Kinetic matrix

Finally, the kinetic matrices h , which are matrix representations of the operator a p, are simply proportional to elements of S ... 

The famous Gershgorin theorem gives estimates of eigenvalues. The estimates of correspondent eigenvectors are not so well-known. In the chapter we use some estimates of eigenvectors of kinetic matrices. Here we formulate and prove these estimates for general matrices. Below A = (a,y) is a complex n x n matrix. Pi — (sums of nondiagonal elements in rows), Qi — (sums of... 

The kinetic matrices (where T = S when T = L and vice versa)... 

The off-diagonal kinetic matrices 11 and are slightly more interesting. The representation (114) operates on M[L, ju,x] giving... 

The one-centre kinetic matrices become particularly simple because of the kinetic matching construction (140) which yields... 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

We also notice that in coordinates weighted by -y/oo/nt the kinetic matrix is universal, that is, independent of the system. 

In general, a relatively direct and straightforward means of analysis may be performed in the case of slow exchange on the chemical shift time-scale by combining the relaxation matrices of the free and bound state with the kinetic matrix to describe the effect of exchange [12]. For the two spin systems described above the expanded relaxation matrix R can be written as ... 

The exchange process is then described by the kinetic matrix K expressed in terms of kon and koff ... 

The kinetic matrix K reflects the fact that only corresponding spins in the free and bound state are connected by exchange. The complete matrix governing relaxation is simply the sum of R and K ... 

Asymptotic formulas allow us to transform kinetic matrix X to a matrix with value of diagonal element could not be smaller than the value of any element from the correspondent column and row. 

Some of their properties are simpler than for general networks. For example, the damping oscillations are impossible, i.e. the eigenvalues of kinetic matrix are real (with probability close to one). If constants are not separated, damped oscillations could exist, for example, if all constants of the cycle are equal, ki — k2 — — kyi — k, then (1 + X/kf — 1 and Xm — k(exp(2nitn/n) — 1) m — the case ni — 0 corresponds to the linear conservation law. 

Furthermore, Eq. 2.42 guarantees that all the eigenvalues of Eq. 2.21 will be real numbers. Also, the quadratic form in Eq. 2.23 together with Eq. 2.16 implies that the kinetic matrix (La ) will be positive definite all the eigenvalues are nonnegative.8... 

Recently, an alternative approach has been developed by Zou [71], where within a given basis set size the kinetic matrix elements can be evaluated to a desired order of accuracy using Stirling s interpolation formula. The kinetic energy matrix elements can then be written in terms of the discretized position space as... 

Here, a is an arbitrary constant. Now, we see that the dissipative part of the kinetic matrix satisfies the Onsager symmetry relation and the positivity of the damping constant c trivially. The constant a can be evaluated as follows. In the case of zero for the dissipative part of the kinetic matrix, these equations must be transformed into Hamilton equations of a simple harmonic oscillator. From this fact, it follows that a is a universal constant and a=1. Also, the final form of the Onsagerian constitutive (kinetic) equation the of damped oscillator is... 

In general, the time evolution inferred from such a model is characterized by A -H 1 characteristic times, or rate coefficients, associated with the nonvanishing eigenvalues of the kinetic matrix K. In our application, however, the energies of the donor and acceptor states 7 = 0, A - -1 are considerably lower than those of the bridge states. Specifically, we assume that the inequalities... 

Having thus lifted himself above animal existence—he must now analyze his influence on the complex ecological rate network, and to practice his added rate-processes highly selectively to avoid undesirable or tragic consequences which, unfortunately are not self-evident in any highly interactive kinetic matrix. 

When dealing with homonuclear multiple-spin systems, polarization transfer in the zero-quantum subspace of the effective dipolar Hamiltonian is advantageous as the spin dynamics of the process may be described by a kinetic matrix. The USEME (unified spin echo and magic echo) and RIL (rotating/laboratory frame) sequences follow this zero-quantum-subspace philosophy and aim at broadband properties of the pulse sequence. 

As mentioned already, this type of purely dipolar average Hamiltonian leads to conservation of the sum polarization. This permits one to describe the polarization-transfer dynamics under the regime of such a Hamiltonian by a simple kinetic matrix K, where the elements of the matrix are proportional to the squared value of the dipolar coupling constant between the individual spins... 

If the kinetic matrix is constant, eq. (12.127) can be solved either analytically by using the Sylvester theorem of the Laplace transformation, or numerically. Finally, the relation between the intrinsic time scale and the laboratory time scale t can be determined from t = u ( )] A constant kinetic matrix is sufficient for... 

In accordance with connection (2.49), the rank of the kinetic matrix in (2.48) is less than n and its determinant is equal to zero. On the other hand, a rank of the kinetic matrix B should be exactly equal to n-1 and a system of the linear homogeneous equations Bx=0 should only have a single solution which can be written as... 

It follows from the properties of the kinetic matrix B that the algebraic additives A -are positive at even-numbered n-1 and are negative in the opposite case. That is why we can conveniently use the values... 

Additional minors of the single groups M, can be founded according to form (2.63) by deleting from the kinetic matrix B the i-line and column ... 

Subsequently, a kinetic matrix B contains only one element reflecting the intensity of a single-top transition into oneself ... 

In accordance with the (2.130) the kinetic matrix has a second order with the elements... 

Ry), are formed via the sequence of elementary reactions 4, 6, 9 and 8, 7, 5, respectively they yield the final chemical transformation 41 + A2 + A3 A kinetic matrix of the third order is as follows ... 

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