Minimization principle

These principles are phrased in the language of the ionic model, but they provide a simpler and more explicit description of stable structures than that given by the ionic model s energy minimization principle. Among the important ideas captured by Pauling s rules are those of local charge neutrality, the definition of electrostatic bond strength, and the rule of parsimony which is closely... 

The corresponding formulation was made by von Neumann2 for quantum mechanics. This entropy-maximizing (or information-minimizing) principle is the most direct path to the canonical distribution and thus to the whole equilibrium theory. It is understood that the extremalizing is conditional, i.e., certain expected values, such as that of the Hamiltonian, are fixed. 

Just by considering equation (4) one may speculate that the NACTs might be similar to the electromagnetic vector potential, S. It is known from classical mechanics that the momentum p of a charged particle in an electromagnetic field changes to p — p + eS - a substitution termed as the minimal principle [1]. Due to the correspondence principle the quantum mechanical minimal principle becomes V—>- V+ i(e/fi)S. However, the NACTs in equation (4), when considering each element separately, do not combine with V (because the... 

T-matrix is an anti-symmetric matrix) and therefore do not fulfill the minimal principle. Nevertheless, to see how the minimal principle can be fulfilled for the NACTs we consider the two-state case ... 

Due to this 2X2 constant transformation the minimal principle is fulfilled V— V it12 and (h/e) t12 is recognized as a vector potential. 

The similarity between the two final expressions as derived in two different frameworks is a surprise because the physical systems are so different. However, the question is if we should be surprised. After all the basic assumption in both frameworks is the minimal principle which states that in order to employ the ordinary SE for charged particles all one has to shift V by V—>> V+ i(e/h)S where S is a vector potential. 

As a final comment we would like to mention that the derivations as presented here apply for time-dependent perturbation caused, e.g. by (strong) electric fields which are characterized by having a zero vector potential. In case the perturbation is caused by magnetic fields [33] the corresponding vector potential has to be included according to the minimal principle. 

There is some evidence in support of the view that an electron-domain s effective volume, if not its shape, is approximately transferable from one system to another. Compress a Sidgwick-type unshared electron on one side and it appears to expand elsewhere, particularly on the opposite (trans) side of the kernel, much as one might expect from the form of the kinetic energy operator and the energy minimization principle, which, taken together, require smooth changes in electron density, within a domain. 

Li, Jinghai, Reh, Lothar, and Kwauk, Mooson. Application of energy minimization principle to hydrodynamics of circulating fluidized beds, in Circulating Fluidized Bed Technology III (P. Basu, M. Horio and M. Hasatani, eds.), p. 163. Pergamon Press, 1991a. 

The strength of Density Functional Theory (DFT) [120-122] is based on the fact that the electronic wavefunction Fo (ri,..., r r) of the electronic ground state of the system can be entirely described only by its electron density n°(r), as stated by the Hohenberg-Kohn theorem [123]. It is based on a minimization principle stating that the ground state electronic density minimizes an energy functional. Thus in principle we have to consider and manipulate a much simpler object, the electronic density, which is simply a function of R. ... 

The main advantage of SVM over other data analysis methods is its relatively low sensitivity to data overfitting, even with the use of a large number of redundant and overlapping molecular descriptors. This is due to its reliance on the structural risk minimization principle. Another advantage of SVM is the ability to calculate a reliability score, R-value, which provides a measure of the probability of a correct classification of a compound [70], The R-value is computed by using the distance between the position of the compound and the hyperplane in the hyperspace. The expected classification accuracy for the compound can then be obtained from the 7 -value by using a chart which shows the statistical relationship between them. As with other methods, SVM requires a sufficient number of samples to develop a classification system and irrelevant molecular descriptors may reduce the prediction accuracies of the SVM classification systems. 

To illustrate the equivalence of the energy minimization principle advanced above and the linear elastic equilibrium equations of section 2.4.2, we resort to evaluating the functional derivative of eqn (2.77) and setting it equal to zero, or in mathematical terms 8Y /8ui = 0. Equivalently, working in indicial notation, we find... 

In 1892 Helmholtz inquired whether we...can cast the empirically known laws of electrodynamics, as they are formulated in Maxwell s equations, in the form of a minimal principle [44]. Indeed such a minimal principle exists in the form of the principle of least action For a system of n degrees of freedom there exists a Lagrangian L qi,qi,t) such that the action integral... 

The path of mechanical systems has been described by extremal principles. We emphasize the principles of Fermat Hamilton. The principle of least action is named after Maupertui but this concept is also associated with Leibnitz, Euler, and Jacobi For details, cf. any textbook of theoretical physics, e.g., the book of Lindsay [16, p. 129]. Further, it is interesting to note that the importance of minimal principles has been pointed out in the field of molecular evolution by Davis [17]. So, in his words... 

The normal vector n, is defined by the vector gradient of Cg, which can be derived from different finite difference approximations which directly influence the accuracy of algorithms. These include Green-Gauss, volume-average, least-squares, minimization principle, and Young s gradients. It is noted that a wide, symmetric stencil for n,y is necessary for a reasonable estimation of the interface orientation. 

In this situation, a generalized Young equation can be arrived at by applying the energy minimization principle to obtain... 

Rigorous application of waste minimization principles can often solve the problems of managing mixed waste. Minimization of mixed waste can be achieved by modifying laboratory processes, improving operations, or using substitute materials. Such efforts are most successful when scientists and environmental health and safety staff work together to evaluate laboratory processes. Examples include the following ... 

The oblique Lanczos algorithm formulated for RPA219 was further improved using the Thouless variational principle (eq D3).22022i resulting stable Lanczos procedure efficiently solves RPA eigenvalue problem and is described below. It it convenient to work in the space of coordinate-momentum variables q = X+ Yand p = X — Y(eq B9) where the Thouless minimal principle eq D3 is given by... 

Intensification, or minimization (principle 1 listed above), consists of reducing the quantities of hazardous chemicals in the plant or combining unit operations (also known as telescoping operations). "What you don t have can t leak" is the bus word. Reduced inventory of methyl isocyante at the Bhopal Union Carbide plant could have reduced the intensity of the accident. 

Like the energy minimization principle, here we have a function W n), the maximum of which identifies the value of n that we are most likely to observe. To maximize W or nW for a scries of coin flips, we compute dW(n)/dn for hxed N, and find the value n = n that causes the derivative to equal zero. The result is that n = N/2. In this case the number of heads n acts like a degree of freedom of the system, n can take on any value from zero to N. Any individual sequence is just as likely as any other. The reason that the 50% heads, 50% tails composition is so strongly favored is that there are more distinguishable sequences of coin flips with that composition than any other. No individual coin flip has any particular tendency, or feels any force. As w e ll see, this is the nature of entropy as a driving force. Systems tend to be found in the states with the highest multiplicity. The maximization of W or In W serves the same predictive role for the composition of coin flips as the minimization of energy serves for the position of the ball in the valley. 

In summary, the previously given form of the minimal principle of entropy production leads to a class of generalized Onsager constitutive theory, which is also direct generalization of the linear Onsager s theory having their dissipation potentials as homogeneous Euler s functions. 

The Minimization Principle associated with Laplace s Equation... 

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