Mathematics Problem Archive

Consider a Hamiltonian system with a center at the origin and Hamiltonian $$H(x,y)=H_{2n}(x,y)+H_m(x,y),\qquad m>2n,$$ where $H_{2n}$ and $H_m$ are ho...

For $$\dot x=P_{2k+1}(x,y),\qquad \dot y=Q_{2\ell+1}(x,y),$$ where $P_{2k+1}$ and $Q_{2\ell+1}$ are homogeneous polynomials of the indicated odd degre...

Let $\mathcal T(n)$ be the maximum number of critical periods that a planar polynomial differential system of degree $n$ can have. Is there a constant...

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be smooth, have a fixed point, and satisfy $$\rho\bigl(|DF(x)|\bigr)<1\quad\text{for every }x\in\mathbb{R}^2.$$ Is...

Let $A_0,\ldots,A_n$ be independent $N(0,1)$ random variables and let $p_n$ be the probability that the zero solution of $$A_nx^{(n)}+A_{n-1}x^{(n-1)}...

Do there exist two irreducible algebraic curves of degrees $n$ and $m$, with $n+m>4$, each having an oval, for which the Poncelet map is well defined ...

Let $\gamma=\{x^2+y^2-1=0\}$ and $\Gamma_\varepsilon=\{p_2(x,y)+\varepsilon p_m(x,y)=0\}$, where $\Gamma_0$ is an ellipse surrounding $\gamma$, the cu...

Let $f(x_1,\ldots,x_n)\in\mathbb{C}[x_1,\ldots,x_n]$ satisfy $$M_m:=\int_0^1\cdots\int_0^1 f(x_1,\ldots,x_n)^m\,dx_1\cdots dx_n=0\qquad(m\geq1).$$ Mus...

Let $f(x)\in\mathbb{C}[x]$ have $k$ monomials. Does there exist $N(k)$ such that if $$M_n:=\int_0^1f(x)^n\,dx=0\qquad(1\leq n\leq N(k)),$$ then $f=0$?...

Define $f:\mathbb{N}\to\mathbb{N}$ by $f(n)=n+\operatorname{rev}(n)$, where $\operatorname{rev}$ reverses the decimal digits. Are there infinitely man...

Determine the structure of the multiple correlation sequences $(\mathcal{C}(n_1,\ldots,n_\ell))$ defined by (source reference E:MultCor). Is it true t...

Is there a sequence that is good for $2$-recurrence of powers but is not good for $2$-recurrence of commuting transformations?...

Give an explicit example of a fast growing sequence that is good for multiple recurrence and convergence of powers and commuting transformations....

Let $\mathcal P$ be an essentially distinct family of integer polynomials, and let $d_{\min}(\mathcal P)$ be the least $d$ for which the Host–Kra fact...

Let $(X,\mathcal X,\mu,T)$ be a system and $f,g\in L^\infty(\mu)$ be functions. If $\Lambda$ is the von Mangoldt function and $\phi$ is a multiplicati...

Let $p_1,\ldots, p_\ell$ be integer valued generalized polynomials. Show that the averages $$ \frac{1}{N}\sum_{n=1}^{N} T_1^{p_1(n)}f_1\cdots T_\ell^{...

Let $(X,\mathcal X,\mu,T_1,\ldots, T_\ell)$ be a system and $\{p_1,\ldots, p_\ell\}$ be a family of intersective integer polynomials. Show that for ev...

Let $(X,\mathcal X,\mu)$ be a probability space, $T_1,\ldots, T_\ell \colon X\to X$ be invertible measure preserving transformations, and $p_1,\ldots,...

Find an example of a function $a\in \mathcal H$ that grows faster than polynomials, meaning, $a(t)/t^k \to\infty$ for every $k\in \mathbb N$, such tha...

Suppose that $n\sigma_n\to\infty$. Show that almost surely the sequence $(a_n(\omega))$ is good for multiple recurrence and convergence of commuting t...

Suppose that $a,b\in(0,1)$ and $a\neq b$. Show that almost surely the following holds: For every system $(X,\mathcal X,\mu,T,S)$ and functions $f, g \...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

Let $(X,\mathcal X,\mu, T_n)$ be a measure preserving system with multiplicative structure and $A\in \mathcal X$ with $\mu(A)>0$. Is it true that ther...

For a smooth dissipative dynamical system with a global attractor, is the supremum of the local Lyapunov dimension on the attractor attained at an equ...

For which classes of ordinary differential equations do the classical and canonically generalized forms of Lyapunov's second method give necessary as ...

In every dimension at least three, is each reversible cellular automaton locally reversible?...

In a regular $n$-gon, call a trajectory preclosed when its endpoints divide their boundary edges into equal-length parts and meet those oriented edges...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

Unfold a billiard trajectory in a regular $n$-gon by reflecting the polygon across each encountered side. A short trajectory joins two vertices and me...

On a regular dodecahedron, unfold a geodesic beginning at a vertex $v$ through successive faces. Call it short if it ends at a vertex and meets no ver...

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...

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...

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...

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...

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...

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...

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...

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...

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...