The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation $\boldsymbol{K} \boldsymbol{M}+\boldsymbol{M} \boldsymbol{K}=\boldsymbol{r}\, \boldsymbol{s}^{T}$ we introduce a scalar function $S^{(i,j)}=\boldsymbol{s}^{T}\, \boldsymbol{K}^j(\boldsymbol{I}+\boldsymbol{M})^{-1}\boldsymbol{K}^i\boldsymbol{r}$ which is defined as same as in discrete case. $S^{(i,j)}$ satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on $\boldsymbol{r}$ and $\boldsymbol{s}$, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of $S^{(i,j)}$ defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property $S^{(i,j)}=S^{(i,j)}$. The solution $\boldsymbol{M}$ provides $\tau$ function by $\tau=|\boldsymbol{I}+\boldsymbol{M}|$. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.Comment: 23 page

Combining Explicit and Implicit Regularization for Efficient Learning in Deep Networks

Works on implicit regularization have studied gradient trajectories during the optimization process to explain why deep networks favor certain kinds of solutions over others. In deep linear networks, it has been shown that gradient descent implicitly regularizes toward low-rank solutions on matrix completion/factorization tasks. Adding depth not only improves performance on these tasks but also acts as an accelerative pre-conditioning that further enhances this bias towards low-rankedness. Inspired by this, we propose an explicit penalty to mirror this implicit bias which only takes effect with certain adaptive gradient optimizers (e.g. Adam). This combination can enable a degenerate single-layer network to achieve low-rank approximations with generalization error comparable to deep linear networks, making depth no longer necessary for learning. The single-layer network also performs competitively or out-performs various approaches for matrix completion over a range of parameter and data regimes despite its simplicity. Together with an optimizer's inductive bias, our findings suggest that explicit regularization can play a role in designing different, desirable forms of regularization and that a more nuanced understanding of this interplay may be necessary

The Effects of Culture on International Advertising Appeals: A Cross-Cultural Content Analysis of U.S. and Japanese Global Brands

An international advertising campaign can be standardized in creative strategy, but localized in execution. As an integral part of the execution, advertising appeals should be tailored to local culture to maximize the effectiveness of international advertising campaigns while minimizing cost. The purpose of this study is to investigate whether advertising appeals mirror predicable differences in cultural values. This study is based on the influential Hofstede model (Hofstede, 2001; Hofstede & Mooij, 2010), which distinguishes cultures according to six dimensions. While individualism and collectivism have been widely discussed in published studies, other dimensions that are also important indicators of advertising appeals have not yet been systematically studied. For example, the sixth and new dimension, “indulgence versus restraint,” was added to Hofstede cultural model in 2010. It has been the least studied cultural dimension, and has not yet been widely adopted in intercultural studies. Therefore, another purpose of this study is to scrutinize these cultural dimensions for a better understanding of the interplay of culture and advertising appeals in cross-national settings. This study pairs the United States and Japan to investigate advertising content across sharply contrasting cultures. A quantitative content analysis with independent samples t-tests is conducted to analyze the variance of sample means