Classification of 3-dimensional integrable scalar discrete equations^{∗}

S.P. Tsarev^{†}

Institut für Mathematik Technische Universität Berlin Berlin, Germany

(tsarev@math.tu-berlin.de  sptsarev@mail.ru)

T. Wolf

Department of Mathematics, Brock University 500 Glenridge Avenue, St.Catharines, Ontario, Canada L2S 3A1

(twolf@brocku.ca)

Abstract. We classify all integrable 3-dimensional scalar discrete affine linear equations Q_{3} = 0 on an elementary cubic cell of the lattice ℤ^{3}. An equation Q_{3} = 0 is called integrable if it may be consistently imposed on all 3-dimensional elementary faces of the lattice ℤ^{4}. Under the natural requirement of invariance of the equation under the action of the complete group of symmetries of the cube we prove that the only nontrivial (non-linearizable) integrable equation from this class is the well-known dBKP-system.

Keywords: integrable systems, discrete equations, large polynomial systems, computer algebra, Reduce, Form, Crack.

Mathematics Subject Classification (2000): 37K10, 52C99

1. Introduction

Discrete differential geometry has been actively studied in the last decade (see e.g. [3, 4]) and has provided much insight into structures that are fundamental both to classical differential geometry and to the theory of integrable PDEs. In addition to such purely mathematical fields, results in discrete differential geometry have a great potential in computer graphics and architectural design (see [9]).

In what follows we consider the elementary combinatorial cubic cell K_{n} with the vertices {(i_{1}, …, i_{n})|i_{s} ∈ {0, 1}}. The field variables f_{i_{1}...i_{n}} ∈ ℂ, i_{s} ∈ {0, 1}, are associated to its 2^{n} vertices. We will use the

∗ Supported by the DFG Research Unit 565 “Polyhedral Surfaces” (TU-Berlin) † On leave from: Krasnoyarsk State Pedagogical University, Russia. SPT ac-

knowledges partial financial support from a grant of Siberian Federal University (NM-project N^{o} 45.2007) and the RFBR grant 06-01-00814.

\begin{matrix}c \\ ◯ \end{matrix} 2009 Springer Science+Business Media, Inc. Manufactured in The Netherlands.

short notation f for the set (f_{00...0}, …, f_{11...1}) of all these 2^{n} variables (see Figures 1A, 1B).

An n-dimensional discrete system of the type considered here is given by an equation of the form

Q_{n}(f) = 0,  (1)

on the field variables on the elementary cubic cell K_{n}. This equation may be extended to the other elementary cubic cells of the cubic lattice ℤ^{n} with vertices at integer points (i_{1}, …, i_{n}), i_{s} ∈ ℤ, n ≥ 2, in the n-dimensional space ℝ^{n}: with each vertex with integer coordinates (i_{1}, …, i_{n}), i_{s} ∈ ℤ, we associate a scalar field variable f_{i_{1}...i_{n}} ∈ ℂ and assume that the equation (1) for every elementary cubic cell of ℤ^{n} is the same, after shifting the indices of f suitably.

In the last two decades the study of special classes of (1) which are “integrable” (in one sense or another) has become very popular. We give below only a brief account of the current state of this field of research, for a more detailed account cf. [1]–[4] and the references given therein. In fact, discrete integrable systems underlie many classical integrable nonlinear PDEs, like the Krichever-Novikov equation and other examples, the latter appear as a continuous limit along some of the discrete directions. Other well known classes of integrable geometric objects (with n = 2 and n = 3), like minimal surfaces, conjugate nets, constant curvature surfaces, Moutard nets, isothermic surfaces, orthogonal curvilinear coordinates etc., are also obtained as some smooth limits along some of the directions of the respective discrete system. The remaining discrete directions automatically provide us with a transformation known in the classical continuous geometric context as Jonas/Ribaucour/Bäcklund transformation between surfaces of the given class. On the other hand, starting from the classical non-linear superposition principles for the aforementioned transformations one can obtain precisely the underlying discrete system. One of the cornerstones of discrete differential geometry (the idea to look for cubic nonlinear superposition formulas of Bäcklund transformations of nonlinear integrable PDEs) was laid down in [5]. The duality between the smooth objects in any of the geometric classes of integrable smooth surfaces mentioned above and their Bäcklund-type transformations is therefore put into a symmetric form of a single discrete n-dimensional system and is encoded as the notion of (n + 1)-dimensional consistency [3]:

An n-dimensional discrete equation (1) is called consistent, if it may be imposed in a consistent way on all n-dimensional faces of a (n + 1)-dimensional cube.

This can also be understood as the possibility to take ℤ^{n+1} and prescribe the n-dimensional equation (1) to hold on every n-dimensional face of every elementary (n+1)-dimensional cube (of size 1, with edges parallel to the coordinate axes) without side relations to appear. For this reason the discrete equation (1) is often called a “face formula”. A precise definition of consistency, suitable for the class of discrete equations treated in this paper, will be formulated in the next section.

This paper is devoted to application of computer algebra systems Reduce and Form ([10]), in particular the Reduce package Crack ([11, 12]), to the classification of 3-dimensional integrable discrete systems.

The paper is organized as follows. In Section 2 we give a brief description of the known results on 2-dimensional integrable scalar discrete equations of type (1) and the precise definition of (n + 1)-dimensional consistency condition for such discrete n-dimensional systems.

Section 3 is devoted to the classification of symmetry types of affine linear equations (1) for dimensions n = 2, 3, 4.

In Section 4 we describe the results of our computations (Theorem 2): the only nontrivial (non-linearizable) integrable scalar affine linear 3d-discrete equation invariant w.r.t. the complete group of symmetries of the cube is given by the formula (5) below.

The technical details of the computations can be found in [13].

2. The setup

The simplest but very important class of 2-dimensional integrable discrete equations was investigated in detail in [1, 2]. They have the form

Q(f_{00}, f_{10}, f_{01}, f_{11}) = 0,  (2)

where f_{ij} are scalar fields attached to the vertices of a square (see Fig. 1A) with two main requirements:

1) Affine linearity. (2) is affine linear w.r.t. every f_{ij}, i.e. Q has degree 1 in any of its four variables:

Q = c_{1}(f_{10}, f_{01}, f_{11})f_{00} + c_{2}(f_{10}, f_{01}, f_{11})

= c_{3}(f_{00}, f_{01}, f_{11})f_{10} + c_{4}(f_{00}, f_{01}, f_{11}) = …

= q_{1111}f_{00}f_{10}f_{01}f_{11} + q_{1110}f_{00}f_{10}f_{01} + q_{1101}f_{00}f_{10}f_{11} + … + q_{0000}.

2) Symmetry. Equation (2) should be invariant w.r.t. the symmetry group of the square or its suitably chosen subgroup.

A few other requirements were given in [1], in particular the formula (2) involved parameters attached to the edges of the square.

f_{00} f_{01}  f_{10} f_{11}  ^{f_{000} f_{001} f_{010} f_{011} f_{100} f_{101}f_{110} f_{111}}
Figure 1.  A. Square K_{2}.  B. Cube K_{3}.

The second requirement of symmetry is obviously very important for the formulation of the condition of 3-dimensional consistency of (2). In fact, suppose we have a 3d-cubic cell of ℤ^{3} (cf. Fig. 1B) and impose (2) to hold on three “initial 2d-faces” {x_{1} =0}: Q(f_{000}, f_{010}, f_{001}, f_{011})=0; {x_{2} = 0}: Q(f_{000}, f_{100}, f_{001}, f_{101}) = 0; {x_{3} = 0}: Q(f_{000}, f_{100}, f_{010}, f_{110}) = 0 (these are used to find f_{011}, f_{101}, f_{110} from f_{000}, f_{100}, f_{010}, f_{001}). Then we impose (2) to hold on the other three “final 2d-faces” {x_{1} = 1}: Q(f_{100}, f_{110}, f_{101}, f_{111}) = 0; {x_{2} = 1}: Q(f_{010}, f_{110}, f_{011}, f_{111}) = 0; {x_{3} = 1}: Q(f_{001}, f_{101}, f_{011}, f_{111}) = 0; so for the last field variable f_{111} we can find 3 (a priori) different rational expressions in terms of the “initial data” f_{000}, f_{100}, f_{010}, f_{001}. The 3d-consistency is the requirement that these three expressions of f_{111} in terms of the initial data should be identically equal. The subtle point of this process consists in the non-uniqueness of the mappings of a given square (Fig. 1A) onto the six 2d-faces of the cube. The requirement of symmetry given above guarantees that we can choose any identification of the vertices of the 2dimensional faces of the 3-dimensional elementary cube (Fig. 1B) with the vertices of the “standard” square where (2) is given; certainly this identification should preserve the combinatorial structure of the square (neighbouring vertices remain neighbouring). In [1] a complete classification of 3d-consistent 2d-discrete equations (in a slightly different setting) was obtained; in [2] a similar classification was given for the case when one does not assume that the formula (2) is the same on all the 6 faces of the 3-dimensional cube.

In the next sections we give a symmetry classification of all possible 3d-discrete equations defined on some “standard” 3-dimensional cube:

Q(f_{000}, f_{100}, f_{010}, f_{001}, f_{110}, f_{101}, f_{011}, f_{111}) = 0  (3)

with respect to the complete symmetry group of the cube. Here, as everywhere in the paper, indices of the field variables f_{ijk} give the coordinates of the corresponding vertices of the standard 3d-cube where our formula (3) is defined.

The requirement of consistency is now formulated similarly to the 2d-case: given a 4d-cube with field values f_{ijkl}, i, j, k, l ∈ {0, 1}, one should impose the formula (3) on every 3d-face of it, by fixing one of the indices i, j, k, l, and making it 0 for the faces which we will call below “initial faces”, or respectively 1 for the faces which we will call “final faces”. One also needs to fix some mapping from the initial “standard” cube (with the vertices labelled f_{ijk}) onto every one of the eight 3dfaces (for example {f_{i1kl}} on {x_{2} = 1}). This can certainly be done using the trivial lexicographic correspondence of the type f_{ijk} ↦ f_{i1jk}. Geometrically this lexicographic correspondence is less natural since it is not invariant w.r.t. the symmetry group of a 4d-cube. On the other hand there is an important example of such a non-symmetric formula corresponding to the discrete BKP equation ([1], equation (76)). Another possibility to avoid this problem is to impose the requirement of symmetry. More precisely, if one applies any one of the transformations from the group of symmetries of the 3d-cube, (3) shall be transformed into an equation with the left hand side proportional to the original expression Q: Q ↦ λ · Q. Since this symmetry group is generated by reflections, one has λ^{2} = 1, so this proportionality multiplier λ may be either (+1) or (−1) for any particular transformation in the complete symmetry group.

From results in [1] we know that there are important 4d-consistent 3d-discrete equations which are preserved under a suitable subgroup of the complete symmetry group of the 3d-cube. No classification of 3d-discrete equations with such restricted symmetry property has beet carried out yet.

3. Symmetry classification

Every n-dimensional discrete equation Q_{n} = 0 which satisfies the requirement of affine linearity has a left hand side of the form

Q_{n} = ∑_{D} q_{D} ∏_{i_{s}=0,1} (f_{i_{1}...i_{n}})^{D_{i_{1}...in}}  (4)

with constant coefficients q_{D}, where the summation is taken over all 2^{2^{n}} many 2^{n}-tuples D = (D_{00...0}, …, D_{11...1}), each power D_{i_{1}...i_{n}} of the respective vertex variable f_{i_{1}...i_{n}} being either 0 or 1. In other words: the 2^{n} indices of q_{D} are the exponents of the 2^{n} vertex field variables f_{i_{1}...i_{n}}, each exponent D_{i_{1}...i_{n}} being 0 or 1. For example, Q_{2} = q_{1111}f_{00}f_{10}f_{01}f_{11} +q_{1110}f_{00}f_{10}f_{01} +q_{1101}f_{00}f_{10}f_{11} +…+q_{0000} has 2^{2^{2}} = 16 terms, Q_{3} has respectively 2^{2^{3}} = 256 terms, and Q_{4} has already 2^{2^{4}} = 65536 terms.

In this Section we classify n-dimensional affine linear equations Q_{n} = 0 for n = 2, 3, 4 that are invariant w.r.t. the complete symmetry group of the respective n-dimensional cube. This problem can be reduced to the enumeration of irreducible representations of this group in the space of polynomials of the form (4). Here this is done in a straightforward way: the group in question is generated by one reflection w.r.t the plane x_{1} = 1/2 and (n − 1) diagonal reflections w.r.t. the planes x_{1} = x_{s}, s = 2, …, n (here x_{k} denote the coordinates in ℝ^{n}).

To every reflection R from this generating set we assign (+) or (−) and require the equality Q(f) = Q(R(f)) (respectively Q(f) = −Q(R(f))) to hold identically in all vertex variables f; this gives us a set of equations for the coefficients q_{D}. Running through all possible choices of the signs for the generating reflections we solve the united sets of simple linear equations for the coefficients q_{D} for every such choice. The main problem consists in the size of the resulting set of equations: for n = 3 we have for each combination of ± for the 3 generating reflections around 770 equation for the 256 coefficients q_{D_{1}...D_{8}}; for n = 4 every set of equations for the coefficients of Q_{4} has around 250,000 equations for the 65536 coefficients q_{D_{1}...D_{16}}. Naturally, not every combination of signs for the generating reflections is possible, most of the resulting sets of equations for q_{D_{1}...D_{(2^{n})}} allow only the trivial solution q_{D_{1}...D_{(2^{n})}} = 0. The results of our computations are formulated in Theorem 1:

THEOREM 1. There are three nonempty symmetry classes of discrete equations Q_{n} for the dimensions n = 2, 3 and four nonempty classes for n = 4. The resulting number of free parameters in the coefficients q_{D} in each of the symmetric discrete equations Q_{n} (including an overall constant multiplier) and the number of nonzero terms are given for each of the nontrivial cases in Table I.

In the notations of Table I for n = 2 the first sign in the second column refers to the reflection on the line x_{1} = 1/2, the next sign stands for the reflection on the line x_{1} = x_{2}. For example, the expressions of the first type (+−) are invariant w.r.t. to the reflection on the line x_{1} = 1/2, and show a change of sign after the reflection on the line x_{1} = x_{2}. The last case (++) consists of expressions which are invariant w.r.t. any element of the complete group of symmetries of the square which corresponds to the choice of the (+) signs for the two generating reflections of the square w.r.t. the lines x_{1} = 1/2 and x_{1} = x_{2}.

Similarly to the case n = 2, there are only three cases (+ + +), (− − −) and (− + +) of nontrivial affine linear Q_{3} for n = 3. The first sign here refers to a reflection on the plane x_{1} = 1/2, the next signs stand for reflections on the planes x_{1} = x_{2}, x_{1} = x_{3} (and x_{1} = x_{4} for n = 4).

Especially remarkable is the totally skew-symmetric case (− − −) for n = 3: it has only one (up to a constant multiplier) nontrivial

Table I. Symmetry classification of the discrete equations w.r.t the complete symmetry group of the cube

n types of symmetry,  number of parameters
number of parameters and terms  in SL_{2}-invariant subcases
2 1) (+−): 1 param.; 4 terms  1) 1 param.; 4 terms
2) (−+): 3 param.; 10 terms  2) none
3) (++): 6 param.; 16 terms  3) 1 param.; 6 terms
3 1) (−−−): 1 param.; 24 terms  1) 1 param.; 24 terms
2) (−++): 13 param.; 186 terms  2) none
3) (+++): 22 param.; 256 terms  3) 3 param.; 114 terms
4 1) (−−−−): 94 param.; 29208 terms 1) 5 param.; 15480 terms
2) (+−−−): 77 param.; 26112 terms 2) none
3) (−+++): 349 param.; 60666 terms 3) 3 param.; 15809 terms
4) (++++): 402 param.; 2^{16} terms  4) 18 param.; 96314 terms

expression

\begin{matrix}Q_{(−−−)} =(f_{100} − f_{001})(f_{010} − f_{111})(f_{101} − f_{110})(f_{011} − f_{000}) − \\ (f_{001} − f_{010})(f_{111} − f_{100})(f_{000} − f_{101})(f_{110} − f_{011}). \end{matrix} (5)

Precisely this expression gives the so called discrete Schwarzian Kadomtsev-Petviashvili (type B) system (dBKP-system) — an integrable discrete system found in [6, 7] and studied in [1] where the fact of its 4d-consistency was first established. The dBKP-system has many equivalent forms and appears in very different contexts. In addition to the known geometric interpretations and a reformulation as Yang-Baxter system ([8]), the dBKP-system may be considered as a nonlinear superposition principle for the classical 2-dimensional Moutard transformations ([5]).

The expression (5) enjoys an extra symmetry property: the equation Q_{(−−−)} = 0 is invariant under the action of the SL_{2}(ℂ) group of fractional-linear transformations f ⟼ (af + b)/(cf + d) (all group parameters a, b, c, d are the same for all the vertices of the cube). This is in fact a direct consequence of the uniqueness of the integrable discrete equation (5) in this class. Since this SL_{2}(ℂ) action obviously preserves the symmetry type of an expression w.r.t. the group action of the cube symmetry group, it is reasonable to find the subclasses of SL_{2}-invariant discrete equations in each symmetry class. This is given in the third column of Table I.

4. Nonexistence of nontrivial integrable discrete equations in other symmetry classes for n = 3

In this section we give a sketch of a computational proof of our main result:

THEOREM 2. Among the three possible symmetry types of 3-dimen-sional affine linear discrete equations given in the second column of Table I, only formula (5) gives a non-trivial (i.e. non-linearizable) 4dcompatible discrete equation. Any 4d-compatible discrete equation in the other two symmetry types may be transformed using the action of the group SL_{2}(ℂ) on the field variables to one of the following linearizable forms:

Q^{(1)} = f_{000}f_{001}f_{010}f_{011}f_{100}f_{101}f_{110}f_{111} − σ,  (6)
Q^{(2)} = f_{001}f_{010}f_{100}f_{111} − σf_{000}f_{011}f_{101}f_{110},  (7)
Q^{(3)} = (f_{001} + f_{010} + f_{100} + f_{111}) − σ(f_{000} + f_{011} + f_{101} + f_{110}), (8)

where σ = ±1.

Technically, in order to find 4d-consistent 3d-discrete equations among the other two 3-dimensional cases (− + +) and (+ + +) (as listed in Table I), one shall run the following algorithmic steps:

Step 1. Take a copy of this Q_{3} formula, map it onto the four initial faces of the 4d-cube (where one of the coordinates x_{i} = 0), solve the mapped equations with respect to f_{0111}, f_{1011}, f_{1101} and f_{1110} leaving the other variables free.

Step 2. Then substitute the obtained rational expressions for f_{0111}, f_{1011}, f_{1101} and f_{1110} into the copies of the discrete equation mapped onto the four final faces (where one of the coordinates x_{i} = 1), finding respectively four different expressions for the last vertex field f_{1111}.

Step 3. Equate these 4 expressions for f_{1111} obtaining three rational equations in terms of 11 free variables f_{0000}, f_{0010}, f_{0101}, … and the parametric coefficients q_{D_{1}...D_{8}} left free in the chosen symmetry class.

Step 4. Removing the common denominators of the equations and splitting the resulting polynomials w.r.t. the 11 free variables f_{ijkl} one obtains a polynomial system of equations for the free coefficients q_{D_{1}...D_{8}}.

Step 5. The latter should be solved, resulting in a complete classification of 4d-consistent affine linear scalar 3d-discrete equations.

This approach, applied in a straightforward way, results in extremely huge expressions. Even building the rational expressions in Step 4 in a straightforward way seems to be unrealistic: for a typical 3d-discrete equation from Table I Step 4 should end up (as our test runs allowed for an estimate) in an expression with around 10^{14} terms, which is beyond the reach of computer algebra systems in the foreseeable future. Even brute force verification of 4d-consistency of the smallest solution (discrete equation (5) which has no free parametric coefficients q_{D_{1}...D_{8}}) results in ≈ 2 · 10^{8} terms (after substituting the expressions for f_{0111}, f_{1011}, f_{1101}, f_{1110}, collecting the terms over the common denominator and expanding the brackets before the cancellation can start in Step 4, cf. [13]). Technically this is explained by the presence of 4 different symbolic denominators of the rational expressions for f_{0111}, f_{1011}, f_{1101}, f_{1110} and their various products. A careful step-by-step substitution and cancellation of like terms in several stages still can be done even on a currently standard PC for this (− − −) case. Using Form (this symbolic computation system was specially designed for large computations), one can prove that all terms finally cancel out for the case of the integrable 3d-discrete equation (5) thus giving a computational proof of its 4d-consistency in 3 min CPU time (3 GHz Intel running Linux SUSE 9.3) and less than 200 Mb disk space for temporary data storage (cf. [13]).

As our preliminary runs had shown, the straightforward approach based on Steps 1–4 is unrealistic for the other two 3-dimensional cases listed in Table I, even when generating the consistency conditions in Step 4.

In order to classify discrete integrable 3d-discrete equations Q_{3} = 0 for the case (− + +) and the hardest case (+ + +) we used a totally different randomized “probing” strategy, explained in detail in [13].

After the computation (cf. [13] for the details) the list of candidate formulas Q_{3} for the case (+ + +) included 5 candidates (before the verification that these formulas, obtained by our “probing” method, really give 4d-consistent 3d-formulas). For the case (− + +) the list of candidate formulas included 3 formulas. All of them include a few free parameters. As one can show, all these formulas can be greatly simplified using the action of the group SL_{2}(ℂ) on the field variables f, resulting in the 4d-consistent 3d-discrete equations (6), (7), (8). The first two formulas (6), (7) can be easily linearized using the logarithmic substitution f̃_{ijk} = log f_{ijk}.

The expressions for the aforementioned candidates, the Form procedures and their logfiles showing the simplification process can be downloaded from

http://lie.math.brocku.ca/twolf/papers/TsWo2007/.

Acknowledgements

For this work facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca) were used.

TW thanks the Konrad Zuse Institut at Freie Universität Berlin and the Technische Universität Berlin where part of the work was done.

References

1. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Classification of integrable equations on quad-graphs. The consistency approach. Comm. Math. Phys., 2003, v. 233, p. 513–543.

2. V.E. Adler, A.I. Bobenko, Yu.B. Suris, Discrete nonlinear hyperbolic equations. Classification of integrable cases. Preprint at www.arxiv.org, nlin.SI/ 0705.1663.

3. Bobenko, A.I., Suris, Yu.B.: Discrete differential geometry. Consistency as integrability. Preliminary version of a book. Preprint at www.arxiv.org, math.DG/0504358 (2005)

4. A.I. Bobenko, Yu.B. Suris, On organizing principles of Discrete Differential Geometry. Geometry of spheres. Russian Math. Surveys, 2007, v. 62, No 1, p. 1–43. Also www.arXiv.org, math/0608291.

5. E.I. Ganzha, S.P. Tsarev, On superposition of the auto-Baecklund transformations for (2+1)-dimensional integrable systems. Russian Math. Surveys, 1996, v. 51, No 6, p. 1200–1202. See also www.arxiv.org, solv-int/9606003.

6. B.G. Konopelchenko, W.K. Schief, Reciprocal figures, graphical statics and inversive geometry of the Schwarzian BKP hierarchy. Stud. Appl. Math., 2002, v. 109, p. 89–124. Also preprint nlin.SI/0107001 at http://www.arXiv.org,

7. J.J.C. Nimmo, W.K. Schief, An integrable discretization of a 2+1-dimensional sine-Gordon equation. Stud. Appl. Math., 1998, v. 100, p. 295–309.

8. V.G. Papageorgiou, A.G. Tongas, A.P. Veselov, Yang-Baxter maps and symmetries of integrable equations on quad-graphs. J. Math. Phys., 2006, v. 47(8), No 083502, 16 pp.

9. Y. Liu, H. Pottmann, J. Wallner, Y. Yang, W. Wang, Geometric modelling with conical meshes and developpable surfaces. ACM Trans. Graphics. 25(3), 681-689 (2006), Proc. SIGGRAPH 2006. See also

http://dmg.tuwien.ac.at/pottmann/

10. J.A.M. Vermaseren, New features of FORM. arXiv:math-ph/0010025, a complete distribution can be downloaded from http://www.nikhef.nl/~form/.

11. T. Wolf, Applications of CRACK in the Classification of Integrable Systems, in: CRM Proceedings and Lecture Notes, v. 37 (2004) pp. 283–300. Also www.arxiv.org, nlin.SI/0301032.

12. T. Wolf, An Online Tutorial for the package Crack.

http://lie.math.brocku.ca/crack/demo

13. Tsarev, S.P., Wolf, T.: Classification of 3-dimensional integrable scalar discrete equations. Preprint at www.arxiv.org, nlin.SI/0706.2464 (2007). The commented Form codes related to this publication can be downloaded from

http://lie.math.brocku.ca/twolf/papers/TsWo2007/

Address for Offprints: Prof. T. Wolf, Department of Mathematics, Brock University 500 Glenridge Avenue, St.Catharines, Ontario, Canada L2S 3A1

twolf@brocku.ca

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