62 problems

Let $R$ be a rational function in $z_1$ with deformation dimension $m=1$, satisfying the conditions of Proposition[necessary condition], and suppose that its first three critical v…

Let $l$ be an odd positive integer, and let $a_l^l$ denote the coefficient in the expansion of the solution referred to as Expansion. Classicality criterion. The solution with odd…

Let $\theta$, $\theta_t$, $\beta$, and $s$ be parameters, let $G$ denote the Barnes $G$-function, and let $\Phi$ and the matrix elements appearing below be the conformal-block data…

Tricolour graph construction. For any tricolour graph there exists a function

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…

Let $y_0\in\mathbb R$ and $y_l\leq y_0$ define a first solution of the first Painlevé equation, and let the second solution be defined by the data $-y_l$ and $-y_0$. Let $\Delta$ b…

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…

Hurwitz–quadrangulation correspondence conjecture. The constants satisfy

Special-case Umemura-polynomial identity. If $b_1=0$, then

Local cohomology conjecture. Under the same notation and assumptions,

Local cohomology conjecture. Under these assumptions,

Dubrovin's universality conjecture. There exists a solution $u=u(x,t;\epsilon)$ to the perturbed equation on the same domain with

Let $Z(t,\nu,\hbar)$ be the perturbative partition function for the Painlevé I system, and let $\mathfrak S$ denote the Stokes automorphism associated with the $B$-periods. Write…

Let $(\nu^\pm,\rho^\pm)$ be two parameter pairs, and let $\mathscr T^\pm(t,\nu^\pm,\rho^\pm,\hbar)$ be analytic $\tau$-functions of $(P_{\rm I})$ defined on domains containing…

Painlevé I limit conjecture. There exists a choice of constant $C = C(P_0,{\bf u})>0$ and a polynomial $Q(t_1,t_2,t_5)$ such that, defining $\hat{\tau}_0:= e^{Q}$, considered as a…

Generating-structure conjecture. For $m\in\mathbb{N}$,

Polynomial-properties conjecture. First, $a=0$ is a first-order zero of $P_{2n-1,m}(a,\tilde{c}_{-1,3})$, while $P_{2n,m}(0,\tilde{c}_{-1,3})\neq0$. Second, when $m\geqslant k+3$,…

Let $q$ be the deformation parameter, $t$ the time parameter, and let $\mathcal{D}$ denote the discrete $q$-lattice … Let $\mathcal{D}$ be the matrix Fredholm determinant defined e…

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…

Let $Z$ denote the zero set of the function $\tau(T,X)$ associated with the critical Benjamin–Ono profile. Nonvanishing conjecture. … i.e., $\tau(T,X)\neq 0$ for all…

Painlevé asymptotic conjecture. For $0\leq\omega<\pi$, up to an order not yet determined,

Let $\beta_{\ell,k}=(0,s_{\ell,k})$ be the points on the $s$-axis, where $s_{\ell,1},s_{\ell,2},\dots$ are the positive zeros of the Bessel function $J_\ell$, and let…

Let $k\geqslant4$, and let $a_{k,j}$ be the coefficients in the asymptotic expansion, with parameters $\alpha$, $\varkappa$, $\varepsilon$, $b$, and $a_{1,\pm1}$ as defined in the…

Let $a_{k,j}$ be the coefficients in the asymptotic expansion indexed by $k\in\mathbb{N}$ and $j\in\mathbb{Z}$ with $|j|\leqslant k$. Parity and extremal-coefficient conjecture. If…