Mathematics Problem Archive

In a symmetric space, is every Killing tensor a sum of symmetric products of Killing vectors? Equivalently, is it true that the algebra of all polynom...

Assume that the exterior of an outer billiard table is foliated by invariant curves. Prove that the table is an ellipse....

{\rm (A. Glutsyuk, S. Tabachnikov )} Given two nested closed convex hypersurfaces, assume that the billiard transformations on the set of oriented lin...

Construct real analytic examples of Nijenhuis operators on closed two-dimensional surfaces whose eigenvalues are real and generically distinct....

Does there exist a Hamiltonian system with a smooth invariant torus on which the induced dynamics is mixing? No such mixing can occur on a two-dimensi...

If positive square matrices $A,B$ are shift equivalent over $\mathbb Q_+$, prove that they are strong shift equivalent over $\mathbb Q_+$....

Let $S\subset\mathbb R$ be a unital subring. Prove that a tuple $\Lambda=(\lambda_1,\ldots,\lambda_k)$ is the nonzero spectrum of a primitive matrix o...

Prove that every expansive automorphism of a one-sided full shift is topologically conjugate to a two-sided full shift....

If $(a(n))$ is good for $\ell$-recurrence of powers, is then $(a(n)^k)$ good for $1$-recurrence for $k=1,\ldots, \ell$?...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are pairwise independent. Show that there exists $d\in \mathbb N$ such that the fact...

Suppose that the polynomials $p_1,\ldots, p_\ell\in \mathbb Z[t]$ are rationally independent. Show that $\mathcal{K}_{rat}(T_1),\ldots, \mathcal{K}_{r...

Is it true that one always has a decomposition $$ \int f_0 \cdot T_1^n f_1 \cdot \ldots \cdot T_\ell^n f_\ell \ d\mu= \psi(n)+e(n) $$ where $(\psi(n))...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Here $\mathcal F=\{a_1,\ldots,a_\ell\}$ is a family of functions of polynomial growth in one Hardy field, and $\operatorname{span}^*(\mathcal F)$ deno...

Let $c$ be a positive non-integer. Show that the sequence $([p_n^c])$ is good for multiple recurrence and convergence of powers....

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Fix an ellipse with center $O$ and foci $f_1,f_2$, and let $P=(P_i)_{i=1}^N$ range over its Poncelet family of $N$-periodic billiard orbits tangent to...

Can a Siegel disk have a boundary that is not a Jordan curve?...

If a rational map $f$ has $J(f)=\widehat{\mathbb C}$, is $\omega(z)=\widehat{\mathbb C}$ for almost every $z$, and is $f$ conservative with respect to...