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fp neural network output for pattern p i input unit index / total number of input units (bias unit excluded) j hidden unit index J total number of hidden units (bias unit excluded) k output unit index K total number of output units l functional link units index L total number of functional link units N(0, a2) Gaussian distribution with 0 mean, and variance a2 netyhP net input signal to hidden unit yj for pattern p net0k^p net input signal to output unit Ojt for pattern p Ok k-th output unit <>k,p output of the k-th output unit for pattern p p pattern index P total number of patterns PG number of patterns in generalization set PT number of patterns in training set PV number of patterns in validation set t time step T total number of delayed time steps for TDNN tp target for pattern p tktp target for k-th output unit for pattern p U(a, b) uniform random number between o and 6 Uki direct weight between output unit Ok and input unit Zj Awfcj vji Avji Vji direct weight change between output unit Ok and input unit weight between hidden unit yj and input unit zi weight change between hidden unit yj and input unit Z{ weight between hidden unit yj and functional link unit hi weight between output unit Ok and hidden unit yj weight change between output unit o* and hidden unit yj epoch counter j-th hidden unit activation of hidden unit yj i-th input unit value of input unit for pattern p value of input unit for pattern p, delayed for time step t

1 1 1 (1,1,1) -1 -1 -1 (1,1,1)

We shall find, in 13, that the way to understand loops, statements that continually change the program variables, is to examine what is left unchanged by the loop the so-called loop invariant Here are some more exercises aimed at getting you to think in terms of invariant properties They all describe some sort of game (sometimes one-person, sometimes two-person) in which every move has to obey the given rules of the game They are all solved by identifying an invariant property of the game, ie a property of the position reached in the game that is true no matter how many moves have been made All the invariant properties are very simple even 'obvious' once pointed out Finding them can be difficult, however, because thinking in terms of invariants is a skill that is rarely taught properly, if at all.

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vector satisfies equation (1.49). Along the [001] direction, the smallest reciprocal lattice vector G is (0,0,2) (in units of 2T/Ao). Substituting into equation (1.49) gives:

Try not to consult the solutions too soon, and have fun! Exercise 121 Several tumblers are placed in a line on a table Some tumblers are upside down, some are the right way up It is required to turn all the tumblers the right way up However, the tumblers may not be turned individually; an allowed move is to turn any two tumblers simultaneously Describe the initial states of the tumblers from which it is possible to turn all the tumblers the right way up, and describe the strategy for doing so D Exercise 122 (a) In this game there are two players A pile of matches is placed on a table; a move is to remove one, two or three matches from the pile The winner is the one who removes the last match Suppose an odd number of matches is placed on the table.

yj yjjp Zi Zitp z i,p(t)

This then implies the electron will be diffracted when:

Who should win and what is the winning strategy (b) Suppose that the players are allowed to remove any number of matches from 1 up to 2m +1, where m is a natural number (fixed by mutual agreement between the players in advance of the game) The initial pile of matches contains an odd number of matches Who should win and what is the winning strategy (c) Suppose that the players are allowed to remove any number of matches from 1 up to m, where m is a natural number (fixed by mutual agreement between.

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