Regular point

Both Poisson s and Laplace s equations describe the behavior of the potential at regular points where the first derivatives of the field exist. To characterize the behavior of the potential at the boundary of media with different densities, let us make use of Equation (1.39) according to which a component of the field along some direction / is equal to the derivative of the potential in this direction ... 

This field of the centrifugal force, unlike the attraction field, is fictitious, and correspondingly, we observe a volume distribution of fictitious sources with a density proportional to co. A summation of the first and second Equations (2.62 and 2.63) gives the system of equations of the gravitational field at regular points... 

Thus, the potential of gravitational field obeys Poisson s equation at regular points inside the earth... 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

The factor space representing these components can be formed by a triangle where each vertex represents a pure component. Measurements of the criteria of mixtures of the three components are made at regular points (according to a simplex lattice design) in the factor space (Figure 4.16). 

Let x be a global minimum of (3.6) at which the gradients of the equality constraints are linearly independent (i.e., x is a regular point). Perturbing the right-hand sides of the equality constraints, we have... 

Now, let us go back to the methods of forecasting conditions for obtaining critical regimes for keroplast flow. As demonstrated in [20], critical tension of melted filled polymer can be calculated as a regular point on the flow curve, i.e. 

The question has often been raised whether greater regularities in b.ps. would be found if these were compared at different pressures, but nothing significant has come of this. The b.p. is no doubt related to the attractive forces between the liquid molecules. Regularities pointed out by Huckel<5 are ... 

On any surface, the principal directions are mutually orthogonal at regular points (recall section 1.3). On minimal surfaces, this is true for asymptotic directions as well. (An asymptotic direction is that along which the normal curvature vanishes.) Orthogonality of the asymptotic directions can be shown... 

Cubic membranes can be involved in curvature-controlled activation of certain enzymes, as well as control of enzyme activity. It is tempting to suggest the latter as a general function of cubic membranes since proteins could conceivably be located at regular points in the lattice. This could enhance transport efficiency of both product and substrate. Such mechanisms are particularly well suited for mass cooperative synthesis, such as those... 

I) at the investigated point, for example x = (0,..., 0) = 0, the function does not have a critical point (such a point will be called regular point) ... 

A regular point of a function of n variables F(xj,..., x ) is defined as the point not being a critical point of this function. In other words, at a regular point, let it be for example the point x = 0, the function gradient (cf. definition (2.2)) does not vanish ... 

In the case of a function of one variable, V(x), it could be represented in the vicinity of a regular point in the form V(x) = x, see equation (2.7a). In the case of a regular critical point of several variables, the function V(xx ) can be simplified in a similar way, without changing its local character nearby this point. 

The transformation x - x is an allowed change of variables when it does not alter the character of the investigated regular point x 0. This is the case when the Jacobian of transformation (2.28) does not vanish nearby the point x = 0... 

As mentioned above, we shall not describe at this point the method of solving the problem of determinacy for functions of two variables (see Appendix, A2). We shall confine ourselves to providing a list of the simplest potential function, having at x = (0, 0) a regular point, a degenerate critical point, for which the problem of determinacy has been solved. In other words, addition of a perturbation to the functions listed in Table 2.4 must not convert a degenerate critical point into another degenerate critical point. Table 2.4 (functions of two variables) is a counterpart to Table 2.1 (functions of one variable). 

The equation has singular points at i = 1, both of which are regular points (sec Sec. 17), so that it is necessary to discuss the indicial equation at each of these points. In order to study the behavior near z = +1, it is... 

The coefficients of F and F possess singularities at the origin, which is a regular point (cf. Sec. 17), so that we again make the substitution... 

The singular points, which are regular points, have now been shifted to the points 0 and 1 of x, so that the indicial equation must be obtained at each of these points. Making the substitution T(x) = x G(x), we find by the procedure of Section 17 that s equals % K — M, while the substitution... 

Condition 2, a very strong constraint, ensures the proper behavior of the slope dP/dp on the phase boundary. Note that when P = Pa, then dP/dp = 0 for all regular points of Equation 1, i.e. Equation 1 yields the correct isothermal behavior in the two-phase transition region of a fluid (see Figures 1 and 2). 

---