25 problems

Let $(n_k)$ be the sequence defined earlier in the paper, and let $\mathcal H(n,1)$ be the associated Hankel determinant. Monotonicity conjecture. The sequence of absolute values…

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$. Under the $p$-toppling rule, each edge incident to a toppled vertex independently sends a particle with pr…

Let $w$ be an even integer with $w \ge 6$ and $w \ne 8,10$, and let $X_{w,1}$ denote the extremal quasimodular form of weight $w$ and depth $1$. Monotonicity conjecture. For all ev…

Let $G$ be a graph, let $k$ be fixed, and let $\st{G}{k}_r$ denote the $r$-restricted graphical Stirling number, where the restricted-set size is indexed by $r$. Restricted-number…

Let $\mathcal{D}_{\rm bulk}$ and $\mathcal{D}_{-\infty}$ denote the parameter sets for which the corresponding conditional entropies are monotone under conditionally mixing channel…

Let $\mathcal{A}_N$ and $\mathcal{C}_N$ be the final states defined in the model's classification, and write $\mathbb{P}_q$ for probability under parameter $q$. Monotonicity conjec…

Neumann monotonicity with volume scaling. The function

Dirichlet monotonicity with volume scaling. The function

Higher-dimensional Neumann monotonicity conjecture. The function

Higher-order ratio monotonicity conjecture. The functions $R_n^{[k]}(x)$ are positive and increasing in $x$, satisfy $R_n^{[k+1]}(x)>R_n^{[k]}(x)$, and obey $R_n^{[k]}(x)<1$ when…

Let $f_b(x):=\frac{1}{1-bx+x^2}$, and let $\beta^+=\frac{3+\sqrt{13}}{4}$. Monotonicity conjecture. 1. $L_n(f_b)$ is neither increasing nor decreasing for all…

Median offset monotonicity conjecture. The function $\alpha_{k,n}-n$ is decreasing in $n$ under one of the following conditions: (i) $k=2$ and $n\ge3\kappa$; (ii) $k\in[3,6]$ and…

Let $P(n)$ denote the number of polyominoes with $n$ cells. Successive-ratio monotonicity conjecture. The sequence … is increasing. The conjecture is motivated by the available val…

Let $P(n)$ be the number of polyominoes with $n$ cells. A polyomino is constructible if it is the concatenation of two polyominoes of smaller sizes, and inconstructible otherwise;…

Auxiliary-function conjectures. (1) The function $f$ is convex on $(0,1)$. (2) The functions $h_{11}$ and $h_{12}$ are both strictly increasing and convex on $(0,1)$, with ranges…

Li's monotonicity conjecture. Along the $\mathbb C^*$-flow, the energy density decreases pointwise as $|t|$ decreases.

Let $X$ be a topological space and let $f:X\to[0,\infty)$ be a continuous function. For $p\in(0,\infty]$, let $\operatorname{{\bf ucat}}^p(f)$ denote the minimum number of unimodal…

Gamma-quotient monotonicity conjecture. The quotient decreases as $c$ increases.

Let $b_n(t)$ and $a_n(t)$ denote the positive solutions and associated quantities introduced in the paper, with $n$ a nonnegative integer. For fixed $t>0$, the expressions…

Logarithmic lower-bound conjecture. Monotonicity testing in the ${\sf EVAL}$ model has query complexity

Higher-order Stirling-number conjecture. The following claims are valid: (1) for fixed $ell\in\mathbb{N}$ and $n\geq \ell+3$, the finite sequence…

The monotonicity conjecture. For every $\tau\geq0$, the function $g_\tau$ is strictly increasing on $(0,\infty)$. This conjecture is one of the monotonicity problems arising from r…

For $a\in(0,1)$, let $J(a,a;\rho)$ denote the Gaussian noise-stability quantity for two sets of Gaussian measure $a$, and let $\mathbb{E}_\rho J(1_A(X),1_A(Y);\rho)$ denote the cor…

Monotonicity conjecture. $F_q$ is decreasing on $[0,\frac{\pi}{2}]$ for $q>2$. For $q\leq 2$, $F_q$ is nondecreasing on $[0,\frac{\pi}{2}]$ when $\left[\frac{(n+1)q}{2}\right]$ is…

Ideal monotonicity conjecture. The function $f_{i,p,s,t}$ is increasing with respect to $x$ if and only if $p\le -(i+1)$, and decreasing with respect to $x$ if and only if…