### Table I. Test Matrices and Their Disciplines Discipline Matrices

2003

Cited by 38

### Table 2: Some formalisms used to specify structured dynamical systems according to the continuous or discrete nature of space, time, and state variables of the components. The heading Numerical Solutions refers to explicit numerical solutions of partial differential equations and systems of coupled ordinary differential equations.

2002

Cited by 1

### Table 2. Numerical results obtained with solution technique 3 using 4 first order differential equations.

"... In PAGE 7: ... In summary, we use three different approaches to the same problem. Results are reported in Table 1, Table2 , and Table 3, respectively. For each computed solution, a numerical integration using a Runge-Kutta scheme is carried out using a control evaluated over a fine grid.... ..."

Cited by 1

### Table 1- Methods for solving differential equations Method Message Description

"... In PAGE 3: ... In the more optimistic synchronization strategy we are currently developing, the objects encapsulating analog behavior may use a sufficient small time step as needed by the numerical integration, while synchronization with other objects will be made only when objects have to communicate events on interface signals. The analog objects have default methods, where the equations and parameters are included, as shown in Table1 . The Start method is responsible for the initialization of parameters, such as the integration step.... ..."

### TABLE IV DIFFERENTIAL EQUATION

1997

Cited by 6

### Table 7. Tariffs and Nontariff Barriers on Mercosur apos;s Imports

"... In PAGE 23: ... World Bank-UNCTAD records show that most of the products listed in Table 4 do not encounter major OECD restrictions (the processed foodstuffs are an exception). In addition, this study shows (see Table7 ) that Mercosur provides sizeable trade preferences on intra-trade. It is also possible that idiosyncracies in demand patterns and in the ability to produce certain varieties of goods made trade between Mercosur countries increase disproportionately fast as a result of the MFN liberalization.... In PAGE 24: ... However, the margins have not changed over the period which is the focus of analysis. In contrast, Table7 in this study shows that preferences Mercosur countries extend to each other are far higher. Tariff differentials set at these levels clearly have the potential to signif.... In PAGE 26: ... A key question is what factors are responsible for this surprising reorientation of trade? Evidence suggests that Mercosur apos;s own trade barriers are the cause. The analysis that follows (see Table7 ) will show that goods, such as those listed in Table 5, generally are protected by higher than average discriminatory trade measures. As a result, local producers would have a strong incentive to seek the higher prices available on sales to Mercosur markets.... In PAGE 29: ...goods experiencing the greatest shift toward intra-trade. Since the goods in these three deciles enjoy well above average levels of protection against third countries (see Table7 ) domestic producers would have a strong incentive to divert trade to local markets (which would cause the RCA indices to decline) in order to profit from the higher prices. Mercosur apos;s regional orientation indices show a fairly consistent tendency to move counter to the revealed comparative advantage measure over the decile ranges.... In PAGE 32: ... 17Moreover, if in fact the bulk of the preferences was not implemented until late 1994 the trade reorientation we reported above would have arisen from partial preferences and so one might expect an even larger effect to follow over 1995 and 1996. 18The 1996 tariff statistics reported in Table7 are drawn from the WTO apos;s Integrated Data Base and are the average duties actually applied by Argentina, Brazil and Uruguay to imports from non-Mercosur sources. They are based on the lower of the following two rates: (i) the legally bound tariff, or (ii) the current MFN applied rate.... In PAGE 33: ... Given the tariff data, however, our thesis that discrimination exists does not depend on these NTB measures. Table7 summarizes statistics relating to these barriers. The data strongly suggest that tariff preferences, and the protection they provide for intra-trade, have been a major factor behind the recent pattern of trade changes.... In PAGE 36: ...Similarly, Table7 shows that nontariff barriers were also structured along lines that would reinforce the trade distorting effects of the agreements preferential tariffs. According to the UNCTAD data, nontariff restrictions were applied to about 21 percent of all tariff line items, which is almost one- half the corresponding ratio (41.... In PAGE 37: ...Table7 . Tariffs and Nontariff Barriers on Mercosur apos;s Imports Average 1988-94 Change in External Tariff (%) Nontariff Barrier Ratio Mercosur Intra-Trade Regional Orientation ($000) Index Change Frequency Trade Decile Range 1988 1996 Ratio Coverage First Decile Products 682,680 54.... In PAGE 38: ...Similarly, Table7 shows that nontariff barriers were also structured along lines that would reinforce the trade distorting effects of the agreements preferential tariffs. According to the UNCTAD data, nontariff restrictions were applied to about 21 percent of all tariff line items, which is almost one- half the corresponding ratio (41.... ..."

### Table 7. Tariffs and Nontariff Barriers on Mercosur apos;s Imports

"... In PAGE 23: ... World Bank-UNCTAD records show that most of the products listed in Table 4 do not encounter major OECD restrictions (the processed foodstuffs are an exception). In addition, this study shows (see Table7 ) that Mercosur provides sizeable trade preferences on intra-trade. It is also possible that idiosyncracies in demand patterns and in the ability to produce certain varieties of goods made trade between Mercosur countries increase disproportionately fast as a result of the MFN liberalization.... In PAGE 24: ... However, the margins have not changed over the period which is the focus of analysis. In contrast, Table7 in this study shows that preferences Mercosur countries extend to each other are far higher. Tariff differentials set at these levels clearly have the potential to signif.... In PAGE 26: ... A key question is what factors are responsible for this surprising reorientation of trade? Evidence suggests that Mercosur apos;s own trade barriers are the cause. The analysis that follows (see Table7 ) will show that goods, such as those listed in Table 5, generally are protected by higher than average discriminatory trade measures. As a result, local producers would have a strong incentive to seek the higher prices available on sales to Mercosur markets.... In PAGE 29: ...goods experiencing the greatest shift toward intra-trade. Since the goods in these three deciles enjoy well above average levels of protection against third countries (see Table7 ) domestic producers would have a strong incentive to divert trade to local markets (which would cause the RCA indices to decline) in order to profit from the higher prices. Mercosur apos;s regional orientation indices show a fairly consistent tendency to move counter to the revealed comparative advantage measure over the decile ranges.... In PAGE 32: ... 17Moreover, if in fact the bulk of the preferences was not implemented until late 1994 the trade reorientation we reported above would have arisen from partial preferences and so one might expect an even larger effect to follow over 1995 and 1996. 18The 1996 tariff statistics reported in Table7 are drawn from the WTO apos;s Integrated Data Base and are the average duties actually applied by Argentina, Brazil and Uruguay to imports from non-Mercosur sources. They are based on the lower of the following two rates: (i) the legally bound tariff, or (ii) the current MFN applied rate.... In PAGE 33: ... Given the tariff data, however, our thesis that discrimination exists does not depend on these NTB measures. Table7 summarizes statistics relating to these barriers. The data strongly suggest that tariff preferences, and the protection they provide for intra-trade, have been a major factor behind the recent pattern of trade changes.... In PAGE 35: ...Table7 . Tariffs and Nontariff Barriers on Mercosur apos;s Imports Average 1988-94 Change in External Tariff (%) Nontariff Barrier Ratio Mercosur Intra-Trade Regional Orientation ($000) Index Change Frequency Trade Decile Range 1988 1996 Ratio Coverage First Decile Products 682,680 54.... In PAGE 36: ...Similarly, Table7 shows that nontariff barriers were also structured along lines that would reinforce the trade distorting effects of the agreements preferential tariffs. According to the UNCTAD data, nontariff restrictions were applied to about 21 percent of all tariff line items, which is almost one- half the corresponding ratio (41.... In PAGE 38: ...Similarly, Table7 shows that nontariff barriers were also structured along lines that would reinforce the trade distorting effects of the agreements preferential tariffs. According to the UNCTAD data, nontariff restrictions were applied to about 21 percent of all tariff line items, which is almost one- half the corresponding ratio (41.... ..."

### Table 2: Bounds on the number of solutions of the FKP for a robot with planar platform (9 unknowns) When there are more than 3 sensors (we always assume that the sensors are not redundant, which means that they actually give information), it is not interesting to build the dialytic matrix. Indeed it is better to solve the non-linear system by taking advantage of its structure when the linear equations have been eliminated. We obtain in this way a better bound. If 6 sensors are used and give information, we obtain a unique solution by solving the linear system corresponding to the 6 equations of type IV given by the sensors and the 3 equations of type III. The cpu times given in Table 3 are the times we needed to obtain the bound with the symbolic method of Section 3.3. They are only indicative. In fact in practice this computation is not done since we only want to compute numerically the result.

"... In PAGE 26: ... Sensors 3 2 1 Before linear elimination 28 After linear elimination 10 15 21 Table 1: Number of monomials present in the equations before and after the resolu tion of the linear equations 5.1 Planar platform Table2 gives bounds on the number of solutions, depending on the number of extra sensors that are added on the robot. They also give the number of unknowns in the initial non-linear system, the number of equations in the square system obtained by... ..."

### 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.... ..."