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Statistical Properties of the T-Exponential of Isotropically Distributed Random Matrices

“…Thus, one cannot restrict the consideration by Gaussian approximation: non-Gaussianity is of crucial importance. The results achieved in this paper allow to calculate the Lyapunov spectrum based on experimental measurements of large-scale velocity statistics, then derive the statistic characteristics of velocity field at small scales in accordance with [13], and compare them to the observations. This work is supported by the RAS program 'Nonlinear dynamics in mathematical and physical sciences'.…”

“…This is very important for applications. In particular, in [13] the observed scaling space distribution of velocity and statistical properties of a turbulent flow were derived from the statistics of velocity deformation tensor at large scales based on a stochastic analog to Euler equation.…”

“…This is just what occurs in the theory of magnetic dynamo (see [6]) and in the theory of turbulence: the third-order correlator is related to the energy dissipation rate ( [7,8]), and the asymmetry of the Lyapunov spectrum is necessary to provide the time anisotropy of a turbulent flow ( [9,10]). Besides, the velocity gradient tensor, which is an important object in the problems related to passive scalar ( [11,12]) and in derivation of scaling exponents ( [9,13]), is proved to have non-Gaussian distribution ([14]). In this paper we propose a functional integration method for calculation the averages of continuous products of N × N random matrices A(t).…”

Formal model of evaluating the reliability of thermal power plants

“…Êàíîíè÷åñêîå ðàçëîaeåíèå (13) èìååò â îáùåì ñëó÷àå íååäèíñòâåííûé âèä è ôàêòè÷åñêè ôîðìàëèçóåò ïðÿìûå è îáðàòíûå ýêâèâàëåíòíûå ïðåîáðàçîâàíèÿ ìàòðèöû [25]. Èñïîëüçîâàíèå êàíîíè÷åñêîãî ðàçëîaeåíèÿ ïîçâîëÿåò ïîëó÷àòü â àíàëèòè÷åñêîì âèäå ìíîaeåñòâî ðåøåíèé, èìåþùèõ ìèíèìàëüíûé ðàíã.…”

“… íàó÷íîì èñòî÷íèêå [23] ïîêàçàíî, ÷òî â ýòîì ñëó÷àå ïðîáëåìà ïîëó÷åíèÿ åäèíñòâåííîãî ðåøåíèÿ çàäà÷è èäåíòèôèêàöèè çàìêíóòûõ îáúåêòîâ áóäåò õàðàêòåðíà äëÿ ëþáûõ âõîäíûõ ñèãíàëîâ íåçàâèñèìî îò íàëè÷èÿ èíôîðìàöèè î ïàðàìåòðàõ ñèñòåìû óïðàâëåíèÿ. Ýòîò ôàêò ïîêàçûâàåò, ÷òî íà ïðàêòèêå ïàðàìåòðè÷åñêèå ìåòîäû èäåíòèôèêàöèè èñïîëüçóþòñÿ ëèáî ñ ó÷åòîì àïðèîðíîé èíôîðìàöèè î ìîäåëè îáúåêòà, ëèáî ñ ïîìîùüþ ïîäà÷è òåñòîâûõ ñèãíàëîâ íåïîñðåäñòâåííî íà ÀÑÓ ÒÏ (îðãàí óïðàâëåíèÿ) [24,25].…”