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Let (M; g) be a Riemannian manifold. Using classical Stokes’ theorem one can show that the equality (dω; η)L2 = (ω; δη)L2 holds for smooth forms ! and η with compact supports, where δ is the formal adjoint of d . There are some examples of Riemannian manifolds for which the above equality does not hold for general forms ! and η i:e: smooth square-integrable forms such taht d! and δη are also squareintegrable. In the case that the above equality holds for such general forms on a Riemannian manifold (M; g) , we say that the L2 - Stokes theorem holds for (M; g) . In 1952, Gaffney showed that the L2 - Stokes theorem holds for complete Riemannian manifolds. But at that time, there was no powerful... 

In this thesis, we first briefly review the basic concepts of stochastic processes. After reviewing and studying the dynamic equation that can explain a stochastic process, we show how one can find on a data-driven basis, the first-, second- and higher-order interactions between different subunits of a complex system by disentangling the dynamics of multivariate time series into stochastic and deterministic parts. Our data-driven approach is to detect different degrees of interactions obtained using conditional moments of Kramers-Moyal coefficients from unconditioned correlation functions and statistical moments of multivariate N-dimensional multivariate time series. Finally, we study the... 

Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy, sociology, and engineering. Critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change in the system exist in many subsystems in complex systems. These thresholds are called tipping points, and these subsystems are called tipping elements. Additionally, many systems with tipping points can be modeled as networks of coupled multistable subsystems. Domino-like tippings are called tipping cascades. Considering that these tipping cascades are primarily unprecedented, it is essential to study the dynamics and control of these...