Subsystem, defined

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

After a brief summary of the molecular and MO-communication systems and their entropy/information descriptors in OCT (Section 2) the mutually decoupled, localized chemical bonds in simple hydrides will be qualitatively examined in Section 3, in order to establish the input probability requirements, which properly account for the nonbonding status of the lone-pair electrons and the mutually decoupled (noncommunicating, closed) character of these localized a bonds. It will be argued that each such subsystem defines the separate (externally closed) communication channel, which requires the individual, unity-normalized probability distribution of the input signal. This calls for the variable-input revision of the original and fixed-input formulation of OCT, which will be presented in Section 4. This extension will be shown to be capable of the continuous description of the orbital(s) decoupling limit, when AO subspace does not mix with (exhibit no communications with) the remaining basis functions. 

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y)... 

The quantity TtlH) is the average kinetic energy of the subsystem defined specifically in terms of the density K(r) (eqns (5.48) and (5.49)) and the virial of the forces exerted on an electron in the subsystem resulting from its instantaneous interactions with the nuclei and other electrons in the system (see eqn (6.19))... 

Environments with stable energetic stresses are frequently divided into nearly decoupled spatial or compositional subsystems. Tliis is true of quasi-stable energetic redox couples at hydrothermal vents and of tlie weakly coupled 6000 K spectrum of solar visible light and 300 K terrestrial tliemial black body [55]. Tlie separate components may constitute internally near-equilibrium subsystems, defined individually by simple ensemble constraints. 

The same kind of arguments can be extended to consider in general the partitioning of the overall system of balance equations into k smaller subsystems (defining k equivalent subproblems). At the /th stage we will have... 

First we perform an energy balance around a subsystem defined by the cooler, throttling valve, and separator. This part of the process exchanges no heat of work with the surroundings and involves three streams, stream 4 going in and streams 7 and 9 coming out. 

Cooler/throttling/separator. The subsystem defined by the precooler, the throttling valve, and the separator exchanges no heat or work with the surroundings and it involves 3 streams (4,7, and 9) whose properties are known. The unknown flow rate x may be determined by application of the energy balance ... 

Figure 2. Pi and sigma subsystems of The pi subsystems define C-H pi bonding and the sigma subsystems C-H and C-C bonding. The signs of the MO overlap integrals within each sigma subsystem are such that the MO s form Mbbius arrays each of which accommodates four electrons.

CMPP is made up of four basic subsystems which form a data base that can be customized for any machine shop. (Fig. l.) The first subsystem defines the part model based on the completed part design, specification input, and raw material description. The user has the option of entering the part data interactively or in batch mode. The second subsystem defines the manufacturing logic based on user input of such items as the order and purpose of the operations. The third subsystem, for defining manufacturing resources, establishes information about shop facilities and shop rules and procedures. The final subsystem creates and displays the process plan. 

---