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294 CHAPTER 13 Ontologies on the Web--putting it all together This example shows a number of features of the role that models play in linking data sources. The structure of the Good Relations ontology gave us a consistent way to represent offerings, prices, amounts, and units of competing businesses. QUDT provides useful services (units conversions) for computations over these data. But as often happens in the data wilderness, these two models were not designed to connect. In particular, the Good Relations model does not use QUDT as its unambiguous reference for units. Fortunately, there is a common reference shared by QUDT and Good Relations--namely, the United Nations UNECE unit names standard. Since they both share a reference to UNECE, the connection can be made (with a single triple in the graph pattern!). The result is a price comparison based on published information, from a single query over the combined data sets. DIMENSION CHECKING IN QUDT There are a lot of services that an engineer or scientist could ask of a system of units; many of these go by the name dimensional analysis. Whole books have been written on the subject; we can barely scratch the surface of this topic here (therefore, see the Further Reading section at the conclusion of the book). We will illustrate enough of the QUDT ontology to show how it can support certain basic operations in dimensional analysis. The basic idea of dimensional analysis is that a quantity has a signature that tells how it relates to basic quantities like length, time, and mass. QUDT defines eight base quantities of this sort, but in this exposition, we will focus on these three. A compound quantity has a signature in these basic quantities. For example, the compound quantity velocity is defined as a ratio between distance and time; this means that its signature in terms of length, time, and mass is length/time. The signature can be written as a vector, with one vector component for each base unit, and the magnitude of the vector in that component being the exponent of the base quantity in the formula for the compound quantity. If we write our vectors in the order [length, mass, time] then the vector for velocity is [1, 0, À1]. The vector for any base quantity will have magnitude 1 in exactly one place; so the vector for mass is [0, 1, 0]. Acceleration is given as a quotient of velocity by time; so its vector is [1, 0, À2]. This vector is called the dimensionality of a compound unit. These vectors can be used to check the validity of a scientific or engineering formula. For example, one of the basic laws of motion is given by the formula F ¼ ma, or Force equals the product of mass times acceleration. Only quantities with identical dimensionality can be meaningfully compared, so this formula makes sense only if the dimensionality of Force is the same as the dimensionality of the product of mass and acceleration. The dimensionality of Force is given by the expression ml=t 2 , that is, mass times length divided by time twice. The dimensionality of acceleration is given as l=t 2 , or length divided by time twice, and mass is given simply by m. So we can check the dimensionality of F ¼ ma by replacing each term with its dimensions, i.e., ml=t 2 ¼ m$ðl=t 2 Þ. Since the dimensions on both sides are the same, the formula has been verified to pass the test of dimensionality. It is important to note that this kind of simple dimensional analysis can uncover certain formulas that are incorrect; a correct dimensional analysis does not guarantee that the formula doesn't have some other problems with it. In terms of dimension vectors, we can do the same calculation using vectors. The dimension vector for Force is [1, 1, À2]. The vector for mass is [0, 1, 0] and the vector for acceleration is [1, 0, À2]. The formula is verified if the vectors add up; that is, if the vector for mass plus the vector for acceleration