Mathematics Problem Archive

Is $(n+m)/2$ the maximum number of limit cycles of $$\dot x=P_n(x,y),\qquad \dot y=Q_m(x,y),$$ where $n\neq m$ and $P_n,Q_m$ are homogeneous polynomia...

(i) For the cubic family $$\begin{cases}\dot x=ax+by,\\ \dot y=cx^3+dx^2y+exy^2+fy^3,\end{cases}$$ is $2$ the maximum number of limit cycles? (ii) If ...

For a general $T$-periodic Riccati equation $$\frac{dx}{dt}=A_2(t)x^2+A_1(t)x+A_0(t),$$ give effective criteria determining whether it has a continuum...

For $$\frac{dx}{dt}=(a_0+a_1\sin t+a_2\cos t)x^3+(b_0+b_1\sin t+b_2\cos t)x^2,$$ is $3$ the maximum number of $2\pi$-periodic limit cycles?...

Given integers $p>q\geq2$ and $m,n\in\mathbb{N}$, find the maximum number of $2\pi$-periodic limit cycles of $$\frac{dx}{dt}=A_m(t)x^p+B_n(t)x^q,$$ wh...

Let $\mathcal M_m$ be the family of planar polynomial vector fields that are linear combinations of $m$ distinct monomial vector fields $(x^{n_j}y^{k_...

Let $f$ be a continuous, nonzero, $T$-periodic function and let $p>0$. Find necessary and sufficient conditions on $f$ for the existence of positive $...

Determine the maximum number $\mathcal{C}_n$ of centers for planar polynomial differential systems of degree $n\geq4$....

Consider $$x_{n+k}=\frac{A_0+A_1x_n+\cdots+A_kx_{n+k-1}}{B_0+B_1x_n+\cdots+B_kx_{n+k-1}},$$ where the coefficients are nonnegative, $\sum A_i,\sum B_i...

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...

For the family of reversible quadratic centers $$\begin{cases}\dot x=-y+xy,\\ \dot y=x+Dx^2+Fy^2,\end{cases}$$ is $2$ the maximum number of critical p...

Is the period function associated with the period annulus of the origin for $$\dot z=iz+(z\bar z)^n z^{k+1},$$ where $n$ and $k$ are positive integers...

Determine the entries currently marked unknown in the following comparison between quadratic systems and planar piecewise-linear systems with a straig...

Let $\mathcal H(n)$ be the maximum number of limit cycles of degree-$n$ planar polynomial systems, and let $\mathcal L(n)$ be the maximum number of cr...

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)}...

Does every triangular billiard have a periodic trajectory?...

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 $f$ be real analytic near the origin, with $f(0,0)=0$, and let $n>1$. Suppose the origin is an isolated equilibrium of $$\dot x=2^n\operatorname{R...

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 $f=0$?...

Find a reasonable, or sharp, upper bound in terms of $m_1,m_2$ for the maximum number of simple positive-coordinate solutions of a real polynomial sys...

Is $(2m_1-1)(2m_2-1)$ the maximum number of simple solutions of a real polynomial system $f_1(x,y)=f_2(x,y)=0$, where $m_i$ is the number of monomials...

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...

Let $\mathcal C_{T,S}$ be the set of sequences $(\int f\,T^ng\,S^nh\,d\mu)_{n\ge1}$ over probability-preserving systems with commuting $T,S$ and bound...

If $(a_1(n)),\ldots, (a_\ell(n))$ are sequences of integers, then show that the following %%three statements are equivalent: • The sequences $(a_1(n))...

Let $(a(n))$ be a sequence that satisfies: • for every connected $\ell$-step nilmanifold $X$ and every irrational nilrotation $b$ in $X$ the sequence ...

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

If a sequence is good for $2$-convergence of powers, then show that it is good for $2$-convergence of commuting transformations....

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...

Suppose that the sequence of $\ell$-tuples of polynomials $(p_{1,N},\ldots, p_{\ell,N})$ is good. Show that for every ergodic system $(X,\mathcal X,\m...

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 $(a(n))$ be the sequence of integers $(p_n)$, where $p_n$ is the $n$-th prime, or $([n^c])$ where $c>0$, or $(2^n)$. Is it true that for every erg...

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^{...

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...

Suppose that the polynomials $p_1,\ldots,p_\ell\in \mathbb Z[t]$ are rationally independent and have zero constant term. Show that for every $A\in \ma...

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 $(X,\mathcal X,\mu, T,S)$ be a system and $f,g\in L^\infty(\mu)$ be functions. Show that the averages $$ \frac1N \sum_{n=1}^N f(T^nx)\cdot g(S^nx)...

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))...

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,...

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 $a,b$ be distinct positive non-integers. Show that for every ergodic system $(X,\mathcal{X},\mu,T)$ and functions $f, g \in L^\infty(\mu)$, we hav...