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Journal articles on the topic 'Differential and difference algebra'

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Published: 4 June 2021
Last updated: 9 February 2022

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1

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

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2

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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3

Zhao, Xiangui, and Yang Zhang. "Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras." Algebra Colloquium 23, no. 04 (September 26, 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-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.
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4

Li, Wei, and Chun-Ming Yuan. "Elimination Theory in Differential and Difference Algebra." Journal of Systems Science and Complexity 32, no. 1 (February 2019): 287–316. http://dx.doi.org/10.1007/s11424-019-8367-x.

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5

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

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6

DOBREV, V. K., H. D. DOEBNER, and C. MRUGALLA. "DIFFERENCE ANALOGUES OF THE FREE SCHRÖDINGER EQUATION." Modern Physics Letters A 14, no. 17 (June 7, 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 differential operators. Our method produces also differential-difference equations in which only space- or time-differentiation is replaced with the corresponding difference operators.
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7

Kakei, Saburo, and Yasuhiro Ohta. "A differential-difference system related to toroidal Lie algebra." Journal of Physics A: Mathematical and General 34, no. 48 (November 28, 2001): 10585–92. http://dx.doi.org/10.1088/0305-4470/34/48/322.

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8

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

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9

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

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10

SMIRNOV, YURI, and ALEXANDER TURBINER. "LIE ALGEBRAIC DISCRETIZATION OF DIFFERENTIAL EQUATIONS." Modern Physics Letters A 10, no. 24 (August 10, 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 introduced.
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11

SUN, XIAN-LONG, DA-JUN ZHANG, XIAO-YING ZHU, and DENG-YUAN CHEN. "SYMMETRIES AND LIE ALGEBRA OF THE DIFFERENTIAL–DIFFERENCE KADOMSTEV–PETVIASHVILI HIERARCHY." Modern Physics Letters B 24, no. 10 (April 20, 2010): 1033–42. http://dx.doi.org/10.1142/s0217984910023098.

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By introducing suitable non-isospectral flows, we construct two sets of symmetries for the isospectral differential–difference Kadomstev–Petviashvili hierarchy. The symmetries form an infinite dimensional Lie algebra.
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12

N'Guérékata, G. M., T. Diagana, and A. Pankov. "Abstract Differential and Difference Equations." Advances in Difference Equations 2010 (2010): 1–2. http://dx.doi.org/10.1155/2010/857306.

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13

LV, NA, JIANQIN MEI, QILONG GUO, and HONGQING ZHANG. "LIE SYMMETRIES OF TWO (2+1)-DIMENSIONAL DIFFERENTIAL-DIFFERENCE EQUATIONS BY GEOMETRIC APPROACH." International Journal of Geometric Methods in Modern Physics 08, no. 01 (February 2011): 79–85. http://dx.doi.org/10.1142/s0219887811004975.

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We obtain the Lie symmetries of two (2+1)-dimensional differential-difference equations based on the extended Harrison and Estabrook's geometric approach that is extended from the continuous differential equations to the differential-difference equations. Moreover, it is shown that both of the two equations possess a Kac–Moody–Virasoro symmetry algebra.
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14

Faber, B. F., and M. Van der put. "Formal theory for differential-difference operators." Journal of Difference Equations and Applications 7, no. 1 (January 2001): 63–104. http://dx.doi.org/10.1080/10236190108808263.

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15

Ohara, Katsuyoshi, and Nobuki Takayama. "Holonomic rank of A-hypergeometric differential-difference equations." Journal of Pure and Applied Algebra 213, no. 8 (August 2009): 1536–44. http://dx.doi.org/10.1016/j.jpaa.2008.11.018.

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16

Lomadze, Vakhtang. "On duality for partial differential (and difference) equations." Journal of Algebra 275, no. 2 (May 2004): 791–800. http://dx.doi.org/10.1016/j.jalgebra.2003.07.022.

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17

Pulita, Andrea. "p-adic confluence of q-difference equations." Compositio Mathematica 144, no. 4 (July 2008): 867–919. http://dx.doi.org/10.1112/s0010437x07003454.

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AbstractWe develop the theory of p-adic confluence of q-difference equations. The main result is the fact that, in the p-adic framework, a function is a (Taylor) solution of a differential equation if and only if it is a solution of a q-difference equation. This fact implies an equivalence, called confluence, between the category of differential equations and those of q-difference equations. We develop this theory by introducing a category of sheaves on the disk D−(1,1), for which the stalk at 1 is a differential equation, the stalk at q isa q-difference equation if q is not a root of unity, and the stalk at a root of unity ξ is a mixed object, formed by a differential equation and an action of σξ.
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18

Feng, Ruyong, Michael F. Singer, and Min Wu. "Liouvillian solutions of linear difference–differential equations." Journal of Symbolic Computation 45, no. 3 (March 2010): 287–305. http://dx.doi.org/10.1016/j.jsc.2009.09.001.

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19

Barkatou, Moulay, Thomas Cluzeau, Alexey Ovchinnikov, Georg Regensburger, and Markus Rosenkranz. "Special issue on computational aspects of differential/difference algebra and integral operators." Advances in Applied Mathematics 72 (January 2016): 1–3. http://dx.doi.org/10.1016/j.aam.2015.09.017.

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20

Grupp, Frieder. "On zeros of functions satisfying certain differential-difference equations." Acta Arithmetica 51, no. 3 (1988): 247–68. http://dx.doi.org/10.4064/aa-51-3-247-268.

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21

GAROUFALIDIS, STAVROS. "DIFFERENCE AND DIFFERENTIAL EQUATIONS FOR THE COLORED JONES FUNCTION." Journal of Knot Theory and Its Ramifications 17, no. 04 (April 2008): 495–510. http://dx.doi.org/10.1142/s0218216508006245.

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The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.
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22

Grupp, F. "On difference-differential equations in the theory of sieves." Journal of Number Theory 24, no. 2 (October 1986): 154–73. http://dx.doi.org/10.1016/0022-314x(86)90099-5.

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23

Zafer, A., L. Berezansky, and J. Diblík. "Recent Trends in Differential and Difference Equations." Advances in Difference Equations 2010 (2010): 1–2. http://dx.doi.org/10.1155/2010/891697.

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24

Lee, Namyoung, and Yasutaka Sibuya. "A note on partial differential-difference equation." Journal of Difference Equations and Applications 7, no. 1 (January 2001): 13–20. http://dx.doi.org/10.1080/10236190108808260.

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25

ZHANG, DA-JUN, JIE JI, and XIAN-LONG SUN. "CASORATIAN SOLUTIONS AND NEW SYMMETRIES OF THE DIFFERENTIAL-DIFFERENCE KADOMTSEV–PETVIASHVILI EQUATION." Modern Physics Letters B 23, no. 17 (July 10, 2009): 2107–14. http://dx.doi.org/10.1142/s0217984909020254.

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This paper first discusses the condition in which Casoratian entries satisfy for the differential-difference Kadomtsev–Petviashvili equation. Then from the Casoratian condition we find a transformation under which the differential-difference Kadomtsev–Petviashvili equation is invariant. The transformation, consisting of a combination of Galilean and scalar transformations, provides a single-parameter invariant group for the equation. We further derive the related symmetry, and the symmetry together with other two symmetries form a closed three-dimensional Lie algebra.
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26

Gao, X. S., J. Van der Hoeven, C. M. Yuan, and G. L. Zhang. "Characteristic set method for differential–difference polynomial systems." Journal of Symbolic Computation 44, no. 9 (September 2009): 1137–63. http://dx.doi.org/10.1016/j.jsc.2008.02.010.

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27

T., V., and John C. Strikwerda. "Finite Difference Schemes and Partial Differential Equations." Mathematics of Computation 55, no. 192 (October 1990): 869. http://dx.doi.org/10.2307/2008454.

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28

BALLESTEROS, ANGEL, FRANCISCO J. HERRANZ, and PREETI PARASHAR. "QUANTUM TWO-PHOTON ALGEBRA FROM NON-STANDARD Uz(sl(2,ℝ)) AND A DISCRETE TIME SCHRÖDINGER EQUATION." Modern Physics Letters A 13, no. 16 (May 30, 1998): 1241–52. http://dx.doi.org/10.1142/s0217732398001315.

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The non-standard quantum deformation of the (trivially) extended sl (2,ℝ) algebra is used to construct a new quantum deformation of the two-photon algebra h6 and its associated quantum universal R-matrix. A deformed one-boson representation for this algebra is deduced and applied to construct a first-order deformation of the differential equation that generates the two-photon algebra eigenstates in quantum optics. On the other hand, the isomorphism between h6 and the (1+1) Schrödinger algebra leads to a new quantum deformation for the latter for which a differential-difference realization is presented. From it, a time discretization of the heat-Schrödinger equation is obtained and the quantum Schrödinger generators are shown to be symmetry operators.
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29

Bachmayr, Annette, and Michael Wibmer. "Algebraic groups as difference Galois groups of linear differential equations." Journal of Pure and Applied Algebra 226, no. 2 (February 2022): 106854. http://dx.doi.org/10.1016/j.jpaa.2021.106854.

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30

Zhang, Yang, and Xiangui Zhao. "Gelfand-Kirillov dimension of differential difference algebras." ACM Communications in Computer Algebra 49, no. 1 (June 10, 2015): 32. http://dx.doi.org/10.1145/2768577.2768642.

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31

Tao, Mengshuang, and Huanhe Dong. "Algebro-Geometric Solutions for a Discrete Integrable Equation." Discrete Dynamics in Nature and Society 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/5258375.

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With the assistance of a Lie algebra whose element is a matrix, we introduce a discrete spectral problem. By means of discrete zero curvature equation, we obtain a discrete integrable hierarchy. According to decomposition of the discrete systems, the new differential-difference integrable systems with two-potential functions are derived. By constructing the Abel-Jacobi coordinates to straighten the continuous and discrete flows, the Riemann theta functions are proposed. Based on the Riemann theta functions, the algebro-geometric solutions for the discrete integrable systems are obtained.
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32

Freiling, G., G. Jank, and H. Abou-Kandil. "Generalized Riccati difference and differential equations." Linear Algebra and its Applications 241-243 (July 1996): 291–303. http://dx.doi.org/10.1016/0024-3795(95)00587-0.

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33

Marsaglia, George, Arif Zaman, and John C. W. Marsaglia. "Numerical solution of some classical differential-difference equations." Mathematics of Computation 53, no. 187 (September 1, 1989): 191. http://dx.doi.org/10.1090/s0025-5718-1989-0969490-3.

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34

Nishioka, Seiji. "Differential transcendence of solutions of difference Riccati equations and Tietze's treatment." Journal of Algebra 511 (October 2018): 16–40. http://dx.doi.org/10.1016/j.jalgebra.2018.06.008.

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35

DUNKL, CHARLES F. "AN INTERTWINING OPERATOR FOR THE GROUP B2." Glasgow Mathematical Journal 49, no. 2 (May 2007): 291–319. http://dx.doi.org/10.1017/s0017089507003709.

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AbstractThere is a commutative algebra of differential-difference operators, acting on polynomials on $\mathbb{R}^{2}$, associated with the reflection group B2. This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4F3-type. These properties are in the form of recurrence and contiguity relations and are proved herein.
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36

TANG, X., X. QIAN, and W. DING. "A differential-difference Kadomtsev–Petviashvili family possesses a common Kac–Moody–Virasoro symmetry algebra." Chaos, Solitons & Fractals 23, no. 4 (February 2005): 1311–17. http://dx.doi.org/10.1016/s0960-0779(04)00382-0.

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37

Abramov, S. A. "A problem in computer algebra connected with solution of linear differential and difference equations." Cybernetics 27, no. 2 (1991): 198–207. http://dx.doi.org/10.1007/bf01068371.

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38

Bergvelt, M. J., and A. P. E. ten Kroode. "Differential‐difference AKNS equations and homogeneous Heisenberg algebras." Journal of Mathematical Physics 28, no. 2 (February 1987): 302–6. http://dx.doi.org/10.1063/1.527658.

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39

T., V., and G. D. Smith. "Numerical Solution of Partial Differential Equations, Finite Difference Methods." Mathematics of Computation 48, no. 178 (April 1987): 834. http://dx.doi.org/10.2307/2007849.

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40

Liu, Jinn Liang. "A finite difference method for symmetric positive differential equations." Mathematics of Computation 62, no. 205 (January 1, 1994): 105. http://dx.doi.org/10.1090/s0025-5718-1994-1208839-5.

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41

Kratsios, Anastasis. "Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras." Mathematics 9, no. 3 (January 27, 2021): 251. http://dx.doi.org/10.3390/math9030251.

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The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.
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42

Malek, Stéphane. "On singularly perturbed small step size difference-differential nonlinear PDEs." Journal of Difference Equations and Applications 20, no. 1 (July 18, 2013): 118–68. http://dx.doi.org/10.1080/10236198.2013.813941.

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43

Zhang, Ning, and Xi-Xiang Xu. "A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields." Discrete Dynamics in Nature and Society 2021 (August 30, 2021): 1–9. http://dx.doi.org/10.1155/2021/9912387.

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Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.
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44

Linchuk, Stepan, and Yuriy Linchuk. "On a class of differential-difference operators in spaces of analytic functions." Operators and Matrices, no. 4 (2017): 1033–46. http://dx.doi.org/10.7153/oam-2017-11-71.

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45

Lewanowicz, S. "On the differential-difference properties of the extended Jacobi polynomials." Mathematics of Computation 44, no. 170 (May 1, 1985): 435. http://dx.doi.org/10.1090/s0025-5718-1985-0777275-2.

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46

Ben Saïd, Salem, Sara al-Blooshi, Maryam al-Kaabi, Aisha al-Mehrzi, and Fatima al-Saeedi. "A Deformed Wave Equation and Huygens’ Principle." Mathematics 8, no. 1 (December 19, 2019): 10. http://dx.doi.org/10.3390/math8010010.

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We consider a deformed wave equation where the Laplacian operator has been replaced by a differential-difference operator. We prove that this equation does not satisfy Huygens’ principle. Our approach is based on the representation theory of the Lie algebra s l ( 2 , R ) .
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47

Akhavizadegan, M., and D. A. Jordan. "Prime ideals of quantized Weyl algebras." Glasgow Mathematical Journal 38, no. 3 (September 1996): 283–97. http://dx.doi.org/10.1017/s0017089500031712.

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The main object of study in this paper is the quantized Weyl algebra which arises from the work of Maltsiniotis [10] on noncommutative differential calculus. This algebra has been studied from the point of view of noncommutative ring theory by various authors including Alev and Dumas [1], the second author [9], Cauchon [3], and Goodearl and Lenagan [5]. In [9], it is shown that has n normal elements zi and, subject to a condition on the parameters, the localization obtained on inverting these elements is simple of Krull and global dimension n. It is easy to show that each of these normal elements generates a height one prime ideal and that these are all the height one prime ideals of . The purpose of this paper is to determine, under a stronger condition on the parameters, all the prime ideals of and to compare the prime spectrum with that of a related algebra . This algebra has more symmetric defining relations than those of but it shares the same simple localization which again is obtained by inverting n normal elements zi. Like the alternative algebra can be regarded as an algebra of skew differential (or difference) operators on the coordinate ring of quantum n-space.
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48

Dunkl, Charles. "Differential-difference operators and monodromy representations of Hecke algebras." Pacific Journal of Mathematics 159, no. 2 (June 1, 1993): 271–98. http://dx.doi.org/10.2140/pjm.1993.159.271.

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49

Zhu, Junyi, and Xianguo Geng. "Algebro–geometric constructions of the ()-dimensional differential–difference equation." Physics Letters A 368, no. 6 (September 2007): 464–69. http://dx.doi.org/10.1016/j.physleta.2007.04.041.

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50

Wu Roeger, Lih-Ing, and Ronald E. Mickens. "Exact finite difference scheme for linear differential equation with constant coefficients." Journal of Difference Equations and Applications 19, no. 10 (October 2013): 1663–70. http://dx.doi.org/10.1080/10236198.2013.771635.

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