shirshov bases for two classes of noncommutative algebras: differential difference algebras and skew solvable polynomial rings. This thesis consists of three manuscripts (Chapters 2--4), an introductory chapter (Chapter 1) and a concluding chapter (Chapter 5). In Chapter 1, we introduce the background and the goals of the thesi
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El-Nakla, Jehad A. H. "Finite difference methods for solving mildly nonlinear elliptic partial differential equations." Thesis, Loughborough University, 1987. https://dspace.lboro.ac.uk/2134/10417.

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This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods
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Chen, Shaoshi. "Quelques applications de l'algébre différentielle et aux différences pour le télescopage créatif." Phd thesis, Ecole Polytechnique X, 2011. http://pastel.archives-ouvertes.fr/pastel-00576861.

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Depuis les années 90, la méthode de création télescopique de Zeilberger a joué un rôle important dans la preuve automatique d'identités mettant en jeu des fonctions spéciales. L'objectif de long terme que nous attaquons dans ce travail est l'obtension d'algorithmes et d'implantations rapides pour l'intégration et la sommation définies dans le cadre de cette création télescopique. Nos contributions incluent de nouveaux algorithmes pratiques et des critères théoriques pour tester la terminaison d'algorithmes existants. Sur le plan pratique, nous nous focalisons sur la construction de télescopeur
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Masmali, Ibtisam Ali. "Hopf algebra and noncommutative differential structures." Thesis, Swansea University, 2010. https://cronfa.swan.ac.uk/Record/cronfa42676.

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In this thesis I will study noncommutative differential geometry, after the style of Connes and Woronowicz. In particular two examples of differential calculi on Hopf algebras are considered, and their associated covariant derivatives and Riemannian geometry. These are on the Heisenberg group, and on the finite group A4. I consider bimodule connections after the work of Madore. In the last chapter noncommutative fibrations are considerd, with an application to the Leray spectral sequence. NOTATION. In this thesis equations are numbered as round brackets (), where (a.b) denotes equation b in ch
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Nagloo, Joel Chris Ronnie. "Model theory, algebra and differential equations." Thesis, University of Leeds, 2014. http://etheses.whiterose.ac.uk/6813/.

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In this thesis, we applied ideas and techniques from model theory, to study the structure of the sets of solutions XII - XV I , in a differentially closed field, of the Painlevé equations. First we show that the generic XII - XV I , that is those with parameters in general positions, are strongly minimal and geometrically trivial. Then, we prove that the generic XII , XIV and XV are strictly disintegrated and that the generic XIII and XV I are ω-categorical. These results, already known for XI , are the culmination of the work started by P. Painlevé (over 100 years ago), the Japanese school an
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Khalid, Abdul Muqeet. "Hypergeometric equation and differential-difference bispectrality." Thesis, University of Leeds, 2018. http://etheses.whiterose.ac.uk/21411/.

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The bispectral problem was posed by Duistermaat and Grünbaum in 1986. Since then, many interesting links of this problem with nonlinear integrable PDEs, algebraic geometry, orthogonal polynomials and special functions have been found. Bispectral operators of rank one are related to the KP equation and have been completely classified by G. Wilson. For rank greater than 1 some large families related to Bessel functions are known, although the classification problem remains open. If one generalises the bispectral problem by allowing difference operators in the spectral variable, then this has a
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Khanizadeh, Farbod. "Symmetry structure for differential-difference equations." Thesis, University of Kent, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655204.

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Having infinitely many generalised symmetries is one of the definition of integrability for non-linear differential-difference equations. Therefore, it is important to develop tools by which we can produce these quantities and guarantee the integrability. Two different methods of producing generalised symmetries are studied throughout this thesis, namely recursion operators and master symmetries. These are objects that enable one to obtain the hierarchy of symmetries by recursive action on a known symmetry of a given equation. Our first result contains new Hamiltonian, symplectic and recursion
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Postell, Floyd Vince. "High order finite difference methods." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/28876.

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Xue, Fei. "Asymptotic solutions of almost diagonal differential and difference systems." Morgantown, W. Va. : [West Virginia University Libraries], 2006. https://eidr.wvu.edu/etd/documentdata.eTD?documentid=4556.

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Books on the topic "Differential and difference algebra"

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Kondratieva, M. V. Differential and Difference Dimension Polynomials. Springer Netherlands, 1999.

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Ashyralyev, Allaberen. New Difference Schemes for Partial Differential Equations. Birkhäuser Basel, 2004.

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Zbigniew, Hajto, ed. Algebraic groups and differential Galois theory. American Mathematical Society, 2011.

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Jacques, Sauloy, and Singer Michael F. 1950-, eds. Galois theories of linear difference equations: An introduction. American Mathematical Society, 2016.

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Grossman, Robert. Hopf-algebraic structure of combinatorial objects and different operators. National Aeronautics and Space Administration, 1989.

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Wolfgang, Kliemann, ed. Dynamical systems and linear algebra. American Mathematical Society, 2014.

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1980-, Blazquez-Sanz David, Morales Ruiz, Juan J. (Juan José), 1953-, and Lombardero Jesus Rodriguez 1961-, eds. Symmetries and related topics in differential and difference equations: Jairo Charris Seminar 2009, Escuela de Matematicas, Universidad Sergio Arboleda, Bogotá, Colombia. American Mathematical Society, 2011.

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Difference algebra. Springer, 2008.

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Levin, Alexander. Difference Algebra. Springer Netherlands, 2008. http://dx.doi.org/10.1007/978-1-4020-6947-5.

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Maximon, Leonard C. Differential and Difference Equations. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-29736-1.

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Book chapters on the topic "Differential and difference algebra"

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Dimension Polynomials in Difference and Difference-Differential Algebra." In Differential and Difference Dimension Polynomials. Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_6.

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Basic Notions of Differential and Difference Algebra." In Differential and Difference Dimension Polynomials. Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_3.

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Kondratieva, M. V., A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. "Some Application of Dimension Polynomials in Difference-Differential Algebra." In Differential and Difference Dimension Polynomials. Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-017-1257-6_7.

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Zuevsky, Alexander. "Geometric Versus Automorphic Correspondence for Vertex Operator Algebra Modules." In Differential and Difference Equations with Applications. Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-75647-9_50.

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Slavyanov, Sergey, and Vladimir Papshev. "Differential and Difference Equations for Products of Classical Orthogonal Polynomials." In Computer Algebra in Scientific Computing. Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/11555964_34.

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Heredero, R., D. Levi, M. Rodríguez, and P. Winternitz. "Symmetries of differential difference equations and Lie algebra contractions." In Bäcklund and Darboux Transformations. The Geometry of Solitons. American Mathematical Society, 2001. http://dx.doi.org/10.1090/crmp/029/20.

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Abramov, S. A., and M. Petkovšek. "On Polynomial Solutions of Linear Partial Differential and (q-)Difference Equations." In Computer Algebra in Scientific Computing. Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-32973-9_1.

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Ayryan, E. A., M. D. Malykh, L. A. Sevastianov, and Yu Ying. "On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds." In Computer Algebra in Scientific Computing. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-26831-2_23.

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Bronstein, Manuel. "Computer Algebra Algorithms for Linear Ordinary Differential and Difference equations." In European Congress of Mathematics. Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8266-8_9.

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Zuevsky, Alexander. "Algebraic Properties of the Semi-direct Product of Kac–Moody and Virasoro Lie Algebras and Associated Bi-Hamiltonian Systems." In Differential and Difference Equations with Applications. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32857-7_1.

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Conference papers on the topic "Differential and difference algebra"

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van Hoeij, Mark. "Closed Form Solutions for Linear Differential and Difference Equations." In ISSAC '17: International Symposium on Symbolic and Algebraic Computation. ACM, 2017. http://dx.doi.org/10.1145/3087604.3087660.

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Levin, Alexander. "Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals." In ISSAC '18: International Symposium on Symbolic and Algebraic Computation. ACM, 2018. http://dx.doi.org/10.1145/3208976.3209008.

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Bao, Wendi, Yongzhong Song, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "A Local RBF-generated Finite Difference Method for Partial Differential Algebraic Equations." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636968.

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Vanderploeg, Martin J., and Jeff D. Trom. "Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations." In ASME 1991 Design Technical Conferences. American Society of Mechanical Engineers, 1991. http://dx.doi.org/10.1115/detc1991-0121.

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Abstract This paper presents a new approach for linearization of large multi-body dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute lineariz
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Zhou, Jianping, and Zhigang Feng. "Transient Response of Distributed Parameter Systems." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4080.

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Abstract A semi-analytic method is presented for the analysis of transient response of distributed parameter systems which are consist of one dimensional subsystems. The system is first divided into one dimensional sub-systems. Within each subsystem, replacing differentials on time t by finite difference, the governing partial differential equations are reduced to difference-differential equations. The solution of derived ordinary differential equations is obtained in an exact and closed form by distributed transfer function method and represented in nodal displacement parameters. Assemling gl
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Kosasih, Engkos Achmad, and Raldi Artono Koestoer. "Determination of Air Temperature Distribution in the Annular Space Using Finite Difference Method." In ASME 1993 International Computers in Engineering Conference and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/cie1993-0057.

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Abstract This papers discusses air temperature distribution in the annular space on forced convection of turbulent air flow, which have been determined using numerical method, and compares the result with experimental data. Partial Differential Equations are presented in the final formulation, whereas turbulent flow model applied the simple algebraic model. These equations are changed into numerical equations by means of Finite Difference Method, in the form of explicit equation systems. The steps for solving these systems will be discussed. The comparison between numerical solutions and exper
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Siami, A., and M. Farid. "Identification and Defect Detection of Continuous Dynamic Systems." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14364.

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This paper presents a systematic and efficient algorithm using a coupled finite element - finite difference - least square method for identification and defect detection of continuous system using dynamic response of such systems. First the governing partial differential equations of motion of continuous systems such as beams are reduced to a set of ordinary differential equations in time domain using finite elements. Then finite difference method is used to convert these equations into a set of algebraic equations. This set of equations is considered as a set of equality constraints of an opt
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Kime, Katherine A. "Effect of the Spatial Extent of the Control in a Bilinear Control Problem for the Schroedinger Equation." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86440.

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We consider control of the one-dimensional Schroedinger equation through a time-varying potential. Using a finite difference semi-discretization, we consider increasing the extent of the potential from a single central grid-point in space to two or more gridpoints. With the differential geometry package in Maple 8, we compute and compare the corresponding Control Lie Algebras, identifying a trend in the number of elements which span the Control Lie Algebras.
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Saghafi, Mehdi, and Harry Dankowicz. "Nondegenerate Continuation Problems for the Excitation Response of Nonlinear Beam Structures." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13115.

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This paper investigates the dynamics of a slender beam subjected to transverse periodic excitation. Of particular interest is the formulation of nondegenerate continuation problems that may be analyzed numerically, in order to explore the parameter-dependence of the steady-state excitation response, while accounting for geometric nonlinearities. Several candidate formulations are presented, including finite-difference (FD) and finite-element (FE) discretizations of the governing scalar, integro-partial differential boundary-value problem (BVP), as well as of a corresponding first-order-in-spac
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Szabó, Zsolt, S. C. Sinha, and Gábor Stépán. "Dynamics of Pipes Containing Pulsative Flow." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-4022.

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Abstract Several mechanical models exist on elastic pipes containing fluid flow. In this paper those models are considered, where the fluid is incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The pipe can be modelled either as a chain of articulated rigid pipes or as a continuum. The dynamic behaviour of the system strongly depends on the different kinds of boundary conditions and on the fact whether the pipe is considered to be inextensible, i.e. the cross-sectional area of the pipe is constan
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Reports on the topic "Differential and difference algebra"

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Malitsky, N., A. Reshetov, and Y. Yan. ZLIB++: Object-oriented numerical library for differential algebra. Office of Scientific and Technical Information (OSTI), 1994. http://dx.doi.org/10.2172/10147641.

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Yan, Yiton T. ZLIB++: Object Oriented Numerical Library for Differential Algebra. Office of Scientific and Technical Information (OSTI), 2003. http://dx.doi.org/10.2172/813295.

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Hyman, J. M., M. Shashkov, M. Staley, S. Kerr, S. Steinberg, and J. Castillo. Mimetic difference approximations of partial differential equations. Office of Scientific and Technical Information (OSTI), 1997. http://dx.doi.org/10.2172/518902.

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R.A. Axford. Construction of Superconvergent Discretizations with Differential-Difference Invariants. Office of Scientific and Technical Information (OSTI), 2005. http://dx.doi.org/10.2172/883452.

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Yan, Y. Applications of differential algebra to single-particle dynamics in storage rings. Office of Scientific and Technical Information (OSTI), 1991. http://dx.doi.org/10.2172/5166998.

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Malitsky, Nikolay. Application of a Differential Algebra approach to a RHIC Helical Dipole. Office of Scientific and Technical Information (OSTI), 1994. http://dx.doi.org/10.2172/1119445.

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Svetlana G. Shasharina. Final report: Efficient and user friendly C++ library for differential algebra. Office of Scientific and Technical Information (OSTI), 1998. http://dx.doi.org/10.2172/761041.

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Berz, M., E. Forest, and J. Irwin. Exact computation of derivatives with differential algebra and applications to beam dynamics. Office of Scientific and Technical Information (OSTI), 1988. http://dx.doi.org/10.2172/7050634.

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Malitsky, N. Application of a Differential Algebra Approach to a RHIC Helical Dipole (12/94). Office of Scientific and Technical Information (OSTI), 1994. http://dx.doi.org/10.2172/1149789.

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Caspi, S., M. Helm, L. J. Laslett, and V. O. Brady. An approach to 3D magnetic field calculation using numerical and differential algebra methods. Office of Scientific and Technical Information (OSTI), 1992. http://dx.doi.org/10.2172/7252409.

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