14 problems

Let $t\in(0,1)$ and $n\in\mathord{\mathbb N}$. Let $\mathcal{D}_n$ be the corresponding local $n$-periodic field, and let $U_{t,n}$ be the associated quantum walk. Its velocity is…

Let $X$ be a graph, and suppose that $X$ admits uniform mixing at time $t$. Mullin's conjecture. Then $e^{it}$ is a root of unity. This conjecture concerns an arithmetic constraint…

Let $X$ be a simple connected unweighted graph on an odd number of vertices. Odd-order Laplacian PST conjecture. Laplacian perfect state transfer does not occur in $X$. This extend…

Let $P_n$ be the path on $n$ vertices, and let $\overset{2}{\uplus}~P_n$ denote its double blow-up, with corresponding vertices $(0,u)$ and $(1,u)$ for each vertex $u$ of $P_n$. Le…

Let $n,m \geq 3$ with $(n,m) \neq (4,4)$. A toroidal $(n,m)$-grid is the toroidal grid with parameters $n$ and $m$, and periodic means periodic under the discrete-time quantum walk…

Let $x$ and $t$ be integers satisfying … Let $a(x,t)$ be the complex amplitude under consideration. Simultaneous nonvanishing conjecture. Then … unless $(x,t)\in\{(-3,11),(5,11)\}$…

Let $Z$ be the graph shown in Figure 6: a longer path $X$ joins the relevant subgraphs, with vertices $a,b,c,d$ labelled as in that figure. No-perfect-state-transfer conjecture. Pe…

Let $b_1(x,t)$ denote the first component of the modified Feynman checker amplitude at position $x$ and time $t$. Modified Feynman checker limiting-probability conjecture. The tota…

Let $a_1(x,t,u)$ denote the first component of the Feynman checker amplitude at position $x$ and time $t$ in the external field $u$, and define $u(x+1/2,t+1/2)=-1$ if both $x$ and…

Let $\mathcal{P}$ be a reversible, ergodic Markov chain with stationary distribution $\uppi$, and let $M$ be a set of marked states. Define … Assume that $p_M<0.5$, and consider Al…

Emms–Hancock–Severini–Wilson conjecture.

Let $G$ and $H$ be strongly regular graphs with the same set of parameters. For a graph $F$, let ${\bf U}(F)$ denote its transition matrix, let ${\bf U}(F)^3$ be its third power, l…

Let $x,y,z\in X$ be three wells in a graph with equal pairwise distances $a=b=c$, and let $c_{uv}$ denote the sum over shortest paths from $u$ to $v$ of the products of inverse squ…

Let $WC$ be the quantum walk operator of a discrete-time quantum walk (DTQW), and let its distribution be the associated position distribution. The operator has only continuous spe…