Another sufficient condition

If all the coefficients c are non-negative, then the kernel will have nonnegative entries and so the mask will also, and the basis function will be non-negative. 

This is a nice trivial result to make very safe design decisions easily. Unfortunately it is so weak that it significantly over-constrains the design space. 

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Another sufficient condition for the convergence of the Jacobi method of iteration follows. Let R be the set of all starting vectors X0 for which the largest Hilbert norm of any matrix B generated by the iterative process has the property that... 

Another efficient and practical method for exact 3D-reconstruction is the Grangeat algorithm [11]. First the derivative of the three-dimensional Radon transfomi is computed from the Cone-Beam projections. Afterwards the 3D-Object is reconstructed from the derivative of the Radon transform. At present time this method is not available for spiral orbits, instead two perpendicular circular trajectories are suitable to meet the above sufficiency condition. 

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. 

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. 

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. 

Compounds with an asymmetric carbon atom When the four groups connected to a carbon center are different from one another, the central carbon is called a chiral center. (However, we must remember that the presence of an asymmetric carbon is neither a necessary nor a sufficient condition for optical activity.)... 

Another feature that we may refer to here is that of structural mimicry (52). It is generally accepted that a necessary and sufficient condition for formation of solid-solution crystals by two or more organic substances is similarity of shapes and sizes of the component molecules (53). Consider the two molecules salicylidene-lV-4-chloroaniline and -4-bromoaniline, which are very similar chemically and, in solution, structurally. Both are dimorphic, with molecules... 

Another possibility is to look at the packing of the top 10 solutions, because this can prove a very discriminating criterion. Even if the solution is well detached, it is still mandatory (and reassuring) to examine carefully the corresponding packing arrangement. This necessary (but not sufficient) condition can actually be implemented in the translation function using analytical expression... 

I believe, however, that the assertion that every autopoietic system is living goes too far and in this section I would like to clarify the limits of this assertion. In doing so, I follow the lines of a paper recently presented in collaboration with Michel Bitbol (Bitbol and Luisi, 2004). The queshon of whether autopoiesis is the necessary and sufficient condition, or only the necessary one, has also been asked by Bourgine and Stewart, (2004) and earlier, in another context, by Weber (2002). 

Cosmic ray particles can work as a trigger of the chain of polymerization of formaldehyde adsorbed (e.g., at the surface of interstellar silicate dust). However, the triggering of the polymerization chain is a necessary but still not sufficient condition for obtaining the interstellar polymers. If the addition of any new link of the chain would require a hit of adsorbed monomer layer by another cosmic ray particle (i.e., if there is no spontaneous growth of chains after they are started by some external factor), the formation of polymers in interstellar space would be highly improbable. Therefore the spontaneous growth of polymer chains near absolute... 

There is, however, another statement of the necessary and sufficient condition of thermodynamic stability of the multicomponent system in relation to mutual diffusion and phase separation that is less stringent than equation (3.20) because it may be fulfilled not for every component of the multicomponent system. For example, in the case of the ternary system biopolymeri + biopolymer2 + solvent, it appears enough to fulfil only two of the inequalities (Prigogine and Defay, 1954)... 

AHM will be zero when the two liquids are identical, and it is obvious that under such conditions the two liquids must be completely miscible. Also, liquids which are very similar to one another and have practically identical values of L, D and K will be miscible. It is, however, not a sufficient condition for solubility that the two molecules have equal heats of evaporation. We now return to the example of water and hexane. The boiling points are not markedly different and consequently their heats of evaporation are of the same order. For hexane AH — A, and D and K are both zero, while for water the K term is so predominant that, as a good approximation, AH — K. AHM, which then has its largest possible value... 

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... 

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 ... 

The refinement procedure utilises the fact that if some query node Q(X) has another node Q(fV) at some specific distance ) ( and/or angle), and if some database node D(Z) matches with Q(W), then there must also be some node D(Y) at the appropriate distance(s) from D(Z) which matches with Q(X) this is a necessary, but not sufficient, condition for a subgraph isomorphism to be present (except in the limiting case of all the query nodes having been matched, when the condition is both necessary and sufficient). The refinement procedure is called before each possible assignment of a database node to a query node and the matched substructure is increased by one node if, and only if, the condition holds for all nodes W, X, Y and Z. The basic algorithm terminates once a match has been detected or until a mismatch has been confirmed [70] it is easy to extend the algorithm to enable the detection of all matches between a query pattern and a database structure, as is required for applications such as those discussed here. 

Krotov has shown [7] that a sufficient condition for conforming with inequality (5.13) proves to be another inequality... 

The results demonstrate that selectivity for one ion over another (stronger complex formed by that ion) is not a sufficient condition for shell-core behavior. This is the case for high selectivity of B ion (Kp /Kk > 1) and high diffiisivity of the A ion (D > Dp). 

We see, therefore, that only those liquids which can be mixed without evolution of heat will give rectilinear vapour pressure curves when dissolved in one another. This condition, however, is not sufficient to ensure the linear variation of the partial... 

However, spectral identity is a necessary but not a sufficient condition for compound identification. If the instrument parameters that determine how spectra are acquired are matched suitably, the spectral match of an unknown to a known spectram can be used as strong indication of compound similarity or confirmation of identity. In a chromatogram, the retention time of a given peak offers an obvious means of preselection of candidate spectra through the application of a retention time window. All spectra from the set of possible candidates that fall in the window are compared to the unknown spec-tram. The candidate with the best match factor above a certain threshold level is then used to identify the compound for the peak. Another possibility is to weigh both the match factor and the similarity in retention times and use the combined information for identification. 

Positivity analysis is riddled with loose sufficient conditions. Another such is the condition that if every step is positive, however different they may be, the sequence will remain positive. Clearly if no step can create a negative value in the net matrix, it remains positive. We can therefore expect that the overall value of the norm of the scheme will be no worse than the largest value of any of the steps. 

Table 21.1 summarizes the relationships between reaction conditions, presence of Ag 8+ cluster, and hydrogen effect on HC-SCR. Sazama et al. and Burch et al. have pointed out that the hydrogen effect cannot be attributed only to the formation of Ag 8 + cluster because of (i) the delay in the disappearance of Ag cluster after H2 cutoff [21], and (ii) the formation of Ag 8 + cluster by other non-effective reductants such as octane and CO [47]. However, most importantly, the hydrogen effect was not observed in the absence of Ag 8 + clusters. It is clearly shown that neither Ag cluster nor hydrogen effect were observed in H2-propane-SCR over 0.5 wt.%Ag/Al203. Therefore, the formation of Ag 8+ cluster is not a sufficient condition, but a necessary condition. There must be another factor for the hydrogen effect than the formation of Ag 8 + cluster, which cannot be achieved by the co-presence of CO or hydrocarbons. 

For to prove instability, we only have to show some circumstance, no matter how special, under which there will be a tendency to fly away from the steady state. To put it in another way, we can say that Eq. (7.5.1) is a sufficient condition for instability, while its converse, dUrldT < dUJdT is a necessary, but not sufficient, condition for stability. 

Gas flows through the SE will be different ( ventilation effect) and will depend on how the oxide SE was applied on the surface of YSZ and how it was heat-treated and sintered afterwards. The vast majority of the modem technologies will allow creating a uniform thickness of SEs (REs). However, the stmctural orientation of an oxide SE and, especially, its surface and bulk porosity will change from one technology to another and from one sintering temperature to another [9]. This fact, in turn, will influence characteristics of the YSZ-based sensor, such as sensitivity, reproducibility of measurements, response and recovery time, and so on. Let s indicate both necessary and sufficient conditions of existence of such a ventilation effect within the SE. 

In the last section it was shown that the existence of zeros in the graph spectrum is a sufficient condition for the biradical nature of the molecule. Using Theorem 3, all even cycles can be reduced to 4- or 6-membered ring systems. Calculations show that there is no zero in the spectrum of a 6-membered cycle (i.e. benzene) and that there are two zeros in the spectrum of a 4-membered cycle [i.e. cyclobutadiene). Another approach to this problem is given by Rouvray 180>. We can also make use of the Sachs Theorem 27>. The coefficient an of the characteristic polynomial is given as ... 

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