Solved in principle

At this stage the problem may be considered to be solved in principle since one can readily calculate the spectrum (7.55) with d ico, V) found from Eq. (7.61). 

It has been assumed above that Eq. (18) has been solved. In principle, the resulting eigenvalues and eigenfunctions can then be substituted in Eq. (19) to yield the first-order corrections, and so on, for higher orders of approximation. 

Whatever the X, the binary kernel gives a convergent contribution18 to the evolution of 9 1(p1, t). This solves, in principle at least, the problem of the divergence of the Born development (Eqs. 56, 57). 

Bifunctional systems In the case of bifunctional systems (or molecules) only two alternatives are possible the bifunctional relationships are either "consonant" or "dissonant" (apart from molecules or systems with functional groups of type A to which we have referred to as "assonant"). In the first case, the synthetic problem will have been solved, in principle, in applying the "heuristic principle" HP-2 that is to say, the molecule will be disconnected according to a retro-Claisen, a retro-aldol or a retro-Mannich condensation, or a retro-Michael addition, proceeding if necessary by a prior adjustment of the heteroatom oxidation level (FGI). 

Although the problem of the liquid membrane potential was solved in principle by Nemst, a discussion developed in the ensuing two decades between Bauer [6], who developed the adsorption theory of membrane potentials, and Beutner [10,11,12], who based his theories on Nernst s work. This problem was finaly solved by Bonhoeffer, Kahlweit and Strehlow [13], and by Karpfen and Randles [49]. The latter authors also introduced the concept of the distribution potential. 

I. Linear differential equations in which only the inhomogeneous term is a random function, such as the Langevin equation. Such equations have been called additive and can be solved in principle. 

This set may be solved in principle to yield P and Xj, each as a function of r, with the knowledge of a value of P and Xj for fixed values of r. The solution is easily obtained on the assumption that the ideal solution laws are applicable and that either the ideal gas equation is followed in the case of a gas phase or the volumes are independent of the pressure. Then, for an ideal liquid solution, Equation (14.27) becomes... 

The control of product purity should cope with flexibility requirements. Excessive purity turns into higher operation costs, but lower purity is unacceptable. The problem may be solved in principle by adequate local control loops. The same is valid for waste minimization. However, when the separators are involved in the recycle loops, both the design and control must take into account the effect of interactions. 

The system (8-42) should be completed with appropriate equations resulting from the boundary conditions and it can be solved, in principle, by the same factorization method of Crout for systems with tri-diagonal matrix. 

While this particular problem of mechanical stability can be solved in principle by means of the Schroedinger equation, exact solutions have never been obtained for any polyatomic molecules except H2 and Hf. Later on we shall consider some of the approximate treatments which have been made. 

Equation [8] can be solved, in principle, to yield the time evolution... 

The Schrodinger equation This equation can be solved, in principle, for the energies and the wave functions for any quantum system of interest. The energy is quantized whenever a particle is confined by a potential. The square of the wave function gives the probability of finding the particle at a particular position in space. Normalization of the wave function by requiring that... 

It is seen from the foregoing results, c.g., equations (24.11) and (24.12), that by combining statistical mechanics with the Boltzmann-Planck equation it is possible to derive a relationship between the molar entropy of any gas, assuming it to behave ideally, and the partition function of the given species. Since the partition function and its temperature coefficient may be regarded as known, from the discussion in Chapter VI, the problem of calculating entropies may be regarded as solved, in principle. In order to illustrate the procedure a number of cases will be considered. 

Eq. (33), together with the usual normalization condition, can readily be solved in principle for the FC S g vector. In practice, however, this is an infinite set of equations that must be truncated (see further below) to obtain a solution. 

The correlation problem can be solved in principle by configuration interaction (Cl) or one of its variants. The exact wavefunction is expanded as a linear combination of Slater determinants ... 

Together with the governing equation and boundary condition, (9-129), the matching condition (9-157) yields a well-posed problem that could be solved, in principle, to obtain... 

The question of the theoretical calculation of gaseous equilibria is thus solved in principle, subject, of course, to the further assumption that formula (125) is also of general validity but this, again, we can hardly doubt in the light of our present knowledge. 

The fundamental point to be noted is that we may regard a simulation problem as solved in principle as... 

These six equations can therefore be solved, in principle, subject to appropriate boundary and initial conditions to yield velocity, pressure, density, and temperature profiles in the atmosphere. 

Equations (16.1) to (16.4) represent six equations for the six unknowns i, U2, M3, p, p, and r. These equations can therefore be solved, in principle, subject to appropriate boundary and initial conditions to yield velocity, pressure, density, and temperature profiles in an ideal gas. Because of the highly coupled nature of (16.1) to (16.4), these equations are virtually impossible to solve analytically. However, we can exploit certain aspects characteristic of the lower atmosphere to simplify them. 

---