66 problems

Let $\mathbb{U}=\{x\in\mathbb{R}^n:|x|<1\}$ be the open unit ball, and let $p^N_{\mathbb{U}}(x,y,t)$ denote its Neumann heat kernel. Laugesen–Morpurgo conjecture. For every fixed…

Let $P$ be a poset on $n$ elements, and let $\Omega(P;t)$ be its order polynomial. Kahn–Saks conjecture. The map … is weakly decreasing for positive integer values of $t$. The conj…

Let $B_k(x)$ be the functions defined for $k\geqslant1$ and $x\in(0,1]$ in the branching-process setting above. Monotonicity conjecture. Every function $x\mapsto B_k(x)$ is decreas…

Chen–Rubin's conjecture. The sequence $m(n)-n$ decreases for integers $n\geq 1$.

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 $\mathcal{D}_{\rm bulk}$ and $\mathcal{D}_{-\infty}$ be the sets of parameter pairs for which the corresponding conditional entropies are monotone under conditionally mixing ch…

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

Global monotonicity conjecture. The map

Let $P$ be a finite partially ordered set, and consider the two notions of monotonicity for continuous-time Markov processes on $P$ studied by Dai Pra et al. An acyclic extension i…

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…

Spekreijse-region conjecture. Leaving the Spekreijse region may cause practical problems, namely failure of monotonicity, in the context of directionally constant flow and flux-spl…

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…

The $9/(4k-1)$ bound conjecture. The pmf decreases monotonically for all $n\geq k$ whenever

The exact-threshold conjecture. For fixed $k\geq 2$,

Parent-number monotonicity conjecture. If the CKP with $M=m_1$ eliminates all errors, then the CKP with $M=m_2$ also eliminates all errors.

Checking-parameter monotonicity conjecture. If a CKP with checking parameters $(p_1,k_1)$ eliminates all errors, then a CKP with parameters $(p_2,k_2)$ also eliminates all errors.

Multi-parent monotonicity conjecture. Monotonicity of error elimination with respect to the checking parameters $p$ and $k$ should hold when nodes can connect to more than one pare…

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…