Geometric space

Clearly, the incorporation of the cut and stick tailoring feature of projection operators into the resolvent-technique formulation, makes it a particularly adaptable modelling tool. Moreover, it enables the atomic structure of the geometric space to be reflected in the so-called site representation of the GF. 

Equivalent v-sites i have the same probability p, to be occupied by a dye molecule. The occupation probability p is equal to the ratio between the occupied and the total number of equivalent sites. The number of unit cells I1C is controlled by the host while ns is determined by the length of the guest, which means that p relies on purely geometrical (space-filling) reasoning and that the dye concentration per unit volume of a zeolite crystal can be expressed as a function of p as follows ... 

Two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe them have the same numerical value (n, = identical or idem). 

Dimensional analysis is a method for producing dimensionless numbers that completely characterize the process. The analysis can be applied even when the equations governing the process are not known. According to the theory of models, two processes may be considered completely similar if they take place in similar geometrical space and if all the dimensionless numbers necessary to describe the process have the same numerical value [2], The scale-up procedure, then, is simple express the process using a complete set of dimensionless numbers, and try to match them at different scales. This dimensionless space in which the measurements are presented or measured will make the process scale invariant. 

In what sense might a thermodynamic variable be considered as a vector, or the associated equilibrium state as a geometrical space of vectors ... 

Let us consider instead the possibility of a geometry-based description. In this framework, a thermodynamic variable is identified with a vector of the geometrical space, designated Ms, describing the equilibrium state S of interest. To distinguish this vector aspect notationally, we employ the Dirac ket symbol F) to denote the abstract geometrical vector that corresponds to (double-headed arrow) the calculus-based descriptors of the variable,... 

Distance-based metrics quantify the diversity of a set of compounds as a function of their pairwise (dis)similarities in a descriptor space. It is important to mention that distance coefficients are analogous to distances in multidimensional geometric space, although they are usually not equivalent to such distances. For a distance coefficient to be described as a metric, it must possess the following four properties (1) Distance values must be nonzero and the distance from an object to itself must be zero. (2) Distance values must be symmetric. (3) Distance values must obey the triangular inequality. (4) Distances between nonidentical objects must be greater than zero. A coefficient containing only the first three properties is dubbed a pseudometric, and one without the third property is a nonmetric. 

To evaluate the effectiveness of SAGE in terms of diversity selection and chemical space coverage, several simulated datasets were used. For instance, in a geometrical space (2D or lOOD), a certain number of cluster centers were generated, which were more than a preset distance away from each other. Some 99 cluster centers were then generated in a 2D space, and 95 cluster centers were generated in a lOOD space. Then, a random number (between 1 and 100) of points for each cluster were generated around each cluster center within a cutoff distance, so that no members from two different clusters could overlap. This led to two simulated data sets one with 951 points distributed in 99 clusters in 2D space, and second with 950 points distributed in 95 clusters in lOOD space. 

Fixing the position of the basis functions to the nuclei allows for a compact basis set, otherwise sets of basis functions positioned at many points in the geometrical space would be needed. When comparing energies at different geometries, however, the nuclear fixed basis set introduces an error. The quality of the basis set is not the same at all geometries, owing to the, fact that the electron density around one nucleus may be... 

A set of numbers representing the states or observables is called a representation. In the geometrical space of three-dimensional vectors r a set of numbers representing r is the set of three coordinates x, y, z in a system of orthogonal axes. We may consider the system of unit vectors x, y, il in the directions of the axes as a basis for the representation of r. A coordinate is the scalar product of r with one of the unit vectors. A different basis is provided by a rotated set of axes. A vector is changed into a new vector by operating with a 3 x 3 matrix. This concept is easily extended to the spaces of quantum mechanics. 

In tableting applications, the process scale-up involves different speeds of production in what is essentially the same unit volume (die cavity in which the compaction takes place). Thus, one of the conditions of the theory of models (similar geometric space) is met. However, there are still kinematic and dynamic parameters that need to be investigated and matched for any process transfer. 

A radius-diameter diagram is defined as a bivariate distribution of the - data set compounds in the space defined by the molecular radius and diameter it provides a summary of the similarities among the molecule chemical shapes in the topological or geometrical space. 

The Hausdorff chirality measure is a chirality measure of the second class [Buda and Mislow, 1992]. Let Q and Q denote two enantiomorphous, nonempty, and bounded sets of points defined in the geometrical space (x,y,z). Let d(q,q ) denotes the distance between two points G Q and qi G Q. Then, the Hausdorff distance h between sets Q and Q is defined as... 

Such weaker relationships are usually represented in the physical sciences by so-called Hasse or chain diagrams. These are constructed by depicting set members, s , as points, a , in some geometrical space and then connecting pairs of points that satisfy the relationship R. Chain diagrams have been widely employed in the chemical context to order members of chemical series, such as... 

The real space of the naturalist coincides with the physical medium, in which the phenomena he deals with take place. It does when he expresses the phenomena geometrically. If a naturalist is talking about a real space of nature, he is talking about the geometrical structure of a physical medium. There is no ideal geometrical space for the naturalist. It would be real for him, if the observations showed that space is isotropic, homogeneous everywhere". 

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