The article is devoted to the research of economic cycles and crises, which are studied within the framework of the theory of long waves by N. D. Kondratiev (K-waves) using the mathematical apparatus of fractional calculus. The object of the study is the mathematical model of S.V. Dubovsky 3 and some of its modifications, which describe the dynamics of the economic system, characterizing the interaction between the efficiency of new technologies and the efficiency of capital productivity. Modifications in the model are determined by the following points: the mathematical model takes into account the dependence of the rate of accumulation on capital productivity, there is an external harmonic influence - the influx of investments and new technological solutions, the economic system has heredity, including time-dependent, which leads to a delayed effect - the system’s response to impact. The property of heredity in the mathematical model is taken into account using fractional derivatives of constant and variable orders. Mathematical model S.V. Dubovsky and its modifications are studied numerically using the Adams-Bashforth-Moulton algorithm. Oscillograms and phase trajectories are constructed, and an interpretation of the simulation results is given.

The aim of this report is to evaluate the segmental rib index (RI) from T1-T12 spinal level in mild and moderate idiopathic scoliosis (IS) curves of thoracic, thoracolumbar and lumbar type by gender. The relationship of segmental RI to the frontal plane radiological deformity presented as the Cobb angle and to the posterior truncal surface deformity presented as the scoliometric readings of Angle of Trunk Rotation (ATR) in these patients is also assessed. Any statistical significant rela-tionship of these parameters would be very important for biomechanical relations of the rib cage (RC) deformity presented as the rib hump deformity (RHD) and the deformity in the spine and it would thus provide valuable information about scoliogeny. The segmental rib index (RI) is presented in 83 boys and girls with mild and moderate IS. The measurements include the scoliometric readings for truncal asymmetry (TA), the Cobb angle assessment and the segmental RI from T1-T12. The statistical package SPSS 23 was used for statistical analysis. The TA is documented and the Cobb angle is presented by gender and curve type. The segmental RI of thoracic, thoracolumbar and lumbar curves are presented for the first time. The correlations of the segmental RI to surface deformity presented as the rib hump deformity (RHD) in all IS patients and particularly in thoracic curves, to Cobb angle by gender and age and the comparison of the segmental RI index of asymmetric but not scoliotic children to the scoliotic peers by curve (in thoracic, thoracolumbar, lumbar curves) in boys and girls are presented. The findings emphasize the significant protagonistic role of the thoracic asymmetry in relation to the spinal deformity mainly in girls for the thoracic and in boys for the thoracolumbar curves. The cut-off point of age of the examined scoliotics is 14 years, that is when the RI shows stronger correlation with the spinal deformity, namely when the thoracic deformity is decisively effective to the developing thoracic spinal deformity, in terms of Cobb angle. In summary, the results of this study provide aetiological implications for IS, as far as the role of thorax is con-cerned.