Yanzhi Zhang

A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose--Einstein condensates (BECs) in a rotational frame with or without long-range dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dyn...

Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation

We propose three Fourier spectral methods, i.e.,źthe split-step Fourier spectral (SSFS), the Crank-Nicolson Fourier spectral (CNFS), and the relaxation Fourier spectral (ReFS) methods, for solving the fractional nonlinear Schrodinger (NLS) equation. All of them are mass conservative and time reversible, and they have the spectral order accuracy in space and the second-order accuracy in time. In addition, the CNFS and ReFS methods are energy conservative. The performance of these methods in simulating the plane wave and soliton dynamics is discussed. The SSFS method preserves the dispersion relation, and thus it is more accurate for studying the long-time behaviors of the plane wave solutions. Furthermore, our numerical simulations suggest that the SSFS method is better in solving the defocusing NLS, but the CNFS and ReFS methods are more effective for the focusing NLS.

An efficient spectral method for computing dynamics of rotating two-component Bose-Einstein condensates via coordinate transformation

In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose-Einstein condensates (BECs) which is described by the coupled Gross-Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in the literature. Numerical results show that to achieve the same accuracy, our method takes much shorter computing time. We also apply our method to study issues such as dynamics of vortex lattices and giant vortices in rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross-Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.

A comparative study on nonlocal diffusion operators related to the fractional Laplacian

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as \alpha \to 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k \in N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \alpha \in (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \delta is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(\delta^{-\alpha}). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \alpha \to 2, it generally provides inconsistent result from that of the fractional Laplacian if \alpha \ll 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem

Abstract In this paper, we develop a novel finite difference method to discretize the fractional Laplacian ( − Δ ) α / 2 in hypersingular integral form. By introducing a splitting parameter, we formulate the fractional Laplacian as the weighted integral of a weak singular function, which is then approximated by the weighted trapezoidal rule. Compared to other existing methods, our method is more accurate and simpler to implement, and moreover it closely resembles the central difference scheme for the classical Laplace operator. We prove that for u ∈ C 3 , α / 2 ( R ) , our method has an accuracy of O ( h 2 ) uniformly for any α ∈ ( 0 , 2 ) , while for u ∈ C 1 , α / 2 ( R ) , the accuracy is O ( h 1 − α / 2 ) . The convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. Additionally, we apply our method to solve the fractional Poisson equation and study the convergence of its numerical solutions. The extensive numerical examples that accompany our analysis verify our results, as well as give additional insights into the convergence behavior of our method.

UNCOVERING A LOCAL TREND IN CONSUMER EYE-TRACKING DATA – APPLICATION OF SINGULAR VALUE DECOMPOSITION IN ANALYZING GAZE SEQUENCE DATA

As visual marketing gains a more critical role in marketing communications, consumer eye-tracking data has been utilized to assess the effectiveness of those marketing efforts (Croll, 2016; Glazer, 2012). With eye-tracking data, researchers can capture consumers’ visual attention effectively and may predict their behavior better than with traditional memory measures (Wedel & Pieters, 2008). However, due to the complexity of data: its volume, velocity and variety, known as 3Vs of Big Data, marketing scholars have been slow in fully utilizing eye-tracking data. These data properties may pose a challenge for researchers to analyze eye-tracking data, especially gaze sequence data, with traditional statistical approaches. Commonly, researchers may analyze gaze sequences by computing average probabilities of gaze transitions from a particular area of interest to another area of interest. When the variance of gaze sequence data in the sample is small, this method would uncover a meaningful “global” trend, a trend consistent across all the individuals. However, when the variance is large, this method may not enable researchers to understand the nature of the variance, or the “messiness” of data. In this paper, first, to overcome this challenge, we propose an innovative method of analyzing gaze sequence data. Utilizing the singular value decomposition, our proposed method enables researchers to reveal a “local” trend, a trend shared by only some individuals in the sample. Second, we illustrate the benefits of our method through analyzing gaze sequence data collected in an advertising study. Finally, we discuss the implications of our proposed method, including its capability of uncovering a hidden “local” trend in “messy” gaze sequence data.

A fast algorithm for solving the space–time fractional diffusion equation

Abstract In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ( − Δ ) s α / 2 results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.

Open-Source Strategy to Enhance Imaginative Intensity and Profits

The Android project is one of the recent most successful open-source projects. In the Android community, the open-source firm (Google) and application developers co-create value through developing applications for the Android operating system provided by the open-source firm, and share profit from the sales of applications. Alternatively, using a closed-source strategy, a firm could keep the operating system proprietary and sell it to end users. As the first paper to apply the concept of imaginative intensity to analyze the benefits of an open-source strategy, we offer a new explanation for a firm’s selection of an open-source strategy over a closed-source strategy. We propose a model to investigate how a for-profit organization utilizes an open-source strategy, in contrast to a closed-source strategy, to enhance imaginative intensity and consequently profit. Our model suggests that an open-source strategy is more effective to manage the diminishing value of ideas than a closed-source strategy.

Understanding B2B Relationships Between an Open-Source Firm and Application Developers: Sharing Profits from Applications and In-Application Advertisements

Recently, an open-source strategy has received considerable attention among for-profit organizations. In open-source projects, such as Google’s Android project, an open-source firm (e.g., Google) takes a portion of profits that its partners (e.g., application developers) earn from selling application and in-application advertisements. However, such a use of profit-sharing scheme and the role of in-application advertisements as a source of profits have not been well-studied in the literature on open-source strategy. Thus, we propose a mathematical model to understand how the profit-sharing percentage that an open-source firm takes from application developers affects profits of the open-source firm from applications and in-application advertisements. Our proposed model shows that (1) the size of the user network (e.g., the number of Android users) plays a more important role in generating profits from advertisements than those from applications and (2) the open-source firm may experience a decline in its profit by increasing its own profit share percentage too much.