### Table 3 Cohomologies for Lie algebras of vector fields

1994

"... In PAGE 10: ... 2]. Table3 describes the cohomology spaces H1(h; C1=M) for each of the Lie algebras and corresponding modules. The rst column indicates the dimension of the cohomology space, and the second column gives a representa- tive cocycle of each nontrivial cohomology class.... ..."

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### Table 2 Finite-dimensional modules for Lie algebras of vector fields

1994

"... In PAGE 10: ... OLVER systems than Lie.) Table2 describes the di erent nite-dimensional modules for each of these Lie algebras. The rst column tells whether the module is necessarily spanned by monomials, i.... In PAGE 10: ... Finally, Table 4 describes the quantization condition resulting from the quasi- exactly solvability assumption that, assuming M = f1g, the Lie algebra admit a nite-dimensional module N. If the cohomol- ogy is trivial, so g is spanned by vector elds and the constant functions, then it automatically satis es the quasi-exactly solvable condition, with the associated nite-dimensional modules being explicitly described in Table2 . The maximal al- gebras, namely Case 11, sl(2) sl(2), Case 15, sl(3), and Case 24, gl(2) n Rr, play an important role in Turbiner apos;s theory of di erential equations in two dimensions with orthogonal polynomial solutions, [33].... ..."

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### Table 4 Additional Primitive Lie Algebras of Vector Fields in R2

1996

"... In PAGE 3: ... For instance, extensive symbolic algebra calculations using the realization of so(3; R) given by Case 4.2 in Table4 failed to yield any real-valued potentials, even after allowing for a possible rescaling of the wave function by a non-zero gauge factor, cf. (7).... In PAGE 15: ... However, the complexi cation (or analytic continuation) of each of these ve additional Lie algebras will, of course, be equivalent, under a complex di eomorphism, to one of the complex normal forms on our list. The complete list of these additional real forms along with their canonical complexi- cations appears in Table4 ; the required changes of variables needed to change them into the canonical form appears later in this section. Therefore, to complete the classi cation of all real Lie algebras of rst-order di erential operators, we need only determine the real cohomology spaces associated with these ve additional real forms, and, further, to determine what values of the cohomology parameters will produce quasi-exactly solvable algebras.... In PAGE 21: ... Theorem 11. Every nite-dimensional Lie algebra of rst order di erential opera- tors with vector eld part given by one of the ve additional primitive real forms listed in Table4 is a subalgebra of one of the pseudo-orthogonal Lie algebras given by (45). 6.... ..."

### Table 1 Transitive, Imprimitive Lie Algebras of Vector Fields in C 2

1996

### Table 2 Primitive Lie Algebras of Vector Fields in C 2

1996

### Table 3 Intransitive Lie Algebras of Vector Fields in C 2

1996

### Table 6 Cohomology for Lie Algebras of Vector Fields in R2

1996

### Table 1. Mean result of our outlier rejection scheme on 10000 random scene vector fields. The last row lists the proportion of outliers in the resulting vector field after ap- plying our method.

### Table 1. Restoration of the height map from various smoothed surface normal vector fields. Surface Normal Number of

2002

"... In PAGE 3: ... The result- ing heights have been smoothed by averaging in a a0 a40 a0 neighbourhood in two passes over the entire image. The MCE between the vector fields obtained from the esti- mated height map and those provided initially are shown in Table1 . We can observe from this table that we obtain a quite accurate representation of the surface normals de- rived from the estimated height map.... ..."

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