General statements, starting with

Equation [43] was first derived in Ref. 15, where we represented V (r) by a generic Jacobian function/(F), and it represents a correct generalization of the Liouville equation to account for the nonvanishing compressibility of phase space. Equation [43] can be derived in many ways. A general approach starts with a statement of continuity valid for a space with any metric (see Appendix 2). One can examine the transformation from one set of phase space coordinates r to another F. The metric determinant transforms according to... 

In order to examine what makes electrostatic stabilization more effective than other feasible factors, it is useful to ask what is required for an effective reduction of AAg +p. We may start from the general statement that an effective catalyst must interact with the changes during the reaction and such... 

In radical chemistry, one often needs to estimate a priori a value of the disproportionation constant. Before attempting to do this, let us start with the following consideration. In general, each closed-shell molecule can be converted to radical ions by a removal or an uptake of an odd number of electrons, and to polyions by a removal or an uptake of an even number of electrons. This statement can be expressed schematically as... 

Perturbation of structural, vibrational, and electronic features of the catalytic center by interaction with probe molecules is the most important experimental approach for understanding the accessibiUty and the reactivity of the site itself. The understanding of the system increases enormously if the experimental results are interpreted on the basis of accurate ab initio modeling. These general statements of course also hold for TS-1 [49,52,64,74-77]. Unfortunately, we do not have the space to enter into a discussion of the abim-dant computational literature published so far on TS-1 catalyst in particular and on titanoshlcates in general. The reader can find an excellent starting point in the Uterature quoted in [49,52,64,74-77,88]. 

Twenty years ago, Bogolubov3 developed a method of generalizing the Boltzmann equation for moderately dense gases. His idea was that if one starts with a gas in a given initial state, its evolution is at first determined by the initial conditions. After a lapse of time—of the order of several collision times—the system reaches a state of quasi-equilibrium which does not depend on the initial conditions and in which the w-particle distribution functions (n > 2) depend on the time only through the one-particle distribution function. With these simple statements Bogolubov derived a Boltzmann equation taking into account delocalization effects due to the finite radius of the particles, and he also established the formal relations that the n-particle distribution function has to obey. 

Chemical process design, as it is commonly known, typically starts with a general problem statement with respect to the chemical product that needs to be produced, its specifications that need to be matched, and the chemicals (raw materials) that may be used to produce it. Based on this information, a series of decisions and calculations are made at various stages of the design process to obtain first a conceptual process design, which is then further developed to obtain a final design, satisfying at the same time, a set of economic and process constraints. The important point to note here is that the identity of the chemical product and its desired qualities are known at the start but the process (flowsheet/operations) and its details are unknown. 

The relative ease of solving the system of non-linear equations for rather complex equilibrium problems, as indicated by the shortness of the function NewtonRaphson. m and by the inconsequentiality of poor initial guesses, is misleading. As we will see shortly, this statement is particularly pertinent to cases of general systems of m equations with m parameters. Solving systems of equations is a common task and we give a short introduction. To start with, we investigate the simple case of one equation with one parameter. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

A Corps of Engineers spec starts with a general statement but quickly becomes very specific ... 

Equilibrium statistical mechanics is a first principle theory whose fundamental statements are general and independent of the details associated with individual systems. No such general theory exists for nonequilibrium systems and for this reason we often have to resort to ad hoc descriptions, often of phenomenological nature, as demonstrated by several examples in Chapters 1 and 8. Equilibrium statistical mechanics can however be extended to describe small deviations from equilibrium in a way that preserves its general nature. The result is Linear Response Theory, a statistical mechanical perturbative expansion about equilibrium. In a standard application we start with a system in thermal equilibrium and attempt to quantify its response to an applied (static- or time-dependent) perturbation. The latter is assumed small, allowing us to keep only linear terms in a perturbative expansion. This leads to a linear relationship between this perturbation and the resulting response. 

When organizing your evidence within the paragraph, start with the most general statements and conclude with the most specific facts. 

To introduce the idea of identifiability, we start with two very simple problems, one from enz3rme kinetics and one from compartmental modeling. In the next section, we classify the parameters section IV then gives a general statement of the identifiability problem. 

The inspection will usually start with a plant tour, during which the inspector will obtain a general overview of the facility and its operations. During this tour, a management representative may accompany the inspector (unless he or she wishes to speak to an employee privately). At the conclusion of the tour, the OSHA officer must inform the plant management of any violations found, along with a statement as to how much time is to be allowed for correction. 

Let us now proceed with the induction. We start with statement (a). Since the expression Z (Pjj) is S3mimetric with respect to inverting the partial order of P, without loss of generality, it is enough to consider only the case (x) < X. Let us show that in this case, A P x) is nonevasive. 

The written policy statement generally starts with a clear, simple expression of your concern for and attitude about employee safety and health. Examples of introductions to policy statements include ... 

From what was said in the preceding part, it is to be expected that properties of polymer blends depend on those of the ingredients and their content in a more-or-less additive way. While this is a good base-line to start with, in reahty the additivity is rarely observed [Utracki, 1991]. There are several mixing rules used in the field. However, it is impossible a priori to predict which one will be observed. Thus, only a very general statement can be made. 

In general, a PLL expression has three parts, identifying code properties relevant for testing 1) one or several property classes like Statement (STM), Relation Operator (RO), Decision (D), Division by 0 (DZ) etc. 2) a unique ID of a specific entity of this property class in the code 3) a specific property value of this entity. One example is D 3 l, where D denotes the decisions in the code, 3 is the unique ID of a decision in the code (IDs are automatically generated by ET) and 1 represents the case that the decision evaluates to true. In case of observers, the PLL expression has two additional parts in the beginning i.e., in total five parts it starts with 0 [OBSID]where [OBSID] is a unique identifier of an observer, e.g., 0 0BS 1 D 3 1. 

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