Actions with Invariants

We saw in Section 3.5, Actions with Invariants, that invariants are implicitly conjoined with action specifications. Static invariants factor those constraints that apply to every state, and effect invariants capture rules about every state change. Both of them simplify action specs by making them fess redundant. 

The most remarkable feature of expression (4.8) is that it does not contain any cross terms 8x 8s. This is a consequence of time-shift invariance of the instanton solution (d s/dt = dVjds, x = 0). This fact can be expressed as invariance of the action with respect to the infinitesimal transformation s s -I- cs, c 0 [cf. eq. (3.42)]. In the new coordinates the determinants break up into longitudinal and transverse parts and (4.4) becomes... 

In Catalysis, the standard pattern for developing a software system or component is to begin with the business context and represent the business goals in terms of actions and invariants. Then we decide how to meet these goals in a more refined view, with the various roles and interactions within the business. 

A class defines an implementation or partial implementation of an object, with program code for localized actions and variables for storage of its state. A class can also be documented with invariants over its variables and pre/post specifications for each operation signature. Some programming languages support these features—notably Eiffel, which provides a testing facility that uses them. 

An input property is an attribute exposed according to a component architecture so that it can be controlled by the output of another component. It can be linked with invariants to other attributes. There is an implied invariant that an input property will be equal to whatever output (of some other component) it may be coupled to. Therefore, although an input property can be used in a postcondition, it doesn t make sense to imply that it is changed by the action ... 

An advantage of the Lagrangian formulation is that it permits the identification of conserved quantities by studying the invariances of the action S. The fundamental result behind this statement is Noether s theorem, which identifies conserved currents associated with invariance of S (and hence L) under infinitesimal continuous transformation. 

It has been amply observed that the thiazide diuretics enhance urinaiy excretion of both Na and H2O by specifically inhibiting Na reabsorption located in the cortical (thick) portion of the ascending limb of Henle s loop and also in the early distal tubules. Besides, they also progressively cause an increase in the excretion of Cl, and HCO3 (to a lesser extent) ions. However, the latter efiect is predominantly by virtue of their mild carbonic anhydrase-inhibitory action. Importantly, due to their site of action, they invariably interfere with the dilution whereas, the concentration of urine is not affected appreciably. 

If the series fiXj converges in a ball Bs of radius 6 centered at the point and the function f(x) = // j b constant on the connectedness components of the intersections OnBs(0 of all the orbits O of the coadjoint action with the ball then such a function b called a local invariant at the point Since the coadjoint action b analytic, it foUows that for any regular point there exbt analytic... 

Hydrochloric acid and sulphuric acid are widely employed in the preparation of standard solutions of acids. Both of these are commercially available as concentrated solutions concentrated hydrochloric acid is about 10.5- 12M, and concentrated sulphuric acid is about 18M. By suitable dilution, solutions of any desired approximate concentration may be readily prepared. Hydrochloric acid is generally preferred, since most chlorides are soluble in water. Sulphuric acid forms insoluble salts with calcium and barium hydroxides for titration of hot liquids or for determinations which require boiling for some time with excess of acid, standard sulphuric acid is, however, preferable. Nitric acid is rarely employed, because it almost invariably contains a little nitrous acid, which has a destructive action upon many indicators. 

The action of 5-HT on brain neuronal systems is complex depending on the neurons involved, it can induce inhibition or excitation. Experiments with iontophoretic application of LSD to neurons have shown that 5-HT-induced excitation is invariably blocked by LSD, whereas LSD mimics inhibition at sites where 5-HT exerts an inhibiting effect (Aghajanian et al. 1987 Martin and Sloane 1986). Tryptamine itself is found in all major regions of... 

Figure 8 displays the escape actions thus obtained for trajectories that react into channel A or B. It confirms, first of all, that all escape actions are positive. Furthermore, they take a maximum in the interior of each reactive island and decrease to zero as the boundaries of the islands are approached. These boundaries therefore coincide with the invariant manifolds that are characterized by 1 = 0. A more detailed study of the island structure [40] reveals in addition that the time-dependent normal form approach is necessary to describe the islands correctly. Neither the harmonic approximation of Section IVB1 nor the earlier autonomous TST described in Section II yield the correct island boundaries. 

A collaboration is a set of related actions between typed objects playing certain roles with respect to others in the collaboration, within a common model of attributes. The actions are grouped into a collaboration so as to indicate that they serve a common purpose. Typically, the actions are used in different combinations to achieve different goals or to maintain an invariant between the participants. Each role is a place for an object and is named relative to the other roles in the overall collaboration. 

You can write in the bottom section of the box an invariant that applies to all the external actions. A static invariant would be anded with all their pre- and postconditions an effect invariant would be anded with all postconditions. This approach is useful for expressing some rule that is always observed when nothing is going on inside the collaboration but that is not observed by the collaborators between themselves. An open collaboration typically cannot list external actions explicitly, because they are usually unknown. Instead, you can use an effect invariant to constrain every external action to conform to specific rules. For example, the external effect invariant in Figure 4.18 states. 

To illustrate how Catalysis lets you choose how detailed or abstract to be, we could have written the overall requirement without mentioning the update action at all, with an even less detailed external effect invariant ... 

Each such aspect can be generalized to be independent of seminar specifics, creating a library of reusable abstract specification frameworks, shown in Figure 11.9 with details of invariants and action specs omitted. 

You should end up with a model showing the types of main interest. They should have static associations and attributes, and they should have action links showing how they interact. The model consists of diagrams, invariants, and a dictionary (see Chapter 5, Effective Documentation). You should be able to describe any significant business event or activity entirely in terms of the model. 

Review model. Delete attributes not used in any action spec add attributes that are needed, or that simplify the specification, and relate them to other attributes by invariants. Delete types not used for any query or action parameter. Update the dictionary with defini-... 

UML has three distinct constructs that remain largely unrelated to one another an attribute, a role in an association, and a state in a state chart. For example, a pie- and postcondition pair cannot refer to the name of a state. In Catalysis these three constructs are closely related. An association defines a pair of inverse attributes that are drawn differently a state defines a Boolean attribute, and the structure of states in a state chart defines invariants on those attributes. Transitions on a state diagram are thus no more than a graphical depiction of action specifications. This approach provides a simpler core set of constructs with different presentations. Actions on UML state transitions are imperative and appear unrelated to pre- and postspecification of operations. 

The mapping preserves the commutation relations (70) of the spin operators. As can be seen from (75d), the image of the 2s + 1)-dimensional spin Hilbert space is the subspace of the two-oscillator Hilbert space with 2s quanmm of excitation— the so-called physical subspace [218, 220], This subspace is invariant under the action of any operator which results by the mapping (75a)-(75d) from an arbitrary spin operator A(5 i, S2, S3). Thus, starting in this subspace the system will always remain in it. As a consequence, the mapping yields the following identity for the matrix elements of an operator A ... 

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