339 problems

For each integer $k\geqslant 2$, let $\gamma_k$ be the constant in the asymptotic growth … of the maximal ranks of at-most-$k$-furcating rooted trees with $n$ leaves. Monotonicity…

Let $calpha>2$ be real, and let $c_{n,k}$ and the associated quantities be those defined in the paper. All-α conjecture. The statement of Theorem cref{t:super4} holds true for ever…

The conjecture asserts an asymptotic formula for a binomial sum arising in the enumeration of $(sn,n)$-Dyck paths and related enumeration problems. The supplied sources do not stat…

For every integer $n\ge 1$, define $P_n(q)=\prod_{j=1}^{n}(1-q^{3j-2})(1-q^{3j-1})$ and $c_{n,m}=[q^{3m+2}]P_n(q)^4$ for $0\le m\le 4n^2-1$. After all zero coefficients are omitted…

Let $p(m,n)$ be the number of partitions of an $m\times n$ rectangle into integer-sided rectangular blocks, where partitions are identified when they consist of the same multiset o…

Let $\kappa_r(A,B)\in\mathbb{Q}[A,B]$ be the $r$-th additive phase constant in the asymptotic phase equation for the zeros of Jacobi polynomials, with $A=\alpha^2$ and $B=\beta^2$.…

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…

Let $B(t)$ be standard Brownian motion, let $M(t)=\sup_{0\leq s\leq t}B(s)$, and let $\mathcal A=\int_0^1 M(t)\,dt$. Write $f_{\mathcal A}$ for the density function of $\mathcal A$…

Let $\rho\in\mathbb{N}$, and assume (A1) obtained from the preceding lemma. Let $(a_k)_{k\geqslant 0}$ be the sequence defined by the two Nesterov equations referred to in the sour…

Zograf's conjecture. For any fixed $n\geq 0$, as $g\to\infty$,

Second-order asymptotic conjecture. Then

Asymptotic ratio conjecture. The limit exists:

Asymptotic conjecture for $(2,k,k)$-numbers. For each $k$,

Brauchart–Hardin–Saff conjecture. The following asymptotic expansion holds:

Let $\mathcal{Z}_{\alpha}(\boldsymbol{z},u)$ and $\mathcal{Z}_{\alpha}(\boldsymbol{x})$ be pure partition functions, and let $\Phi(\boldsymbol{x},\boldsymbol{\xi},u)$ and…

Fast convergence rate conjecture. The convergence satisfies

Borot–Eynard–Majumdar–Nadal conjecture. The left-tail asymptotic expansion is

Let $({\overline{\partial}}_E,\varphi)$ be a Higgs bundle in the regular locus $\mathcal M'$, and let $g_{L^2}$ and $g_{\mathrm{sf}}$ denote Hitchin's $L^2$-metric and the semiflat…

Let $\{p_n(x)\}_{n=m}^{\infty}$ be a family of exceptional orthogonal polynomials with weight function … where $W_0(x)$ is the weight function of a classical orthogonal polynomial…

Let ${\mathcal S}\subset\mathbb{Z}^2$ be a non-singular step set, let $Q(1,1;t)$ be the corresponding generating function, and let … Let $(x^*,y^*)$ be the unique point minimizing…

Strong ratio-limit conjecture. Locally uniformly for $(x,y)\in M\times M$,

Chalker–Mehlig conjecture. (i) For any two points $z_1,z_2$ such that $|z_1|<1$, $|z_2|<1$ and $|z_1-z_2|>0$,

Assume the asymptotic expansion conjecture holds for a closed oriented $3$-manifold $M$. Let $\mathcal{M}(M)$ be the moduli space of flat $\operatorname{SU}(N)$-connections, and le…

Let $f^\beta(x)$ be the unique solution of equation (2) with $f^\beta(0)=1$. Asymptotic growth conjecture. As $x\to+\infty$, the function $f^\beta$ should be asymptotic to … for a…

Let $u_r$ be the symmetric homogeneous polynomial coordinates in $N$ variables constructed in the main theorem. For a partition $\sigma\vdash r$ and a differential operator of orde…