Integral rules

The value of the vanishing integral rule is that it allows the matrix H to be block diagonalized. This occurs if... 

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

Furthermore, knowledge of the irreducible representations to which the vibrational wavefunctions belong coupled with the vanishing integral rule tells us a good deal about the infra-red and Raman spectra of the molecule under consideration. 

Consequently, the functions ifyf also form a basis for P and hence by invoking the vanishing integral rule, the integral... 

The velocity at time step n -1-1 is obtained by using Simpson s integration rule ... 

The closed Newton-Cotes integral rules are given by ... 

Closed Newton-Cotes differential methods were produced from the integral rules. For Table 1 we have the following differential methods ... 

The above method has been obtained by the simplest Open Newton-Cotes integral rule. 

In some advanced implementations of the modified pseudo-Voigt function, an asymmetric peak can be constructed as a convolution of a symmetric peak shape and a certain asymmetrization function, which can be either empirical or based on the real instrumental parameters. For example, as described in section 2.9.1, and using the Simpson s multi-term integration rule this convolution can be approximated using a sum of several (usually 3 or 5) symmetric Bragg peak profiles ... 

In a customary interpretation the Reynolds transport theorem provides the link between the system and control volume representations, while the Leibnitz s theorem is a three dimensional version of the integral rule for differentiation of an integral. There are several notations used for the transport theorem and there are numerous forms and corollaries. 

The Leibnitz s integral rule gives a formula for differentiation of an integral whose limits are functions of the differential variable [2, 36, 8, 16, 9, 3, 28, 33, 18]. The formula is also known as differentiation under the integral sign. 

In Sect. 3.2, an integrator tool for the coupling of the process simulator Aspen Plus [518] and the CAE system Comos PT [745] is described. The integration is driven by integration rules, which have been partially derived from the CLiP partial models Processing Subsystem and Mathematical Model, as explained in [15]. See also Sect. 6.3, where a more recent approach to derive such integration rules is described. 

Like CLiP, OntoCAPE can be used to derive integration rules for the integrator tools developed by subproject B2 (cf. Sect. 3.2). Section 6.3 describes the approach in detail. 

Integration rules An integrator tool is driven by rules defining which object patterns may be related to each other. There must be support for defining and applying these rules. Rules may be interpreted or hardwired into software. 

Mode of interaction While an integrator tool may operate automatically in simple scenarios, it is very likely that user interaction is required. On the one hand, user interaction can be needed to resolve non-deterministic situations when integration rules are conflicting. On the other hand, there can be situations where no appropriate rule exists and parts of the integration have to be corrected or performed manually. 

Adaptability An integrator tool must be adaptable to a specific application domain. Adaptability is achieved by defining suitable integration rules and controlling their application (e.g., through priorities). In some cases, it must be possible to modify the rule base on the fly. 

During integration, the integrator core is controlled by integration rules, which can be provided following different approaches ... 

Context increments are needed when the execution of a rule depends on increments belonging to an already existing link that was created by the application of another rule. Context is used for instance to embed newly created edges between already transformed patterns. Owned increments can be further divided into dominant and normal increments. Dominant increments play a special role in the execution of integration rules (see Subsect. 3.2.4). Each link can have at most one dominant increment in each document. A link can relate an arbitrary number of normal increments. 

There is additional information stored with a link, e.g. its state and information about possible rule applications. This information is needed by the integration algorithm but not for the definition of integration rules. 

---