### 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 3: Commutative division algebras.

"... In PAGE 32: ... Proposition 2.2 [1, Theorem 3] An algebra given by Table3 is a division algebra if and only if d2 lt; 4b a b c d . First, we consider the case when A has exactly one idempotent.... In PAGE 33: ... Lemma 2.4 An algebra determined by Table3 has exactly one idempotent if and only if either (2a d)2 lt; 4c(1 2b) or d = 2a; b = 1 2. Taking into account Proposition 2.... ..."

### Table 3: Commutative division algebras.

2004

Cited by 6

### Table 3: Commutative division algebras.

2004

Cited by 6

### Table 3: The semantic rules of the equivalent AG for the Fibonacci series problem example 3.3 (FLP paradigm).

"... In PAGE 11: ... replace all the remaining arguments with sai(Re g) Table 2: Transformation Table for Functional Logic Programming. X f0 F1F2 X f0 F1F2 f1 XF2 f1 XF2 b b b f2 XF1 f2 XF1 Figure 5: Dependency graph for AG in Table3 (solid lines). fib(X,F1+F2) :- fib(X-2,F2), fib(X-1, F1).... In PAGE 11: ... Functional arguments are prioritized in the uni cation procedure (the uni cation procedure becomes matching procedure since we are dealing with interpreted functional terms), so that when we have to unify a variable argument which is in the argument list of a functional argument we prefer to unify the latter and discard the former. This can easily be seen in Table3 , where the equivalent AG is given after the use of transformation Table 1 in conjunction with the transformation Table 2. Fig.... In PAGE 11: ... Fig. 5 shows the dependency graph induced by the equivalent grammar in Table3 (solid lines). The idea behind these transformations looks familiar.... ..."

### Table 5. Laws for Commuting and Distributing Update Connectives

2006

"... In PAGE 55: ...Schema Variables Table5 . Modi ers for Schema Variables Modi er Applicable to rigid \term A \formula Terms or formulae that can syntactically be identi ed as rigid strict \term A Terms of type A (and not of proper subtypes of A) list \program t Sequences of program entities.... In PAGE 106: ...xample 2. We continue Example 1 and assume the same vocabulary/algebra. a := 1 ; f(a) := 2 a := 1 j f(1) := 2 valS; (a := 1 ; f(a) := 2) = fhai 7! 1; hf; (1)i 7! 2g valS; (a := 1 ; (a := 3 j f(a) := 2)) = fhai 7! 3; hf; (1)i 7! 2g We normalise the update in the second line using the given rewriting rules: a := 1 ; (a := 3 j f(a) := 2) (R45) ! a := 1 j fa := 1g (a := 3 j f(a) := 2) (R48) ! a := 1 j (fa := 1g a := 3 j fa := 1g f(a) := 2) (R47) ! a := 1 j (a := fa := 1g 3 j f(fa := 1g a) := fa := 1g 2) (R2); (R12) ! a := 1 j (a := 3 j f(non-rec(a := 1; a; ())) := 2) (R11) ! a := 1 j (a := 3 j f(if true then 1 else a) := 2) The last expression can be simpli ed further using rules for conditional terms, which are, however, beyond the scope of this paper. Further, using (R54) in Table5 , it is possible to eliminate the assignment a := 1, which is overridden by a := 3. 8 Soundness and Completeness of Update Application The following two lemmas state that the rewriting rules from Sect.... In PAGE 111: ...ewriting rules for update application (than the ones given in Sect. 5). This has been done for the implementation of updates in KeY. Table5 gives, besides others, identities that enable to establish form (1) by turning sequential composition into parallel composition, distributing if and for through parallel composition and commuting if and for. Another impor-... ..."

### Table 1. (Continued.) Algebra Non-zero commutation relations Invariants

"... In PAGE 8: ...V Boyko et al Table1 . (Continued.... ..."

### Table 2: Commuting distance

"... In PAGE 12: ...5 minutes for workers who do not work from home).23 Further, they have a shorter commuting distance (see Table2 ). The preferred measure of the length of the commute is commuting distance, because commuting time is influenced by the endogenously chosen speed which may differ between employees and self-employed.... In PAGE 14: ... In line with the theoretical model, we find that the excess commute is larger in less urban areas. As can be see in column (3) of Table2 , the elasticity of address density on the excess commute is 0.... In PAGE 24: ...046 (0.008) 17 sectors Included Occupations (83) Included Log Likelihood -10880 N 33902 Note: The explanatory variable Log (commuting distance) is the midpoint of the commuting distance class as reported in Table2... ..."

### Table 9: Fixed effects estimates of commuting times Dependent Variable : log (travel-to-work time) (standard errors in parentheses)

"... In PAGE 19: ... The use of these two binary variables as instruments for the wage results in the wage being insignificant, but this is likely to reflect insufficient time-series movement in these indicator variables to allow the isolation of a well- determined commuting/wage relationship, in a fixed effects framework13. In the light of this, and in order to focus on the robustness of the other individual characteristics, Table9 reports the fixed effects results with the endogenous wage term omitted. The influence of several of the variables, such as age and race, is now subsumed into the fixed effects.... ..."

### Table 4: Regressions on Transit Commuters

"... In PAGE 12: ... These are shown in a number of tables which are addressed one by one in the following subsections. Table 3 records the regressions on time, speed, and distance of 8651 individual automobile commuters across the country; Table4 which looks at 627 individual transit commuters; Charts 1,2, and 3, which summarize the regressions of individual automobile commuters in each of 39 cities, and Table 5 which looks at transit commuters in New York. In general these regressions, because they are performed using as observations the behavior of individuals, have a lower R-square than a regression against aggregates (such as mean metropolitan commuting time, distance or speed) would have.... In PAGE 14: ...November 5, 1996 Transit Commuters: Nationally Table4 shows the regression for transit users across the country. While in the previous section we were able to use each density class as an independent variable, because of the smaller sample of transit commuters, we had to aggregate the density variable to attain meaningful results.... ..."