Discrete properties

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. 

Polydispersity arises in systems composed of particles characterized by a property (e.g., particle diameter) that spans a continuum of values. Small molecules exhibit discrete properties, so they do not form polydisperse mixtures. Only at the level of macromolecules and colloidal aggregates does polydispersity become an issue. Here variations in particle size are known to influence the ordering into a solid phase. Experimentally it has been observed that colloidal systems will not form a solid phase if the size polydispersity (as measured by the standard deviation of the particle-size distribution) is greater than about 5% to 10% of the average diameter [252]. 

Gradient Material (FGM) presenting discrete properties variation on its thickness. It is then necessary to identify the requirements of each layer in order to define an architecture in agreement with membrane functionalities ... 

Nonlocal electrostatics takes into consideration discrete properties of solvent molecules. It assumes that fluctuations of solvent polarization are correlated in space. This means that the average polarization at each point is correlated with the electric displacement at all other points and therefore, uses the solvent dielectric... 

Both the basic types of linguistic unit, the phoneme and the word, are discrete. Phonemes are distinct from one another and form a finite and fixed set. The set of words is less fixed (we can create new words), but nevertheless all words are distinct from one another and are countable. What is remarkable about the discreteness property of language is that the primary signal of language, speech, is not discrete and hence there is a mismatch between form and signal. Speech signals are continuous functions of pressure over time (see Chapter 10) and so speaking is a process of... 

Par Parameswaran, V., Shukla, A. Dynamic fracture of a functionally gradient material having discrete property variation. J. Mater. Sci. 33 (1998) 3303-3311. 

One unique property of platelets is their stickiness [30]. Platelets adhere to each other and to the endothelium when discontinuity in the endothelium occurs. Stickiness must be distinguished from viscous metamorphosis and clot formation. Stickiness is a more discrete property of platelets, and it is not unlikely to be important in priming the homeostatic process and the formation of thrombi. 

In our further work performed in collaboration with Grosberg [76] we have considered the same hypothetical model system using the approach of equilibrium statistical mechanics that takes into account the discrete properties of the system. It was demonstrated that the puzzling result obtained in [75] is a consequence of two factors ... 

Figure 3 Structural alignments with discrete properties. Methods are based on discrete properties using the DG algorithm (1) or clique-detection (11) as implemented in distance comparisons (DISCO), and Apex-3D. The structure representation, based on discrete properties, resorts to one atomic descriptor (I), usually the atom type, or multiple atomic or site descriptors (II). In the first method (I), the conformational analysis is restricted to the generation of molecular geometries which allow a common arrangement of selected phaimacophoric moieties present in a rigid compound used as template. In the second method (II), the conformational analysis procedure may involve a systematic enumeration of all the possible conformadons for each ligand. The search similarity is directed towards the confirmation of a predefined pharmacophore postulated by the modeler or from some classical SAR in the case of the active analog approach (1), or the automated identification of pharmacophores and bioacdve conformations (II)...

More recently, the preparation of dendrimer 18, which contains three interior azobenzene units radially configured about the core unit, was reported. All four possible states ( , Z, ZZ, and ZZZ) with regard to the azobenzene chromophore were detected in solution and exhibited different physical properties. Thus, the dendrimer undergoes multiple discrete property changes upon an external stimulus of the interior groups. 

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