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If m is odd, doing nothing is not valid and we must seek a property of division-bytwo that will allow us to proceed. The appropriate property is, of course, that for an odd positive integer m, m-r-2 = (m- 1H2 and m-1 is even. Moreover, we are told that the operation dec calculates m-1 when m is a strictly positive number (see (10.14)). All odd, positive integers are strictly positive, so it is thus the case that { 0 ^ m = M A odd.m } m := dec.m (10.20) { O ^ m A even.m A m-r2 = M-^2 } . (Use the assignment axiom to check the claim, making sure that no details have been overlooked.) The final step is to put (10.19) and (10.20) together using the conditional rule given in Section 10.4. Every number is either even or odd. So { 0^m = M } if even.m < skip n odd.m > m := dec.m

Figure 3.31 Electric field strength from an infinite plane of charge of volume density d(z) and thickness 8z

1, based on the formal definition of a decision system Section 212 shows how rough sets treat vagueness by forming a boundary region, while Section 213 discusses the treatment of uncertainty in the implementation of the rough membership function..

A even.m A m-r-2 = M-^2 } .

Given that the potential profiles, VCB(Z) for example, are one-dimensional, then they will also produce a one-dimensional charge distribution. In addition, remembering that the quantum wells are assumed infinite in the x-y plane then any charge density p(z] can be thought of as an infinite plane, i.e. a sheet, with areal charge density a(z) and thickness 'Sz, as shown in Fig. 3.31(a). Such an infinite plane of charge produces an electric field perpendicular to it, and with a strength:

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The construction of S satisfying (10.15) is thereby solved. To complete the construction, we combine the above with (10.17) using the rule of sequential composition to get

Note that as the sheet is infinite in the plane, then the field strength is constant for all distances from the plane. The total electric field strength due to many of these planes

The discussion on rough sets will be with reference to a decision system Firstly, an information system is formally defined as an ordered pair A = (U, A), where U is the universe of discourse and A is a non-empty set of attributes The universe of discourse is a set of objects (or patterns, examples), while the attributes define the characteristics of a single object Each attribute a G A is a function a : U * Va, where Va is the range of values for attribute a A decision system is an information system for which the attributes are grouped into disjoint sets of condition attributes and decision attributes The condition attributes represent the input parameters, and the decision attributes represent the class The basic idea upon which rough sets rests is the discernibility between objects.

It is beyond the scope of this book to derive the Hamiltonian that describes the valence band states, so its form will be stated and it will be employed in the calculation of quantised states within the valence band of nanostructures in order to illustrate how it is used. In many cases good accuracy can be obtained by using the so-called 6x6 Hamiltonian, although its shortened version, the 4x4 Hamiltonian is frequently just as good it all depends on the energy range of interest. As mentioned above, a valid basis set can be the atomic p-like orbitals with x, y, and z-like spatial symmetry, here denoted as | X } , |Y}, | Z ) , each also having the spin projection along the z-axis equal to either +1/2 or 1/2, denoted as | or | respectively. However, common practice

{ 0 < m A even.m A m+2 = M-^-2 }

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