Chain rule matrix

When the variational energy is a functional of the reference state well-defined for nonintegral occupation numbers. This implies two chain rules, for m 0 ... 

Let us now consider the effective CS of fragments of M for a partitioning of M into two mutually closed, complementary subsystems, M = (A B), e.g., reactants in the donor-acceptor systems. Of interest in this case are the elements of the condensed hardness matrix, e.g., ijJ b = (df /dNA). Using the chain rule transformation we obtain ... 

Although both fim and Hm (Sm) are dependent upon the choice of the atom m, the predicted responses d N are independent of such a choice the same is true in the case of other response quantities, e.g., the internal softness matrix including all m atoms, S s (dN/d/i)N. This can be explicitly demonstrated by means of the following chain rule transformation ... 

Inspection of equation (9.151) shows that h depends on bj, 2 4nd y at a fixed T. However, we know that the dependent variable y also depends on b and 2 Thus, sensitivity equations (Jacobian matrix elements dyjdb and dyjdb ) can be obtained by differentiating both sides of equation (9.151) with respect to b by using the chain rule ... 

The chain rule for the differentiation of products has the matrix generalization... 

The first term on the right hand side is recognized precisely as the matrix K J introduced earlier. The term that is linear in X may be evaluated via the chain rule as... 

The hardness matrix of equation (78a) can be expressed in terms of the subsystem hardness kernels using the following chain-rule transformation ... 

The explicit formulae for all terms in the multipole expansion up to R 5, which includes the quadrupole-quadrupole, octupole-dipole and hexadecapole-charge terms, have been published (Price et al., 1984 Stone, 1991). The chain-rule type formalism for the associated forces, torques and second derivatives has also been established, along with the derivatives with respect to the strain matrix which defines the unit cell shape, by Willock et al. (1995), with related analyses by Popelier and Stone (1994). 

We note that the Willson type B matrix [16] for the reference point (dummy atom) representation can be evaluated without specific reference to Cartesian coordinates of the dummy atoms, by the chain rule ... 

A further derivation of Grad E to any gives a two-dimensional field of combinations of d Ejdx dx with i and k. It is usually arranged in a matrix, named the Hessian matrix (after Otto Hesse, 1811-1874). In general, co- or contravariant characteristic cannot be assigned to the only partial derivatives of this matrix under a coordinate transformation, because there are mixed terms coming out of the chain rule. The new terms are connected with the coordinate system and can be compressed in special symbols. The matrix... 

We now proceed to a consideration of the molecular Hessian - that is, the matrix of second derivatives of the molecular electronic energy with respect to geometrical distortions. Differentiating the molecular gradient in the form of equation (18), we obtain from the chain rule... 

This rule of thumb does not apply to all polymers. For certain polymers, such as poly (propylene), the relationship is complicated because the value of Tg itself is raised when some of the crystalline phase is present. This is because the morphology of poly(propylene) is such that the amorphous regions are relatively small and frequently interrupted by crystallites. In such a structure there are significant constraints on the freedom of rotation in an individual molecule which becomes effectively tied down in places by the crystalhtes. The reduction in total chain mobility as crystallisation develops has the effect of raising the of the amorphous regions. By contrast, in polymers that do not show this shift in T, the degree of freedom in the amorphous sections remains unaffected by the presence of crystallites, because they are more widely spaced. In these polymers the crystallites behave more like inert fillers in an otherwise unaffected matrix. 

Working with Markov chains, confusion is bound to arise if the indices of the Markov matrix are handled without care. As stated lucidly in an excellent elementary textbook devoted to finite mathematics,24 transition probability matrices must obey the constraints of a stochastic matrix. Namely that they have to be square, each element has to be non-negative, and the sum of each column must be unity. In this respect, and in order to conform with standard rules vector-matrix multiplication, it is preferable to interpret the probability / , as the probability of transition from state. v, to state s (this interpretation stipulates the standard Pp format instead of the pTP format, the latter convenient for the alternative 5 —> Sjinterpretation in defining p ), 5,6... 

It is straightforward to show by applying Cramer s rule recursively to chains of increasing N that the determinant of matrix [A] is 1, with det[A] = —1 for N... 

The Ufson-Roig matrix theory of the helix-coil transition In polyglycine is extended to situations where side-chain interactions (hydrophobic bonds) are present both In the helix and in the random coil. It is shown that the conditional probabilities of the occurrence of any number and size of hydrophobic pockets In the random coil can be adequately described by a 2x2 matrix. This is combined with the Ufson-Roig 3x3 matrix to produce a 4 x 4 matrix which represents all possible combinations of any amount and size sequence of a-helix with random coil containing all possible types of hydrophobic pockets In molecules of any given chain length. The total set of rules is 11) a state h preceded and followed by states h contributes a factor wo to the partition function 12) a state h preceded and followed by states c contributes a factor v to the partition function (3) a state h preceded or followed by one state c contributes a factor v to the partition function 14) a state c contributes a factor u to the partition function IS) a state d preceded by a state other than d contributes a factor s to the partition function 16) a state d preceded by a state d contributes a factor r to the partition function. 

It is necessary to note that the matrix K does not deal with the hold-ups at all. The same rule for its construction was suggested in [8,9] on the basis of particle fractions flows balance only, without referring to Markov chains models. Here it was obtained as a particular case of the developed model, which is presented by the matrix P and allows describing also the evolution of hold-ups in general case provided the smaller matrices of P are known. 

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