Dimensionality definition

These two equations are a special case of the corresponding tliree-dimensional definition, connnon in XRD, with the surface nonnal n replacing the third lattice vector c. 

This is essentially the multi-dimensional definition of slope. It describes how changes in u depend on changes in x, y, and z. Note that we use du to examine systematic errors but (du) to examine random errors. 

Lab-supports should promote the development in industry of commercial supports with better dimensional definition and surface quality, so that the extra development steps needed on the high-quality lab-supports can be omitted. 

It is therefore natural to generalize the nonrelativistic three-dimensional definition of velocity v = dr/dtby the so-called 4-velocity given by... 

There are many different definitions and dimensions of quality to be found in books and academic literature. We will present three of these definitions selected from published literature and propose a three-dimensional definition of quality. 

As it has been shown lately, the notion nanoparticle (nanocluster) gets well over the limits of purely dimensional definition and means substance state specific character in sizes nanoscale. The nanoparticles, sizes of which are within the range of order of 1 100 nm, are already not classical macroscopic objects. They represent themselves the boundary state between macro and microworld and in virtue of this they have specific features number, to which the following ones are attributed ... 

Key to the microscopic description of ET used in several MD simulations is a definition of the reaction coordinate. Because solvent fluctuations control the probability of electron transfer, this coordinate must be a function of solvent configuration. A useful one-dimensional definition, used extensively in simulations, is the energy gap when the system is in either of the two electronic... 

It is useful to define the tenns coverage and monolayer for adsorbed layers, since different conventions are used in the literature. The surface coverage measures the two-dimensional density of adsorbates. The most connnon definition of coverage sets it to be equal to one monolayer (1 ML) when each two-dimensional surface unit cell of the unreconstructed substrate is occupied by one adsorbate (the adsorbate may be an atom or a molecule). Thus, an overlayer with a coverage of 1 ML has as many atoms (or molecules) as does the outennost single atomic layer of the substrate. 

It is thus tempting to define the first saturated layer as being one monolayer, and this often done, causing some confiision. One therefore also often uses tenns like saturated monolayer to indicate such a single adsorbate layer that has reached its maximal two-dimensional density. Sometimes, however, the word saturated is omitted from this definition, resulting m a different notion of monolayer and coverage. One way to reduce possible confiision is to use, for contrast with the saturated monolayer, the tenn fractional monolayer for the tenn that refers to the substrate unit cell rather than the adsorbate size as the criterion for the monolayer density. 

The traditional definition of a zeolite refers to microporous, crystalline, hydrated aluminosilicates with a tliree-dimensional framework consisting of comer-linked SiO or AlO tetrahedra, although today the definition is used in a much broader sense, comprising microporous crystalline solids containing a variety of elements as tetrahedral building units. The aluminosilicate-based zeolites are represented by the empirical fonmila... 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

We must now mention, that traditionally it is the custom, especially in chemo-metrics, for outliers to have a different definition, and even a different interpretation. Suppose that we have a fc-dimensional characteristic vector, i.e., k different molecular descriptors are used. If we imagine a fe-dimensional hyperspace, then the dataset objects will find different places. Some of them will tend to group together, while others will be allocated to more remote regions. One can by convention define a margin beyond which there starts the realm of strong outliers. "Moderate outliers stay near this margin. 

Pastor M, G Cruciani and S dementi 1997. Smart Region Definition A New Way to Improve tl Predictive Ability and Interpretability of Three-Dimensional Quantitative Structure-Activi Relationships. Journal of Medicinal Chemistry 40 1455-1464. 

Proc., 65(3), 45 (1986)] is preferred. To use this and alternate models, dimensional characteristics of structured packing must be defined. Figure 14-51 shows nomenclature and definitions of key dimensions. Not shown, but also important, is the angle the corrugations make with the horizontal (usu y 45 or 60°). Then the Rocha et al. predictive equation is ... 

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

In vertical downward flow as well as in upward and downward inclined flows, the flow patterns that can be observed are essentially similar to those described above, and the definitions used can be applied. Experimental data on flow patterns and the transition boundaries are usually mapped on a two dimensional plot. Two basic types of coordinates are generally used for this mapping - one that uses dimensional coordinates such as superficial velocities, mass superficial velocities, or momentum flux and another that uses dimensionless coordinates in which some kind of dimensionless groups are used as coordinates. The dimensional coordinates maps are inherently limited to the range of data and flow conditions under which the experiments were conducted. In spite of this limitation, it is widely used because of its simplicity and ease of use. Figure 24 provides an example of such a map. 

Frequency with the dimensions of per unit time, ranges from zero to infinity and means the number of occurrences per time interval. Probability is dimensionless, ranges from zero to one, and has several definitions. The confusion between frequency and probability arises from the need to determine the probability that a given system will fail in a year. Such a calculation of probability explicitly considers the time interval and, hence, is frequency. However, considerable care must be used to ensure that calculations are dimensionally correct as well as obeying the appropriate algebra. Three interpretations of the meaning of probability are ... 

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