Take a $\mathcal C^2$ potential $V:\mathbb R^d\to \mathbb R$, and assume that it is bounded from below (say $\min V=0$ for simplicity, so that $V\geq 0$). Consider the autonomous gradient-flow $$ \dot X_t=-\nabla V(X_t) $$ and let $\Phi(t,X_0)$ be the corresponding flow. It is well-known that if ...

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is thi...

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid Lewin (Generalized Interval Systems and Transformations), Rudolf Wille (discussed also by Jan Thom...

Let $\alpha $ be a positive rational number and $\theta$ be a real number, I have tried to solve the following equation: $$(\cos {\theta})\alpha^{(1/\cos \theta)}+({1/\cos \theta})\alpha^{\cos \theta}=2 \alpha$$ in rational numbers regarding values of $\theta$, I have got only $\theta =0$ ? Are ...