Records in fractal stochastic processes

Abstract

The records statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes, otherwise, it has no impact on the records statistics.

Hausdorff dimension of the record set of a fractional Brownian motion

“…Thus, the expected number of boxes containing a record scales as M ε ≈ N ε ε 1−H = ε −H , suggesting a fractal dimension H of the record set. This scaling is verified numerically in figure 2.1, as well as in [3].…”

“…suggesting a fractal dimension H of the record set. This scaling is verified numerically in figure 2.1, as well as in [3].…”

Effects of Antioxidant Supplements on the Survival and Differentiation of Stem Cells

“…Cryopreservation of spermatogonial stem cells, in the presence of catalase (CAT) and α -tocopherol ( α -TCP), promoted cell viability and suppressed apoptosis through inducing the expression of the antiapoptotic BcL-2 gene and inhibiting the expression of the proapoptotic BAX gene [74]. In other studies, cryopreservation with antioxidants could promote cell enrichment and increase the efficiency of colony formation in isolated SSCs [75, 76].…”

“…In other studies, cryopreservation with antioxidants could promote cell enrichment and increase the efficiency of colony formation in isolated SSCs [75, 76]. Spermatogonia-derived colonies showed increased SSC marker activity, enhanced expression of self-renewal genes, such as promyelocytic leukemia zinc finger (Plzf) protein and DNA-binding protein inhibitor ID4, and suppressed expression of the proto-oncogene (c-kit) in both CAT and α -TCP treated groups [74]. This technique can increase the possibility of SSCs banking for men with malignant diseases and promote the resumption of spermatogenesis in SCCs recipients.…”

Correspondence between a noisy sample-space-reducing process and records in correlated random events

“…In a recent Ref. [26], the records for fractional Brownian motion and fractional Gaussian noise have been studied. We note that the records statistics in these cases also behave in a similar manner to that of the survival time statistics for the noisy SSR or strictly SSR process, respectively.…”