Separation constants equations

Since the first term contains no y-dependence and the second contains no x-dependence, both must actually be constant (these two constants are denoted Ex and Ey, respectively), which allows two separate Schrodinger equations to be written ... 

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

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

The advantage of the MTC model, as opposed to the CRK model, is that the exchange rate constant kp is no longer an empirical constant, but is now defined in terms of more fundamental processes that can be separately modeled. Equations 3.1 and 3.12 are linked via the equality... 

Since F(t) is only a function of time t, and P(qj) is only a function of the spatial coordinates qj, and because the left hand and right hand sides must be equal for all values of t and of qj, both the left and right hand sides must equal a constant. If this constant is called E, the two equations that are embodied in this separated Schrodinger equation read as follows ... 

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... 

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

For reaction networks with well-separated constants coordinates of left eigenvectors Z are close to 0 or 1. We can use the left eigenvectors for coordinate change. For the new coordinates z,- = Vc (eigenmodes) the simplest equations hold Zi = 2 Z . The zero-one law for left eigenvectors means that the eigenmodes are (almost) sums of some components z, = some sets of numbers Vj. 

Since dhk and L are constant, Equation (100) predicts that A is inversely proportional to dhk, with A the distance of the diffraction spot from the spot produced by the primary. Figure 9.14 shows that dhk is largest for planes of low Miller indices since A varies inversely with dhk, it follows that spots nearest the primary spot are due to low Miller index planes. Likewise, more distant spots are due to planes of higher index. There is a reciprocal relationship between the location of the spot on the photographic plate and the separation of the planes responsible for the spot. 

The only way this equation can be true is for both sides to equal a constant. If we call this separation constant E, we can write... 

In the theory of differential equations, the quantity e is called a separation constant. Here it is equal to the energy of the system. The latter equation can be instantly solved to give... 

Since equation (2.40i) which is similar to equation (2.38) does not contain y, it can be solved separately from equation (2.402). It is clear that under constant temperature and pressure conditions the thickness of the ApBq layer asymptotically tends with passing time to the maximally possible value (see Section 2.5)... 

Sturm-Liouville differential equations, resulting from a separation of variables have been known since the middle of the 19th century. Separation constants, subject to boundary conditions, yield sets of characteristic, or eigenvalue, solutions. 

We saw in section 3.3 that neither the orbital (TiL = R a P) nor the spin (1/2)7/a angular momenta commute with the Dirac Hamiltonian and are not therefore separate constants of motion, although their sum is. However, we can construct the mean orbital angular momentum and mean spin angular momentum operators in the same way as in equation (3.129). These operators are, respectively,... 

Here cti(/z) and ct2(F) are, respectively, the surface charge densities of plates 1 and 2, which are not constant but depend on the value of plate separation h. Equation (10.58), which was derived by Hogg et al., is called the Hogg-Healy-Fuerste-nau (HHF) formula. 

Here, s(P) is playing the role of the usual separation constant in this method of solving partial differential equations. The solution of Eq. (18) is immediately found to have the Gaussian form... 

In the course of developing models for the impedance response of physical systems, differential equations are commonly encountered that have complex variables. For equations with constant coefficients, solutions may be obtained using the methods described in the previous sections. For equations with variable coefficients, a numerical solution may be required. The method for numerical solution is to separate the equations into real and imaginary parts and to solve them simultaneously. 

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