Intuitive methods

The axiomatic way of teaching quantum mechanics (QM) is anaiyzed in the light of its effectiveness in making students ready to understand and use QM. A more intuitive method of teaching QM is proposed. An outline of how a course implementing that method could be structured is presented. 

The most common and intuitive method for the determination of this number of eigenvalues is called Cross Validation. The idea is to remove one (or several samples) from the calibration set, use what is left for the computation of a new calibration, and use it to predict the quality of the removed sample(s). Each prediction is compared with the actual quality that is known as the removed sample really is part of the total calibration set. In a loop all samples are removed either one by one or in groups and after recalibration with the reduced calibration set their qualities are predicted and compared with the true values. In order to determine the best number of eigenvectors this procedure is repeated in a big loop systematically trying all numbers of eigenvectors. This complete procedure is called Cross Validation. 

How then can an exact match be made In this case there must be some outside knowledge supplied and an examination of the X-ray fluorescence spectra would indicate the presence of only Co and O in the unknown powder. When this information is used, the unknown is identified as Co304, card 9-418. However, when the card is examined the weak line 2.13 A remains unidentified in the unknown pattern. X-ray intensity is proportional to the amount of the diffracting material present, so it is natural to suspect that a contaminant phase will only display its most intense lines. The problem in this case is that there is only one line remaining so that the Hanawalt manual is useless to help in the identification. The intuitive method must be used. The key question to be answered is, what contaminants might be found in a sample containing only Co and O The two possibilities that come to mind are elemental Co or another oxide of Co. When these two possibilities are checked, it is quickly found that the most intense line of CoO, card 9-402, is indeed 2.13 A, and the identification is complete. 

As a next step we combine the Lagrangian Equation (24-1) of an in-situ production/ consumption process, R(Ct), with the Eulerian view (Eq. 24-4) of transport and reaction. There are two ways to analyze the resulting situation, a more intuitive method and a formal approach. We elaborate on the former and explain the latter in Box 24.1. 

This concludes a discussion of exactly solvable second-order processes. As one can see, only a very few second-order cases can be solved exactly for their time dependence. The more complicated reversible reactions such as 2Apt C seem to lead to very complicated generating functions in terms of Lame functions and the like. This shows that even for reasonably simple second- and third-order reactions, approximate techniques are needed. This is not only true in chemical kinetic applications, but in others as well, such as population and genetic models. The actual models in these fields are beyond the scope of this review, but the mathematical problems are very similar. Reference 62 contains a discussion of many of these models. A few of the approximations that have been tried are discussed in Ref. 67. It should also be pointed out at this point that the application of these intuitive methods to chemical kinetics have never been justified at a fundamental level and so the results, although intuitively plausible, can be reasonably subject to doubt. 

An intuitive method for controlling the motion of a wave packet is to use a pair of pump-probe laser pulses, as shown in Fig. 13. This method is called the pump-dump control scenario, in which the probe is a controlling pulse that is used to create a desired product of a chemical reaction. The controlling pulse is applied to the system just at the time when the wave packet on the excited state potential energy surface has propagated to the position of the desired reaction product on the ground state surface. In this scenario the control parameter is the delay time r. This type of control scheme is sometimes referred to as the Tannor-Rice model. 

Occasionally the qualitative and intuitive methods of risk assessment fail, and something better is required. For example, a designer may have taken steps to address some hazard, but be unsure whether these are sufficient. There is also the possibility of failure of equipment, control systems or operating procedures which may reduce margins of safety. In these circumstances, quantitative risk assessment may be considered. This is an attempt to put numbers to the risks so that we can judge them objectively. 

Del Re (1958) has developed a chemically intuitive method, in which all the parameters are determined in a self-consistent manner. The basic feature of this treatment is the non-overlap of the localized orbitals, which interact through inductive effects. This method, modified for use in conjugated systems by Berthed and Pullman (1965), has been used by Denis and Pullman (1967). 

The first step led soon afterwards to a fundamentally new insight into the constitution of benzene and other aromatic molecules. While in general a correct conception regarding the molecular constitution had been obtained quite early in organic chemistry by inductive intuitive method this was not the case with the aromatic molecules. It was the quantum mechanical theory which first gave the solution of the benzene problem. The same ideas also laid the foundation for a better understanding of the very important problem... 

One of the early problems with multilayer perceptrons was that it was not clear how to train them. The perception training rule doesn t apply directly to networks with hidden layers. Fortunately, Rumelhart and others (Rumelhart et al 1986) devised an intuitive method that quickly became adopted and revolutionized the field of artificial neural networks. The method is called back-propagation because it computes the error term as described above and propagates the error backward through the network so that weights to and from hidden units can be modified in a fashion similar to the delta rule for perceptions. 

It is easy to verify that the expressions (2.14) and (2.15) represent the effective solutions by inserting them in eqn (2.13). The expression (2.14) must go to zero for n = 0 and this is possible if b is large enough. Duxbury et al (1987) gave several arguments in favour of (2.14). Here we choose a more intuitive method to select the convenient solution. 

These assumptions and procedures are certainly not desirable. As a result his expression (4.24) does not reduce to the Debye formula. Nevertheless his intuitive method seems to be essentially correct. 

Wq is the molecular weight of a segment, is Avogadro s number and the unit of the intrinsic viscosity corresponding to Eq. (4.27) is dl Eq. (4.26) agrees in its general form with what Peterlin (5) obtained by an intuitive method, as we shall discuss in the next section. However, since the constants A and B are different we have determined them from the plot illustrated in Fig. 1. The data here are those on polystrene... 

As we mentioned before, Eq. (4.26) agrees in form with what Peterlin obtained by an intuitive method. However, irrespective of the similarity in the theoretical results there are important differences between his theory and ours. The main differences are ... 

By Brown s intuitive method (which is an adaptation of the 1905 arguments of Einstein)... 

The nature of the cholinergic receptor has been the subject of considerable study for over 75 years. Aspects of its shape, conformations, and active sites have, over the years, slowly and enticingly revealed themselves to our prying. The efforts were initially by empirical-intuitive methods, then by SAR analogies, receptor mapping, and other indirect methods. Generous applications of X-ray crystallography and molecular receptor probes followed, and most recently, the actual isolation of the active receptor preparations were achieved. 

Chapter 4 Types of Chemical Reactions and Solution Stoichiometry," now gives a more qualitative and intuitive method for balancing oxidation-reduction equations, which is based... 

Data mining methods are widely available and can often be highly sophisticated algorithms that use advanced techniques from computer science and artificial intelligence. However, simple and intuitive methods can often work well, without much loss in predictive ability. With small datasets, where the focus is developing interpretable models, these simple methods may be the best first approach, perhaps as part of a conscious elfort to explore the data. In any case it is useful to have some benchmark result against which the performance of more complex, computationally expensive and difficult to interpret methods can be compared. The simplest naive model is prediction by the mean of the dataset, which is in elfect prediction without using a model, and this can serve as a useful reality check and comparator. 

In a previous study of 18 aromatic trlfluoroketones, their inhibitory activity against JHE was correlated to the structure using four substituent parameters (H). The descriptors were chosen more or less arbitrarily and the effect of the different substituent positions could be determined seml-emplrlcally, mainly by examining how the various data-polnts fit the equation. Important conclusions could, however, be drawn based on the empirical and intuitive method the authors found that meta- and para-substituted compounds fit the regression line better and offered better selectivity against JHE. 

In this section we describe the process of data analysis using the intuitive method of determining the backscattering ampHtude and the phase-shift functions experimentally and then to use these functions to determine the structural parameters of the sample under investigation. Differences to data analysis using theoretical functions will be explained. 

Industrial applications are related to obtaining the 3D CAD model of existing components because the most intuitive method of process planning is the physical interaction with the model. In conjunction with rapid prototyping systems CAD models of new products can be quickly updated with modifications on prototypes proposed during the Concurrent Engineering process. Moreover when the model is required to have... 

We have derived the general Inversion theorem for pole singularities using Cauchy s Residue theory. This provides the fundamental basis (with a few exceptions, such as /s) for inverting Laplace transforms. However, the useful building blocks, along with a few practical observations, allow many functions to be inverted without undertaking the formality of the Residue theory. We shall discuss these practical, intuitive methods in the sections to follow. Two widely used practical approaches are (1) partial fractions, and (2) convolution. 

In a complex supply chain network, multiple business units including retailers, distribution centers and factories may be considered in an ATP search. These business units can be dispersed geographically. Hence, an intuitive method for ATP search is based on distance (or lead-time) between inventory location and customer order location. The search process may also be based on a pre-assigned location priority scheme, or on the echelon-level of the business units, e.g. retailer level, distribution center level, manufacturer level. Considering a complex supply chain with multiple distribution centers and factories, Jeong et al. (2002) proposed a global search model to find avail-... 

Utilizing concentrated electrolyte solutions is an intuitive method to increase ionic conductivity and amount of active material present in the system. This approach certainly raises operating efficiencies and improves the kinetic parameters of electrode activity, as expected [6]. There is a concurrent drawback, however, that increasing the amount of electrochemically active species present in the electrolyte might negatively influence diffusion parameters due to the resultant increase in kinematic viscosities of the solution [37, 38]. As is the case of the zinc half-cell, the kinetics and behavior of the Br2/Br redox reaction in the bromine half-cell are also influenced by the presence and concentrations of supporting electrolytes in solution [39]. 

The relation between a and the thermodynamic functions may be rigorously established by means of the general partition function or by making use of Guggenheim s intuitive method, based on the principle of detailed equilibrium. According to this principle if a system is in equilibrium, each single process has to be balanced by its reverse. The processes that must be considered in these circumstances are evaporation of an r-mer molecule from solution followed by condensation of r monomer molecules in the r sites previously occupied and the reverse process, i.e. evaporation of r monomer molecules from these sites followed by condensation of one r-mer molecule on the sites considered. The rate of the first process is proportional to f(rp) and the vapour pressure of the monomer at an exponent r, p and that of the second is proportional to /(rS) and the vapour pressure of r-mer pg, which is negligible. The ratio may then be vritten as ... 

---