Waterloo Maple Software

Maple V, Release 2, Waterloo Maple Software, University of Waterloo, Ontario, Canada. 

Clearly for larger matrices the situation becomes quite complicated and, while the principle ought to be understood, computer programs are available for the numerical as well as for the symbolic evaluation of large determinants. I find the programs MapleVl for Windows (Waterloo Maple Software, 160 Columbia Street West, Waterloo, Ontario, Canada N2L... 

Finally, the apparent foam modulus can be obtained through combination and numerical integration of Equ.(l-3,5-7). For this. Maple 9.5 (Waterloo Maple Software) was used and the results are discussed next. 

The Routh-Hurwitz conditions are well known and can be used to determine, in principle, the stability properties of the steady state of any n-variable system. This advantage is, however, balanced by the fact that in practice their use is very cumbersome, even for n as small as 3 or 4. The evaluation, by hand, of all the coefficients Cl of the characteristic polynomial and the Hurwitz determinants A constitutes a rather arduous task. It is for this reason that in the past this tool of linear stability analysis could hardly be found in the literature of nonlinear dynamics. The situation changed with the advent of computer-algebra systems or symbolic computation software. Software such as Mathematica (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario) makes it easy to obtain exact, analytical expressions for the coefficients C/ of the characteristic polynomial (1.12) and the Hurwitz determinants A . 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Thus, David Radcliffe says, 10 ends in 2.5 x 10 - r zeros, where r is between 1 and 1,000. He used a software package called MAPLE (University of Waterloo), which supports high-precision arithmetic, to determine the exact value of r, and finds the number of trailing zeros to be 2.5 x 10 - 143. 

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