### Table 2: A Scienti c Basis for Computational Science 5.1. Computer algebra A very familiar, generic task in science is solving for a di erential equation. An equation of the same form is solved identically, whether it arises in population biology, mathematical psychology, or elsewhere. Standard symbolic computation systems can solve the di erential equation y0 = f(x) over restricted classes of elementary functions. This is, of course, also known as inde nite integration: y = R f(x)dx. Solving inde nite integrals is not an isolated case: there also exist algorithms for solving higher-order di erential equations, nding roots of polynomials symbolically, and factoring multivariate polynomials over various coe cient domains. Kaltofen [16] and Davenport et al. [8] provides overviews of the eld of computer algebra.

"... In PAGE 5: ... Rather than slicing up the computa- tional sciences horizontally into computational volcanology, physics, etc. as in Table 1, we propose many vertical slices as in Table2 . The next few sections will develop our meaning by examining in some detail various generic tasks from science.... ..."

### Table 3: Timing breakdown under ESOLID, without PRECISE, for the examples in figure 5. The number of curve-curve intersections is given. The number of algebraic numbers found as roots of univariate polynomials is shown, along with the maximum number of bits of precision used to represent these algebraic numbers. The total time is shown, along with the percentage of time spent in curve-curve intersection, the major component of the boundary evaluation algorithm. The percentage of total time spent in the two major components of curve-curve intersection, resultant computations and Sturm computations (generation and evaluation of Sturm sequences), is also shown.

### Table 1. Average execution time in ms of the algorithm for derivations defined by dense polynomials (100 derivations for each degree)

"... In PAGE 10: ...entium 4 HT processor of 2.8 GHz, with 512 MB of primary memory. The first test we performed calculated the average time taken by the algorithm to show that a generic derivation of a given degree, defined by a pair of randomly chosen dense polynomials does not have an algebraic solution. Table1 summarizes the output of a program that randomly generates 100 pairs of dense polynomials for each degree and computes the average CPU time taken to check that the deriva- tion defined by each of the pairs does not have an algebraic solution. In this first test, none of the derivations tested caused the algorithm to fail.... ..."

### 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 2. Speedup in Worst-Case Execution Time for Optimized Virtual Table Algorithm

"... In PAGE 5: ... However, for the OVTA, the optimiza- tion over VTA depends completely on the characteristics of the generator polynomial chosen. Table2 shows the improvement over the VTA for several different polyno- mials (refer to Section 4 for a description of CRC32sub8 and CRC32sub16) . Note that for the particular CRC24 and CRC32 polynomials we used for our experiments, the OVTA has no improvement at all over the VTA.... ..."

### Table 1: Existence of Polynomial Time Learning Algorithms

1993

"... In PAGE 25: ...quivalence queries consist of arbitrary read-once formulas. Q.E.D. 9 Summary and remarks Table1 summarizes what is known of the computational di culty of learning monotone and arbitrary read-once formulas according to six types of learning protocols. The entries are discussed in order below.... ..."

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### Table 1: Various algebras built using Albert.

1992

"... In PAGE 6: ... This command merely causes Albert to check if the polynomial expands to zero in the free algebra. Testing The results of several experiments are shown in Table1 . All times re ect computations made on a Sun SPARCstation1+, and are approximate.... In PAGE 6: ... A theorem of Hall apos;s [4] describes a basis for free Lie rings in terms of a set of \standard monomials quot;. The dimensions for Lie algebras, shown in Table1 , are precisely the numbers predicted by Hall apos;s theorem. Another test we employed was to make use of Artin apos;s theorem ([14], p 29) that states that any alternative algebra generated bytwo elements is associative.... In PAGE 10: ... In fact, these problems are beyond the scope of Albert, given the speed and memory sizes of present computers. The computations in Table1 were performed done on a computer with 12 mb memory. Albert can do useful work, however, with much less than this.... In PAGE 10: ... Albert can do useful work, however, with much less than this. As Table1 demonstrates, the system tends to do well when the number of generators is kept small, usually two or three. (For example note that the dimension of the degree 8 Lie algebra with 4 a apos;s and 4 b apos;s is only 39, and yet the degree 7 Lie algebra having 3 a apos;s, 2 b apos;s, 1 c,and1d is 232.... ..."

Cited by 7

### Table 1: Various algebras built using Albert.

"... In PAGE 6: ... This command merely causes Albert to check if the polynomial expands to zero in the free algebra. Testing The results of several experiments are shown in Table1 . All times re ect computations made on a Sun SPARCstation1+, and are approximate.... In PAGE 6: ... A theorem of Hall apos;s [4] describes a basis for free Lie rings in terms of a set of \standard monomials quot;. The dimensions for Lie algebras, shown in Table1 , are precisely the numbers predicted by Hall apos;s theorem. Another test we employed was to make use of Artin apos;s theorem ([14], p 29) that states that any alternative algebra generated by two elements is associative.... In PAGE 10: ... In fact, these problems are beyond the scope of Albert, given the speed and memory sizes of present computers. The computations in Table1 were performed done on a computer with 12 mb memory. Albert can do useful work, however, with much less than this.... In PAGE 10: ... Albert can do useful work, however, with much less than this. As Table1 demonstrates, the system tends to do well when the number of generators is kept small, usually two or three. (For example note that the dimension of the degree 8 Lie algebra with 4 a apos;s and 4 b apos;s is only 39, and yet the degree 7 Lie algebra having 3 a apos;s, 2 b apos;s, 1 c, and 1 d is 232.... ..."

### Table 1: Polynomial Real Root Calculation poly-

1996

"... In PAGE 13: ... The experiment is carried out on a Sun-UltraSPARC 1 using the SACLIB [2] library of computer algebra programs. Table1 shows that real root isolation for A5 takes 170 microseconds. The polynomial has only one real root.... ..."

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