Gramian

The Hessian of the internal energy is the Gramian of a metric space. 

Both Hessian and Gramian aspects of M(c+2) contribute to the richness of Ms ... 

Table 11.3 displays a variety of geometrical descriptors for each fluid (extending results presented in Sidebar 11.3 for a monatomic ideal gas). These descriptors include the lengths of various thermodynamic vectors ( T, P, S, V ), 0Sy coupling angle (=0TP cf. Fig. 11.2), Gramian M, and minor eigenvalue e2 of the metric M. Sidebar 11.5 describes numerical evaluation of e2. 

The general features of critical dimensional collapse in Ms are fairly evident. As previously remarked [cf. (10.21)], the key feature of M is its Gramian character, i.e., its composition from scalar products RjR7 = (M j of underlying vectors RJ. Sidebar 11.7 outlines the general mathematical theory relating these Gramian vectors... 

The equations relating Gramian vectors R to eigenvectors U have a simple geometrical interpretation. One can show [see, e.g., G. E. Shilov. An Introduction to the Theory of... 

Linear Spaces (Prentice-Hall, Englewood Cliffs, NJ, 1961), pp. 167-73] that the Gramian determinant... 

In theorem 12 of P it was shown that the first term is that of a positive definite Gramian matrix. The second is clearly the general term of a positive semi-definite matrix we cannot assert that it is definite since one or more of the A Hi might be zero. But the sum of a positive definite and a positive semi-definite matrix is definite and so the Jacobian nowhere vanishes. 

Hahn, J., Edgar, T. F., An improved method for nonlinear model reduction using balancing of empirical gramians, Computers Chem. Eng. 2002, 26 1379— 1397. 

Determine the so-called controllability and observability gramians, the matrices P and Q. These are solutions to the equations ... 

This is a transformation of the linearized system, and can be derived by substituting xb = Txi in the linearized system. The gramians Pb and Qb of the balanced system satisfy ... 

The matrix X7X (7 x 7) is Gramian (symmetric and positive (semi-)deflnite) and can be decomposed as [Schott 1997]... 

Gramian determinants and their ratios play a key role in the dimensional scaling treatment of many-electron atoms (and other many-body problems), so we summarize in this appendix several properties of these objects. 

Consider a set of N unit vectors f,- rooted at a common point, which can be taken to be the origin. Their Gramian determinant is defined as [27]... 

Now consider N vectors r,- of arbitrary lengths r,= rj. Although Gramian determinants are sometimes defined for non-unit vectors, it is more convenient for electronic structure applications to continue to keep the radial and angular aspects of the problem separated. Thus, we define the Gramian determinant for these vectors as above, with f = r,/r,-. The iV-dimensional volume of the parallelotope (t.e., generalized parallelepiped) defined by the r,- is clearly just rir2 ... 

The Gramian determinant ratios F d/r occurring in the centrifugal potential can also be expanded out in analogy with Eq. (34),... 

In the treatment of larger atoms, the evaluation of the Gramian determinants is the most time-consuming step. In the absence of assumptions regarding symmetry, it would undoubtedly be best to use elimination methods (requiring operations) rather than explicit expansions ( n operations) for the evaluations. However, when the matrices are highly structured, explicit expansions become more useful. This is the case, for example, if one assumes that all electrons within a shell are equivalent. Suppose that the atom has S shells, with N electrons in shell a. Then the Gramian determinant has the form... 

The Gramian determinant P is obtained by removing one row and one column from the matrix for F. All atoms in the ring (that is, the chain with periodic boimdary conditions) were equivalent, but it is simplest to eliminate the atom labeled 1 or N. This gives the Huckel determinant for a linear conjugated polyene [20], Cn-iH2n ... 

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