Two-dimensional similarity

Chemical Structures on the Basis of Two-Dimensional Similarity Measures. 

We present here one result obtained by this method. The system is two dimensional, similar to the system described in Section 6.11. Here, we have waterlike particles at a number density of = 0.65 and a simple solute with a number density of qs = 0 05. The other molecular parameters for this particular computation are... 

Betelh S.I. and Diez J.A. (1999). A two-dimensional similarity solution for capillary driven flows. Physica D, 126,136-140. 

In order to ensure perpendicular beam incidence on the cylindrical specimen, the circular B-scan profiles were acquired by high frequency (narrow beam) transducers in a synthetic circular aperture array. From these profiles two-dimensional reflection tomograms were reconstructed using a filtered backprojection technique. Straight line propagation was assumed. Several artificial discontinuity types in a cylindrical Plexiglas (Perspex) specimen were compared with similar artificial discontinuities in a cylindrical A/Si-alloy [2]. Furthermore, examples of real discontinuities (an inclusion and a feed head) in the cylindrical AlSi-alloy are presented. 

It must be remembered that, in general, the constants a and b of the van der Waals equation depend on volume and on temperature. Thus a number of variants are possible, and some of these and the corresponding adsorption isotherms are given in Table XVII-2. All of them lead to rather complex adsorption equations, but the general appearance of the family of isotherms from any one of them is as illustrated in Fig. XVII-11. The dotted line in the figure represents the presumed actual course of that particular isotherm and corresponds to a two-dimensional condensation from gas to liquid. Notice the general similarity to the plots of the Langmuir plus the lateral interaction equation shown in Fig. XVII-4. 

The Ag (100) surface is of special scientific interest, since it reveals an order-disorder phase transition which is predicted to be second order, similar to tire two dimensional Ising model in magnetism [37]. In fact, tire steep intensity increase observed for potentials positive to - 0.76 V against Ag/AgCl for tire (1,0) reflection, which is forbidden by symmetry for tire clean Ag(lOO) surface, can be associated witli tire development of an ordered (V2 x V2)R45°-Br lattice, where tire bromine is located in tire fourfold hollow sites of tire underlying fee (100) surface tills stmcture is depicted in tlie lower right inset in figure C2.10.1 [15]. 

The model consists of a two dimensional harmonic oscillator with mass 1 and force constants of 1 and 25. In Fig. 1 we show trajectories of the two oscillators computed with two time steps. When the time step is sufficiently small compared to the period of the fast oscillator an essentially exact result is obtained. If the time step is large then only the slow vibration persists, and is quite accurate. The filtering effect is consistent (of course) with our analytical analysis. Similar effects were demonstrated for more complex systems [7]. 

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

SONNIA can be employed for the classification and clustering of objects, the projection of data from high-dimensional spaces into two-dimensional planes, the perception of similarities, the modeling and prediction of complex relationships, and the subsequent visualization of the underlying data such as chemical structures or reactions which greatly facilitates the investigation of chemical data. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

Using a irocedure similar to the formulation of two-dimensional forms the working equations of the U-V - P scheme in axisymmetric coordinate systems are derived on the basis of Equations (4.10) and (4.11) as... 

The situation is similar for a linear curve fit, except that now the data set is two-dimensional and the number of degrees of freedom is reduced to (n — 2). The analogs of the one-dimensional variance )/( 1) the standard... 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

Other Individual Country Databases and Auxiliary Files. The USPatents files on ORBIT, supphed by Derwent, are similar ia their contents to the CLAIMS-Bibliographic files, including all the front page information and the full claims language. These files do not iaclude the two-dimensional stmctures provided by IFI, nor do they have IFI s standardization of assignee names. Citation searching is available at a cost considerably less than that for the portion of the CLAIMS-Citation file that covers the same period. 

Similar treatments can be used for all sorts of two-dimensional problems for calculating the plastic collapse load of structures of complex shape, and for analysing metal-working processes like forging, rolling and sheet drawing. 

The radial-inflow turbine has many eomponents similar to those of a eentrifugal eompressor. Flowever, the names and funetions differ. There are two types of radial-inflow turbines the eantilever radial-inflow turbine and the mixed-flow radial-inflow turbine. Cantilever blades are often two-dimensional and use nonradial inlet angles. There is no aeeeleration of the... 

Figure 18.20 The two-dimensional NMR spectrum shown in Figure 18.17 was used to derive a number of distance constraints for different hydrogen atoms along the polypeptide chain of the C-terminal domain of a cellulase. The diagram shows 10 superimposed structures that all satisfy the distance constraints equally well. These structures are all quite similar since a large number of constraints were experimentally obtained. (Courtesy of P. Kraulis, Uppsala, from data published in P. Kraulis et ah. Biochemistry 28 7241-7257, 1989, by copyright permission of the American Chemical Society.)...

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