Separation constants

This is satisfied only when the separation constant is equal to an integer m = 0, 1, 2,. .. 

The mechanistic analysis of the rate of polymerization and the fact that the separate constants individually follow the Arrhenius equation means that... 

Constants and factors are often supplied by the manufacturer. Separate constants may be given for converting the stress and shear rate terms to the correct quantities and units. 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

Heat Balance—Adjacent Key Systems with Sharp Separations, Constant Molal Overflow... 

Leaving air temperatures should be above 35°C to avoid cold drafts. Where mixed systems of radiators and fan convectors are installed it is advisable to supply fan-assisted units on a separate constant-temperature circuit to avoid the above problems. To minimize stratiflcation, leaving air temperature above 50°C should be avoided. ... 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

The left-hand side of equation (2.28) is a function only of t, while the right-hand side is a function only of x. Since x and t are independent variables, each side of equation (2.28) must equal a constant. If this were not true, then the left-hand side could be changed by varying t while the right-hand side remained fixed and so the equality would no longer apply. For reasons that will soon be apparent, we designate this separation constant by E and assume that it is a real number. 

For conservation of mass it is required that 6 = 2. For the scaled equation (56) to be time-invariant, the separation constant (w) given by... 

The solution for T is rather similar to X with the constant c2A replacing A. Boundary conditions are used to select acceptable solutions from the infinite set. It can happen that one or more of these boundary conditions can be satisfied only when the separation constant takes on some special values. The subset so generated contains only permissible values, or eigenvalues, for the problem. The corresponding solutions are called eigenfunctions. 

The angle-dependent equation can be separated into two equations by introducing the separation constant m2,... 

The typical choices for the predefined trajectories or forces are constant velocity, including zero velocity, constant separation, constant forces, and/or... 

The x-dependent separation constant is the y-component of the wave-vector, and acts as an effective index profile, cf. figure 5,... 

Since the coefficients do not decrease with k for large k, this series will diverge for z = 1 unless it truncates at finite order. This truncation only happens if the separation constant X... 

Systems with well-separated constants and monotone relaxation 118... 

Ensembles with well-separated constants, formal approach 123... 

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

It is also proven that for cycles with well-separated constants damped oscillations are impossible, and spectrum of the matrix of kinetic coefficients is real. For general reaction networks with well-separated constants this property is proven in Section 4. 

In this section, we use some mathematics to define the multiscale ensembles with well-separated constants. This is necessary background for the analysis of systems with limitation, and technical consequences are rather simple. We need... 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

The log-uniform identical distribution of independent constants ki,...,k with sufficiently big interval of distribution ji—a)/ ) gives us the first example of ensembles with well-separated constants any two constants are connected by relation or with probability close to one. Such systems (not only cycles, but much more complex networks too) could be studied analytically "up to the end". 

If we consider one-parametric families of systems, then appearance of systems with two comparable constants may be unavoidable. Let us imagine a continuous path fc,(s) (s e [0,1], s is a parameter along the path) in the space of systems, which goes from one system with well-separated constants (s = 0) to another such system (s = 1). On this path fc,(s) such a point s that fc,(s) = fcy(s) may exist, and this existence may be stable, that is, such a point persists under continuous perturbations. This means that on a path there may be points where not all the constants are well separated, and trajectories of some constants may intersect. 

For systems with well-separated constants this expression can be simplified if ki kj then k] ki+ and if fc kj then k ki+,k i/kj. The first case, fc, ) > fc, is limitation in the small cycle (of length two) A, <- A,+] by the inverse reaction A, A,+]. The second case, fc, kj, means the direct reaction is the limiting step in this small cycle. 

So, influence of a single inverse reaction on the irreversible catalytic cycle with well-separated constants is determined by relations of three constants ki,kj and fc,+]. If kj is much smaller than at least one of fc,+i, then there is no influence in the main order. If kj k and k ki+ then the relaxation... 

In previous section, ensembles with well-separated constants appear. We represented them by a log-uniform distribution in a sufficiently big interval log ke[a, jS], but we were not interested in most of probability distribution properties, and did not use them. The only property we really used is if fcj >fcy, then ki/kj 1 (with probability close to one). It means that we can assume that ki/kj a for any preassigned value of a that does not depend on k values. One can interpret this property as an asymptotic one for a — co,p- co. 

The last transformation is convenient for estimation of the product for well-separated constants (compare to Equation (4)) ... 

---