983 problems

Let $\beta>0$, $k\in\mathbb{N}$, and $s\in\mathbb{R}_+$. Let $\mathsf{M}^{(\beta)}_N(k;s)$ denote the corresponding C$\beta$E field moment, and let $\mathfrak{c}^{(\beta)}(k;s)$ be…

Guo–Yang polynomiality conjecture. For each $k$, the coefficient of $g^{-k}$ in this expansion is a polynomial in $n$ and in the multiplicities of the arguments, and only the multi…

For a Fisher–Hartwig determinant of size $N$, the relevant singularities are expected to produce additional logarithmic terms in its asymptotic expansion. Fisher–Hartwig's conjectu…

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

Let $\{P_n(x)\}_{n\geq 0}$ be monic polynomials orthogonal on $\mathbb{R}$ with respect to the weight … where $\rho>-1$ and $m>0$. Write $\beta_n$ for the recurrence coefficients i…

Let $E(s)$ denote the probability that the interval $(x^*-s,x^*+s)$ is empty, and suppose that the density of states behaves as $|x-x^*|^{\alpha}$ as $x\to x^*$. Forrester–Chen–Eri…

Let $p_d(n)$ denote the number of $d$-dimensional partitions of $n$. For solid partitions, $d=4$, and the relevant normalization is $n^{1-1/d}$. MacMahon's numbers are the numbers…

Let $({\mathbb S}^2)^n$ be the configuration space of $n$ points on the unit sphere, and let $\mathcal{E}_{\log}(n)$ denote the minimum of … over $({\mathbb S}^2)^n$, where…

Logarithmic-gap conjecture. One has

Fast convergence rate conjecture. The convergence satisfies

Let $w_M$ be a weight supported on $(-1,1)$ such that $w_M(x)/(1+x)^{\beta}$ has a positive finite limit as $x\to -1$, and $w_M(x)/[-(1-x)^{\alpha}\log(1-x)]$ has a positive finite…

Let $q(x;\alpha,\kappa)$ be a Clarkson–McLeod solution of the fourth Painlevé equation, with $\alpha-\frac{1}{2}\notin\mathbb{Z}$, and let $\kappa^*>0$ denote the critical value se…

Logarithmic-phase asymptotic conjecture. For all $n>0$,

Let $0<s<1$, let $\mathcal{C}_1,\mathcal{C}_2,\ldots$ be independent events with probabilities $\mathbf{P}_s(\mathcal{C}_n)=1-e^{-ns}$, and let $A_k$ be the event that there is no…

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

Necessity conjecture. This condition is not only sufficient but also necessary: the Flamant solution should not apply for models with quadratic growth in other asymptotic regimes.

Let $E(n)$ denote the fluctuation term in the decomposition of $Q(n)$ into its main term and fluctuation. Sublinear fluctuations conjecture. The fluctuation term satisfies … This i…

Exner's conjecture. For small radii, the bound-state energy depends on $a$ as

Dunkl kernel asymptotic conjecture. We conjecture that

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…

Let $\Omega$ be a domain with fractal boundary, and let fractal curvatures be the limiting curvature quantities associated with parallel sets of the boundary, with scaling exponent…

Consider the singularly perturbed boundary-value problem for $A(z)$ … with $A(z)\to 0$ as $\operatorname{Re}[z]\to\pm\infty$ along $\operatorname{Im}[z]=0$. At each order, the asym…

Sharp asymptotic rate conjecture. As $t\to\infty$, for some positive constant $c_\alpha$ depending on $K(\cdot)$,

Let solutions of the Lifshitz–Slyozov–Wagner (LSW) model evolve from admissible initial data. LSW self-similar attractor conjecture. The large-time behavior of solutions to the LSW…

Barequet–Barequet–Rote's conjecture.