New coordinates

The total mass of the molecule is Af = Ma + Mb + mn. The components of the centre-of-mass position vector are  

we decide to abandon this coordinate system (SFCS). Instead of the old coordinates, we will choose a new set of 3n + 6 coordinates (see Appendix I on p. 971, choice II)  

---

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

As noted above, the coordinate system is now recognized as being of fimdamental importance for efficient geometry optimization indeed, most of the major advances in this area in the last ten years or so have been due to a better choice of coordinates. This topic is seldom discussed in the mathematical literature, as it is in general not possible to choose simple and efficient new coordinates for an abstract optimization problem. A nonlmear molecule with N atoms and no... 

To transform the Hamiltonian H into these new coordinates, we use the identities... 

Pragmatically, the procedure considers only one atom at a lime, computiiig the 3x3 Hessian matrix associated with that atom and the 3 compon en IS of Ihe gradien t for that atom and then inverts the 3x3 matrix and obtains new coordinates for the atom accord-ingto the Newton-Raphson form u la above. It then goes on lothe next atom and moves it in the same way. using first and second derivatives for the second atom that include any previous nioiioii of atom s. 

I le u. . erlap between the 2pK and 2p orbitals in this new coordinate system is j(2p — 2p ). If tile zei o differential overlap approximation were applied, then different results would be oliLiined for the two coordinate systems unless the overlap in the new, transformed svntem was also ignored. 

It turns out that the htppropriate X matrix" of the eigenvectors of A rotates the axes 7t/4 so that they coincide with the principle axes of the ellipse. The ellipse itself is unchanged, but in the new coordinate system the equation no longer has a mixed term. The matrix A has been diagonalized. Choice of the coordinate system has no influence on the physics of the siLuatiun. so wc choose the simple coordinate system in preference to the complicated one. 

The new coordinates are found by rotation of the old ones in the x-y plane such that they lie along the principal axes of the ellipse. 

The new coordinates (x) and accelerations (a) are computed at in tegral tim es an d the velocities (v) at half in tegral times. The tim e step At entered by the user is the time between evaluations of a, i.e., At= tj - tg. The temperatures reported at integral times are the averages of the values on either side, determined from and... 

In constant pattern analysis, equations are transformed into a new coordinate system that moves with the wave. Variables are changed from (, Ti) to — Ti, Ti). The new variable — Ti is equal to zero at the stoichiometric center of the wave. Equation (16-130) for a bed... 

In some cases, where the nearest rating of the fuse itself is too high for the rated current, a larger cable is recommended. The thermal (/ - t) characteristics of all such components will vary from one manufacturer to another and may not be readily available with a design or a field engineer, while making the selection. The manufacturers of such components therefore as standard practice, perform this coordination for their products and make such data readily available for the user lo make a quick selection. It may be noted that OCR and fuses at least, of different brands, will require a new coordination. 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

In the new coordinates the action, expanded up to quadratic terms, reads... 

The most remarkable feature of expression (4.8) is that it does not contain any cross terms 8x 8s. This is a consequence of time-shift invariance of the instanton solution (d s/dt = dVjds, x = 0). This fact can be expressed as invariance of the action with respect to the infinitesimal transformation s s -I- cs, c 0 [cf. eq. (3.42)]. In the new coordinates the determinants break up into longitudinal and transverse parts and (4.4) becomes... 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

If the particle leaves the box through the left border, jumping to cell -2, for instance, then again applying the bitwise AND between new coordinate and mask (the mask is given simply by the number of the last cell in the box, 7 in this example) yields the correct new periodic position, i.e., in cell 6. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

A linear coordinate transformation may be illustrated by a simple two-dimensional example. The new coordinate system is defined in term of the old by means of a rotation matrix, U. In the general case the U matrix is unitary (complex elements), although for most applications it may be chosen to be orthogonal (real elements). This means that the matrix inverse is given by transposing the complex conjugate, or in the... 

In the new coordinate system A is a diagonal matrix, and the (normalized) e vector is a new coordinate axis. The diagonal elements in A are therefore directly the eigenvalues, and since e = U e, the columns in the U matrix are the eigenvectors. 

Equation (13.20) corresponds to a symmetrical orthogonalization of the basis. The initial coordinate system, (the basis functions %) is non-orthogonal, but by multiplying with a matrix such as S the new coordinate system has orthogonal axes. 

Whatever boundary conditions are appropriate for a given problem may, of course, themselves be easily transcribed into the new coordinate system. 

Taylor Series. Most of us have practiced how to approximate a curve over a bounded region as a series of power terms y = a<, + a,x + a2X1 2. ... But, we probably never realized that each power term x, x2,. .., can be considered as a new coordinate axis, and each coefficient ao, a . .., is simply the new coordinate on its respective axis. 

Very often, the axes of the new coordinate system, or factor space are chosen to be mutually orthogonal, but this is not an absolute requirement. Of the above examples, the axes chosen for 3 and S are generally not mutually orthogonal. 

If we could find a pair of axes that lay in this plane, we could use these axes as the basis of a new coordinate system. We could then simply specify each spectrum in terms of its distance along each of the two axes of our new coordinate system. Notice that this doesn t change the data at all. The data points do not move when we change coordinate systems. This is no different than deciding to define a point in space in terms of its polar coordinates rather than its rectangular coordinates. We would no longer need to provide 100 individual numbers to identify each spectrum by its 100-dimensional spectrum. We could, instead specify each spectrum of this 2 component system by just two numbers, the distances along each new coordinate axis. By extension, the spectra of a 3 component system would require three numbers, a 4 component system 4 numbers, etc. 

So we have found a pair of axes that we can use as the basis of a new coordinate system. And since each axis spans the maximum possible amount of variance in the data, we can be assured that there are no axes that can serve as a more efficient frame of reference than these two. Each axis is a factor or principal component of the data. Together, they comprise the basis space of this data set. 

As we saw in the last chapter, by discarding the noise eigenvectors, we are able to remove a portion of the noise from our data. We have called the data that results after the noise removal the regenerated data. When we perform principal component regression, there is not really a separate, explicit data regeneration step. By operating with the new coordinate system, we are automatically regenerating the data without the noise. 

We compute a PCR calibration in exactly the same way we computed an ILS calibration. The only difference is the data we start with. Instead of directly using absorbance values expressed in the spectral coordinate system, we use the same absorbance values but express them in the coordinate system defined by the basis vectors we have retained. Instead of a data matrix containing absorbance values, we have a data matrix containing the coordinates of each spectrum on each of the axes of our new coordinate system. We have seen that these new coordinates are nothing more than the projections of the spectra onto the basis vectors. These projections are easily computed ... 

Now, we are ready to apply PCR to our simulated data set. For each training set absorbance matrix, A1 and A2, we will find all of the possible eigenvectors. Then, we will decide how many to keep as our basis set. Next, we will construct calibrations by using ILS in the new coordinate system defined by the basis set. Finally, we will use the calibrations to predict the concentrations for our validation sets. 

In addition to the set of new coordinate axes (basis space) for the spectral data (the x-block), we also find a set of new coordinate axes (basis space) for the concentration data (the y-block). 

---