Space dimension

Table 7.4 Oi dor of phase transition and threshold probabilities versus space dimension for rules R, ...Rn, as determined by mean-field theory and numerical calculation ([bidaux89a], [bidaux89b]).

Overall boiler design, including steam and water space dimensions, boiler rating, pressure, and separation equipment arrangements. 

To show [115] that Liouville s theorem holds in any number of phase-space dimensions it is useful to restate some special features of Hamilton s equations,... 

In one space dimension, neglecting temperature gradients relative to concentration gradients, this can be rewritten... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

Palmes, E. D., and M. Lippmann. Influence of respiratory air space dimensions on aerosol deposition. In W. H. Walton, Ed. Inhaled Particles IV. Proceedings of the Fourth International Symposium on Inhaled Particles and Vapors, British Occupational Hygiene Society, Edinburgh. September, 1975. London Pergamon Press, Ltd. (in press)... 

Equations (if.4) and (ff.S) are solved, along with the continuity equation (which does not change upon nondimensionalization), in a Cartesian coordinate system using the Fourier-Galerkin (spectral) technique under periodic boundary conditions in all three space dimensions. The scheme is similar to that used by Orszag [8] for direct solution of the incompressible Navier-Stokes equations. More details can be found in [9] and [7], and the scheme may be considered to be pseudospectral. ... 

There are inherent scale limitations in the time and space dimensions covered by laboratory studies. The applicability of the near field geochemical models derived from laboratory observations have to be applied to long-term, large-scale situations like the ones involved in the safety assessment of nuclear waste repositories. Hence, there is a need to test the models developed from laboratory investigations in field situations that are related to the ones to be encountered in repository systems. 

The analysis of plane waves is straightforward in several respects. As soon as we begin to consider waves varying in more than one space dimension, however, we will encounter new phenomena that further complicate the analysis. This also applies to the superposition of elementary modes to form wavepackets. In this section an attempt is made to investigate dissipation-free axially symmetric modes in presence of a nonzero electric field divergence... 

The equivalent result in 3-space dimensions has been given by Evans and Jeffers... 

In this chapter, we will give a brief introduction to the basic concepts of chemoinformatics and their relevance to chemical library design. In Section 2, we will describe chemical representation, molecular data, and molecular data mining in computer we will introduce some of the chemoinformatics concepts such as molecular descriptors, chemical space, dimension reduction, similarity and diversity and we will review the most useful methods and applications of chemoinformatics, the quantitative structure-activity relationship (QSAR), the quantitative structure-property relationship (QSPR), multiobjective optimization, and virtual screening. In Section 3, we will outline some of the elements of library design and connect chemoinformatics tools, such as molecular similarity, molecular diversity, and multiple objective optimizations, with designing optimal libraries. Finally, we will put library design into perspective in Section 4. 

In other words, the natural representation of G on Hoiiv (V. W) is trivial. Still, ( V, W) does carry important information. In Section 6.4 we will find the vector space dimension of HomoCV, W) to be useful. 

Equation (316) should be compared with eqn. (44). It is second order because it involves the second space derivative V2, partial because of the three space dimensions and time (independent variables), inhomogeneous because the term J (r, t) is taken to be independent of p(r, t), linear because only first powers of the density p appear, and self adjoint in efic/p(r, t), the importance of which we shall see in the next section [491, 499]. The homogeneous equation corresponding to eqn. (316) has a solution p0 (r, t), which satisfies the same boundary conditions as p... 

For the arbitrary space dimension d the relation e0 oc D/l2 is still valid. Taking into account that V oc ld, (2.1.105) could be generalized ... 

A more refined approach is based on the local description of fluctuations in non-equilibrium systems, which permits us to treat fluctuations of all spatial scales as well as their correlations. The birth-death formalism is applied here to the physically infinitesimal volume vo, which is related to the rest of a system due to the diffusion process. To describe fluctuations in spatially extended systems, the whole volume is divided into blocks having distinctive sizes Ao (vo = Xd, d = 1,2,3 is the space dimension). Enumerating these cells with the discrete variable f and defining the number of particles iVj(f) therein, we can introduce the joint probability of arbitrary particle distribution over cells. Particle diffusion is also considered in terms of particle death in a given cell accompanied with particle birth in the nearest cell. 

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