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as the equation of motion If moved from its equilibrium position by a distance x0 and released from rest, the system illustrated in Fig 111 will execute simple harmonic motion and move in a sinusoidal trajectory given by x = A cos(w n t + f ) (114)

Many processes of nature and many mathematical applications involve logarithmic and exponential functions For example, if we examine a population of bacteria, we notice that the rate at which the population grows is proportional to the number of bacteria present To see that this makes good sense, suppose that a bacterium reproduces itself every 4 hours If we begin with 5 thousand bacteria then after 4 hours there are 10 thousand bacteria after 8 hours there are 20 thousand bacteria after 12 hours there are 40 thousand bacteria after 16 hours there are 80 thousand bacteria etc The point is that each new generation of bacteria also reproduces, and the older generations reproduce as wellAsketch (Fig 612) of the bacteria population against time shows that the growth is certainly not linear indeed the shape of the curve appears to be of exponential form

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// Format an enumeration using System; class EnumFmtDemo { enum Direction { North, South, East, West } [Flags] enum Status { Ready=0x1, OffLine=0x2, Waiting=0x4, TransmitOK=0x8, ReceiveOK=0x10, OnLine=0x20 } static void Main() { Direction d = DirectionWest; ConsoleWriteLine("{0:G}", d); ConsoleWriteLine("{0:F}", d);

22:

for some amplitude A, frequency wn, and phase shift f Using this expression in the equation of motion gives

ConsoleWriteLine("{0:D}", d); ConsoleWriteLine("{0:X}", d); Status s = StatusReady | StatusTransmitOK; ConsoleWriteLine("{0:G}", ConsoleWriteLine("{0:F}", ConsoleWriteLine("{0:D}", ConsoleWriteLine("{0:X}", } } s); s); s); s);

Notice that when the number of bacteria is large, then different generations of bacteria will be reproducing at different times So, averaging out, it makes sense to hypothesize that the growth of the bacteria population varies continuously as in Fig 613 Here we are using a standard device of mathematical analysis: even though the number of bacteria is always an integer, we represent the graph of the population of bacteria by a smooth curve This enables us to apply the tools of calculus to the problem

The output is shown here:

2 mx = - mAw n cos(w n t + f ) = - kA cos(w n t + f )

Meaning Displays the value as a decimal integer Same as D Displays the name of the value However, if the value can be created by ORing together two or more values defined by the enumeration, then the names of each part of the value will be displayed This applies whether or not the Flags attribute has been specified Same as F Displays the name of the value If the enumeration is preceded by the Flags attribute, then all names that are part of the value will be displayed (assuming a valid value) Same as G Displays the value as a hexadecimal integer Leading zeros will be added to ensure that at least eight digits are shown Same as X

lthough C# contains many exciting features, one of its most powerful is its built-in support for multithreaded programming A multithreaded program contains two or more parts that can run concurrently Each part of such a program is called a thread, and each thread defines a separate path of execution Thus, multithreading is a specialized form of multitasking Multithreaded programming relies on a combination of features defined by the C# language and by classes in the NET Framework Because support for multithreading is built into C#, many of the problems associated with multithreading in other languages are minimized or eliminated

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