Representation theorems for

We use the representation theorem for the basis states j) and form the matrix element with the bra vector (i. ... 

The representation theorems for such linear functions (A.67), (A.58), (A.68) from Appendix A.2 then gives for scalar functions... 

We also note that the vector or tensor responses (3.187), (3.189) depend only on the vector or tensor driving forces respectively. This fact is known in linear irreversible thermodynamics as the Curie principle [36, 80, 88, 89] (cf. discussion in [34, 38]). Present theory shows however, that this property follows from the isotropy of constitutive functions and from the representation theorems of such linear functions, see Appendix A.2, Eqs.(A.ll)-(A.13) and (A.57)-(A.59). But representation theorems for nonlinear isotropic constitutive functions [64, 65] show that the Curie principle is not valid generally. 

Because this is vaiid for aii Q e O, we can see that / is a scalar isotropic function (A. 14) of two vectors u and w. According to Cauchy representation theorem for such a function (proved above), the dependence J on these vectors must be expressed through scalar products... 

Recalling the role of scalar parameters, we can also consider (A.28)-(A.31) as the definitions of isotropic functions (vector or tensor functions up to fourth order) of scalar variables only and relations (A.32)-(A.35) are representation theorems for such functions then of course, scalars rj, a, (3, and 7 are (not specified) functions of scalar variables (of course, some of these results follow as a special case of a more general representation theorem below). 

The goal of all MTI methods is to use observed values of ground displacement to infer properties of the souree, as characterized by the moment tensor. By using the representation theorem for seismic sources (Aki Richards 1980) and assuming a point source, the displacement field recorded at a receiver k is given by ... 

Many of the problems formulated in connection with the question of reducing one science to another may be formulated as a series of problems using the notion of a representation theorem for the models of a theory. For instance, the thesis that psychology may be reduced to physiology would be for many people appropriately established if one could show that, for any model of a psychological theory, it was possible to construct an isomorphic model within physiological theory. (Suppes 1967, 59)... 

Suppose that the equilibrium stress field resulting from the relaxation step shown in part (c) of Figure 6.23 is denoted by alj(x,y). An expression for this stress field can be written immediately by appeal to the representation theorem for stress in terms of the elastic Green s function for a concentrated force under plane strain conditions. Suppose a stress field ijk(x, y) can be found which, for fixed k, is the stress field in the plane due to a concentrated force of unit magnitude applied at the origin x = Q, y = Q and acting in the A —th direction. This singular solution is known, so the solution for any distribution of concentrated forces can be constructed by... 

Our presentation is focused on the analysis of the scattered field in the far-held region. We begin with a basic representation theorem for electromagnetic scattering and then introduce the primary quantities which dehne the single-scattering law the far-held patterns and the amplitude matrix. Because the measurement of the ampUtude matrix is a comphcated experimental problem, we characterize the scattering process by other measurable quantities as for instance the optical cross-sections and the phase and extinction matrices. 

Representation theorems for electromagnetic helds have been given by Stratton and Chu [216]. If Eg, Hg is a radiating solution to Maxwell s equations in Dg, then we have the Stratton-Chu formulas... 

The Stratton-Chu representation theorem for the scattered field in Dg, yields the expansion of the approximate scattered field in the exterior of a sphere enclosing the particle... 

For this purpose we choose a sufficiently large auxiliary surface S enclosing O and Oi (Fig. 2.6) and in each coordinate system we use the Stratton-Chu representation theorem for the incident field in the interior of S. We obtain... 

To compute the T matrix of the two-particles system and to derive a scattered-field expansion centered at the origin O of the global coordinate system we use the Stratton-Chu representation theorem for the scattered field Eg in Dg. In the exterior of a sphere enclosing the particles, the expansion of the approximate scattered field in terms of radiating vector spherical wave functions reads as... 

The Stratton-Chu representation theorem for the incident and scattered fields in Di, where Di = I4ipU )i 2US i2 = R —Dg, together with the boundary conditions (2.154) and (2.155), lead to the general null-field equation... 

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