Theorems on Linear Systems*

Graphical Evaluation of Blood-to-Brain Transfer Constants from Multiple-Time Uptake Data

“…9, V:K is a row vector whose components are the K column sums, which are equal to -(k p i + kbi) (Hearon, 1963). Since the ith component of V I ;G is kbb the ith element of V I ; (K + G) is -k p i• As has been discussed by Hearon (1963), A and K-l consist of positive and negative elements, respectively. Therefore, the second term of Eq.…”

“…Figure 1 illustrates this model system. The necessary equations of this tracer distribu tion system are the standard ones (see Hearon, 1963). If t is the time, A is an (n X 1) vector of the amounts of model solute in each of the exchangeable compartments, K is the (n x n) matrix of the rate constants (kij), Q is the (n x 1) vector of the rate constants from the plasma to the exchangeable compartments (kip), and Cp(t) is the tracer concen tration in the plasma, then the following relationship describes tracer accumulation in the reversible compartment:…”

“…5 by parts, the insertion of Eqs. 3 and 4, and rearranging yields If Cp is constant, A will approach a finite limit as t � 00, since the real parts of the eigenvalues of K are negative (Hearon, 1963). Thus, for the case when t � 00:…”

“…18 produces (19) Because the elements Ao and G are positive but those of K-l are negative (Hearon, 1963), Eq. 19 indicates that …”

Bistable Photoconduction in Semiconductors

“…Therefore, it will be sufficient to test the sign of the eigenvalues of T S : the restriction to the space S of the linear operator defined by T ij . To this aim, we recall two conclusions concerning sign-symmetric and irreducible matrices, reached by Hearon (Hearon, 1963) and more recently surveyed in the context of general M-matrix theory (see chapter 6 of Berman & Plemmons (1994)):…”