Formula vector

The entry for each species (in file order listed) of file formula vector formed by file subscripts to the elements (in file order listed) ... 

Brinkley (4 postulated C species at equilibrium, p species, referred to as "components," were selected to have linearly independent formula vectors, where p is the rank of the atom matrix, (mjk), and Yj is the formula vector for the jth species, [mj, mj2f...mjE]. Given the choice of components, the stoichiometric coefficients for an independent set of chemical reactions are computed ... 

The nonnegative integers Z(a, m) make up the atomic matrix Z with the columns called formula vectors. A (complex chemical) reaction among atomic components as above is an atomic reaction. 

Now let us return to the independence of elementary reactions in this general setting. A set of elementary reactions is said to be independent if there is no way of expressing any of the formula vectors as a linear combination of the others. In the opposite case the components are said to be dependent. From this definition it is clear that the number of independent components is the number of independent columns of Z. But this number is the rank of Z rank(Z). In our example (3.1)-(3.2) this number is equal to 3. 

The Champ-Sons model is based upon this approximation. It results into a modified Rayleigh integral where specific terms appear. The resulting formula for the refracted field (e.g., displacement vector-field), is given by... 

The decoupled set of equations in system (20) can be solved for all the Qi and associated velocities V<.i by closed-form formulas that depend on the eigenvalues [71]. The harmonic position and velocity vectors at time nAt can then obtained from the expressions ... 

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

Here t = (—n2,ni) is the tangential unit vector at T. Integrating by parts, one can obtain the Green formula (Temam, 1983 Khludnev, Sokolowski, 1997)... 

Here the values correspond to the positive and negative directions of respectively. Denote by = 0 the tangent components of the vectors and make use of the formulae... 

To conclude the section we write the formula (4.159) in the form which does not contain the function 9. To this end, consider a neighbourhood Sl of the set L with a smooth boundary T l assuming that 9 = 1 on Sl- Denote by vi,V2,i 3) the unit external normal vector to T. Integrating by parts in (4.159) we obtain... 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

In order to And rjopt, we minimise an expression of the total CPU time, T, required for the Ewald sums. We assume that T is proportional to the number of vectors used in both the reciprocal and direct space sums, G /(2j ) and R /a . and to t, and t<, the CPU times required for the evaluation of a single term in each series. In formulae. 

Still another interpretation can be made by taking A22 to be a scalar, hence A21 a row vector and A12 a column vector. Suppose A1X has been inverted or factored as before. Then L21, R12, and A22 are obtainable, the two triangular matrices are easily inverted, and their product is the inverse of the complete matrix A. This is the basis for the method of enlargement. The method is to start with aai which is easily inverted apply the formulas to... 

In quantum mechanics, angular momenta other than orbital make their appearance. Their structure is not revealed by the simple considerations leading to (7-8). That formula, in fact, arises also from the general transformation properties of vectors under rotation, as will now be shown. 

The L and S values are those from which the / value was formed via the vector coupling rule. These formulae strictly apply only for the magnetism of free-ion levels. They provide a good aproximation for the magnetism of lanthanide complexes, as we shall note in Chapter 10, but provide no useful account of the magnetic properties of d block compounds. 

The difference problem under consideration will be completely posed if the subsidiary information is available on the vector for j — 0 with the initial condition y = Uq- The value is successively calculated by the explicit formula... 

In light of the special structure of the diagonal matrices k, whose blocks are lower triangle, the components s = 1,2,..., z, of the unknown vector yjQ.) are to be determined successively by the elimination method in passing from a to qH- 1 and from s to s- 1. By the elimination formulas for a three-point equation we constitute in a term-by-term fashion the vectors a = 1,2,..., p. Moving in reverse order from a -f 1 to a and from s -b 1 to s the vectors y(p+i), >y(2p) recovered from the system... 

For example, the ZN theory, which overcomes all the defects of the Landau-Zener-Stueckelberg theory, can be incorporated into various simulation methods in order to clarify the mechanisms of dynamics in realistic molecular systems. Since the nonadiabatic coupling is a vector and thus we can always determine the relevant one-dimensional (ID) direction of the transition in multidimensional space, the 1D ZN theory can be usefully utilized. Furthermore, the comprehension of reaction mechanisms can be deepened, since the formulas are given in simple analytical expressions. Since it is not feasible to treat realistic large systems fully quantum mechanically, it would be appropriate to incorporate the ZN theory into some kind of semiclassical methods. The promising semiclassical methods are (1) the initial value... 

In standard high level language programming the dimension of the NSS n, signals the number of nested do loops which are necessary to reproduce the structure in a computational environment. But the mathematical usefulness of this entity can be easily recognized when the particular characteristic of this symbolic unit is analyzed the involved vector parameters could be chosen with arbitrary and variable dimensions. There are many scientific and mathematical formulae which will benefit of this property, when written in a paper or computationally implemented. 

This relationship is called the second Green s formula and it represents Gauss s theorem when the vector X is given by Equation (1.98). In particular, letting ij/ — constant we obtain the first Green s formula ... 

The relationships between scores, loadings and latent vectors can be written in a compact way by means of the so-called transition formulae. ... 

One can define transition formulae for the two sets of generalized latent vectors in A and B (see also Section 31.1.6) ... 

These transition formulae express one set of generalized latent vectors (A or B) in terms of the other set (B or A). They follow readily from the definition of the generalized SVD problem which has been stated above. 

Scales (1986) recommends the Polak Ribiere version because it has slightly better convergence properties. Scales also gives an algorithm which is used for both methods that differ only in the formula for the updating of the search vector. 

---