Discrete sets

The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

The energies of the various contributions are quantised , i.e., in a given state the isolated molecule may possess one of a discrete set of values these values are often referred to as energy levels. When a molecule absorbs light, its energy is momentarily increased by an amount equal to that of the photon. The energy is related to the wave length (X) and frequency (v) by the equation ... 

Dcri c appropriate reference doses for each discrete set of conditions. 

Comparisons of LC and SFC have also been performed on naphthylethylcar-bamoylated-(3-cyclodextrin CSPs. These multimodal CSPs can be used in conjunction with normal phase, reversed phase, and polar organic eluents. Discrete sets of chiral compounds tend to be resolved in each of the three mobile phase modes in LC. As demonstrated by Williams et al., separations obtained in each of the different mobile phase modes in LC could be replicated with a simple CO,-methanol eluent in SFC [54]. Separation of tropicamide enantiomers on a Cyclobond I SN CSP with a modified CO, eluent is illustrated in Fig. 12-4. An aqueous-organic mobile phase was required for enantioresolution of the same compound on the Cyclobond I SN CSP in LC. In this case, SFC offered a means of simplifying method development for the derivatized cyclodextrin CSPs. Higher resolution was also achieved in SFC. 

It has been suggested (Shull and Lowdin 1955) that, instead of the set III.56, one should use the complete discrete set... 

On the theoretical side, Marcelja [26] was first to account explicitly for flexible tail chains in nematic ordering, using the Maier-Saupe model potential (Eq. 1) for each segment of the molecule. More complex models were proposed by Samulski et al. [27] and Emsley et al. [28]. In these approaches alkyl chains are assumed to exist in a discrete set of conformers described by... 

Within the framework of the present chapter the domain G -f F of continuous variation of an argument (point) is replaced by some discrete set of points (nodes) known as a grid. 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

In general, the expectation value A) of the observable A may be written for a discrete set of eigenfunctions as... 

If the eigenkets ia) constitute a discrete set, we may expand the state vector W) as... 

Equation (53) has to be solved numerically. The field < )(r) at time t is represented by the discrete set < ), -)t(t) on a lattice, and the time evolution is obtained by performing iterations over t. The initial conditions are generated by assigning the average volume fraction < )0 to each lattice point, and the interface is defined as < )(r) < )0. An example of the interface evolution for symmetric... 

Producing a reasonably good accuracy for analytically defined surfaces, this scheme of calculation is very inaccurate when the field is specified by the discrete set of values (the lattice scalar field). The surface in this case is located between the lattice sites of different signs. The first, second, and mixed derivatives can be evaluated numerically by using some finite difference schemes, which normally results in poor accuracy for discrete lattices. In addition, the triangulation of the surface is necessary in order to compute the integral in Eq. (8) or calculate the total surface area S. That makes this method very inefficient on a lattice in comparison to the other methods. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

The main goal of this report is to present a phenomenon of highly general nature manifested in various dynamical systems. We present the occurrence of peculiar quantization by the parameter of intensity of the excited oscillations and we show that given unchanging conditions it is possible to excite oscillations with a strictly defined discrete set of amplitudes the rest of the amplitudes being forbidden . The realization of oscillations with a specific amplitude from the permitted discrete set of amplitudes is determined by the initial conditions. The occurrence of this unusual property is predetermined by the new general initial conditions, i.e. the nonlinear action of the external excited force with respect to the coordinate of the system subjected to excitation. 

The performed analysis shows that the continuous wave having a frequency much larger than the frequency of a given oscillator can excite in it oscillations with a frequency close to its natural frequency and an amplitude belonging to a discrete set of possible stable amplitudes. 

The phenomenon of continuous oscillation excitation with an amplitude belonging to a discrete set of stationary amplitudes has been demonstrated on the basis of a common model - an oscillator under wave influence. It is shown that the conditions necessary for the manifestation of this phenomenon are realized in a natural way in an oscillator system interacting with a continuous fall wave. 

A stochastic program is a mathematical program (optimization model) in which some of the problem data is uncertain. More precisely, it is assumed that the uncertain data can be described by a random variable (probability distribution) with sufficient accuracy. Here, it is further assumed that the random variable has a countable number of realizations that is modeled by a discrete set of scenarios co = 1,..., 2. 

The effect of DP-2) is to produce a virtual pulse whose length is the width of the central lobe. Of course, this is never completely perfect since it does have side-lobes, but waveforms have been described for which the performance in this respect is excellent. DP-3) means that the Doppler frequency shift is being sampled at a discrete set of time points. If the sampling rate is faster than the Nyquist of the Doppler frequency shift, then the Doppler can be unambiguously extracted. 

As the dynamics of the system are removed from the model, it is no longer necessary to allow the molecules to live in a continuous space. Instead, the use of lattices - discrete sets of coordinates on to which the molecules are restricted - is popular. Digital computers are of course much more efficient with discrete space than with continuum space. The use of a lattice implies that one removes all properties that occur on shorter length scales than the lattice spacing from the model. This is no problem if the main interest is in phenomena that are larger than this length scale. 

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