Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. We begin with the conservative Crank-Nicolson finite difference (CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size $h$ and time step $\tau$ as well as the small parameter $0<\varepsilon\le 1$. Based on the error bound, in order to obtain `correct' numerical solutions in the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the CNFD method requests the $\varepsilon$-scalability: $\tau=O(\varepsilon^3)$ and $h=O(\sqrt{\varepsilon})$. Then we propose and analyze two numerical methods for the discretization of the nonlinear Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their $\varepsilon$-scalability is improved to $\tau=O(\varepsilon^2)$ and $h=O(1)$ when $0<\varepsilon\ll 1$ compared with the CNFD method. Extensive numerical results are reported to confirm our error estimates.Comment: 35 pages. 1 figure. arXiv admin note: substantial text overlap with arXiv:1504.0288

Modeling Emotion Influence from Images in Social Networks

Images become an important and prevalent way to express users' activities, opinions and emotions. In a social network, individual emotions may be influenced by others, in particular by close friends. We focus on understanding how users embed emotions into the images they uploaded to the social websites and how social influence plays a role in changing users' emotions. We first verify the existence of emotion influence in the image networks, and then propose a probabilistic factor graph based emotion influence model to answer the questions of "who influences whom". Employing a real network from Flickr as experimental data, we study the effectiveness of factors in the proposed model with in-depth data analysis. Our experiments also show that our model, by incorporating the emotion influence, can significantly improve the accuracy (+5%) for predicting emotions from images. Finally, a case study is used as the anecdotal evidence to further demonstrate the effectiveness of the proposed model

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime

We propose and rigourously analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation with a dimensionless parameter $\varepsilon\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. $0<\varepsilon\ll 1$, the solution exhibits highly oscillatory propagating waves with wavelength $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively. Due to the rapid temporal oscillation, it is quite challenging in designing and analyzing numerical methods with uniform error bounds in $\varepsilon\in(0,1]$. We present the MTI-FP method based on properly adopting a multiscale decomposition of the solution of the Dirac equation and applying the exponential wave integrator with appropriate numerical quadratures. By a careful study of the error propagation and using the energy method, we establish two independent error estimates via two different mathematical approaches as $h^{m_0}+\frac{\tau^2}{\varepsilon^2}$ and $h^{m_0}+\tau^2+\varepsilon^2$, where $h$ is the mesh size, $\tau$ is the time step and $m_0$ depends on the regularity of the solution. These two error bounds immediately imply that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for all $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\lesssim \tau$. Numerical results are reported to demonstrate that our error estimates are optimal and sharp. Finally, the MTI-FP method is applied to study numerically the convergence rates of the solution of the Dirac equation to those of its limiting models when $\varepsilon\to0^+$.Comment: 25 pages, 1 figur

The Role of Visual Identity Design in the Evolution of Luxury Fashion Brand Strategy

Fashion luxury brands have taken longer than other retail segments to understand and adapt to consumer behavior changes created by the digital revolution over the past decade. Digital and social media has created a fundamental shift in consumers expectations of their shopping experiences. The big debate in the luxury fashion industry has been whether the integration of ‘off-line’ and ‘on-line’ actually creates value or reduces the exclusivity associated with the brands. In the past, luxury brands have relied on traditional marketing channels such as print and direct mail to engage their high-value customers. Additionally, the industry has placed an enormous focus on the in-store experience as an opportunity to differentiate through enhanced service, ambiance, and customization. Data, however, shows that there really is no debating the benefits of going on-line any longer. The impact on performance by digital is clear, for brands that have chosen to invest in enhancing their shopping experience using digital and/or social media channels and platforms. It is not one versus the other. Instead, the two mediums need to come together to create a relevant customer journey. The challenge is that there is not a simple repeatable roadmap to succeeding in the digital landscape. Brands need to invest in testing, adapting, and finally innovating their on-line offering to win a share of consumers’ minds. This thesis explored the role that visual identity design plays in social media – the most heavily leveraged platform by luxury fashion brands to create their on-line experience. The thesis ideates and iterates on opportunities through design to create a seamless browse-to-purchase experience for luxury customers. The specific experiments conducted were through the lens of a fashion re-seller called ‘TELLER BLANC’