Degenerate solutions

The reduced costs also indicate if there are multiple optima. Let all Cj 0 and let ck = 0 for some nonbasic variable xk. Then, if the constraints allow that variable to be made positive, no change in/results, and there are multiple optima. It is possible, however, that the variable may not be allowed by the constraints to become positive this may occur in the case of degenerate solutions. We consider the effects of degeneracy later. A corollary to these results is the following ... 

The first equation is scalar, and has a wave solution with velocity Vi = -J c /p). This is the longitudinal wave of eqn (6.7). It is sometimes called an irrotational wave, because V x u = 0 and there is no rotation of the medium. The second equation is vector, and has two degenerate orthogonal solutions with velocity v = s/(cu/p)- These are the transverse or shear waves of eqn (6.6) the degenerate solutions correspond to perpendicular polarization. They are sometimes called divergence-free waves, because V u = 0 and there is no dilation of the medium. Waves in fluids may be considered as a special case with C44 = 0, so that the transverse solutions vanish, and C = B, the adiabatic bulk modulus. 

Now consider the situation when there are degenerate solutions to the equation Hv Ev (8-2.16)... 

Note that any given value of o (or prescribed difference between the two diagonal elements) results in two different sets of degenerate solutions in Eq. (9), one of which should be unphysical (for more on this see Ref. [14]). 

Now, to have degenerate solutions (i.e., an unavoided crossing), the discriminant must vanish, and it is necessary to satisfy two independent conditions ... 

For each final energy Ef = Ei + hu there are several possible dissociation channels represented by degenerate solutions of the nuclear Schro-dinger equation the corresponding wavefunctions are distinguished by the boundary conditions at large distances. 

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

The index 5 here denotes a particular degenerate solution, specified by boundary conditions, at given total energy E. The iV-electron function p is a target state with... 

If there are n0 open channels at energy E, there are n linearly independent degenerate solutions of the Schrodinger equation. Each solution is characterized by a vector of coefficients aips, for i = 0,1, defined by the asymptotic form of the multichannel wave function in Eq. (8.1). The rectangular column matrix a consists of the two n0 x n0 coefficient matrices ao, < i Any nonsingular linear combination of the column vectors of a produces a physically equivalent set of solutions. When multiplied on the right by the inverse of the original matrix a0, the transformed a-matrix takes the canonical form... 

Persistent degeneracies can also be caused by poor preprocessing. In such a case, degenerate solutions are observed even for a (too) low number of components. Extracting too many components can also give a degeneracy-like problem because the noise components can be correlated. Some details about degeneracies can be found in Section 6.3. 

An indication of degenerate solutions can thus be obtained by monitoring the correlation between all pairs of components. For three-way PARAFAC, the measure in Equation (5.22) can be calculated as... 

Finally, some results are presented for Tucker3 models and the problem of degenerate PARAFAC solutions is discussed. Such degenerate solutions can be encountered in practice and sometimes have to do with postulating the wrong model. 

Usually, several low energy solutions will remain, but it is often possible to design experiments that can differentiate between them. On the other hand, if there are many degenerate solutions to the model, we can conclude that the model is underdetermined. Finally, distance geometry s random sampling of models often produces surprises, which occasionally lead to new ideas. 

Chemistry teachers are encouraged to accept this vague picture at face value and to believe that Px,Py,Pz occurs as a three-fold degenerate solution of Schrodinger s equation for the H electron. It does not. 

To obtain a similar expansion of the matrix elements // (q) of ft we pick as an electronic basis set the two wave functions M1 61 qo) and j(rel q0) which are degenerate solutions of (5) at q = q0 and therefore diagonalize at this value of q but not, in general, at other values. It should be noted that these functions differ from those used in... 

These linear combinations still have the same energy as the original complex wavefunctions. This is a general property of degenerate solutions of the Hamiltonian operator. The reason why they are labelled 2p and 2p is that in polar coordinates the Cartesian coordinates x, y and z have the same angular dependence as the orbitals in Figure 2.3 ... 

General Case of n-fold Degenerate Solutions of the Srhrodingcr Equation 113... 

Degenerate solutions to the critical DSR conditions given by Equation 7.19a and 7.19b occur when a DSR equilibrium point is reached (when this occurs, r(C, T) and v are collinear and Equation 7.19a and 7.19b produce the null vector). These points represent solutions to a nonisothermal CSTR equation. However, one must also account for CSTR point as starting points for PER and critical DSR trajectories. Additional conditions must be satisfied in this situation. In order for a CSTR to be a connector on the AR boundary for a nonisothermal system in IR, it may be shown that the tangent vector of the CSTR locus be coplanar with either the mixing or rate vector as follows ... 

Like all other meshless methods, the first step in GFD is to scatter nodal points in the computational domain and along the boimdary. To each node (point), a collection of neighboring nodes are associated which is called star. The number and the position of nodes in each star are decisive factors affecting the finite difference approximation. Particular node patterns can lead to ill-conditioned situations and ultimately degenerated solutions. Using the Taylor s series expansion, the value of any sufficiently differentiable smooth function u at the central node of star, uq, can be expressed in terms of the value of the same function at the rest of nodes, with i = 1,. .N where N is the total number of neighboring nodes in the star and is one less than the total number of nodes in it. In two dimensions, a second-order accurate Taylor series expansion can be written as... 

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