External coordinates discrete

The disperse phase is constituted by discrete elements. One of the main assumptions of our analysis is that the characteristic length scales of the elements are smaller than the characteristic length scale of the variation of properties of interest (i.e. chemical species concentration, temperature, continuous phase velocities). If this hypothesis holds, the particulate system can be described by a continuum or mean-field theory. Each element of the disperse phase is generally identified by a number of properties known as coordinates. Two elements are identical if they have identical values for their coordinates, otherwise elements are indistinguishable. Usually coordinates are classified as internal and external. External coordinates are spatial coordinates in fact, the position of the elements in physical space is not an internal property of the elements. Internal coordinates refer to more intimate properties of the elements such as their momenta (or velocities), their enthalpy... 

Abstract The metal-driven construction of multi-porphyrin assemblies, which exploits the formation of coordination bonds between exocyclic donor site(s) on the porphyrins and metal centers, has recently allowed the design and preparation of sophisticated supra-molecular architectures whose complexity and function begin to approach the properties of naturally occurring systems. Within this framework, meso-pyridyl/phenyl porphyrins (PyPs), or strictly related chromophores, can provide geometrically well-defined connections to as many as four metal centers by coordination of the pyridyl peripheral groups. Several discrete assemblies of various nuclearities, in which the pyridylporphyrins are connected through external coordination compounds, have been constructed in recent years. In this review, we summarize recent work in this field from our and other laboratories. The photophysical properties of some ruthenium-mediated assemblies of porphyrins prepared by our group are also described. 

The particles of interest to us have both internal and external coordinates. The internal coordinates of the particle provide quantitative characterization of its distinguishing traits other than its location while the external coordinates merely denote the location of the particles in physical space. Thus, a particle is distinguished by its internal and external coordinates. We shall refer to the joint space of internal and external coordinates as the particle state space. One or more of either the internal and/or external coordinates may be discrete while the others may be continuous. Thus, the external coordinates may be discrete if particles can occupy only discrete sites in a lattice. There are several ways in which the internal coordinates may be discrete. A simple example is that of particle size in a population of particles, initially all of uniform size, undergoing pure aggregation, for in this case the particle size can only vary as integral multiples of the initial size. For a more exotic example, let the particle be an emulsion droplet (a liquid) in which a precipitation process is carried out producing a discrete number of precipitate particles. Then the number of precipitate particles may serve to describe the discrete internal coordinate of the droplet, which is the main entity of population balance. 

Continuous variables may be encountered more frequently in population balance analysis. They often arise as a natural solution to dealing with indefinite or variable discreteness. For example, a particle-splitting process where the products of splitting could conceivably have any size smaller than the parent particle is most naturally handled by assigning particle size as a continuous variable. The external coordinates denoting the position vector of (the centroid of) a particle describing continuous motion through space represent continuous variables. The temperature of a particle in a fluidized bed is another example of a continuous variable. 

One of the most detailed discrete simulations in the literature is that of Zeitlin and Tavlarides (1972a,b,c), which addressed the evolution of drop size distributions in mechanically agitated liquid-liquid distributions produced in batch, semibatch, and continuous flow vessels. In these simulations, the flow vessel was divided into distinct spatial regions among which individual droplets commuted with specified velocities. Thus, both internal and external coordinates characterized the individual particle state. Drop... 

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities the phenomenon of discrete oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. 

The main goal of this report is to present a phenomenon of highly general nature manifested in various dynamical systems. We present the occurrence of peculiar quantization by the parameter of intensity of the excited oscillations and we show that given unchanging conditions it is possible to excite oscillations with a strictly defined discrete set of amplitudes the rest of the amplitudes being forbidden . The realization of oscillations with a specific amplitude from the permitted discrete set of amplitudes is determined by the initial conditions. The occurrence of this unusual property is predetermined by the new general initial conditions, i.e. the nonlinear action of the external excited force with respect to the coordinate of the system subjected to excitation. 

In the field of supramolecular chemistry, metals are known as key components of self-assembled molecular architectures with discrete structures as well as metal-based chemical and physical properties. Metal coordination is directional noncovalent bonding, and hence plays an important role in determining the relative arrangement of ligands. Furthermore, its reversible nature allows us to design molecular switching or motion that shows a dramatic change in structure and hence function by external perturbation. 

Compared with the very limited capacity of the more or less discrete control electronics of classic TCSPC systems, the use of FPGAs led to a breakthrough in functionality. Advanced TCSPC devices use a multidimensional histogramming process. They record the photon density not only as a function of the time in the signal period, but also of other parameters, such as wavelength, spatial coordinates, location within a scanning area, the time from the start of the experiment, or other externally measured variables. It is actually the multidimensionality and flexibility in the data acquisition process that makes new TCSPC techniques really advanced . 

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