Eigenvalue positive, simple

In the special positive cases just described, multiplicative processes have a unique positive eigenvector Ai, a vector being called positive when all its components are positive. The associated eigenvalue is simple, and larger (in magnitude, for discrete multiplicative processes, and in real part, for continuous multiplicative processes) than any other eigenvalue. Hence, we have the asymptotic formulas ... 

Theorem 2. The largest in modulus) eigenvalue A of T is positive, simple, and its corresponding unique eigenvector T can be chosen to have non-negative components. Furthermore, for any arbitrary positive vector ij>o, the iteration procedure... 

Flows that produce an exponential increase in length with time are referred to as strong flows, and this behavior results if the symmetric part of the velocity gradient tensor (D) has at least one positive eigenvalue. For example, 2D flows with K > 0 and uniaxial extensional flow are strong flows simple shear flow (K = 0) and all 2D flows with K < 0 are weak flows. 

To demonstrate the quality of this simple approach we show in Figures 6 and 7 the PE spectra of 1,3-butadiene 2, (3 )-hexa-l,3,5-triene 161, (3E, 5 )-ocla-1,3,5,7-tetraene 234, [3]radialene 173, 3,4-dimethylidenecyclobutene 174 and fulvene 175. The observed positions I j1 of the tt-bands are collected in the third column of Table 2, and the eigenvalues xj obtained from standard HMO models in the fourth. A least-squares calculation yields the linear regression... 

The evaluation of the contribution of cycle diagrams to the activity coefficient is formally complete once the eigenvalues of SI have been found. Let us write the result explicitly in terms of the Fourier transforms for the important case of two defects a and / each of which is allowed to occupy a particular one of the two positions in the unit cell. In this simple system the labelling of positions in the unit cell by superscripts on Fourier transforms becomes redundant and can be omitted. The result is... 

Figure 41. Evaluating the stability of the simple example pathway shown in Fig. 40 Metabolic states and the corresponding saturation parameters are sampled randomly and their stability is evaluated. For each sampled model, the largest positive part within the spectrum of eigenvalues is recorded. Shown is the probability of unstable models, as a function of (A) The size of the system. Here the number of regulatory interactions increases proportional to the length of pathway (number of metabolites). Other parameters are maximal reversibility ymax 1 and p 0.5. (B) An increasing number of regulatory interactions. The number of metabolites m 100 is constant. Maximal reversibility ymax 1 and p 0.5. (C) Maximal reversibility v /v . and p 0.5. Other...

Of course, similar results for perturbations of zero eigenvalue are valid for more general ergodic chemical reaction network with positive steady state, and not only for simple cycles, but for cycles we get simple explicit estimates, and this is enough for our goals. 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

We assume that the equations (7.200) have a simple hysteresis type static bifurcation as depicted by the solid curves in Figures 10 to 12 (A-2). The intermediate static dashed branch is always unstable (saddle points), while the upper and lower branches can be stable or unstable depending on the position of eigenvalues in the complex plane for the right-hand-side matrix of the linearized form of equations (7.198) and (7.199). The static bifurcation diagrams in Figures 10 to 12 (A-2) have two static limit points which are usually called saddle-node bifurcation points. 

A symmetric matrix A is said to be positive definite if all of the eigenvalues are positive it is said to be negative definite if all of the eigenvalues are negative. Semidefinite is similarly defined. There is a simple test to determine if a symmetric matrix is positive or negative definite. 

For exclusively real eigenvalues of fV the time dependence of the average excess production is determined by the choice of initial conditions. As shown in Appendix 5, optimization of (t) is restricted to initial conditions in the positive orthant [yt(0) >0 k = 0,1,.. ., n]. These initial conditions are not difficult to fulfil, and they will apply to many cases in reality. We should keep in mind, nevertheless, that there are other choices of initial conditions, such as the start with a pure master sequence, for which the simple principle does not hold. For one particular type of choice, yi (0) > 1 and y/j(0) < 0 for all /c 1, the average excess production decreases monotonically. 

A transition from a definite to a non-definite matrix occurs even for the simple case of the harmonic oscillator as a function of the total time. In general, the shorter the time, the smaller the term with the potential derivatives and the matrix as a whole is more positive. At sufficiently long time we expect some of the eigenvalues to reverse their sign and become negative, making the matrix indefinite. 

For purposes of comparison, it is possible to classify the various types of potential functions which may be represented by the functional form used in Eq. (3.32) with a few simple considerations. The restrictions we shall make are always to locate the origin in the minimum, or if more than one, in the deepest minimum second minima or inflection points are restricted to negative values of the coordinate Z and the positive values of Z always represent the most rapidly rising portion of the function. These restrictions do not eliminate any unique shape of potential function. Any other functions described by Eq. (3.32) are related to those already included by a simple translation of the origin or by rotation about the vertical axis. These operations, at most, change the eigenvalues by an additive constant. The different types of potential functions are summarized in Table 3.1. 

Point 1 is the intersection of the tangent at the points A and V of the curve of measurement. It can be determined in any case. Because of the affine transformation it is always valid that the curve of measurement intersects the triangle A with the ratio of the two eigenvalues. If it is known that a simple consecutive reaction 1 takes place, the points V and C coincide. In addition if k, > 2, the point 1 relates to the comer B. However, if k, < 2, then point B will coincide with point 2. The position of this point can always be determined using the absorbance diagram with... 

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