U domains

The collocation analysis is carried out in the U] - domain and U2 - domain for Section 1 and Section 2, respectively. We choose N interior collocation points in Section 1, and for simplicity we also choose the same number of interior collocation points in Section 2. 

The mass balance equation (A9.1-4) is valid at any point within the u domain. Thus, evaluating that equation at the i-th interior collocation point we get ... 

Protein sequence based on intron/exon modeling performed by Todd Richmond (hHp //cellwall. stanford.edu/php/structure.php). Black boxes = putative transmembrane domains White boxes = conserved U domains Grey boxes = hydrophobic regions manually. 

This procedure has been employed in the construction of the diabatic potential matrix for the l A and 2 A electronic states of H3 with conical intersection by Abrol and Kuppermann," in which the diabatization angle 7 (a function of internal nuclear coordinates) was calculated by solving the three-dimensional Poisson equation (with an optimal set of boundary conditions) for the entire U domain of nuclear configuration space bearing important significance for reactive scattering. The procedure was also employed by Mota and Varandas in their construction of the double many-body expansion (DMBE) diabatic potential matrix for the l A and 2 A states of the HN2 system, with a newly proposed diabatization scheme where the diabatization angle is represented by some specific functions. Earlier construction of the DK (Dobbyn and Knowles)... 

Let us consider a domain U e R, representing the three-dimensional flaw imbedded in a homogeneous conductive media, with electric conductivity uo and permeability The flawed region D is assumed to be inhomogeneous, and characterized by the relative real conductivity ... 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

Let V be some known function defined in the domain fic- If v and the boundary dflc = F U F+ U Fj are sufficiently smooth, then we can define values of v at the boundary (the exact smoothness conditions are studied in Section 1.4). In particular, having the values u p+ and u p-, we introduce the jump of v at Fc by the formula... 

Now consider a domain fl containing a surface Tc, whose properties are described in Section 1.1.7. Denote Sc = Tc clTc, flc = fl Tg. Introduce the unit normal n to Tc and define the opposite faces T of the surface Tg. The signs fit positive and negative directions of n, respectively. Then we denote the boundary of flc by dflc = T U T. We assume that there exists a closed extension S of Tc dividing the domain fl into two subdomains Di, D2 with boundaries dfli,dfl2 and such that Tc C S. It is assumed that = S , 80,2 = S+ U r. We say that the boundary dOc belongs to the class if 80,1,80,2 belong to G . ... 

Consider the domain flc with the boundary dflc described in the previous subsection. Let a function u G be given. We assume that dflc... 

Extend Sc up to the boundary F such that fl is divided into two domains with Lipschitz boundaries dfli,dfl2- Assume that mens (F n 90 ) > 0, i = 1,2. In each of these domains, for u G the second Korn... 

To estimate the third-order derivatives of the function w with respect to y, we make use of the following fact (see Duvaut, Lions, 1972). Let O d E be a bounded domain with smooth boundary and let u be a distribution on O such that u, Du G Then u G L 0) and there is a constant c,... 

Let a point x be interior with respect to i.e. there exists a neighbourhood U of the point x such that U C We choose a smooth function X = (W, w) in the domain flc such that a support of x belongs to U and... 

In the domain ft we consider the problem of finding a function u such that... 

The perturbed problem corresponding to (4.101)-(4.103) is as follows. In the domain fig, we want to find a function = (u such that... 

The equilibrium problem for an elastic body occupying the domain fig can be formulated as follows. We want to find a function W = u,v,w)... 

Sandig A.M., Richter U., Sandig R. (1989) The regularity of boundary value problem for the Lame equations in a polygonal domain. Rostock. Math. Kolloq. 36, 21-50. 

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