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Refraction was now nearly as well understood as re ection. To take the nal step of providing a theory that predicted the action of both took one of the greatest mathematical minds and egotists the world has ever known. Pierre de Fermat has had a spell of recent fame thanks to the drama surrounding Fermat s last theorem . This seemingly simple mathematical proposition was nally cracked by the English mathematician Andrew Wiles in 1993. The actual problem dated back to the ancient Greeks, but Fermat, in a typically offhand style, made it a challenge that would intrigue and infuriate generations of mathematicians. The speci cs of the problem could interest no one but a maths buff, but Fermat ensured (and subsequent books and TV series have proved) that the chase itself would be fascinating. Fermat loved to tease others with his wisdom by setting challenges. In a copy of the Greek writer Diophantus s Arithmetica, he added a note of his own, stating that while it is possible for the square of one number to be the sum of two other squares, a similar relationship isn t possible for

1. Journal of Orofacial Pain. Quintessence. Telephone 800-621-0387. 2. Cranio: The Journal of Craniomandibular Practice. Chroma. Telephone 800-624-4141. 3. TMJ Update. Anadem. Telephone 800-633-0055.

Stage 1: Observation (a)

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When z ) w as in the Fraunhofer approximation, the term involving w2 in Eq. (7.3-5) is skipped. The resulting approximations will be referred to as the near eld approximation (NFA) and the far eld approximation (FFA) below. The general case covering both NFA and FFA will be referred to as NFFA.

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Here are the steps to perform this refactoring. 1. Build a page transition diagram that depicts the links between your pages. While this step is not strictly required, it is very helpful when you are trying to make sure that all page transitions are happening properly. As an alternative, you can use HTTPUnit or Cactus (testing tools based on JUnit) to build tests asserting the current navigational structure of the application and run them periodically as the pages are refactored. 2. Create a Servlet that will act as your Traffic Cop and will front the requests made to the application. The Traffic Cop will have all the navigational information for your application and will use that information to correctly redirect each request made. 3. Initialize this Traffic Cop with the navigational information captured earlier in your diagram. Using a HashMap is an easy and quick means to do this. The keys are just strings, and the values are the names of the JSP pages. 4. Name each of the arcs or lines between the pages. Action-oriented names work best, such as login or showPendingPayments. These are the names of the processes your application will be performing for the user. 5. Replace any direct references to other JSPs with the names of the arcs. Look for jsp:include, jsp:forward, and/or href attributes. Do no more than can be easily verified in each iteration through this process. You want to be able to easily verify that what you have done still works when you are done so, do the refactoring in manageable chunks.

Assume further that Jk = 0 k f ( ) d < , for 1 k 5. Denote by X, Xs , Xt , Xss , Xst , Xtt the values of X and the rst and second partial derivatives at the point (s, t) and X = (Xs , Xt )T and X the matrix of

The chief refactoring required here is to move the infrastructure-related functionality from client into service. So, in Cross-Tier Refactoring, found later in this chapter, the client code that performs data validation, security checking, and so on is moved to or replicated in some form down to the service implementation. Note that it may be still acceptable to leave implementation in the client as well to support better performance, but the underlying services should be able to be used in a standalone way. For example, if data validation is being performed in Javascript (or in JSP scriptlet, custom tags, or servlets), then some similar implementation needs to be part of the service implementation. As discussed in the Stovepipe Service AntiPattern, the ideal approach is to define another service that generically performs these actions for multiple services. For instance, if one of the processes is to validate that entity-based data elements are present and in the correct format, then metadata descriptions and rules about the entities can be used in conjunction with an Entity Validation service, so that any service that receives entity instance data can utilize that service to validate before moving on. Likewise, fine-grained security checking (that needs to ensure that a user/role can access, read, or edit some entity property) can be facilitated by a generic security service that utilizes security metadata about entities to perform its checking.

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