Algebraic language

This statement is equivalent, in the algebraic language, to the statement that the spherical harmonics, Y/ M((0, (j>), form a basis for the rotation group. 

Chapter 13 Writing Equations Using Algebraic Language... 

A compact formulation of stationary perturbation theory in the non-relativistic theory has been given [72] in a Lie-algebraic language. One of the essential messages of stationary perturbation theory is that under a certain condition the essential theorems of exact perturbation theory, (e.g. that the first order energy correction is equal to the expectation value of the perturbation with the unperturbed wave function) remain valid. The condition is that all perturbation corrections are formulated in terms of a variational group , with respect to which the unperturbed energy expectation value is stationary. 

J.W. Backus, "The Syntax and Semantics of the Proposed International Algebraic Language of the Zurich ACM-GAMM Conference, Information Frocessing, Froceedings of ICIF Paris, UNESCO, Paris 1960, pp. 125-32. 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

This may not seem to be a real difficulty for people acquainted with the algebraic language, but it becomes a problem when one wants to translate this into another language. One is obliged to clarify the complexity scale by creating new levels in the object s hierarchy for solving this difficulty. 

GRAPH 9.3 The evolution operator in algebraic language and in Formal Graph language, allowing one to write the duration as an integration of this variable ... 

G RAPH 11.13 Difference between the processes of transfer (left) and relaxation (right) in Formal Graph language and in algebraic language. The operators of the individual paths are identical in both cases (evolution and system constitutive properties), but their combination is different Weighted composition (product) for the transfer and addition for the relaxation. (The case offree relaxation corresponds to 6 = 0 and the case... 

In addition to define strictly the type system, NEREUS, like algebraic languages, allows finding instance models that satisfy metamodel specification. Semiformal metamodels such as MOF do not find instance models and only detect constraint violations. 

Some topics covered in this volume are more easily described by mathematical equations than by words, while others can only be described in algebraic language. An attempt is made in this chapter to state the mathematical principles which are the foundation of kinetic behaviour and to present the methods used to derive the equations which describe this behaviour. It is not essential to understand the contents of this chapter to benefit from the rest of this volume, but it is difficult to avoid mistakes in kinetic investigations without it. 

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