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where xi = (xi , yi ) and the thin-plate energy functional E, a measure of the amount of bending in the surface, is minimized. The thin-plate energy functional is de ned as E=

If you needed to define a relational database in a single phrase, you could say that it involves a place for everything, and everything in its place. Everything, in this case, is the collection of entities, or real-world objects, that your application uses and manipulates, and places are the database tables that store them.

2 2 2 gxx (x) + 2gxy (x) + gxx (x) dx,

If you begin your database-engineering project by considering realworld objects for what they are and how they behave throughout their life cycles, you gain a better insight into the details of how your software must work. The same cannot be said if you are simultaneously considering table definitions, data types, and more as you try to conceptualize your database. Figure 8-1 is a top-level view of the entities that we use in the database model in this chapter.

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The model of an entity consists of the name of that entity, the attributes that describe it, and other behaviors that we discuss. The model defines how each of those entities is structured like a mold from which multiple objects of that type can be manufactured. Figure 8-2 shows this data model expanded to display the attributes of each entity.

(11.8)

Table 18.5 Limiting states for spread footing design Limiting state Settlement Bearing capacity Sliding Overturning Overall stability or structural capacity Limit state equation i n Q Rn FS Q Rn FS Resultant load w/in B/6 of centerline T RFS ASD Factor of safety 2.0 3.0 2.0 3.0 1.25 2.0

Each individual object is considered an instance of that entity. But after you manufacture all these similarly structured instances, how do you tell one from the other The same way that you do any individual thing in nature: by using one or more key attributes that best describe it. Each instance of an entity must be uniquely identified by a primary key, without which that instance cannot be referenced or operated on. This primary key value is what your software uses to retrieve, update, and delete this instance. Figure 8-3 further expands the data model by displaying above the dividing line those attributes that serve as keys for the entities that they describe. This is what we ve done: SalesOrder, InventoryItem, and OrderItem all have attributes that initially seem like natural choices for keys, so we choose them. None of the Customer entity s attributes can be used to uniquely describe each customer instance, so we add a CustomerNumber attribute to act as its key. The Category, Newsletter, and CustomerType entities have only single attributes that describe them, so we add an attribute to act as the key for each of these entities. We didn t add a key attribute to the Subscription entity because it is a special case, in that it is identified not by a value of its own, but through its relationships with other entities, as you will see.

To refine our discussion of keys, we need to consider the relationships between the entities in our diagram. When we created these entities, we already had a basic idea of how they related to one another. For instance:

where is the region of interest (face region, in our case). Using the method of radial basis function, the interpolating function that minimizes the energy functional can be shown to take the form

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