Sufficient conditions state

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

The written directives of a quality control program are a necessary, but not a sufficient, condition for obtaining and maintaining an analysis in a state of statistical control. Although quality control directives explain how an analysis should be properly conducted, they do not indicate whether the system is under statistical control. This is the role of quality assessment, which is the second component of a quality assurance program. 

The following theorem - stated without proof (see [jen86a]) - gives necessary and sufficient conditions for CA rules to generate constant temporal sequences ... 

We conclude this section by deriving an important property of jointly gaussian random variables namely, the fact that a necessary and sufficient condition for a group of jointly gaussian random variables 9i>- >< to be statistically independent is that E[cpjCpk] = j k. Stated in other words, linearly independent (uncorrelated),46 gaussian random variables are statistically independent. This statement is not necessarily true for non-gaussian random variables. 

Poincar6-Bendixson (P.B.) Theorem.—This theorem gives the necessary and sufficient conditions for the existence of a cycle. Unfortunately, it requires a preliminary knowledge of the character of integral curves, which often makes its application difficult. The theorem states ... 

The magnitude on the left is the heat absorbed in the isothermal change, and of the two expressions on the right the first is dependent only on the initial and final states, and may be called the compensated heat, whilst the second depends on the path, is always negative, except in the limiting case of reversibility, and may be called the uncompensated heat. From (3) we can derive the necessary and sufficient condition of equilibrium in a system at constant temperature. 

Another important aspects of solubilization are the physical state of the dissolved polymer as well as the thermo-chemistry and kinetics of the dissolution reaction. It is known that a clear cellulose solution is a necessary, but not sufficient condition for the success of derivatization. The reason is that the polymer may be present as an aggregate, as will be discussed below. Additionally, dissolution of activated cellulose requires less time at low temperature, e.g., 2 h at 40 °C, and more than 8 h at 70 °C [106]. These aspects will be commented on below. 

It is clear from Equation (12.19) that t/Xmax > 1 is necessary to avoid washout. However, it is not sufficient. The sufficient condition is found from the requirement that Sout < Sm at steady state. This gives... 

The specific explanation structure for the flowshop problem is given in Fig. 10. In the example we have assumed that the sufficient condition is satisfied by having all the end-times of x less than or equal to those of y. Thus the proof begins by selecting the appropriate variable set, and proceeds to prove that each variable is more loosely constrained in x than in y. The intersituational variables in the flowshop problem are the start-times of the next state. 

The polarization state p = 1 is necessary but not, however, a sufficient condition for a nonzero P coefficient in the angular distribution. Recalling the general CGC symmetry relation... 

The two conditions stated above do not assure the occurrence of gelation. The final and sufficient condition may be expressed in several ways not unrelated to one another. First, let structural elements be defined in an appropriate manner. These elements may consist of primary molecules or of chains as defined above or they may consist of the structural units themselves. The necessary and sufficient condition for infinite network formation may then be stated as follows The expected number of elements united to a given element selected at random must exceed two. Stated alternatively in a manner which recalls the method used in deriving the critical conditions expressed by Eqs. (7) and (11), the expected number of additional connections for an element known to be joined to a previously established sequence of elements must exceed unity. However the condition is stated, the issue is decided by the frequency of occurrence and functionality of branching units (i.e., units which are joined to more than two other units) in the system, on the one hand, as against terminal chain units (joined to only one unit), on the other. 

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows V /(x ) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. 

There are two important consequences of this equality for computer simulations of many-body systems. First, it means that statistically averaged properties of these systems are accessible from simulations that are aimed at generating trajectories -e.g., molecular dynamics, or ensemble averages such as Monte Carlo. Furthermore, for sufficiently long trajectories, the time-averaged properties become independent of the initial conditions. Stated differently, it means that for almost all values of qo, Po, the system will pass arbitrarily close to any point x, p, in phase space at some later time. 

Strictly speaking, all steps in the model (72) have a quantum mechanical nature. The measured rate is determined by the relative values of the kinetic parameters and a number of situations can be envisaged. The rate limiting step for the forward direction, defined from left to the right in Eq.(72), may be located at any level depending of course on the nature of the species. There is, however, a necessary and sufficient condition for the process to occur. This is related to the relaxation time of ASC into quantum states of P1-P2. This relaxation time must be finite. 

However, it is indeed fortunate that the IV-representability problem for the electron density p(r) greatly simplifies itself. In fact, the necessary and sufficient conditions that a given p(r) be /V-representable are actually given by Equation 4.5 above. Nevertheless, question remains Can the single-particle density contain all information about a many-electron system, at least in its ground state An affirmative answer to this question can be given from Kato s cusp condition for a nuclear site in the ground state of any atom, molecule, or solid, viz.,... 

At this point, it is necessary to say a few words about the v-representability of the electron density. An electron density is said to be v-representable if it is associated with the antisymmetric wave function of the ground state, corresponding to an external potential v(r), which may or may not be a Coulomb potential. Not all densities are v-representable. Furthermore, the necessary and sufficient conditions for the v-representability of an electron density are unknown. Fortunately, since the /V-representability (see Section 4.2) of the electron density is a weaker condition than v-representability, one needs to formulate DFT only in terms of /V-representable densities without unduly worrying about v-representability. 

There is, of course, a chemical effect in carbon monoxide flames. This point was mentioned in the discussion of carbon monoxide explosion limits. Studies have shown that CO flame velocities increase appreciably when small amounts of hydrogen, hydrogen-containing fuels, or water are added. For 45% CO in air, the flame velocity passes through a maximum after approximately 5% by volume of water has been added. At this point, the flame velocity is 2.1 times the value with 0.7% H20 added. After the 5% maximum is attained a dilution effect begins to cause a decrease in flame speed. The effect and the maximum arise because a sufficient steady-state concentration of OH radicals must be established for the most effective explosive condition. 

R = 0 leads back to the problem of interpolation by spline functions. It should be noted at this point that the condition stated by eq. (4) is not sufficient for the construction of calibration curves and additional considerations have to take effect. A reformulation of the problem stated in Equations (2) and (3) gives us with 8y. = 1 for all i calibration points another look at the problem that clarifies the role of the integral in Equation (2) as balanced against a value of R. Find S (x) to... 

Alcoba [80] reported four theorems showing that the cancellation of these types of terms is a sufficient condition to guarantee that these matrices correspond to eigenstates of the system. In particular, his first theorem states the following. 

In this section we review the known theorems that relate entanglement to the ranks of density matrices [52]. The rank of a matrix p, denoted as rank(p), is the maximal number of linearly independent row vectors (also column vectors) in the matrix p. Based on the ranks of reduced density matrices, one can derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement [53]. For convenience, let us introduce the following definitions [54—56]. A pure state p of N particles Ai, A2,..., is called entangled when it cannot be written... 

As already mentioned, Bourgine and Stewart (2004) arrive at the same conclusion, based on an elaborate and elegant mathematical treatment. They state in fact that autopoiesis is not a necessary and sufficient condition for hfe. For the hner differences between these two treatments, see Bitbol and Fuisi (2004), where broader implications of the definition of cognition are also discussed. 

It is sometimes said that the opposite of a profound truth is another profound truth. The social sciences offer a number of illustrations of this profound truth. They can isolate tendencies, propensities and mechanisms and show that they have implications for behavior that are often surprising and counterintuitive. What they are more rarely able to do is to state necessary and sufficient conditions under which the various mechanisms are switched on. This is another reason for emphasizing mechanisms rather than laws. Laws by their nature are general and do... 

Only light which is absorbed by one of the reacting substances can be chemically active, but absorption, although a necessary, is not a sufficient condition for reaction. Of frequencies corresponding to absorption bands in different parts of the spectrum some may be effective and some not. The experimental investigation of this matter is still in a very rudimentary state. Examples suitable for experiment are not easy to find. 

It is important to note at this point that the foregoing sketch of the time development of the excited molecular state has only taken into account the nonradiative decay. That is, we have implicitly assumed that the radiative decay can be treated wholly independently, and that the relaxation processes are in no way coupled to one another. Clearly, this simplifying view is not in general valid. Gelbart and Rice15 discuss this question with specific reference to the model molecular system considered in Section XII-B. They establish the following necessary and sufficient conditions for their independent treatment of the two molecular decay processes ... 

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