Continuum matrix

The simplest (though approximate) solution of Eq. (10.11) is obtained by assuming that all the continua are flat, that is, that the bound-continuum matrix elements vary slowly with energy and can be replaced by their value at some average energy, say El = Ei + fi(oh... 

Thus, interference effects between bound-continuum matrix elements will play an important role in determining the magnitude of the susceptibility. However, since these terms vary slowly with energy, the tuning range for harmonic generation will still be very large. 

So far, the combination of a large susceptibility with a wide region of wavelength tunability over the ionization continuum seems to be unique to magnesium vapor. Two factors are important. First, there is constructive interference in the bound-continuum matrix elements, which reflect the continuum electronic structure. Second, configuration interaction with the even parity doubly excited np series brings extra two-photon transition... 

Using the model of Eq. (15) for phofoionization, Greenland and Lane [115] studied the effect of laser flucfuafions (i.e., fluctuations in discrete to continuum matrix elements). They obtained for the decay rate the usual result, InW-, averaged over the bandwidth and also argued, based on arguments similar to those in Ref. [13], fhat laser fluctuations eliminate the post-exponential region unless the transition is fast on the timescale of the fluctuations. 

Heat Transfer from Nanoparticles to the Continuum Matrix... 

Here, the atomistic-continuum model is applied to the study of plastic deformation of BPA-PC. First, the elastic constants of BPA-PC are calculated by atomistic simulation. These values are used as the elastic constants for the matrix throughout the simulated deformation. Then, the system, i.e., the continuum matrix with its atomistic inclusion, is deformed stepwise up to a strain of about 0.2. The overall system is constrained to exactly follow a predescribed deformation sequence, but the atomistic inclusion is free to follow any strain path consistent with the misfit stresses acting between it and the matrix. The result is a new look at the behavior of a glassy polymeric inclusion deformed plastically in a surrounding elastic medium. 

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. 

Many kinds of transition probabilities depend on DOs. Photoionization cross sections, are proportional to the absolute squares of matrix elements between DOs and continuum orbitals, or... 

It can be noted that other approaches, based on irreversible continuum mechanics, have also been used to study diffusion in polymers [61,224]. This work involves development of the species momentum and continuity equations for the polymer matrix as well as for the solvent and solute of interest. The major difficulty with this approach lies in the determination of the proper constitutive equations for the mixture. Electric-field-induced transport has not been considered within this context. 

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

Choosing the continuum inhomogeneity differently yields the alternative K -matrix normalized wave functions ... 

The continuum electron-phase shifts induced by the short-range scattering off the chiral molecular potential are most conveniently introduced by a third choice of continuum function, obtained by diagonalizing the K-matrix by a transformation U, resulting in a set of real eigenchannel functions (apart from normalization) [41] ... 

At a fundamental level, it has been shown that PECD stems from interference between electric dipole operator matrix elements of adjacent continuum f values, and that consequently the chiral parameters depend on the sine rather than the cosine of the relative scattering phases. Generally, this provides a unique probe of the photoionization dynamics in chiral species. More than that, this sine dependence invests the hj parameter with a greatly enhanced response to small changes in scattering phase, and it is believed that this accounts for an extraordinary sensitivity to small conformational changes, or indeed to molecular substitutions, that have only a minimal impact on the other photoionization parameters. 

The K-matrix method is essentially a configuration interaction (Cl) performed at a fixed energy lying in the continuum upon a basis of "unperturbed funetions that (at the formal level) includes both diserete and eontinuous subsets. It turns the Schrodinger equation into a system of integral equations for the K-matrix elements, which is then transformed into a linear system by a quadrature upon afinite L basis set. 

Since these formal bases, which are supposed to describe the true continuum background, will be represented upon finite sets, all the qnantities which must be interpolated from these representations (i.e. matrix elements and phaseshifts) must be smooth functions of the energy index this reqnires a snitable redefinition of the channel hamiltonian Hp if this supports narrow shape resonances. 

Finally, another alternative to continuum regression has been put forward by Wise and de Jong [18]. Their continuum power-PLS (CP-PLS) method modifies the matrix X = USV into X " = i.e. the singular values are raised to a... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Correlation matrix, linear thermodynamics, regression theorem, 17-20 Coupled cluster (CC), ab initio calculations, P,T-odd interactions, 254-259 Coupled continuum, two-pathway excitation, coherence spectroscopy isolated resonances, 168-169 structureless excitation, 167 CPT theorem ... 

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