Canonical interpretation

In chapter 5,1 argue that the category of quality can be systematically derived from form. According to what I call the canonical interpretation, the category of quality divides into four main species. I argue... 

Although Ackrill s criticisms of Aristotle are not unreasonable, in this chapter, I argue that there is an alternative interpretation to the canonical interpretation, what I will call the regimented interpretation, that can not only go some way toward removing the dissatisfaction that he and others have had with it but can indeed be incorporated into the thesis that the categorial scheme is derivable from hylomor-phism. 

It should be clear that the structure represented by this diagram is far more systematic and symmetrical than the structure involved in the canonical interpretation. It can also answer one of the criticisms of Aristotle raised by Ackrill. Recall that Ackrill says ... 

It is sometimes argued that proteins fold in solvent, where the solvent serves as a heat bath. This would provide a fixed canonical temperature such that the canonical interpretation is sufficient to imderstand the transition. However, the solvent-protein interaction is actually implicitly contained in the heteropolymer model and, nonetheless, the microcanonical analysis reveals this effect which is simply lost by integrating out the energetic fluctuations in the canonical ensemble (see Fig. 9.14). 

Formulations The equations of flow through porous media are generally too complicated to have exact solutions in analjdical form and numerical methods must be used. There are special cases, however, of great theoretical interest that are valuable for benchmarking numerical methods and validating computer implementations. As part of the process of deriving a numerical method it is sometimes useful to reformulate the differential equations to clarify the mathematical structure. It is also useful to relate the theory to diffusion, convection or wave propagation, for which there are models with canonical interpretations. To illustrate this idea, and to derive results needed in a later section, the reformulation of incompressible, two-phase flow will be developed. For simplicity, the case without sources is studied. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

The canonical MOs are convenient for the physical interpretation of the Lagrange multipliers. Consider the energy of a system with one electron removed from orbital number Ic, and assume that the MOs are identical for the two systems (eq. (3.32)). 

There is, then, the problem of determining for each discipline what it does in the way of diseovery and proof, what eriteria it uses for measuring the quality of its data, how strietly it can apply canons of evidence, and in general, of determining the route or pathway by whieh the discipline moves from its raw data through a longer or shorter proeess of interpretation to its conclusion . 

Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

C.J.F. ter Braak, Interpreting canonical correlation analysis through biplots of structure correlations and weights. Psychometrika, 55 (1990) 519-531. 

The technique allows immediate interpretation of the regression equation by including the linear and interaction (cross-product) terms in the constant term (To or stationary point), thus simplifying the subsequent evaluation of the canonical form of the regression equation. The first report of canonical analysis in the statistical literature was by Box and Wilson [37] for determining optimal conditions in chemical reactions. Canonical analysis, or canonical reduction, was described as an efficient method to explore an empirical response surface to suggest areas for further experimentation. In canonical analysis or canonical reduction, second-order regression equations... 

We want to know something more. We want to know not just that WHILE programs can compute every partial recursive function semehow or other but that the class of WHILE schemes is a canonical form. Notice that this does not follow from the previous theorem since we want to know about behavior under all interpretations. 

The foregoing unitary transformations may be interpreted as the analogues of canonical transformations in classical mechanics. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

The Sherlock Holmes stories often are read as the triumph of the scientific method of deduction. There is, though, another, darker interpretation to be advanced. That is the frustration of science and technology in changing everyday social practice. Throughout the canon, Holmes is consistently thwarted by the standard operating procedures of the Scotland Yard "regulars." He also is annoyed—and even disturbed—by the inability of Victorian... 

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