Equality stationary point

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

If each value of/, as given by Eq. (4-150), is multiplied by the same arbitrary scale factor, Eq. (4-149) is still satisfied actually Eq. (4-146) is also independent of a scale factor in the fr Substituting Eq. (4-150) into (4-146), we note that the two sums are equal and the stationary point value of E,EST simplifies to... 

Now consider the imposition of inequality [g(x) < 0] and equality constraints 7i(x) = 0] in Fig. 3-55. Continuing the kinematic interpretation, the inequality constraints g(x) < 0 act as fences in the valley, and equality constraints h(x) = 0 act as "rails. Consider now a ball, constrained on a rail and within fences, to roll to its lowest point. This stationary point occurs when the normal forces exerted by the fences [- Vg(x )] and rails [- V/i(x )] on the ball are balanced by the force of gravity [— Vfix )]. This condition can be stated by the following Karush-Kuhn-Tucker (KKT) necessary conditions for constrained optimality ... 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

The location of the predicted optimum pH may be found by differentiating the fitted model with respect to pH. Setting the derivative of Equation 11.47 equal to zero gives the location of the stationary point (in this case, a maximum). 

Anticipating a later section on canonical analysis of second-order polynomial models, we will show that the first-order term can be made to equal zero if we code the model using the stationary point as the center of the symmetrical design. For this new system of coding, c, = 10 2/3 and (see Section 8.5). 

To find the coordinates of the stationary point, we first differentiate the full second-order polynomial model with respect to each of the factors and set each derivative equal to zero. For two-factor models we obtain... 

Rotation of the translated factor axes is an eigenvalue-eigenvector problem, the complete discussion of which is beyond the scope of this presentation. It may be shown that there exists a set of rotated factor axes such that the off-diagonal terms of the resulting S matrix are equal to zero (the indicates rotation) that is, in the translated and rotated coordinate system, there are no interaction terms. The relationship between the rotated coordinate system and the translated coordinate system centered at the stationary point is given by... 

The stationary points on the energy surface (4 1) are obtained as solutions to the equations 3E/dpj = 0. They can be approximately solved by starting from the expansion (4 2), truncated at second order. Setting the derivatives of E in (4 2) equal to zero leads to the system of linear equations ... 

To determine the stationary points of the quadratic model we differentiate the model and set the result equal to zero. We obtain a linear set of equations... 

To summarize, in the RF approach we make the quadratic model bounded by adding higher-order terms. This introduces n+1 stationary points, which are obtained by diagonalizing the augmented Hessian Eq. (3.22). The figure below shows three RF models with S equal to unity, using the same function and expansion points as for the linear and quadratic models above. Each RF model has one maximum and one minimum in contrast to the SO models that have one stationary point only. The minima lie in the direction of the true minimizer. 

Canonical analysis achieves this geometric interpretation of the response surface by transforming the estimated polynomial model into a simpler form. The origin of the factor space is first translated to the stationary point of the estimated response surface, the point at which the partial derivatives of the response with respect to all of the factors are simultaneously equal to zero (see Section 10.5). The new factor... 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

A point Q0 = QJ0 where all gradient matrix elements are equal to zero is referred to as a stationary point ... 

Any trajectory can end when p - I at a stationary point (SP), in which all the right-hand parts of equations (5.2) equal zero. In the case of the terminal model (2.8) all such SPs are those solutions of the non-linear set of the algebraic equations (4.13) which have a physical meaning. Inside m-simplex one can find no more than one SP, the location of which is determined by the solution of the linear equations (4.14). In addition to such an inner azeotrope of the m-simplex, azeotropes can also exist on its boundaries which are n-simplexes (2 S n m - 1). For each of these boundary azeotropes (m — n) components of vector X are equal to zero, so it is found to be an inner azeotrope in the system of the rest n monomers. Moreover, the equations (4.13) always have m solutions x( = 8is (where 8js is the Cronecker Delta-symbol which is equal to 1 when i = s and to 0 when i =(= s) corresponding to each of the homopolymers of the monomers Ms (s = 1,. ..,m). Such solutions together with all azeotropes both inside m-simplex and on its boundaries form a complete set of SPs of the dynamic system (5.2). 

The stationary point of the system (5.2) is by a definition stable one, if all the roots of its characteristic equation (5.11) have the negative real parts. The Routh-Hurwitz criteria presented in Ref. [206] permits escaping the calculations of these roots to establish the simple relations between the coefficients ock, which allow to point out simple stability conditions. For instance, in the case of terpolymerization the positivity of both coefficients oq and oc2 is regarded to be a criteria of such stability, and as for four-component copolymerization the following non-equality a3 < oq < ot2 has also to be hold. At arbitrary number m of the components the positivity of all ak is regarded to be necessary (but not sufficient) stability condition. For the stability of the boundary SP of m-component system located inside the certain boundary 1-subsimplex of monomers Mk, M2,. .., M, the stability of the above SP in such subsimplex and negativity of all values of X, Xl+1,..., vm x (5.13) are needed. 

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