25 problems

Let $T>0$, let $a:[0,T]\to\mathbb{R}$ be a two-step indefinite weight, and let $\gamma\in\mathbb{R}\setminus\{-1,0,1\}$. The conjecture asserts that the boundary-value problem…

For a given sensitivity parameter $\lambda>0$, determine a function $u\in C^2([0,1])$ satisfying the nonlinear boundary-value problem $u''(x)=\lambda\sinh(\lambda u(x))$ for…

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 $X=(P_1,\dots,P_n)$ 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…

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…

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 $\mathcal{E}=\{e_i\}_{i\in I}\subset \mathcal{F}_{\mathbb{M}}^{\mathbb{M}}$…

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 $x_0 otin\mathbb R$, and consider the linear homogeneous ordinary differential equation … Suppose that this equation admits three linearly in…

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…