The Origin Problem

The dipole contribution to the nuclear spin-spin coupling constants in Section 8.3 comes from the more general expression 

There are also the contributions from the delta function part of the field due 

The electron coordinates change to 7 == f — G, the nuclear coordinates to Rgc = Rg — G, while the electron coordinates fg relative a given nucleus remain the same. This means that the spin-spin coupling tensor is invariant to the change of origin. 

From the equation of motion of a (excitation) propagator A B))e, it follows that 

Consideration of the magnetic susceptibility tensor and the observation that the identity 

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Vd satisfying PdVd = Vd, which means that Vd is an approximation of an invariant measure. For an invariant measure, any numerical discretization may be interpreted as a stochastic perturbation of the original problem. 

Reevaluate the data in problem 24 in Chapter 4 using the same significance level as in the original problem. ... 

The initial research effort may prove to be a broad spectmm of apphcations or solutions to the original problem that in turn provide any number of inventions. When efforts move toward reducing the invention to practice and refining the invention so that it proves to be commercially marketable, certain apphcations may prove to be unfeasible or commercially impractical. As a result, only one apphcation, eg, the creation of a given pattern on the surface of the automobile tire, may ultimately prove commercially marketable. However, ah the solutions which are developed and considered over the research and development process may comprise inventions that are worthy of disclosure and claiming in a patent. An apphcation which is not commercially viable today may become viable within the seventeen-year lifetime of a patent. 

In the original problem one usually has m < n. Thus, the vertices of the region of solution lie on the coordinate planes. This follows from the fact that, generally, in n dimensions, n hyperplanes each of dimension (n — 1) intersect at a point. The dual problem defines a polytope in m-dimensional space. In this case not all vertices need lie on the coordinate planes. 

On taking the scalar product with x° and recalling that for a linear programming problem the values of the objective function of the original problem and of the dual coincide at the solution points, we conclude that whenever xf > 0 we must have j8, = 0. 

We now turn to the question of approximations of boundary and initial conditions on a solution of the original problem. This question is intimately connected with the statement of a difference problem. 

With these relations established, we conclude that if the scheme is stable and approximates the original problem, then it is convergent. In other words, convergence follows from approximation and stability and the order of accuracy and the rate of convergence are connected with the order of approximation. 

Let us stress once again that in order to estimate the order of accuracy of a scheme, it is necessary to estimate its order of accuracy only on a solution of the original problem. 

The gain in accuracy provided by refining the step h is limited by requirements of economy. Such an approach is equivalent to minimizing the execution time necessary in this connection in obtaining the solution. But if the solution of the original problem u and / both are smooth functions of X, the accuracy of numerical solution can be increased by performing calculations for the same problem (12) on a sequence of grids, , < h ... 

In this way, the third kind difference boundary-value problem (2)-(4) of second-order approximation on the solution of the original problem is put in correspondence with the original problem (1). 

The original problem. The heat diffusion process on a straight line is described by the heat conduction equation... 

For this, we proceed as usual. This amounts to evaluating the error z — Ai — where Ai is a solution of the original problem (l)-(3) and... 

When providing current manipulations, the solution u = u(x,t) and the available data of the original problem are preassumed to be smooth enough and sufficient for the existence of the asymptotic expansion... 

The original problem. In a common setting it is required to find in the rectangle... 

Of course, the words arbitrary domain cannot be understood in a literal sense. Before giving further motivations, it is preassumed that the boundary F is smooth enough to ensure the existence of a smooth solution u = u x,t) of the original problem (l)-(2). In the accurate account of the approximation error and accuracy we always take for granted that the solution of the original problem associated w ith the governing differential equation exists and possesses all necessary derivatives which do arise in the further development. 

In this regard, we should take into account that auxiliary schemes are not obliged to approximate the original problem the approximation here is ensured by summarizing all the residuals obtained. 

As a matter of experience, the estimation of the nearness of a solution of the difference problem amounts to the proximity between a solution of the original problem (5) and a solution of the chain of problems (6)-(7). The main idea behind this approach is connected with the obvious relation... 

Along these lines, both chains generate approximations on the solution u = u x,t) of the original problem (5). Indeed, it is straightforward to verify for problem (6)-(7) with the aid of the relation Va u = Va that... 

Further comparison of the solution v tj) of problem (55)-(57) with the solution M(tj) of the original problem allows to cite without proofs several interesting remarks. 

Before we can learn new control information, we must formally specify what it is we are trying to learn and how what we learn is to be applied within the original problem-solving framework (Section ID. 

The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. 

Equation 69-15 is the same as equation 69-8. Thus we have demonstrated that the equations generated from calculus, where we explicitly inserted the least square condition, create the same matrix equations that result from the formalistic matrix manipulations of the purely matrix-based approach. Since the least-squares principal is introduced before equation 69-8, this procedure therefore demonstrates that the rest of the derivation, leading to equation 69-10, does in fact provide us with the least squares solution to the original problem. 

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