Dual form

We will say that the map % S3 x Ri. S 2, given by a scalar field %( r. t), generates an electromagnetic field if the corresponding pullback of the area form in S2 y a, or its dual form % cr, verifies the Maxwell equations in empty space. 

Let. .MA" V. o be the intersection of. t(A with the plane x 0. We note that its restriction to the plane x is a manifold of the states that we denote xeth(y ) and call equilibrium states. These are the states for which cp reaches its extremum if considered as a function of x. We shall denote the manifold of equilibrium states by the symbol Meth- Restriction of Mm x, 0 to the plane y is the manifold representing y expressed in terms of x (i.e. the function (4)), and its restriction to the plane (i/. yj is the Gibbs-Legendre manifold Af representing the dual form s s (y ) of the fundamental thermodynamic relation s = s(y) in N that is implied by the fundamental thermodynamic relation h = h(x) in M. This completes our presentation of the passage h(x) —> s(y). 

In consequence of its dual form the compound can act both as a nitrosating (diazo-tizing) and as a nitrating agent. 

Axens Octanizing Aromizing Semiregenerative CCR dual forming for conventional process revamp Over 100 licensed units... 

Figure 10.4 also shows the existenee of two sets of refieetions for the thiek-est sample direet evidenee of polymorphism. The seeond form indexed with (100) appears at higher Q y = 0.525 A eompared with (100) Qj, y = 0.405 A ). This reflects an interplanar distance of = 12.0 A compared with the main form with the tf = 15.6 A. The appearance of higher-order peaks such as (200) and (300) in addition to (200) and (300) gives evidence for well-pronounced ordering of both polymorphs. Interestingly, such dual form is not observed for thinner films. Such polymorphism was also found in powder samples [17]. 

Methods that directly operate on the original (Primal) problem are called Primal Methods to distinguish them from other methods that operate on the dual form... 

Abduction with trick can be considered as a dual form of abduction, which is also the summary of a kind of EBR. Abduction can be used to explain that the symptoms of the patients result from specific diseases, while abduction with trick can be used to exclude some possibili-... 

Equation (13.7) represents the dual form of the regression model, since in it the activity is predicted by considering similarity measures of a test compound in relation to the training set compounds M. In order to obtain the traditional primal form of the 3D-QSAR model, which involves an explicit consideration of molecular descriptors and regression coefficients, one can make the substitution of Eqs. (13.1) and (13.3-13.6) to Eq. (13.7) to obtain ... 

The ensemble average of a phase function A(D at time t can now be written in dual forms,... 

C. Miquel, J.P. Paz, M. Saraceno, E. Knill, R. Laflamme, C. Negrevergne, Interpretation of tomography and spectroscopy as dual forms of quantum computation. Nature 418 (2002) 59. 

The White-Johnson-Dantzig method may be regarded as one of non-linear programming. Dorn" " has shown, that every problem that is based on minimalisation of a convex function has a dual form. Mutual linkage of the two formulations of the dual theorem can only be used to determine the overall free enthalpy in the individual steps, and thus also to establish the accuracy of the solution achieved. When minimalisation is expressed as... 

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