Analytic sufficiency condition

Appendix A The Jahn-Teller Model and the Longuet-Higgins Phase Appendix B The Sufficient Conditions for Having an Analytic Adiabatic-to-Diabatic Transformation Matrix I. Orthogonality II. Analyticity... 

APPENDIX B THE SUFFICIENT CONDITIONS FOR HAVING AN ANALYTIC ADIABATIC-TO-DIABATIC TRANSFORMATION MATRIX... 

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... 

A multiscale system where every two constants have very different orders of magnitude is, of course, an idealization. In parametric families of multiscale systems there could appear systems with several constants of the same order. Hence, it is necessary to study effects that appear due to a group of constants of the same order in a multiscale network. The system can have modular structure, with different time scales in different modules, but without separation of times inside modules. We discuss systems with modular structure in Section 7. The full theory of such systems is a challenge for future work, and here we study structure of one module. The elementary modules have to be solvable. That means that the kinetic equations could be solved in explicit analytical form. We give the necessary and sufficient conditions for solvability of reaction networks. These conditions are presented constructively, by algorithm of analysis of the reaction graph. 

A frequent error encountered in evaluating the performance of an analytical system is to confuse the concepts of sensitivity and detectability. Although both concepts address facets of a system s response, they are not identical, but rather complementary. Sensitivity relates to the ability of the system to respond to changes in analyte concentration and is most typically reflected as the slope of the method s response function. Detectability, as has been noted previously, is the ability of the method to distinguish between two responses (those responses that arise in the presence and absence of the analyte in the sample matrix of interest). It is possible to have a very sensitive method that has a relatively poor detection limit, especially if the method is very unselective or prone to high blanks. However, an insensitive method is very unlikely to exhibit a low LOD. Thus sensitivity is a necessary— but not sufficient—condition for the achievement of a low LOD. 

All in all, the conclusions of this stage of the analysis are that even quite simple physical models can account for many of the properties of the coexistence of solidlike and liquidlike clusters. But let us recall that all this was carried out assuming that the two forms could exist in thermodynamic equilibrium. The question was posed by Natanson et al. ° of what would be the necessary and sufficient conditions for that hypothesis to be valid. Answering that question and pursuing its immediate consequences, even at a qualitative or semiquantitative level, constituted the second stage of the analytic study. 

In the framework of the equilibrium theory, the mass balance equation can be solved analytically in the case of competitive Langmuir isotherm behavior for a TMB column with constant initial and boxmdary conditions [22], Applying the method discussed earlier allows the derivation of the following necessary and sufficient conditions that the flow rate ratios, mj, must satisfy for a complete separation to be achieved [28,38—40] ... 

Takoudis et al. (1981) proposed a model for a bimolecular Langmuir-Hinshelwood surface reaction with two empty sites in its reaction step. The two chemisorbed species were assumed to adsorb competitively on the surface. The two dimensional model with reaction rates as parameters were shown to exhibit oscillations. Bifurcation of this model was also discussed. Takoudis et al. (1982) described a procedure for obtaining necessary and sufficient conditions for the existence of periodic solutions in surface reactions with constant temperature. An analytic method for the analysis of bifurcation to periodic solutions was developed. 

The third major class of analytical techniques may be called morphological methods. This identification consists of comparing the form of particles captured with the morphology of particles of known composition. It goes without saying that morphological similarity is a necessary but not always sufficient condition for compositional identity. In spite of this problem this procedure is widely employed mainly in clean atmosphere, since even Aitken size particles can be identified morphologically (A. Meszaros and Vissy, 1974 Butor, 1976). 

G-spinors satisfy the analytic boundary conditions (137) for jc < 0 and (138) for tc > 0. A G-spinor basis set consists of functions of the form of (147-149) with suitably chosen exponents Xm, m = 1,2,..., d - The choice of sequences Xfn which ensure linear independence of the G-spinors and a form of completeness is discussed in [86]. It is often sufficient to use the GTO exponents from nonrelativistic calculations, of which there are many compilations in the literature perhaps augmented with one or two functions with a larger value of A to improve the fit around the nucleus. 

The third requirement of Mangasarian gives rise to inequalities of the form E(y, u) > 0 [see Inequality (3.52), p.82]. Proving it becomes a new optimal control problem in which zero needs to be estabhshed as the minimum of L for all admissible u. This task cannot be accomplished analytically except in rare, simple cases where the Hessian of L could be easily checked for positive definiteness. Because of this situation, the application of any sufficient condition in optimal control is very limited. 

The stationary state of the network of reactors is stable, if all roots A, of (13.8) have a negative real part. The necessary and sufficient conditions for this to hold are the Routh-Hurwitz conditions, see Theorem 1.2. The stationary state of the network, W, undergoes a stationary instability if = 0, see (1.36), and an oscillatory instability if = 0, together with > 0, A > 0, Z = 1,..., m — 2, see (1.38). The Routh-Hurwitz analysis can be used to determine, in principle, the stability properties of the steady state of any network, even inhomogeneous networks. This advantage is, however, balanced by the fact that it is a computationally expensive task to evaluate all the coefficients C of the characteristic polynomial and the Hurwitz determinants A . In our studies of instabilities in arrays of coupled reactors, we used symbolic computation software, namely Mathematica (Wolfram Research, Inc., Champaign, IL, 2002) and Maple (Waterloo Maple Inc., Waterloo, Ontario, 2002), to obtain exact, analytical expressions for the coefficients C of the characteristic polynomial (13.8) and the Hurwitz determinants A/ for arrays of up to six coupled reactors. 

In Section 2.6 we discussed the nature of cause and effect, and in Section 7.2 we noted that empirical rules are obtained inductively from observations and experiences regarding real structures from three broad categories of situation, success, failure and the near-miss or narrowly avoided failure. Information about successful projects enables us to identify sufficient conditions for success failures tell us the necessary conditions for success. What we would like, of course, are the conditions which are necessary and sufficient but we will never be in that position, for we will never know that there are no unknown phenomena which could occur. Our technical knowledge enables us usually to describe, at least approximately, the technical causal chain in any success or failure. The variancy (Section 2.6) involves all the factors not accounted for in our analytical models, whether theoretical or physical, and it is from this that we learn our lessons for the future. In effect the engineer s judgement based upon his experience is the result of his synthesis of these factors in a teleological explanation. 

The links are factuai or material, or physical hypotheses, and asscat that existence of a state B in the primary science is sufficient (or necessary and sufficient) condition for the state of affairs designated by A . In this scenario, the meanings of A and B are not related analytically. 

Figure 3.11 summarizes such key experimental points. As a first point, we have to choose the appropriate ionization method for the detection of small metabolites, we have alternative choices other than MALDI, such as secondary ion mass spectrometry (SIMS) [15], nanostructure-initiator mass spectrometry (NIMS) [20,21], desorption/ionization on silicon (DIOS) [22], nanoparticle-assisted laser desorptiopn/ ionization (nano-PALDI) [23], and even laser desorption/ionization (LDI) [24,25]. We consider that MALDI is stiU the most versatile method, particularly due to the soft ionization capability of intact analyte. However, other methods each have unique advantages for example, SIMS and nano-PALDI have achieved higher spatial resolution than conventional MALDI-IMS, and above aU, these mentioned alternative methods are all matrix-free methods, and thus can exclude the interruption of the matrix cluster ion. Next, if MALDI is chosen, experimenters should choose a suitable matrix compound, solvent composition, and further matrix application method for their target analyte. All these factors are critical to obtain sufficient sensitivity because they affect efficiency of analyte extraction, condition of cocrystallization, and, above all, analyte-ionization efficiency. In addition, based on the charge state of the analyte molecule, suitable MS polarity (i.e., positive/ negative ion detection mode) should be used in MS measurement. Below, we shall describe the key experimental points for MALDI-IMS applications of representative metabolites. 

The gradual development and prohferation of the new analytical technique by means of solvents was by no means a sufficient condition for chemists interest in and grouping together of proximate principles of plants. The great attention the compound proximate components of plants received by the middle of the eighteenth century was also spurred by developments in natural history and in the chemical theory of compounds and composition. 

Nowadays, all analytical laboratories are aware of the cmdal importance of proper quality assurance activities. Although quality control directives explain how an analysis should be conducted, they do not indicate whether the system is under statistical control. Therefore, the written guidelines of a quality control program are a necessary, but not a sufficient, condition for obtaining and maintaining proper statistical control. This is the role of quality assessment, which includes internal methods coordinated within the laboratory and external methods organized and maintained by an outside agency. 

PyMS is a mass spectrometric technique in which a flash pyrolysis device is coupled directly or indirectly to a mass spectrometer. Total PyMS experiments can be performed in a few minutes. Off-line PyMS of polymers was first reported in 1948 [671, 672] and on-line PyMS of polymers in 1953 [673]. In the ideal experimental design the pyrolytic fragments of macromolecules are generated under non-isothermal conditions, escape sufficiently fast from the dissociating matrix so that overheating and further rearrangement of the pyrolysis products are prevented, and are analysed without further wall contact by soft ionisation MS techniques. The ideal conditions are most closely met when pyrolysis takes place inside the ionisation chamber, but in practice the analytical PyMS conditions are often quite different. 

Fault Tree Analysis employs an analytical tree to display the results of an analysis (Suokas and Rouhiainen, 1993). It starts with the top event (injury or damage). The analysis proceeds backwards in order to identify all events and conditions that have caused the injury or damage. Logical relations (necessary and/or sufficient conditions) are estabhshed. Fault-tree analysis is not an accident model per se and gives limited support in the identification of causal factors. 

A sufficient condition for convergence of Eq. 1.29) to the root x is dial g fjc) < I for all X in the search interval. Fig. 1,2b shows the case when this condition is not valid and the method diverges. This analytical test is often difficult in practice. In a computer program it is easier to determine whether x - jTjI < jjCj - jc, and, therefore, the. successive x values converge. The advantage of this method is that it can be started with only a single point, without the need for calculating the derivative of the function. 

A.L. Bukhgeim (1983) [4] proved necessary and sufficient conditions for the solvability of the two-dimensional inverse kinematic problem with partial local data in class of real analytic functions, and... 

It was indicated that the original method can be extended on systems where two or three analytes can be determined from a single titration curve. The shifts DpH affected by j-th PT addition should be sufficiently high it depends on pH value, a kind and concentration of the buffer chosen and its properties. The criterion of choice of the related conditions of analysis has been proposed. A computer program (written in MATLAB and DELPHI languages), that enables the pH-static titration to be done automatically, has also been prepared. 

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