Returns vector

The returned vector p is obviously the characteristic polynomial. The matrix ql is really the first column of the transfer function matrix in Eq. (E4-30), denoting the two terms describing the effects of changes in C0 on Ci and Cj Similarly, the second column of the transfer function matrix in (E4-30) is associated with changes in the second input Q, and can be obtained with ... 

The returned vector c contains the coefficients of the polynomial. In this example, the result should be y = x3 + 2x2 + 3x + 1. We can check and see how good the fit is. In the following statements, we generate a vector xf it so that we can draw a curve. Then, we calculate the corresponding yf it values, plot the data with a symbol and the fit as a line. 

If Bsub returns Ires=l, then DDAPLUS evaluates the current column of B from the returned vector fpj as follows ... 

Thus, if an ensemble can be prepared that is at equilibrium, then one Metropolis Moni Carlo step should return an ensemble that is still at equilibrium. A consequence of this that the elements of the probability vector for the limiting distribution must satisfy ... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

Consider Equations (6-10) that represent the CVD reactor problem. This is a boundary value problem in which the dependent variables are velocities (u,V,W), temperature T, and mass fractions Y. The mathematical software is a stand-alone boundary value solver whose first application was to compute the structure of premixed flames.Subsequently, we have applied it to the simulation of well stirred reactors,and now chemical vapor deposition reactors. The user interface to the mathematical software requires that, given an estimate of the dependent variable vector, the user can return the residuals of the governing equations. That is, for arbitrary values of velocity, temperature, and mass fraction, by how much do the left hand sides of Equations (6-10) differ from zero ... 

The task of the problem-independent chemistry software is to make evaluating the terms in Equations (6-10) as straightforward as possible. In this case subroutine calls to the Chemkin software are made to return values of p, Cp, and the and hk vectors. Also, subroutine calls are made to a Transport package to return the ordinary multicomponent diffusion matrices Dkj, the mixture viscosities p, the thermal conductivities A, and the thermal diffusion coefficients D. Once this is done, finite difference representations of the equations are evaluated, and the residuals returned to the boundary value solver. 

Spin-lattice relaxation time Time constant for reestablishing equilibrium, involving return of the magnetization vector to its equilibrium position along the z-axis. 

Returning to the unit-ceU, we can also utilize the vector method to derive the origin of the Miller Indices. The general equation for a plane in the lattice is ... 

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

MATLAB will return the vector [0 1.29], meaning that K, = 0, and K2 = 1.29, which was the proportional gain obtained in Example 7.5A. Since K, = 0, we only feedback the controlled variable as analogous to proportional control. In this very simple example, the state space system is virtually the classical system with a proportional controller. 

MATLAB returns the results in a column vector. Most functions in MATLAB take either row or column vectors and we usually do not have to worry about transposing them. 

MATLAB will return the coefficients in a, the corresponding poles in b and whatever is leftover in k, which should be nothing in this case. Note that [] denotes an empty matrix or vector. 

The primary difficulty in using the SOM, which we will return to in the next chapter, is the computational demand made by training. The time required for the network to learn suitable weights increases linearly with both the size of the dataset and the length of the weights vector, and quadratically with the dimension of a square map. Every part of every sample in the database must be compared with the corresponding weight at every network node, and this process must be repeated many times, usually for at least several thousand cycles. This is an incentive to minimize the number of nodes, but as the number of nodes needed to properly represent a dataset is usually unknown, trials may be needed to determine it, which requires multiple maps to be prepared with a consequent increase in computer time. 

Vector events =(Vector) instructorSchedule.get (p) for (Enumeration e = events.elementsQ e.hasMoreElements () ) Event ev = (Event) e.nextElement () if (ev.overlaps (d1,d2)) return false ... 

We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eb the squared length l2 of a n-vector t> is the quadratic form given by... 

In most cases of interest, however, the system represented by equation (5.3.3) is overdetermined and we must enforce the closure condition with a different method. Let us return to a standard mass-balance least-square problem, such as, for instance, calculating the mineral abundances from the whole-rock and mineral chemical compositions. If xu x2,.. -,x are the mineral fractions, which may be lumped together in a vector x, the closure condition... 

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