12 problems

Let $(X,\mathcal{X},\mu,T_1,\ldots,T_\ell)$ be a system, meaning a standard probability space with commuting invertible measure-preserving transformations, and let…

Fractional-polynomial prime-pair conjecture. There exists $B=B(f,A)$ such that

Let $A\in\Gamma_n$, where $\Gamma_n$ denotes the class of accretive $n\times n$ matrices, and let $t$ satisfy $0<t<1$. Fractional-power numerical-radius conjecture. … This conjectu…

Let $k\in\mathbb{N}$ and let $c$ be a positive real number. Write $\mathbb{P}$ for the set of primes. Frantzikinakis's conjecture. For infinitely many pairs…

Fractional-polynomial prime-progression conjecture. There exists $B=B(f_1,\ldots,f_k,A)$ such that

Let $c$ be a positive non-integer, and let $p_n$ denote the $n$-th prime. Frantzikinakis's recurrence and convergence conjecture. The sequence $\lfloor p_n^c\rfloor$ should be good…

The sequences under consideration are integer parts of fractional powers, such as $\lfloor n^{3/2}\rfloor$ and $\lfloor n^{\sqrt{2}}\rfloor$, together with their linear combination…

Let $A$ be a densely defined operator with non-empty resolvent set $\rho(A)$, and suppose that the extension problem … has a unique bounded solution for every $x\in\mathcal{D}(A)$.…

Let $A$ be a densely defined sectorial operator in a Banach space $X$, and let $T_\alpha$ be the (generalised) Dirichlet-to-Neumann operator associated with the corresponding exten…

Weighted subelliptic extension conjecture. The theory developed by Sawyer and Wheeden for subelliptic operators should extend to weighted subelliptic operators with $A_{2}$ weights…

Consider the equation … Here $c>1$, $N$ is an integer, $m_1$ and $m_2$ are natural numbers, and $mathcal P_2$ denotes the set of natural numbers having at most two prime factors co…

Let $\Omega$ be the spatial domain, let $\Gamma$ be its boundary, and let $A_\alpha$ denote the operators associated with the strongly damped wave equation for $\alpha\in[0,1]$. Fo…