Quadratic form

The vibraiimial rroqueiicics are tlenved lioin the harmonic approximation, which assiiiines that the potential surface has a quadratic form. 

A cubic bond-stretching potential passes through a maximum but gives a better approximation to the Morse e close to the equilibrium structure than the quadratic form. 

However, this form becomes repulsive for r - rg > 2/3 CS, so it is unsuitable for bond lengths significantly larger than equilibrium. In the MM2 program, this is avoided by temporarily setting CS to zero, reverting to a quadratic form, which leads to discontinuities in the potential surface that are unacceptable for geometry optimi-... 

If V is a function of more than one variable, then more complex criteria for determining maxima and minima are obtained. Generally, but not always, the second partial derivatives of the function with respect to all its variables are sufficient to determine the character of a stationary value of V. For such functions, the theory of quadratic forms as described by Langhaar [B-1] should be examined. 

When the MFA is used in absence of the external field (J,- = 0) the Lagrange multipliers //, are assumed to give the actual density, p, known by construction. In presence of the field the MFA gives a correction Spi to the density p,. By using the linear response theory we can establish a hnear functional relation between J, and 8pi. The fields Pi r) can be expressed in term of a new field 8pi r) defined according to Pi r) = pi + 8pi + 8pi r). Now, we may perform a functional expansion of in terms of 8pi f). If this expansion is limited to a quadratic form in 8pj r) we get the following result [32]... 

This calculation shows explicitly the correction to the MFA. With the quadratic form considered for and the local approximation, the calculation of k T n det 0A) can be performed exactly [39]. The change in the free energy by unit of volume, A.Fcharging when we switch on the charge is given by... 

Finally, we mention an approach based on using a different energy (or difference) function. The quadratic form used above (equation 10.58) is simple to use but it is not the only form possible. In fact, we could go through the same derivation steps as above by using any function /(Of, Sf) of the net s output Of and actual output... 

To round off this section we note a few unusual applications of Polya s Theorem an application to telecommunications network [CatK75], and one to the enumeration of Latin squares [JucA76]. In pure mathematics there is an application in number theory [ChaC82], and one to the study of quadratic forms [CraT80], being the enumeration of isomorphism types of Witt rings of fields. Finally, we note a perhaps unexpected, but quite natural, application in music theory to the enumeration of chords and tone rows for an n-note scale [ReiD85]. In the latter paper it is shown that for the usual chromatic scale of 12 semitones there are 80 essentially different 6-note chords, and 9,985,920 different tone rows. 

CraT80 Craven, T. C. An application of Polya s theory of counting to an enumeration problem arising in quadratic form theory. J. Combinatorial Theory A 29 (1980) 174-181. 

It relates the space time coordinates xf of an event as labeled by an observer 0, to the space-time coordinates of the same event as labeled by an observer O . The most general homogeneous Lorentz transformation is the real linear transformation (9-8) which leaves invariant the quadratic form... 

As a first example we consider a system bounded periodically in two coordinates and by thermal walls in the other coordinate. The two thermal walls are at rest and maintained at the same temperature, T. The system is subjected to an acceleration field which gives rise to a net flow in the direction of one of the periodic coordinates. For this system, the hydrodynamic equations yield solutions of quadratic form for the velocity and quartic for the temperature. 

These identities may be used to obtain an angle-action equivalent of the quadratic form... 

A simple quadratic form of Eq. (34.10) is due to an identical parabolic form of the free-energy surfaces f/, and U. Since the dependence of the activation free energy on AF is nonhnear, the symmetry factor a may be introduced by a differential relationship,... 

The most general quadratic form for the second entropy is [2]... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

The first energy moment of the isolated system is not conserved and it fluctuates about zero. According to the general analysis of Section IIB, the entropy of the isolated system may be written as a quadratic form,... 

If A is a square matrix and AT is a column matrix, the product AX is a so a column. Therefore, the product XAX is a number. This matrix expression, which is known as a quadratic form, arises often in both classical and quantum mechanics (Section 7.13). In the particular case in which A is Hermitian, the product XxAX is called a Hermitian form, where the elements of X may now be complex. 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

Both quadratic forms in Eq. (A 1.1) can be diagonalized simultaneously by changing to new (normal) coordinates x ... 

It should be clear that the set of all real orthogonal matrices of order n with determinants +1 constitutes a group. This group is denoted by 0(n) and is a continuous, connected, compact, n(n — l)/2 parameter9 Lie group. It can be thought of as the set of all proper rotations in a real n-dimensional vector space. If xux2,. ..,xn are the orthonormal basis vectors in this space, a transformation of 0(n) leaves the quadratic form =1 x invariant. 

The status of H can be used to identify the character of extrema. A quadratic form <2(x) = xrHx is said to be positive-definite if Q(x) > 0 for all x = 0, and said to be positive-semidefinite if Q(x) > 0 for all x = 0. Negative-definite and negative-semidefinite are analogous except the inequality sign is reversed. If Q(x) is positive-definite (semidefinite), H(x) is said to be a positive-definite (semidefinite) matrix. These concepts can be summarized as follows ... 

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... 

Quadrics associated with symmetric matrices Given A x a symmetric matrix, the quadratic form S = xTAx can be rewritten as... 

Figure 2.5 The ellipse 2x2 — 2xy + 2y2 = 1. //, and u2 are the eigenvectors associated with the quadratic form, 3 and 1 the corresponding eigenvalues.

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