274 problems

Let $\mathscr{E}_1,\dots,\mathscr{E}_n$ be independent exponential random variables, and consider weighted sums of their centered versions,…

Yang's conjecture. The coefficients satisfy

Let $K\subset\mathbb{R}^n$ be a convex body, and let $V$ denote mixed volume, with $K[k]$ denoting $k$ copies of $K$. Godbersen's conjecture. For each convex body…

Let $q(n)$ denote the number of partitions of $n$ into distinct parts. A sequence is log-concave when $q(n)^2\geq q(n-1)q(n+1)$, and the third-order Turán inequalities are the corr…

Let $E(r)$ be the complete elliptic integral of the second kind, let $r^{\prime}=\sqrt{1-r^{2}}$, and let $A_{p}(a,b)=\left((a^{p}+b^{p})/2\right)^{1/p}$ for $p\neq 0$, with…

Let $\varphi(n)$, $\psi(n)$ and $\sigma(n)$ denote the Euler totient, Dedekind psi and sum-of-divisors functions, respectively. For integers $k\geq1$ and $n\geq2$, use…

Let $x\in(0,1)$ and let $\mathcal{K}$ denote the complete elliptic integral of the first kind. Anderson–Vamanamurthy–Vuorinen's conjecture. The inequality … holds for all…

Bennett's conjecture. The inequality holds with

Wishart Gaussian product inequality conjecture.

Threshold conjecture. For $1\leq m\leq 11$ and $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the Laguerre inequality of order $m$ whenever…

Let $n\geq2$ and let $\mu,\lambda\in\mathbb{N}^{m}$ satisfy $|\mu|=|\lambda|=d$. For the term-normalized complete homogeneous symmetric functions $H_{n,\mu}$ and $H_{n,\lambda}$, w…

Gaunt–Merkle median bounds conjecture. For $r>0$,

For a partition of $n$, let its Dyson rank be its largest part minus its number of parts, and let $N(a,c,n)$ denote the number of partitions of $n$ whose rank is congruent to $a$ m…

Let $c_{\rm LO}(1,3)$ be the smallest constant for which the Lieb–Oxford inequality is valid in dimension $d=3$ with Coulomb interaction $w(\mathbf r)=|\mathbf r|^{-1}$. Lieb–Oxfor…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d$, and let $I$ and $J$ be $\mathfrak{m}$-primary ideals. For $0\leq i\leq d$, let $e_i(I\mid J)$ denote their mixed…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d\geq2$, and let $I$ and $J$ be $\mathfrak{m}$-primary ideals. Define $e_i=e_i(I\mid J)$ for $0\leq i\leq d$. Teissi…

Let $p(n)$ denote the partition function and let ${\mathrm{spt}}(n)$ denote the smallest-parts function, which counts the smallest parts among the partitions of $n$. Chen's conject…

Granath's conjecture. It holds that

Bennett–Grosse-Erdmann conjecture. The displayed inequality, and respectively its reverse inequality, remain valid with the same best possible constant $(\alpha p+1)^p$ when…

AVV conjecture. The function $H$ is increasing from $(1,\infty)$ into $(1/100,1/30)$. This conjecture is attributed to the authors' earlier work and is motivated by Ramanujan's rec…

Let $R_n=\sum_{k=1}^n\frac{1}{k}-\log\left(n+\frac{1}{2}\right)$, $h(n)=R_n-\gamma$, and $H(n)=n^2h(n)$ for $n\geqslant1$, where $\gamma$ is the Euler–Mascheroni constant. Extendin…

Half-Eulerian inequality-generation conjecture. Every linear form that is nonnegative on the flag vectors of all half-Eulerian posets is the sum of a linear form that is nonnegativ…

Conjecture 2. Under these assumptions, the inequality stated immediately after this parameter condition holds.

Conjecture 1. Theorem LowerGeneral is true for all $\sigma>0$ and

Taikov-type conjecture. The inequality of Taikov type should also hold when $G$ is an $NCVD$ kernel; moreover, the corollary identified in the source as Corollary should also remai…