16 problems

Let $f\in\mathcal{G}_{sh}$ be an analytic function of the form given by equation (1), with Taylor coefficients $a_n$. Coefficient bound conjecture. One should have … The preceding…

Let $g_1,\dots,g_k:\mathbb{R}\to(1,\infty)$ and $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions. Let $a_1,\dots,a_k,b_1,\dots,b_k:\mathbb{N}\to\mathbb{R}$ be sequence…

Let $I\subset\mathbb{R}$ be a compact interval and let $k\in\mathbb{N}$. For each $i=1,\dots,k$, let $(u_{i,n})_{n\in\mathbb{N}}$ be a sequence of functions in $C^1(I,\mathbb{R})$…

Let $(u_n)_{n\in\mathbb{N}}$ satisfy the conditions of the cited Koksma theorem, and let $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions such that $1,f_1,\dots,f_k$ a…

Let $g:\mathbb{R}\to(1,\infty)$ be a nonconstant analytic function, let $b:\mathbb{N}\to\mathbb{R}$ tend to infinity and be scattered, let $k\in\mathbb{N}$, let…

Let $gamma(t)$ be a gradient flow line for an analytic function, and suppose that it has a limit point. Arnold--Thom gradient conjecture. The limit … exists. This is a stronger, fi…

Let $n\ge 2$ and let $I\subseteq\mathbb{R}$ be an open interval containing $0$. Let $\mathscr{L}_{n,n}$ denote the set of Liouville matrices, and let $\textswab{f}_k$ and…

Regular-supershift characterization conjecture. The function $\psi$ is a $\mathbb C$-regular supershift if and only if $\psi$ is the restriction to the real line of an entire funct…

Regular-sampling supershift conjecture. The function $\psi$ is a $\mathbb C$-regular supershift if and only if there is an entire function $F\colon\mathbb C\to\mathbb C$ such that

Let $\mu$ be a finite measure with $r_\mu=1$, let $p>0$, and let $\bm{F}(z)=\sum_{n=0}^\infty a_n\zeta_n z^n$ be a Gaussian analytic function as above. Write $A^{p/2}(\mu)$ for the…

Let $f$ be an analytic function on the bidisk with positive real part and $f(0)=1$, whose Cayley transform is a rational inner function. The authors construct families of such func…

Let a theory of a generic function be a theory whose function has transcendence properties analogous to the trivial Ax–Schanuel property discussed in the source. An analytic model…

Let $\mathcal{F}_3$ be the class of functions considered in the paper, and let $R_{\mathcal{SL}}$ and $R_{\mathcal{M}(\beta)}$ denote the radii for the corresponding classes, with…

Let $\mathcal{F}_2$ be the function class considered in the paper, and let $R_{\mathcal{S}_{\mathcal{P}}}$ and $R_{\mathcal{S}^*(\alpha)}$ denote the corresponding radius constants…

Monotonicity conjecture. $F_q$ is decreasing on $[0,\frac{\pi}{2}]$ for $q>2$. For $q\leq 2$, $F_q$ is nondecreasing on $[0,\frac{\pi}{2}]$ when $\left[\frac{(n+1)q}{2}\right]$ is…

Let $N_1,N_2,\ldots,N_k\ge2$ be integers with $\Phi_{\mathbf{N}}\ge4$, and let ${\bf z}_{\bf N}(q)$ be the mirror map with associated function ${\bf q}_{\bf N}(z)$ and constant…