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

Creating the Royal Society's Sylvester Medal

Following the death of James Joseph Sylvester in 1897, contributions were collected in order to mark his life and work by a suitable memorial. This initiative resulted in the Sylvester Medal, which is awarded triennially by the Royal Society for the encouragement of research into pure mathematics. Ironically the main advocate for initiating this medal was not a fellow mathematician but the chemist and naturalist Raphael Meldola. Religion, not mathematics, provided the link between Meldola and Sylvester; they were among the very few Jewish Fellows of the Royal Society. This paper focuses primarily on the politics of the Anglo-Jewish community and why it, together with a number of scientists and mathematicians, supported Meldola in creating the Sylvester Medal