Separate variables theorem

Since the vibrational Hamiltonian is a sum of terms, each of which involves only one coordinate, the separation-of-variables theorem of Sec- ... 

If a solution is found for the initial and boundary conditions, there is a uniqueness theorem that justifies the assumption. Whether a solution can be found using separation of variables depends on whether the boundary conditions follow the symmetry of the separation variables. 

This method relies on the theorem of separate variables, which is stated as follows. 

The method applies the theorem of separated variables, which is defined as if two curves representing function y with variable x obtained for two values of parameter a are subtracted from each other by an orthogonal affinity of axis Ox, orientation Oy and ratio k in the expression of the function, variable x is separated. Hence, if A is independent from x, we can write ... 

In an analogous fashion to the atomic Hartree-Fock equations, the angular variables can be separated and integrated out using the Wigner-Eckart theorem in the Dirac equation to yield a set of coupled differential equations depending on r (29). 

Proof We first note that this theorem is simply a restatement of the stationarity of the action with respect to the density matrix expressed now in terms of the variables of our interest upon using the relation Eq. (53). By separating the operator H, into parts that contain external field terms and those that do not, by referring to Eq. (49) and using the definitions given in Eq. (41), we reexpress the action as a functional of the variables introduced above. 

Separability theorem, 309 SHAKE algorithm, 385 SHAPES force field, 40 Simulated Annealing (SA), global optimization, 342 Simulation methods, 373 Supidfiidiil, iulcs, 3j6 Susceptibility, 237 Symbolic variables, for optimizations, 416 Symmetrical orthogonalization of basis sets, 314 Symmetry adapted functions, 75 Symmetry breaking, of wave functions, 76 ... 

Even though the variables of the differential Equation 6 are not separable in the exact sense, a condition for separability can be imposed by application of the variation theorem... 

In Figure 2, the distribution of each variable for each group is plotted along with a bivariate scatter plot of the data and it is clear that the two groups form distinct clusters. However, it is equally evident that it is necessary for both variables to be considered in order to achieve a clear separation. The problem facing us is to determine the best line between the data clusters, the discriminant function, and this can be achieved by consideration of probability and Bayes theorem. 

Consider that one of the main advantages of the Laplace transform technique is that it can be used for time dependent boundary conditions, also. The separation of variables technique cannot be directly used and one has to use DuhameFs superposition theorem[l] for this purpose. Consider the modification of example 8.7 ... 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

When we deal with a dynamical system in which a sensitive state occurs (assumptions of the Grobman-Hartman theorem are not fulfilled), it may turn out that the sensitive state is associated only with a part of state variables. The variables related to the sensitive state may then be separated and the catastrophes occurring in a system dependent on a smaller number of state variables examined. 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

In the above equations s is the scaled streamwise coordinate, n the scaled normal coordinate, A=An/An with An denoting the normalized separation between streamlines constituting a streamtube, is the normalized radius of curvature of the streamlines,6 is the direction of flow velocity, and the rest of the variables are the same as those of the previous section. Moreover, for use in nearly frozen flows the momentum theorem. ... 

Dimensional Analysis The flow phenomena in an ejector are complex, and hence, it is not possible to predict the flow rate of the secondary fluid a priori. In conventional chemical engineering, such phenomena are treated by a dimensional analysis of the dependent and independent variables based on the Buckingham % theorem. In the present case, the dependent variable is the rate of entrainment of the secondary fluid, and the independent variables are the physical properties, ejector configuration, and operating parameters. The latter are defined mainly by the flow rate of the primary fluid. Ben Brahim et al. (1984) gave the following separate dependences for the primary and secondary fluids ... 

The theorem may be generalized to more than two variables, that is, if the Hamiltonian is a sum of three or more terms, each acting on functions of a single variable. Variable 1 is separated first and subsequently variable 2, from the others. 

Prom here on it makes sense to consider systems which are separable in some point coordinate, so that the Schrodinger eigenfunctions can be written as products of functions which depend on only one variable each. If the system remains separable under a change of coordinates, then the following theorem is valid the energy is a monotonous function of each of the quantum numbers, when the quantum numbers are interpreted as the number of zeros of the factors of the eigenfunctions which depend on only one coordinate. FVom this it follows immediately that terms of which the separate quantum numbers differ by 1 do not cross for the adiabatic changes considered here. 

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