### 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].... ..."

Cited by 6

### Table 2 Runtime comparison of linear algebra modules

1997

### Table 2 Finite-dimensional modules for Lie algebras of vector fields Monomials? Generators Rules 1. No xie xg(y) (i; ; g) ?! (i ? 1; ; g);

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, [32].... ..."

Cited by 6

### TABLE 6. Spatiotemporal operations currently imple- mented in the ST algebra module.

### Table 1. Ore algebras

1996

"... In PAGE 4: ...elds. We specify commutative ring or commutative eld when necessary. Moreover, all rings under consideration in this paper are of characteristic 0. Table1 gives examples of the type of operators we consider. All these operators share a very simple commutation rule of the variable @ with polynomials in x.... In PAGE 5: ...3 Examples of skew polynomial rings are given in Table1 . In all the cases under consideration in this table, A is of either form K[x] or K(q)[x] with K a eld.... In PAGE 6: ...lgebra F of functions, power series, sequences, distributions, etc. Then Eq. (1) extends to the following Leibniz rule for products 8f; g 2 F @i(fg) = i(f)@i(g) + i(f)g: (6)This makes F an S-algebra. The actions of the operators corresponding to important Ore algebras are given in Table1 . In the remainder of this article, we use the word \function quot; to denote any object on which the elements of an Ore algebra act.... In PAGE 8: ... Then O is left Noetherian and a non-commutative version of Buchberger apos;s algorithm terminates. As can be seen from Table1 , this theorem implies that many useful Ore algebras are left Noe-... In PAGE 12: ... Then the annihilating ideal for any product fg where f is annihilated by I and g is annihilated by K is also @- nite. As can be seen from Table1 , this hypothesis does not represent a severe restriction on the class of Ore algebras we consider. Again, f and g in this lemma need not be interpreted as functions but as generators of the O-mod-... ..."

### Table 1: THE ALGEBRA OF BINARY FLOWNOMIALS .

1998

"... In PAGE 4: ... cation). In Table1 we use some particular cases of the rami cation and identi cation constants, namely ^0; ^2; _0; _2 denoted by ?; ^; gt;; _, respectively. In the standard version presented in [Ste94] there are three groups of algebraic equations (see Table 1): (A) a large group of algebraic equations for owgraphs modulo graph{isomorphism B1{B10, A1{A19, R1{R5 and F1{F5; (S) some critical algebraic equations S1{S4 for the data ow nodes for rami cation and identi cation; (Z) an axiom scheme ENZ, presented as a conditional equation.... In PAGE 5: ...Table1 are valid in the semantical models IRel(D). Suppose D is a set of value-vectors which represent the memory states of a computing device.... In PAGE 6: ... More details of the model(s) are presented in [BeS94, BrS96, BeS95]. Let us recall that not all the axioms in Table1 are valid in this model, but at least the BNA axioms hold. In the following we will use an extension of this setting to the many sorted case.... ..."

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### Table 4.2 Maximal eigenvalues illustrating the efficiency of diagonal scaling and algebraic multigrid

2007

### Table 7.1. Application. tmax is the maximal Abelian subalgebra of the corresponding simple algebra gi. No:

### Table 4.2 Maximal eigenvalues illustrating the efficiency of diagonal scaling and algebraic multigrid

### Table 1: The axioms for boolean algebras.

1999

"... In PAGE 8: ... 3. Data speci cations In the theory pCRL, the data must be given by means of a many-sorted equational speci cation D = h ; Ei that contains the axioms of boolean algebra (see Table1 ); such an equational speci cation is called a data speci cation. For easier explanation of our ideas we shall focuss on a certain kind of data speci cations that have a nice correspondence with theories of rst-order predicate logic.... In PAGE 8: ... We shall occasionally refer to the function declarations of the rst kind as relations, and to those of the second kind as functions. Let us x a data speci cation D = h ; Ei of this kind; we denote by Eb the set of axioms of sort b (it contains the axioms in Table1 ), and by Ed the set of axioms of sort d. We may regard as the rst-order language that has the relations of as relation symbols and the functions of as operation symbols.... ..."