Two-dimensional scalar

Nicolaides D and Bruce A D 1988 Universal configurational structure in two-dimensional scalar models J. 

Figure 1.4 Representations of a two-dimensional scalar field are at the left and middle.

Consider a two-dimensional, scalar conservation law of the form (1) with solution u(t,x). Within the finite volume framework, each discrete value of the function u is viewed as a cell average U( over a cell. The advantage of the finite volume approach is, that any kind of mesh can be used, i.e., the shape of the control volume can be chosen arbitrarily. Here, we work with a conforming triangulation T with cells Te T, = for which the... 

There are very few examples of scalar-mixing cases for which an explicit form for (e, 0) can be found using the known constraints. One of these is multi-stream mixing of inert scalars with equal molecular diffusivity. Indeed, for bounded scalars that can be transformed to a mixture-fraction vector, a shape matrix can be generated by using the surface normal vector n( ) mentioned above for property (ii). For the mixture-fraction vector, the faces of the allowable region are hyperplanes, and the surface normal vectors are particularly simple. For example, a two-dimensional mixture-fraction vector has three surface normal vectors ... 

The effort to solve Eqs.(l) evidently depends on the refractive index profile. For isotropic media in a one-dimensional refractive index profile the modes are either transversal-electric (TE) or transversal-magnetic (TM), thus the problem to be solved is a scalar one. If additionally the profile consists of individual layers with constant refractive index, Eq.(l) simplifies to the Flelmholtz-equation, and the solution functions are well known. Thus, by taking into account the relevant boundary conditions at interfaces, semi-analytical approaches like the Transfer-Matrix-Method (TMM) can be used. For two-dimensional refractive index profiles, different approaches can be... 

Bodenhausen ° developed a pattern-recognition programme (MARCO POLO) in order to extract coupling pathways in COSY spectra. He subsequently described a recursive deconvolution technique for the measurement of couplings, but this seems to offer more benefits to the measurement of scalar couplings in two-dimensional spectra than in one-dimensional spectra. 

Stonehouse and Keeler developed an intriguing method for the accurate determination of scalar couplings even in multiplets with partially convoluted peaks (one- or two-dimensional). They recognized that the time domain signal is completely resolved and that convolution of the frequency domain spectrum is a consequence of the Fourier transform of the signal decay. The method requires that the multiplet be centred about zero frequency and this was achieved by the following method ... 

Whilst scalar couplings are readily identified in two-dimensional spectra, their measurement from cross-peak multiplets poses special problems. 

Identification of constitutive monosaccharides two-dimensional homonuclear NMR techniques such as DQF-COSY and TOCSY are used to assign chemical-shift values for all C-bonded protons in each individual monosaccharide (96). One-dimensional NMR spectra provide useful information about the chemical shifts and scalar couplings of such well-resolved signals as methyl groups for 6-deoxy monosaccharides (fucose, quinovose, and rhamnose) at 6 1.1-1.3 ppm. 

QSAR methods can be divided into several categories dependent on the nature of descriptors chosen. In classical one-dimensional (ID) and two-dimensional (2D) QSAR analyses, scalar, indicator, or topological variables are examples of descriptors used to explain differences in the dependent variables. 3D-QSAR involves the usage of descriptors dependent on the configuration, conformation, and shape of the molecules under consideration. These descriptors can range from volume or surface descriptors to HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) energy values obtained from quantum mechanics (QM) calculations. 

The problem is treated as a two-dimensional one, with x-coordinate parallel to the surface and z-coordinate normal to the surface, with the z-axis negative into the solid. Since 3/3y and uy both vanish, the only non-zero component of the vector potential is fy. Suppose that there is a solution whose longitudinal and shear components each decay exponentially away from the surface, and that these are described by the scalar and vector potentials respectively. Then the potentials may be written... 

For the two-dimensional problem the body force must be purely in the two-dimensional plane. Therefore Vxf must be purely orthogonal to the plane for example, in the r-6 problem, it must point in the z plane. It can be shown that the vortex-stretching term vanishes under these conditions. As a result the vorticity-transport equation is a relatively straightforward scalar parabolic partial differential equation,... 

In order to obtain the ratio V = —fi/A of coefficients from the vector equation (12.12) in its most general form, let A) and B) represent arbitrary vectors in the two-dimensional space, corresponding to thermodynamic variables A, B, respectively. The successive scalar products of these vectors with (12.12) then lead to the linear equations... 

We have considered scalar, vector, and matrix molecular properties. A scalar is a zero-dimensional array a vector is a one-dimensional array a matrix is a two-dimensional array. In general, an 5-dimensional array is called a tensor of rank (or order) s a tensor of order s has ns components, where n is the number of dimensions of the coordinate system (usually 3). Thus the dipole moment is a first-order tensor with 31 = 3 components the polarizability is a second-order tensor with 32 = 9 components. The molecular first hyperpolarizability (which we will not define) is a third-order tensor. 

Thus the scalar product of vectors A and B in two-dimensional space is equal to the sum of the products of their components with no cross terms (e.g., AxBy). This result is actually only a special case of the general rule in p-dimensional space ... 

The general antidynamo theorem of Zeldovich is related to the fact that in the two-dimensional, singly-connected case, a field of divergence 0 is given by a scalar which is invariantly related to it (a streamline function or Hamilton function ). If the field is frozen into the fluid then the corresponding scalar is carried with the flow and, in particular, the integral of its square is conserved at D = 0 and decreases for D > 0, which is in fact why a dynamo is impossible. 

In two-dimensional motion, in the plane and spherical cases with scalar resistivity, a dynamo is impossible, and in the general two-dimensional case, if it is possible, it will prove to be slow (with a characteristic time tending to infinity for fixed l, v and for Rm — oo). For small Rm an effective dynamo is possible for two-dimensional motion as well (excluding the plane and spherical cases). 

Two-dimensional (2D) spectroscopy is used to obtain some kind of correlation between two nuclear spins 7 and J, for instance through scalar or dipolar connectivities, or to improve resolution in crowded regions of spectra. The parameters to obtain 2D spectra are nowadays well optimized for paramagnetic molecules, and useful information is obtained as long as the conditions dictated by the correlation time for the electron-nucleus interaction are not too severe. Sometimes care has to be taken to avoid that the fast return to thermal equilibrium of nuclei wipes out the effects of the intemuclear interactions that are sought through 2D spectroscopy. 

The symmetry group of a linear molecule of different atoms is CKV. It is well-known that its irreducible representations are all two-dimensional except the two unidimensional ones that contain scalars and pseudo-scalars. The basis functions for each two-dimensional representations Ek are 4> k = e lk[Pg.51]

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