Mathematical dimension

Two-Dimensional Descriptors are calculated on two different properties of a molecule, each of which is represented in a single mathematical dimension. The result is numerically a matrix and graphically a two-dimensional function, visualized in three dimensions. 

In an lOS approach the application of both the energy sudden and the centrifugal sudden" decoupling scheme reduces the exact Schroedinger equation for the atom-diatom scattering into a set of fixed angle of approach (7) two mathematical dimension ones. In the Body Fixed formulation these equations read as... 

Lieb E H and Mattis D C 1966 Mathematical Physics in One Dimension (New York Academic)... 

Molecules are usually represented as 2D formulas or 3D molecular models. WhOe the 3D coordinates of atoms in a molecule are sufficient to describe the spatial arrangement of atoms, they exhibit two major disadvantages as molecular descriptors they depend on the size of a molecule and they do not describe additional properties (e.g., atomic properties). The first feature is most important for computational analysis of data. Even a simple statistical function, e.g., a correlation, requires the information to be represented in equally sized vectors of a fixed dimension. The solution to this problem is a mathematical transformation of the Cartesian coordinates of a molecule into a vector of fixed length. The second point can... 

One of the most effective ways to think about optimization is to visualize how a system s response changes when we increase or decrease the levels of one or more of its factors. A plot of the system s response as a function of the factor levels is called a response surface. The simplest response surface is for a system with only one factor. In this case the response surface is a straight or curved line in two dimensions. A calibration curve, such as that shown in Figure 14.1, is an example of a one-factor response surface in which the response (absorbance) is plotted on the y-axis versus the factor level (concentration of analyte) on the x-axis. Response surfaces can also be expressed mathematically. The response surface in Figure 14.1, for example, is... 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

The geometrical problem. This involves evaluating the geometrical effect in item (2). It requires calculation of the volume of the overlapping regions as a function of d and the coil dimensions, say, r. The mathematics of this step are tedious and add little to the polymer aspects of the theory. 

Spray Correlations. One of the most important aspects of spray characterization is the development of meaningful correlations between spray parameters and atomizer performance. The parameters can be presented as mathematical expressions that involve Hquid properties, physical dimensions of the atomizer, as well as operating and ambient conditions that are likely to affect the nature of the dispersion. Empirical correlations provide useful information for designing and assessing the performance of atomizers. Dimensional analysis has been widely used to determine nondimensional parameters that are useful in describing sprays. The most common variables affecting spray characteristics include a characteristic dimension of atomizer, d Hquid density, Pjj Hquid dynamic viscosity, ]ljj, surface tension. O pressure, AP Hquid velocity, V gas density, p and gas velocity, V. ... 

Equations (12-31), (12-32), and (12-33) hold only for a slab-sheet solid whose thickness is small relative to the other two dimensions. For other shapes, reference should be made to Crank The Mathematics of Diffusion, Oxford, London, 1956). 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

On the base of the developed mathematical models was developed regression model of the atomizer efficiency via main design pai ameters such as linear dimensions and operation temperatures. 

The introduction of Lagrangian coordinates in the previous section allows a more natural treatment of a continuous flow in one dimension. The derivation of the jump conditions in Section 2.2 made use of a mathematical discontinuity as a simplifying assumption. While this simplification is very useful for many applications, shock waves in reality are not idealized mathematical... 

The reader is encouraged to use a two-phase, one spatial dimension, and time-dependent mathematical model to study this phenomenon. The UCKRON test problem can be used for general introduction before the particular model for the system of interest is investigated. The success of the simulation will depend strongly on the quality of physical parameters and estimated transfer coefficients for the system. 

Therefore, iiweknowthecompactedbulkdensity,thenitispossibletocomputethemass in thebed using the mathematical statement for the conservation of mass. In this case the reactoranditsphysical dimensions define thecontrol volume. The rateofcatalyst delivery is a constant that we will call min The rate of mass flow out of the reactor is zero, that is,... 

The complete mathematical definition of a particle size distribution is often cumbersome and it is more convenient to use one or two single numbers representing say the mean and spread of the distribution. The mean particle size thus enables a distribution to be represented by a single dimension while its standard deviation indicates its spread about the mean. There are two classes of means ... 

The theory of rate measurements by electrochemistry is mathematically quite difficult, although the experimental measurements are straightforward. The techniques are widely applicable, because conditions can be found for which most compounds are electroactive. However, many questionable kinetic results have been reported, and some of these may be a consequence of unsuitable approximations in applying theory. Another consideration is that these methods are mainly applicable to aqueous solutions at high ionic strengths and that the reactions being observed are not bulk phase reactions but are taking place in a layer of molecular dimensions near the electrode surface. Despite such limitations, useful kinetic results have been obtained. 

In recent years the mechanism of crevice has been mathematically modelled and a more thorough understanding of the corrosion processes has been evolved . From such mathematical modelling it is feasible to predict critical crevice dimensions to avoid crevice corrosion determined with relatively simple electrochemical measurements on any particular stainless steel. 

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