### Table 1. Prime cycles for the binary symbolic dynamics up to length 9.

### Table 1. Symbolic dynamics in mathematics, physics, and engineering. The germ of the ideas is the same in each discipline, but the goals and implementations can be very di erent.

### Table 2: The number of nonzeros obtained by the static symbolic factorization and the dynamic approach in SuperLU.

1998

"... In PAGE 6: ... We have compared the number of nonzeros obtained by the static approach and the number of nonzeros obtained by SuperLU for these matrices. The results in Table2 show that the over-estimation usually leads to about 30 ? 50% more nonzeros, which is acceptable. The extra nonzeros do imply additional computational cost.... ..."

Cited by 32

### Table 2: The number of nonzeros obtained by the static symbolic factorization and the dynamic approach in SuperLU.

1998

"... In PAGE 6: ... We have compared the number of nonzeros obtained by the static approach and the number of nonzeros obtained by SuperLU for these matrices. The results in Table2 show that the over-estimation usually leads to about 30{50% more nonzeros, which is acceptable. The extra nonzeros do imply additional computational cost.... ..."

Cited by 32

### Table 2: The number of nonzeros obtained by the static symbolic factorization and the dynamic approach in SuperLU.

1998

"... In PAGE 6: ... We have compared the number of nonzeros obtained by the static approach and the number of nonzeros obtained by SuperLU for these matrices. The results in Table2 show that the over-estimation usually leads to about 30 ? 50% more nonzeros, which is acceptable. The extra nonzeros do imply additional computational cost.... ..."

Cited by 32

### Table 5: Time, in secs., to compute the symbolic derivative of in- verse dynamic function, D* vs. Mathematica.

### Table 3.1: Symbols used for common rigid body dynamics quantities

### Table 3. Number of prime cycles for various alphabets and grammars up to length 10. The rst column gives the cycle length, the second the formula (11) for the number of prime cycles for complete N-symbol dynamics, columns three through ve give the numbers for N = 2; 3 and 4.

### Table 2: Rotor geometric and dynamic properties (Ref. [22]) Symbol Property baseline AFC blade

2001

"... In PAGE 8: ... It is representative of a large class of medium weight helicopter rotors. The basic properties of the rotor are described in Table2 . Two analytical models have been studied in this paper, both based on the main properties of the mentioned rotor.... ..."

Cited by 7

### Table 1. Equations of convective radiation hydrodynamics in di erential form. D=Dt denotes the comoving or substantial derivative with respect to time. Refer to Table 6 for a comprehensive list of symbols. Gas dynamics

"... In PAGE 2: ... The remaining inte- gration of the inner mass mr over the density structure is performed by solving Eq. 1 in Table1 , where we employ mr as the independent variable. Further details concerning the equations of hydrody- namics can be found e.... In PAGE 2: ....2. Radiative transfer Time-dependent radiative transfer is considered in the grey approximation by solving the rst two frequency- integrated moment equations of the radiation eld rep- resenting energy conservation (radiation energy equation, Eq. 5 in Table1 ) and momentum conservation (radiation momentum equation, Eq. 6 in Table 1).... In PAGE 2: ... 5 in Table 1) and momentum conservation (radiation momentum equation, Eq. 6 in Table1 ). As independent variables we use the zeroth moment of the radiation eld J and the rst moment of the radiation eld H, which are proportional to the radiative energy density and the radiative ux, respectively (e.... In PAGE 3: ...ecko et al. (1998) and Koll ath et al. (1998). The basic equation (Eq. 8 in Table1 ) of the Kuh- fu model is a conservation law for the turbulent kinetic energy density !, which serves as the independent variable of the turbulent eld. The essential term is the turbulent driving through buoyancy forces (S !).... In PAGE 3: ... The basic quantity entering S ! is the entropy gradient @s=@r which is related to the Schwarzschild criterion by Eq. 12 ( Table1 ). Note that S ! can be evaluated even if the Schwarzschild crite- rion indicates convective stability.... In PAGE 4: ...6. The model equations In Table1 the complete set of model equations for gas dynamics, radiative transfer and turbulent convection in di erential form are compiled. With regard to the numer- ical method of solution using a conservative volume dis- cretization on an adaptive mesh (cf.... In PAGE 4: ... The corresponding total internal energy equation (Eq. 4 in Table1 ) can be derived by adding the respective energy equations of gas, radiation and convection (Eqns. 4,5 and 8 in Table 1).... In PAGE 4: ... Numerically speaking the initial model has to lie within the convergence radius of the implicit solution scheme. Consequently we use the hydrostatic and local limit of the nonlinear equations given in Table1 to compute the structure of the initial model. This limit is obtained by omitting all time derivatives, setting the gas velocity iden- tically to zero (u 0), and neglecting turbulent pressure and overshooting.... In PAGE 10: ...urb. energy eq. Table 4. Hydrostatic and local limit of the convective radiation hydrodynamics equations given in Table1 . See text for details.... ..."