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Given prime number p, we let exp.n denote the p-exponent of n. It is defined by the Galois connection: for all natural numbers m and n, exp.n = pm\n . (Read 'exp.n is the greatest m such that pm divides n'.) With these preliminaries we can now pose the following questions. (a) Show that exp.p = 1. (b) Derive an operator e such that exp.gcd(m,n) = exp. we exp.n . (c) Derive an operator such that exp.(raxn) = exp. m exp.n . (You will need to use existential quantification see Section 11. 3.2 to complete this final part of the exercise. Specifically, for prime number p,



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1 Generics facilitate a strongly typed programming model, preventing data types other than those explicitly intended by the members within the parameterized class In Listing 117, the parameterized stack class restricts you to the Cell data type for all instances of Stack<Cell> (The statement pathPush("garbage") produces a compile-time error indicating that there is no overloaded method for SystemCollectionsGenericStack<T>Push(T) that can work with the string garbage, because it cannot be converted to a Cell) 2 Compile-time type checking reduces the likelihood of InvalidCastException type errors at runtime





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3 Using value types with generic class members no longer causes a cast to object; they no longer require a boxing operation (For example, pathPop() and pathPush() do not require an item to be boxed when added or unboxed when removed) 4 Generics in C# reduce code bloat Generic types retain the benefits of specific class versions, without the overhead (For example, it is no longer necessary to define a class such as CellStack) 5 Performance increases because casting from an object is no longer required, thus eliminating a type check operation Also, performance increases because boxing is no longer necessary for value types 6 Generics reduce memory consumption by avoiding boxing and thus consuming less memory on the heap 7 Code becomes more readable because of fewer casting checks and because of the need for fewer type-specific implementations 8 Editors that assist coding via some type of IntelliSense work directly with return parameters from generic classes There is no need to cast the return data for IntelliSense to work At their core, generics offer the ability to code pattern implementations and then reuse those implementations wherever the patterns appear Patterns describe problems that occur repeatedly within code, and templates provide a single implementation for these repeating patterns

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pk \ mxn = (3i,j : k = i+j : pl\m A pj \n} .)

these requirements are bound to include features that weren t predicted when setting up the original product line Fortunately, both problems can be mitigated by the same basic approach: separating out the parts that vary from the parts that remain constant, thereby making changes easier And where the changeable parts form complex structures, DSLs may be created to express their instances

Type Parameter Naming Guidelines Just as when you name a method parameter, you should be as descriptive as possible when naming a type parameter Furthermore, to distinguish the parameter as being a type parameter, its name should include a T prefix For example, in defining a class such as EntityCollection<TEntity> you use the type parameter name TEntity The only time you would not use a descriptive type parameter name is when the description would not add any value For example, using T in the Stack<T> class is appropriate, since the indication that T is a type parameter is sufficiently descriptive; the stack works for any type In the next section, you will learn about constraints It is a good practice to use constraint-descriptive type names For example, if a type parameter must implement IComponent, consider a type name of TComponent

Generic Interfaces and Structs C# 20 supports the use of generics extensively within the C# language, including interfaces and structs The syntax is identical to that used by classes To define an interface with a type parameter, place the type parameter in angle brackets, as shown in the example of IPair<T> in Listing 118

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static class Listable { public static void List( this IListable[] items, string[] headers) { int[] columnWidths = DisplayHeaders(headers); for (int itemCount = 0; itemCount < itemsLength; itemCount++) { string[] values = items[itemCount]ColumnValues; DisplayItemRow(columnWidths, values); } } // count < valuesLength; count++) { ConsoleWrite( "{0}{1,-" + columnWidths[count] + "}", tab, values[count]); { tab = "\t"; } } ConsoleWriteLine(); }

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