Nonseparable problems

The Global Optimum Search GOS aimed at exploiting and invoking special structure for nonconvex nonseparable problems of the type (6.2). 

By this way, Aizermann successfully converted a linearly nonseparable problem to a very simple linearly separable problem to take the difference of electric fields at every point. It is easy to understand that potential functions other than Coulomb potential function are also applicable in this method. Aizermann also suggested the following function for the evaluation of the field strength around every sample point instead of Coulomb potential function ... 

The most general problem should be that of a particle in a nonseparable potential, linearly coupled to an oscillator heat bath, when the dynamics of the particle in the classically accessible region is subject to friction forces due to the bath. However, this multidimensional quantum Kramers problem has not been explored as yet. 

Now, if the many-body (electron) problem can be arranged in such a way that the many-body, nonseparable wave function is expressed in terms of a separable wave function, which depends on N single-particle wave functions (Hartree approximation), i.e.,... 

The orientation of bonds is strongly affected by local molecular motions, and orientation CF reflect local dynamics in a very sensitive way. However, the interpretation of multimolecular orientation CF requires the knowledge of dynamic and static correlations between particles. Even in simple liquids this problem is not completely elucidated. In the case of polymers, the situation is even more difficult since particules i and j, which are monomers or parts of monomers may belong to the same chain or to different Chains. Thus, we believe that the molecular interpretation of monomolecular orientation experiments in polymer melts is easier, at least in the present early stage of study. Experimentally, the OACF never appears as the complicated nonseparated function of time and orientation given in expression (3), but only as correlation functions of spherical harmonics... 

Making nonpermselective membranes for nonseparative applications is not a problem at all. Depending on the particular application, different properties are required for the membrane. First of all, the nonpermselective membrane can be either inert or catalyti-cally activated. 

However, in real life, many nonseparable (linear or nonlinear) classification problems occur, which practically means that distributions between two classes are overlapping. This implies that misclassihcations should be tolerated. Therefore, a set of slack variables > 0) is introduced in the margin minimization approach used for the linearly separable case, allowing some samples inside the margin. For this purpose. Equation 13.11 is replaced by Equation 13.12. 

By contrast, the strong electric field problem has appeared (perhaps prematurely) to be well understood. This view is reinforced by the fact that the Schrodinger equation for an atom in a strong electric field, although nonseparable in spherical polar coordinates (n and i are not good quantum numbers) does turn out to be separable in parabolic cylinder coordinates, given by... 

The decision problem is visualized in Figure 5.33 for a separable (a) and a nonseparable case (b), where the decision boundary... 

Figure 533 Separable case (a) and nonseparable (overlap) case (b) of the decision problem.

Taking the dimension of space as a variable has become a customary expedient in statistical mechanics, in field theory, and in quantum optics [12,17,18,85-87]. Typically a problem is solved analytically for some unphysical dimension D 3 where the physics becomes much simpler, and perturbation theory is employed to obtain an approximate result for D = 3. Most often the analytic solution is obtained in the D oo limit, and 1/D is used as the perturbation parameter. In quantum mechanics, this method has been extensively applied to problems with one degree of freedom, as reviewed by Chatterjee [60], but such problems are readily treated by other methods. Much more recalcitrant are problems involving two or more nonseparable, strongly- coupled degrees of freedom, the chief focus of the methods presented in this book. 

Here is the generalized Laplace operator, defined by equation (37), while R is the hyperradius (equation (5)). In a later chapter of this book. Professor Fano will discuss the application of the hyperspherical method to nonseparable dynamical problems. Here we shall only note that if mass-weighted coordinates axe used, the Schrodinger equation for any system interacting through Coulomb forces can be written in the form ... 

Although the H2 problem for D-dimensions is separable in spheroidal coordinates, just as for D = 3, since we want to examine the nonseparable situation, we employ cylindrical coordinates. In these coordinates the nuclei are located on the z-axis at —i /2 and -t-iJ/2, respectively, and the electron is at p,z). Dimensional scaling is introduced by using units of jZ bohr radii for distance and hartrees for energy, with Z the nuclear charge and k = D — l)/2. The scaled Schrodinger equation for H then takes a simple form. 

Although the starting point is clear, theoretical development is not easy since calculation of any average from eqn (5.9) involves a highly nonseparable integration which is common to many body problems. However, under certain conditions we can progress by using collective coordinates. ... 

The optimization problem from Eq. [20] represents the minimization of a quadratic function under linear constraints (quadratic programming), a problem studied extensively in optimization theory. Details on quadratic programming can be found in almost any textbook on numerical optimization, and efficient implementations exist in many software libraries. However, Eq. [20] does not represent the actual optimization problem that is solved to determine the OSH. Based on the use of a Lagrange function, Eq. [20] is transformed into its dual formulation. All SVM models (linear and nonlinear, classification and regression) are solved for the dual formulation, which has important advantages over the primal formulation (Eq. [20]). The dual problem can be easily generalized to linearly nonseparable learning data and to nonlinear support vector machines. 

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