13,918 research outputs found
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 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 and 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 and time step as well as the small parameter . Based on the error bound, in order to obtain `correct' numerical solutions
in the nonrelativistic limit regime, i.e. , the CNFD method
requests the -scalability: and
. 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 -scalability is improved to and
when 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 which is inversely proportional to the speed of
light. In the nonrelativistic limit regime, i.e. , the
solution exhibits highly oscillatory propagating waves with wavelength
and 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 . 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
and ,
where is the mesh size, is the time step and 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 for all and optimally with
quadratic convergence rate at in the regimes when either
or . 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 .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’
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