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14.19. (a) Find the Fourier sine transform of e x , x A 0 1 x sin mx b Show that dx e m ; m > 0 by using the result in a : 2 x2 1 0 Ans. (c) Explain from the viewpoint of Fourier s integral theorem why the result in (b) does not hold for m 0. p (a) 2= = 1 2 8 <1 Y x sin xt dx 2 : 0

3 628 800

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EXAMPLE 39 When defining piecewise functions, one must use . For example,

g[x_] x2 /; x 0 g[x_] x /; x < 0 g[3] 9 g[ 3] 9

and verify the solution by direction substitution. Ans. Y x 2 2 cos x 4 cos 2x = x PARSEVAL S IDENTITY 14.21. Evaluate 1 dx x2 dx ; b by use of Parseval s identity. 2 2 2 2 0 x 1 0 x 1 [Hint: Use the Fourier sine and cosine transforms of e x , x > 0.] Ans: a =4; b =4 1 (a) (a) 1 1 cos x 2 dx ; x 2 0 1 b

Using = would cause trouble, as Mathematica cannot determine which branch should be taken until a value of x is supplied.

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EXAMPLE 40 You may think that the assignment is more general and can be safely used in any given situation. This is true to a certain extent, but there are times when one should use =. As an extreme, but reasonable, example, let us define

applications from a number of different operating system environments (Figure 3.1). In contrast to the previous chapter's bottom-up discussion of the UNIX OS layers, we will view Windows NT from the top down. At the top layer of the NT architecture stack reside user mode services. These include a Virtual DOS Machine (VDM) for running legacy DOS applications, a 16-bit Windows subsystem for Windows 3.x programs, a POSIX subsystem, an OS/2 subsystem, and the Win32 subsystem. Applications executing within a particular subsystem are managed as separate protected virtual memory spaces. User mode services interact with lower-level kernel-mode services through the executive services buffer. This buffer is a code section that partitions and protects user-mode and kernel-mode operations. Immediately below this buffer are the NT Executive Managers, which regulate object and process life cycles; virtual memory; procedure calls; I/O functions; and monitor access security. A small fixed microkernel sits below the executive manager layer. The microkernel's role is limited to dispatching and synchronizing threads and managing system interrupts. The base of the operating system is made up of the Hardware Abstraction Layer (HAL), whose job is to hide hardware and device-driver dependencies from the rest of the operating system. This simplifies, secures, and generalizes the API for developing applications that use hardware services.

MISCELLANEOUS PROBLEMS 14.24. (a) Solve @U @2 U 2 2 , U 0; t 0; U x; 0 e x ; x > 0; U x; t is bounded where x > 0; t > 0. @t @x (b) Give a physical interpretation. U x; t 2 1

Each time a value of F is computed, Mathematica performs several integration by parts evaluations. Now imagine that many different values of F are needed, for example in the instruction

v = Table[2k 1, {k, 1, 8}]

3, 5, 7, 9, 11, 13, 15}

Ans:

12.9 Prove that the cross product of two vectors in 3 is orthogonal to each of the vectors that form it.

Windows NT comes complete with almost all the network protocols and interfaces one could ask for (Figure 3.2). The usual suspects include TCP/IP, NetBIOS/NetBEUI, and IPX/SPX. The base protocols and interfaces are augmented by additional communication services to support Virtual Private Networking (VPN) and Routing and Remote Access Services (RRAS), as well as gateway support among Microsoft LAN Manager, OS2 LAN Manager, Banyan Vines, Novell, and the Internet. For our purposes, we will narrow the set down to NetBIOS and TCP/IP, which provide the transport for the Server Message Block (SMB) protocol between Samba and Windows clients and servers.

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