### Table 1: Left-plane stability properties of polynomial P (r)

1995

"... In PAGE 10: ... First, a polynomial P (r) is simple if 8r fP (r) = 0 ) d P d r (r) 6 = 0g ; that is, the roots of the polynomial are simple. Some standard left-half plane conditions are given in Table1 while some stability conditions for the unit circle are given in Table 2. In the next section, linear fractional maps are used to change the location of the roots of polynomials.... In PAGE 20: ...ameters. This is just where QE shows some of its strengths. So now assume that the coe cients of the system of ODEs depend on the parameters ~ and that C( ; ~ ) is the characteristic polynomial for the system. Then the sys- tem of ODEs is asymptotically stable for given values of the parameters ~ if and only if 8 2 C fC( ; ~ ) = 0 ) lt; lt; 0g ; that is, the polynomial is a Hurwitz polynomial (see Table1 ). The system of ODEs is unstable for a given ~ if 9 2 C fC( ; ~ ) = 0 ^ lt; gt; 0g : The possibility of multiple roots of the characteristic polynomial make... ..."

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### Table 1. Stability Analysis.

"... In PAGE 5: ... Therefore it is important to understand the stability of this system (for more details see [15]), and incorporate the geometry of the environment in the low-level behaviors. Table1 shows... ..."

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### Table 1: Example of common functions of devices: Same functions are mapped to the same gesture; similar functions may be mapped to the same gesture if this is intuitive and no other function is overloaded.

1997

"... In PAGE 2: ... 1). Further control is ac- cording to Table1 , where one gesture is used for each line. Every gesture is mapped to several similar tasks from di erent devices2, which reduces the number of gestures and makes the dialogue more intuitive.... ..."

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### Table 1. Input data for stability analysis of the Porsuk Dam.

"... In PAGE 4: ... It is used for irrigation, flood control, and domestic and industrial water supply. Most of the inputs and properties of the Porsuk Dam are presented in Table1 . Apart from the available data for the software to be run, some of the inputs are obtained by combining the available data with the related information present in other references.... ..."

### TABLE I ANALYSIS OF EACH TERM OF THE DERIVATIVE LYAPUNOV CANDIDATE FUNCTION (13) TO SHOW IT IS STRICTLY NEGATIVE.

### Table 1. Commands for optimal control of hybrid systems.

"... In PAGE 13: ... The WL P T L handles a piecewise linear DPWLE system as an object. The basic commands for building a PWL system are listed in Table1 . Having partitioned the state space and used the functions for entering data into MATLAB,the system is aggregated into a single record that is passed on to functions for analysis and simulations.... In PAGE 98: ... 3. Understanding the Tools The commands available for solving the control problem are listed in Table1 . There are three main groups of programs: a group of four commands that in various ways approximate the value function of a hybrid optimal control problem, one command for deriving a control signal from the value function, and four commands for simulating hy- brid systems.... In PAGE 105: ... Command Reference This section describes the commands in detail. Being very similar to each other, some of the commands of Table1 are grouped into the same entry on the following pages. The commands ohsf and ohsfe are not found in this section, since they are of little interest to the standard user.... ..."

### Table 3: Orthogonal functions related to exactly solvable chaos Index Pl(x) Fl(x) Lyapunov Exponents

"... In PAGE 15: ...chaos-based Monte Carlo simulations for a family of integrands B(x1; ; xs) = ac + L X m=0 amPpm 1 (x1) Ppm s (xs); (57) where an each dynamical variable Xi;j associated with xi is computed by Xi;j+1 = Fpi(Xi;j), are supere cient if the condition L X m=0 am = 0 (58) is satis ed. Table3 lists these transformations fFlgl=1;2;3 and their dual transfor- mations fF l gl=1;2;3 over I and their related orthogonal functions fPl(x)gl=1;2;3 and fP l (x)gl=1;2;3 which were used for numerical simulations to attain the su- pere ciency over the in nite support. Figure 6(a) shows the graphs of er- godic transformations at = 1.... In PAGE 15: ... In Fig. 7, we show that supere cient Monte Carlo computations of one-dimensional integrands B(x) = P2(x) ? P1(x) = 1?x2 1+x2 ? sgn(x) p1+x2 and B(x) = P 2 (x) ? P 1 (x) = x2?1 x2+1 ? x p1+x2 (see Table3 ) are carried out by the corresponding second-order ergodic map- pings F2(X) and F 2 (X) at = 1. In Fig.... In PAGE 15: ... In Fig. 8, we show that the supere ciency of chaos-based Monte Carlo computation is carried out for a two-dimensional integral of an integrand B(x; y) = P2(x)P3(y)?P1(x)P1(y) = (1 ? x2)(1 ? 3y2)sgn(y) (x2 + 1)(1 + y2)3 2 ? sgn(x)sgn(y) p1 + x2p1 + y2 (59) (see also Table3 ) over the in nite square 2 = (?1; +1) (?1; 1) when we use two di erent chaotic dynamical systems Xn+1 = F2(Xn) and Yn+1 = F3(Yn) at = 1 for the corresponding variables x and y of the two-dimensional integrand B(x; y). Hence, it is shown that chaos-based Monte Carlo computations can be performed to the integration problems over the in nite support by utilizing ergodic transformations over I and supere cient Monte Carlo computation can also be attained if the supere ciency condition is satis ed as before.... In PAGE 22: ...l (y) are derived from the Chebyshev mappings Tl(y) via the topological conjugacy relation h F l (y) = Tl h (y): (82) In this case, F l is equivalent to a multiplication formula of j cot( )j 1 sgn[cos( )] as F l (j cot( )j 1 sgn[cos( )]) = j cot(l )j 1 sgn[cos(l )]: (83) We thus obtain a series of ergodic transformations fF l (y)g such that F 2 (y) = j1 2(jyj ? 1 jyj )j 1 sgn(jyj? 1 jyj); F 3 (y) = jjyj (3 ? jyj2 ) 1 ? 3jyj2 j 1 sgn[y(jyj2 ?3)]; : (84) Correspondingly, orthogonal functions fP l (y)g related to the transformations fF l (y)gl=0; are derived as follows: P 0 (y) = 1; P 1 (y) = h (y); P 2 (y) = jyj2 ? 1 jyj2 + 1; P 3 (y) = jyj (jyj2 ? 3) (1 + jyj2 )32 sgn(y); : (85) In the construction of the above ergodic mappings, there exists a duality such that Fl(y)F l (z) = 1; for yz = 1: (86) Table3 summarizes the analytical results here for the ergodic transformations and the related orthogonal functions, which are necessary for producing the nu- merical results of the supere cient chaos Monte Carlo computations over the in nite support I in Section 7. The graphs of ergodic transformations over I are depicted in Fig.... In PAGE 28: ... Fig. 7 The log-log plot of the expectation value of the square of the er- ror for the chaos Monte Carlo computations of the one-dimensional integrals B(x) = P2(x) ? P1(x) = 1?x2 1+x2 ? sgn(x) p1+x2 and B(x) = P 2 (x) ? P 1 (x) = x2?1 x2+1 ? x p1+x2 over the in nite support (?1; 1) where the second-order non-Gaussian chaos mappings F2(x) and F 2 (x) at = 1 in Table3 are utilized as non-uniform random-number generators over (?1; 1). Supere cient Monte Carlo compu- tations are carried out for the corresponding second-order non-Gaussian chaos maps F2(X) and F 2 (X) respectively.... ..."

### Table 1. Description of the functions in a virtual switch Methods Functions

"... In PAGE 8: ... In our current simulation platform, the VPI/VCI are emulated by the real ATM links using AAL5 protocol. Table1 lists some important control functions that are used to manipulate the routing tables and to setup or tear down the ATM connections. Table 1.... ..."

### Table 1. Description of the functions in a virtual switch Methods Functions

"... In PAGE 8: ... In our current simulation platform, the VPI/VCI are emulated by the real ATM links using AAL5 protocol. Table1 lists some important control functions that are used to manipulate the routing tables and to setup or tear down the ATM connections. Table 1.... ..."