Ordered/1 problem

The idea is simple. We take the zero-order problem as the HF (or HF-LCAO) one, where each electron moves in an average field due to the nuclei and the... 

They are also correct wavefunctions for the zero-order problem, dents can of course be chosen to normalize each result. 

The identification of the few genes regulated by hormones, among the multitude of the genes that are expressed in each cell, is a first-order problem. What makes the identification possible is the existence of some short specific sequences of DNA, situated in the promoter region of each gene, that are recognized by the dimer of the hormone receptor. These sequences are called hormone response elements (HRE) (Seiler-Tuyns et al. 1986 Tzukerman et al. 1994). 

The model is oversimplified in the sense that we have not attempted to specify what effects are incorporated in u. We will, however, consider the main effects from the vacuum fluctuations as well as other possible perturbations needed to produce the degeneracy above as well as, if necessary, considering the weak energy dependence in the Hamiltonian referred to in Eq. (11). To see how the CPT theorem affects our formulation we note that our zero order problem is an irreducible representation of... 

If the zero-order problem is degenerate, one may generalize the formulation with minor modifications to the multidimensional case with the p-dimensional manifold / = <7 (orthonormal for simplicity) and... 

Long Range Ordering Problems 11.3.1 Polytypism in Layered Structures... 

S. Gonzalez-Pinto, S. Perez-Rodriguez and R. Rojas-Bello, Efficient iterations for Gauss methods on second-order problems. Journal of Computational and Applied Mathematics,... 

Note that in Eq. (3) the parameter space has also been transformed. In the next step, we partition the vector x such that x = [xf, xJ f, where Xj comprises the first p (the order of the reduced model) components ofx. The matrices in Eq. (3) are also partitioned according to the partition of the state vector. The reduced order problem can then result from the deletion of the last n-p states, x, and the corresponding matrix blocks, as follows ... 

In this local basis approach the zero-order problem is defined in terms of states localized on the left side of the barrier (the L continuum), the right side (the... 

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

The next order problem is 0(k). At this level, the interface concentration is still independent of position, as we can see by applying the expansions (7-303) to (7 298), from which we see that... 

The method is presently under development, but the possible fields of applications are many and diverse, including order-N methods and alloy ordering problems outside of the quantum corrals. 

One observes that a, b, a, c and b, c are subsets of a, b, c and therefore of lower relative importance and analogously for the singlecomponent subsets a, b, c. The above example represents one of the simplest cases of relative importance ordering problems, which finds chemical applications (section Relative importance of Kekule Structures of Benzenoid Hydrocarbons Chain ordering ). 

To set the stage for the examples, the most simple rank-one second-order problem is described in mathematical terms and its relation to three-way analysis is established. The standard sample (i.e. the sample that contains a known amount of analyte) is denoted Xi(/ x K) and the mixture (i.e. the unknown sample containing an unknown amount of analyte and possibly some interferents) is called X2 (J x K). It is assumed that the first instrumental direction (e.g. the liquid chromatograph) is sampled at J points, and the second instrumental direction (e.g. the spectrometer) at K points. Then the rank-one second-order calibration problem for the case of one analyte and one interferent can be presented as... 

An illustration of the generalized rank annihilation solution of a second-order problem is given to show some of its properties. Suppose that the problem is the following ... 

In cases where order relation problems presented themselves, correction estimates were obtained using a simple averaging of the points in the region of the problem. This is consistent with the procedure recommended by Isaaks(lO) and the results are nearly identical to more sophisticated averaging algorithms. Once these ordering problems are resolved, several types of data presentations are possible. 

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