### Table 5: Algorithm Algebra Connection.

### 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: Six types of problems in Elementary Algebra

2006

"... In PAGE 4: ... A simple example of a precedence relation between problems is illustrated by Fig. 1, which displays a plausible precedence diagram pertaining to the six types of algebra prob- lems illustrated in Table1 . Note in passing that we distinguish between a type of problem and an instance of that type.... In PAGE 5: ...recedence relation proposed by the diagram of Fig. 1 is a credible one. For example, if a student responds correctly to an instance of Problem (f), it is highly plausible that the same student has also mastered the other five problems. Note that this particular precedence relation is part of a much bigger one, representing a comprehensive coverage of all of Beginning Algebra, starting with the solution of simple linear equations and ending with problem types such as (f) in Table1 . An example of such a larger precedence relation is represented by the diagram of Fig.... In PAGE 8: ... 2 or 3 obtained?) This question and other critical ones are considered later on in this article. For the time being, we focus on the miniature example of Table1 which we use to introduce and illustrate the basic ideas. The knowledge states.... ..."

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### Table II An Algebraic Representation of the MD Problem

1996

Cited by 2

### Table II An Algebraic Representation of the MD Problem

1996

Cited by 2

### Table 1. Comparison of results between grids with and without diagonals. New results

1994

"... In PAGE 2: ... For two-dimensional n n meshes without diagonals 1-1 problems have been studied for more than twenty years. The so far fastest solutions for 1-1 problems and for h-h problems with small h 9 are summarized in Table1 . In that table we also present our new results on grids with diagonals and compare them with those for grids without diagonals.... ..."

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### Table 2: Iteration counts, using the modi ed algebraic preconditioner, for the sequences of matrices obtained from piecewise linear discretizations of Problems 1 and 2.

in A New Parallel Domain Decomposition Preconditioner II: Generalization to a Mesh-Free Parallel Solver

2000

"... In PAGE 13: ... Some signi cant improvements in performance may easily be obtained however. Table2 shows equivalent results to those in Ta- ble 1 but using a slightly di erent (less aggressive) coarsening algorithm. Here, step 6(a) has been modi ed to Set f := 2((`?1)=2) (where integer division is used in the exponent) and dmax has been doubled (to 16).... ..."

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### Table 2 Progol and HR results for 818 algebraic discrimination problems

"... In PAGE 34: ... In order to determine the extra-logical settings for Progol, we experimented until it could solve the problem of discriminating between two groups of size 6, one of which is Abelian and one of which is not (note that this is not one of the 818 discrimination problems in the main experiments). The settings determined in this manner were as follows: :- set(nodes,2000)? :- set(inflate,800)? :- set(c,2)? :- set(h,100000)? :- set(r,100000)? The results from these experiments are given in Table2 . For an initial application, the results are very promising: Progol solved 558 of the 818 discrimination problems (68%) compared to HR which achieved 96%.... ..."

### Table 2 Progol and HR results for 818 algebraic discrimination problems

"... In PAGE 34: ... In order to determine the extra-logical settings for Progol, we experimented until it could solve the problem of discriminating between two groups of size 6, one of which is Abelian and one of which is not (note that this is not one of the 818 discrimination problems in the main experiments). The settings determined in this manner were as follows: :- set(nodes,2000)? :- set(inflate,800)? :- set(c,2)? :- set(h,100000)? :- set(r,100000)? The results from these experiments are given in Table2 . For an initial application, the results are very promising: Progol solved 558 of the 818 discrimination problems (68%) compared to HR which achieved 96%.... ..."

### Table III A Modified Algebraic Representation of the MD Problem

1996

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