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which ensures that the electric field, and hence the potential, go to zero at large distances from the charge distribution. For the case of a doped semiconductor, there would be two contributions to the charge density a(z), where the first would be the ionised impurities and the second the free charge carriers themselves. While the former would be known from the doping density in each semiconductor layer, as defined at growth time, the latter would be calculated from the probability distributions of the carriers in the heterostructure. Thus if d(z] defines the volume density of the dopants at position z, where the planes are separated by the usual step length 6z, then the total number of carriers, per unit cross-sectional area, introduced into the heterostructure is given by:

It is now possible to find all dispensible, or redundant, attributes. An attribute a e B C A is dispensible if IND(B) = IND(B - {a}). Using the definition of dispensibility, a reduct of B C A is the set of attributes B C B such that all a e B - B' are dispensible, and IND(B) = IND(B'). The reduct of B is denoted by RED(B), while RED(E, B) denotes the relative reduct of B for equivalence class E. A relative reduct contains sufficient information to discern objects in one class from all other classes.

P := Pxx + b ;

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where q is the charge on the extrinsic carriers. The step length 6z selects the proportion of the carriers that are within that 'slab' and converts the volume density of dopant, d(z) into an areal density. If the charge carriers are distributed over more than one subband, then the contribution to the charge density cr(z) would have to be summed over the relevant subbands, i.e.

Figure 16.7 On-line computation of cyclic codes. This increases the degree of r by one. Consequently, it may falsify degree.r < degree.Q . If it does, the degree of r after the assignment must be equal to the degree of Q. The invariant can thus be re-established by adding Q to r . If it does not, no further action needs to be taken. So the assignments to r that we have to implement are

where Eni=1 Ni = N. Fig. 3.32 shows the areal charge density along the growth axis for a 100 A GaAs well, n-typedopedto2x 1018cm 3, surrounded by 200 A undoped Gao.sAlo.2 As barriers. The ionised donors yield a constant contribution to a within the well of d(z]8z =

Vagueness in rough set theory, where vagueness is with reference to concepts (e.g. a tall person), is based on the definition of a boundary region. The boundary region is defined in terms of an upper and lower approximation of the set under consideration. Consider the set X C [7, and the subset of attributes B C A. The lower approximation of X with regard to B is defined as BX = U{E U/IND(B)\E and the upper approximation of X, BX = (J{E 6 U/IND(B)\E n X / 0} C X}

r := rxx + b ;

Figure 332 Areal charge density a for a 100 A GaAs well, n-type doped to 2xl0 18 cm 3 , surrounded by undoped Gao8Alo2 As barriers 2x 1024m-3 x lA= +2x 1014m 2, in each of the 1 A thick slabs Hence, the total number N of electrons in the quantum well is 100x2x 1014m 2 = 2x 1012cm 2 By assuming that the electrons introduced by such doping all occupy the ground state of the quantum well, then the curve on top of the ionised impurity background clearly resembles u * u , as expected from the mathematics The discontinuities in a occur at the edges of the doping profiles and are of magnitude 2x 1014m 2, again as expected There are a number of points to note about Fig 3.

A direct hardware implementation of the conditional statement would be inefficient. However, by observing that the assignment r := rxx + b falsifies degree.r < degree.Q exactly when rm-\ = 1, we see that it may be replaced by the unconditional assignment r := r + Qxr m _i . The two assignments can now be combined resulting in the program in Figure 16.7. The assignment to r does, indeed, have the form of the assignment in (16.2). (The 'modx m ' can be dropped because the resulting value of r is guaranteed to have degree less than m.) The coefficient, c, in (16.2) takes the value r m _i, the bit of the remainder that is 'shifted out' of the shift register by the operation r := rxx + b. So, the assignment tor can be implemented by a simple feedback loop. Figure 16.8 illustrates the circuit for the particular generator polynomial

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