42 problems

Let $A,B\in\mathbb{H}_n$ be Hermitian matrices, and let $H$ and $R_k$ denote the operators defined in the paper from the coefficients of $e^{(A+B)t}$ and $e^{At}e^{Bt}$, respective…

Let $\mathcal{M}$ be a semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$, and let $\mathcal{P}$ be the positive cone of the space of all $\tau$-…

Let $x,y,d\in\mathbb R^n$ with $d\geq 0$ be given. A matrix $A\in\mathbb R_+^{n\times n}$ is column-stochastic when $\mathbbm e^\top A=\mathbbm e^\top$; it fixes $d$ when $Ad=d$. T…

Rayleigh–Ritz mixed-type majorization error bounds conjecture. The following weak majorization inequalities hold:

Let ${\mathcal X}$ and ${\mathcal Y}$ be subspaces of a Hilbert space ${\mathcal H}$ with ${{\rm \dim}} \mathcal X={{\rm \dim}} \mathcal Y$, and let $A$ be Hermitian. For a vector…

Majorization conjecture. If $T\in\mathcal{A}'$ and $n\ge 1$, then for every $P\in\mathcal{C}_n$ at least one of the relations

Let $(\mathcal K,\mathcal K')$ be a family pair, with relative Laplacian spectrum $\mathbf{s}(\mathcal K,\mathcal K')$ and transposed relative generalized degree sequence…

Complex spectral-order conjecture. If $P\in\mathcal{C}_{n}$, then

Noncommutative alpha-geometry conjecture. Suppose $n\geq2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then…

Commutative alpha-geometry conjecture. Suppose $n>2$. If $p\in(1,2)$, then $\operatorname{Scal}_p$ is strictly Schur-decreasing; if $p\in(2,+\infty]$, then $\operatorname{Scal}_p$…

Let $K$ be an $r$-dimensional simplicial complex. Using the spectrum $\bm{\lambda}_{r-1}(K)$ of the up-Laplacian and the conjugate vertex-degree sequence $\mathbf d0^top(K)$, with…

Duval–Reiner conjecture. The eigenvalue sequence is majorized by the conjugate vertex degree sequence:

Le's conjecture. Then $\operatorname{diag}(A)\in\Gamma(P)$ and

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. Let $P_\lambda(x;q,t)$ be the Macdonald polynomial, $\mathcal M_S(\mathbb F_{\geqslant0})$ the Muirhead semiring over…

Let $\lambda$ and $\mu$ be partitions, and let $\mathcal J_C^\sigma(\mathbb R_{\geqslant0})$ be the Jack cone generated by normalized Jack differences with parameter $\sigma$. Fix…

Fix $\tau_0\in[0,\infty]$, and let $\lambda$ and $\mu$ be partitions. Let $\mathcal M_S(\mathbb R_{\geqslant0})$ denote the Muirhead semiring over the nonnegative reals. Muirhead-s…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$. For Jack polynomials $P_\lambda(x;\tau)$, let $\mathbb F_{\geqslant0}^{\mathbb R}$ denote the real cone of rational fu…

Nonlinear majorization monotone conjecture. If there exists a blockwise bistochastic matrix $\bm{B}\in\mathcal{B}_{n,d}$ such that $\bm{Q}=\bm{B}\*\bm{P}$, then for any ordering of…

Let … where $a_1\leq\cdots\leq a_m$ and $b_1\leq\cdots\leq b_m$. Two-sided majorization conjecture. For every $\ell$, one has … Equivalently, the multiset of numerator parameters w…

Exact catalytic entropy conjecture. Such a density matrix $csigma$ and unitary $U$ exist if and only if $H(\rho)<H(\rho')$ and…

Let $P$ be any degree $k$ lpm-polynomial, and let $\lambda,\mu\in\mathbb{R}^k$ with $\lambda$ majorizing $\mu$. Hyperbolic Horn conjecture. There exists a symmetric matrix $X$ such…

Let $f,g:T^2\to\mathbb R_+$, where $g$ is superadditive in each variable and … Let $\lambda\geq 10\delta$. Small-energy majorization conjecture. There exists…

Let $A,B\in M_{n\times n}(\mathbb{C})$, and let $\sigma_\uparrow(A)$ denote the singular-value vector of $A$ arranged in increasing order, while $\sigma_\downarrow(B)$ is arranged…

Let $A,B\in M_{n\times n}(\mathbb{C})$, and let $\sigma_\downarrow(A)$ and $\sigma_\downarrow(B)$ denote their singular-value vectors arranged in decreasing order. For…

Decreasing-density conjecture. For $\Lambda\in\mathcal Q_n$ and $\boldsymbol{F}\in\mathcal M_D^n$, one has