23 problems

Linearizability conjecture. For $n\geq 3$, the equation is linearizable if and only if $F_y=0$; in that case, it is already a linear equation. The claim extends the verified analys…

Let $Y(t;\kappa)$ denote the solution of … Denote the eigenvalues of $Y(t;\kappa)$ by $\lambda_n(t;\kappa)$, ordered so that each $\lambda_n$ is continuous as a function of $t$. Le…

A two-point boundary value problem (BVP) for an ordinary differential operator is called dissipative when its associated operator is dissipative, and Birkhoff-regular when its boun…

Let $x_1<x_2<0$, and let $X(x)$ and $X_{\min}(x)$ denote the corresponding extremal location functions for real solutions of the first Painlevé equation. Comparison conjecture. … Th…

Let $y_{\max}(x_0;x)$ and $y_{\min}(x_0;x)$ be the two real solutions with pole at $x_0\leq0$, with $x$ in the interval of existence of $y_{\min}(x_0;x)$. Ordering conjecture. … The i…

Uniqueness conjecture. For any $x_0$, the solution with a pole at $x_0$ and minimum at $\Xi_{\min}(x_0)$ is unique. The claim is presented as a numerical and heuristic extension of…

Consider an ODE of the form of equation with a bistable function $f(y)$. Five-solution conjecture. Such an ODE has at most five $1$-periodic solutions. This conjecture concerns the…

Let $\beta\geq 5/3$ and define … Assume there is an efficient procedure for sampling from the probability distribution induced by the normalized tensor $c/J$. Consider the quadratic…

Let $(\mathcal{F}_{\mathbb{M}}^{\mathbb{M}},{\Greekmath 011A})$ be a Hilbert function space. Assume that $\mathcal{E}=\{e_i\}_{i\in I}\subset \mathcal{F}_{\mathbb{M}}^{\mathbb{M}}$…

Let $X=(P_1,\dots,P_n)$ be an autonomous $C^1$-vector field on $\mathbb{R}^n$ with a unique singularity at the origin. For each $x\in\mathbb{R}^n$, let $DX(x)$ denote its Jacobian…

Consider the full constrained 3D system studied in the paper, rather than either of its restrictions to an invariant hypersurface. The Painlevé property means that its solutions ha…

Apparent-singularity conjecture. The point $0$ is an apparent singularity if and only if $B=D_0=D=0$; equivalently, this polynomial system has only the trivial solution $B=D_0=D=0$…

Multitype coalescent speed conjecture. Provided the conditions of the theorem establishing coming down from infinity hold, the ordinary differential equation

Let $A$ be a system matrix sampled from the Ginibre ensemble, and let $\mathbf{x}_{0}$ be an initial condition whose coordinates $x_{0,i}$ have second-order moments bounded by a co…

Infinite-order resolution ODE conjecture. Under certain regularity conditions on $g(z,s)$ and $s$, including, for example, that $g(z,s)$ is infinitely differentiable and that $s$ i…

Let $a,b,c,d$ be analytic functions at $x0 otinmathbb R$, and consider the linear homogeneous ordinary differential equation … Suppose that this equation admits three linearly inde…

Let … be a linear ordinary homogeneous differential equation of order $n$ with real-valued continuous coefficients, disconjugate on an interval $I$. For a positive integer…

Consider the system $\dot{x}=f(x)$, and suppose that $x=0$ is an asymptotically stable equilibrium. Let $\psi(x_0,t)$ denote its solution with initial condition $x_0$, and let…

Rumanov auxiliary-system uniqueness conjecture. The system has a unique smooth solution $(q_2,\alpha)$ satisfying these conditions as $t\to+\infty$. Such uniqueness would make the…

The uniqueness conjecture. Such a solution $u(t)$ cannot possess another solution $v(t)$ satisfying these initial conditions.

Let $M>0$, and consider the third-order equation … Here $p(x,E)$ is the potential in the conformal-limit equation, and $x$ is understood on a suitable cover of the punctured comple…

General-order linear ODE symmetry-generator conjecture. The vector field $X^0$ is the infinitesimal generator of $G_c$.

Consider the differential equation … . This finiteness would justify the use of a numerical solver to determine the number of global solutions for a given M-shaped $\phi$, and the…