###### On the integrability of the Hamiltonian systems with homogeneous polynomial potentials

1 Introduction In the theory of ordinary differential equations and in particular in the theory of Hamiltonian systems the existence of first integrals is important, because they allow to lower the dimension where the Hamiltonian system is defined. Furthermore, if we know a sufficient number of first integrals, these allow to solve the Hamiltonian system explicitly, and we say that the system is integrable. Almost until the end of the 19th century the major part of mathematicians and physicians believe that the equations of classical mechanics were

###### EMR-related problems at the interface between the crystal field Hamiltonians and the zero-field splitting Hamiltonians

. (1992). Electron paramagnetic resonance of d transition-metal compounds. Amsterdam: Elsevier. 9. Misra, S. K. (Ed.) (2011). Multifrequency electron paramagnetic resonance. Weinheim: Wiley-VCH. 10. Boča, R. (1999). Theoretical foundations of molecular magnetism. Amsterdam: Elsevier. 11. Buschow, K. H. J., & de Boer, F. R. (2003). Physics of magnetism and magnetic materials . New York: Kluwer Academic. 12. Boča, R. (2006). Magnetic parameters and magnetic functions in mononuclear complexes beyond the spin-Hamiltonian formalism. Struct

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Spin-Hamiltonian Parameters For Cr_{Li}
^{3+} Doped In LiNbO_{3}

## Abstract

By using crystal field theory, the spin-Hamiltonian parameters [zero-field splitting D, the giromagnetic factors g (*g*
_{||} and *g*
_{⊥}) and the first excited state splitting *δ*(^{2}
*E*)] for the Cr_{Li}
^{3+} doped in LiNbO_{3} have been calculated from the higher-order perturbation formulas. The method used is based on the two-spin-orbit coupling parameter model, in a cluster approach. The g parameters were also calculated as a second derivative of the energy, method implemented into ORCA computer program. The results were discussed and good agreement with experimental data was demonstrated.

###### Analysis of fractional factor system for data transmission in SDN

graph for every independent set I of G , a graph is a fractional independent-set-deletable ( a,b,m )-deleted graph (shortly, fractional ID-( a,b,m )-deleted graph). If a = b = k , then a fractional ID-( a,b,m )-deleted graph is a fractional ID-( k,m )-deleted graph. If m = 0, then a fractional ID-( a,b,m )-deleted graph is just a fractional ID-( a,b )-factor-critical graph. If G has a fractional ( g, f )-factor containing a Hamiltonian cycle, it is said that G includes a Hamiltonian fractional ( g, f )-factor. A graph G is called an ID-Hamiltonian

######
Electronic Levels Of Cr^{2+} Ion Doped In II-VI Compounds Of ZnS – Crystal Field Treatment

## Abstract

The aim of present paper is to report the results on the modeling of the crystal field and spin-Hamiltonian parameters of Cr^{2+} doped in II-VI host matrix ZnS and simulate the energy levels scheme of such system taken into account the fine interactions entered in the Hamiltonian of the system. All considered types of such interaction are expected to give information on the new peculiarities of the absorption and emission bands, as well as of non-radiative transitions between the electronic states of impurity ions. The obtained results were disscused, compared with similar obtained results in literature and with experimental data.

###### Poisson and symplectic reductions of 4–DOF isotropic oscillators. The van der Waals system as benchmark

1 Introduction The use of computer algebra systems for normal forms computations is considered at present a routine operation. As a general reference see e.g. Sanders et al . [ 36 ] and Meyer et al . [ 32 ]. Nevertheless when we deal with special classes of differential equations, like Poisson or Hamiltonians systems which is our case, it is advisable to employ specific transformations as well as tailored variables for those problems [ 32 ], mostly connected with the symmetries that those systems might possess. More precisely we are interested in

######
EMR study and superposition model analysis of Cr^{3+} and Fe^{3+} impurity ions in mullite powders used in aerospace industry

.1002/3527607358. 4. Rager, H., Schneider, H., & Graetsch, H. (1990). Chromium incorporation in mullite. Am. Miner. , 75 , 392–397. 5. Abragam, A., & Bleaney, B. (1986). Electron paramagnetic resonance of transition ions . New York: Dover. 6. Rudowicz, C. (1987). Concept of spin Hamiltonian, forms of zero-field splitting and electronic Zeeman Hamiltonians and relations between parameters used in EPR. A critical review. Magn. Reson. Rev., 13 , 1–89; (1988) Erratum, ibidem 13 , 335. 7. Rudowicz, C., & Misra, S. K. (2001). Spin-Hamiltonian formalisms in electron

###### Finding Electron-Hole Interaction in Quantum Kinetic Framework

## Abstract

The article presents a quantum kinetic framework to study interacting quan¬tum systems. Having the constituting model Hamiltonians of two-band semiconductor and the photoexcited electron-hole pair, their quantum kinetic evolution has been revi-sited. Solution to this nonlinear problem of electron-hole interaction is obtained making use of the self-consistency loop between the densities of photoexcited electrons and holes and the pairwise interaction terms in the constituting model Hamiltonians. In the leading order, this approach supports the required isomorphism between the pairwise interaction and the birth and annihilation operators of the photoexcited electrons and holes as a desirable property. The approach implies the Hilbert space and the tensor product mathematical techniques as an appropriate generalization of the noninteracting electron-hole pair toward several-body systems.

###### Eenergy Extraction from air or Water by Vibrations

://en.wikipedia.org/wiki/Lev_Pontryagin http://en.wikipedia.org/wiki/Pontryagin%27s_maximum_principle http://en.wikipedia.org/wiki/Hamiltonian_%28control_theory%29 Boltyanskii, V.G., Gamkrelidze, R.V., & Pontryagin, L.S. (1956). On the theory of optimum processes. Dokl. AN SSSR , 110, (1), 7-10 (in Russian). Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., & Mischenko, E.F. (1961). The Mathematical Theory of Optimal Processes , Moscow: Fizmatgiz (in Russian). Pontryagin, L.S., Boltyanskii, V

###### Note On The Schrödinger Equation

## Abstract

A second-order formalism leading to an equation describing the same dynamics as the Schrödinger one is developed under some compatible initial conditions.