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The main simplification algorithm works as follows: Constants and variables are passed through verbatim. negate is utilized when simplifying a negation, which assumes that the expression at hand no longer contains differences and that sums were flattened (see the next item in this list). Sums are traversed and nested sums are flattened, at the same time as collecting and adding up all constants. This reduces the complexity of further simplification considerably. Differences are converted to sums; for instance, A-B-C is converted to A+(-B)+(-C). Thus, the first element is preserved without negation. When simplifying a product, you first simplify its factors, and then you remove identity operations (multiplying by zero or one) and reduce products of constants. Fractions are handled similarly. Zero divided by anything is 0, anything divided by 1 is itself, and multiline fractions can be collapsed if you find numeric denominators or numerators. The rest of the match cases deal with simplifying subexpressions.

In 2005, Jesse James Garrett of Adaptive Path coined the term Ajax to describe an asynchronous method of loading page content. Traditionally, web applications involved a lot of page refreshes. The user made a choice, that information was sent back to the server, and then the server responded with a new page based on the user s actions. Even if the user saw only a small change to one area of the page, a whole new page was served with all the branding, navigation, header, and footer elements around the obligatory change. Using Ajax, only the appropriate portion of the page changes. Everything else the branding, the navigation, header, and footer remains the same. The user still makes a choice, but this time, a small change appears within the page that is already loaded in the browser, rather than the full page being served again. The advantage of using Ajax is that with Ajax, the request to the server occurs asynchronously to the page. Instead of serving up a whole new page every time the user sends a request, the browser can process requests in the background while the user continues to view and interact with the page. This allows your scripts to retrieve and create new content without loading a whole new page and interrupting the user experience With Ajax, web applications can offer content rich interfaces and a much more interactive, desktop-like experience, as you can see in applications such as Google Maps. Like any new technology however, Ajax must be used appropriately. Ajax relies on JavaScript and therefore won t work in some browsers or search engine spiders.

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The Csla.Server.SimpleDataPortal object (discussed later in the chapter) will ultimately invoke methods on business objects based on the MethodInfo object returned from GetMethod(). To support this, MethodCaller implements two different CallMethod() overloads: public static object CallMethod( object obj, string method, params object[] parameters) { MethodInfo info = GetMethod(obj.GetType(), method, parameters); if (info == null) throw new NotImplementedException( method + " " + Resources.MethodNotImplemented); return CallMethod(obj, info, parameters); }

Applying symbolic differentiation is a straightforward translation of the mathematical rules of differentiation into code. We could have used local functions that act as constructors and perform local simplifications, but with the simplification function described earlier this is no longer needed. Listing 12-7 shows the implementation of symbolic differentiation for the Expr type. Note how beautifully and succinctly the code follows the math behind it, and the essence of the symbolic processing is merely 20 lines of code! Listing 12-7. Symbolic Differentiation for Algebraic Expressions let Differentiate v e = let rec diff v = function | Num num -> Num 0N | Var v' when v'=v -> Num 1N | Var v' -> Num 0N | Neg e -> diff v (Prod ((Num -1N), e)) | Add exprs -> Add (List.map (diff v) exprs) | Sub (e1, exprs) -> Sub (diff v e1, List.map (diff v) exprs) | Prod (e1, e2) -> Add [Prod (diff v e1, e2); Prod (e1, diff v e2)] | Frac (e1, e2) -> Frac (Sub (Prod (diff v e1, e2), [Prod (e1, diff v e2)]), Pow (e2, 2N)) | Pow (e1, num) -> Prod (Prod(Num num, Pow (e1, num - 1N)), diff v e1) | Sin e -> Prod (Cos e, diff v e) | Cos e -> Neg (Prod (Sin e, diff v e)) | Exp (Var v') as e when v'=v -> e | Exp (Var v') as e when v'<>v -> Num 0N | Exp e -> Prod (Exp e, diff v e) diff v e

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