Sungbin Lim

We study some Gaussian models for anomalous diffusion, which include the time-rescaled Brownian motion, two types of fractional Brownian motion, and models associated with fractional Brownian motion based on the generalized Langevin equation. Gaussian processes associated with these models satisfy the anomalous diffusion relation which requires the mean-square displacement to vary with t(alpha), 0<alpha<2. However, these processes have different properties, thus indicating that the anomalous dif

Data augmentation is an essential technique for improving generalization ability of deep learning models. Recently, AutoAugment \cite{cubuk2018autoaugment} has been proposed as an algorithm to automatically search for augmentation policies from a dataset and has significantly enhanced performances on many image recognition tasks. However, its search method requires thousands of GPU hours even for a relatively small dataset. In this paper, we propose an algorithm called Fast AutoAugment that find

Abstract. We derive rigorously explicit formulas of the Casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by zeta regularization method. We study both the low and high temperature expansions of the free energy. In each case, we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms. We show that the free energy is always a decreasing functio

The relationship between standard fractional Brownian motion (FBM) and FBM based on the Riemann-Liouville fractional integral (or RL-FBM) is clarified. The absence of stationary property in the increment process of RL-FBM is compensated by a weaker property of local stationarity, and the stationary property for the increments of the large-time asymptotic RL-FBM. Generalization of RL-FBM to the RL-multifractional Brownian motion (RL-MBM) can be carried out by replacing the constant Hölder exponen

We study some of the basic properties of a generalized Cauchy process indexed by two parameters. The application of the Lamperti transformation to the generalized Cauchy process leads to a self-similar process which preserves the long-range dependence. The asymptotic properties of spectral density of the process are derived. Possible application of this process to model relaxation phenomena is considered.

We consider three types of generalized Ornstein–Uhlenbeck processes: the stationary process obtained from the Lamperti transformation of fractional Brownian motion, the process with stretched exponential covariance and the process obtained from the solution of the fractional Langevin equation. These stationary Gaussian processes have many common properties, such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency lim

A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so‐obtained can be expressed explicitly in terms of multivariate Mittag‐Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag‐Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain