Bilinear

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) = normal component of the induced magnetic field. This system is non hnear (bilinear) because the product of the two unknowns /(r) and E(r) is present. 

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Now we consider a two-dimensional solid occupying a bounded domain fl C with a smooth boundary T. Let the bilinear form B be introduced by the formula... 

Let the brackets ( , ) denote the integration over Introduce the two bilinear forms used in previous sections,... 

Let C be a bounded domain with the smooth boundary L, which has an inside smooth curve Lc without self-intersections. We denote flc = fl Tc. Let n = (ni,ri2) be a unit normal vector at L, and n = ( 1,1 2) be a unit normal vector at Lc, which defines a positive and a negative surface of the crack. We assume that there exists a closed continuation S of Lc dividing fl into two domains the domain fl with the outside normal n at S, and the domain 12+ with the outside normal —n at S (see Section 1.4). By doing so, for a smooth function w in flc, we define the traces of w at boundaries 912+ and, in particular, the traces w+ and the jump [w] = w+ — w at Lc. Let us consider the bilinear form... 

The bilinear form Bg -, ) describing the bending properties of the shell is as follows ... 

It is easily seen that further indifferent stress rates result if terms are added which are bilinear in dand s. From (A.l 1), (A.42), and (A.69)... 

Figure 8.32. Cumulative number distribution data for lead impact experiment and comparison with bilinear exponential distribution.

ME Ochs, RS Stoyanova, E Arias-Mendoza, TR Brown. A new method for spectral decomposition using a bilinear Bayesian approach. I Magn Reson 137 161-176, 1999. 

The first-order term in this expansion renormalizes the potential V Q) while the bilinear term is analogous to the last term in (5.38). This is the linear-response theory for the bath. In fact, it shows... 

The role of two-phonon processes in the relaxation of tunneling systems has been analyzed by Silbey and Trommsdorf [1990]. Unlike the model of TLS coupled linearly to a harmonic bath (2.39), bilinear coupling to phonons of the form Cijqiqja was considered. In the deformation potential approximation the coupling constant Cij is proportional to (y.cUj. There are two leading two-phonon processes with different dependence of the relaxation rate on temperature and energy gap, A = (A Two-phonon emission prevails at low temperatures, and it is... 

Tustin s Rule Tustin s rule, also called the bilinear transformation, gives a better approximation to integration since it is based on a trapizoidal rather than a rectangular area. Tustin s rule approximates the Laplace transform to... 

The bilinear form in the argument of the exponential must be definite positive to have a well defined physics. This leads to constraints on the elements of the matrix A, as we shall see below. 

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

The indefinite bilinear form (XF, TB) as defined by Eq. (9-687) can be expressed in terms of the Hilbert space scalar product (Y,x)h 85 follows ... 

In effect the scalar product in (9-688), which makes the vector space into a Hilbert space, omits the factor ( —1) from the bilinear form (9-687). We shall always work with the indefinite bilinear form (9-687). Thus, for example, one verifies that with this indefinite metric... 

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