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  • Expansion is a canonical white noise expansion

    2018-11-09

    Expansion is a canonical white-noise expansion in terms of a wavelet basis, i.e., it is a sum of random quantities multiplied by determinate time-varying functions, in this case, wavelet ones. Amiantov\'s study proved, in particular, the assumption that the probability density functional takes the form for a flat spectrum of a stationary random process. It becomes apparent from analyzing the derivation of formula in Ref. by reading it in reverse order that a flat spectrum can be obtained provided the process energy is constant over a large period of time and the elements of the correlation matrix are uncorrelated . It is important for the matrix to be constructed for the cross-sections of the random process taken at the instants of time corresponding to Kotelnikov\'s sampling theorem. When solving practical problems (computer simulation of random processes), it is expedient to use those wavelets that are well-localized both by time and by frequency. These wavelets were discussed in Refs. , and some others. In order to expand real random processes, the real part of the Morlet wavelet: (where ω is the parameter) or the real part of the Paul wavelet: (where is the parameter) can be taken as mother wavelets. Let us also note that if we take interferences of the type examined by Kotelnikov in Ref. , expansion can be written in the following way:
    Formula is similar to the one obtained by Kotelnikov , but differs in that the trigonometric basis has been replaced by a wavelet one, and by the double summation present that can be substituted (taking into account the well-known properties of enumerable sets known in mathematics) with summation over one index.
    Introduction The data on the photoabsorption and radiation spectra of atoms offer clues to understanding their structure and the processes occurring when they interact with the electromagnetic field. It should be noted that buy UM 171 shell structure can be described only through a theory that would adequately interpret the experimental data. The emergence of quantum mechanics, the formulation of the Schrödinger wave equation and its generalizations to many-electron atoms in the form of the Hartree–Fock equations (HF) all once played a key role in understanding the structure of electrons and molecules [1–3]. The HF single-particle approximation allows computing wave functions and energies of the ground and the excited states of many-electron systems. However, this approach has significant disadvantages. The eigenvalues of the Hartree–Fock Hamiltonian mean the ionization energies of the respective electron shells, but, as the computational results indicate, these energies are always substantially different from the corresponding experimental values. Additionally, the excited states are computed in the combined field of the frozen core and the vacancy formed. This means that the wave function of an excited electron is computed without taking into account the rearrangement of the self-consistent field due to a hole appearing in the structure of the electron core and as well as the polarization of the electron core by the knock-on electron. We should note that the methods presented in this study can be applied to describing the optical properties of the individual many-electron atoms as well as of the more complex structures. The atomic system of units is used in this study:
    Theoretical approach A photoabsorption cross-section of an atom is described by the following formula [2]: where F is the highest occupied level (the Fermi level) of the system in its ground state,is the matrix element of the atomic electron transition from the state i to the state v under the effect of the external electromagnetic field in a dipole approximation. Random Phase Approximation (RPAE) [2,3,5,6] turned out to be one of the most effective approaches to constructing the operator of the interaction between the atom and the external field; this method allows taking into account many-electron correlations. The photoabsorption cross-sections were first successfully computed in this approximation for a considerable number of atoms with a good agreement between the numerical results and the experimental data. RPAE is based on the assumption that an excited electron is not immediately transferred to the final excited state but instead passes through a number of intermediate short-lived (virtual) ones. The virtual excited states of the electron–hole type interact with the real excitation by the Coulomb field with the exchange interaction allowed for.