"... In PAGE 4: ... Mohr et al. (1957) first used the striation thickness (see Table1 ), defined as one-half the spacing between layer midplanes in a lamellar structure, as a measure of mixing. Striation thickness is related to the specific area of a lamellar mixture by = 1 SV (2.... In PAGE 5: ....2.2 Example Area Tensors When the mixture has one discrete and one continuous phase, the area tensor provides information about the shape and size of the discrete-phase domains. Table1 shows the area tensors for three example mixture morphologies. The area tensor is triaxial (isotropic) for spherical domains, biaxial (transversely isotropic) for cylindrical domains, and uniaxial for lamellar structures.... In PAGE 5: ... We can then define a local characteristic length scale Lc for the discrete phase as the ratio of the total volume Vd of the discrete phase within V to the total interfacial area Sd within V Lc Vd=Sd = =SV (2.8) Table1 gives the characteristic length scales for the three example morphologies. The characteris- tic radius of the lamellar structure is related to the striation thickness by r = .... In PAGE 14: ... This approximation is generated by choosing a functional form, constraining the form, and fitting the constrained function to data generated from the exact closure. The constraints force the closure to obey geometric symmetries, give exact results in the three limiting cases of Table1 , and have correct asymptotic behavior near those limits (Wetzel and Tucker, 1997). Exact data for ^ A (1), ^ A (2), and ^ A (3) as a function of ^ A(1) and ^ A(2) were generated using Eqns.... In PAGE 22: ... (2.8) and Table1 with a dispersed-phase volume fraction = 0:10, this tensor represents a lamellar morphology with an average sheet... ..."

"... In PAGE 10: ...ames. These typically involve degenerate cases, such as copying from one register to another. They exploit the fact that integer register $31 is always 0. Table9 lists some typical cases and their translationsinto actual Alpha instructions. No-op instructions nop are commonly used to pad code to meet specified alignment requirements.... In PAGE 10: ... Note that the native C compiler CC makes use of these special names. The GCC compiler, on the other hand, generates the implementation form shown in the right hand column of Table9 , as does the disassembler dis. Understanding these special cases is important to being able to read the code generated by these programs.... ..."

"... In PAGE 10: ...ames. These typically involve degenerate cases, such as copying from one register to another. They exploit the fact that integer register $31 is always 0. Table9 lists some typical cases and their translations into actual Alpha instructions. No-op instructions nop are commonly used to pad code to meet specified alignment requirements.... In PAGE 10: ... Note that the native C compiler CC makes use of these special names. The GCC compiler, on the other hand, generates the implementation form shown in the right hand column of Table9 , as does the disassembler dis. Understanding these special cases is important to being able to read the code generated by these programs.... ..."

"... In PAGE 12: ... Table2 illustrates the importance of checking for special cases. These include in- variants, syntactically safe properties, bounded LTL properties, and liveness properties of the form F p, where p is a propositional formula.... In PAGE 12: ...ng properties. The column labeled k has the same meaning as in Table 1. If the value of k is 0, the corresponding property is an inductive invariant. In Table2 , the general method is slower. There are two reasons for that: The rst is that using the termination criteria of Theorem 1 and (3a00) generate more clauses for a given value of k.... ..."