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In accordance with (21) the frequencies of collective modes of oscillations of dot magnetic moments, are defined by a rather cumbersome formulae 

<img src='content_image/350669.jpg'>

but an analysis of the dispersion relation can be done numerically. The dependence of ω ( ⃗ q ) for the two branches, ω$_{(}$$_{−}$$_{)}$ ( ⃗ q ) and ω$_{(+)}$ ( ⃗ q ), for specific values of a magnetic field are presented in Fig. 5. 

<img src='content_image/350671.jpg'>

The given expression is essentially simplified at symmetrical points of the Brillouin zone (in the zone center and on the edges), where the expression 

becomes zero at these symmetrical point. 

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