Page 205 of the book Classical Groups and Algebraic K-Theory defines a symplectic module to be an arbitrary quadratic module $(M,h,q)$ over a form ring $(R,\Lambda)$ with $(J,\varepsilon)$ where $J=\operatorname{id}_R$, $\varepsilon=-1$ and $\Lambda=R$. In particular, this implies that $R$ is com...

According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the backward heat equation $u_t+\Delta u=0$. So what is the formula or does the fundamental solution e...

A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form: $$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n+x}}{(1)_{n+x}} \right) \sum_{k=0}^{m} a_k (n+x)^k z_0^{n+x} = \frac{\sqrt{(-1)^m \chi}}{\pi^m}....

A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form: $$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n+x}}{(1)_{n+x}} \right) \sum_{k=0}^{m} a_k (n+x)^k z_0^{n+x} = \frac{\sqrt{(-1)^m \chi}}{\pi^m}....

A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form: $$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n+x}}{(1)_{n+x}} \right) \sum_{k=0}^{m} a_k (n+x)^k z_0^{n+x} = \frac{\sqrt{(-1)^m \chi}}{\pi^m}....