Practical uniqueness

Unfortunately, the number of uniqueness theorems for geophysical inverse problems is relatively small. These theorems cover very specific geophysical models. In practical situations we should rely on a more simple but important property of inverse problem solution. Following Hjelt (1992), we call this property practical uniqueness. It can be described using the following simple considerations. 

Consider a case where we would like to determine some function of n variables from the observed data. To obtain the solution of the inverse problem in this case, we obviously have to measure a function of the same or greater number of variables. For example, if the model parameters are given in 3-D space m = m(x, y, z) then the data 

Then the observed data will be the function of four parameters  

We can expect in this situation that the practical inverse problem would have a unique solution, because the space of the four-dimensional data parameters is bigger than the space of the three-dimensional model parameters. 

Another way to increase the dimension of the data parameter space is to take into account the data dependence on time or frequency  

---

Each chapter starts with a description of the topic covered in the chapter. This is followed by a short example highlighting a reported incident involving a batch reaction system. The case study is followed by a listing of key issues and process safety practices unique to the topic. The issues and concerns presented in this book, as well as potential design solutions and sources of additional information are presented in the tables. This format concisely conveys the necessary and relevant information in a familiar and convenient format. The organization of the tables is described below. 

Furthermore, the use of silica-based ceramic matrices as functional supports has a number of practical, unique advantages including stability towards harsh conditions, low swelling and consistent binding sites... 

As it follows from the above-said, nowadays any study of the autowave processes in chemical systems could be done on the level of the basic models only. As a rule, they do not reproduce real systems, like the Belousov-Zhabotinsky reaction in an implicit way but their solutions allow to study experimentally observed general kinetic phenomena. A choice of models is defined practically uniquely by the mathematical formalism of standard chemical kinetics (Section 2.1), generally accepted and based on the law of mass action, i.e., reaction rates are proportional just to products of reactant concentrations. 

Huisgen and coworkers have also described the cycloaddition behavior of the munchnones , unstable mesoionic A2-oxazolium 5-oxides with azomethine ylide character.166 Their reactions closely parallel those of the related sydnones. These mesoionic dipoles are readily prepared by cyclodehydration of N-acyl amino acids (216) with reagents such as acetic anhydride. The reaction of munchnones with alkynic dipolarophiles constitutes a pyrrole synthesis of broad scope.158-160 1,3-Dipolar cycloaddition of alkynes to the A2-oxazolium 5-oxide (217), followed by cycloreversion of carbon dioxide from the initially formed adduct (218), gives pyrrole derivative (219 Scheme 51) in good yield. Cycloaddition studies of munchnones with other dipolarophiles have resulted in practical, unique syntheses of numerous functionalized monocyclic and ring-annulated heterocycles.167-169... 

I would like to emphasize, at the end of this section, that practical uniqueness has nothing in common with theoretical uniqueness. The requirement that the observations should be a function of the same or greater number of parameters than the model provides a qualitative background for the uniqueness only. Theoretically, even in this situation, the inverse problem could have several equivalent solutions. However, this requirement serves as a useful practical guide to ways of obtaining a unique solution for a practical problem. 

It is important that the reader have a clear mental picture of an actual chromatographic column before proceeding in this section. Packed columns are similar to those discussed in Chapter 21. Open tubular columns are practically unique to GC, and the reader may wish to go forward to the section on columns first before... 

The Pik is large, when k and I belong to the atoms forming a chemical bond (if compared to two non-bonded atoms see Appendix S available at booksite.elsevier.com/978-0-444-59436-5, p. el43) therefore, the dipole moments related to pairs of atoms come practically uniquely from chemical bonds. The contribution of the lone pairs of the atom A is hidden in the second term of and may be quite large (cf. Appendix T available at booksite. elsevier.com/978-0-444-59436-5 on p. el49). 

The above challenges in the audit process made the conviction strong to implement the system and improve the system continuously based on the feedback received from all stake holders. The objectives of the audit are shared with both students and faculty to seek their full acceptance and cooperation. This increases the chances of success and sustainability and whole-hearted participation of both faculty and students to make this practice unique and meaningful. 

The contrast between the practical uniqueness and utility of rubber and the tarnished image of plastics in their early years effectively disguised from the ordinary person the fact that plastics and rubbers are sub-groups of the same class of materials—organic high polymers—differing only in detail in their molecular structures. 

The efforts of the experts from Pivdenny have made it possible for Ukraine to become firmly established in the first three of space powers (after USA and Russia). More than 400 earth satellites developed in Pivdenny have been in space In recent years experts have developed the unique camer-rockets Zenith and Cyclone, capable of taking 4 and 14 tons into orbit, respectively. No other carrier-rockets of this type exist anywhere in the world, so they were selected for the international project Sea Start and Globalstar The NDT experts from Pivdenny have made a great contribution to these development, as practically all the parts and components of the carrier-rockets are subjected to thorough control. 

While not unique, the Scluodinger picture of quantum mechanics is the most familiar to chemists principally because it has proven to be the simplest to use in practical calculations. Hence, the remainder of this section will focus on the Schrodinger fomuilation and its associated wavefiinctions, operators and eigenvalues. Moreover, effects associated with the special theory of relativity (which include spin) will be ignored in this subsection. Treatments of alternative fomuilations of quantum mechanics and discussions of relativistic effects can be found in the reading list that accompanies this chapter. 

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

Representative Method The best way to appreciate the importance of the theoretical and practical details discussed in the previous section is to carefully examine the procedure for a typical precipitation gravimetric method. Although each method has its own unique considerations, the determination of Mg + in water and waste-water by precipitating MgNH4P04 6H2O and isolating Mg2P20y provides an instructive example of a typical procedure. 

The goal of a collaborative test is to determine the expected magnitude of ah three sources of error when a method is placed into general practice. When several analysts each analyze the same sample one time, the variation in their collective results (Figure 14.16b) includes contributions from random errors and those systematic errors (biases) unique to the analysts. Without additional information, the standard deviation for the pooled data cannot be used to separate the precision of the analysis from the systematic errors of the analysts. The position of the distribution, however, can be used to detect the presence of a systematic error in the method. 

This unique reference makes some of the esoteric aspects of this important area of analysis more readily comprehensible to those who deal with analytical instruments but who have not been trained in mass spectrometry. It also serves as a refresher for practicing mass spectroscopists by clarifying principles to afford a better appreciation and application of this technology. 

The separation of fmctose from glucose illustrates the interaction between the framework stmcture and the cation (Fig. 5) (50). Ca " is known to form complexes with sugar molecules such as fmctose. Thus, Ca—Y shows a high selectivity for fmctose over glucose. However, Ca—X does not exhibit high selectivity. On the other hand, K—X shows selectivity for glucose over fmctose. This polar nature of faujasites and their unique shape-selective properties, more than the molecular-sieving properties, make them most useful as practical adsorbents. 

Forensic science is an applied science having a focus on practical scientific issues that come up during criminal investigations or at trial. Some components ate unique to the field because it is conducted within the legal arena. Forensic science issues in chemistry and biochemistry in criminal investigations are discussed herein. There are a host of other forensic science areas, eg, forensic medicine, forensic dentistry, forensic anthropology, forensic psychiatry, and forensic engineering, any of which may overlap with forensic chemistry. 

The quahty manual should be organized to faciUtate referral to the quahty system standard. It should be brief and refer to other documentation for more detail. The manual should be under document control, ie, each page is uniquely identified as to date or revision number and its preparer. It is common practice to offer customers a copy of the manual upon request, ie, the manual should not be proprietary. 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

For practical reasons, the blast furnace hearth is divided into two principal zones the bottom and the sidewalls. Each of these zones exhibits unique problems and wear mechanisms. The largest refractory mass is contained within the hearth bottom. The outside diameters of these bottoms can exceed 16 or 17 m and their depth is dependent on whether underhearth cooling is utilized. When cooling is not employed, this refractory depth usually is determined by mathematical models these predict a stabilization isotherm location which defines the limit of dissolution of the carbon by iron. Often, this depth exceeds 3 m of carbon. However, because the stabilization isotherm location is also a function of furnace diameter, often times thermal equiHbrium caimot be achieved without some form of underhearth cooling. 

The Shawinigan process uses a unique reactor system (36,37). The heart of the process is the fluohmic furnace, a fluidized bed of carbon heated to 1350—1650°C by passing an electric current between carbon electrodes immersed in the bed. Feed gas is ammonia and a hydrocarbon, preferably propane. High yield and high concentration of hydrogen cyanide in the off gas are achieved. This process is presently practiced in Spain, AustraUa, and South Africa. 

---