Abstract:
Complex systems are sometimes subject to non Gaussian alpha stable Levy fluctuations. A new method is devised to estimate this uncertain parameter and other system parameters, using observations on either mean exit time or escape probability for the system evolution. It is based on solving an inverse problem for a deterministic, non-local partial differential equation via numerical optimization. The existing methods for estimating parameters require observations on system state sample paths for long time periods or probability densities at large spatial ranges. The method proposed here, instead, requires observations on mean exit time or escape probability only for an arbitrarily small spatial domain. This new method is beneficial to systems for which mean exit time or escape probability is feasible to observe.

This study aimed to explore Chinese teachers’ emotion
regulation goals and strategies used before, in, and after classroom teaching.
Thirty-four teachers from elementary, middle and high schools were interviewed
with semi-structure questionnaire. Chinese teachers’ goals for regulating
emotions included achieving instructional goals, decreasing the negative impact
of emotions on student learning, confirming the professional and ethical norms,
maintaining teachers’ and students’ mental health, keeping positive emotional
images, and nurturing good teacher-student relationships. Teachers used various
antecedent-focused and response-focused strategies to control their emotions
before, in, and after class. In general, Chinese teachers used
response-modulation most frequently, followed by cognitive changes in and after
classroom teaching. These findings have implications for productive delivery of
education service, teacher training and policy-making.

Abstract:
The research on the propagation of seismic waves in anisotropic media is an important topic in the research of seismic exploration. In this paper, the optimal nearly-analytic discrete method of high precision and low dispersion was adopted in vertical transverse isotropic (VTI) medium of two-dimensional elastic wave equation numerical simulation. Based on the elastic wave equations in two dimensional VTI of the medium, the paper began with a discussion of the significance of the physical parameters of Thomsen, and then a different equation of elastic wave equation was established with the optimal nearly-analytic discrete method (ONAD). Finally, through the method of numerical simulation of a uniform and layered VTI model, the author focused on the research of snapshots and ground seismic records of different Thomsen parameters. In this paper, not only was the impact of the value of Thomsen parameters on the elastic wave propagation analyzed, but also the propagation laws of elastic wave in VTI medium were discussed, which have provided some reference and basis for practical seismic exploration.

Abstract:
In bacteriorhodopsin, the order of molecular events that control the cytoplasmic or extracellular accessibility of the Schiff bases (SB) are not well understood. We use molecular dynamics simulations to study a process involved in the second accessibility switch of SB that occurs after its reprotonation in the N intermediate of the photocycle. We find that once protonated, the SB C15 = NZ bond switches from a cytoplasmic facing (13-cis, 15-anti) configuration to an extracellular facing (13-cis, 15-syn) configuration on the pico to nanosecond timescale. Significantly, rotation about the retinal’s C13 = C14 double bond is not observed. The dynamics of the isomeric state transitions of the protonated SB are strongly influenced by the surrounding charges and dielectric effects of other buried ions, particularly D96 and D212. Our simulations indicate that the thermal isomerization of retinal from 13-cis back to all-trans likely occurs independently from and after the SB C15 = NZ rotation in the N-to-O transition.

Abstract:
A goal of data assimilation is to infer stochastic dynamical behaviors with available observations. We consider transition phenomena between metastable states for a stochastic system with (non-Gaussian) $\alpha-$stable L\'evy noise. With either discrete time or continuous time observations, we infer such transitions by computing the corresponding nonlocal Zakai equation (and its discrete time counterpart) and examining the most probable orbits for the state system. Examples are presented to demonstrate this approach.

Abstract:
Due to lack of scientific understanding, some mechanisms may be missing in mathematical modeling of complex phenomena in science and engineering. These mathematical models thus contain some uncertainties such as uncertain parameters. One method to estimate these parameters is based on pathwise observations, i.e., quantifying model uncertainty in the space of sample paths for system evolution. Another method is devised here to estimate uncertain parameters, or unknown system functions, based on experimental observations of probability distributions for system evolution. This is called the quantification of model uncertainties in the space of probability measures. A few examples are presented to demonstrate this method, analytically or numerically.

Abstract:
The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian $\alpha-$stable symmetric L\'evy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.

Abstract:
This paper is devoted to exploring the effects of non-Gaussian fluctuations on dynamical evolution of a tumor growth model with immunization, subject to non-Gaussian {\alpha}-stable type L\'evy noise. The corresponding deterministic model has two meaningful states which represent the state of tumor extinction and the state of stable tumor, respectively. To characterize the lifetime for different initial densities of tumor cells staying in the domain between these two states and the likelihood of crossing this domain, the mean exit time and the escape probability are quantified by numerically solving differential integral equations with appropriate exterior boundary conditions. The relationships between the dynamical properties and the noise parameters are examined. It is found that in the different stages of tumor, the noise parameters have different influence on the lifetime and the likelihood inducing tumor extinction. These results are relevant for determining efficient therapeutic regimes to induce the extinction of tumor cells.

Abstract:
The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian alpha-stable type Levy motions. Both deterministic quantities are characterized by differential-integral equations(i.e.,differential equations with non local terms) but with different exterior conditions. The non-Gaussianity of noises manifests as nonlocality at the level of mean exit time and escape probability. An objective of this paper is to make mean exit time and escape probability as efficient computational tools, to the applied probability community, for quantifying stochastic dynamics. An accurate numerical scheme is developed and validated for computing the mean exit time and escape probability. Asymptotic solution for the mean exit time is given when the pure jump measure in the Levy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of alpha in the alpha-stable Levy jump measure. The mean exit time can measure which of the two competing factors in alpha-stable Levy motion, i.e. the jump frequency or the jump size, is dominant in helping a process exit a bounded domain. The escape probability is shown to vary with the underlying vector field(i.e.,drift). The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the process is subject to certain deterministic potential and the value of alpha is in (0,1).

Abstract:
Effects of non-Gaussian $\alpha-$stable L\'evy noise on the Gompertz tumor growth model are quantified by considering the mean exit time and escape probability of the cancer cell density from inside a safe or benign domain. The mean exit time and escape probability problems are formulated in a differential-integral equation with a fractional Laplacian operator. Numerical simulations are conducted to evaluate how the mean exit time and escape probability vary or bifurcates when $\alpha$ changes. Some bifurcation phenomena are observed and their impacts are discussed.