Necessary and sufficient condition

A minimum is strong when, in its neighborhood, any direction increases the function. 

A minimum is weak if the function is constant along at least one direction. 

If the function, and the first and second derivatives are continuous, it can be expanded by Taylor series in the neighborhood of the generic point x  

If d is sufficiently small, the Taylor expansion can be stopped at the first term  

The relation (3.4) is the necessary but not sufficient condition to identify a local minimum it is a necessary and sufficient condition to have a stationary point (minimum, maximum, or saddle point). 

A symmetric ideal (SI) solution is defined as a solution for which the chemical potential of each component obeys relation (6.8.1) in the entire range of compositions, keeping T, P constant, i.e., 

For most practical applications, it is also useful to require that (6.8.2) hold also in a small neighborhood of T and P so that we may differentiate (6.8.2) with respect to these variables. There is no need to require the validity of (6.8.2) for all T, P, since no real mixture is expected to fulfill such an exaggerated requirement. 

Since p is presumed to be nonzero, (6.8.6) implies (6.8.5) and hence (6.8.5) is also a necessary condition for SI solutions. 

I The case of very low densities, p - 0, will be discussed separately in section 6.10. Here, we are interested in solutions at liquid densities. (Note, however, that here p is not an independent variable it is determined by r, P, Xa. ) Also, we shall assume throughout that all the Gap are finite quantities. 

The condition (6.8.5) is very general for SI solutions. It should be recognized that this condition does not depend on any modelistic assumption for the solution. For instance, within the lattice models of solutions we find a sufficient condition for SI solutions of the form 

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Smith B.D. Image reconstruction from cone-beam projections necessary and sufficient conditions and reconstruction methods., IEEE Trans. Med. Imaging, V. 4, 1985, p. 14-28. 

Theorem 5. The transpose of is a complete B-matrrx of equation 13. It is advantageous if the dependent variables or the variables that can be regulated each occur in only one dimensionless product, so that a functional relationship among these dimensionless products may be most easily determined (8). For example, if a velocity is easily varied experimentally, then the velocity should occur in only one of the independent dimensionless variables (products). In other words, it is sometimes desirable to have certain specified variables, each of which occurs in one and only one of the B-vectors. The following theorem gives a necessary and sufficient condition for the existence of such a complete B-matrix. This result can be used to enumerate such a B-matrix without the necessity of exhausting all possibilities by linear combinations. 

If (a) above is satisfied, then the necessary and sufficient condition for stability is either... 

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

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

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. 

It is possible, of course, to use some sort of blank symbol to separate code words, but this would really imply that a tertiary alphabet was available. One easy way to guarantee that the code words can be separated from each other is to use a prefix code. We define a prefix code as a code in which no code word is the same as the initial part, or prefix, of another code word. More precisely, for any i and j, if % < np then vt must not be equal to the first nt digits of v. The following theorem now gives necessary and sufficient conditions on the set of lengths %, , n that can be used in a prefix code. 

Theorem 4-4 (Szilard, Kraft). A necessary and sufficient condition on the lengths n, 1 < j J, of the J code words of a binary prefix code is that... 

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 ... 

This condition is thus the necessary and sufficient condition for the existence of a stable stationary solution (oscillation) of the differential equation (6-127). 

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. 

If the only external force is a normal, uniform, and constant pressure p, the necessary and sufficient condition for equilibrium is that for all virtual isothermal-isopiestic changes ... 

The necessary (and sufficient) condition for a three-dimensional Pfaffian to be inexact but integrable is... 

The translational periodicity of the potential is the necessary and sufficient condition for describing the wavefunction as a linear combination of Bloch functions... 

The first analysis is connected with the case when A is a constant self-adjoint positive operator A = A > 0. As we have shown in Section 2, a necessary and sufficient condition for the stability of the weighted scheme (47) with respect to the initial data is... 

Let now // be a complex space and S be a non-self-adjoint operator. Then a necessary and sufficient condition for the stability in the space Ha with respect to the initial data of the scheme... 

Along these lines, it is worth noting that we are still in the framework of the general stability theory outlined in Chapter 6, Section 2 for difference schemes like (22) asserting that a necessary and sufficient condition for the estimate < pWllkWo valid for any 0 < / < 1 and any operator... 

A necessary and sufficient condition for a two-layer scheme to be stable can be written as the operator inequality... 

Goldratt has outlined the necessary and sufficient conditions for a technology to confer a benefit [23]. To generate productivity from a new technology he advocates answering a set of questions about the technology. Let s use them in detail to examine software solutions in the pharmaceutical industry. Goldratt s questions are as follows ... 

Chapman s work produced the following theorem, which provides the necessary and sufficient conditions for guaranteeing the truth of any given statement in a partial plan, if all operations are modeled by STRIPS-like operators. 

Formally, we define the dominance condition by specifying a set of necessary and sufficient conditions, which it has to satisfy (Ibaraki, 1978) ... 

The necessary and sufficient conditions for equivalence are repeated here for convenience. 

Sufficiency. Given the example, the theory we employ to deduce the conditions must guarantee that the conditions on equivalence or dominance will be satisfied, since the example itself is simply an instance of the particular problem structure, and may not capture all the possible variations. The sufficient theory will thus end up being specialized by the example, but since we have asserted the need for validity of this process, it will still be guaranteed to satisfy the abstract necessary and sufficient conditions on dominance and equivalence. 

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. 

If the characteristic polynomial passes the coefficient test, we then construct the Routh array to find the necessary and sufficient conditions for stability. This is one of the few classical techniques that we do not emphasize and the general formula is omitted. The array construction up to a fourth order polynomial is used to illustrate the concept. 

The proposed model to explain OCS is schematized in Fig. 7.2. Several agents, induced or not for estrogen deficiency, stimulate the expression of RANKL on stromal/OB cells. The binding of RANKL with its receptor RANK on osteoclastic precursors, together with M-CSF, is a necessary and sufficient condition to carry out all the steps in the formation and differentiation of the osteoclasts. Undoubtedly all this is much more complex than what is described here since at least 24 genes that positively and negatively regulate OCS have been described (Boyle et al. 2003). 

As before, the necessary and sufficient condition that thermal equilibrium depends on equal temperatures identifies 6 = kT. By similar arguments it follows that equilibrium across a semi-permeable wall requires that (ni/9) =... 

Equation (1.37) is actually the necessary and sufficient condition that any one-electron property52 (P) (such as electron density, dipole moment, kinetic energy,...) could be written in the resonance-averaged form (1.36). The goal of a quantitative resonance theory is to find the resonance weights wa (if any) for which Eq. (1.37) is most accurately satisfied. 

Details concerning the necessary and sufficient conditions for minimization are presented in Chapter 4. 

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