Definition of the Problem

Perovskite-type titanates (e.g. SrTiCL, BaTiOi) constitute an important class of electroceramic materials and are, for example, used in PTC (positive temperature coefficient) resistors, capacitors, varistors or sensors [324-326]. A degradation process that 

2 Bulk Conductivities in SrTiOs Measured by Means of Microelectrodes 

3 Conductivity Profiles in SrTiC 3 Single Crystals after High Field Stress 

The electrocoloration (high-field stress) experiments were performed on Fe-doped SrTiC 3 single crystals (0.22 mol % Fe). Circular microelectrodes (10 pm in diameter and 20 or 30 pm in distance) were again prepared by a lithographic lift-off process from an evaporated 20 nm Cr/200 nm Au film. Two further Cr/Au electrode stripes were used to achieve the resistance degradation (see Fig. 32a). At 493 K, an electrical 

Before a large dc field was applied to a single crystal, a homogeneous conductivity could be detected [273]. After a high field stress (103 V cm-1 for 90 minutes), however, a distinct conductivity profile has developed in the sample (Fig. 33a). It consists of four characteristic regions i) an enhanced conductivity region at the anode with a relatively smooth drop towards the center of the sample ii) a sharp 

---

This ratio cannot exceed unity by definition of the problem. 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

In order to get at least a formal definition of the problem, we will write the exact solution to the Schrodinger equation (Eq. II. 1) in the form... 

Steps 1, 2, and 3 deal with the mathematical definition of the problem, that is, identification of variables, specification of the objective function, and statement of the constraints. We devote considerable attention to problem formulation in the remainder of this chapter, as well as in Chapters 2 and 3. If the process to be optimized is very complex, it may be necessary to reformulate the problem so that it can be solved with reasonable effort. 

The first involves the proper definition of the problem and hence the goals and objectives of the study. 

The PBPK model development for a chemical is preceded by the definition of the problem, which in toxicology may often be related to the apparent complex nature of toxicity. Examples of such apparent complex toxic responses include nonlinearity in dose-response, sex and species differences in tissue response, differential response of tissues to chemical exposure, qualitatively and/or quantitatively difference responses for the same cumulative dose administered by different routes and scenarios, and so on. In these instances, PBPK modeling studies can be utilized to evaluate the pharmacokinetic basis of the apparent complex nature of toxicity induced by the chemical. One of the values of PBPK modeling, in fact, is that accurate description of target tissue dose often resolves behavior that appears complex at the administered dose level. 

In this first attempt at a systematic definition of the problem it is recognized explicitly that there may be a multiplicity of chemically distinct chain-carriers growing simultaneously in the same reaction mixture (enieidic polymerisation). The fact that these may include paired and unpaired ions is considered from the point of view of conventional ionic equilibria, and a warning is given that there may be tight and solvent-separated ion-pairs to be considered. This idea, taken over from the theory of anionic polymerisations, was shown much later to be inappropriate for cationic polymerisations 154. ... 

This contradiction bears a close resemblance to Luxemburg s posing of the question of how new capital goods can be produced in the absence of sufficient demand to satisfy the new capacity. Sufficient demand, to meet the requirements of a balanced growth in capacity, is unlikely to be forthcoming from within the Domar model, from within the reproduction schema. Joan Robinson s interpretation of Luxemburg has some resonance with the Domar definition of the problem ... 

We start with a definition of the problem and based on this, we identify the candidates (such as, molecules, mixtures and formulations) through expert knowledge, database search, model-based search, or a combination of all. The next step is to perform experiments and/or model-based simulations (of product behavior) to identify a feasible set of candidates. At this stage, issues related to process design are introduced and a process-product match is obtained. The final test is related to product quality and performance verification. Other features, such as life cycle assessment could also be introduced at this stage. 

Therefore, Eq, (4.27) together with the boundary conditions of Eqs. (4.28-4.31) provide a definition of the problem of pressure P(x, y, t) with an unknown boundary Xo(y, t) provided that function q(t) specifying the pressure at the exit from a point gat into a cavity is known. In practice, function P0(t) is known having determined the mass balance function q(t), the final formulation takes the form ... 

The guiding principles for the selection or development of speciation procedures are similar to those recommended for other forms of chemical analysis. For example, the initial step should be careful definition of the problem, including listing of the analytical specifications (e.g. type of analysis, concentration range, potential sources of error). This step can be followed by selection of a suitable measurement procedure, nomination of a selective separation procedure (if required) and organisation of the total protocol. 

A good general approach in the case of presumed malodor problems is to urge the client to conduct exploratory consumer research to find out whether the odor is truly perceived as objectionable by the consumer and, if so, at what stages of product use—for example, when opening the bottle, during product use, or after use. A clear definition of the problem, in consumer terms, places the perfumer in the best possible condition to solve it. 

A more fundamental approach to process selection first requires a clear definition of the problem and size enlargement objectives. This is followed by comparison with the capabilities of the available processes as catalogued in Table 1.3 and in greater detail in later chapters. Promising methods can then be selected and the clearly unsuitable methods ruled out. Factors to be considered in this comparison include ... 

The general definition of the problem is the following inside the uniform regular lattice of obstacles with the elementary cell in a form of equal-sided triangle with the side of length c, is placed a polymer chain with /V-links having a segment... 

It is usually fairly easy to conclude that there is a problem, more difficult to decide exactly what it is and very hard to identify the appropriate solution (there are always a great many possible solutions). A simple illustration of this point, for conventional livestock production, is disease. Most diseases could, of course, be totally eradicated from a farming system but this might be quite uneconomic. Thus the problem for a veterinarian is commonly not how to prevent or cure a disease but to do so in such a way that the farmer s profit is increased (or at least he is not made bankrupt by the solution). All such requirements have to be part of the definition of the problem. In the case of organic farming, the solution has to be consistent with the principles of organic production. So the relevant applied research may be different. 

We must have an appreciation for the problems of our curator colleagues and work in collaboration with them. We hope they in turn will be concerned for and become knowledgeable in matters of preservation. It is not necessary for either curator or conservator to become a scientist, but understanding the concepts, methods, and language of science will help them both not only to work with scientists but also to know how to ask the right kind of questions or define with precision the particular problems they may wish them to investigate. The more precise the definition of the problem, the more likelihood there is that the questions will lead to answers that can be relevant to actual conservation practice. 

Before going into a detailed description, a mathematical definition of the problem is useful. If we have conserved quantities, like the total angular momentum J and its projection J, the eigenvalues of the linearized motion matrix M has now the following eigenvalue structure. Let us have, as usual, n degrees of freedom and k < n —2 conserved quantities. Then we have... 

---